Preface

The purpose of this book is to study the interaction of electromagnetic waves, and the application of direct current signals in different media, with topics that have very important applications in science and technology today. The media where the interaction occurs are various, such as dispersive media, conductors, biological tissues of animals, and other media. The constitutive relationships that link the electric and magnetic fields with the densities of electric and magnetic flux are used, and the concepts of electrical conductivity and permittivity, electric field, magnetic field, voltage, power, energy, and heat are also covered.

It is recommended that the reader be a graduate of engineering, physics, or an equivalent subject, where they have dealt with the topics of mechanics, physics, heat, electromagnetism, and mathematical analysis, which make advanced study of the subjects essential. To understand this text, it is necessary to have knowledge of the laws of electromagnetism and the electromagnetism equations or Maxwell's equations.

The book consists of seven chapters that are interconnected by means of concepts and can be read independently.

In Chapter 1 we begin with the resolution of Maxwell's equations with adequate edge conditions to obtain the electric and magnetic fields, and the rest of the parameters of interest in an electromagnetic engineering problem. It is explained that the physical models of electrical permittivity and magnetic permeability must comply with the Kramers-Kronig equations, which obey the physical principle of causality.

Chapter 2 deals with electromagnetic propagation in various dispersive media that is of interest for its technological applications. The electromagnetic propagation in this case has different speeds for each wave excitation frequency, and there will also be attenuation to the amplitude of the electric and magnetic fields in the dispersive media, which will increase with frequency. The phenomena of plasmonic dispersion, dispersion in conductive media, modal dispersion, chromatic dispersion, and intramodal dispersion are explained.

Chapter 3 deals with the electrical properties of solids such as electrical permittivity and electrical conductivity, in the first part the fundamental concepts, the properties of transmission lines with losses, and their associated parameters such as characteristic impedance and propagation constant, are explained and speed of propagation, where the time domain and frequency methods are used, and finally experimental results are presented for the case of dry sand. It is observed that there are interesting applications to agriculture, geophysics, and engineering.

Thermoelectric properties of the Chalcogenide System are presented in Chapter 4.

The first part explains the Seebeck, Peltier thermoelectric phenomena, electrical conductivity and power factor. Electrical conductivity and Seebeck coefficient measurements are explained. Experimental results of the ternary and quaternary Tellurium Telluride chalcogenides, Tl10-x-yAxByTe6 nanoparticles, with different types of dopants (A = Sb, and B = Sn) and with different concentration of Sn are presented.

Chapter 5 discusses electrical conductivity of molten salts and ionic conduction in electrolyte solutions. A microscopic description for the partial DC conductivities in molten salts has been discussed, starting from a Langevin equation. The obtained results for concentration dependency of electrical conductivity are basically represented as a function of the square root of concentration. The electrophoretic effect and the relaxation effect are discussed from a microscopic view point.

In Chapter 6 we study the effects of ultraviolet radiation on the body of an animal.

In order to properly protect and control the effects of electromagnetic radiation on the human body, it is necessary to know and understand the process of absorption and conversion of electromagnetic radiation falling on the surface of the body. The material contains the original results of experimental studies electromagnetic radiation transmission through a sample of quasi-vital skin samples from pigs of different ages.

Chapter 7 covers the application of square-wave electric impulses of 10 ms and 100 V/cm of field force, produced by an impulse generator device, and the proliferation patterns of different animal cells. The discussion is about the influence of one or three square impulses with field force of 100 V/cm on different cells growing in a monolayer and the influence of one or three square impulses with a field force of 100 V/cm on the cells that grow in suspensions.

In this book, the authors aimed to provide material of important topics for the researcher, because novel experimental results are presented, or with a theoretical work, so that many people can apply these results. The applications that are found are diverse and we hope that they will be useful to the researchers in the field of engineering and sciences.

My sincere thanks to Dr. Patricia Larocca and Dr. Adrian César Razzitte from Universidad de Buenos Aires who have helped me in the task of reviewing and evaluating the chapters of the book, and both have been a great help.

> **Dr. Ing. Walter Gustavo Fano** Professor, "Electromagnetic Radiation Laboratory", Facultad de Ingenieria, Electronic Department, Universidad de Buenos Aires, Argentina

### **Adrian César Razzitte and Patricia Larocca**

University of Buenos Aires, Argentina **Chapter 1**

*Walter Gustavo Fano*

**1. Fundamental concepts**

specific topic.

**2. Electromagnetic model of a material**

Introductory Chapter: Causal

and Magnetic Permeability

Models of Electrical Permittivity

The electricity and magnetism theory was formulated by a series of experimental physical laws, such as the Gauss's law of electrostatics, Ampere's law, Biot and Sabart's law, and Faraday's law, with the concepts of charges and electric currents that were used up to the middle of the 1800s. Since James C. Maxwell's *Treatise on Electricity and Magnetism*, with his contribution in the year 1873 [1], it was essential to formulate electromagnetic theory. This electromagnetic theory considers the addition of the displacement current to the conduction current to obtain the total current, which was a fundamental contribution to consider all the physical laws including the law of conservation of charge. Maxwell's equations are generally expressed differentially and are used considering the constitutive relationships, which are the relationships between the vectors of the electric and magnetic fields, which when applying the boarder conditions and the initial conditions, allow obtaining the solutions. These solutions are usually the electric and magnetic fields, since with these vector fields, the electric current, the electric potentials, the power, and other physical parameters of technological utility can be obtained. An issue that has been important in solving the cases that are found experimentally has been the electric and magnetic potentials, which allow the fields to be obtained many times in a simplified form. In electromagnetic theory, the so-called simple media are commonly used, whose characteristics are homogeneous, isotropic, and linear [2]. Here the properties of the media such as the electrical permittivity and the magnetic permeability of the constitutive relationships can be represented as complex numbers, where the electrical and magnetic losses are considered in the imaginary part. For cases of ferrous magnetic materials, for example, with losses, it is necessary to consider nonlinear behavior, although it will not be of interest in our study. Furthermore, the usual treatment of electromagnetic theory is done from the macroscopic point of view, although materials with electric or magnetic dipole moments are considered, because the treatment of quantum electromagnetism is already a

Consider a material medium with an excitation of one electromagnetic wave, whose electric and magnetic fields vary over the time, it is considered that the input variables will be the electric or magnetic fields and the output variables will be the

#### **Chapter 1**

## Introductory Chapter: Causal Models of Electrical Permittivity and Magnetic Permeability

*Walter Gustavo Fano*

#### **1. Fundamental concepts**

The electricity and magnetism theory was formulated by a series of experimental physical laws, such as the Gauss's law of electrostatics, Ampere's law, Biot and Sabart's law, and Faraday's law, with the concepts of charges and electric currents that were used up to the middle of the 1800s. Since James C. Maxwell's *Treatise on Electricity and Magnetism*, with his contribution in the year 1873 [1], it was essential to formulate electromagnetic theory. This electromagnetic theory considers the addition of the displacement current to the conduction current to obtain the total current, which was a fundamental contribution to consider all the physical laws including the law of conservation of charge. Maxwell's equations are generally expressed differentially and are used considering the constitutive relationships, which are the relationships between the vectors of the electric and magnetic fields, which when applying the boarder conditions and the initial conditions, allow obtaining the solutions. These solutions are usually the electric and magnetic fields, since with these vector fields, the electric current, the electric potentials, the power, and other physical parameters of technological utility can be obtained. An issue that has been important in solving the cases that are found experimentally has been the electric and magnetic potentials, which allow the fields to be obtained many times in a simplified form. In electromagnetic theory, the so-called simple media are commonly used, whose characteristics are homogeneous, isotropic, and linear [2]. Here the properties of the media such as the electrical permittivity and the magnetic permeability of the constitutive relationships can be represented as complex numbers, where the electrical and magnetic losses are considered in the imaginary part. For cases of ferrous magnetic materials, for example, with losses, it is necessary to consider nonlinear behavior, although it will not be of interest in our study. Furthermore, the usual treatment of electromagnetic theory is done from the macroscopic point of view, although materials with electric or magnetic dipole moments are considered, because the treatment of quantum electromagnetism is already a specific topic.

#### **2. Electromagnetic model of a material**

Consider a material medium with an excitation of one electromagnetic wave, whose electric and magnetic fields vary over the time, it is considered that the input variables will be the electric or magnetic fields and the output variables will be the

vectors of electric flux density and magnetic, respectively. The material can be considered as a system, with a specific transfer function, and this system is usually considered causal in physics, and from the point of view of the study of signals, it is called linear and time independence (LTI) [3]. These causal systems are important, because the Kramers-Kronig relations can be applied, which relate the real and imaginary part of the electrical permittivity and the magnetic permeability. The theoretical model of electrical permittivity and magnetic permeability of each media can be tested by mean of the Kramers-Kronig relations and Hilbert transform [3, 4]:

$$
\epsilon'(\boldsymbol{\omega}) - \mathbf{1} = \frac{1}{\pi} P \Big|\_{ - \infty }^{ \approx } \frac{\epsilon''(\boldsymbol{\omega})}{\boldsymbol{\omega} - \boldsymbol{\omega}} d\boldsymbol{\omega} \tag{1}
$$

infrared band have materials that have losses and dispersion that must be

*Introductory Chapter: Causal Models of Electrical Permittivity and Magnetic Permeability*

temperature.

**4. Organization of the book**

*DOI: http://dx.doi.org/10.5772/intechopen.92313*

as tellurium telluride chalcogenide nano-materials.

**3**

considered. In the case of an alternating current flowing in the soil, it will also be necessary to consider the electrical properties of the soil as the electrical conductivity for the various applications in electrical engineering. The application of heat to a junction of two metals or two semiconductors produces a potential difference at the ends; this phenomenon is called the Seebeck effect, which in metals the potential differences obtained are very small. For this reason, new composite materials that can obtain a higher Seebeck coefficient are investigated. In metals the Seebeck coefficient is generally of the order of 1*μV=C*; it increases greatly in cases where a metal is measured with a semiconductor, for example. Currently, a technological application of this effect is thermocouples, which are used to measure

In chapter I of the book, the physical sense of the phenomenon of dispersion of electromagnetic waves is discussed; the group speed is obtained. Then from the Lorentz force, the plasma model and the dispersion in the plasma, and in a conductive medium, are discussed. Dispersion topics that are of greatest technological interest are discussed, such as modal, chromatic, and intramodal dispersion. Chapter II studies electromagnetic propagation through the soil, where historically it was used for telegraphic transmissions, in the transmission of surface waves in the AM bands; the knowledge of the electrical properties of soil are applied to the study of agriculture and archeology, which have become very relevant these days. In this chapter the different methods of measuring the electrical properties of the soil are discussed. A widely used technique is time domain reflectometry, which studies the response of the reflected pulses in the time domain to obtain the electrical properties of the soil. Another way to obtain the electrical properties of the soil is by measuring the impedance in the frequency domain of a transmission line known and built for this purpose. In this chapter the own experimental results obtained by the author are presented. In chapter III the electrical conductivity in direct current in molten salts ("Electrical Conductivity of Molten Salts") is studied from a microscopic point of view using the Langevin equation, which implies a time-dependent memory function *γ*ð Þ*t* in relation to the friction forces acting on the constituent ions under the electric field. The properties of the ionic liquid transport phenomenon are important for industry and technological applications. Ionic liquids are divided into two main groups: molten salts and electrolytic solutions. Chapter IV deals with the interaction of electromagnetic waves with the biological tissues of human beings and the skin of animals. Electromagnetic waves can come from the sun, and frequencies range from very low frequencies to gamma ray frequencies. As it is wellknown, the atmosphere filters the highest energy frequencies such as gamma rays, X rays, or ultraviolet rays. This work deals with and studies the transmission, reflection, and reflection coefficients in the skin of humans and animals of electromagnetic waves. In chapter V, we work with the Seebeck effect, which is about two metals or semiconductors to which different temperatures are applied, and a potential difference is produced. The reverse effect is called Peltier and consists of applying a potential difference to the conductors/semiconductors, and heating or cooling occurs at the junction. These thermoelectric properties have technological applications that are being used such as thermocouple temperature sensing and Peltier effect cooling systems. Here we present new thermoelectric materials tested

$$
\varepsilon''(\boldsymbol{\omega}) = -\frac{1}{\pi} P \int\_{-\infty}^{\infty} \frac{\varepsilon'(\boldsymbol{x}) - 1}{\boldsymbol{x} - \boldsymbol{\omega}} d\boldsymbol{x} \tag{2}
$$

where P is the Cauchy principal value.

The fundamental assumption is known as the causality condition. The most primitive and intuitive one can be formulated as follows: the effect cannot precede the cause [5]. The numerical techniques now can allow the computation of Hilbert transform in order to test the electric permittivity model of the material.

#### **3. Electromagnetic wave propagation**

The interaction of electromagnetic waves with matter is an interesting topic to study several applications. Maxwell's equations allow to solve propagation problems in different media together with the boarder solutions that allow to obtain solutions in each application. In electromagnetic theory it is the development of the wave equation or D'Alembert's equation that is in the time domain, and as a function of frequency, we work with the Helmholtz equation, which, in the case of monochromatic sources, provides the two wave solutions, the wave vector and the propagation constant that allow to study the propagation in the different media, which are usually studied as perfect dielectrics or dielectrics without losses, perfect conductors, and dielectrics with losses. This last case of dielectric with losses is the one that has application to the topics of electromagnetic engineering, optoelectronic engineering, RF engineering, and communications engineering. The frequency of electromagnetic waves in technological applications is ranging from low-frequency waves, radio frequencies, microwaves, optical frequencies, infrared, ultraviolet, and even higher to high-energy frequencies. The energy associated with the electromagnetic wave is proportional to the propagation frequency using the Plank constant. The electromagnetic waves that affect an interface from the air to the dielectric medium that usually has losses will be reflected energy and transmitted to the medium under study, dissipating heat in the medium, and it will attenuate the amplitude of the electromagnetic wave that propagates and causes the dispersion effect. This means that the propagating signal will have different propagation speeds for different frequencies, causing a distortion of the propagating signal as it moves through the dispersive medium. Knowledge of the electrical and magnetic properties, which are intrinsic properties of matter, such as the response of materials to be used in the electronic industry, are essential for the design and construction of electrical and electronic devices. The materials in the transmission lines, waveguides, and fiber optics where an electromagnetic wave propagates in the

*Introductory Chapter: Causal Models of Electrical Permittivity and Magnetic Permeability DOI: http://dx.doi.org/10.5772/intechopen.92313*

infrared band have materials that have losses and dispersion that must be considered. In the case of an alternating current flowing in the soil, it will also be necessary to consider the electrical properties of the soil as the electrical conductivity for the various applications in electrical engineering. The application of heat to a junction of two metals or two semiconductors produces a potential difference at the ends; this phenomenon is called the Seebeck effect, which in metals the potential differences obtained are very small. For this reason, new composite materials that can obtain a higher Seebeck coefficient are investigated. In metals the Seebeck coefficient is generally of the order of 1*μV=C*; it increases greatly in cases where a metal is measured with a semiconductor, for example. Currently, a technological application of this effect is thermocouples, which are used to measure temperature.

#### **4. Organization of the book**

vectors of electric flux density and magnetic, respectively. The material can be considered as a system, with a specific transfer function, and this system is usually considered causal in physics, and from the point of view of the study of signals, it is called linear and time independence (LTI) [3]. These causal systems are important, because the Kramers-Kronig relations can be applied, which relate the real and imaginary part of the electrical permittivity and the magnetic permeability. The theoretical model of electrical permittivity and magnetic permeability of each media can be tested by mean of the Kramers-Kronig relations and Hilbert

*ε*0

where P is the Cauchy principal value.

*Electromagnetic Field Radiation in Matter*

**3. Electromagnetic wave propagation**

<sup>ð</sup>*ω*Þ � <sup>1</sup> <sup>¼</sup> <sup>1</sup>

transform in order to test the electric permittivity model of the material.

*<sup>ε</sup>*"ð*ω*޼� <sup>1</sup>

*π P* ð<sup>∞</sup> �∞

> *ε*0 ð*x*Þ � 1

*π P* ð<sup>∞</sup> �∞

The fundamental assumption is known as the causality condition. The most primitive and intuitive one can be formulated as follows: the effect cannot precede the cause [5]. The numerical techniques now can allow the computation of Hilbert

The interaction of electromagnetic waves with matter is an interesting topic to study several applications. Maxwell's equations allow to solve propagation problems in different media together with the boarder solutions that allow to obtain solutions in each application. In electromagnetic theory it is the development of the wave equation or D'Alembert's equation that is in the time domain, and as a function of frequency, we work with the Helmholtz equation, which, in the case of monochromatic sources, provides the two wave solutions, the wave vector and the propagation constant that allow to study the propagation in the different media, which are usually studied as perfect dielectrics or dielectrics without losses, perfect conductors, and dielectrics with losses. This last case of dielectric with losses is the one that has application to the topics of electromagnetic engineering, optoelectronic engineering, RF engineering, and communications engineering. The frequency of electromagnetic waves in technological applications is ranging from low-frequency waves, radio frequencies, microwaves, optical frequencies, infrared, ultraviolet, and even higher to high-energy frequencies. The energy associated with the electromagnetic wave is proportional to the propagation frequency using the Plank constant. The electromagnetic waves that affect an interface from the air to the dielectric medium that usually has losses will be reflected energy and transmitted to the medium under study, dissipating heat in the medium, and it will attenuate the amplitude of the electromagnetic wave that propagates and causes the dispersion effect. This means that the propagating signal will have different propagation speeds for different frequencies, causing a distortion of the propagating signal as it moves through the dispersive medium. Knowledge of the electrical and magnetic properties, which are intrinsic properties of matter, such as the response of materials to be used in the electronic industry, are essential for the design and construction of electrical and electronic devices. The materials in the transmission lines, waveguides, and fiber optics where an electromagnetic wave propagates in the

*ε*"ð*x*Þ

*<sup>x</sup>* � *<sup>ω</sup> dx* (1)

*<sup>x</sup>* � *<sup>ω</sup> dx* (2)

transform [3, 4]:

**2**

In chapter I of the book, the physical sense of the phenomenon of dispersion of electromagnetic waves is discussed; the group speed is obtained. Then from the Lorentz force, the plasma model and the dispersion in the plasma, and in a conductive medium, are discussed. Dispersion topics that are of greatest technological interest are discussed, such as modal, chromatic, and intramodal dispersion. Chapter II studies electromagnetic propagation through the soil, where historically it was used for telegraphic transmissions, in the transmission of surface waves in the AM bands; the knowledge of the electrical properties of soil are applied to the study of agriculture and archeology, which have become very relevant these days. In this chapter the different methods of measuring the electrical properties of the soil are discussed. A widely used technique is time domain reflectometry, which studies the response of the reflected pulses in the time domain to obtain the electrical properties of the soil. Another way to obtain the electrical properties of the soil is by measuring the impedance in the frequency domain of a transmission line known and built for this purpose. In this chapter the own experimental results obtained by the author are presented. In chapter III the electrical conductivity in direct current in molten salts ("Electrical Conductivity of Molten Salts") is studied from a microscopic point of view using the Langevin equation, which implies a time-dependent memory function *γ*ð Þ*t* in relation to the friction forces acting on the constituent ions under the electric field. The properties of the ionic liquid transport phenomenon are important for industry and technological applications. Ionic liquids are divided into two main groups: molten salts and electrolytic solutions. Chapter IV deals with the interaction of electromagnetic waves with the biological tissues of human beings and the skin of animals. Electromagnetic waves can come from the sun, and frequencies range from very low frequencies to gamma ray frequencies. As it is wellknown, the atmosphere filters the highest energy frequencies such as gamma rays, X rays, or ultraviolet rays. This work deals with and studies the transmission, reflection, and reflection coefficients in the skin of humans and animals of electromagnetic waves. In chapter V, we work with the Seebeck effect, which is about two metals or semiconductors to which different temperatures are applied, and a potential difference is produced. The reverse effect is called Peltier and consists of applying a potential difference to the conductors/semiconductors, and heating or cooling occurs at the junction. These thermoelectric properties have technological applications that are being used such as thermocouple temperature sensing and Peltier effect cooling systems. Here we present new thermoelectric materials tested as tellurium telluride chalcogenide nano-materials.

*Electromagnetic Field Radiation in Matter*

#### **Author details**

Walter Gustavo Fano Electromagnetic Radiation Laboratory, Facultad de Ingeniería, Universidad de Buenos Aires, Buenos Aires, Argentina

**References**

[1] Stratton JA. Electromagnetic Theory. Hoboken, New Jersey, USA: McGraw-

*DOI: http://dx.doi.org/10.5772/intechopen.92313*

*Introductory Chapter: Causal Models of Electrical Permittivity and Magnetic Permeability*

[2] Trainotti V, Fano WG. Ingenieria Electromagnetica. 1st ed. Vol. 1. Buenos Aires, Argentina: Nueva Libreria; 2004

[3] Fano WG, Boggi S, Razzitte AC. Causality study and numerical response of the magnetic permeability as a function of the frequency of ferrites using Kramers-Kronig relations. Physica B. 2008;**403**:526-530

[4] Landau LD, Lifchitz EM.

Academic Press; 1972

**5**

Electrodynamics of Continuous Media. Boston, USA: Addison Wesley; 1981

[5] Nussenzveig HM. Causality and Dispersion Relations. New York:

Hill Book Company; 2007

\*Address all correspondence to: gfano@fi.uba.ar

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Introductory Chapter: Causal Models of Electrical Permittivity and Magnetic Permeability DOI: http://dx.doi.org/10.5772/intechopen.92313*

#### **References**

[1] Stratton JA. Electromagnetic Theory. Hoboken, New Jersey, USA: McGraw-Hill Book Company; 2007

[2] Trainotti V, Fano WG. Ingenieria Electromagnetica. 1st ed. Vol. 1. Buenos Aires, Argentina: Nueva Libreria; 2004

[3] Fano WG, Boggi S, Razzitte AC. Causality study and numerical response of the magnetic permeability as a function of the frequency of ferrites using Kramers-Kronig relations. Physica B. 2008;**403**:526-530

[4] Landau LD, Lifchitz EM. Electrodynamics of Continuous Media. Boston, USA: Addison Wesley; 1981

[5] Nussenzveig HM. Causality and Dispersion Relations. New York: Academic Press; 1972

**Author details**

**4**

Walter Gustavo Fano

Buenos Aires, Buenos Aires, Argentina

*Electromagnetic Field Radiation in Matter*

provided the original work is properly cited.

\*Address all correspondence to: gfano@fi.uba.ar

Electromagnetic Radiation Laboratory, Facultad de Ingeniería, Universidad de

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Chapter 2**

**Abstract**

as well.

**7**

fiber optics

**1. Introduction**

*Emeka Ikpeazu*

Fields in Dispersive Media

It is not just the case that matter affects the propagation of light—or more specifically electromagnetic (EM) radiation—it is also the case that light affects the matter through which it propagates. Conversely, this affects the propagation of light through the medium, but in a much more specific way; this effect is a function of the properties of both the material and the incident EM radiation. We will additionally discuss the effects of dispersion in confined (bounded) media, i.e., where the dispersion is a function of the arrangement of certain materials and unbounded media where EM radiation is free to propagate undisturbed. This will be important when we discuss the propagation electric field signals of such media

**Keywords:** refraction, materials, dispersion, electromagnetics, wave propagation,

Incident electromagnetic (EM) radiation excites the molecules in a material, and these molecules become polarized; they respond according to the direction of the electric field. **Figure 1** shows the initial step of the process of material polarization. It should be adequately noted that there are *several* ways that material response to light-matter interaction can be analyzed. For the purposes of relevance, this chapter will analyze the mechanical response of the atom—as illustrated in

In this chapter we will discuss polarization at the level of the interacting material, the effects of dispersion on a train of pulses, various types of polarization, and methods for reducing polarization to maintain the integrity of optical signals. But first, some words on non-dispersion in unbounded media, a more general term for

Unbounded media are the baseline of understanding EM wave propagation. In

Additionally, unbounded media are generally non-dispersive. This is to say that the speed of energy propagation is orthogonal to the frequency of the said propagating energy. In the previous paragraph, it was said the waves in unbounded media

unbounded media, waves are free to propagate unperturbed. Examples of unbounded media include the ocean, the air, and outer space. An unbounded medium would be the ideal location for an isotropic antenna as the radiation would be free to propagate in all directions, only weakening in accordance with the inverse square law. Such is ideal for radio towers which produce low-frequency EM waves

**Figure 2**—in response to incident EM radiation waves.

which can propagate for kilometers and reach many people.

what would normally be called "free space."

## **Chapter 2** Fields in Dispersive Media

*Emeka Ikpeazu*

### **Abstract**

It is not just the case that matter affects the propagation of light—or more specifically electromagnetic (EM) radiation—it is also the case that light affects the matter through which it propagates. Conversely, this affects the propagation of light through the medium, but in a much more specific way; this effect is a function of the properties of both the material and the incident EM radiation. We will additionally discuss the effects of dispersion in confined (bounded) media, i.e., where the dispersion is a function of the arrangement of certain materials and unbounded media where EM radiation is free to propagate undisturbed. This will be important when we discuss the propagation electric field signals of such media as well.

**Keywords:** refraction, materials, dispersion, electromagnetics, wave propagation, fiber optics

#### **1. Introduction**

Incident electromagnetic (EM) radiation excites the molecules in a material, and these molecules become polarized; they respond according to the direction of the electric field. **Figure 1** shows the initial step of the process of material polarization.

It should be adequately noted that there are *several* ways that material response to light-matter interaction can be analyzed. For the purposes of relevance, this chapter will analyze the mechanical response of the atom—as illustrated in **Figure 2**—in response to incident EM radiation waves.

In this chapter we will discuss polarization at the level of the interacting material, the effects of dispersion on a train of pulses, various types of polarization, and methods for reducing polarization to maintain the integrity of optical signals. But first, some words on non-dispersion in unbounded media, a more general term for what would normally be called "free space."

Unbounded media are the baseline of understanding EM wave propagation. In unbounded media, waves are free to propagate unperturbed. Examples of unbounded media include the ocean, the air, and outer space. An unbounded medium would be the ideal location for an isotropic antenna as the radiation would be free to propagate in all directions, only weakening in accordance with the inverse square law. Such is ideal for radio towers which produce low-frequency EM waves which can propagate for kilometers and reach many people.

Additionally, unbounded media are generally non-dispersive. This is to say that the speed of energy propagation is orthogonal to the frequency of the said propagating energy. In the previous paragraph, it was said the waves in unbounded media

Since Belt B is traveling faster than Belt A, the distance between BB�<sup>1</sup> and RB�<sup>1</sup> will decrease. From the perspective of Belt A, Belt B is moving with a velocity of 0.5 cm/s. At time *t* ¼ 5, the center of BB�<sup>1</sup> is aligned with that of RB�1; however, in that time the balls on Belt B have moved a net 15 cm, while those on Belt A have moved a net 12.5 cm. However, if one were to see the conveyor belts as a group, the belts would appear to have moved 2.5 cm (measuring from location of alignment) within that time, giving the conveyor belts a "group velocity" of 0.5 m/s even

The situation described is analogous to the behavior of a wave propagating in free space. As the individual waves travel with their respective phase velocities along the guide, the two peaks become disaligned for a period [2]. Because the center of RB�<sup>1</sup> will have traveled a distance *λ*RB þ Δ*z* with a phase velocity *vz*RB and the center of BB�<sup>1</sup> will have traveled a distance *λ*BB þ Δ*z* with phase velocity *vz*BB,

> <sup>Δ</sup>*<sup>t</sup>* <sup>¼</sup> *<sup>λ</sup>RB* � *<sup>λ</sup>BB vz*,*RB* � *vz*,*BB*

<sup>Δ</sup>*<sup>z</sup>* <sup>¼</sup> *<sup>λ</sup>*RB*vz*,BB � *<sup>λ</sup>*BB*vz*,RB *vz*,RB � *vz*,BB

> <sup>¼</sup> *<sup>f</sup> BB* � *<sup>f</sup> RB* 1 *<sup>λ</sup>BB* � <sup>1</sup> *λRB*

*λRB* þ Δ*z* ¼ *vz*,*RB*Δ*t* (1) *λBB* þ Δ*z* ¼ *vz*,*BB*Δ*t* (2)

> <sup>¼</sup> *<sup>λ</sup>RBλBBf BB* � *<sup>λ</sup>BBλRBf RB λRB* � *λBB*

> > ! *<sup>ω</sup>BB* � *<sup>ω</sup>RB* 2*π* <sup>1</sup> *<sup>λ</sup>BB* � <sup>1</sup> *<sup>λ</sup>RB*

(3)

(4)

(5)

though the individual components are moving faster.

one can then write a set of equations as

*DOI: http://dx.doi.org/10.5772/intechopen.91432*

In the same way, one can express Δ*z* as

Finally, one can express the group velocity as

<sup>¼</sup> *<sup>λ</sup>BBλRB <sup>f</sup> BB* � *<sup>f</sup> RB λRB* � *λBB*

*The separation of white light into different colors by prismatic dispersion.*

<sup>¼</sup> *vz* <sup>¼</sup> *<sup>λ</sup>RBvz*,*BB* � *<sup>λ</sup>BBvz*,*RB λRB* � *λBB*

We can then write Δ*t* as

*Fields in Dispersive Media*

Δ*z* Δ*t*

**Figure 4.**

**9**

**Figure 1.**

**Figure 2.**

*The mechanical response of the material to incoming monochromatic EM radiation. The vibrational response of the molecule is a function of its properties. This same principle is also related to the color of the flame that a molecule produces [1].*

are free to propagate unperturbed. This is not *completely* true. Natural disturbances *within* the air or sea can interfere with the propagation of energy therein. However, these natural impediments are not necessarily treated as features of the media itself. It is for this reason that direct or *line-of-sight* propagation is key in facilitating communication between antennas and cell towers.

#### **2. Understanding dispersion**

Dispersion can be difficult to understand. There are picture balls on two parallel infinite conveyor belts running at different speeds, Belt A at the top and Belt B at the bottom. The conveyor belts in **Figure 3** carry balls at 2.5 cm/s and 3.0 cm/s, respectively. Additionally, the centers of the balls on each belt are separated by 10 cm and 12.5 cm, respectively. At time *t* ¼ 0, the centers of Red Ball 0 (RB0) and Blue Ball 0 (BB0) are aligned. However, the center of BB�<sup>1</sup> trails that of RB�<sup>1</sup> by a distance of 2.5 cm.

**Figure 3.** *A mechanical kinematic illustration of the principle of the dispersion.*

#### *Fields in Dispersive Media DOI: http://dx.doi.org/10.5772/intechopen.91432*

Since Belt B is traveling faster than Belt A, the distance between BB�<sup>1</sup> and RB�<sup>1</sup> will decrease. From the perspective of Belt A, Belt B is moving with a velocity of 0.5 cm/s. At time *t* ¼ 5, the center of BB�<sup>1</sup> is aligned with that of RB�1; however, in that time the balls on Belt B have moved a net 15 cm, while those on Belt A have moved a net 12.5 cm. However, if one were to see the conveyor belts as a group, the belts would appear to have moved 2.5 cm (measuring from location of alignment) within that time, giving the conveyor belts a "group velocity" of 0.5 m/s even though the individual components are moving faster.

The situation described is analogous to the behavior of a wave propagating in free space. As the individual waves travel with their respective phase velocities along the guide, the two peaks become disaligned for a period [2]. Because the center of RB�<sup>1</sup> will have traveled a distance *λ*RB þ Δ*z* with a phase velocity *vz*RB and the center of BB�<sup>1</sup> will have traveled a distance *λ*BB þ Δ*z* with phase velocity *vz*BB, one can then write a set of equations as

$$
\lambda\_{RB} + \Delta \mathbf{z} = \boldsymbol{\upsilon}\_{\boldsymbol{x},RB} \Delta t \tag{1}
$$

$$
\lambda\_{BB} + \Delta \mathbf{z} = \boldsymbol{\nu}\_{\mathbf{z},BB} \Delta t \tag{2}
$$

We can then write Δ*t* as

are free to propagate unperturbed. This is not *completely* true. Natural disturbances *within* the air or sea can interfere with the propagation of energy therein. However, these natural impediments are not necessarily treated as features of the media itself. It is for this reason that direct or *line-of-sight* propagation is key in facilitating

*The mechanical response of the material to incoming monochromatic EM radiation. The vibrational response of the molecule is a function of its properties. This same principle is also related to the color of the flame that a*

*An EM traveling wave is incident on a molecule of a particular material. This induces a* unique *response in*

Dispersion can be difficult to understand. There are picture balls on two parallel infinite conveyor belts running at different speeds, Belt A at the top and Belt B at the bottom. The conveyor belts in **Figure 3** carry balls at 2.5 cm/s and 3.0 cm/s, respectively. Additionally, the centers of the balls on each belt are separated by 10 cm and 12.5 cm, respectively. At time *t* ¼ 0, the centers of Red Ball 0 (RB0) and Blue Ball 0 (BB0) are aligned. However, the center of BB�<sup>1</sup> trails that of RB�<sup>1</sup> by a

communication between antennas and cell towers.

*A mechanical kinematic illustration of the principle of the dispersion.*

**2. Understanding dispersion**

distance of 2.5 cm.

**Figure 3.**

**8**

**Figure 1.**

**Figure 2.**

*molecules of the material.*

*Electromagnetic Field Radiation in Matter*

*molecule produces [1].*

$$
\Delta t = \frac{\lambda\_{RB} - \lambda\_{BB}}{\upsilon\_{x,RB} - \upsilon\_{x,BB}} \tag{3}
$$

In the same way, one can express Δ*z* as

$$
\Delta z = \frac{\lambda\_{\rm RB} v\_{x,\rm BB} - \lambda\_{\rm BB} v\_{x,\rm RB}}{v\_{x,\rm RB} - v\_{x,\rm BB}} \tag{4}
$$

Finally, one can express the group velocity as

$$\frac{\Delta z}{\Delta t} = \nu\_x = \frac{\lambda\_{RB}\nu\_{x,BB} - \lambda\_{BB}\nu\_{x,RB}}{\lambda\_{RB} - \lambda\_{BB}} = \frac{\lambda\_{RB}\lambda\_{BB}f\_{BB} - \lambda\_{BB}\lambda\_{RB}f\_{RB}}{\lambda\_{RB} - \lambda\_{BB}}\tag{5}$$

$$=\frac{\lambda\_{RB}\lambda\_{RB}(f\_{BB} - f\_{RB})}{\lambda\_{RB} - \lambda\_{BB}} = \frac{f\_{BB} - f\_{RB}}{\frac{1}{\lambda\_{RB}} - \frac{1}{\lambda\_{RB}}} \longrightarrow \frac{\alpha\_{BB} - \alpha\_{RB}}{2\pi\left(\frac{1}{\lambda\_{RB}} - \frac{1}{\lambda\_{RB}}\right)}$$

**Figure 4.** *The separation of white light into different colors by prismatic dispersion.*

We know that the wavevector *k* ¼ 2*π=λ*, so one express the group velocity as

$$v\_{\mathfrak{g}} = \frac{dw}{dk} \tag{6}$$

As a result, the *effective* permittivity is expressed in (11) as

*<sup>ϵ</sup>* <sup>¼</sup> *<sup>ϵ</sup>*<sup>0</sup> <sup>1</sup> � *Nq*<sup>2</sup>

are rare. Altogether we can express the dispersion relation thusly

*ϵμ* <sup>¼</sup> *<sup>ϵ</sup>*0*μ*<sup>0</sup> *<sup>ω</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup>

This means that the group velocity is expressed as

*vg* <sup>¼</sup> *kc*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *k*2 *c*<sup>2</sup> þ *ω*<sup>2</sup> *p* <sup>q</sup> <sup>¼</sup> *<sup>c</sup>*

to measure the frequency-dependent group velocity of light therein as

frequency, the permittivity is negative, the *k*-vector is imaginary, and,

*The group velocity of light through germanium as a function of frequency.*

In a material like germanium (Ge) the intrinsic carrier density *N*Ge is 2*:*5 � 1019m�<sup>3</sup> [6]; the rest of the variables are known constants. This gives germanium a plasma frequency of *f <sup>p</sup>*,Ge ¼ 44*:*8 GHz. We apply the relation in (13) to germanium

Below the plasma frequency, the value of the *k*-vector becomes purely imaginary, meaning that the electric field is purely evanescent—and non-propagating at that point. The group velocity is the slope of the dispersion curve in **Figure 6**. For both curves, the lowest allowable frequency is the plasma frequency. Below this

where the plasma frequency *<sup>ω</sup><sup>p</sup>* <sup>¼</sup> *<sup>q</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>k</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91432*

*Fields in Dispersive Media*

demonstrated in **Figure 5**.

**Figure 5.**

**11**

*ϵ*0*mω*<sup>2</sup> � �

*N=ϵ*0*m* p . The plasma frequency is the natural oscillation frequency of the displaced electrons in a neutral plasma [4, 5] of free electrons where it is assumed that collisions

> *p* � � ! *<sup>ω</sup>*ð Þ¼ *<sup>k</sup>*

<sup>¼</sup> *<sup>ϵ</sup>*<sup>0</sup> <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

*p ω*<sup>2</sup> !

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> *p ω*2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *k*2 *c*<sup>2</sup> þ *ω*<sup>2</sup> *p*

vu ! ut (13)

, (11)

(12)

This is to say the group velocity is the degree change in the temporal frequency for every change in the spatial frequency. What this means is that the value of the frequency is on some level a function of the *k*-vector and ultimately a function of the wavelength. As illustrated in **Figure 4**, this has implications for the propagation of light through various media as different wavelengths will travel at different velocities and take various pathways therein [3].

#### **3. Plasmonic dispersion**

The Lorentz force is the force on a point charge due to EM fields. As discussed in the Introduction, EM radiation causes motion among the particles in a material as shown in **Figure 2**. The direction (vector) of this force is called the polarization. Assuming a sea of free particles in vacuum [4], the strength of the Lorentz force on a point charge is expressed as

$$
\overrightarrow{\mathbf{F}} = q \left( \overrightarrow{\mathbf{E}} + \overrightarrow{\mathbf{v}} \times \overrightarrow{\mathbf{B}} \right),
\tag{7}
$$

where *F* ! , *q*, **E** ! , *v* !, and **B** ! are the force, the charge, the electric field strength, the particle velocity, and the magnetic flux density, respectively. For our purposes, we consider an isotropic material where the electric permittivity is a simple scalar *ϵ*. This means that the Lorentz force vector will be unidirectional. In birefringent and anisotropic, the permittivity would be expressed as a tensor, and the Lorentz force vector would be multidirectional.

We additionally assume a dielectric unmagnetized material such that the value of **B** ! is assumed to be zero. This makes the response of the atoms into a simple harmonic oscillator. In this case we can express the Lorentz force as simply

$$
\overrightarrow{\mathbf{F}} = q\overrightarrow{\mathbf{E}} = m\overrightarrow{a} = m\left(\frac{d^2\overrightarrow{\mathbf{d}}}{dt^2}\right) = m(j\omega)^2\overrightarrow{\mathbf{d}} = -m\omega^2\overrightarrow{\mathbf{d}},\tag{8}
$$

where *m*, *ω*, and *d* ! are the mass, the angular frequency, and dipole vector, respectively. We can simplify to get the dipole vector as

$$
\overrightarrow{\mathbf{d}} = -\frac{q\overrightarrow{\mathbf{E}}}{m o^2} \rightarrow \overrightarrow{\mathbf{P}} = Nq\overrightarrow{\mathbf{d}} = -\frac{Nq^2\overrightarrow{\mathbf{E}}}{m o^2},\tag{9}
$$

where **P** ! is the polarization vector and *N* is the dipole density. The electric displacement field **D** ! is the permittivity *ϵ* multiplied by the electric field **E** ! , but in dielectric materials, the electric field induces a response in the material known as the polarization. This effect is added to the original displacement field to make it so that

$$
\overrightarrow{\mathbf{D}} = \epsilon\_0 \overrightarrow{\mathbf{E}} + \overrightarrow{\mathbf{P}} = \left(\epsilon\_0 - \frac{Nq^2}{ma^2}\right)\overrightarrow{\mathbf{E}}\tag{10}
$$

We know that the wavevector *k* ¼ 2*π=λ*, so one express the group velocity as

*vg* <sup>¼</sup> *<sup>d</sup><sup>ω</sup>*

This is to say the group velocity is the degree change in the temporal frequency for every change in the spatial frequency. What this means is that the value of the frequency is on some level a function of the *k*-vector and ultimately a function of the wavelength. As illustrated in **Figure 4**, this has implications for the propagation of light through various media as different wavelengths will travel at different

The Lorentz force is the force on a point charge due to EM fields. As discussed in the Introduction, EM radiation causes motion among the particles in a material as shown in **Figure 2**. The direction (vector) of this force is called the polarization. Assuming a sea of free particles in vacuum [4], the strength of the Lorentz force on

> *F* !

¼ *q E* ! þ *v* ! � *<sup>B</sup>* � �!

particle velocity, and the magnetic flux density, respectively. For our purposes, we consider an isotropic material where the electric permittivity is a simple scalar *ϵ*. This means that the Lorentz force vector will be unidirectional. In birefringent and anisotropic, the permittivity would be expressed as a tensor, and the Lorentz force

We additionally assume a dielectric unmagnetized material such that the value

¼ *Nqd* !

is the polarization vector and *N* is the dipole density. The electric

dielectric materials, the electric field induces a response in the material known as the polarization. This effect is added to the original displacement field to make it so that

is the permittivity *ϵ* multiplied by the electric field **E**

<sup>¼</sup> *<sup>ϵ</sup>*<sup>0</sup> � *Nq*<sup>2</sup>

*mω*<sup>2</sup> � �

*E* !

<sup>¼</sup> *m j* ð Þ *<sup>ω</sup>* <sup>2</sup>

are the mass, the angular frequency, and dipole vector,

*d* !

¼ � *Nq*<sup>2</sup>*<sup>E</sup>*

!

¼ �*mω*<sup>2</sup>*<sup>d</sup>* !

, (8)

*<sup>m</sup>ω*<sup>2</sup> , (9)

! , but in

(10)

is assumed to be zero. This makes the response of the atoms into a simple

harmonic oscillator. In this case we can express the Lorentz force as simply

*d*2 *d* ! *dt*<sup>2</sup> !

<sup>¼</sup> *ma*! <sup>¼</sup> *<sup>m</sup>*

respectively. We can simplify to get the dipole vector as

*D* ! ¼ *ϵ*0*E* ! þ *P* !

¼ � *<sup>q</sup><sup>E</sup>* ! *<sup>m</sup>ω*<sup>2</sup> ! *<sup>P</sup>* !

*d* !

velocities and take various pathways therein [3].

*Electromagnetic Field Radiation in Matter*

**3. Plasmonic dispersion**

a point charge is expressed as

vector would be multidirectional.

*F* ! ¼ *qE* !

!

!

where *m*, *ω*, and *d*

where **P** !

**10**

displacement field **D**

where *F* ! , *q*, **E** ! , *v* !, and **B** !

of **B** ! *dk* (6)

, (7)

are the force, the charge, the electric field strength, the

As a result, the *effective* permittivity is expressed in (11) as

$$
\varepsilon = \varepsilon\_0 \left( \mathbf{1} - \frac{Nq^2}{\varepsilon\_0 m a^2} \right) = \varepsilon\_0 \left( \mathbf{1} - \frac{a \mathbf{o}\_p^2}{a^2} \right),
\tag{11}
$$

where the plasma frequency *<sup>ω</sup><sup>p</sup>* <sup>¼</sup> *<sup>q</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *N=ϵ*0*m* p .

The plasma frequency is the natural oscillation frequency of the displaced electrons in a neutral plasma [4, 5] of free electrons where it is assumed that collisions are rare. Altogether we can express the dispersion relation thusly

$$\kappa^2 = a^2 \epsilon \mu = \epsilon\_0 \mu\_0 \left( a^2 - a\_p^2 \right) \to a(k) = \sqrt{k^2 c^2 + a\_p^2} \tag{12}$$

This means that the group velocity is expressed as

$$v\_{\mathcal{g}} = \frac{kc^2}{\sqrt{k^2c^2 + {\alpha\_p^2}^2}} = c\sqrt{1 - \left(\frac{\alpha\_p^2}{\alpha^2}\right)}\tag{13}$$

In a material like germanium (Ge) the intrinsic carrier density *N*Ge is 2*:*5 � 1019m�<sup>3</sup> [6]; the rest of the variables are known constants. This gives germanium a plasma frequency of *f <sup>p</sup>*,Ge ¼ 44*:*8 GHz. We apply the relation in (13) to germanium to measure the frequency-dependent group velocity of light therein as demonstrated in **Figure 5**.

Below the plasma frequency, the value of the *k*-vector becomes purely imaginary, meaning that the electric field is purely evanescent—and non-propagating at that point. The group velocity is the slope of the dispersion curve in **Figure 6**. For both curves, the lowest allowable frequency is the plasma frequency. Below this frequency, the permittivity is negative, the *k*-vector is imaginary, and,

**Figure 5.** *The group velocity of light through germanium as a function of frequency.*

**Figure 6.** *The dispersion relation for germanium where ω is function of k.*

consequently, the field simply evanesces into the material and propagation completely ceases.

#### **4. Dispersion in conductive media**

In the previous section, we derived plasmonic dispersion from understanding the Lorentz force law as it pertains to electrons in a dielectric material. Dispersion can also occur in a conducting material where the charges therein are unbound. When we apply our knowledge of electromagnetics and electrostatics to a conductor, we discover a few things:


Additionally, in a conductor the free charges are always pushed to the surface so that electrostatic equilibrium is maintained. This means that the gradient of the electric field **∇** ! � **E** ! equals zero. Things are different in the presence of an electric field, e.g., **Figure 8**, where the electrons are pulled in the opposite direction of the electric field.

With regard to conductive media, we can do the same thing through an understanding and application of Faraday's law which states that a time-varying magnetic flux density *always* accompanies a nonconservative electric field, i.e., a field **E** ! for which **∇** ! � **E** ! ¼ **0**. This is demonstrated by the equation

∇ ! � *E* !

generates a time-varying electric field such that

**Figure 7.**

**Figure 8.**

**13**

*conduction current.*

*equilibrium where Q***<sup>Σ</sup>** ¼ **0***.*

*Fields in Dispersive Media*

*DOI: http://dx.doi.org/10.5772/intechopen.91432*

¼ � *<sup>∂</sup><sup>B</sup>* !

*The distribution of freely moving protons (red) and electrons (blue) in a conductive material in electrostatic*

We can combine this with Ampère's law where the curl of the magnetic field

*The kinetic behavior of electrons in the presence of an electric field produces a steady-state current known as the*

*<sup>∂</sup><sup>t</sup>* ¼ �*μ*<sup>0</sup>

*∂H* !

*<sup>∂</sup><sup>t</sup>* (14)

**Figure 7.**

consequently, the field simply evanesces into the material and propagation

In the previous section, we derived plasmonic dispersion from understanding the Lorentz force law as it pertains to electrons in a dielectric material. Dispersion can also occur in a conducting material where the charges therein are unbound. When we apply our knowledge of electromagnetics and electrostatics to a conduc-

1.There is zero net charge within a conductor, although charges can still

2.All the charge rests on the surface of the conductor, perpendicular to the surface.

Additionally, in a conductor the free charges are always pushed to the surface so that electrostatic equilibrium is maintained. This means that the gradient of the

equals zero. Things are different in the presence of an electric

! for

3.The presence of an electric field induces the motion of electrons, creating a

field, e.g., **Figure 8**, where the electrons are pulled in the opposite direction of the

flux density *always* accompanies a nonconservative electric field, i.e., a field **E**

¼ **0**. This is demonstrated by the equation

With regard to conductive media, we can do the same thing through an understanding and application of Faraday's law which states that a time-varying magnetic

randomly move around therein as seen in **Figure 7**.

completely ceases.

**Figure 6.**

**4. Dispersion in conductive media**

*Electromagnetic Field Radiation in Matter*

*The dispersion relation for germanium where ω is function of k.*

tor, we discover a few things:

conduction current [7].

! � **E** !

electric field **∇**

electric field.

which **∇** ! � **E** !

**12**

*The distribution of freely moving protons (red) and electrons (blue) in a conductive material in electrostatic equilibrium where Q***<sup>Σ</sup>** ¼ **0***.*

#### **Figure 8.**

*The kinetic behavior of electrons in the presence of an electric field produces a steady-state current known as the conduction current.*

$$
\overrightarrow{\nabla} \times \overrightarrow{\mathbf{E}} = -\frac{\partial \overrightarrow{\mathbf{B}}}{\partial t} = -\mu\_0 \frac{\partial \overrightarrow{H}}{\partial t} \tag{14}
$$

We can combine this with Ampère's law where the curl of the magnetic field generates a time-varying electric field such that

$$
\overrightarrow{\nabla} \times \overrightarrow{\mathbf{H}} = \overrightarrow{f}\_f + \epsilon \frac{\partial \overrightarrow{\mathbf{E}}}{\partial t} = \sigma \overrightarrow{\mathbf{E}} + \frac{\partial \overrightarrow{\mathbf{E}}}{\partial t} = (\sigma + j\alpha\epsilon)\overrightarrow{\mathbf{E}} = j\alpha\epsilon\_{\text{eff}}\overrightarrow{\mathbf{E}},\tag{15}
$$

2 ffiffiffiffiffi *ϵ*0 *μ*0*ϵ* q

values of *ϵ* and *σ* are listed in **Table 1** for some good conductors [9].

*ϵω* 2 � � <sup>þ</sup> *<sup>σ</sup>*

*σ*2

*DOI: http://dx.doi.org/10.5772/intechopen.91432*

*Fields in Dispersive Media*

relaxation time is 66.1 picoseconds.

incoming EM radiation.

*Conductivities of some good conductors.*

*Conductivities of some good dielectrics.*

**Table 1.**

**Table 2.**

**15**

*<sup>ϵ</sup>*2*<sup>ω</sup>* <sup>þ</sup> <sup>2</sup>*<sup>ω</sup>* � � *cos tan* �<sup>1</sup> *<sup>σ</sup>*

*<sup>ω</sup>*<sup>2</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup> *ϵ*2 � �

q

Additionally, the attenuation is a new phenomenon resulting from the fact that the *k*-vector is complex. When we take the reciprocal of this coefficient get another quantity known as the *skin depth* or penetration depth *δp*, the extent to which the field penetrates the material. In a good conductor, *δ<sup>p</sup>* takes a value ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ω</sup>*<sup>4</sup> <sup>þ</sup> *<sup>σ</sup>*2*ω*<sup>2</sup> *ϵ*2 � � <sup>4</sup> *sin tan* �<sup>1</sup> *<sup>σ</sup> ϵω* 2

� � (19)

<sup>2</sup>*=σμ*0*<sup>ω</sup>* <sup>p</sup> . The

*ϵ*

Good conductors are characterized by an abundance of free electrons. Good conductors (good dielectrics), on the other hand, are characterized by a paucity of such electrons. Examples of good dielectrics are enumerated in **Table 2** [10].

germanium where the skin depth is roughly a centimeter and the dielectric

Conductive media have more free electrons that *move* in the presence of an electric field than those that simply vibrate. Conductive media produce a loss (attenuation) component in propagation of incoming EM radiation. Additionally, the drift movement of electrons in conductive media causes heat to which the electrical energy from the incoming field is transferred. This energy conversion causes the dissipation of the field. Nevertheless, in the context of the phenomenon of dispersion, the degree of this attenuation is a function of the frequency of the

**Material Conductivity (S/m) @** *T* ¼ **293 K**

**Material Permittivity (F/m)** Calcium <sup>2</sup>*:*<sup>685</sup> � <sup>10</sup>�<sup>11</sup> Marble <sup>7</sup>*:*<sup>08</sup> � <sup>10</sup>�<sup>11</sup> Silicon dioxide (SiO2) <sup>3</sup>*:*<sup>452</sup> � <sup>10</sup>�<sup>11</sup> Slate <sup>3</sup>*:*<sup>54</sup> � <sup>10</sup>�<sup>11</sup> Polytetrafluoroethylene (PTFE) <sup>1</sup>*:*<sup>77</sup> � <sup>10</sup>�<sup>11</sup>

Silver <sup>6</sup>*:*<sup>1</sup> � <sup>10</sup><sup>7</sup> Copper <sup>5</sup>*:*<sup>8</sup> � <sup>10</sup><sup>7</sup> Gold <sup>4</sup>*:*<sup>1</sup> � <sup>10</sup><sup>7</sup> Aluminum <sup>3</sup>*:*<sup>5</sup> � <sup>10</sup><sup>7</sup> Brass <sup>1</sup>*:*<sup>5</sup> � <sup>10</sup><sup>7</sup>

**Figure 9** shows the interaction between an electric field and an good conductor (e.g., gold) at a wavelength of 1.55μm. The field decays very rapidly as gold has a skin depth of 5.66 nm and a dielectric relaxation time of 1.49 attoseconds (10�<sup>18</sup> s) at this wavelength. This contrasts with a lossy dielectric where the skin depth is much larger, and there is thus more field penetration as displayed in **Figure 10** with

where **J** ! *<sup>f</sup>* is the free current density vector produced in the presence of an electric field as exemplified in **Figure 8** and *σ* is the conductivity. We can calculate the effective permittivity this way such that *<sup>ϵ</sup>eff* <sup>¼</sup> *<sup>ϵ</sup>* <sup>1</sup> � *<sup>j</sup><sup>σ</sup> ωϵ* � �. In this way we can write the dispersion relation.

$$k = \alpha \sqrt{\mu\_0 \epsilon\_{\mathcal{H}}(\omega)} = \alpha \sqrt{\frac{\mu\_0 \epsilon}{\epsilon\_0}} \sqrt{1 - \frac{j\sigma}{\alpha \epsilon}} = \sqrt{\frac{\mu\_0 \epsilon}{\epsilon\_0}} \sqrt{\alpha^2 - \frac{j\sigma \alpha}{\epsilon}}$$

$$= \sqrt{\frac{\mu\_0 \epsilon}{\epsilon\_0}} \sqrt{\alpha^4 + \left(\frac{\sigma^2 \alpha^2}{\epsilon^2}\right)} e^{-i\left(\frac{\tan^{-1}(\frac{\omega}{\alpha \epsilon})}{2}\right)}\tag{16}$$

This wavenumber *k* is also complex where *k* ¼ *k*<sup>0</sup> � *jk*<sup>00</sup>, where

$$\begin{split}k' &= \sqrt{\frac{\mu\_0 \varepsilon}{\varepsilon\_0}} \sqrt[4]{w^4 + \left(\frac{\sigma^2 w^2}{\varepsilon^2}\right)} \cos\left(\frac{\tan^{-1}\frac{\sigma}{\varepsilon \omega}}{2}\right) \leftrightarrow\\k'' &= \sqrt{\frac{\mu\_0 \varepsilon}{\varepsilon\_0}} \sqrt[4]{w^4 + \left(\frac{\sigma^2 w^2}{\varepsilon^2}\right)} \sin\left(\frac{\tan^{-1}\frac{\sigma}{\varepsilon \omega}}{2}\right). \end{split} \tag{17}$$

The real part of the wavenumber *k*<sup>0</sup> represents wavenumber inside the material and the imaginary part *k*<sup>00</sup> represents the attenuation coefficient inside the material.

$$\begin{split} v\_{\xi} &= \left(\frac{dk}{d\alpha}\right)^{-1} \\ &= \sqrt{\frac{\varepsilon\_{0}}{\mu\_{0}c}} \left\{ \frac{\left(\frac{\sigma^{2}}{\varepsilon^{2}} + 2\alpha^{2}\right) \cos\left(\frac{\tan^{-1}\frac{\sigma}{\varepsilon\alpha}}{2}\right)}{2\alpha\left(\frac{\sigma^{2}}{\varepsilon\_{0}^{2}} + \alpha^{2}\right)} + \frac{\frac{\sigma}{c}\sqrt[4]{\alpha^{4} + \left(\frac{\sigma^{2}\alpha^{2}}{\varepsilon^{2}}\right)}}{\left(\alpha^{2} + \frac{\sigma^{2}}{\varepsilon^{2}}\right)} \sin\left(\frac{\tan^{-1}\frac{\sigma}{\alpha}}{2}\right) \right\}^{-1} \\ &= \frac{2\sqrt{\frac{\varepsilon\_{0}}{\mu\_{0}c}} \left(\alpha^{2} + \frac{\sigma^{2}}{\varepsilon^{2}}\right)}{\left(\frac{\sigma^{2}}{\varepsilon^{2}w} + 2\alpha\right) \cos\left(\frac{\tan^{-1}\frac{\sigma}{\alpha}}{2}\right) + \frac{\sigma}{c}\sqrt[4]{\alpha^{4} + \left(\frac{\sigma^{2}\alpha^{2}}{\varepsilon^{2}}\right)} \sin\left(\frac{\tan^{-1}\frac{\sigma}{\alpha}}{2}\right)} \end{split} \tag{18}$$

Of course, there are various conditions that could simplify Eq. (16). The *σ=ωϵ* component in (15) is known as the dielectric *loss tangent* tan *δ* [8]*.* This figure *δ* describes a materials inherent ability to dissipate electromagnetic energy within it. For example, a material with a high conductivity will absorb—dissipate—EM radiation more quickly. Such a material has would have a high loss tangent compared to a good dielectric.

We can express the dielectric relaxation time *τ<sup>d</sup>* as *ϵ=σ*. In cases that *δ* ≪ 1 or *τ*�<sup>1</sup> *<sup>d</sup>* ≪ *ω*—as would be the case in a lossy dielectric—we can approximate the permittivity as *<sup>ϵ</sup>eff* <sup>≈</sup><sup>1</sup> � *<sup>j</sup><sup>σ</sup>* <sup>2</sup>*ωϵ* using the binomial expansion theorem. The material behaves like a standard dielectric as *vg* ¼ *c=n*, where *n* is the index of refraction, and *<sup>k</sup>*<sup>00</sup> <sup>¼</sup> *ση=*2, where *<sup>η</sup>* is the impedance of the material ffiffiffiffiffiffiffiffiffi *<sup>μ</sup>*0*=<sup>ϵ</sup>* <sup>p</sup> .

In cases where 1 ≪ *δ* or *ω* ≪ *τ*�<sup>1</sup> *<sup>d</sup>* , the case of a good conductor, *k*≈ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *σμ*0*ω=*<sup>2</sup> <sup>p</sup> ð Þ <sup>1</sup> � *<sup>j</sup>* and *vg* <sup>≈</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8*ω=σμ*<sup>0</sup> p . In the latter case, the field penetrates only slightly into the material as <sup>O</sup> *<sup>k</sup>*<sup>0</sup> � � ¼ O *<sup>k</sup>*<sup>00</sup> � �.

*Fields in Dispersive Media DOI: http://dx.doi.org/10.5772/intechopen.91432*

∇ ! � *H* ! ¼ *J* ! *<sup>f</sup>* þ *ϵ ∂E* ! *<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>σ</sup><sup>E</sup>* ! þ *∂E* !

*Electromagnetic Field Radiation in Matter*

write the dispersion relation.

*k* ¼ *ω*

q

*k*<sup>0</sup> ¼

*k*<sup>00</sup> ¼

*<sup>ϵ</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ω*<sup>2</sup> � �

*<sup>ϵ</sup>*2*<sup>ω</sup>* <sup>þ</sup> <sup>2</sup>*<sup>ω</sup>* � � *cos tan* �<sup>1</sup> *<sup>σ</sup>*

2*ω <sup>σ</sup>*<sup>2</sup> *ϵ*2 0 þ *ω*<sup>2</sup> � � þ

> 2 ffiffiffiffiffi *ϵ*0 *μ*0*ϵ* q

*ϵω* 2 � �

*<sup>k</sup>*<sup>00</sup> <sup>¼</sup> *ση=*2, where *<sup>η</sup>* is the impedance of the material ffiffiffiffiffiffiffiffiffi

8*ω=σμ*<sup>0</sup>

*vg* <sup>¼</sup> *dk*<sup>0</sup> *dω* � ��<sup>1</sup>

> ffiffiffiffiffiffiffi *ϵ*0 *μ*0*ϵ* <sup>r</sup> *<sup>σ</sup>*<sup>2</sup>

< :

¼

¼

*τ*�<sup>1</sup>

**14**

*k*≈ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*σ*2

compared to a good dielectric.

In cases where 1 ≪ *δ* or *ω* ≪ *τ*�<sup>1</sup>

*σμ*0*ω=*<sup>2</sup> <sup>p</sup> ð Þ <sup>1</sup> � *<sup>j</sup>* and *vg* <sup>≈</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

slightly into the material as <sup>O</sup> *<sup>k</sup>*<sup>0</sup> � � ¼ O *<sup>k</sup>*<sup>00</sup> � �.

permittivity as *<sup>ϵ</sup>eff* <sup>≈</sup><sup>1</sup> � *<sup>j</sup><sup>σ</sup>*

where **J** ! *<sup>∂</sup><sup>t</sup>* <sup>¼</sup> ð Þ *<sup>σ</sup>* <sup>þ</sup> *<sup>j</sup>ωϵ <sup>E</sup>*

¼

�*<sup>i</sup> tan* �<sup>1</sup> *<sup>σ</sup>* ð Þ *ϵω* 2 � �

> tan �<sup>1</sup> *<sup>σ</sup> ϵω* 2

tan �<sup>1</sup> *<sup>σ</sup> ϵω* 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ω</sup>*<sup>4</sup> <sup>þ</sup> *<sup>σ</sup>*2*ω*<sup>2</sup> *ϵ*2 � � <sup>4</sup>

> *sin tan* �<sup>1</sup> *<sup>σ</sup> ϵω* 2

<sup>2</sup>*ωϵ* using the binomial expansion theorem. The material

*<sup>d</sup>* , the case of a good conductor,

*<sup>μ</sup>*0*=<sup>ϵ</sup>* <sup>p</sup> .

p . In the latter case, the field penetrates only

*<sup>ω</sup>*<sup>2</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup> *ϵ*2 � � sin

*e*

cos

sin

The real part of the wavenumber *k*<sup>0</sup> represents wavenumber inside the material and the imaginary part *k*<sup>00</sup> represents the attenuation coefficient inside the material.

> *σ ϵ*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ω</sup>*<sup>4</sup> <sup>þ</sup> *<sup>σ</sup>*2*ω*<sup>2</sup> *ϵ*2 � � <sup>4</sup>

Of course, there are various conditions that could simplify Eq. (16). The *σ=ωϵ* component in (15) is known as the dielectric *loss tangent* tan *δ* [8]*.* This figure *δ* describes a materials inherent ability to dissipate electromagnetic energy within it. For example, a material with a high conductivity will absorb—dissipate—EM radiation more quickly. Such a material has would have a high loss tangent

We can express the dielectric relaxation time *τ<sup>d</sup>* as *ϵ=σ*. In cases that *δ* ≪ 1 or

behaves like a standard dielectric as *vg* ¼ *c=n*, where *n* is the index of refraction, and

*<sup>d</sup>* ≪ *ω*—as would be the case in a lossy dielectric—we can approximate the

� � <sup>8</sup>

q

0 @

0 @

*<sup>f</sup>* is the free current density vector produced in the presence of an electric field as exemplified in **Figure 8** and *σ* is the conductivity. We can calculate

the effective permittivity this way such that *<sup>ϵ</sup>eff* <sup>¼</sup> *<sup>ϵ</sup>* <sup>1</sup> � *<sup>j</sup><sup>σ</sup>*

¼ *ω*

s

<sup>r</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ω</sup>*<sup>4</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup>*ω*<sup>2</sup> *ϵ*2

<sup>r</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ω</sup>*<sup>4</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup>*ω*<sup>2</sup> *ϵ*2

cos tan �<sup>1</sup> *<sup>σ</sup>*

þ *σ ϵ*

*ϵω* 2 � �

> *<sup>ω</sup>*<sup>2</sup> <sup>þ</sup> *<sup>σ</sup>*<sup>2</sup> *ϵ*2 � �

> > q

� � <sup>4</sup>

� � <sup>4</sup>

ffiffiffiffiffiffiffi *μ*0*ϵ ϵ*0

This wavenumber *k* is also complex where *k* ¼ *k*<sup>0</sup> � *jk*<sup>00</sup>

s

s

ffiffiffiffiffiffiffi *μ*0*ϵ ϵ*0

ffiffiffiffiffiffiffi *μ*0*ϵ ϵ*0

ffiffiffiffiffiffiffi *μ*0*ϵ ϵ*0

� � <sup>4</sup>

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ω</sup>*<sup>4</sup> <sup>þ</sup> *<sup>σ</sup>*2*ω*<sup>2</sup> *ϵ*2

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>j</sup><sup>σ</sup> ωϵ*

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *μ*0*ϵff*ð Þ *ω*

¼

!

*ωϵ* � �

> ffiffiffiffiffiffiffi *μ*0*ϵ ϵ*0

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ω</sup>*<sup>2</sup> � *<sup>j</sup>σω ϵ*

, where

1 A \$

1 A*:*

> tan �<sup>1</sup> *<sup>σ</sup> ϵω* 2

� � (18)

9 = ;

�1

r

¼ *jωϵeff E* !

, (15)

(16)

(17)

. In this way we can

$$\frac{2\sqrt{\frac{\epsilon\alpha}{\mu\_0\epsilon}} \left(\alpha^2 + \frac{\sigma^2}{\epsilon^2}\right)}{\left(\frac{\sigma^2}{\epsilon^2\alpha} + 2\alpha\right)\cos\left(\frac{\tan^{-1}\frac{\sigma}{\alpha\epsilon}}{2}\right) + \frac{\sigma}{\epsilon}\sqrt[4]{\alpha^4 + \left(\frac{\sigma^2\alpha^2}{\epsilon^2}\right)}\sin\left(\frac{\tan^{-1}\frac{\sigma}{\alpha\epsilon}}{2}\right)}\tag{19}$$

Additionally, the attenuation is a new phenomenon resulting from the fact that the *k*-vector is complex. When we take the reciprocal of this coefficient get another quantity known as the *skin depth* or penetration depth *δp*, the extent to which the field penetrates the material. In a good conductor, *δ<sup>p</sup>* takes a value ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*=σμ*0*<sup>ω</sup>* <sup>p</sup> . The values of *ϵ* and *σ* are listed in **Table 1** for some good conductors [9].

Good conductors are characterized by an abundance of free electrons. Good conductors (good dielectrics), on the other hand, are characterized by a paucity of such electrons. Examples of good dielectrics are enumerated in **Table 2** [10].

**Figure 9** shows the interaction between an electric field and an good conductor (e.g., gold) at a wavelength of 1.55μm. The field decays very rapidly as gold has a skin depth of 5.66 nm and a dielectric relaxation time of 1.49 attoseconds (10�<sup>18</sup> s) at this wavelength. This contrasts with a lossy dielectric where the skin depth is much larger, and there is thus more field penetration as displayed in **Figure 10** with germanium where the skin depth is roughly a centimeter and the dielectric relaxation time is 66.1 picoseconds.

Conductive media have more free electrons that *move* in the presence of an electric field than those that simply vibrate. Conductive media produce a loss (attenuation) component in propagation of incoming EM radiation. Additionally, the drift movement of electrons in conductive media causes heat to which the electrical energy from the incoming field is transferred. This energy conversion causes the dissipation of the field. Nevertheless, in the context of the phenomenon of dispersion, the degree of this attenuation is a function of the frequency of the incoming EM radiation.


#### **Table 1.**

*Conductivities of some good conductors.*


#### **Table 2.**

*Conductivities of some good dielectrics.*

modal dispersion [11]. This is when the velocity of various modes changes as a function of the input wavelength. As displayed in **Figure 11**, the different modes that propagate at different group velocities and angles, which themselves are functions of an input wavelength. These modes that arise in a waveguide are the different forms that the electric field can take in a geometrically confined structure. These forms are functions of satisfying the Helmholtz equation for the geometry of the confining structure. Only a finite number of fields do this and produce the appropriate wavevector in the *z*-direction, *β*. Consequently, only a finite number of

The angle at which certain modes propagate within makes a difference as to the speed of propagation because shallower angles have less deviation from the center

In a rectangular waveguide, the total distance a mode must travel is expressed as

*cos θ<sup>x</sup> cos θ<sup>y</sup>*

where *θ<sup>x</sup>* and *θ<sup>y</sup>* are the angles of deviation in the *x-* and *y*-directions, respectively. The higher the order of the mode, the steeper the angle of the travel, and the more distance the mode must traverse through the waveguide. In **Figure 13**, the same principle holds true in an optical fiber. The rays of the fundamental modes (TM and TE) take a meridional form (a) passing through origin as they propagate and the hybrid modes—HE and EH—spiral in a helical form taking a twisted path through the fiber. Additionally, modes that have a shallower angle of interaction with the corecladding boundary will have higher group velocities. A higher-order mode has a

(20)

*<sup>L</sup>* <sup>¼</sup> *<sup>L</sup>*

*The various modes within a step-index waveguide—where h, nc, and ncl are the height and the core and*

*A mode in a rectangular waveguide propagates in the x- and y-directions at various angles.*

modes will be allowed in the structure.

*DOI: http://dx.doi.org/10.5772/intechopen.91432*

equilibrium as shown in **Figure 12**.

*Fields in Dispersive Media*

**Figure 11.**

**Figure 12.**

**17**

*cladding indices, respectively.*

**Figure 9.** *The rapid decay of an electric field upon impact interaction with gold.*

**Figure 10.** *The comparatively slow decay of an electric field upon impact interaction with germanium.*

#### **5. Modal dispersion**

So far, this chapter has covered the form that dispersion takes in media (i.e., material dispersion). However, there are other types of dispersion that exist which ultimately stem from the properties of the material. One of these other ways is

modal dispersion [11]. This is when the velocity of various modes changes as a function of the input wavelength. As displayed in **Figure 11**, the different modes that propagate at different group velocities and angles, which themselves are functions of an input wavelength. These modes that arise in a waveguide are the different forms that the electric field can take in a geometrically confined structure. These forms are functions of satisfying the Helmholtz equation for the geometry of the confining structure. Only a finite number of fields do this and produce the appropriate wavevector in the *z*-direction, *β*. Consequently, only a finite number of modes will be allowed in the structure.

The angle at which certain modes propagate within makes a difference as to the speed of propagation because shallower angles have less deviation from the center equilibrium as shown in **Figure 12**.

In a rectangular waveguide, the total distance a mode must travel is expressed as

$$L = \frac{L}{\cos \theta\_{\text{x}} \cos \theta\_{\text{y}}} \tag{20}$$

where *θ<sup>x</sup>* and *θ<sup>y</sup>* are the angles of deviation in the *x-* and *y*-directions, respectively. The higher the order of the mode, the steeper the angle of the travel, and the more distance the mode must traverse through the waveguide. In **Figure 13**, the same principle holds true in an optical fiber. The rays of the fundamental modes (TM and TE) take a meridional form (a) passing through origin as they propagate and the hybrid modes—HE and EH—spiral in a helical form taking a twisted path through the fiber.

Additionally, modes that have a shallower angle of interaction with the corecladding boundary will have higher group velocities. A higher-order mode has a

#### **Figure 11.**

*The various modes within a step-index waveguide—where h, nc, and ncl are the height and the core and cladding indices, respectively.*

**5. Modal dispersion**

**Figure 10.**

**16**

**Figure 9.**

*The rapid decay of an electric field upon impact interaction with gold.*

*Electromagnetic Field Radiation in Matter*

So far, this chapter has covered the form that dispersion takes in media (i.e., material dispersion). However, there are other types of dispersion that exist which ultimately stem from the properties of the material. One of these other ways is

*The comparatively slow decay of an electric field upon impact interaction with germanium.*

**Figure 13.** *Rays taking meridional (a) and helical (b) forms in an optical fiber [12].*

lower *β* value as its *k*-vectors in other directions are higher, leading to steeper interaction angles with the core-cladding boundary.

The way that the modes propagate through the fiber at different velocities makes the waveguide an effective multipath propagation channel where the received signal is a sum of the various scaled "echoes" of the original signal carried by the various modes that propagate therein.

As a result, this dispersion distorts the pulse and corrupts the information. As shown in **Figures 14** and **15**, the pulses deform from their respective Gaussian and rectangular forms as a result of the different path delays, a phenomenon known as *delay distortion* [13]. Calculation of modal delays is used in order to set various bit rates for optical pulses.

One way to ameliorate modal (intermodal) dispersion is to use a single-mode waveguide where the only one mode is carried through. Another way is to make use of waveguides with a parabolic index of refraction, producing a quasi-sinusoidal pathway through the waveguide as demonstrated in **Figure 16**.

> The ray switches paths as it passes the region of highest index, meaning that it spends less of its time in the region of lowest index. This is to say that the phase velocity of the wave changes within the waveguide. From a ray optics perspective, the crest of the wave has a position-varying velocity, effectively making continuous

*The intermodally dispersive time evolution of the more commonly used rectangular pulse in a multimode*

The dispersion characteristic for graded-index (GRIN) waveguides is quite different as well. **Figure 17** shows that higher wavelengths are correlated with less dispersion for GRIN waveguides. The step-index, on the other hand, maintains a

It should be noted that modal dispersion is more of a function of the *arrangement* of media, not the sole medium itself. Changes in size, cladding material, and wave-

Changes in wavelength are important in understanding how light will propagate. As shown in **Figure 18**, the way these *β* values vary with the wavelength produces various group velocities for different modes. The higher-order modes have a lower group velocity as they travel. The index of the fused silica core in the waveguide

switches through "different materials."

**Figure 15.**

**Figure 16.**

*rectangular waveguide.*

*Fields in Dispersive Media*

*DOI: http://dx.doi.org/10.5772/intechopen.91432*

**6. Chromatic dispersion**

**19**

positive wavelength dispersion characteristic.

length can affect the propagation of light therein.

*The pathway through the waveguide is sinusoidal [14] as it is bound to its axis.*

**Figure 14.** *The intermodally dispersive time evolution of a Gaussian optical pulse in a multimode rectangular waveguide.*

*Fields in Dispersive Media DOI: http://dx.doi.org/10.5772/intechopen.91432*

lower *β* value as its *k*-vectors in other directions are higher, leading to steeper

The way that the modes propagate through the fiber at different velocities makes the waveguide an effective multipath propagation channel where the received signal is a sum of the various scaled "echoes" of the original signal carried

As a result, this dispersion distorts the pulse and corrupts the information. As shown in **Figures 14** and **15**, the pulses deform from their respective Gaussian and rectangular forms as a result of the different path delays, a phenomenon known as *delay distortion* [13]. Calculation of modal delays is used in order to set various bit

One way to ameliorate modal (intermodal) dispersion is to use a single-mode waveguide where the only one mode is carried through. Another way is to make use of waveguides with a parabolic index of refraction, producing a quasi-sinusoidal

*The intermodally dispersive time evolution of a Gaussian optical pulse in a multimode rectangular waveguide.*

pathway through the waveguide as demonstrated in **Figure 16**.

interaction angles with the core-cladding boundary.

*Rays taking meridional (a) and helical (b) forms in an optical fiber [12].*

by the various modes that propagate therein.

*Electromagnetic Field Radiation in Matter*

rates for optical pulses.

**Figure 14.**

**18**

**Figure 13.**

*The intermodally dispersive time evolution of the more commonly used rectangular pulse in a multimode rectangular waveguide.*

The ray switches paths as it passes the region of highest index, meaning that it spends less of its time in the region of lowest index. This is to say that the phase velocity of the wave changes within the waveguide. From a ray optics perspective, the crest of the wave has a position-varying velocity, effectively making continuous switches through "different materials."

The dispersion characteristic for graded-index (GRIN) waveguides is quite different as well. **Figure 17** shows that higher wavelengths are correlated with less dispersion for GRIN waveguides. The step-index, on the other hand, maintains a positive wavelength dispersion characteristic.

It should be noted that modal dispersion is more of a function of the *arrangement* of media, not the sole medium itself. Changes in size, cladding material, and wavelength can affect the propagation of light therein.

#### **6. Chromatic dispersion**

Changes in wavelength are important in understanding how light will propagate. As shown in **Figure 18**, the way these *β* values vary with the wavelength produces various group velocities for different modes. The higher-order modes have a lower group velocity as they travel. The index of the fused silica core in the waveguide

**Figure 17.**

*The fundamental mode (TE0) dispersion characteristic of GRIN slab waveguide vs. that of a step-index slab waveguide.*

The plot in **Figure 18** can be reconfigured to show that each mode has an effective index of refraction in the core that varies with its wavelength. This means that different "colors" of light will have different refractive indices in the wave-

**Figure 19** shows that there is an added index of refraction that comes from the dispersion that contributes the delay of the modes; this is also known as the group

Dispersion can occur even in the context of a single-mode waveguide (of whatever geometry). In the previous section, we demonstrated a difference in *neff* as it pertains to various wavelengths stimulating a particular mode. However, we can

In the Introduction, we discussed how one can model the plasmonic impulse response of a material in the presence of an electric field as a mechanical system for

> *E*0*e jωt*

, (21)

guide. **Figure 19** shows that as *λ* increases, the group velocity decreases.

observe a similar phenomenon in a single-mode fiber.

*Dispersion creates a higher refractive index for the modes as λ increases.*

*x*

!ðÞ¼ *<sup>t</sup>* �*e=<sup>m</sup> ω*<sup>2</sup> � *ω*<sup>2</sup> 0 <sup>þ</sup> *<sup>j</sup>γω*

where *E*0, *ω*0, *e*, *m*, and *γ* are the electric field amplitude, the resonant frequency, the electric charge, the mass of the electron, and the damping constant, respectively [11]. We can multiply the quantity described in (21) be *e* to get the dipole moment vector. We additionally multiply by the dipole density *N* to get the bulk polarization. We see in (9) that the displacement field is equal to the added

index *Ng*.

**Figure 19.**

*Fields in Dispersive Media*

*DOI: http://dx.doi.org/10.5772/intechopen.91432*

which

**21**

**7. Intramodal dispersion**

#### **Figure 18.**

*The group velocities of the transverse electric (TE) modes in a fused silica rectangular waveguide as a function of the wavelengths in the visible spectrum.*

described in **Figure 18** is 1.5, meaning that light will travel roughly **<sup>2</sup> <sup>10</sup><sup>8</sup>** m/s therein. However, the contribution from dispersion increases the *effective* index *neff* "experienced" by each mode at a particular wavelength.

#### *Fields in Dispersive Media DOI: http://dx.doi.org/10.5772/intechopen.91432*

**Figure 19.** *Dispersion creates a higher refractive index for the modes as λ increases.*

The plot in **Figure 18** can be reconfigured to show that each mode has an effective index of refraction in the core that varies with its wavelength. This means that different "colors" of light will have different refractive indices in the waveguide. **Figure 19** shows that as *λ* increases, the group velocity decreases.

**Figure 19** shows that there is an added index of refraction that comes from the dispersion that contributes the delay of the modes; this is also known as the group index *Ng*.

#### **7. Intramodal dispersion**

Dispersion can occur even in the context of a single-mode waveguide (of whatever geometry). In the previous section, we demonstrated a difference in *neff* as it pertains to various wavelengths stimulating a particular mode. However, we can observe a similar phenomenon in a single-mode fiber.

In the Introduction, we discussed how one can model the plasmonic impulse response of a material in the presence of an electric field as a mechanical system for which

$$\overrightarrow{\dot{\mathbf{x}}}(t) = \frac{-e/m}{\left(\alpha^2 - \alpha\_0^2\right) + j\gamma\rho} E\_0 e^{j\alpha t},\tag{21}$$

where *E*0, *ω*0, *e*, *m*, and *γ* are the electric field amplitude, the resonant frequency, the electric charge, the mass of the electron, and the damping constant, respectively [11]. We can multiply the quantity described in (21) be *e* to get the dipole moment vector. We additionally multiply by the dipole density *N* to get the bulk polarization. We see in (9) that the displacement field is equal to the added

described in **Figure 18** is 1.5, meaning that light will travel roughly **<sup>2</sup> <sup>10</sup><sup>8</sup>** m/s therein. However, the contribution from dispersion increases the *effective* index *neff*

*The group velocities of the transverse electric (TE) modes in a fused silica rectangular waveguide as a function of*

*The fundamental mode (TE0) dispersion characteristic of GRIN slab waveguide vs. that of a step-index slab*

"experienced" by each mode at a particular wavelength.

**Figure 17.**

*Electromagnetic Field Radiation in Matter*

*waveguide.*

**Figure 18.**

**20**

*the wavelengths in the visible spectrum.*

effect of the field and the polarization response of the material. However, we get a different value because we are not dealing in the plasmonic regime. The relative permittivity or the square of the index *n* is expressed as

$$m^2(\boldsymbol{\alpha}) = \frac{\epsilon(\boldsymbol{\alpha})}{\epsilon\_0} = \mathbf{1} + \frac{Ne^2/\epsilon\_0 m}{\left(\boldsymbol{\alpha}^2 - \boldsymbol{\alpha}\_0^2\right) + j\gamma \boldsymbol{\alpha}}\tag{22}$$

*ϕN*ð Þ¼ *ν*

convert this line-shape function to one of the *k*-vector*k* where

*<sup>ϕ</sup>N*ð Þ¼ *<sup>k</sup>* <sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91432*

*P v*ð Þ¼

temperature, and the velocity, respectively.

where *ν*<sup>0</sup> is the center frequency.

ffiffiffiffiffiffiffiffiffiffiffiffiffi *m* 2*πkBT*

*e*

�*mc*<sup>2</sup> *<sup>ν</sup>*�*<sup>ν</sup>* ð Þ <sup>0</sup> <sup>2</sup> 2*ν*2

Doppler broadening *both* result in intramodal pulse broadening.

components, which *broaden* the temporal profile of the signal.

*ν* which can be expressed as

shown in (28):

*Fields in Dispersive Media*

*k*-domain like so

**23**

*P*ð Þ¼ *ν*

*c ν*0 r

*π*

where *k*<sup>0</sup> is the center spatial frequency.

*kc* <sup>2</sup>*<sup>π</sup>* � *<sup>k</sup>*0*<sup>c</sup>* 2*π* � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi *m* 2*πkBT*

r

1 *π*

*ΔνN=*2

a continuum—of *k*-vectors (and thus wavelengths) that is allowed.

<sup>N</sup> ð Þ 0, *kBT=<sup>m</sup>* while the energy *<sup>E</sup>* undergoes a chi-squared distribution *<sup>χ</sup>*<sup>2</sup>

*e* � *mv*<sup>2</sup>

<sup>þ</sup> *Δν<sup>N</sup>* 2

The spectrum described in (28) shows that there is an uncountable infinity—i.e.,

In the case of a gas, the same can be done with the Boltzmann relation that is used to compute the statistical distribution of velocities (and thus *k*-vectors) under Doppler [18] broadening where the velocity component *v* is normally distributed

<sup>2</sup>*kBT* \$*<sup>E</sup>*¼*mv*2*=*<sup>2</sup>

Because of the Doppler effect, the radiation will be shifted to a higher frequency

*v c*

*k*0

r

ffiffiffiffiffiffiffiffiffiffiffiffiffi *m* 2*πkBT*

where *m*, *kB*,*T*, and *v* are the particle's mass, Boltzmann's constant, the

*ν* ¼ *ν*<sup>0</sup> 1 þ

With (30) in mind, we can rewrite (28) in terms of *ν* and convert to the

<sup>0</sup>*kBT* \$ *P k*ð Þ¼ *<sup>c</sup>*

What this means is that in the case of a realistic laser, dispersion within a confined medium will result even if the medium allows for only *one* mode to be activated. This is to say that Lorentzian homogenous broadening and Gaussian

What is interesting is the counterintuitive connection between the frequency broadening and the temporal pulse broadening as their behaviors mirror each other. However, this is because we are dealing with *two* separate systems that happen to be connected. The first system is that of the laser linewidth and the decaying atoms and spontaneous thermal processes that diminish monochromaticity. These processes spread the spectrum out and a broader range of frequency (and wavelength) contribution to the spectrum. The second system is that of the delays accrued within the fiber-waveguide channel according to the wavelength contributions of the input spectrum. The input laser spectrum allows for a larger contribution to delays that will be "experienced" by the larger range of frequency and wavelength

*ΔνN=*2

<sup>2</sup> <sup>þ</sup> *Δν<sup>N</sup>* 2

� �<sup>2</sup> <sup>¼</sup> <sup>2</sup>*πΔνN=c*<sup>2</sup> ð Þ *k* � *k*<sup>0</sup>

> *P E*ð Þ¼ <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>4</sup>*πkBT* <sup>p</sup>

� �<sup>2</sup> , (27)

� �<sup>2</sup> *:* (28)

<sup>1</sup>ð Þ *E* as

p (29)

<sup>2</sup> <sup>þ</sup> *πΔν<sup>N</sup> c*

> *e* � *<sup>E</sup> kBT* ffiffiffi *E*

� � (30)

*e*

�*mc*<sup>2</sup> *<sup>k</sup>*�*<sup>k</sup>* ð Þ <sup>0</sup> <sup>2</sup> 2*k*2

<sup>0</sup>*kBT* (31)

ð Þ *ν* � *ν*<sup>0</sup>

where Δ*ν<sup>N</sup>* is the FWHM frequency point or the natural linewidth [17]. This is the spectral profile of a typical laser centered around a frequency *ν* ¼ *ν*0. We can

We can further imagine that there are a certain number of mechanical systems operating simultaneously. This means that for the *i*th system, there will be a certain fraction of atoms *fi* in that system; i.e., the effective dipole density for the *i*th system is *fi N*.

$$n^2(o) = \sum\_{i}^{Z} \mathbf{1} + \frac{f\_i\left(\frac{N\epsilon^2}{mc\_0}\right)}{\left(\omega^2 - \omega\_i^2\right) + j\gamma\_i o} = \sum\_{i}^{Z} \mathbf{1} + \frac{f\_i\left(\frac{N\epsilon^2}{mc\_0}\right)}{\left(\frac{4\pi^2 c^2}{\lambda^2} - \frac{4\pi^2 c^2}{\lambda\_i^2}\right) + \frac{j2\pi\gamma}{\lambda}}$$

$$= \sum\_{i}^{Z} \mathbf{1} + \frac{f\_i\left(\frac{N\epsilon^2}{mc\_0}\right)\lambda^2}{\left(4\pi^2 c^2 - \frac{4\pi^2 c^2 \lambda^2}{\lambda\_i^2}\right) + j2\pi c\gamma\lambda} = \sum\_{i}^{Z} \mathbf{1} + \frac{f\_i\lambda\_i^2 \left(\frac{N\epsilon^2}{mc\_0}\right)\lambda^2}{\left(4\pi^2 c^2 \lambda\_i^2 - 4\pi^2 c^2 \lambda^2\right) + j2\pi c\gamma\lambda\_i^2 \lambda} \tag{23}$$

where *c* is the speed of light.

We then make the assumption that *γ*, the damping constant, goes to zero for the assumption of a simple harmonic oscillator.

$$m^2(\lambda) = \sum\_{i}^{Z} \mathbf{1} + \frac{f\_i \lambda\_i^2 \left(\frac{Ne^2}{4\pi^2 c^2 m c\_0}\right) \lambda^2}{\left(\lambda\_i^2 - \lambda^2\right) + \frac{jr\lambda\_i^2}{2\pi c} \lambda} \approx \sum\_{i}^{Z} \mathbf{1} + \frac{f\_i \lambda\_i^2 \left(\frac{Ne^2}{4\pi^2 c^2 m c\_0}\right) \lambda^2}{\left(\lambda\_i^2 - \lambda^2\right)}\tag{24}$$

We can make further simplifications to say that

$$m^2(\lambda) = \sum\_{i}^{Z} \mathbf{1} + \frac{G\_i \lambda^2}{\left(\lambda\_i^2 - \lambda^2\right)},\tag{25}$$

where *λ<sup>i</sup>* is the *i*th resonant wavelength and the coefficient *Gi* is the strength of that resonance. The coefficient *Gi* is expressed as

$$G\_i = \left(\frac{Ne^2}{4\pi^2c^2m\epsilon\_0}\right)f\_i\lambda\_i^2\tag{26}$$

These coefficients are known as Sellmeier coefficients [15]. These coefficients play a role in determining the group velocity of a particular wavelength in a fiber.

One of the misconceptions brought about by our knowledge of lasers is the "myth of monochromaticity." The light sources we use—even for optical fibers are never fully monochromatic. Rather they emit a spectrum of frequencies (and wavelengths) concentrated around a particular center [16]. This spectrum will generate respectively scaled eigenmodes with their own group velocities and delays through the fiber. This causes pulse broadening even when only a single mode is propagating.

For a typical laser with a homogenous linewidth undergoing natural line broadening due to atomic decay, we can express the line-shape function as a typical Lorentzian for which

*Fields in Dispersive Media DOI: http://dx.doi.org/10.5772/intechopen.91432*

effect of the field and the polarization response of the material. However, we get a different value because we are not dealing in the plasmonic regime. The relative

¼ 1 þ

We can further imagine that there are a certain number of mechanical systems operating simultaneously. This means that for the *i*th system, there will be a certain fraction of atoms *fi* in that system; i.e., the effective dipole density for the *i*th

> <sup>¼</sup> <sup>X</sup> *Z*

*i* 1 þ

We then make the assumption that *γ*, the damping constant, goes to zero for the

X *Z*

*i* 1 þ *fi λ*2 *i*

*Giλ*<sup>2</sup> *λ*2

> *fi λ*2

*Ne*2*=ϵ*0*m*

(22)

*i λ*

(23)

*ω*<sup>2</sup> � *ω*<sup>2</sup> 0 � � <sup>þ</sup> *<sup>j</sup>γω*

*Z*

*fi Ne*<sup>2</sup> *mϵ*<sup>0</sup> � �

*fi λ*2 *i Ne*<sup>2</sup> *mϵ*<sup>0</sup> � � *λ*2

*Ne*<sup>2</sup> 4*π*2*c*2*mϵ*<sup>0</sup> � �

*<sup>i</sup>* � *<sup>λ</sup>*<sup>2</sup> � � , (25)

*<sup>i</sup>* (26)

*λ*2

*λ*2

*<sup>i</sup>* � *<sup>λ</sup>*<sup>2</sup> � � (24)

*<sup>i</sup>* � <sup>4</sup>*π*<sup>2</sup>*c*<sup>2</sup>*λ*<sup>2</sup> � � <sup>þ</sup> *<sup>j</sup>*2*πcγλ*<sup>2</sup>

<sup>þ</sup> *<sup>j</sup>*2*πc<sup>γ</sup> λ*

4*π*2*c*<sup>2</sup> *<sup>λ</sup>*<sup>2</sup> � <sup>4</sup>*π*2*c*<sup>2</sup> *λ*2 *i* � �

4*π*<sup>2</sup>*c*<sup>2</sup>*λ*<sup>2</sup>

*i* 1 þ

permittivity or the square of the index *n* is expressed as

ð Þ¼ *<sup>ω</sup> ϵ ω*ð Þ *ϵ*0

> *fi Ne*<sup>2</sup> *mϵ*<sup>0</sup> � �

þ *j*2*πcγλ*

*Ne*<sup>2</sup> 4*π*2*c*2*mϵ*<sup>0</sup> � �

*<sup>i</sup>* � *<sup>λ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>j</sup>γλ*<sup>2</sup>

*λ*2

*i* <sup>2</sup>*π<sup>c</sup> λ* ≈|{z} *γ*!0

ð Þ¼ *<sup>λ</sup>* <sup>X</sup> *Z*

*i* 1 þ

*Gi* <sup>¼</sup> *Ne*<sup>2</sup>

where *λ<sup>i</sup>* is the *i*th resonant wavelength and the coefficient *Gi* is the strength of

4*π*<sup>2</sup>*c*<sup>2</sup>*mϵ*<sup>0</sup> � �

These coefficients are known as Sellmeier coefficients [15]. These coefficients play a role in determining the group velocity of a particular wavelength in a fiber. One of the misconceptions brought about by our knowledge of lasers is the "myth of monochromaticity." The light sources we use—even for optical fibers are never fully monochromatic. Rather they emit a spectrum of frequencies (and wavelengths) concentrated around a particular center [16]. This spectrum will generate respectively scaled eigenmodes with their own group velocities and delays through the fiber. This causes pulse broadening even when only a single mode is

For a typical laser with a homogenous linewidth undergoing natural line broad-

ening due to atomic decay, we can express the line-shape function as a typical

*<sup>ω</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> *i* � � <sup>þ</sup> *<sup>j</sup>γi<sup>ω</sup>* <sup>¼</sup> <sup>X</sup>

*n*2

system is *fi*

<sup>¼</sup> <sup>X</sup> *Z*

*i* 1 þ

*n*2

propagating.

**22**

Lorentzian for which

*N*.

*n*2

ð Þ¼ *<sup>ω</sup>* <sup>X</sup> *Z*

*Electromagnetic Field Radiation in Matter*

where *c* is the speed of light.

ð Þ¼ *<sup>λ</sup>* <sup>X</sup> *Z*

*i* 1 þ *fi λ*2 *i*

*i* 1 þ

> *fi Ne*<sup>2</sup> *mϵ*<sup>0</sup> � � *λ*2

<sup>4</sup>*π*<sup>2</sup>*c*<sup>2</sup> � <sup>4</sup>*π*2*c*2*λ*<sup>2</sup> *λ*2 *i*

assumption of a simple harmonic oscillator.

*λ*2

We can make further simplifications to say that

that resonance. The coefficient *Gi* is expressed as

*n*2

� �

$$\phi\_N(\nu) = \frac{1}{\pi} \frac{\Delta \nu\_N / 2}{\left(\nu - \nu\_0\right)^2 + \left(\frac{\Delta \nu\_N}{2}\right)^2},\tag{27}$$

where Δ*ν<sup>N</sup>* is the FWHM frequency point or the natural linewidth [17]. This is the spectral profile of a typical laser centered around a frequency *ν* ¼ *ν*0. We can convert this line-shape function to one of the *k*-vector*k* where

$$\phi\_N(k) = \frac{1}{\pi} \frac{\Delta\nu\_N/2}{\left(\frac{kc}{2\pi} - \frac{k\nu c}{2\pi}\right)^2 + \left(\frac{\Delta\nu\_N}{2}\right)^2} = \frac{2\pi\Delta\nu\_N/c^2}{\left(k - k\_0\right)^2 + \left(\frac{\pi\Delta\nu\_N}{c}\right)^2}.\tag{28}$$

where *k*<sup>0</sup> is the center spatial frequency.

The spectrum described in (28) shows that there is an uncountable infinity—i.e., a continuum—of *k*-vectors (and thus wavelengths) that is allowed.

In the case of a gas, the same can be done with the Boltzmann relation that is used to compute the statistical distribution of velocities (and thus *k*-vectors) under Doppler [18] broadening where the velocity component *v* is normally distributed <sup>N</sup> ð Þ 0, *kBT=<sup>m</sup>* while the energy *<sup>E</sup>* undergoes a chi-squared distribution *<sup>χ</sup>*<sup>2</sup> <sup>1</sup>ð Þ *E* as shown in (28):

$$P(\nu) = \sqrt{\frac{m}{2\pi k\_B T}} \ e^{-\frac{mv^2}{2k\_B T}} \underset{E = m v^2/2}{\longleftrightarrow} P(E) = \frac{1}{\sqrt{4\pi k\_B T}} \frac{e^{-\frac{E}{k\_B T}}}{\sqrt{E}} \tag{29}$$

where *m*, *kB*,*T*, and *v* are the particle's mass, Boltzmann's constant, the temperature, and the velocity, respectively.

Because of the Doppler effect, the radiation will be shifted to a higher frequency *ν* which can be expressed as

$$
\omega = \nu\_0 \left( \mathbf{1} + \frac{\nu}{c} \right) \tag{30}
$$

where *ν*<sup>0</sup> is the center frequency.

With (30) in mind, we can rewrite (28) in terms of *ν* and convert to the *k*-domain like so

$$P(\nu) = \frac{c}{\nu\_0} \sqrt{\frac{m}{2\pi k\_B T}} \ e^{-\frac{m^2 (\nu - \nu\_0)^2}{2k\_0^2 k\_B T}} \leftrightarrow P(k) = \frac{c}{k\_0} \sqrt{\frac{m}{2\pi k\_B T}} \ e^{-\frac{m^2 (k - k\_0)^2}{2k\_0^2 k\_B T}}\tag{31}$$

What this means is that in the case of a realistic laser, dispersion within a confined medium will result even if the medium allows for only *one* mode to be activated. This is to say that Lorentzian homogenous broadening and Gaussian Doppler broadening *both* result in intramodal pulse broadening.

What is interesting is the counterintuitive connection between the frequency broadening and the temporal pulse broadening as their behaviors mirror each other. However, this is because we are dealing with *two* separate systems that happen to be connected. The first system is that of the laser linewidth and the decaying atoms and spontaneous thermal processes that diminish monochromaticity. These processes spread the spectrum out and a broader range of frequency (and wavelength) contribution to the spectrum. The second system is that of the delays accrued within the fiber-waveguide channel according to the wavelength contributions of the input spectrum. The input laser spectrum allows for a larger contribution to delays that will be "experienced" by the larger range of frequency and wavelength components, which *broaden* the temporal profile of the signal.

As noise is intrinsic to a lasing system so is dispersion when that system is directed into a medium; this is especially true in the context of confined or *bounded* media and their applications in modern communication networks.

**References**

*Fields in Dispersive Media*

[1] Le Bourhis E. Appendix 1. Glass; Mechanics and Technology. 1st ed. Weinheim: Wiley-VCH; 2007

*DOI: http://dx.doi.org/10.5772/intechopen.91432*

[online]. 2010. Available from: https:// www.engineeringtoolbox.com/relativepermittivity-d\_1660.html [Accessed: 09

Optoelectronics. 1st ed. Chicago: Irwin;

[12] Mitschke F. Fiber Optics: Physics and Technology. Heidelberg, Dordrecht, London, New York: Springer; 2009.

[13] Mitschke F. Fiber Optics: Physics and Technology. Heidelberg, Dordrecht, London, New York: Springer. 2009.

Optoelectronics. 1st ed. Chicago: Irwin;

[15] Sellmeier W. Über die durch die Aetherschwingungen erregten

[16] Kasap SO. Optoelectronics and Photonics–Principles and Practices. 1st ed. Upper Saddle River, NJ: Prentice

[17] Peach G. Theory of the pressure broadening and shift of spectral lines. Advances in Physics. 1981;**30**(3):367-474

[18] Siegman AE. Lasers. 1st ed. Mill Valley, CA: University Science Books;

Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien (II. Theil). Annalen der Physik und Chemie.

[14] Pollock CR. Graded-index Waveguides. Fundamentals of

1872;**223**(11):386-403

Hall; 2001. p. 78

1986

[11] Pollock CR. Dispersion in Waveguides. Fundamentals of

October 2019]

2003

p. 22

p. 21

2003

[2] Rao NN. Waveguide Principles. Fundamentals of Electromagnetics for Electrical and Computer Engineering. 1st ed. Upper Saddle River, NJ: Pearson/ Prentice Hall; 2009. pp. 290-338. DOI:

[3] Isaac N. Opticks. London: Royal Society; 1704. ISBN: 0-486-60205-2

[4] Staelin DH. Electromagnetics and Applications. Cambridge, MA: Department of Electrical Engineering and Computer Science, Massachusetts

[5] Charles K. Introduction to Solid State Physics. 8th ed. Hoboken, NJ: Wiley;

Photoconductivity of germanium. In: Seitz F, Turnbull D, editors. Solid State Physics. Vol. 8. New York: Academic

[7] Rao NN. Waveguide Principles. Fundamentals of Electromagnetics for Electrical and Computer Engineering. 1st ed. Upper Saddle River, NJ: Pearson/

Prentice Hall; 2009. p. 142. DOI: 10.1002/978047974704.ch8

VV, Varadan VK. Microwave

Wiley; 2004

**25**

[8] Chen LF, Ong CK, Neo CP, Varadan

Electronics: Measurement and Materials Characterization. 1st ed. Chichester:

[9] Rao NN. Waveguide Principles. Fundamentals of Electromagnetics for Electrical and Computer Engineering. 1st ed. Upper Saddle River, NJ: Pearson/

Prentice Hall; 2009. p. 143. DOI: 10.1002/978047974704.ch8

[10] Engineering ToolBox. Relative Permittivity—The Dielectric Constant

10.1002/978047974704.ch8

Institute of Technology; 2011

[6] Newman R, Tyler WW.

Press; 1959. pp. 49-107

2005

#### **8. Conclusion**

Dispersion has been discussed in this chapter as it relates to electric fields in various media. We first begin with an understanding of electron responses to EM radiation; i.e., the light-matter interaction as understood in classical electromagnetics. When then used this concept to understand how the velocity of a signal through a medium is a function of the frequency of that signal.

Additionally, we explained and developed the concept of a group velocity with a simple analogy and then moved on to discuss that concept at length in various media. We contrasted group velocity with phase velocity and developed dispersion relations which contrasted the two.

We took the discussion of dispersive materials to light-confining structures. Optical waveguides and fibers function as dispersive materials not only because of their material properties but also because these materials are arranged. The arrangement of these materials—i.e., core-cladding indices, size, index distribution, etc.—all plays a role in the phenomenon of dispersion. This also influences the allowable forms of light that propagate in one of these confined media; the properties of the medium affect these modes of propagation. We additionally learned about ways in which this effect can be eliminated for engineering applications.

We finally ended this chapter with a discussion of effect of intramodally dispersive media on the frequency spectrum and time evolution of electric field signals. This was demonstrated by first dismantling the "myth of monochromaticity" and understanding light itself as a spectrum. This helped us better understand how a single mode can undergo dispersion with itself. It most importantly demonstrated the degree to which dispersion in inherent in EM propagation through all media.

#### **Author details**

Emeka Ikpeazu University of Virginia, Charlottesville, Virginia, United States of America

\*Address all correspondence to: eikpeazu@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

As noise is intrinsic to a lasing system so is dispersion when that system is directed into a medium; this is especially true in the context of confined or *bounded*

Dispersion has been discussed in this chapter as it relates to electric fields in various media. We first begin with an understanding of electron responses to EM radiation; i.e., the light-matter interaction as understood in classical electromagnetics. When then used this concept to understand how the velocity of a signal

Additionally, we explained and developed the concept of a group velocity with a

simple analogy and then moved on to discuss that concept at length in various media. We contrasted group velocity with phase velocity and developed dispersion

We took the discussion of dispersive materials to light-confining structures. Optical waveguides and fibers function as dispersive materials not only because of their material properties but also because these materials are arranged. The

arrangement of these materials—i.e., core-cladding indices, size, index distribution, etc.—all plays a role in the phenomenon of dispersion. This also influences the allowable forms of light that propagate in one of these confined media; the properties of the medium affect these modes of propagation. We additionally learned about ways in which this effect can be eliminated for engineering applications.

We finally ended this chapter with a discussion of effect of intramodally dispersive media on the frequency spectrum and time evolution of electric field signals. This was demonstrated by first dismantling the "myth of monochromaticity" and understanding light itself as a spectrum. This helped us better understand how a single mode can undergo dispersion with itself. It most importantly demonstrated the degree to which dispersion in inherent in EM propagation through all media.

University of Virginia, Charlottesville, Virginia, United States of America

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: eikpeazu@gmail.com

provided the original work is properly cited.

media and their applications in modern communication networks.

through a medium is a function of the frequency of that signal.

**8. Conclusion**

**Author details**

Emeka Ikpeazu

**24**

relations which contrasted the two.

*Electromagnetic Field Radiation in Matter*

[1] Le Bourhis E. Appendix 1. Glass; Mechanics and Technology. 1st ed. Weinheim: Wiley-VCH; 2007

[2] Rao NN. Waveguide Principles. Fundamentals of Electromagnetics for Electrical and Computer Engineering. 1st ed. Upper Saddle River, NJ: Pearson/ Prentice Hall; 2009. pp. 290-338. DOI: 10.1002/978047974704.ch8

[3] Isaac N. Opticks. London: Royal Society; 1704. ISBN: 0-486-60205-2

[4] Staelin DH. Electromagnetics and Applications. Cambridge, MA: Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology; 2011

[5] Charles K. Introduction to Solid State Physics. 8th ed. Hoboken, NJ: Wiley; 2005

[6] Newman R, Tyler WW. Photoconductivity of germanium. In: Seitz F, Turnbull D, editors. Solid State Physics. Vol. 8. New York: Academic Press; 1959. pp. 49-107

[7] Rao NN. Waveguide Principles. Fundamentals of Electromagnetics for Electrical and Computer Engineering. 1st ed. Upper Saddle River, NJ: Pearson/ Prentice Hall; 2009. p. 142. DOI: 10.1002/978047974704.ch8

[8] Chen LF, Ong CK, Neo CP, Varadan VV, Varadan VK. Microwave Electronics: Measurement and Materials Characterization. 1st ed. Chichester: Wiley; 2004

[9] Rao NN. Waveguide Principles. Fundamentals of Electromagnetics for Electrical and Computer Engineering. 1st ed. Upper Saddle River, NJ: Pearson/ Prentice Hall; 2009. p. 143. DOI: 10.1002/978047974704.ch8

[10] Engineering ToolBox. Relative Permittivity—The Dielectric Constant [online]. 2010. Available from: https:// www.engineeringtoolbox.com/relativepermittivity-d\_1660.html [Accessed: 09 October 2019]

[11] Pollock CR. Dispersion in Waveguides. Fundamentals of Optoelectronics. 1st ed. Chicago: Irwin; 2003

[12] Mitschke F. Fiber Optics: Physics and Technology. Heidelberg, Dordrecht, London, New York: Springer; 2009. p. 22

[13] Mitschke F. Fiber Optics: Physics and Technology. Heidelberg, Dordrecht, London, New York: Springer. 2009. p. 21

[14] Pollock CR. Graded-index Waveguides. Fundamentals of Optoelectronics. 1st ed. Chicago: Irwin; 2003

[15] Sellmeier W. Über die durch die Aetherschwingungen erregten Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien (II. Theil). Annalen der Physik und Chemie. 1872;**223**(11):386-403

[16] Kasap SO. Optoelectronics and Photonics–Principles and Practices. 1st ed. Upper Saddle River, NJ: Prentice Hall; 2001. p. 78

[17] Peach G. Theory of the pressure broadening and shift of spectral lines. Advances in Physics. 1981;**30**(3):367-474

[18] Siegman AE. Lasers. 1st ed. Mill Valley, CA: University Science Books; 1986

**Chapter 3**

**Abstract**

of the media inside.

**1. Introduction**

**27**

The Electrical Properties of Soils

The electric properties of the soils are very important for several sciences like telecommunications, electrical engineering, geophysics, and agriculture. There are semiempirical dielectric models for soils, which represent the real and imaginary part of the dielectric permittivity as the function of the frequency. The measurement methods to obtain the dielectric properties of soils are described for different bands of frequencies from some kHz to several GHz. The parallel plate capacitors are widely used to measure dielectric properties. The transmission line method of a coaxial transmission line can be used in frequency domain and time domain. The time domain technique with transmission lines is usually called time-domain reflectometry (TDR), because it is based on the voltage measurement as a function of time of pulses. The frequency domain technique with transmission lines is based

on the reflection coefficient measurement of the transmission line. The

transmission line method is described with short load and open-circuit load because it is useful in obtaining the characteristic impedance and the electric permittivity

**Keywords:** soil, dielectric properties, permittivity, models, measurement method

The knowledge of electrical properties of soils in physics and electrical engineering are important for many applications. The long-distance electromagnetic telegraph systems from 1820 are used, with two or more wires to carry the signal and the return currents. It was discovered that the earth could be used as a return path to complete the circuit, making the return wire unnecessary [1]. However, during dry weather, the earth connection often developed a high resistance, requiring water on the earth electrode to enable the telegraph to ring [1].

An important radio propagation and engineering problem has been solved in 1909 by A. Sommerfeld. He has solved the general problem of the effect of the finite conductivity of the ground on the radiation from a short vertical antenna at the surface of a plane earth. The surface wave propagation is produced over real ground for the medium frequency AM radio service, where the attenuation of the electric

with Their Applications

and Engineering

*Walter Gustavo Fano*

to Agriculture, Geophysics,

#### **Chapter 3**

## The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics, and Engineering

*Walter Gustavo Fano*

### **Abstract**

The electric properties of the soils are very important for several sciences like telecommunications, electrical engineering, geophysics, and agriculture. There are semiempirical dielectric models for soils, which represent the real and imaginary part of the dielectric permittivity as the function of the frequency. The measurement methods to obtain the dielectric properties of soils are described for different bands of frequencies from some kHz to several GHz. The parallel plate capacitors are widely used to measure dielectric properties. The transmission line method of a coaxial transmission line can be used in frequency domain and time domain. The time domain technique with transmission lines is usually called time-domain reflectometry (TDR), because it is based on the voltage measurement as a function of time of pulses. The frequency domain technique with transmission lines is based on the reflection coefficient measurement of the transmission line. The transmission line method is described with short load and open-circuit load because it is useful in obtaining the characteristic impedance and the electric permittivity of the media inside.

**Keywords:** soil, dielectric properties, permittivity, models, measurement method

#### **1. Introduction**

The knowledge of electrical properties of soils in physics and electrical engineering are important for many applications. The long-distance electromagnetic telegraph systems from 1820 are used, with two or more wires to carry the signal and the return currents. It was discovered that the earth could be used as a return path to complete the circuit, making the return wire unnecessary [1]. However, during dry weather, the earth connection often developed a high resistance, requiring water on the earth electrode to enable the telegraph to ring [1].

An important radio propagation and engineering problem has been solved in 1909 by A. Sommerfeld. He has solved the general problem of the effect of the finite conductivity of the ground on the radiation from a short vertical antenna at the surface of a plane earth. The surface wave propagation is produced over real ground for the medium frequency AM radio service, where the attenuation of the electric

field depends on the dielectric properties of the soil, mainly of the dielectric losses [2]. Considering the word "Soil" means the uppermost layer of the earth's crust, it contains the organic as well as mineral matter. From 1936 up to 1941, Norton, Van der Pol, and Bremmer made the computation of the field strengths at distant points on the flat and spherical Earth's surface [3, 4].

Δ*p* ! <sup>¼</sup> <sup>X</sup> *m pm*

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics…*

*P* ! ¼ P *<sup>m</sup> pm* �! Δ*v*

In **Figure 1**, the external electric field applied to a dielectric material and the

and when the electric field applied to the dielectric is not high powered, the relationship between the polarization and the electric field will be linear, as is the case

> *P* ! ¼ *χE* !

*D* !

*D* !

and the electric field vector *E*

*D* ! ¼ ϵ*E* !

means of the Gaussian law with the free charges; therefore

It is convenient to define the electric displacement, because it allows to relate by

¼ ϵ0*E* þ *P*

¼ ϵ0*E* þ *χE*

The electrical permittivity is defined as the relationship between the electric

!

!

!

, and thus

From the macroscopic point of view in most of the dielectric material, when the electric field is canceled, the polarization in the material will be nullified. In addition, the polarization of the material will vary as the electric field varies, i.e.,

. The variable *χ*ð Þ *E* is called electrical susceptibility of the material,

And the polarization

*P E*ð Þ ��! <sup>¼</sup> *<sup>χ</sup>*ð Þ *<sup>E</sup> <sup>E</sup>*

of a soil [10]:

Then

**Figure 1.**

**29**

displacement vector *D*

!

*Polarization applying external electric field E, to a dielectric material.*

!

resulting polarization can be observed.

*DOI: http://dx.doi.org/10.5772/intechopen.88989*

�! (3)

(4)

(5)

(6)

(7)

(8)

In agriculture applications, the electrical resistivity methods have been introduced by Conrad Schlumberger in France and Frank Wenner in the United States, for the evaluation of ground electrical resistivity. In saline soils, the electric conductivity measured is high, and the effects of salinity are manifested in the loss of stand, reduced rates of plant growth, reduced yields, and in severe cases, total crop failure [5].

The applications like the protection of electrical generating plant are necessary to provide earth connections with low electrical resistance. The radio transmitting and receiving stations for broadcasting is generally covered by radiation transmitted directly along the ground [6]. In electrical engineering, "ground" is the reference point in an electrical circuit from which voltages are measured.

For archeology, geophysics, engineering, and military applications, the so-called ground-penetrating radar (GPR) is a technique widely used. The radar signal is an electromagnetic wave that propagates through the earth, and its signal is reflected when an object appears or there is a change in the properties of the earth. In order to determine the depth of an object under the ground, it is necessary to know the electrical properties of the soil [7].

#### **2. Fundamental concepts**

The equations that relate the electric field (E) and magnetic field (H) are based on the electromagnetic theory formulated by James Clerk Maxwell in 1864, whose validity has allowed great advances in diverse areas, such as telecommunications, electricity, electronics, and materials [8].

Regarding the behavior of the materials under the action of an electric field, in the conductive materials, the charges can move freely, meaning that the electrons are not associated with an atomic nucleus. In the case of dielectric materials, the charges are associated with an atom or specific molecule [9]. There are two main mechanisms where the electric field distorts the distribution of charge in a dielectric, stretching and rotation. The relationship between the electric dipole moment inducted under the action of an applied electric field is called atomic electric polarizability *p* ! and can be written as

$$
\vec{p} = a\vec{E}\tag{1}
$$

In a material with an applied electric field, a convenient definition is to consider the contributions of the dipole moment per unit volume; this parameter is called polarization, which is a macroscopic definition instead of a molecular or atomic definition [9, 10]:

$$
\overrightarrow{P} = \frac{\Delta \overrightarrow{p}}{\Delta v} \tag{2}
$$

It is evident that the contributions of the electric dipole moment in a volume element Δ*v* are given by the sum of the microscopic contributions *pm* !; therefore you can write:

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics… DOI: http://dx.doi.org/10.5772/intechopen.88989*

$$
\Delta \overrightarrow{p} = \sum\_{m} \overrightarrow{p\_{m}} \tag{3}
$$

And the polarization

field depends on the dielectric properties of the soil, mainly of the dielectric losses [2]. Considering the word "Soil" means the uppermost layer of the earth's crust, it contains the organic as well as mineral matter. From 1936 up to 1941, Norton, Van der Pol, and Bremmer made the computation of the field strengths at distant

In agriculture applications, the electrical resistivity methods have been introduced by Conrad Schlumberger in France and Frank Wenner in the United States, for the evaluation of ground electrical resistivity. In saline soils, the electric conductivity measured is high, and the effects of salinity are manifested in the loss of stand, reduced rates of plant growth, reduced yields, and in severe cases, total

The applications like the protection of electrical generating plant are necessary to provide earth connections with low electrical resistance. The radio transmitting and receiving stations for broadcasting is generally covered by radiation transmitted directly along the ground [6]. In electrical engineering, "ground" is the reference

For archeology, geophysics, engineering, and military applications, the so-called ground-penetrating radar (GPR) is a technique widely used. The radar signal is an electromagnetic wave that propagates through the earth, and its signal is reflected when an object appears or there is a change in the properties of the earth. In order to determine the depth of an object under the ground, it is necessary to know the

The equations that relate the electric field (E) and magnetic field (H) are based on the electromagnetic theory formulated by James Clerk Maxwell in 1864, whose validity has allowed great advances in diverse areas, such as telecommunications,

Regarding the behavior of the materials under the action of an electric field, in the conductive materials, the charges can move freely, meaning that the electrons are not associated with an atomic nucleus. In the case of dielectric materials, the charges are associated with an atom or specific molecule [9]. There are two main mechanisms where the electric field distorts the distribution of charge in a dielectric, stretching and rotation. The relationship between the electric dipole moment inducted under the action of an applied electric field is called atomic electric

> *p* ! ¼ *αE* !

*P* ! <sup>¼</sup> <sup>Δ</sup>*<sup>p</sup>* ! Δ*v*

element Δ*v* are given by the sum of the microscopic contributions *pm*

It is evident that the contributions of the electric dipole moment in a volume

In a material with an applied electric field, a convenient definition is to consider the contributions of the dipole moment per unit volume; this parameter is called polarization, which is a macroscopic definition instead of a molecular or atomic

(1)

(2)

!; therefore

points on the flat and spherical Earth's surface [3, 4].

*Electromagnetic Field Radiation in Matter*

point in an electrical circuit from which voltages are measured.

crop failure [5].

electrical properties of the soil [7].

electricity, electronics, and materials [8].

! and can be written as

**2. Fundamental concepts**

polarizability *p*

definition [9, 10]:

you can write:

**28**

$$
\overrightarrow{P} = \frac{\sum\_{m} \overrightarrow{p\_{m}}}{\Delta v} \tag{4}
$$

In **Figure 1**, the external electric field applied to a dielectric material and the resulting polarization can be observed.

From the macroscopic point of view in most of the dielectric material, when the electric field is canceled, the polarization in the material will be nullified. In addition, the polarization of the material will vary as the electric field varies, i.e., !

*P E*ð Þ ��! <sup>¼</sup> *<sup>χ</sup>*ð Þ *<sup>E</sup> <sup>E</sup>* . The variable *χ*ð Þ *E* is called electrical susceptibility of the material, and when the electric field applied to the dielectric is not high powered, the relationship between the polarization and the electric field will be linear, as is the case of a soil [10]:

$$
\overrightarrow{P} = \chi \overrightarrow{E} \tag{5}
$$

It is convenient to define the electric displacement, because it allows to relate by means of the Gaussian law with the free charges; therefore

$$
\overrightarrow{D} = \epsilon\_0 E + \overrightarrow{P} \tag{6}
$$

Then

$$
\vec{D} = \epsilon\_0 E + \chi \vec{E} \tag{7}
$$

The electrical permittivity is defined as the relationship between the electric displacement vector *D* ! and the electric field vector *E* ! , and thus

$$
\overrightarrow{D} = \epsilon \overrightarrow{E} \tag{8}
$$

**Figure 1.** *Polarization applying external electric field E, to a dielectric material.*

Result

$$
\epsilon = \epsilon \mathbf{0} + \chi \tag{9}
$$

at the surface of the solid constituents, like in clay soil; the electrical charges located at the surface of the clay particles lead to greater electrical conductivity than in

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics…*

There are evidences that for compacted soils of clay, it exhibits an anisotropic behavior in the resistivity measured in the horizontal and vertical directions [14]. The literature contains the measurement of the dielectric properties of soils at different frequencies with slotted lines and time-domain reflectometry (TDR)

The measured variations of the electric permittivity of soils with fractions of sand, silt, and clay and with volumetric moisture content have been studied for

The coaxial probe technique terminated in the material under test has been used to measure the dielectric properties of the vegetation. The dielectric data reported are based on measurements of the amplitude and phase of the reflection coefficient

The transmission line method has been used to measure the dielectric properties [19, 20]. These transmission lines are coaxial, quasi-coaxial, and two-wire transmission lines. Consider a transmission line with a homogenous dielectric material inside, and the propagation is transverse electromagnetic mode (TEM), where the electric and magnetic field are perpendicular to the propagation direction; this can

The separation between the conductive cylinders that form the coax transmission line should be much lower than the wavelength of the signal that propagates, so the transmission line will not be affected by the propagation modes of high orders,

Coaxial transmission lines are widely used for the transmission of radiofrequency signals and its application in radiocommunications and for broadcasting [21]. The transmission lines allow the connection between a generator or emitter and a load or antenna. The air coaxial transmission line consists of two cylindrical conductors, with air between both conductors. These metallic conductors are those that impose the boarder conditions that must comply with the electric and magnetic

fields of the electromagnetic wave that travel inside the line. The coaxial

*Section of the two-wire transmission line with the electric and magnetic fields.*

frequency of 440 MHz used by the radar observations [16].

coarse-textured soils [13].

*DOI: http://dx.doi.org/10.5772/intechopen.88989*

of a coaxial probe [17, 18].

**3. Transmission line fundamentals**

be observed in **Figures 3** and **4**.

such as the *TE*<sup>11</sup> [19].

**Figure 3.**

**31**

methods [15].

It is convenient to define [11]:

$$\chi = \epsilon \,\%\,\tag{10}$$

Result

$$
\epsilon = \epsilon\_0 (\mathbf{1} + \boldsymbol{\chi}\_r) \tag{11}
$$

The electric properties of the material are completely defined by means of *ε* or *χ* [10].

In problems with electromagnetic fields, four vectors are defined: E and B; D and H. These vectors are assumed to be finite throughout the entire field, and at all ordinary points to be continuous functions of position and time, with continuous derivatives [12]. The constitutive relations link the vectors of the fields *B* ! with *H* ! and *D* ! with *E* ! , usually dependent on the frequency [8]:

$$\begin{aligned} \overrightarrow{B} &= \mu(w)\overrightarrow{H} \\ \overrightarrow{D} &= \varepsilon(w)\overrightarrow{E} \\ \overrightarrow{J} &= \sigma(w)\overrightarrow{E} \end{aligned} \tag{12}$$

For the electromagnetic propagation in soils, the parameters *μ*, ϵ, and *σ* must be determined. The soils are usually nonmagnetic media; therefore the magnetic permeability will be that of the vacuum *μ* ¼ *μ*0, and the variables to be determined will be the electric permittivity ϵ and the electric conductivity *σ*.

The electrical resistivity obtained by soil mapping exhibits a large range of values from 1Ω*=m* for saline soil to several 10<sup>5</sup> Ω*=m* for dry soil overlaying crystalline rocks [13]. In **Figure 2**, the resistivity of different soils can be observed. The electrical conductivity is related to the particle size by the electrical charge density

**Figure 2.** *Table of electric resistivity* ½ � *Ω=m and electric conductivity* ½ � *σ=m of soils (Ref. Samoulian et al.) [13].*

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics… DOI: http://dx.doi.org/10.5772/intechopen.88989*

at the surface of the solid constituents, like in clay soil; the electrical charges located at the surface of the clay particles lead to greater electrical conductivity than in coarse-textured soils [13].

There are evidences that for compacted soils of clay, it exhibits an anisotropic behavior in the resistivity measured in the horizontal and vertical directions [14].

The literature contains the measurement of the dielectric properties of soils at different frequencies with slotted lines and time-domain reflectometry (TDR) methods [15].

The measured variations of the electric permittivity of soils with fractions of sand, silt, and clay and with volumetric moisture content have been studied for frequency of 440 MHz used by the radar observations [16].

The coaxial probe technique terminated in the material under test has been used to measure the dielectric properties of the vegetation. The dielectric data reported are based on measurements of the amplitude and phase of the reflection coefficient of a coaxial probe [17, 18].

#### **3. Transmission line fundamentals**

Result

Result

and *D* !

**Figure 2.**

**30**

with *E* !

It is convenient to define [11]:

*Electromagnetic Field Radiation in Matter*

ϵ ¼ ϵ<sup>0</sup> þ *χ* (9)

*χ* ¼ ϵ0*χ<sup>r</sup>* (10)

ϵ ¼ ϵ<sup>0</sup> 1 þ *χ<sup>r</sup>* ð Þ (11)

!

Ω*=m* for dry soil overlaying crystal-

with *H* !

(12)

The electric properties of the material are completely defined by means of *ε* or *χ* [10]. In problems with electromagnetic fields, four vectors are defined: E and B; D and H. These vectors are assumed to be finite throughout the entire field, and at all ordinary points to be continuous functions of position and time, with continuous

> ¼ *μ ω*ð Þ*H* !

¼ ϵð Þ *ω E* !

¼ *σ ω*ð Þ*E* !

For the electromagnetic propagation in soils, the parameters *μ*, ϵ, and *σ* must be determined. The soils are usually nonmagnetic media; therefore the magnetic permeability will be that of the vacuum *μ* ¼ *μ*0, and the variables to be determined will

The electrical resistivity obtained by soil mapping exhibits a large range of

line rocks [13]. In **Figure 2**, the resistivity of different soils can be observed. The electrical conductivity is related to the particle size by the electrical charge density

*Table of electric resistivity* ½ � *Ω=m and electric conductivity* ½ � *σ=m of soils (Ref. Samoulian et al.) [13].*

derivatives [12]. The constitutive relations link the vectors of the fields *B*

*B* !

*D* !

*J* !

be the electric permittivity ϵ and the electric conductivity *σ*.

values from 1Ω*=m* for saline soil to several 10<sup>5</sup>

, usually dependent on the frequency [8]:

The transmission line method has been used to measure the dielectric properties [19, 20]. These transmission lines are coaxial, quasi-coaxial, and two-wire transmission lines. Consider a transmission line with a homogenous dielectric material inside, and the propagation is transverse electromagnetic mode (TEM), where the electric and magnetic field are perpendicular to the propagation direction; this can be observed in **Figures 3** and **4**.

The separation between the conductive cylinders that form the coax transmission line should be much lower than the wavelength of the signal that propagates, so the transmission line will not be affected by the propagation modes of high orders, such as the *TE*<sup>11</sup> [19].

Coaxial transmission lines are widely used for the transmission of radiofrequency signals and its application in radiocommunications and for broadcasting [21]. The transmission lines allow the connection between a generator or emitter and a load or antenna. The air coaxial transmission line consists of two cylindrical conductors, with air between both conductors. These metallic conductors are those that impose the boarder conditions that must comply with the electric and magnetic fields of the electromagnetic wave that travel inside the line. The coaxial

**Figure 3.** *Section of the two-wire transmission line with the electric and magnetic fields.*

transmission lines are used to measure the electrical properties of a dielectric material located inside the coaxial transmission line, as shown in **Figure 4**.

By analyzing the circuit model of a transmission line, the currents and voltages that propagate along it can be determined, using the circuit theory [22]. The equivalent circuit model of a transmission line can be seen in **Figure 5**. According to the equivalent circuit model of a transmission line, the characteristic impedance *Z*<sup>0</sup> and the propagation constant *γ* can be expressed thus [21]:

$$Z\_0 = \sqrt{\frac{R + j\alpha L}{G + j\alpha C}}\tag{13}$$

The input impedance of a transmission line, with a material inside considering

where *γ* is the propagation constant 1½ � *=m* ; *ZL* is the load impedance ½ � Ω ; *l* is the length of the transmission line from the load ½ � *m* ; *Z*<sup>0</sup> is the characteristic impedance

**4. Time-domain measurement method of dielectric permittivity and**

ering the soil like a nonmagnetic media with low dielectric loss is [26, 27]:

The time-domain reflectometry (TDR) is a well-known technique used to find the interruption point of the transmission lines in a CATV installation and is also

The time-domain reflectometry uses a step generator and an oscilloscope; a fast edge is launched into the transmission line under investigation, where the incident and reflected voltage waves on the transmission line are monitored by the oscilloscope. This method shows the losses and the characteristic impedance of the line: resistive, inductive, or capacitive [25]. The TDR method is based on the velocity of the electromagnetic wave that propagates through the soil, and the velocity of the wave depends on the water content of the soil. If a pulse is applied to a no-loss transmission line, the time domain graphic can be shown like in **Figure 7**. Consid-

*ZL* þ *Z*<sup>0</sup> *tanh* ð Þ *γl*

*<sup>Z</sup>*<sup>0</sup> <sup>þ</sup> *ZL tanh* ð Þ *<sup>γ</sup><sup>l</sup>* (17)

the material with dielectric losses, can be expressed thus [23]:

useful to determine the dielectric permittivity (see **Figure 6**).

of the transmission line ½ � Ω .

**Figure 6.**

**33**

*Setup of the dielectric measurement by the TDR method [24].*

**conductivity of soils (TDR)**

*DOI: http://dx.doi.org/10.5772/intechopen.88989*

*Zi* ¼ *Z*<sup>0</sup>

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics…*

$$\chi = \sqrt{(\mathbf{R} + j\alpha \mathbf{L})(\mathbf{G} + j\alpha \mathbf{C})} \tag{14}$$

where R is the series resistance per unit length ½ � Ω*=m* ; L is the series inductance per unit length ½ � *H=m* ; C is the parallel capacity per unit length ½ � *F=m* ; G is the parallel conductance per unit length ½ � *S=m* .

If the transmission line has no losses, it means that R = 0 and G = 0; then the characteristic impedance can be reduced as follows:

$$Z\_0 = \sqrt{\frac{L}{C}}\tag{15}$$

$$
\gamma = j a \sqrt{L \bar{C}} \tag{16}
$$

**Figure 4.** *Section of coaxial transmission lines and the electric and magnetic fields.*

**Figure 5.** *Distributed parameters of the transmission line.*

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics… DOI: http://dx.doi.org/10.5772/intechopen.88989*

The input impedance of a transmission line, with a material inside considering the material with dielectric losses, can be expressed thus [23]:

$$Z\_i = Z\_0 \frac{Z\_L + Z\_0 \tanh\left(\gamma l\right)}{Z\_0 + Z\_L \tanh\left(\gamma l\right)}\tag{17}$$

where *γ* is the propagation constant 1½ � *=m* ; *ZL* is the load impedance ½ � Ω ; *l* is the length of the transmission line from the load ½ � *m* ; *Z*<sup>0</sup> is the characteristic impedance of the transmission line ½ � Ω .

#### **4. Time-domain measurement method of dielectric permittivity and conductivity of soils (TDR)**

The time-domain reflectometry (TDR) is a well-known technique used to find the interruption point of the transmission lines in a CATV installation and is also useful to determine the dielectric permittivity (see **Figure 6**).

The time-domain reflectometry uses a step generator and an oscilloscope; a fast edge is launched into the transmission line under investigation, where the incident and reflected voltage waves on the transmission line are monitored by the oscilloscope. This method shows the losses and the characteristic impedance of the line: resistive, inductive, or capacitive [25]. The TDR method is based on the velocity of the electromagnetic wave that propagates through the soil, and the velocity of the wave depends on the water content of the soil. If a pulse is applied to a no-loss transmission line, the time domain graphic can be shown like in **Figure 7**. Considering the soil like a nonmagnetic media with low dielectric loss is [26, 27]:

**Figure 6.** *Setup of the dielectric measurement by the TDR method [24].*

transmission lines are used to measure the electrical properties of a dielectric mate-

By analyzing the circuit model of a transmission line, the currents and voltages that propagate along it can be determined, using the circuit theory [22]. The equivalent circuit model of a transmission line can be seen in **Figure 5**. According to the equivalent circuit model of a transmission line, the characteristic impedance *Z*<sup>0</sup> and

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *R* þ *jωL G* þ *jωC*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *R* þ *jωL* ð Þ *G* þ *jωC*

> ffiffiffi *L C* r

<sup>p</sup> (16)

where R is the series resistance per unit length ½ � Ω*=m* ; L is the series inductance

If the transmission line has no losses, it means that R = 0 and G = 0; then the

*<sup>γ</sup>* <sup>¼</sup> *<sup>j</sup><sup>ω</sup>* ffiffiffiffiffiffi *LC*

per unit length ½ � *H=m* ; C is the parallel capacity per unit length ½ � *F=m* ; G is the

*Z*<sup>0</sup> ¼

(13)

(14)

(15)

rial located inside the coaxial transmission line, as shown in **Figure 4**.

*Z*<sup>0</sup> ¼

q

s

the propagation constant *γ* can be expressed thus [21]:

parallel conductance per unit length ½ � *S=m* .

*Electromagnetic Field Radiation in Matter*

**Figure 4.**

**Figure 5.**

**32**

*Distributed parameters of the transmission line.*

characteristic impedance can be reduced as follows:

*Section of coaxial transmission lines and the electric and magnetic fields.*

*γ* ¼

**Figure 7.**

*Propagation of the pulses in the time domain graphic with dielectric air [24].*

**Figure 8.**

*Picture of the voltage as a function of time for the probe is in the soil [26].*

$$v = \frac{c}{\sqrt{\mathfrak{e}}}\tag{18}$$

where *V*<sup>0</sup> is the amplitude of the TDR pulse; *Vs* is the amplitude after reflection from the start of the probe; *Ve* is the amplitude after reflection from the end of the

> 1 � *ρ* 1 þ *ρ*

*<sup>ρ</sup>* <sup>¼</sup> *Vf* � *<sup>V</sup>*<sup>0</sup> *V*<sup>0</sup>

*K* is a geometric constant of the probe, and it is experimentally determined by immersing the probe in solutions of known electrical conductivity Nt at

*<sup>K</sup>* <sup>¼</sup> *EC* � *RL* � *ft*�<sup>1</sup> ð Þ *<sup>m</sup>* �<sup>1</sup>

**5. Measurement method of dielectric permittivity and conductivity**

The vector network analyzer can measure the scattering coefficient of a two-port passive network where the reflection coefficient in voltage Γ is *S*<sup>11</sup> [34]. The probe impedance with the material inside is related with the reflected

> *Zp Z*0

Temperature correction *fT* ¼ 1 þ *KT T*ð Þ � 25 *KT* depends on the used solution.

This method is based on the measurement of the reflection coefficient by means of the vector network analyzer (VNA) on the frequency domain of a coaxial trans-

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>Γ</sup>

where *Zp* is the impedance of the probe at the load and *Z*<sup>0</sup> is the characteristic

*<sup>c</sup>*<sup>ϵ</sup> <sup>∗</sup> <sup>p</sup> *cotangh <sup>ω</sup><sup>L</sup>* ffiffiffiffiffiffi

where *L* is the electric length of the probe; *c* is the speed of light; *L* is the coaxial

Then the complex electric permittivity for frequencies lower than 50 MHz can

!�<sup>1</sup>

<sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>*L*<sup>2</sup> 3*c*<sup>2</sup>

*L* 2 *log <sup>B</sup> A* 1 *ZTω<sup>j</sup>* � *Cs* <sup>ϵ</sup> <sup>∗</sup> <sup>p</sup>

� � (22)

<sup>1</sup> � <sup>Γ</sup> (23)

*<sup>c</sup> <sup>j</sup>* (24)

(25)

*<sup>σ</sup>* <sup>¼</sup> *<sup>K</sup> Z*0

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics…*

*Z*<sup>0</sup> is the characteristic impedance of the transmission line.

mission line in the soil; this can be observed in **Figure 9** [30–32].

The impedance of the probe can be calculated thus:

*Zp* <sup>¼</sup> <sup>2</sup> *log <sup>B</sup> <sup>A</sup>* ffiffiffiffiffiffiffiffi

probe of length; *A* is the inner diameter; *B* is the outer diameter.

ϵ ¼

probe; *Vf* is the reflected signal after a very long time. Also the conductivity can expressed thus [29]:

*DOI: http://dx.doi.org/10.5772/intechopen.88989*

where:

temperature:

coefficient Γ [31]:

impedance of the probe.

be approximated thus [31]:

**35**

*KT* ¼ 0*:*0191 for a 0.01 M KCl solution.

**of soils in frequency domain**

where *c m*½ � *=s* is the light velocity; ϵ is the electric permittivity of the soil under test.

The time interval Δ*t* between the received pulse and incident pulse can be observed in **Figure 8**, and the velocity can be expressed thus:

$$v = \frac{c}{\sqrt{\epsilon}} = \frac{2L}{\Delta t} \tag{19}$$

where *L* is the probe length. Then

$$e = \left(\frac{c\Delta t}{2L}\right)^2\tag{20}$$

Usually the transmission line probes have a minimum length of 15 cm, because the incident electromagnetic wave takes a time of 1 ns in air in order to return to the input of the line. This time is too short to be measured.

The conductivity of the soil can be determined computing the reflected pulses in the probe in the time domain graphic (see **Figure 8**) [26, 28]. Numerous methods have been proposed by researchers; one of these is the procedure of Dalton et al. (1984) [26]:

$$
\sigma\_{\text{dalton}} = \left(\frac{\sqrt{\epsilon}}{120\pi L} \ln\left(\frac{V\_f}{V\_\epsilon - V\_f}\right)\right) \tag{21}
$$

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics… DOI: http://dx.doi.org/10.5772/intechopen.88989*

where *V*<sup>0</sup> is the amplitude of the TDR pulse; *Vs* is the amplitude after reflection from the start of the probe; *Ve* is the amplitude after reflection from the end of the probe; *Vf* is the reflected signal after a very long time.

Also the conductivity can expressed thus [29]:

$$
\sigma = \frac{K}{Z\_0} \left( \frac{1 - \rho}{1 + \rho} \right) \tag{22}
$$

where:

*<sup>v</sup>* <sup>¼</sup> *<sup>c</sup>* ffiffi

where *c m*½ � *=s* is the light velocity; ϵ is the electric permittivity of the soil under

The time interval Δ*t* between the received pulse and incident pulse can be

<sup>ϵ</sup> <sup>¼</sup> *<sup>c</sup>*Δ*<sup>t</sup>* 2*L* � �<sup>2</sup>

Usually the transmission line probes have a minimum length of 15 cm, because the incident electromagnetic wave takes a time of 1 ns in air in order to return to the

The conductivity of the soil can be determined computing the reflected pulses in the probe in the time domain graphic (see **Figure 8**) [26, 28]. Numerous methods have been proposed by researchers; one of these is the procedure of Dalton et al.

> *Vf Ve* � *Vf*

! !

ffiffi ϵ p <sup>120</sup>*π<sup>L</sup> ln*

*<sup>v</sup>* <sup>¼</sup> *<sup>c</sup>* ffiffi <sup>ϵ</sup> <sup>p</sup> <sup>¼</sup> <sup>2</sup>*<sup>L</sup>*

observed in **Figure 8**, and the velocity can be expressed thus:

*Picture of the voltage as a function of time for the probe is in the soil [26].*

*Propagation of the pulses in the time domain graphic with dielectric air [24].*

*Electromagnetic Field Radiation in Matter*

input of the line. This time is too short to be measured.

*σdalton* ¼

where *L* is the probe length.

test.

**Figure 8.**

**Figure 7.**

Then

(1984) [26]:

**34**

<sup>ϵ</sup> <sup>p</sup> (18)

<sup>Δ</sup>*<sup>t</sup>* (19)

(20)

(21)

$$
\rho = \frac{V\_f - V\_0}{V\_0}
$$

*Z*<sup>0</sup> is the characteristic impedance of the transmission line.

*K* is a geometric constant of the probe, and it is experimentally determined by immersing the probe in solutions of known electrical conductivity Nt at temperature:

$$K = E \mathbf{C} \cdot \mathbf{R} \mathbf{L} \cdot \mathbf{f} \mathbf{f}^{-1} \left( m \right)^{-1}$$

Temperature correction *fT* ¼ 1 þ *KT T*ð Þ � 25 *KT* depends on the used solution. *KT* ¼ 0*:*0191 for a 0.01 M KCl solution.

#### **5. Measurement method of dielectric permittivity and conductivity of soils in frequency domain**

This method is based on the measurement of the reflection coefficient by means of the vector network analyzer (VNA) on the frequency domain of a coaxial transmission line in the soil; this can be observed in **Figure 9** [30–32].

The vector network analyzer can measure the scattering coefficient of a two-port passive network where the reflection coefficient in voltage Γ is *S*<sup>11</sup> [34]. The probe impedance with the material inside is related with the reflected coefficient Γ [31]:

$$\frac{Z\_p}{Z\_0} = \frac{\mathbf{1} + \Gamma}{\mathbf{1} - \Gamma} \tag{23}$$

where *Zp* is the impedance of the probe at the load and *Z*<sup>0</sup> is the characteristic impedance of the probe.

The impedance of the probe can be calculated thus:

$$Z\_p = \frac{2\log\frac{B}{A}}{\sqrt{c\varepsilon\ast}} \text{cotangh}\frac{oL\sqrt{\varepsilon\ast}}{c}j\tag{24}$$

where *L* is the electric length of the probe; *c* is the speed of light; *L* is the coaxial probe of length; *A* is the inner diameter; *B* is the outer diameter.

Then the complex electric permittivity for frequencies lower than 50 MHz can be approximated thus [31]:

$$\varepsilon = \left(\frac{\frac{L}{2\log\frac{R}{\lambda}}}{\frac{1}{Z\_{T}\alpha\dot{\jmath}} - C\_{t}} + \frac{\alpha^{2}L^{2}}{3c^{2}}\right)^{-1} \tag{25}$$

*γ* ¼ *α* þ *jβ* (28)

(29)

(30)

(33)

(34)

(31)

(32)

where *α* is the attenuation constant [Neper/m] and *β* is the phase constant

*l*

*atanh <sup>Z</sup>*<sup>0</sup> *Zi*

*atanh Zi Z*0

ð Þ *x*

<sup>2</sup>*<sup>l</sup> ln <sup>Z</sup>*<sup>0</sup> <sup>þ</sup> *Zi Z*<sup>0</sup> � *Zi*

<sup>2</sup>*<sup>l</sup> ln Zi* <sup>þ</sup> *<sup>Z</sup>*<sup>0</sup> *Zi* � *Z*<sup>0</sup>

<sup>2</sup>*<sup>l</sup> ln <sup>ρ</sup>*1*<sup>e</sup>*

<sup>2</sup>*<sup>l</sup> ln <sup>ρ</sup>*2*<sup>e</sup>*

*ϕ*1

*ϕ*2

<sup>þ</sup> *<sup>j</sup>*2*k<sup>π</sup> <sup>k</sup>* <sup>¼</sup> 0, 1, 2, 3*:*… (35)

<sup>þ</sup> *<sup>j</sup>*2*k<sup>π</sup> <sup>k</sup>* <sup>¼</sup> 0, 1, 2, 3*:*… (36)

*<sup>γ</sup>*j*ZL*!<sup>∞</sup> <sup>¼</sup> <sup>1</sup>

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics…*

*ZL*¼<sup>0</sup> <sup>¼</sup> <sup>1</sup> *l*

*ZL*!<sup>∞</sup> <sup>¼</sup> <sup>1</sup>

*ZL*¼<sup>0</sup> <sup>¼</sup> <sup>1</sup>

*<sup>γ</sup>*j*ZL*!<sup>∞</sup> <sup>¼</sup> <sup>1</sup>

*<sup>γ</sup>*j*ZL*¼<sup>0</sup> <sup>¼</sup> <sup>1</sup>

<sup>2</sup>*<sup>l</sup> ln <sup>ρ</sup>*<sup>1</sup> ð Þþ *ln ej<sup>ϕ</sup>*<sup>1</sup>

<sup>2</sup>*<sup>l</sup> ln <sup>ρ</sup>*<sup>2</sup> ð Þþ *ln e<sup>j</sup>ϕ*<sup>2</sup>

*γ*j

*γ*j

*γ*j

Using the relation between *ln x*ð Þ and *th*�<sup>1</sup>

[rad/m].

Using Eqs. (26) and (27)

*DOI: http://dx.doi.org/10.5772/intechopen.88989*

The argument of the ln

Replacing the Ln

<sup>2</sup>*<sup>l</sup> ln <sup>ρ</sup>*1*<sup>e</sup>*

<sup>2</sup>*<sup>l</sup> ln <sup>ρ</sup>*2*<sup>e</sup>*

*ϕ*1 <sup>¼</sup> <sup>1</sup>

*ϕ*2 <sup>¼</sup> <sup>1</sup>

*Input impedance of the transmission line for ZL* ¼ *0 and ZL* ! ∞*.*

*<sup>γ</sup>*j*ZL*!<sup>∞</sup> <sup>¼</sup> <sup>1</sup>

*ZL*¼<sup>0</sup> <sup>¼</sup> <sup>1</sup>

*γ*j

**Figure 10.**

**37**

**Figure 9.**

*Dielectric measurement by the coaxial transmission line method. (a) Setup of the measurement experiment; (b) section of the transmission line and the material under test [33].*

#### **6. Dielectric measurement by the characteristic impedance of a transmission line in frequency domain**

Some references of these measurement methods by means of characteristic impedance have been developed [35, 36]. This methods is shown in **Figure 10**.

The input impedance can be computed by Eq. (17) for two different loads' impedance:

(a) Open circuit in the load *ZL* ! ∞

$$\left. Z\_{i} \right|\_{Z\_{L} = \infty} = \frac{Z\_{0}}{\tanh\left(\eta l\right)} = Z\_{0} \tanh^{-1}(\eta l) \tag{26}$$

(b) Short circuit in the load *ZL* ¼ 0

$$\left. Z\_i \right|\_{Z\_L=0} = Z\_0 \tanh\left(\eta l\right) \tag{27}$$

where, in general, the material inside the transmission line could be a dielectric loss; the propagation constant can be written thus:

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics… DOI: http://dx.doi.org/10.5772/intechopen.88989*

$$
\gamma = a + j\beta \tag{28}
$$

where *α* is the attenuation constant [Neper/m] and *β* is the phase constant [rad/m].

Using Eqs. (26) and (27)

$$\left.\gamma\right|\_{Z\_{\rm L}\to\infty} = \frac{1}{l} atanh\left(\frac{Z\_0}{Z\_i}\right) \tag{29}$$

$$\left.\gamma\right|\_{Z\_{\mathbb{Z}}=0} = \frac{1}{l}atamb\left(\frac{Z\_i}{Z\_0}\right) \tag{30}$$

Using the relation between *ln x*ð Þ and *th*�<sup>1</sup> ð Þ *x*

$$\left.\gamma\right|\_{Z\_{\mathbb{L}}\to\infty} = \frac{1}{2l}\ln\frac{Z\_0 + Z\_i}{Z\_0 - Z\_i} \tag{31}$$

$$\gamma|\_{Z\_{\rm L}=0} = \frac{1}{2l} \ln \frac{Z\_i + Z\_0}{Z\_i - Z\_0} \tag{32}$$

The argument of the ln

$$\left.\gamma\right|\_{Z\_{\mathbb{Z}}\to\infty} = \frac{1}{2l}\ln\left(\rho\_1 e^{\phi\_1}\right) \tag{33}$$

$$\left.\gamma\right|\_{Z\_L=0} = \frac{1}{2l} \ln\left(\rho\_2 e^{\phi\_2}\right) \tag{34}$$

Replacing the Ln

$$\left. \gamma \right|\_{Z\_{\mathbb{Z}} \to \infty} = \frac{1}{2l} \ln \left( \rho\_1 e^{\phi\_1} \right) = \frac{1}{2l} \left[ \ln \left( \rho\_1 \right) + \ln \left( e^{j\phi\_1} \right) + j2k\pi \right] \ k = 0, 1, 2, 3... \tag{35}$$

$$\left. \begin{array}{c} \gamma \right|\_{Z=0} = \frac{1}{2l} \left[ \ln \left( \rho\_2 e^{\phi\_2} \right) \right] = \frac{1}{2l} \left[ \ln \left( \rho\_2 \right) + \ln \left( e^{i\phi\_2} \right) + j2k\pi \right] \left. k = 0, 1, 2, 3... \right] \end{array} \tag{36}$$

#### **Figure 10.**

*Input impedance of the transmission line for ZL* ¼ *0 and ZL* ! ∞*.*

**6. Dielectric measurement by the characteristic impedance of a**

Some references of these measurement methods by means of characteristic impedance have been developed [35, 36]. This methods is shown in **Figure 10**. The input impedance can be computed by Eq. (17) for two different loads'

*Dielectric measurement by the coaxial transmission line method. (a) Setup of the measurement experiment; (b)*

*tanh* ð Þ *<sup>γ</sup><sup>l</sup>* <sup>¼</sup> *<sup>Z</sup>*<sup>0</sup> *tanh* �<sup>1</sup>

where, in general, the material inside the transmission line could be a dielectric

ð Þ *γl* (26)

*ZL*¼<sup>0</sup> <sup>¼</sup> *<sup>Z</sup>*<sup>0</sup> *tanh* ð Þ *<sup>γ</sup><sup>l</sup>* (27)

**transmission line in frequency domain**

*section of the transmission line and the material under test [33].*

*Electromagnetic Field Radiation in Matter*

(a) Open circuit in the load *ZL* ! ∞

(b) Short circuit in the load *ZL* ¼ 0

*Zi*j

loss; the propagation constant can be written thus:

*ZL*¼<sup>∞</sup> <sup>¼</sup> *<sup>Z</sup>*<sup>0</sup>

*Zi*j

impedance:

**36**

**Figure 9.**

Then the propagation constant can be written thus:

$$\begin{aligned} \left. a \right|\_{Z\_{\perp} \to \infty} &= \frac{1}{2l} \ln \left( \rho\_{1} \right) \\ \left. \beta \right|\_{Z\_{\perp} \to \infty} &= \frac{1}{2l} [\phi\_{1} + 2k\pi] \left. k = 0, 1, 2, 3 \dots \\ \left. a \right|\_{Z\_{\perp} = 0} &= \frac{1}{2l} \ln \left( \rho\_{2} \right) \\ \left. \beta \right|\_{Z\_{\perp} = 0} &= \frac{1}{2l} [\phi\_{2} + 2k\pi] \left. k = 0, 1, 2, 3 \dots \right. \end{aligned} \tag{37}$$

Results

*Rin* ffi <sup>10</sup>�<sup>3</sup>

*<sup>σ</sup>* <sup>¼</sup> <sup>2</sup>*αβ ωμ* <sup>ϵ</sup> <sup>¼</sup> *<sup>β</sup>*<sup>2</sup> � *<sup>α</sup>*<sup>2</sup> *ω*2*μ*

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics…*

The series resistance of the conductor of the coaxial transmission line used is

1.The input impedances are measured for a load impedance at short circuit *ZL* ¼

3.The electric permittivity ϵ and the electrical conductivity *σ* are calculated using

In this way, a practical method of measurement is available to determine the parameters of dielectric materials, using coaxial transmission lines, in the frequency range from 1 to 1000 MHz. A problem that appears when measuring dielectric materials is the connector that establishes the link between the coaxial transmission

Therefore, the study and correction of the mentioned error in the section will be

Three coaxial transmission lines of General Radio (GR) Type 874, with air dielectric, have been used with a length of 100, 200, and 300 mm. The main characteristics of the General Radio coaxial transmission lines, type 874, are the

*N connector and its equivalent of a transmission line with dielectric of air.*

line and the vector impedance meter. A systematic error in the impedance

2.The attenuation constant *α* and phase *β* are calculated with the equations

has no effect in the measured attenuation constant *α*.

0 and open circuit *ZL* ! ∞ (see Eqs. (26) and (27)).

**6.2 Measurement procedure of the ϵ and** *σ*

*DOI: http://dx.doi.org/10.5772/intechopen.88989*

found ad hoc (see Eq. (37)).

measured is introduced.

**6.3 Transmission lines used**

carried out.

following:

**Figure 11.**

**39**

the equations found in Section 6.1.

Ω*=m f*ð Þ ¼ 100*MHz ZL* ¼ 0. This resistance can be neglected, because it

(45)

By these last equations, the attenuation constant and the phase constant can be calculated with *Zi*j *ZL*¼<sup>0</sup> or *ZL* ! <sup>∞</sup>.

#### **6.1 Determining ϵ and** *σ* **from the propagation constant**

The propagation constant *γ* can be written thus [11]:

$$\gamma = \sqrt{j a \mu (\sigma + j a \epsilon \overline{\epsilon})} \tag{38}$$

From Eqs. (38) and (28)

$$
\gamma^2 = j a \mu (\sigma + j a \epsilon') = -a^2 \mu \epsilon' + j a \mu \sigma \tag{39}
$$

$$
\gamma^2 = a^2 - \beta^2 + 2ja\beta = \left(a^2 + \beta^2\right)e^{j2atan\left(\frac{\rho}{a}\right)}\tag{40}
$$

Equating real and imaginary parts of *γ*<sup>2</sup> of Eqs. (39) and (40)

$$\begin{aligned} -\alpha^2 \mu \epsilon' &= \left( a^2 + \beta^2 \right) \cos \left( 2 \atanlash \left( \frac{\beta}{a} \right) \right) \\ \alpha \mu \sigma &= \left( a^2 + \beta^2 \right) \sin \left( 2 \atanlash \left( \frac{\beta}{a} \right) \right) \end{aligned} \tag{41}$$

ϵ and *σ* can be obtained:

$$\begin{aligned} \varepsilon' &= \frac{\left(a^2 + \beta^2\right)\cos\left(2\operatorname{atan}\left(\frac{\beta}{a}\right)\right)}{-a^2\mu} \\ \sigma &= \frac{\left(a^2 + \beta^2\right)\sin\left(2\operatorname{atan}\left(\frac{\beta}{a}\right)\right)}{a\mu} \end{aligned} \tag{42}$$

Another expression of ϵ and *σ* is using Eqs.(39) and (40):

$$\begin{aligned} \chi^2 &= j a \mu (\sigma + j a \epsilon^\prime) \\ \chi^2 &= a^2 - \beta^2 + 2j a \beta \end{aligned} \tag{43}$$

Equating real and imaginary part of Eq. (43)

$$\begin{aligned} a\mu\sigma &= 2a\beta\\ -a^2\mu\epsilon &= a^2 - \beta^2 \end{aligned} \tag{44}$$

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics… DOI: http://dx.doi.org/10.5772/intechopen.88989*

Results

Then the propagation constant can be written thus:

*ZL*!<sup>∞</sup> <sup>¼</sup> <sup>1</sup>

*ZL*!<sup>∞</sup> <sup>¼</sup> <sup>1</sup> 2*l*

*ZL*¼<sup>0</sup> <sup>¼</sup> <sup>1</sup>

*ZL*¼<sup>0</sup> <sup>¼</sup> <sup>1</sup> 2*l*

**6.1 Determining ϵ and** *σ* **from the propagation constant**

The propagation constant *γ* can be written thus [11]:

*γ* ¼

Equating real and imaginary parts of *γ*<sup>2</sup> of Eqs. (39) and (40)

�*ω*<sup>2</sup>

ϵ<sup>0</sup> ¼

*σ* ¼

Equating real and imaginary part of Eq. (43)

Another expression of ϵ and *σ* is using Eqs.(39) and (40):

q

*<sup>γ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>j</sup>ωμ σ* <sup>þ</sup> *<sup>j</sup>ω*ϵ<sup>0</sup> ð Þ¼�*ω*<sup>2</sup>

*<sup>γ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>α</sup>*<sup>2</sup> � *<sup>β</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*jαβ* <sup>¼</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> � �*e<sup>j</sup>*2*atan <sup>β</sup>*

*<sup>μ</sup>*ϵ<sup>0</sup> <sup>¼</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> � � *cos* <sup>2</sup>*atan <sup>β</sup>*

*<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> � � *cos* <sup>2</sup>*atan <sup>β</sup>*

*<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> � � *sin* <sup>2</sup>*atan <sup>β</sup>*

*<sup>γ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>j</sup>ωμ σ* <sup>þ</sup> *<sup>j</sup>ω*ϵ<sup>0</sup> ð Þ

*ωμσ* ¼ 2*αβ*

�*ω*<sup>2</sup>

�*ω*<sup>2</sup>*μ*

*ωμ*

*ωμσ* <sup>¼</sup> *<sup>α</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> � � *sin* <sup>2</sup>*atan <sup>β</sup>*

*ZL*¼<sup>0</sup> or *ZL* ! <sup>∞</sup>.

<sup>2</sup>*<sup>l</sup> ln <sup>ρ</sup>*<sup>1</sup> ð Þ

<sup>2</sup>*<sup>l</sup> ln <sup>ρ</sup>*<sup>2</sup> ð Þ

½ � *ϕ*<sup>1</sup> þ 2*kπ k* ¼ 0, 1, 2, 3*:*…

(37)

(38)

(42)

*μ*ϵ<sup>0</sup> þ *jωμσ* (39)

*α* � � � �

� � � � (41)

*α*

*α* � � � �

*α* � � � �

*<sup>γ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>α</sup>*<sup>2</sup> � *<sup>β</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*jαβ* (43)

*<sup>μ</sup>*<sup>ϵ</sup> <sup>¼</sup> *<sup>α</sup>*<sup>2</sup> � *<sup>β</sup>*<sup>2</sup> (44)

ð Þ*<sup>α</sup>* (40)

½ � *ϕ*<sup>2</sup> þ 2*kπ k* ¼ 0, 1, 2, 3*:*…

By these last equations, the attenuation constant and the phase constant can be

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *jωμ σ*ð Þ þ *jω*ϵ

*α*j

*Electromagnetic Field Radiation in Matter*

*β*j

*α*j

*β*j

calculated with *Zi*j

From Eqs. (38) and (28)

ϵ and *σ* can be obtained:

**38**

$$\begin{aligned} \sigma &= \frac{2a\beta}{a\mu} \\ \varepsilon &= \frac{\beta^2 - a^2}{a^2\mu} \end{aligned} \tag{45}$$

The series resistance of the conductor of the coaxial transmission line used is *Rin* ffi <sup>10</sup>�<sup>3</sup> Ω*=m f*ð Þ ¼ 100*MHz ZL* ¼ 0. This resistance can be neglected, because it has no effect in the measured attenuation constant *α*.

#### **6.2 Measurement procedure of the ϵ and** *σ*


In this way, a practical method of measurement is available to determine the parameters of dielectric materials, using coaxial transmission lines, in the frequency range from 1 to 1000 MHz. A problem that appears when measuring dielectric materials is the connector that establishes the link between the coaxial transmission line and the vector impedance meter. A systematic error in the impedance measured is introduced.

Therefore, the study and correction of the mentioned error in the section will be carried out.

#### **6.3 Transmission lines used**

Three coaxial transmission lines of General Radio (GR) Type 874, with air dielectric, have been used with a length of 100, 200, and 300 mm. The main characteristics of the General Radio coaxial transmission lines, type 874, are the following:

**Figure 11.**

*N connector and its equivalent of a transmission line with dielectric of air.*

Characteristic impedance *Z*<sup>0</sup> ¼ 50Ω. Input and output connector GR874

$$r\_1 = 12 \cdot 10^{-3} [m]$$

$$r\_2 = 5.2 \cdot 10^{-3} [m]$$

$$\sigma \cong 5.810^{\overline{\!\!\!/}} [\text{S/}m]$$

#### **6.4 Correction error produced by the connector of the transmission line**

It is important to perform the correction of the impedance introduced by the connector of the transmission line used. This connector is shown in **Figure 11**, and it is composed by a dielectric of very low dielectric losses and has a length of 10 mm (**Figure 12**). The characteristic impedance of the connector is practically *Z*<sup>0</sup> ¼ 50Ω with no losses [36]:

$$
\gamma = a + j\beta \cong j\beta \tag{46}
$$

Considering the connector with no losses

*DOI: http://dx.doi.org/10.5772/intechopen.88989*

**6.5 Results and discussion**

*6.5.1 Method of measurement*

evident.

**Figure 13.**

**41**

*transmission lines: 100, 200, and 300 mm.*

*α* ¼ 0

ffiffiffiffiffiffiffiffiffi *μ*0ϵ<sup>0</sup>

p (48)

*ZL*¼<sup>0</sup> <sup>¼</sup> *jZ*0*contg*ð Þ *<sup>β</sup><sup>x</sup>* (49)

� � (50)

*β* ¼ *ω*<sup>0</sup>

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics…*

where *x* is the length of the connector of the equivalent transmission line. The length *x* of this equivalent transmission line can be written thus:

*tg*�<sup>1</sup> *Zicon*<sup>j</sup>

The experimental results of the electric conductivity and the dielectric permit-

In **Figure 14**, the relative electric permittivity as a function of the frequency, by means of the capacitive method, and the three types of transmission line lengths

*Electric conductivity as a function of the frequency for dry sand samples, using a capacitive method and three*

tivity measurement of the dry sand can be observed in **Figures 13** and **14**. In **Figure 13**, the electric conductivity as a function of the frequency, by means of the capacitive method, and the three types of transmission line lengths have been measured: *L* = 100, 200, and 300 mm; the convergence of all measurements are

*ZL*¼0 *jZ*0*con*

Then the input impedance of the connector with *ZL* ¼ 0 is

*Zi*j

*<sup>x</sup>* <sup>¼</sup> <sup>1</sup> *β*

The input impedance to the connector can be written thus:

$$Z\_i = Z\_{0con} \frac{Z\_L + jZ\_{0con} \text{tg}(\beta \infty)}{Z\_{0con} + jZ\_L \text{tg}(\beta \infty)} \tag{47}$$

where *x* is the length of the transmission line [m].

The electric permittivity of the dielectric of the connector is unknown; then it is easy to assume a transmission line with air equivalent to the connector with ϵ ¼ ϵ0, *μ* ¼ *μ*0, and *Z*0*con* ¼ 50Ω.

**Figure 12.** *Equivalent length of the transmission line of the connector GR874.*

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics… DOI: http://dx.doi.org/10.5772/intechopen.88989*

Considering the connector with no losses

$$a = \mathbf{0}$$

$$\beta = a\_0 \sqrt{\mu\_0 \epsilon\_0} \tag{48}$$

Then the input impedance of the connector with *ZL* ¼ 0 is

$$\left. Z\_i \right|\_{Z\_L=0} = jZ\_{0con} \text{tg}\left(\beta \mathbf{x}\right) \tag{49}$$

where *x* is the length of the connector of the equivalent transmission line. The length *x* of this equivalent transmission line can be written thus:

$$\infty = \frac{1}{\beta} \text{tg}^{-1} \left( \frac{\mathbf{Z}\_{icon}|\_{\mathbf{Z}\_{\mathcal{L}}=0}}{j \mathbf{Z}\_{0con}} \right) \tag{50}$$

#### **6.5 Results and discussion**

Characteristic impedance *Z*<sup>0</sup> ¼ 50Ω. Input and output connector GR874

*Electromagnetic Field Radiation in Matter*

with no losses [36]:

*μ* ¼ *μ*0, and *Z*0*con* ¼ 50Ω.

**Figure 12.**

**40**

*<sup>r</sup>*<sup>1</sup> <sup>¼</sup> <sup>12</sup> � <sup>10</sup>�<sup>3</sup>

*<sup>r</sup>*<sup>2</sup> <sup>¼</sup> <sup>5</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>3</sup>

*<sup>σ</sup>* ffi <sup>5</sup>*:*810<sup>7</sup>

**6.4 Correction error produced by the connector of the transmission line**

The input impedance to the connector can be written thus:

*Zi* ¼ *Z*0*con*

where *x* is the length of the transmission line [m].

*Equivalent length of the transmission line of the connector GR874.*

It is important to perform the correction of the impedance introduced by the connector of the transmission line used. This connector is shown in **Figure 11**, and it is composed by a dielectric of very low dielectric losses and has a length of 10 mm (**Figure 12**). The characteristic impedance of the connector is practically *Z*<sup>0</sup> ¼ 50Ω

½ � *m*

½ � *m*

*γ* ¼ *α* þ *jβ* ffi *jβ* (46)

(47)

½ � *S=m*

*ZL* þ *jZ*0*contg*ð Þ *βx Z*0*con* þ *jZLtg*ð Þ *βx*

The electric permittivity of the dielectric of the connector is unknown; then it is easy to assume a transmission line with air equivalent to the connector with ϵ ¼ ϵ0,

#### *6.5.1 Method of measurement*

The experimental results of the electric conductivity and the dielectric permittivity measurement of the dry sand can be observed in **Figures 13** and **14**. In **Figure 13**, the electric conductivity as a function of the frequency, by means of the capacitive method, and the three types of transmission line lengths have been measured: *L* = 100, 200, and 300 mm; the convergence of all measurements are evident.

In **Figure 14**, the relative electric permittivity as a function of the frequency, by means of the capacitive method, and the three types of transmission line lengths

#### **Figure 13.**

*Electric conductivity as a function of the frequency for dry sand samples, using a capacitive method and three transmission lines: 100, 200, and 300 mm.*

have been measured: *L* = 100, 200, and 300 mm; there is a convergence of all measurements. It is important to note that the shorter transmission line has a wider bandwidth of measurement. The transmission line length of *L* ¼ 300*mm* shows the useful results up to 30 MHz; with a TL length of *L* ¼ 200*mm*, the useful results are up to 50 MHz; and with a TL length of *L* ¼ 100*mm*, the useful results are up to 300 MHz.

The expected value of the dielectric permittivity measured for the dry sand by means of a parallel plate capacitor and the three transmission lines used are shown in **Table 1**. The standard deviation of the three measurements shows a good agreement up to the vicinity of the resonant frequency of each transmission line. In **Table 2**, the electric conductivity of the dry sand can be observed. These curves

**T. line 200 mm**

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics…*

 1 1.1 1.3 1.2 1.15 0.13 1.6 2.0 1.7 1.7 1.75 0.17 2.1 2.6 2.3 2.2 2.30 0.22 3.2 3.1 3.3 3.1 3.20 0.096 4.4 5 4.3 4 4.40 0.42 6.0 6.5 5.5 5.1 5.80 0.6 10 11 9.4 8.4 9.70 1.08 14.4 14.6 12.7 11.1 13.20 1.60 21.9 21.5 20.8 16 20.00 2.7

**T. line 300 mm** **Expected value**

**Std. dev**

The values of the electrical conductivity and the electrical permittivity are very

Apparent soil electrical conductivity (ECa) to agriculture has its origin in the measurement of soil salinity, in arid-zone problem, which is associated with irrigated agricultural land. ECa is a quick, reliable, easy-to-take soil measurement that often relates to crop yield. For these reasons, the measurement of ECa is among the most frequently used tools in precision agriculture research for the spatiotemporal characterization of edaphic and anthropogenic properties that influence crop yield [37]. There are portable instruments for measuring the electrical conductivity of the soil by the method of electromagnetic induction and by the method of the four conductors, which are installed in the agricultural machinery to obtain a map of the

For geophysics applications, the solar disturbances (flares, coronal mass ejections) create variations of the Earth's magnetic field. These geomagnetic variations induce a geoelectric field at the Earth's surface and interior. The geoelectric field in turn drives geomagnetically induced currents, also called telluric currents along electrically conductive technological networks, such as power transmission lines, railways, and pipelines [38]. This geomagnetically induced currents create condi-

tions where enhanced corrosion may occur. Earth conductivity can create

useful to evaluate the propagation of surface waves in real ground, where the attenuation depends mostly on the conductivity of the soil. Such is the case that AM transmitters include radials, which consist of metallic conductors, placed at the base of the monopole antenna to increase conductivity, and in this way the losses due to Joule effect on the earth's surface are reduced. When the conductivity of the soil is perfect, the electric field vector that propagates will be perpendicular to the earth's surface; however, in real soils the electric field vector tilts and partly spreads into the earth, which dissipates power and transforms into heat [2]. This constitutes

have the same slope and show a good convergence.

*<sup>m</sup> <sup>10</sup><sup>5</sup> .*

**Capacitor T. line**

*DOI: http://dx.doi.org/10.5772/intechopen.88989*

**100 mm**

soil, before carrying out the work of tilling the earth.

*6.5.2 Applications*

*Electric conductivity of dry sand σ <sup>S</sup>*

**freq. (MHz)**

**Table 2.**

losses on earth.

**43**

#### **Figure 14.**

*The relative dielectric permittivity as a function of the frequency for dry sand samples has been measured, using a capacitive method and three transmission lines: 100, 200, and 300 mm.*


#### **Table 1.** *Relative electric permittivity of dry sand.*


*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics… DOI: http://dx.doi.org/10.5772/intechopen.88989*

#### **Table 2.**

have been measured: *L* = 100, 200, and 300 mm; there is a convergence of all measurements. It is important to note that the shorter transmission line has a wider bandwidth of measurement. The transmission line length of *L* ¼ 300*mm* shows the useful results up to 30 MHz; with a TL length of *L* ¼ 200*mm*, the useful results are up to 50 MHz; and with a TL length of *L* ¼ 100*mm*, the useful results are up to

*The relative dielectric permittivity as a function of the frequency for dry sand samples has been measured, using*

**T. line 200 mm**

 2.80 2.86 2.65 2.65 2.74 0.034 2.60 2.76 2.62 2.75 2.68 0.021 2.60 2.70 2.60 2.70 2.65 0.010 2.50 2.61 2.57 2.65 2.58 0.012 2.50 2.54 2.55 2.62 2.55 0.0075 2.50 2.51 2.50 2.60 2.52 0.0073 2.40 2.45 2.47 2.57 2.47 0.0153 2.40 2.42 2.46 2.57 2.46 0.0173 2.40 2.40 2.46 — 2.41 0.0028 — 2.37 — — —— — 2.35 — — ——

**T. line 300 mm** **Expected value**

**Std. dev**

*a capacitive method and three transmission lines: 100, 200, and 300 mm.*

**100 mm**

**Capacitor T. line**

300 MHz.

*Electromagnetic Field Radiation in Matter*

**Figure 14.**

**freq. (MHz)**

**Table 1.**

**42**

*Relative electric permittivity of dry sand.*

#### *Electric conductivity of dry sand σ <sup>S</sup> <sup>m</sup> <sup>10</sup><sup>5</sup> .*

The expected value of the dielectric permittivity measured for the dry sand by means of a parallel plate capacitor and the three transmission lines used are shown in **Table 1**. The standard deviation of the three measurements shows a good agreement up to the vicinity of the resonant frequency of each transmission line. In **Table 2**, the electric conductivity of the dry sand can be observed. These curves have the same slope and show a good convergence.

#### *6.5.2 Applications*

The values of the electrical conductivity and the electrical permittivity are very useful to evaluate the propagation of surface waves in real ground, where the attenuation depends mostly on the conductivity of the soil. Such is the case that AM transmitters include radials, which consist of metallic conductors, placed at the base of the monopole antenna to increase conductivity, and in this way the losses due to Joule effect on the earth's surface are reduced. When the conductivity of the soil is perfect, the electric field vector that propagates will be perpendicular to the earth's surface; however, in real soils the electric field vector tilts and partly spreads into the earth, which dissipates power and transforms into heat [2]. This constitutes losses on earth.

Apparent soil electrical conductivity (ECa) to agriculture has its origin in the measurement of soil salinity, in arid-zone problem, which is associated with irrigated agricultural land. ECa is a quick, reliable, easy-to-take soil measurement that often relates to crop yield. For these reasons, the measurement of ECa is among the most frequently used tools in precision agriculture research for the spatiotemporal characterization of edaphic and anthropogenic properties that influence crop yield [37]. There are portable instruments for measuring the electrical conductivity of the soil by the method of electromagnetic induction and by the method of the four conductors, which are installed in the agricultural machinery to obtain a map of the soil, before carrying out the work of tilling the earth.

For geophysics applications, the solar disturbances (flares, coronal mass ejections) create variations of the Earth's magnetic field. These geomagnetic variations induce a geoelectric field at the Earth's surface and interior. The geoelectric field in turn drives geomagnetically induced currents, also called telluric currents along electrically conductive technological networks, such as power transmission lines, railways, and pipelines [38]. This geomagnetically induced currents create conditions where enhanced corrosion may occur. Earth conductivity can create

geomagnetically induced current variations, in particular where a pipeline crosses a highly resistive intrusive rock. It is important to make pipeline surveys once a year to measure the voltage at test posts to ensure that pipe-to-soil potential variations are within the safe range, impressed by cathodic protection systems [38].

**References**

IEEE Press; 2009

Libreria; 2006

**25**:1192-1202

[1] Joffe E, Kai-Sang L. Grounds for Grounding: A Circuit to System

[2] Trainotti V, Fano WG, Dorado L. Ingenieria Electromagnetica, Vol. 2. 1st ed. Buenos Aires, Argentina: Nueva

[3] Norton KA. The physical reality of space and surface waves in the radiation field of radio antennas. Proceedings of the Institute of Radio Engineers. 1937;

[4] Angulo I, Barclay L, Chernov Y, Deminco N, Fernandez I, Gil U, et al.

[5] Corwin DL, Lesch SM. Application of soil electrical conductivity to precision agriculture. American Society of Agronomy. 2003;**95**(3):455-471

[6] Smith-Rose RL. The electrical properties of soil for alternating currents at radio frequencies. Proceedings of the Royal Society of

[7] Jol HM. Ground Penetrating Radar Theory and Applications. Oxford, UK:

[8] Ramo S, Whinnery J, Van Duzer T. Fields and Waves in Communication Electronics. India: Wiley; 2008

[10] Reitz JR, Milford FJ, Christy RW. Foundations of Electromagnetic Theory.

[9] Griffiths DJ. Introduction to Electrodynamics. 4th ed. Boston, MA: Pearson; 2013. Re-published by Cambridge University Press in 2017

London. 1933;**140**:359-377

Elsevier Science; 2009

**45**

Handbook on Ground Wave Propagation. Geneva, Switzerland: Radiocommunication Bureau, ITU; 2014

Handbook. Piscataway, NJ, USA: Wiley,

*DOI: http://dx.doi.org/10.5772/intechopen.88989*

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics…*

4th ed. USA: Addison-Wesley Publishing Company; 2008

[11] Trainotti V, Fano WG. Ingenieria Electromagnetica, Vol. 1. 1st ed. Buenos Aires, Argentina: Nueva Libreria; 2004

[13] Samoulian A, Cousin I, Tabbagh A,

[12] Stratton JA. Electromagnetic Theory. Hoboken, New Jersey, USA: IEEE Press, Wiley-Interscience; 2007

Bruand A, Richard G. Electrical resistivity survey in soil science: A review. Soil and Tillage Research. 2005;

[14] Abu-Hassanein ZS, Benson CH, Blotz LR. Electrical resistivity of compacted clays. Journal of

Geotechnical Engineering. 1996;**122**(5):

[15] Hoekstra P, Delaney A. Dielectric properties of soils at uhf and microwave frequencies. Journal of Geophysical Research. 1974;**79**:11, 1699-1708

Dobson MC. Dielectric properties of soils in the 0.3–1.3-GHz range. IEEE Transactions on Geoscience and Remote

[17] El-rayes MA, Ulaby FT. Microwave dielectric spectrum of vegetation—Part I: Experimental observations. IEEE Transactions on Geoscience and Remote

[18] Ulaby FT, El-rayes MA. Microwave dielectric spectrum of vegetation—Part

Transactions on Geoscience and Remote

[19] Hipp JE. Frequency dispersion of complex permeability in mnzn and nizn

[16] Peplinski NR, Ulaby FT,

Sensing. 1995;**33**:803-807

Sensing. 1987;**GE-25**:541-549

II: Dual-dispersion model. IEEE

Sensing. 1987;**GE-25**:550-557

**83**(2):173-193

397-406

### **Author details**

Walter Gustavo Fano Faculty of Engineering, Universidad de Buenos Aires, Buenos Aires, Argentina

\*Address all correspondence to: gustavo.gf2005@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics… DOI: http://dx.doi.org/10.5772/intechopen.88989*

#### **References**

geomagnetically induced current variations, in particular where a pipeline crosses a highly resistive intrusive rock. It is important to make pipeline surveys once a year to measure the voltage at test posts to ensure that pipe-to-soil potential variations

are within the safe range, impressed by cathodic protection systems [38].

*Electromagnetic Field Radiation in Matter*

Faculty of Engineering, Universidad de Buenos Aires, Buenos Aires, Argentina

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: gustavo.gf2005@gmail.com

provided the original work is properly cited.

**Author details**

**44**

Walter Gustavo Fano

[1] Joffe E, Kai-Sang L. Grounds for Grounding: A Circuit to System Handbook. Piscataway, NJ, USA: Wiley, IEEE Press; 2009

[2] Trainotti V, Fano WG, Dorado L. Ingenieria Electromagnetica, Vol. 2. 1st ed. Buenos Aires, Argentina: Nueva Libreria; 2006

[3] Norton KA. The physical reality of space and surface waves in the radiation field of radio antennas. Proceedings of the Institute of Radio Engineers. 1937; **25**:1192-1202

[4] Angulo I, Barclay L, Chernov Y, Deminco N, Fernandez I, Gil U, et al. Handbook on Ground Wave Propagation. Geneva, Switzerland: Radiocommunication Bureau, ITU; 2014

[5] Corwin DL, Lesch SM. Application of soil electrical conductivity to precision agriculture. American Society of Agronomy. 2003;**95**(3):455-471

[6] Smith-Rose RL. The electrical properties of soil for alternating currents at radio frequencies. Proceedings of the Royal Society of London. 1933;**140**:359-377

[7] Jol HM. Ground Penetrating Radar Theory and Applications. Oxford, UK: Elsevier Science; 2009

[8] Ramo S, Whinnery J, Van Duzer T. Fields and Waves in Communication Electronics. India: Wiley; 2008

[9] Griffiths DJ. Introduction to Electrodynamics. 4th ed. Boston, MA: Pearson; 2013. Re-published by Cambridge University Press in 2017

[10] Reitz JR, Milford FJ, Christy RW. Foundations of Electromagnetic Theory. 4th ed. USA: Addison-Wesley Publishing Company; 2008

[11] Trainotti V, Fano WG. Ingenieria Electromagnetica, Vol. 1. 1st ed. Buenos Aires, Argentina: Nueva Libreria; 2004

[12] Stratton JA. Electromagnetic Theory. Hoboken, New Jersey, USA: IEEE Press, Wiley-Interscience; 2007

[13] Samoulian A, Cousin I, Tabbagh A, Bruand A, Richard G. Electrical resistivity survey in soil science: A review. Soil and Tillage Research. 2005; **83**(2):173-193

[14] Abu-Hassanein ZS, Benson CH, Blotz LR. Electrical resistivity of compacted clays. Journal of Geotechnical Engineering. 1996;**122**(5): 397-406

[15] Hoekstra P, Delaney A. Dielectric properties of soils at uhf and microwave frequencies. Journal of Geophysical Research. 1974;**79**:11, 1699-1708

[16] Peplinski NR, Ulaby FT, Dobson MC. Dielectric properties of soils in the 0.3–1.3-GHz range. IEEE Transactions on Geoscience and Remote Sensing. 1995;**33**:803-807

[17] El-rayes MA, Ulaby FT. Microwave dielectric spectrum of vegetation—Part I: Experimental observations. IEEE Transactions on Geoscience and Remote Sensing. 1987;**GE-25**:541-549

[18] Ulaby FT, El-rayes MA. Microwave dielectric spectrum of vegetation—Part II: Dual-dispersion model. IEEE Transactions on Geoscience and Remote Sensing. 1987;**GE-25**:550-557

[19] Hipp JE. Frequency dispersion of complex permeability in mnzn and nizn spinel ferrites and their composite materials. Proceedings of the IEEE. 1974;**62**:98-103

[20] Fano WG, Trainotti V. Dielectric properties of soils. In: 2001 Annual Report Conference on Electrical Insulation and Dielectric Phenomena (Cat. No.01CH37225); 2001. pp. 75-78

[21] Karakash J. Transmission Lines and Filter Networks. New York, USA: Macmillan; 1950

[22] Peterson A, Durgin DG. Transient signals on transmission lines: An introduction to non-ideal effects and signal integrity issues in electrical systems. Synthesis Lectures on Computational Electromagnetics. 2008;**3**(1):1-144

[23] Collin R. Foundations for Microwave Engineering. 2nd ed. New York, USA: Wiley India Pvt. Limited; 2007

[24] Olchawa O, Kumor M. Time domain reflectometry (TDR) measuring dielectric constant of polluted soil to estimate diesel oil content. Archives of Hydro-Engineering and Environmental Mechanics. 2008;**55**(1–2):55-62

[25] Keysight. Time domain reflectometry theory. Keysight Technologies; Application Note; 2015

[26] Zante P. Water content and soil bulk electrical conductivity measurements. TDR Technology Applied to Soils. 2002;**49**(1):1-27

[27] Robinson DA, Jones SB, Wraith JM, Friedman SP. A review of advances in dielectric and electrical conductivity measurement in soils using time domain reflectometry. Vadose Zone Journal. Soil Science Society of America. 2003;**2**:444-475

[28] Evett SR. Coaxial multiplexer for time domain reflectometry

measurement of soil water content and bulk electrical conductivity. Transactions of the ASAE. 1998;**41**(2): 361

[37] Corwin D, Lesch S. Apparent soil electrical conductivity measurements in agriculture. Computers and Electronics in Agriculture. 2005;**46**(1):11-43

*DOI: http://dx.doi.org/10.5772/intechopen.88989*

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics…*

Boteler DH, Trichtchenko L, Larocca P. Earth conductivity structures and their effects on geomagnetic induction in pipelines. Annales Geophysicae. 2007;

[38] Fernberg PA, Samson C,

**25**(1):207-218

**47**

[29] Ebrahimi-Birang N, Maul CP, Morley WA. Calibration of a TDR instrument for simultaneous measurements of soil water and soil electrical conductivity. Transactions of the ASABE (American Society of Agricultural and Biological Engineers). 2006;**49**(1):75-82

[30] Seyfried MS, Murdock MD. Measurement of soil water content with a 50-MHz soil dielectric sensor. Soil Science Society of America Journal. 2004;**68**:394-403

[31] Campbell JE. Dielectric Properties of Moist Soils at rf and Microwave Frequencies. New Hampshire, USA: Dartmouth Digital Library Collections; 1988

[32] ZChaudhari HC, Shinde VJ. Dielectric properties of soils at x-band microwave frequency. Indian Journal of Pure and Applied Physics. 2012;**50**(1): 64-66

[33] La Gioia A, Porter E, Merunka I, Shahzad A, Salahuddin S, OHalloran M. Open-ended coaxial probe technique for dielectric measurement of biological tissues: Challenges and common practices. Diagnostics. 2018;**8**(2):2-38

[34] Pozar D. Microwave Engineering. 4th ed. Hoboken, NJ, USA: Wiley; 2011

[35] Kirkscether EJ. Ground constant measurements using a section of balanced two-wire transmission line. IRE Transactions on Antennas and Propagation. 1960:307-312

[36] Fano W. Interacción de la radiación electromagnética Con Materiales dieléctricos y magnéticos. Aplicaciones. PHD Thesis. Universidad de Buenos Aires, Buenos Aires, Argentina; 2004

*The Electrical Properties of Soils with Their Applications to Agriculture, Geophysics… DOI: http://dx.doi.org/10.5772/intechopen.88989*

[37] Corwin D, Lesch S. Apparent soil electrical conductivity measurements in agriculture. Computers and Electronics in Agriculture. 2005;**46**(1):11-43

spinel ferrites and their composite materials. Proceedings of the IEEE.

*Electromagnetic Field Radiation in Matter*

measurement of soil water

2006;**49**(1):75-82

2004;**68**:394-403

1988

64-66

361

content and bulk electrical conductivity. Transactions of the ASAE. 1998;**41**(2):

[29] Ebrahimi-Birang N, Maul CP, Morley WA. Calibration of a TDR instrument for simultaneous measurements of soil water and soil electrical conductivity. Transactions of the ASABE (American Society of Agricultural and Biological Engineers).

[30] Seyfried MS, Murdock MD.

Moist Soils at rf and Microwave Frequencies. New Hampshire, USA: Dartmouth Digital Library Collections;

[32] ZChaudhari HC, Shinde VJ. Dielectric properties of soils at x-band microwave frequency. Indian Journal of Pure and Applied Physics. 2012;**50**(1):

[33] La Gioia A, Porter E, Merunka I, Shahzad A, Salahuddin S, OHalloran M. Open-ended coaxial probe technique for dielectric measurement of biological tissues: Challenges and common practices. Diagnostics. 2018;**8**(2):2-38

[34] Pozar D. Microwave Engineering. 4th ed. Hoboken, NJ, USA: Wiley; 2011

[35] Kirkscether EJ. Ground constant measurements using a section of balanced two-wire transmission line. IRE Transactions on Antennas and

[36] Fano W. Interacción de la radiación electromagnética Con Materiales dieléctricos y magnéticos. Aplicaciones. PHD Thesis. Universidad de Buenos Aires, Buenos Aires, Argentina; 2004

Propagation. 1960:307-312

Measurement of soil water content with a 50-MHz soil dielectric sensor. Soil Science Society of America Journal.

[31] Campbell JE. Dielectric Properties of

[20] Fano WG, Trainotti V. Dielectric properties of soils. In: 2001 Annual Report Conference on Electrical Insulation and Dielectric

Phenomena (Cat. No.01CH37225);

[21] Karakash J. Transmission Lines and Filter Networks. New York, USA:

Transient signals on transmission lines: An introduction to non-ideal effects

electrical systems. Synthesis Lectures on Computational Electromagnetics.

Microwave Engineering. 2nd ed. New York, USA: Wiley India Pvt. Limited;

[24] Olchawa O, Kumor M. Time domain

1974;**62**:98-103

2001. pp. 75-78

Macmillan; 1950

2008;**3**(1):1-144

2007

[22] Peterson A, Durgin DG.

and signal integrity issues in

[23] Collin R. Foundations for

reflectometry (TDR) measuring dielectric constant of polluted soil to estimate diesel oil content. Archives of Hydro-Engineering and Environmental

Mechanics. 2008;**55**(1–2):55-62

[26] Zante P. Water content and soil

[27] Robinson DA, Jones SB, Wraith JM, Friedman SP. A review of advances in dielectric and electrical conductivity measurement in soils using time domain reflectometry. Vadose Zone Journal. Soil Science Society of America.

[28] Evett SR. Coaxial multiplexer for time domain reflectometry

[25] Keysight. Time domain reflectometry theory. Keysight Technologies; Application Note; 2015

bulk electrical conductivity measurements. TDR Technology Applied to Soils. 2002;**49**(1):1-27

2003;**2**:444-475

**46**

[38] Fernberg PA, Samson C, Boteler DH, Trichtchenko L, Larocca P. Earth conductivity structures and their effects on geomagnetic induction in pipelines. Annales Geophysicae. 2007; **25**(1):207-218

**Chapter 4**

**Abstract**

gradient techniques.

power factor

**49**

**1. Introduction**

Thermoelectric Properties of

*Wiqar Hussain Shah and Waqas Muhammad Khan*

We will discuss the development of a new ternary and quaternary tellurium telluride chalcogenide nanoparticles used for efficient thermo-electric waste heat energy convertor called thermo-electric generator. Nanoparticles-based tellurium telluride chalcogenide nanoparticles, which will be used for thermoelectric generator, will eventually solve an important issue of the energy crises, that is, conversion of waste heat into useful electrical energy. By injecting charge carriers in the host matrix of Tl10-x-yAxByTe6 nanomaterials system, different types of dopants (A = Pb, Sn, Ca and B = Pb, Sb Sr, etc.), with x = 0–2.5 and y = 0–2.5 on tellurium telluride has been introduced to synthesize new materials by Co-precipitation techniques and also by solid state reaction techniques followed by Ball-Milling for the fabrication of nanomaterials. We will study the effect of reduction of charge carriers in thermal and transport properties using different dopants contents by replacing host atoms. The charge carrier's concentration will affect the ratio of electron-hole concentration which in turns increases the electron scattering in these chalcogenide nanoparticles, which will affect the electrical conductivity and thermo-power. The prime purpose of doping with different ionic radii and different concentration is to enhance the power factor for the tellurium telluride nanosystem. At the end one will be able to control different physical parameters such as, thermally assisted electrical conductivity, and thermopower. Different characterization technique will be applied, for example, X-Ray diffraction techniques will be used for structural analysis, SEM will shows the morphological structure of the particles at 100 nm and energy dispersive x-rays spectroscopy will be used for elemental analysis. The electrical conductivity will be measured by four-probe resistivity measurement techniques, and Seebeck coefficient will be measured by standard temperature

**Keywords:** effect of doping, Seebeck coefficient, electrical conductivity,

Energy storage and conversion devices (**Figure 1**) continue to be rich areas for scientific and engineering studies that incorporate novel features and functions in intelligent and interactive modes, represent a radical advance in consumer products, such as wearable electronics, healthcare devices, artificial intelligence, electric vehicles, smart household, and space satellites. However, there are still grand challenges in fundamental research and understanding to accelerate energy storage and

Chalcogenide System

#### **Chapter 4**

## Thermoelectric Properties of Chalcogenide System

*Wiqar Hussain Shah and Waqas Muhammad Khan*

#### **Abstract**

We will discuss the development of a new ternary and quaternary tellurium telluride chalcogenide nanoparticles used for efficient thermo-electric waste heat energy convertor called thermo-electric generator. Nanoparticles-based tellurium telluride chalcogenide nanoparticles, which will be used for thermoelectric generator, will eventually solve an important issue of the energy crises, that is, conversion of waste heat into useful electrical energy. By injecting charge carriers in the host matrix of Tl10-x-yAxByTe6 nanomaterials system, different types of dopants (A = Pb, Sn, Ca and B = Pb, Sb Sr, etc.), with x = 0–2.5 and y = 0–2.5 on tellurium telluride has been introduced to synthesize new materials by Co-precipitation techniques and also by solid state reaction techniques followed by Ball-Milling for the fabrication of nanomaterials. We will study the effect of reduction of charge carriers in thermal and transport properties using different dopants contents by replacing host atoms. The charge carrier's concentration will affect the ratio of electron-hole concentration which in turns increases the electron scattering in these chalcogenide nanoparticles, which will affect the electrical conductivity and thermo-power. The prime purpose of doping with different ionic radii and different concentration is to enhance the power factor for the tellurium telluride nanosystem. At the end one will be able to control different physical parameters such as, thermally assisted electrical conductivity, and thermopower. Different characterization technique will be applied, for example, X-Ray diffraction techniques will be used for structural analysis, SEM will shows the morphological structure of the particles at 100 nm and energy dispersive x-rays spectroscopy will be used for elemental analysis. The electrical conductivity will be measured by four-probe resistivity measurement techniques, and Seebeck coefficient will be measured by standard temperature gradient techniques.

**Keywords:** effect of doping, Seebeck coefficient, electrical conductivity, power factor

#### **1. Introduction**

Energy storage and conversion devices (**Figure 1**) continue to be rich areas for scientific and engineering studies that incorporate novel features and functions in intelligent and interactive modes, represent a radical advance in consumer products, such as wearable electronics, healthcare devices, artificial intelligence, electric vehicles, smart household, and space satellites. However, there are still grand challenges in fundamental research and understanding to accelerate energy storage and

In 1834, The French watch maker and a part time physicist Jeane Chaarlese Athanaese Peltiere (1785–1845) published new research article in the French journal "Annal. Phy. Che," where Peltiere printed that by applying of electrical current in the direction of two dissimilar conductors, which are linked in series, disclosed that temperature changes by the side of joint of conductor. He recognized this specific

In upper formulation, the term "Q" representing degree by which warmth remains flow away per unit of time, while "*П*" in formula is the "Peltiere coefficient" while the term "*I*" in upper formulation is electrical current which is formed

Wiliam Thomoson in 1851 presented his well-known effect, which is named Thomson effect. "By means of creating a change in temperature "*ΔT*" crossways a conductor, wherein he displays association among Seebeck and Peltier effect."

In the above equation "*K*" is Thomson's coefficient and "*J*" is the density of

Once material is being heated through one side, then due to thermal gradient the thermoelectric voltage rises, since charge carriers (holes/electrons) drift from hot to

> *<sup>S</sup>* <sup>¼</sup> *<sup>Δ</sup><sup>V</sup> ΔT*

Among materials two (2) edges is recognized as per coefficient of Seebeck. Its

When taking a Seebeck coefficient into its constituent, then written as below,

*Bm*<sup>∗</sup> 3*eh*<sup>2</sup>

The above equation is Seebeck coefficient of degenerate semiconductors and metals. Above equation comprises of three (3) variables: Variable "T" signifies temperature, variable "n" signifies charge carrier concentration, besides variable

*π* 3*n* <sup>2</sup>*=*<sup>3</sup>

ward cold. An induced voltage amount over a gradient of temperature

*<sup>S</sup>* <sup>¼</sup> *<sup>T</sup>* <sup>8</sup>*π*<sup>2</sup>*k*<sup>2</sup>

The Seebeck, The Peltiere and The Thomsen's coefficients are connected

*dq*

*Q* ¼ П*I* (5)

*<sup>I</sup>* ½ � *Watt=Ampere* (6)

*dt* ¼ �*KJΔ<sup>T</sup>* (7)

П ¼ *TS* (8)

(9)

(10)

This particular effect can be mathematically written as,

<sup>П</sup> <sup>¼</sup> *<sup>Q</sup>*

effect as a "Peltier effect."

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

in thermoelectric material (TEM).

current.

through the equation,

**1.1 The Seebeck Coefficient**

unit is volt per kelvin (v/k).

"*m*<sup>∗</sup> " signifies "effective mass"

**51**

Mathematically it can be written as,

conversion devices to commercial reality, which include new materials and structures with high ionic conductivity, tailored mixed electron/ion conductivity, novel interface engineering methodologies, new device concepts, efficient and scalable techniques for materials and system-level integrations. This research study is intended to provide information for those working in energy storage and conversion devices from materials, characterizations, devices and system integrations to communicate recent progress on current technologies and to exchange ideas about next-generation solutions. In this research work, we will design and develop a new tellurium telluride chalcogenide materials used for efficient thermo-electric waste heat energy convertor called thermo-electric generator. Nanostructures based on tellurium telluride chalcogenide materials, which are used for efficient thermoelectric generator will eventually solve large issues of the energy crises.

Thomas john Seebeck discovered the effect of Seebeck on 14 December 1820, at famous science Academy of "Berlin" via detecting the deflection of magnetic compass needle nearby close ring of conducting wire that one adjacent is linked to metal of a low temperature while the second side is linked with metal of a high temperature. This effect recently verified that magnetic compass needle is bounced by reason of electrical current movement in wire and term "Thermo-magnetism" is changed by means of "the effect of thermoelectric". This electrical current/emf are similarly recognizing via means of "Seebeck emf." Thermo power "S" or Seebeck effect is the variation in electrical potential

$$(\Delta \mathbf{V} = \mathbf{V}\_{\text{hot}} - \mathbf{V}\_{\text{cold}}) \tag{1}$$

Divided by the thermal gradient

$$\mathbf{T}\left(\Delta \mathbf{T} = \mathbf{T}\_{\text{-hot}} - \mathbf{T}\_{\text{-cold}}\right),\tag{2}$$

Mathematically we can write it as,

$$\mathbf{S} = \Delta \mathbf{V} / \Delta \mathbf{T} \tag{3}$$

$$\mathbf{S} = (\mathbf{V}\_{\text{-cold}} - \mathbf{V}\_{\text{-hot}})/(\mathbf{T}\_{\text{-hot}} - \mathbf{T}\_{\text{-cold}}) [\mathbf{V}/\mathbf{K}] \tag{4}$$

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

In 1834, The French watch maker and a part time physicist Jeane Chaarlese Athanaese Peltiere (1785–1845) published new research article in the French journal "Annal. Phy. Che," where Peltiere printed that by applying of electrical current in the direction of two dissimilar conductors, which are linked in series, disclosed that temperature changes by the side of joint of conductor. He recognized this specific effect as a "Peltier effect."

This particular effect can be mathematically written as,

$$Q = \Pi I \tag{5}$$

$$
\Pi = \frac{Q}{I} \text{ [Watt/Ampere]} \tag{6}
$$

In upper formulation, the term "Q" representing degree by which warmth remains flow away per unit of time, while "*П*" in formula is the "Peltiere coefficient" while the term "*I*" in upper formulation is electrical current which is formed in thermoelectric material (TEM).

Wiliam Thomoson in 1851 presented his well-known effect, which is named Thomson effect. "By means of creating a change in temperature "*ΔT*" crossways a conductor, wherein he displays association among Seebeck and Peltier effect."

Mathematically it can be written as,

$$\frac{dq}{dt} = -K\mathcal{J}\Delta T\tag{7}$$

In the above equation "*K*" is Thomson's coefficient and "*J*" is the density of current.

The Seebeck, The Peltiere and The Thomsen's coefficients are connected through the equation,

$$
\Pi = T\mathbf{S} \tag{8}
$$

#### **1.1 The Seebeck Coefficient**

Once material is being heated through one side, then due to thermal gradient the thermoelectric voltage rises, since charge carriers (holes/electrons) drift from hot to ward cold. An induced voltage amount over a gradient of temperature

$$\left(\mathcal{S} = \frac{\Delta V}{\Delta T}\right) \tag{9}$$

Among materials two (2) edges is recognized as per coefficient of Seebeck. Its unit is volt per kelvin (v/k).

When taking a Seebeck coefficient into its constituent, then written as below,

$$S = T \frac{8\pi^2 k\_B^2 m^\*}{3eh^2} \left(\frac{\pi}{3n}\right)^{2/3} \tag{10}$$

The above equation is Seebeck coefficient of degenerate semiconductors and metals. Above equation comprises of three (3) variables: Variable "T" signifies temperature, variable "n" signifies charge carrier concentration, besides variable "*m*<sup>∗</sup> " signifies "effective mass"

conversion devices to commercial reality, which include new materials and structures with high ionic conductivity, tailored mixed electron/ion conductivity, novel interface engineering methodologies, new device concepts, efficient and scalable techniques for materials and system-level integrations. This research study is intended to provide information for those working in energy storage and conversion devices from materials, characterizations, devices and system integrations to communicate recent progress on current technologies and to exchange ideas about next-generation solutions. In this research work, we will design and develop a new tellurium telluride chalcogenide materials used for efficient thermo-electric waste heat energy convertor called thermo-electric generator. Nanostructures based on tellurium telluride chalcogenide materials, which are used for efficient thermoelec-

Thomas john Seebeck discovered the effect of Seebeck on 14 December 1820, at famous science Academy of "Berlin" via detecting the deflection of magnetic compass needle nearby close ring of conducting wire that one adjacent is linked to metal of a low temperature while the second side is linked with metal of a high temperature. This effect recently verified that magnetic compass needle is bounced by reason of electrical current movement in wire and term "Thermo-magnetism" is changed by means of "the effect of thermoelectric". This electrical current/emf are similarly recognizing via means of "Seebeck emf." Thermo power "S" or Seebeck

ð Þ ΔV ¼ Vhot � Vcold (1)

ð Þ ΔT ¼ T\_hot � T\_cold , (2)

S ¼ ð Þ V\_cold � V\_hot *=*ð Þ T\_hot � T\_cold ½ � V*=*K (4)

S ¼ ΔV*=*ΔT (3)

tric generator will eventually solve large issues of the energy crises.

effect is the variation in electrical potential

**Figure 1.**

*Energy storage and conversion devices.*

*Electromagnetic Field Radiation in Matter*

Divided by the thermal gradient

Mathematically we can write it as,

**50**

#### **1.2 The electrical conductivity**

Charge movement in matter is named conductivity, it is represented by "*σ*". The movement of holes or free electrons in a particular way reasons an electrical current in matter. This electrical current in matter which in a particular way, is the consequence of movement of charges via smearing the potential change crossways a semiconductor or conductor. The equation for current density "*J*" is;

$$J = nq\nu\_d\tag{11}$$

**1.4 Objectives**

Briefly, the specific objectives are as follows:

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

thermos-chemical treatment.

its effect on TE performance.

chemical reduction.

temperature TE applications.

phase of the materials.

conductivity.

**53**

synthesis to obtain nanomaterials. Specifically:

1.Fabrication of chalcogenides based materials via cost effective chemical

2.Optimization of critical SPS parameters (such as sintering temperature, applied pressure, holding time and heating rates) for chalcogenides while consolidating these materials to preserve the nanostructure, to reduce thermal

3.Bottom-up chemical synthesis and detailed characterization for low

followed by detailed physiochemical characterizations.

c. Study the effect of ternary and quaternary tellurium telluride chalcogenide nanoparticles as nanoinclusions/grain boundary.

5.Optimization of SPS critical parameters (such as sintering temperature, applied pressure, holding time and heating rates) while consolidating these materials to preserve the nanostructure and obtain the desired phases.

**Figure 2** shows the measurement system that was used to measure the electrical properties of the samples. **Figure 2(a)** displays the photograph of the commercial ZEM-3 system and **Figure 2(b)** shows a sample mounted in the ZEM-3 apparatus for electrical resistivity and Seebeck coefficient measurement. In the ZEM-3 system, electrical resistivity was measured using a four probe technique and electrical conductivity was calculated from the electrical resistivity. The four probe technique for measuring the resistivity simply accounts for the contact resistance between metal electrodes and the semiconducting samples. **Figure 3(a)** displays a schematic diagram of the four probe used by the ZEM-3 system. As shown in **Figure 3(a)**, in the four probe technique current I was passed through one set of probes (blue blocks) and the voltage difference (ΔV) was measured using another set of probes (small red spheres). These four probes were connected to four thermocouples. The voltage and

**1.5 Electrical conductivity and Seebeck coefficient measurements**

4.Fabrication of silicide based TE materials through mechanical alloying (topdown approach). Specifically: a. n-type Tl5Te3 by ball milling for an optimized reaction time and followed by materials' characterizations to identify the

a. Doping of Sb and Pb in n-type Tl5Te3 nanomaterials and to investigate

b. Fabrication of p-type HMS via ball milling by utilizing optimized react

a. Nanostructured n-type Tl5Te3 via solution co-precipitation and

b. Nanostructured p-type Tl5Te3 via solution co-precipitation and fast

In above equation, "*n*" signifies carrier concentration, the q in equation signifies charge carrier which is equivalents for holes to <sup>þ</sup>1*:*<sup>602</sup> � <sup>10</sup>�19*<sup>C</sup>* while �1*:*<sup>602</sup> � 10�19*C* for e's, and "*νd*" is called electron drift velocity. By means of putting value of drift velocity, *<sup>ν</sup><sup>d</sup>* <sup>¼</sup> *Ee<sup>τ</sup> <sup>m</sup>*<sup>∗</sup> in Eq. (14), we obtained the relation which is used in the existence of constant electric field for current density, and this is as well-known as law of ohm.

$$J = \frac{ne^2 \pi E}{m^\*} \tag{12}$$

In terms of electrical conductivity and electric field, the current density *J* is,

$$J = \sigma E \tag{13}$$

$$
\sigma = \frac{ne^2 \pi}{m^\*} = ne\mu \tag{14}
$$

Where

$$
\mu = \frac{e\tau}{m^\*} \tag{15}
$$

While "*μ*" is charge transporter movement and having dimension of *cm*<sup>2</sup> *Vs* h i. "*τ*" is the scattering time while"*m*<sup>∗</sup> " is electron effective mass. The scattering time "*τ*" can be described as, "it is the amount of time in which the charge transporter their momentum is places besides turn out to be in balance after the elimination of exterior electric field. The scattering time takes near relative through the electronegativity of an element.

#### **1.3 Power factor**

Through study of power factor (PF), the success of a cooler of thermoelectric cooler (TEC) besides generator (TEG) is resolute through study of power factor; it is represented via "PF," and calculated through Seebeck coefficient square multiplied through the electrical conductivity at precise temperature.

Mathematically it can be written as,

$$PF = \mathbb{S}^2 \sigma \tag{16}$$

In the above equation, *PF* is Power Factor, *S* is Seebeck coefficient while *σ* is Electrical conductivity.

Thermoelectric devices having values of Seebeck coefficient (S) and electrical conductivities high, gives high power factor (PF) and too charitable high electrical power.

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

#### **1.4 Objectives**

**1.2 The electrical conductivity**

*Electromagnetic Field Radiation in Matter*

drift velocity, *<sup>ν</sup><sup>d</sup>* <sup>¼</sup> *Ee<sup>τ</sup>*

law of ohm.

Where

electronegativity of an element.

Mathematically it can be written as,

**1.3 Power factor**

Electrical conductivity.

power.

**52**

Charge movement in matter is named conductivity, it is represented by "*σ*". The movement of holes or free electrons in a particular way reasons an electrical current in matter. This electrical current in matter which in a particular way, is the consequence of movement of charges via smearing the potential change crossways a

In above equation, "*n*" signifies carrier concentration, the q in equation signifies charge carrier which is equivalents for holes to <sup>þ</sup>1*:*<sup>602</sup> � <sup>10</sup>�19*<sup>C</sup>* while �1*:*<sup>602</sup> � 10�19*C* for e's, and "*νd*" is called electron drift velocity. By means of putting value of

existence of constant electric field for current density, and this is as well-known as

*<sup>J</sup>* <sup>¼</sup> *ne*<sup>2</sup>*τ<sup>E</sup>*

In terms of electrical conductivity and electric field, the current density *J* is,

*<sup>μ</sup>* <sup>¼</sup> *<sup>e</sup><sup>τ</sup>*

While "*μ*" is charge transporter movement and having dimension of *cm*<sup>2</sup>

the scattering time while"*m*<sup>∗</sup> " is electron effective mass. The scattering time "*τ*" can be described as, "it is the amount of time in which the charge transporter their momentum is places besides turn out to be in balance after the elimination of exterior electric field. The scattering time takes near relative through the

Through study of power factor (PF), the success of a cooler of thermoelectric cooler (TEC) besides generator (TEG) is resolute through study of power factor; it

*PF* <sup>¼</sup> *<sup>S</sup>*<sup>2</sup>

In the above equation, *PF* is Power Factor, *S* is Seebeck coefficient while *σ* is

Thermoelectric devices having values of Seebeck coefficient (S) and electrical conductivities high, gives high power factor (PF) and too charitable high electrical

is represented via "PF," and calculated through Seebeck coefficient square multiplied through the electrical conductivity at precise temperature.

*<sup>σ</sup>* <sup>¼</sup> *ne*<sup>2</sup>*<sup>τ</sup>*

*<sup>m</sup>*<sup>∗</sup> in Eq. (14), we obtained the relation which is used in the

*J* ¼ *nqν<sup>d</sup>* (11)

*<sup>m</sup>*<sup>∗</sup> (12)

*J* ¼ *σE* (13)

*<sup>m</sup>*<sup>∗</sup> <sup>¼</sup> *ne<sup>μ</sup>* (14)

*<sup>m</sup>*<sup>∗</sup> (15)

*σ* (16)

*Vs* h i

. "*τ*" is

semiconductor or conductor. The equation for current density "*J*" is;

Briefly, the specific objectives are as follows:

	- a. Nanostructured n-type Tl5Te3 via solution co-precipitation and thermos-chemical treatment.
	- b. Nanostructured p-type Tl5Te3 via solution co-precipitation and fast chemical reduction.
	- a. Doping of Sb and Pb in n-type Tl5Te3 nanomaterials and to investigate its effect on TE performance.
	- b. Fabrication of p-type HMS via ball milling by utilizing optimized react followed by detailed physiochemical characterizations.
	- c. Study the effect of ternary and quaternary tellurium telluride chalcogenide nanoparticles as nanoinclusions/grain boundary.

#### **1.5 Electrical conductivity and Seebeck coefficient measurements**

**Figure 2** shows the measurement system that was used to measure the electrical properties of the samples. **Figure 2(a)** displays the photograph of the commercial ZEM-3 system and **Figure 2(b)** shows a sample mounted in the ZEM-3 apparatus for electrical resistivity and Seebeck coefficient measurement. In the ZEM-3 system, electrical resistivity was measured using a four probe technique and electrical conductivity was calculated from the electrical resistivity. The four probe technique for measuring the resistivity simply accounts for the contact resistance between metal electrodes and the semiconducting samples. **Figure 3(a)** displays a schematic diagram of the four probe used by the ZEM-3 system. As shown in **Figure 3(a)**, in the four probe technique current I was passed through one set of probes (blue blocks) and the voltage difference (ΔV) was measured using another set of probes (small red spheres). These four probes were connected to four thermocouples. The voltage and

The Seebeck coefficient is simply defined as the ratio of an open-circuit potential

<sup>S</sup> <sup>¼</sup> <sup>Δ</sup>*<sup>V</sup>*

For Seebeck coefficient measurement, the voltage and temperatures were measured simultaneously by the same thermocouple probe (small red spheres) as shown in **Figure 3(a)**. Then, the voltage difference (Δ*V*) was measured for a set of temperature differences (Δ*T*) between the two probes and the Seebeck coefficient was

Thallium antimony telluride TlSbTe2 nanoparticles have been prepared by coprecipitation techniques. We have investigated that the electrical resistivity is high and the thermal conductivity is low as compared to Sintered Bi2Te3 and TAGS "(GeTe1-x(AgSbTe2)x" material. The Seebeck coefficient of TlSbTe2 is 224 μvk�<sup>1</sup> at 666 k which is positive in the whole temperature range showing p-type behavior.

low as compared the power factor of current thermoelectric devices, that is, in the

Prepared a new low-valent thallium silicon telluride Tl6Si2Te6 and compared there results crystal and electronic structure and there electronic properties with Tl6Ge2Te6, they observed the same crystal structure of Tl6Si2Te6 with Tl6Ge2Te6,

was found with a single Si-Si bond, the weak bond exist among Tl-Tl and irregularly coordinated by 5 or 6 Te atoms, the black color was observed for both compounds exhibiting a small band gap of the order **0***:***9 ev** for Tl6Si2Te6 and **0***:***5 ev** for

Tl6Ge2Te6 compounds. The electrical conductivity and Seebeck coefficient investi-

Prepared the samples of polycrystalline Ag9TlTe5, the different nominal compo-

dimensionless thermoelectric figure of merit investigated for Ag9TlTe5.0 is very low approximately 0*:*08 while for X≥5*:*0 is high approximately 1*:*0, which shows that all the physical properties are changing by changing tellurium content in Ag9TlTex [3]. Herman et al. [4] uses the concept of electronic density of states to increase the thermoelectric figure of merit in lead telluride PbTe, the Seebeck coefficient was enhanced by deforming the electronic density of states, leads to double the thermoelectric figure of merit and he further explained that in nanostructured material

Synthesized Tl4MTe4 ð Þ M ¼ Zr, Hf and investigated their crystal structure and

thermoelectric properties, for investigating their crystal structure, the X-ray diffraction technique were used, they found that the crystal structure of Tl4MTe4 is octahedral with a space group *R*3, the unit cell dimension for Tl4ZrTe4 is

**cm**�**<sup>1</sup>** and <sup>þ</sup>**<sup>65</sup> <sup>μ</sup>vK**�**<sup>1</sup>**

sition by them are: Ag9TlTexð Þ *X* 5*:*0, 5*:*05, 5*:*1, 5*:*2, 5*:*3, 5*:*5, 5*:*7, 6*:*0 , the Ag9TlTex samples were made by heating the Ag2Te, Tl2Te, Tl2Te1.2 and Te with a proper quantity in sealed quartz tube, the ball-milling and hot-press techniques were used for construction of required samples. The X-Ray Diffraction (XRD) technique were used for analysis of phase relation, the electrical resistivity is calculated which is decreasing with increasing temperature except for Ag9TlTe5.0, the investigated Seebeck coefficient is almost independent of temperature except for Ag9TlTe5.0, the power factor is quite high of the order of 0*:*<sup>3</sup> � <sup>0</sup>*:*<sup>4</sup> � <sup>10</sup>�<sup>3</sup> Wm�1K�<sup>2</sup> while the power factor for Ag9TlTe5.0 is quite low of the order of0*:*<sup>05</sup> � <sup>10</sup>�<sup>3</sup> Wm�1K�<sup>2</sup>

ð Þ *at* **300** *K* [2].

the quantitative results for Tl6Si2Te6. The demerit ½ � *Si***6***Te***<sup>6</sup>**

<sup>σ</sup>) found for TlSbTe2 is 8*:*<sup>9</sup> � <sup>10</sup>�<sup>4</sup> Wm�1k�<sup>2</sup> at 576 K which is

. The figure of merit (ZT) of the order of 0.87 was found at

<sup>Δ</sup>*<sup>T</sup>* (19)

**<sup>2</sup>**� units crystal structure

. The

ð Þ *at* **300** *K* , while for Tl6Gd2Te6

difference (Δ*V*) to a temperature gradient (Δ*T*),

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

calculated from the slope of Δ*V*- Δ*T* plot.

The power factor (S<sup>2</sup>

range 10�<sup>3</sup> Wm�1k�<sup>2</sup>

715 k for TlSbTe2 [1].

gated for Tl6Si2Te6 is **5***:***5 Ω**�**<sup>1</sup>**

**cm**�**<sup>1</sup>** and <sup>þ</sup>**<sup>150</sup> <sup>μ</sup>vK**�**<sup>1</sup>**

it may give us further good results [4].

is **3 Ω**�**<sup>1</sup>**

**55**

#### **Figure 2.**

*(a) A commercial ZEM-3 system and (b) magnified sample holder region [indicated by red circle in (a)] with a sample mounted for measurement.*

**Figure 3.**

*(a) Displays a schematic diagram of the four probe used by the ZEM-3 system and (b) a typical I-V curve for resistance measurement.*

current control, data acquisition, and interpretation were fully automated and computer controlled. The electrical resistivity was found from the relation,

$$
\rho = \frac{A}{l} \left(\frac{\Delta v}{\Delta I}\right) \tag{17}
$$

Where (*ΔV/ΔI*) is the slope of the I-V curve as shown in **Figure 3(b)**, *A* is the cross-sectional area of the sample and *l* is the distance between the voltage probes. The electrical conductivity was then calculated as the reciprocal of the resistivity.

$$
\sigma \ll \frac{1}{\rho} \tag{18}
$$

During the resistivity measurement, the temperatures at both probes were kept constant to minimize the Seebeck voltage. The same ZEM-3 system (**Figure 2**) was used for Seebeck coefficient measurement.

The Seebeck coefficient is simply defined as the ratio of an open-circuit potential difference (Δ*V*) to a temperature gradient (Δ*T*),

$$\mathbf{S} = \frac{\Delta V}{\Delta T} \tag{19}$$

For Seebeck coefficient measurement, the voltage and temperatures were measured simultaneously by the same thermocouple probe (small red spheres) as shown in **Figure 3(a)**. Then, the voltage difference (Δ*V*) was measured for a set of temperature differences (Δ*T*) between the two probes and the Seebeck coefficient was calculated from the slope of Δ*V*- Δ*T* plot.

Thallium antimony telluride TlSbTe2 nanoparticles have been prepared by coprecipitation techniques. We have investigated that the electrical resistivity is high and the thermal conductivity is low as compared to Sintered Bi2Te3 and TAGS "(GeTe1-x(AgSbTe2)x" material. The Seebeck coefficient of TlSbTe2 is 224 μvk�<sup>1</sup> at 666 k which is positive in the whole temperature range showing p-type behavior. The power factor (S<sup>2</sup> <sup>σ</sup>) found for TlSbTe2 is 8*:*<sup>9</sup> � <sup>10</sup>�<sup>4</sup> Wm�1k�<sup>2</sup> at 576 K which is low as compared the power factor of current thermoelectric devices, that is, in the range 10�<sup>3</sup> Wm�1k�<sup>2</sup> . The figure of merit (ZT) of the order of 0.87 was found at 715 k for TlSbTe2 [1].

Prepared a new low-valent thallium silicon telluride Tl6Si2Te6 and compared there results crystal and electronic structure and there electronic properties with Tl6Ge2Te6, they observed the same crystal structure of Tl6Si2Te6 with Tl6Ge2Te6, the quantitative results for Tl6Si2Te6. The demerit ½ � *Si***6***Te***<sup>6</sup> <sup>2</sup>**� units crystal structure was found with a single Si-Si bond, the weak bond exist among Tl-Tl and irregularly coordinated by 5 or 6 Te atoms, the black color was observed for both compounds exhibiting a small band gap of the order **0***:***9 ev** for Tl6Si2Te6 and **0***:***5 ev** for Tl6Ge2Te6 compounds. The electrical conductivity and Seebeck coefficient investigated for Tl6Si2Te6 is **5***:***5 Ω**�**<sup>1</sup> cm**�**<sup>1</sup>** and <sup>þ</sup>**<sup>65</sup> <sup>μ</sup>vK**�**<sup>1</sup>** ð Þ *at* **300** *K* , while for Tl6Gd2Te6 is **3 Ω**�**<sup>1</sup> cm**�**<sup>1</sup>** and <sup>þ</sup>**<sup>150</sup> <sup>μ</sup>vK**�**<sup>1</sup>** ð Þ *at* **300** *K* [2].

Prepared the samples of polycrystalline Ag9TlTe5, the different nominal composition by them are: Ag9TlTexð Þ *X* 5*:*0, 5*:*05, 5*:*1, 5*:*2, 5*:*3, 5*:*5, 5*:*7, 6*:*0 , the Ag9TlTex samples were made by heating the Ag2Te, Tl2Te, Tl2Te1.2 and Te with a proper quantity in sealed quartz tube, the ball-milling and hot-press techniques were used for construction of required samples. The X-Ray Diffraction (XRD) technique were used for analysis of phase relation, the electrical resistivity is calculated which is decreasing with increasing temperature except for Ag9TlTe5.0, the investigated Seebeck coefficient is almost independent of temperature except for Ag9TlTe5.0, the power factor is quite high of the order of 0*:*<sup>3</sup> � <sup>0</sup>*:*<sup>4</sup> � <sup>10</sup>�<sup>3</sup> Wm�1K�<sup>2</sup> while the power factor for Ag9TlTe5.0 is quite low of the order of0*:*<sup>05</sup> � <sup>10</sup>�<sup>3</sup> Wm�1K�<sup>2</sup> . The dimensionless thermoelectric figure of merit investigated for Ag9TlTe5.0 is very low approximately 0*:*08 while for X≥5*:*0 is high approximately 1*:*0, which shows that all the physical properties are changing by changing tellurium content in Ag9TlTex [3].

Herman et al. [4] uses the concept of electronic density of states to increase the thermoelectric figure of merit in lead telluride PbTe, the Seebeck coefficient was enhanced by deforming the electronic density of states, leads to double the thermoelectric figure of merit and he further explained that in nanostructured material it may give us further good results [4].

Synthesized Tl4MTe4 ð Þ M ¼ Zr, Hf and investigated their crystal structure and thermoelectric properties, for investigating their crystal structure, the X-ray diffraction technique were used, they found that the crystal structure of Tl4MTe4 is octahedral with a space group *R*3, the unit cell dimension for Tl4ZrTe4 is

current control, data acquisition, and interpretation were fully automated and computer controlled. The electrical resistivity was found from the relation,

*(a) Displays a schematic diagram of the four probe used by the ZEM-3 system and (b) a typical I-V curve for*

*(a) A commercial ZEM-3 system and (b) magnified sample holder region [indicated by red circle in (a)] with*

<sup>ρ</sup> <sup>¼</sup> *<sup>A</sup> l*

Δ*v* Δ*I* 

Where (*ΔV/ΔI*) is the slope of the I-V curve as shown in **Figure 3(b)**, *A* is the cross-sectional area of the sample and *l* is the distance between the voltage probes. The electrical conductivity was then calculated as the reciprocal of the resistivity.

> σ α 1 *ρ*

used for Seebeck coefficient measurement.

**Figure 2.**

**Figure 3.**

**54**

*resistance measurement.*

*a sample mounted for measurement.*

*Electromagnetic Field Radiation in Matter*

During the resistivity measurement, the temperatures at both probes were kept constant to minimize the Seebeck voltage. The same ZEM-3 system (**Figure 2**) was

(17)

(18)

*a* ¼ 14*:*6000 5ð Þ *Å* and *c* ¼ 14*:*189 1ð Þ *Å*, and Tl4HfTe4 the unit cell dimension is *a* ¼ 14*:*594 1ð Þ *Å* and *c* ¼ 14*:*142 3ð Þ *Å*. Linear muffin-tin orbital (LMTO) methods were used for the calculation of electronic structure clearing that Tl4MTe4 exhibit semiconducting behavior exhibiting an indirect band gap of0*:*3 ev. They observed that the electrical resistivity and Seebeck coefficient decreased while the thermal conductivity increased temperature, the thermoelectric figure of merit (ZT) for Tl4ZrTe4 compound increased from 0*:*14 to 0*:*1 between the temperature 370 K and 420 K while decreasing when the temperature increased from 420 K, for Tl4HfTe4 the ZT varies from 0*:*05 to 0*:*09 between temperature 370 K and 540 K [5].

values are 0*:*22 at 550 K were achieved for the stoichiometric compound on cold

Fabricate and improve the thermoelectric properties of alumina nanoparticledispersed Bi0.5Sb1.5Te3 matrix composites, the nanoparticles were fabricated by ball milling process and followed by spark plasma sintering process. The p-type bismuth antimony telluride (BST) nanopowder prepared from the mechano-chemical process, were mixed with 1*:*0, 0*:*5, and 0*:*3 vol*:*% with Al2O3 nanoparticles by ball milling process. They studied the surface morphology and investigated the size of the nanoparticles. The electrical resistivity is increasing with temperature from <sup>1</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>5</sup> <sup>Ω</sup>m at 293 K to 2*:*<sup>5</sup> � <sup>10</sup>�<sup>5</sup> <sup>Ω</sup>m at 473 K, and the seebeck coefficient is increases from <sup>þ</sup><sup>205</sup> *<sup>μ</sup>VK*�<sup>1</sup> to <sup>þ</sup><sup>210</sup> *<sup>μ</sup>VK*�<sup>1</sup> from temperature 293 K to 473 K, respectively, shows p-type semiconducting behavior. The highest Seebeck coefficient of the order of 235 μV*K*�<sup>1</sup> at 373 K was observed from 0*:*3vol*:*% Al2O3/BST nanocomposite. They show that the increasing volume fraction of Al2O3 increases the carrier density which affects the Seebeck coefficient, carrier mobility and electrical resistivity of Al2O3/BST nanocomposites. The observed power factor is 1*:*7

decreased by the addition of Al2O3 nanoparticles, for pure BST is 0*:*8 Wm�1K�<sup>1</sup> and

Studied the thermoelectric properties of Indium doped SnTe (InxSnx-1Te) nano-

Investigated the thermoelectric properties of indium doped PbTe1�ySey alloys, solid state method was used for the synthesis. They showed that the carrier concentration and electrical resistivity increases with increasing temperature in n-type indium doped PbTe1�ySey alloys affecting the Seebeck coefficient and as a result the power factor of PbTe1�ySey alloys. The bipolar effect was observed at high temperature which restricts the thermoelectric figure of merit to 0*:*66 at 800 K with 30% content in sample; they further concluded that for the enhancement of thermoelec-

0*:*3vol*:*% Al2O3/BST composite at 373 K which is higher from pure BST [9].

greater than 1 is at temperature 873 K in for In0.0025Sn0.9975Te [10].

tric properties, the increased carrier concentration must be reduced at high

Optimized the thermoelectric properties of Tl10�x�ySnxBiyTe6, a quaternary telluride series has been studied. The crystal structure was investigated by X-Ray diffraction which belongs to Tl5Te3 type structure and the volume is increases with increasing the Sn concentration in Tl10-x-ySnxBiyTe6, the electronic structure calculation revealed that Tl8.5SnBi0.5Te6 is a narrow band gap p-type intrinsic semiconductor and Tl9Sn0.5 Bi0.5Te6 is a p-type and narrow band gap extrinsic semiconductor. The electrical conductivity is decreasing with increasing temperature for Tl9Sn1�yBiyTe6, Tl8.67Sn1�<sup>y</sup> BiyTe6 and Tl8.33Sn1.12Bi55Te6. The low increase of the

structured compound. They prepared InxSnx-1Te by ball milling and hot press techniques, and investigated their thermal conductivity, diffusivity and electrical conductivity decreases with temperature ranging from 300 to 900 K, while the power factor and Seebeck coefficient is increases and the specific heat is almost constant a little increase was seen in InxSnx�1Te. The sample is also prepared by ball mill and hand mill method but the result is the same. They observed the relationship of carrier concentration vs. Seebeck effect which shows that In doped SnTe shows abnormal behavior with increasing carrier concentration, they get the SEM, TEM, and HRTEM images of InxSnx-1Te which clearly shows the sample is consist of both small and large grain boundaries with a good crystallinity which effects the thermal conductivity of the sample. The thermal electric figure of merit is observed which

cm at 393 K for 0*:*3vol*:*% Al2O3/BST

cm. The thermal conductivity is

was observed in the thermal conductivity of

. The thermoelectric figure of merit (ZT) is 1.5 for

press pellets [8].

times higher from pure BST, that is, 33 μWK�<sup>2</sup>

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

nanocomposites, and for pure BST is22 μWK�<sup>2</sup>

for Al2O3/BST is0*:*7 Wm�1K�<sup>1</sup>

temperature [11].

**57**

order of 0*:*4 Wm�1K�<sup>1</sup> to 1*:*4 Wm�1K�<sup>1</sup>

In the other research, prepared the ternary compound Tl2ZrTe3 and compared it with Tl2SnTe3 and investigated the different properties such as structural, physical and thermal properties. Tl2ZrTe3 compound exhibits a simple cubic structure with a lattice parameter *a* ¼ 19*:*118 1ð Þ *Å Z*ð Þ ¼ 36 . They investigated the electronic properties which clears that the Tl2ZrTe3 exhibits semiconducting behavior. The band gap observed for Tl2ZrTe3 is 0*:*7 ev which is higher from the band gap of Tl2SnTe3 ð Þ *Eg* ¼ 0*:*4 ev , the electrical conductivity is independent of temperature ranging from room temperature to 450 K, when the temperature rises from 450 K the electrical conductivity falls abruptly, while for Tl2SnTe3 compound the electrical conductivity decreases from 22 Ω�<sup>1</sup> cm�<sup>1</sup> to 15 Ω�<sup>1</sup> cm�<sup>1</sup> with increasing temperature from room temperature to 515 K, from 373 K to 450 K the thermal conductivity decreases from 0*:*39 Wm�1K�<sup>1</sup> to 0*:*30 Wm�1K�<sup>1</sup> for Tl2ZrTe3 and for Tl2SnTe3 the thermal conductivity decreases from 0*:*24 Wm�1K�<sup>1</sup> at 420 K to 0*:*20 Wm�1K�<sup>1</sup> at 450 K. For Tl2ZrTe3 the calculated Seebeck coefficient is generally same in the range of 373 K to450 K, but decreases with temperature greater than450 K, the peak experiential value of Seebeck coefficient is120 μVK�<sup>1</sup> . For Tl2SnTe3 the Seebeck coefficient increases from 240 μVK�<sup>1</sup> to 330 μVK�<sup>1</sup> with increasing temperature from room temperature to 450 K. The calculated power factor for Tl2ZrTe3 is almost same changes from 0*:*35 μWcm�1K�<sup>1</sup> to 0*:*41 μWcm�1K�<sup>1</sup> while for Tl2SnTe2 the power factor is increasing with temperature. The ZT value of 0*:*18 at 450 K was observed for Tl2ZrTe3 which is less than ZT value of Tl2SnTe3 ð Þ ZT ¼ 0*:*31 at 500 K [6].

Studied the thermoelectric properties of ternary thallium chalcogenides TlGdQ <sup>2</sup> (Q = SE, Te) and Tl9GdTe6. They found that TlGdQ <sup>2</sup> is isostructural with TlSbQ <sup>2</sup> and Tl9GdTe6 is isostructural with Tl9BiTe6. They found the high Seebeck coefficient and low electrical conductivity of TlGdQ 2. The low thermal conductivity of the order of 0*:*5 Wm�1K�<sup>1</sup> was investigated at room temperature for TlGdTe2. In the studies of Tl9GdTe6 they found the low power factor due to the high electrical conductivity of 850 Ω�<sup>1</sup> cm�<sup>1</sup> and low seebeck coefficient of 27 μVK�<sup>1</sup> at550 K. In the whole study they found the good thermoelectric property for TlGdTe2, the dimensionless figure of merit is 0*:*5 at 500 K [7].

Prepared the thallium lanthanide telluride Tl10�xLnxTe6 with Ln = CE, Pr, Nd, Sm, Gd, Tb, Dy, Ho and Er and 0*:*25≤ *x*≤1*:*32 by hot press method. He found that the crystal structure of Tl10�xLnxTe6 is isostructural to Tl9BiTe6 and the volume of unit cell is increases with increasing lanthanide content. They investigated that the electrical and thermal conductivity decreases with increasing the lanthanide content while the Seebeck coefficient is increases. In this series for **Tl8.97Ce1.03 Te6, Tl8.92Pr1.08Te6 and Tl8.99Sm1.01Te6** the electrical and thermal conductivity increases and Seebeck coefficient value decreases due to the discontinuity in the band gap as compared to Tl9LnTe6 compounds. The power factor is increases with increasing the lanthanide content "x" in the Tl9-xLnxTe6 resulting the increase in the dimensionless thermoelectric figure of merit, the best thermoelectric figure of merit *a* ¼ 14*:*6000 5ð Þ *Å* and *c* ¼ 14*:*189 1ð Þ *Å*, and Tl4HfTe4 the unit cell dimension is *a* ¼ 14*:*594 1ð Þ *Å* and *c* ¼ 14*:*142 3ð Þ *Å*. Linear muffin-tin orbital (LMTO) methods were used for the calculation of electronic structure clearing that Tl4MTe4 exhibit semiconducting behavior exhibiting an indirect band gap of0*:*3 ev. They observed that the electrical resistivity and Seebeck coefficient decreased while the thermal conductivity increased temperature, the thermoelectric figure of merit (ZT) for Tl4ZrTe4 compound increased from 0*:*14 to 0*:*1 between the temperature 370 K and 420 K while decreasing when the temperature increased from 420 K, for Tl4HfTe4 the ZT varies from 0*:*05 to 0*:*09 between temperature 370 K and 540 K [5].

In the other research, prepared the ternary compound Tl2ZrTe3 and compared it with Tl2SnTe3 and investigated the different properties such as structural, physical and thermal properties. Tl2ZrTe3 compound exhibits a simple cubic structure with a lattice parameter *a* ¼ 19*:*118 1ð Þ *Å Z*ð Þ ¼ 36 . They investigated the electronic properties which clears that the Tl2ZrTe3 exhibits semiconducting behavior. The band

gap observed for Tl2ZrTe3 is 0*:*7 ev which is higher from the band gap of

ature from room temperature to 515 K, from 373 K to 450 K the thermal

cal conductivity decreases from 22 Ω�<sup>1</sup>

*Electromagnetic Field Radiation in Matter*

value of Tl2SnTe3 ð Þ ZT ¼ 0*:*31 at 500 K [6].

dimensionless figure of merit is 0*:*5 at 500 K [7].

the order of 0*:*5 Wm�1K�<sup>1</sup>

conductivity of 850 Ω�<sup>1</sup>

**56**

Tl2SnTe3 ð Þ *Eg* ¼ 0*:*4 ev , the electrical conductivity is independent of temperature ranging from room temperature to 450 K, when the temperature rises from 450 K the electrical conductivity falls abruptly, while for Tl2SnTe3 compound the electri-

conductivity decreases from 0*:*39 Wm�1K�<sup>1</sup> to 0*:*30 Wm�1K�<sup>1</sup> for Tl2ZrTe3 and for Tl2SnTe3 the thermal conductivity decreases from 0*:*24 Wm�1K�<sup>1</sup> at 420 K to 0*:*20 Wm�1K�<sup>1</sup> at 450 K. For Tl2ZrTe3 the calculated Seebeck coefficient is generally same in the range of 373 K to450 K, but decreases with temperature greater than450 K, the peak experiential value of Seebeck coefficient is120 μVK�<sup>1</sup>

Tl2SnTe3 the Seebeck coefficient increases from 240 μVK�<sup>1</sup> to 330 μVK�<sup>1</sup> with increasing temperature from room temperature to 450 K. The calculated power

0*:*41 μWcm�1K�<sup>1</sup> while for Tl2SnTe2 the power factor is increasing with temperature. The ZT value of 0*:*18 at 450 K was observed for Tl2ZrTe3 which is less than ZT

studies of Tl9GdTe6 they found the low power factor due to the high electrical

the whole study they found the good thermoelectric property for TlGdTe2, the

while the Seebeck coefficient is increases. In this series for **Tl8.97Ce1.03 Te6, Tl8.92Pr1.08Te6 and Tl8.99Sm1.01Te6** the electrical and thermal conductivity increases and Seebeck coefficient value decreases due to the discontinuity in the band gap as compared to Tl9LnTe6 compounds. The power factor is increases with increasing the lanthanide content "x" in the Tl9-xLnxTe6 resulting the increase in the dimensionless thermoelectric figure of merit, the best thermoelectric figure of merit

Prepared the thallium lanthanide telluride Tl10�xLnxTe6 with Ln = CE, Pr, Nd, Sm, Gd, Tb, Dy, Ho and Er and 0*:*25≤ *x*≤1*:*32 by hot press method. He found that the crystal structure of Tl10�xLnxTe6 is isostructural to Tl9BiTe6 and the volume of unit cell is increases with increasing lanthanide content. They investigated that the electrical and thermal conductivity decreases with increasing the lanthanide content

Studied the thermoelectric properties of ternary thallium chalcogenides TlGdQ <sup>2</sup> (Q = SE, Te) and Tl9GdTe6. They found that TlGdQ <sup>2</sup> is isostructural with TlSbQ <sup>2</sup> and Tl9GdTe6 is isostructural with Tl9BiTe6. They found the high Seebeck coefficient and low electrical conductivity of TlGdQ 2. The low thermal conductivity of

was investigated at room temperature for TlGdTe2. In the

cm�<sup>1</sup> and low seebeck coefficient of 27 μVK�<sup>1</sup> at550 K. In

factor for Tl2ZrTe3 is almost same changes from 0*:*35 μWcm�1K�<sup>1</sup> to

cm�<sup>1</sup> to 15 Ω�<sup>1</sup>

cm�<sup>1</sup> with increasing temper-

. For

values are 0*:*22 at 550 K were achieved for the stoichiometric compound on cold press pellets [8].

Fabricate and improve the thermoelectric properties of alumina nanoparticledispersed Bi0.5Sb1.5Te3 matrix composites, the nanoparticles were fabricated by ball milling process and followed by spark plasma sintering process. The p-type bismuth antimony telluride (BST) nanopowder prepared from the mechano-chemical process, were mixed with 1*:*0, 0*:*5, and 0*:*3 vol*:*% with Al2O3 nanoparticles by ball milling process. They studied the surface morphology and investigated the size of the nanoparticles. The electrical resistivity is increasing with temperature from <sup>1</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>5</sup> <sup>Ω</sup>m at 293 K to 2*:*<sup>5</sup> � <sup>10</sup>�<sup>5</sup> <sup>Ω</sup>m at 473 K, and the seebeck coefficient is increases from <sup>þ</sup><sup>205</sup> *<sup>μ</sup>VK*�<sup>1</sup> to <sup>þ</sup><sup>210</sup> *<sup>μ</sup>VK*�<sup>1</sup> from temperature 293 K to 473 K, respectively, shows p-type semiconducting behavior. The highest Seebeck coefficient of the order of 235 μV*K*�<sup>1</sup> at 373 K was observed from 0*:*3vol*:*% Al2O3/BST nanocomposite. They show that the increasing volume fraction of Al2O3 increases the carrier density which affects the Seebeck coefficient, carrier mobility and electrical resistivity of Al2O3/BST nanocomposites. The observed power factor is 1*:*7 times higher from pure BST, that is, 33 μWK�<sup>2</sup> cm at 393 K for 0*:*3vol*:*% Al2O3/BST nanocomposites, and for pure BST is22 μWK�<sup>2</sup> cm. The thermal conductivity is decreased by the addition of Al2O3 nanoparticles, for pure BST is 0*:*8 Wm�1K�<sup>1</sup> and for Al2O3/BST is0*:*7 Wm�1K�<sup>1</sup> . The thermoelectric figure of merit (ZT) is 1.5 for 0*:*3vol*:*% Al2O3/BST composite at 373 K which is higher from pure BST [9].

Studied the thermoelectric properties of Indium doped SnTe (InxSnx-1Te) nanostructured compound. They prepared InxSnx-1Te by ball milling and hot press techniques, and investigated their thermal conductivity, diffusivity and electrical conductivity decreases with temperature ranging from 300 to 900 K, while the power factor and Seebeck coefficient is increases and the specific heat is almost constant a little increase was seen in InxSnx�1Te. The sample is also prepared by ball mill and hand mill method but the result is the same. They observed the relationship of carrier concentration vs. Seebeck effect which shows that In doped SnTe shows abnormal behavior with increasing carrier concentration, they get the SEM, TEM, and HRTEM images of InxSnx-1Te which clearly shows the sample is consist of both small and large grain boundaries with a good crystallinity which effects the thermal conductivity of the sample. The thermal electric figure of merit is observed which greater than 1 is at temperature 873 K in for In0.0025Sn0.9975Te [10].

Investigated the thermoelectric properties of indium doped PbTe1�ySey alloys, solid state method was used for the synthesis. They showed that the carrier concentration and electrical resistivity increases with increasing temperature in n-type indium doped PbTe1�ySey alloys affecting the Seebeck coefficient and as a result the power factor of PbTe1�ySey alloys. The bipolar effect was observed at high temperature which restricts the thermoelectric figure of merit to 0*:*66 at 800 K with 30% content in sample; they further concluded that for the enhancement of thermoelectric properties, the increased carrier concentration must be reduced at high temperature [11].

Optimized the thermoelectric properties of Tl10�x�ySnxBiyTe6, a quaternary telluride series has been studied. The crystal structure was investigated by X-Ray diffraction which belongs to Tl5Te3 type structure and the volume is increases with increasing the Sn concentration in Tl10-x-ySnxBiyTe6, the electronic structure calculation revealed that Tl8.5SnBi0.5Te6 is a narrow band gap p-type intrinsic semiconductor and Tl9Sn0.5 Bi0.5Te6 is a p-type and narrow band gap extrinsic semiconductor. The electrical conductivity is decreasing with increasing temperature for Tl9Sn1�yBiyTe6, Tl8.67Sn1�<sup>y</sup> BiyTe6 and Tl8.33Sn1.12Bi55Te6. The low increase of the order of 0*:*4 Wm�1K�<sup>1</sup> to 1*:*4 Wm�1K�<sup>1</sup> was observed in the thermal conductivity of Tl10�x�ySnxBiyTe6 type materials due to thermal conductivity of electron *κel*, and for Tl9Sn0.2Bi0.8Te6 the thermal conductivity decreases with increasing temperature due to the increase in lattice vibration. The thermoelectric figure of merit, Seebeck coefficient and power factor was increased with increasing temperature, The high power factor S<sup>2</sup> <sup>σ</sup> <sup>¼</sup> <sup>8</sup>*:*<sup>1</sup> <sup>μ</sup>Wcm�1K�<sup>2</sup> was observed for Tl9(Sn, Bi)Te6 type system, they identified that with low Sn concentration the Seebeck coefficient and power factor is high and at low temperature the power factor is decreases. Yi et al. were study that bulk crystalline ingots into nanopowders by ball milling and hot pressing, in nanostructured bulk bismuth antimony telluride had achieved high figure of merit. He obtained a high value of ZT� 1.3 in the temperature range of 75 and 100°*C*. The improvement of ZT is mostly due by the lower of thermal conductivity. TEM observation of microstructure show that the lower thermal conductivity due to the increased of phonon scattering, and increasing of phonon scattering due to the increasing of grain boundaries of nanograins, nanodots, precipitates and defects. When ingot was used as the starting material, the highest ZT value of � 0.7 was obtained for bismuth antimony telluride [12].

**3. Results and discussions**

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

Various concentrations of *Sn* doped Tl10�xSnxTe6 compounds series were synthesized, and their physical properties were studied for x = 1, 1.25, 1.50, 1.75, 2.00. The powder x-rays diffraction pattern which is measured at room temperature for all these compounds is presented in **Figure 4**. It is found that the tetragonal single phase Tl8Sn2Te6 is obtained in the present study. The tetragonal lattice parameters observed at room temperature are *a* = 8.8484 nm, and c = 13.0625 nm in table. The materials are iso-structural with the binary tellurium telluride Tl*10*Te6, and the crystal structure of Tl9Sb1Te6 was determined with the experimental formula, possessing the same space group 14/mcm as Tl*5*Te3 and Tl8Sb2Te6, in contrast to Tl9Sb1Te6 that adopt the space group 14/m. **Figure 5** shows the SEM and EDX images. The SEM shows the morphology of the ternary compound at 100 nm scale.

We were interested in the effects of Sn doping in parent composition on the border line of semi-conductor and metallic. The experiment was conducted in a commercial oxford instrument cryostat with temperature control better than 0*:*5 *K*. the contacts were standard 4-probe, and were made using high quality silver point, contacts resistance was checked at room temperature and experiments were only carried out if it was satisfactorily low, typical current used were of the rate of 0*:*1 *mA*. The electrical properties of tin doped thallium telluride nanostructural system has been investigated under dependency of temperature varying from room temperature 300 to 650*K* by four probe resistivity technique. It has been founded that

dopant increases from *x* ¼ 1*:*0 to *x* ¼ 2*:*0, an electrical conductivity decreases, for

1*:*0 10% ð Þ as shown in **Figure 6**. The electrical conductivity, σ, for the Tl10-xSnxTe6 samples with *x*< 2*:*0 decreases with increasing temperature, **Figure 6**, which clarifies that this decrease in the electrical conductivity is due to the high charge carrier

*cm*�<sup>1</sup> at *<sup>x</sup>* <sup>¼</sup> <sup>2</sup>*:*0 20% ð Þ and *<sup>σ</sup>* <sup>¼</sup> <sup>1629</sup>*:*21*Ω*�<sup>1</sup>

*cm*�1, and when Sn

*cm*�<sup>1</sup> at *<sup>x</sup>* <sup>¼</sup>

Tl8Sn2Te6 has the lowest electrical conductivity of 471*:*6*Ω*�<sup>1</sup>

*X-ray diffractometery of Tl10*�*xSnxTe6 doping Sn* ¼ 1, 1*:*25, 1*:*50, 1*:*75 *and* 2*.*

**3.1 Ternary system**

*3.1.1 Structural analysis*

**3.2 Physical properties**

example, *<sup>σ</sup>* <sup>¼</sup> <sup>471</sup>*:*<sup>68</sup> *<sup>Ω</sup>*�<sup>1</sup>

**Figure 4.**

**59**

*3.2.1 Electrical conductivity measurements*

It is very important for the control of carrier concentration, which is good for thermoelectric properties. Another approach like nanostructuring, the dimensionless figure-of-merit is increased by the reduction of lattice thermal conductivity.

#### **2. Experiment**

For the preparation of Tl10�x�yAxByTe6 compounds of different types of dopants (x = Pb, Sn, Ca and y = Pb, Sb, Sr, etc.), with different concentration on tellurium telluride has been prepared by solid state reactions in evacuated sealed silica tubes. The purpose of this study were mainly for discovering new type of ternary and quaternary compounds by using Tl+1, Sn+3,Pb+3 and etc. Te�<sup>2</sup> elements as the starting materials [13]. Direct synthesis of stoichiometric amount of high purity elements, that is, 99.99% of different compositions have been prepared for a preliminary investigation. Since most of these starting materials for solid state reactions are sensitive to oxygen and moistures, they were weighing stoichiometric reactants and transferring to the silica tubes in the glove box which is filled with Argon. Then, all constituents were sealed in a quartz tube. Before putting these samples in the resistance furnace for the heating, the silica tubes was put in vacuum line to evacuate the argon and then sealed it. This sealed power were heated up to 650 Co at a rate not exceeding 1 k/mint and kept at that temperature for 24 hours. The sample was cooled down with extremely slow rate to avoid quenching, dislocations, and crystals deformation. The nanoparticles have been prepared by ball-milling techniques.

Structural analysis of all these samples was carried out by x-rays diffraction, using an Intel powder diffractometer with position-sensitive detector and CuKα radiation at room temperature. No additional peaks were detected in any of the sample discussed here. X-ray powder diffraction patterns confirm the single phase composition of the compounds.

The temperature dependence of Seebeck coefficient was measured for all these compounds on a cold pressed pellet in rectangular shape, of approximately 5 � 1 � 1 mm<sup>3</sup> dimensions. The air sensitivity of these samples was checked (for one sample) by measuring the thermoelectric power and confirmed that these samples are not sensitive to air. This sample exposes to air more than a week, but no appreciable changes observed in the Seebeck values. The pellet for these measurements was annealed at 400 C<sup>0</sup> for 6 hours. For the electrical transport measurements (electrical conductivity) four-probe resistivity technique was used and the pellets were cut into rectangular shape with approximate dimension of 5 � <sup>1</sup> � 1 mm<sup>3</sup> .

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

#### **3. Results and discussions**

#### **3.1 Ternary system**

Tl10�x�ySnxBiyTe6 type materials due to thermal conductivity of electron *κel*, and for Tl9Sn0.2Bi0.8Te6 the thermal conductivity decreases with increasing temperature due to the increase in lattice vibration. The thermoelectric figure of merit, Seebeck coefficient and power factor was increased with increasing temperature, The high

they identified that with low Sn concentration the Seebeck coefficient and power factor is high and at low temperature the power factor is decreases. Yi et al. were study that bulk crystalline ingots into nanopowders by ball milling and hot pressing, in nanostructured bulk bismuth antimony telluride had achieved high figure of merit. He obtained a high value of ZT� 1.3 in the temperature range of 75 and 100°*C*. The improvement of ZT is mostly due by the lower of thermal conductivity. TEM observation of microstructure show that the lower thermal conductivity due to the increased of phonon scattering, and increasing of phonon scattering due to the increasing of grain boundaries of nanograins, nanodots, precipitates and

defects. When ingot was used as the starting material, the highest ZT value of � 0.7

It is very important for the control of carrier concentration, which is good for thermoelectric properties. Another approach like nanostructuring, the dimensionless

For the preparation of Tl10�x�yAxByTe6 compounds of different types of dopants (x = Pb, Sn, Ca and y = Pb, Sb, Sr, etc.), with different concentration on tellurium telluride has been prepared by solid state reactions in evacuated sealed silica tubes. The purpose of this study were mainly for discovering new type of ternary and quaternary compounds by using Tl+1, Sn+3,Pb+3 and etc. Te�<sup>2</sup> elements as the starting materials [13]. Direct synthesis of stoichiometric amount of high purity elements, that is, 99.99% of different compositions have been prepared for a preliminary investigation. Since most of these starting materials for solid state reactions are sensitive to oxygen and moistures, they were weighing stoichiometric reactants and transferring to the silica tubes in the glove box which is filled with Argon. Then, all constituents were sealed in a quartz tube. Before putting these samples in the resistance furnace for the heating, the silica tubes was put in vacuum line to evacuate the argon and then sealed it. This sealed power were heated up to 650 Co at a rate not exceeding 1 k/mint and kept at that temperature for 24 hours. The sample was cooled down with extremely slow rate to avoid quenching, dislocations, and crystals defor-

figure-of-merit is increased by the reduction of lattice thermal conductivity.

mation. The nanoparticles have been prepared by ball-milling techniques.

into rectangular shape with approximate dimension of 5 � <sup>1</sup> � 1 mm<sup>3</sup>

Structural analysis of all these samples was carried out by x-rays diffraction, using an Intel powder diffractometer with position-sensitive detector and CuKα radiation at room temperature. No additional peaks were detected in any of the sample discussed here. X-ray powder diffraction patterns confirm the single phase

The temperature dependence of Seebeck coefficient was measured for all these compounds on a cold pressed pellet in rectangular shape, of approximately 5 � 1 � 1 mm<sup>3</sup> dimensions. The air sensitivity of these samples was checked (for one sample) by measuring the thermoelectric power and confirmed that these samples are not sensitive to air. This sample exposes to air more than a week, but no appreciable changes observed in the Seebeck values. The pellet for these measurements was annealed at 400 C<sup>0</sup> for 6 hours. For the electrical transport measurements (electrical conductivity) four-probe resistivity technique was used and the pellets were cut

.

was obtained for bismuth antimony telluride [12].

<sup>σ</sup> <sup>¼</sup> <sup>8</sup>*:*<sup>1</sup> <sup>μ</sup>Wcm�1K�<sup>2</sup> was observed for Tl9(Sn, Bi)Te6 type system,

power factor S<sup>2</sup>

*Electromagnetic Field Radiation in Matter*

**2. Experiment**

composition of the compounds.

**58**

#### *3.1.1 Structural analysis*

Various concentrations of *Sn* doped Tl10�xSnxTe6 compounds series were synthesized, and their physical properties were studied for x = 1, 1.25, 1.50, 1.75, 2.00. The powder x-rays diffraction pattern which is measured at room temperature for all these compounds is presented in **Figure 4**. It is found that the tetragonal single phase Tl8Sn2Te6 is obtained in the present study. The tetragonal lattice parameters observed at room temperature are *a* = 8.8484 nm, and c = 13.0625 nm in table. The materials are iso-structural with the binary tellurium telluride Tl*10*Te6, and the crystal structure of Tl9Sb1Te6 was determined with the experimental formula, possessing the same space group 14/mcm as Tl*5*Te3 and Tl8Sb2Te6, in contrast to Tl9Sb1Te6 that adopt the space group 14/m. **Figure 5** shows the SEM and EDX images. The SEM shows the morphology of the ternary compound at 100 nm scale.

#### **3.2 Physical properties**

#### *3.2.1 Electrical conductivity measurements*

We were interested in the effects of Sn doping in parent composition on the border line of semi-conductor and metallic. The experiment was conducted in a commercial oxford instrument cryostat with temperature control better than 0*:*5 *K*. the contacts were standard 4-probe, and were made using high quality silver point, contacts resistance was checked at room temperature and experiments were only carried out if it was satisfactorily low, typical current used were of the rate of 0*:*1 *mA*.

The electrical properties of tin doped thallium telluride nanostructural system has been investigated under dependency of temperature varying from room temperature 300 to 650*K* by four probe resistivity technique. It has been founded that Tl8Sn2Te6 has the lowest electrical conductivity of 471*:*6*Ω*�<sup>1</sup> *cm*�1, and when Sn dopant increases from *x* ¼ 1*:*0 to *x* ¼ 2*:*0, an electrical conductivity decreases, for example, *<sup>σ</sup>* <sup>¼</sup> <sup>471</sup>*:*<sup>68</sup> *<sup>Ω</sup>*�<sup>1</sup> *cm*�<sup>1</sup> at *<sup>x</sup>* <sup>¼</sup> <sup>2</sup>*:*0 20% ð Þ and *<sup>σ</sup>* <sup>¼</sup> <sup>1629</sup>*:*21*Ω*�<sup>1</sup> *cm*�<sup>1</sup> at *<sup>x</sup>* <sup>¼</sup> 1*:*0 10% ð Þ as shown in **Figure 6**. The electrical conductivity, σ, for the Tl10-xSnxTe6 samples with *x*< 2*:*0 decreases with increasing temperature, **Figure 6**, which clarifies that this decrease in the electrical conductivity is due to the high charge carrier

**Figure 4.** *X-ray diffractometery of Tl10*�*xSnxTe6 doping Sn* ¼ 1, 1*:*25, 1*:*50, 1*:*75 *and* 2*.*

**Figure 5.** *SEM and EDX image of Tl8Sn2Te6.*

maximum of *<sup>S</sup>* <sup>¼</sup> <sup>79</sup>*:*<sup>77</sup> <sup>þ</sup> *<sup>μ</sup>V:K*�<sup>1</sup> at 300*<sup>K</sup>* and *<sup>S</sup>* ¼ þ157*:*931*μV:K*�<sup>1</sup> at 550 *<sup>K</sup>*. The lowest Seebeck coefficient *<sup>S</sup>* ¼ þ33*:*<sup>15</sup> *<sup>μ</sup>VK*�<sup>1</sup> at 300*<sup>K</sup>* has been observed for *<sup>x</sup>* <sup>¼</sup> <sup>1</sup>*:*0 which is increases to*<sup>S</sup>* ¼ þ65*:*84*μV:K*�<sup>1</sup> at 550 K, it has been declared that when Sn content increase in host sample the Seebeck coefficient *S*, also increases, for example, for *<sup>x</sup>* <sup>¼</sup> <sup>1</sup>*:*0 the Seebeck coefficient has been observed is *<sup>S</sup>* ¼ þ33*:*15*μVK*�<sup>1</sup> to *<sup>S</sup>* ¼ þ79*:*77*μV:K*�<sup>1</sup> for at 300 *<sup>K</sup>* and the Seebeck coefficient has been observed is *<sup>S</sup>* ¼ þ65*:*<sup>844</sup> *<sup>μ</sup>VK*�<sup>1</sup> to *<sup>S</sup>* ¼ þ157*:*937*μV:K*�<sup>1</sup> for *<sup>x</sup>* <sup>¼</sup> <sup>2</sup>*:*0 at 550 *<sup>K</sup>*, respectively as shown in **Figure 7**. The power factor of Sn doping is increases as the temperature is

**<sup>Ω</sup>**�**1cm**�**<sup>1</sup>** ð Þ **at 300K**

Tl9Sn1Te6 1629.21 939.137 Tl8.75Sn1.25Te6 1550.997 841.481 Tl8.50Sn1.50Te6 1310.326 795.66 Tl8.25Sn1.75Te6 1287.64 781.313 Tl8Sn2Te6 471.68 292.102

The power factor investigations show that it increases because of increasing behavior in Seebeck coefficient with temperature (**Table 3**). The calculated power factor is directly proportional to the square of the Seebeck coefficient and the electrical conductivity. The lowest power factor 1*:*9*μWcm*�<sup>2</sup>*K*�<sup>2</sup> has been observed for *Sn* <sup>¼</sup> 1 and the highest power factor 7*:*579*μWcm*�<sup>2</sup>*K*�<sup>2</sup> has been observed for *Sn* <sup>¼</sup> <sup>1</sup>*:*75 at temperature 300*K*, while for *Sn* <sup>¼</sup> 2 the power factor is 3*:*639*μWcm*�<sup>2</sup>*K*�<sup>2</sup> as shown in **Table 4**, which is increases with increasing temperature; this low power factor has been observed due to extremely low electrical conductivity of Tl8Sn2Te6

which play a man role in the investigation of power factor.

increase as shown in **Figure 8**.

**Sample Crystallite size,**

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

*Crystallite size, lattice constant, and volume of unit cell.*

**Sample Electrical conductivity**

*Electrical conductivity of Tl10-xSnxTe6*ð Þ 1 ≤*x* ≤2 *at 300 K and 650 K.*

**Table 1.**

**Table 2.**

**61**

**D** ¼ **0***:***9λ***=***βcosθ** ð Þ **nm**

Tl9Sn1Te6 62.919 a = b = 8.7930

Tl8.75Sn1.25Te6 63.965 a = b = 8.8450

Tl8.50Sn1.50Te6 66.2833 a = b = 8.8250

Tl8.25Sn1.75Te6 59.820 a = b = 8.8100

Tl8Sn2Te6 56.793 a = b = 8.8484

**Lattice constant a, b, c** ¼ **Å** 

c = 13.0050

c = 13.0755

c = 13.0000

c = 13.0010

c = 13.0625

**Volume (Å3**

1005.505

1022.948

1012.44

1009.086

1022.717

**Electrical conductivity <sup>Ω</sup>**�**1cm**�**<sup>1</sup>** ð Þ **at 650K**

**)**

*3.2.3 Power factor analysis*

**Figure 6.** *Electrical conductivity measurements at different concentration and high temperature.*

concentration. The highest value has been observed for Tl9Sn1Te6 and the lowest value has been observed in sample Tl8.0Sn2.0Te6 at 300*K*. **Figure 6**, shows that the increases the concentration of the dopant decreases the electrical conductivity with temperature of the ternary compounds.

#### *3.2.2 Seebeck coefficient measurements*

The positive Seebeck coefficient, S, observed in all samples of Tl10-xSnxTe6 as shown in **Table 1**, which increases smoothly with increasing temperature with 1*:*0≤*x*≤2*:*0, for p-type semiconductors having high charge carrier concentration (**Table 2**). The Seebeck curve of the sample with x = 2.0 exhibits a clear

#### *Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*


#### **Table 1.**

*Crystallite size, lattice constant, and volume of unit cell.*


#### **Table 2.**

*Electrical conductivity of Tl10-xSnxTe6*ð Þ 1 ≤*x* ≤2 *at 300 K and 650 K.*

maximum of *<sup>S</sup>* <sup>¼</sup> <sup>79</sup>*:*<sup>77</sup> <sup>þ</sup> *<sup>μ</sup>V:K*�<sup>1</sup> at 300*<sup>K</sup>* and *<sup>S</sup>* ¼ þ157*:*931*μV:K*�<sup>1</sup> at 550 *<sup>K</sup>*. The lowest Seebeck coefficient *<sup>S</sup>* ¼ þ33*:*<sup>15</sup> *<sup>μ</sup>VK*�<sup>1</sup> at 300*<sup>K</sup>* has been observed for *<sup>x</sup>* <sup>¼</sup> <sup>1</sup>*:*0 which is increases to*<sup>S</sup>* ¼ þ65*:*84*μV:K*�<sup>1</sup> at 550 K, it has been declared that when Sn content increase in host sample the Seebeck coefficient *S*, also increases, for example, for *<sup>x</sup>* <sup>¼</sup> <sup>1</sup>*:*0 the Seebeck coefficient has been observed is *<sup>S</sup>* ¼ þ33*:*15*μVK*�<sup>1</sup> to *<sup>S</sup>* ¼ þ79*:*77*μV:K*�<sup>1</sup> for at 300 *<sup>K</sup>* and the Seebeck coefficient has been observed is *<sup>S</sup>* ¼ þ65*:*<sup>844</sup> *<sup>μ</sup>VK*�<sup>1</sup> to *<sup>S</sup>* ¼ þ157*:*937*μV:K*�<sup>1</sup> for *<sup>x</sup>* <sup>¼</sup> <sup>2</sup>*:*0 at 550 *<sup>K</sup>*, respectively as shown in **Figure 7**. The power factor of Sn doping is increases as the temperature is increase as shown in **Figure 8**.

#### *3.2.3 Power factor analysis*

The power factor investigations show that it increases because of increasing behavior in Seebeck coefficient with temperature (**Table 3**). The calculated power factor is directly proportional to the square of the Seebeck coefficient and the electrical conductivity. The lowest power factor 1*:*9*μWcm*�<sup>2</sup>*K*�<sup>2</sup> has been observed for *Sn* <sup>¼</sup> 1 and the highest power factor 7*:*579*μWcm*�<sup>2</sup>*K*�<sup>2</sup> has been observed for *Sn* <sup>¼</sup> <sup>1</sup>*:*75 at temperature 300*K*, while for *Sn* <sup>¼</sup> 2 the power factor is 3*:*639*μWcm*�<sup>2</sup>*K*�<sup>2</sup> as shown in **Table 4**, which is increases with increasing temperature; this low power factor has been observed due to extremely low electrical conductivity of Tl8Sn2Te6 which play a man role in the investigation of power factor.

concentration. The highest value has been observed for Tl9Sn1Te6 and the lowest value has been observed in sample Tl8.0Sn2.0Te6 at 300*K*. **Figure 6**, shows that the increases the concentration of the dopant decreases the electrical conductivity with

*Electrical conductivity measurements at different concentration and high temperature.*

The positive Seebeck coefficient, S, observed in all samples of Tl10-xSnxTe6 as

with 1*:*0≤*x*≤2*:*0, for p-type semiconductors having high charge carrier concentration (**Table 2**). The Seebeck curve of the sample with x = 2.0 exhibits a clear

shown in **Table 1**, which increases smoothly with increasing temperature

temperature of the ternary compounds.

*3.2.2 Seebeck coefficient measurements*

**Figure 5.**

**Figure 6.**

**60**

*SEM and EDX image of Tl8Sn2Te6.*

*Electromagnetic Field Radiation in Matter*

**3.3 Quaternary System**

*3.3.1 Structural analysis*

**3.4 Physical properties**

**Table 4.**

**Figure 9.**

**63**

composition of the compound in **Figure 10**.

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

*3.4.1 Electrical conductivity measurements*

*Power factor of Tl10*�*xSnxTe6*ð Þ 1≤*x*≤2 *at* 300 K *and* 550 K*.*

*XRD data of Tl9(SnSb)1Te6 with Sn = 0.01, 0.025 and 0.05.*

X-ray diffraction is the greatest and significant method for the investigation of crystal structure of nanomaterials. With the purpose, to check the purities of different phases of compound peaks in XRD figures, as per revealed in **Figure 9**. It is authenticated that the XRD design of all these samples are fine unchanging with the literature and has been recognized that the crystal structure scheme is isostructural with reference data of Tl9GdTe6 and Tl9BiTe6 having tetragonal crystal structure with the space group symbol of 4/cm. The SEM shows, morphological structure at the 100 nm scale. The energy dispersion X-ray diffractometer show the concentrate

The temperature variations of electrical conductivity of quaternary compounds are revealed in **Figure 11**. The conductivity experiential for the entire samples are studied here, decreases with increasing temperature, representing the degenerate semiconductor performance because of positive temperature coefficient, subsequent from the phonons scattering of charge carriers and grains boundaries effects.

**Sample Power factor (***μWcm*�**<sup>2</sup>***K*�**<sup>2</sup>**Þ**at 300 K Power factor (***μWcm*�**<sup>2</sup>***K*�**<sup>2</sup>**<sup>Þ</sup> **at 550 K**

Tl9Sn1Te6 1.9 4.505 Tl8.75Sn1.25Te6 2.637 9.234 Tl8.50Sn1.50Te6 6.755 9.579 Tl8.25Sn1.75Te6 7.574 10.135 Tl8Sn2Te6 3.638 9.777

**Figure 7.** *Seebeck coefficient measurements of Tl10xSnxTe6* ð Þ 1 ≤*x*≤2 *at 300K and 550K.*

**Figure 8.** *The power factor (*PF*) of Tl10*�*xSnxTe6 with x* ¼ 1, 1*:*25, 1*:*50, 1*:*75, *and* 2*:*


#### **Table 3.**

*Thermoelectric properties of Tl10*�*xSnxTe6*ð Þ 1≤*x*≤2 *at 300 K and 550 K.*

#### **3.3 Quaternary System**

#### *3.3.1 Structural analysis*

X-ray diffraction is the greatest and significant method for the investigation of crystal structure of nanomaterials. With the purpose, to check the purities of different phases of compound peaks in XRD figures, as per revealed in **Figure 9**. It is authenticated that the XRD design of all these samples are fine unchanging with the literature and has been recognized that the crystal structure scheme is isostructural with reference data of Tl9GdTe6 and Tl9BiTe6 having tetragonal crystal structure with the space group symbol of 4/cm. The SEM shows, morphological structure at the 100 nm scale. The energy dispersion X-ray diffractometer show the concentrate composition of the compound in **Figure 10**.

#### **3.4 Physical properties**

#### *3.4.1 Electrical conductivity measurements*

The temperature variations of electrical conductivity of quaternary compounds are revealed in **Figure 11**. The conductivity experiential for the entire samples are studied here, decreases with increasing temperature, representing the degenerate semiconductor performance because of positive temperature coefficient, subsequent from the phonons scattering of charge carriers and grains boundaries effects.


#### **Table 4.**

**Figure 8.**

**Figure 7.**

**Table 3.**

**62**

*The power factor (*PF*) of Tl10*�*xSnxTe6 with x* ¼ 1, 1*:*25, 1*:*50, 1*:*75, *and* 2*:*

*Seebeck coefficient measurements of Tl10xSnxTe6* ð Þ 1 ≤*x*≤2 *at 300K and 550K.*

*Electromagnetic Field Radiation in Matter*

*Thermoelectric properties of Tl10*�*xSnxTe6*ð Þ 1≤*x*≤2 *at 300 K and 550 K.*

**Sample Seebeck coefficient <sup>μ</sup>VK**�**<sup>1</sup>** ð Þ **at 300K Seebeck coefficient <sup>μ</sup>VK**�**<sup>1</sup>** ð Þ **at 550K**

Tl9Sn1Te6 33.15 65.844 Tl8.75Sn1.25Te6 39.953 99.035 Tl8.50Sn1.50Te6 69.207 103.419 Tl8.25Sn1.75Te6 73.879 110.958 Tl8Sn2Te6 79.77 157.931

*Power factor of Tl10*�*xSnxTe6*ð Þ 1≤*x*≤2 *at* 300 K *and* 550 K*.*

**Figure 9.** *XRD data of Tl9(SnSb)1Te6 with Sn = 0.01, 0.025 and 0.05.*

*3.4.2 Seebeck coefficient (S) analyses*

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

coefficient and effective mass.

**Figure 12.**

**65**

*3.4.3 Power factor calculation and analysis*

To improve the power factor (PF = *S*<sup>2</sup>

To examine the influence of decrease of the charge carriers in thermal and transport features, Sn content was increased in Tl9Sb1xSnxTe6 (x = 0.01, 0.025, and 0.05) by means of replacing Sb atoms conferring to the formula. The temperature variation as a function of the Seebeck coefficient (S) for the Tl9Sb1-xSnxTe6 (x = 0.01, 0.025 and 0.05) compounds are revealed in **Figure 12**. The Seebeck coefficient was measured in the temperature gradient of 1 K. The positive Seebeck coefficient increases easily with increasing temperature from 300 K to 550 K, for all compounds in mainly for p-type semiconductors having high charge carrier concentration. It is understandable that all the samples display positive Seebeck coefficient for the whole temperature range, signifying that the p-type (hole) carrier's conduction controls the thermoelectric transportation in these compounds. When the amount of Sn increased from 0.01 to 0.05, the Sn doping is supposed to increase the carrier's density. Though, the smaller grains upon Sn doping are thought to be talented to improve the electron scattering, yielding an increase of the Seebeck

decouple the electrical conductivity from the Seebeck coefficient, typically inversely proportional to each other in these systems. The key contribution in the "PF" originates from the Seebeck coefficient, so we must design the materials such that their "*S*" should be improved. The power factors calculated from the electrical conductivity "σ" and the square of Seebeck coefficient "S," gotten for Tl9Sb1-xSnxTe6 compounds with x = 0.01, 0.025 and 0.05 are showed in **Figure 13**. The power factor increases with increasing temperature for all these compounds. The doping concentration demonstrations a systematic effect on the power factor as

increasing the doping concentration, the power factor is increases. The

*Seebeck coefficient measurements at different concentration and high temperature (0.01, 0.025, 0.05).*

σ) for these compounds, we require to

**Figure 10.** *SEM and EDX image of Tl9(SnSb)1Te6.*

**Figure 11.** *Electrical conductivity measurements at different high temperature.*

An increasing "*x"* value (i.e. increasing the Sn deficiency) is predictable to increase the number of holes, which is experimental detected. The smaller temperature need may be produced by (less temperature dependence) more grain boundary scattering. No systematic trend was found in the variation of the electrical resistivity for samples Tl9Sb1xSnxTe6 (x = 0.01, 0.025, and 0.05) with "Sn" concentration. The low electrical conductivity in the pressure less sintered sample may be caused by means of the oxide impurity phase in the grain boundary and the number of the grain boundary. The *Sn* doping level and grain boundary resistance may play significant part for increasing electrical conductivity.

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

#### *3.4.2 Seebeck coefficient (S) analyses*

To examine the influence of decrease of the charge carriers in thermal and transport features, Sn content was increased in Tl9Sb1xSnxTe6 (x = 0.01, 0.025, and 0.05) by means of replacing Sb atoms conferring to the formula. The temperature variation as a function of the Seebeck coefficient (S) for the Tl9Sb1-xSnxTe6 (x = 0.01, 0.025 and 0.05) compounds are revealed in **Figure 12**. The Seebeck coefficient was measured in the temperature gradient of 1 K. The positive Seebeck coefficient increases easily with increasing temperature from 300 K to 550 K, for all compounds in mainly for p-type semiconductors having high charge carrier concentration. It is understandable that all the samples display positive Seebeck coefficient for the whole temperature range, signifying that the p-type (hole) carrier's conduction controls the thermoelectric transportation in these compounds. When the amount of Sn increased from 0.01 to 0.05, the Sn doping is supposed to increase the carrier's density. Though, the smaller grains upon Sn doping are thought to be talented to improve the electron scattering, yielding an increase of the Seebeck coefficient and effective mass.

#### *3.4.3 Power factor calculation and analysis*

To improve the power factor (PF = *S*<sup>2</sup> σ) for these compounds, we require to decouple the electrical conductivity from the Seebeck coefficient, typically inversely proportional to each other in these systems. The key contribution in the "PF" originates from the Seebeck coefficient, so we must design the materials such that their "*S*" should be improved. The power factors calculated from the electrical conductivity "σ" and the square of Seebeck coefficient "S," gotten for Tl9Sb1-xSnxTe6 compounds with x = 0.01, 0.025 and 0.05 are showed in **Figure 13**. The power factor increases with increasing temperature for all these compounds. The doping concentration demonstrations a systematic effect on the power factor as increasing the doping concentration, the power factor is increases. The

**Figure 12.** *Seebeck coefficient measurements at different concentration and high temperature (0.01, 0.025, 0.05).*

An increasing "*x"* value (i.e. increasing the Sn deficiency) is predictable to increase the number of holes, which is experimental detected. The smaller temperature need may be produced by (less temperature dependence) more grain boundary scattering. No systematic trend was found in the variation of the electrical resistivity for samples Tl9Sb1xSnxTe6 (x = 0.01, 0.025, and 0.05) with "Sn" concentration. The low electrical conductivity in the pressure less sintered sample may be caused by means of the oxide impurity phase in the grain boundary and the number of the grain boundary. The *Sn* doping level and grain boundary resistance may play sig-

nificant part for increasing electrical conductivity.

*Electrical conductivity measurements at different high temperature.*

**Figure 10.**

**Figure 11.**

**64**

*SEM and EDX image of Tl9(SnSb)1Te6.*

*Electromagnetic Field Radiation in Matter*

All samples exhibited positive S values, increasing Sn-filling, the Seebeck coefficient increased due to increase in its metallic behavior and low thermal conductivity. By increasing the temperature, the Seebeck coefficient was increased and the

quently, power factor was enhanced and increased with high Sn concentration up to Sn <sup>¼</sup> <sup>1</sup>*:*75 and the maximum power factor *PF* <sup>¼</sup> <sup>7</sup>*:*579*μWcm*�2*K*�<sup>2</sup> was observed for Tl8.25Sn1.75Te6. The reduction in the power factor for Tl8Sn2Te6 is due to their low electrical conductivity. The thermopower is positive in the whole temperature range studied here, which is increasing with increase in temperature, representing that the nanoparticles under study is hole conduction dominated. For higher concentrations of *Sn*, the Seebeck coefficient of the doped tellurium telluride is decreasing because of increasing the holes concentration which in turns increasing the electron scattering in this doped chalcogenide system. However, the smaller grains upon Sn concentrations will improve the electron scattering, resulting increase in thermos-power. Therefore, power factor was improved and increased with high "Sn" concentration up to Sn = 0.05 and the maximum power factor

factor will improve the thermoelectric efficiency and results decent thermoelectric applications, which is the key goal of this study. At the end, we are going to

accomplish that this work is the finest example of enhancing dopants concentration to attain required thermoelectric properties in "*Sn"* doped Tl9Sb1-xSnxTe6 chalcogenide system. To understand our results, we start by final the basic understanding of the metallic long-range interactions due to do the injections of charge carriers concentration, and semiconducting frustration effects fore-most to metallic like

Department of Physics, Faculty of Basic and Applied Sciences, International Islamic

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

) was observed for Tl9Sb0.95Sn0.05Te6. This enhanced power

. Conse-

highest Seebeck coefficient was observed for Tl8Sn2Te6; *<sup>S</sup>* ¼ þ157*μVK*�<sup>1</sup>

(*PF* = 8.37 *μWcm*�<sup>1</sup>

**Author details**

**67**

University, Islamabad, Pakistan

provided the original work is properly cited.

Wiqar Hussain Shah\* and Waqas Muhammad Khan

\*Address all correspondence to: wiqar.hussain@iiu.edu.pk

*K*�<sup>2</sup>

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

conduct in these chalcogenides.

**Figure 13.** *The power factor (*PF*) of Tl9Sb1*�*xSnxTe6 with x* ¼ 0*:*01, 0*:*025, 0*:*05*.*

Tl9Sb0.95Sn0.05Te6 compound showed the highest value 8.37 (μWtt-cm�<sup>1</sup> -K�<sup>2</sup> ) of "PF" at 550 K and 3.75 (μWtt-cm�<sup>1</sup> -K�<sup>2</sup> ) at 295 K. The lowest "PF" were experiential for Tl9Sb0.99Sn0.01Te6 compound which have values of 5.55 (μWtt-cm�<sup>1</sup> -K�<sup>2</sup> ) at 550 K and 2.56 (μWtt-cm�<sup>1</sup> -K�<sup>2</sup> ) at 295 K. As deliberated earlier, an increasing the "Sn" contents are probable to increase the number of holes and the dominant charge carriers.

#### **4. Conclusion**

In this study, the ternary and quaternary Tellurium Telluride chalcogenides, Tl10-x-yAxByTe6 nanoparticles, with different types of dopants (A = Sb, and B = Sn) and with different concentration of Sn has been introduced to synthesize new materials by co-precipitation techniques and explored their structural, electrical and thermal properties has been analyzed in details. The structural investigation revealed that Tl10-xSnxTe6 is isostructural with Tl5Te3 with a same space group I4/mcm. All peaks are corresponding to their respective element, and no extra peaks are observed, which shows that we got a correct crystal structure for our design materials and also shows that no impurities or dislocation in the sample has been observed. An energy dispersive X-ray spectroscopy was used for the confirmation of elemental and compositional ratio of all the samples studied here. The electrical characterizations shows that parent compounds behaves like a semiconductor, but increasing the Sn contents, this materials tend toward the metallic properties, which show that increasing the temperature the electrical conductivity will decreases. The electrical characterizations show that parent compounds behaves like a semiconductor, but increasing the *Sn* contents, this nanomaterials tend to metallic phase, which display that increasing the temperature the electrical conductivity will decreases at higher temperature.

*Thermoelectric Properties of Chalcogenide System DOI: http://dx.doi.org/10.5772/intechopen.93248*

All samples exhibited positive S values, increasing Sn-filling, the Seebeck coefficient increased due to increase in its metallic behavior and low thermal conductivity. By increasing the temperature, the Seebeck coefficient was increased and the highest Seebeck coefficient was observed for Tl8Sn2Te6; *<sup>S</sup>* ¼ þ157*μVK*�<sup>1</sup> . Consequently, power factor was enhanced and increased with high Sn concentration up to Sn <sup>¼</sup> <sup>1</sup>*:*75 and the maximum power factor *PF* <sup>¼</sup> <sup>7</sup>*:*579*μWcm*�2*K*�<sup>2</sup> was observed for Tl8.25Sn1.75Te6. The reduction in the power factor for Tl8Sn2Te6 is due to their low electrical conductivity. The thermopower is positive in the whole temperature range studied here, which is increasing with increase in temperature, representing that the nanoparticles under study is hole conduction dominated. For higher concentrations of *Sn*, the Seebeck coefficient of the doped tellurium telluride is decreasing because of increasing the holes concentration which in turns increasing the electron scattering in this doped chalcogenide system. However, the smaller grains upon Sn concentrations will improve the electron scattering, resulting increase in thermos-power. Therefore, power factor was improved and increased with high "Sn" concentration up to Sn = 0.05 and the maximum power factor (*PF* = 8.37 *μWcm*�<sup>1</sup> *K*�<sup>2</sup> ) was observed for Tl9Sb0.95Sn0.05Te6. This enhanced power factor will improve the thermoelectric efficiency and results decent thermoelectric applications, which is the key goal of this study. At the end, we are going to accomplish that this work is the finest example of enhancing dopants concentration to attain required thermoelectric properties in "*Sn"* doped Tl9Sb1-xSnxTe6 chalcogenide system. To understand our results, we start by final the basic understanding of the metallic long-range interactions due to do the injections of charge carriers concentration, and semiconducting frustration effects fore-most to metallic like conduct in these chalcogenides.

#### **Author details**

Tl9Sb0.95Sn0.05Te6 compound showed the highest value 8.37 (μWtt-cm�<sup>1</sup>


*The power factor (*PF*) of Tl9Sb1*�*xSnxTe6 with x* ¼ 0*:*01, 0*:*025, 0*:*05*.*


tial for Tl9Sb0.99Sn0.01Te6 compound which have values of 5.55 (μWtt-cm�<sup>1</sup>

"Sn" contents are probable to increase the number of holes and the dominant charge

In this study, the ternary and quaternary Tellurium Telluride chalcogenides, Tl10-x-yAxByTe6 nanoparticles, with different types of dopants (A = Sb, and B = Sn) and with different concentration of Sn has been introduced to synthesize new materials by co-precipitation techniques and explored their structural, electrical and

thermal properties has been analyzed in details. The structural investigation revealed that Tl10-xSnxTe6 is isostructural with Tl5Te3 with a same space group I4/mcm. All peaks are corresponding to their respective element, and no extra peaks are observed, which shows that we got a correct crystal structure for our design materials and also shows that no impurities or dislocation in the sample has been observed. An energy dispersive X-ray spectroscopy was used for the confirmation of elemental and compositional ratio of all the samples studied here. The electrical characterizations shows that parent compounds behaves like a semiconductor, but increasing the Sn contents, this materials tend toward the metallic properties, which show that increasing the temperature the electrical conductivity will decreases. The electrical characterizations show that parent compounds behaves like a semiconductor, but increasing the *Sn* contents, this nanomaterials tend to metallic phase, which display that increasing the temperature the electrical conductivity will

"PF" at 550 K and 3.75 (μWtt-cm�<sup>1</sup>

*Electromagnetic Field Radiation in Matter*

550 K and 2.56 (μWtt-cm�<sup>1</sup>

decreases at higher temperature.

**66**

carriers.

**Figure 13.**

**4. Conclusion**


> -K�<sup>2</sup> ) at

) at 295 K. The lowest "PF" were experien-

) at 295 K. As deliberated earlier, an increasing the

Wiqar Hussain Shah\* and Waqas Muhammad Khan Department of Physics, Faculty of Basic and Applied Sciences, International Islamic University, Islamabad, Pakistan

\*Address all correspondence to: wiqar.hussain@iiu.edu.pk

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Kurosaki K, Uneda H, Muta H, Yamanaka S. Thermoelectric properties of thallium antimony telluride. Journal of Alloys and Compounds. 2004:43-48

[2] Assoud A, Soheilnia N, Kleinke H. Crystal structure, electronic structure and physical properties of the new lowvalent thallium silicon telluride Tl6Si2Te6 in comparison to Tl6Ge2Te6. Journal of Solid State Chemistry. 2006; **179**:2707-2713

[3] Kurosaki K, Goto K, Muta H, Yamanaka S. Fabrication and thermoelectric properties of Ag9TlTeX (X=5:06:0). Materials Transactions. 2007;**48**(8):2083-2087

[4] Heremans JP, Jovovic V, Toberer ES, Saramat A, Kurosaki K, Charoenphakdee A, et al. Enhancement of thermoelectric efficiency in PbTe by distortion of the electronic density of states. Science. 2008;**321**:554-557

[5] Sankar CR, Sanasy SB, Assoud A, Kleinke H. Syntheses, crystal structures and thermoelectric properties of two new thallium tellurides: Tl4ZrTe4 and Tl4HfTe4. Journal of Materials Chemistry. 2010;**20**:7485-7490

[6] Sankar CR, Guch M, Assoud A, Kleinke H. Structural, thermal, and physical properties of the thallium zirconium telluride Tl2ZrTe3. Chemistry of Materials. 2011;**23**: 3886-3891

[7] Raj C, Savitree BS, Holger K. Thermoelectric properties of TlGdQ2(Q = Se, Te) and Tl9GdTe6. Journal of Electronic Materials. 2012;**41**(6): 1662-1666

[8] Sanasy SB. Thermoelectric properties of Tl10-xLnxTe6, with Ln=Ce, Pr, Nd, Sm, Gd, Tb, Dy, and Er, and 0.25<x< 1.32. Journal of Alloys and Compounds. 2013:126-134

[9] Kim KT, Ha GH. Fabrication and enhanced thermoelectric properties of alumina nanoparticle-dispersed Bi0.5Sb1.5Te3 matrix composites. Journal of Nanomaterials. 2013;**2013**:1-6 **Chapter 5**

**Abstract**

functions Z<sup>σ</sup>

discussed.

**1. Introduction**

namely, σ<sup>+</sup>

**69**

Langevin equation, MD simulation

(DC)/σ(DC) = m/m<sup>+</sup> [4].

group [5–9]. Detailed procedure will be shown in what follows.

been developed from 1960s by several researchers [10, 11].

Electrical Conductivity of Molten

*Shigeru Tamaki, Shigeki Matsunaga and Masanobu Kusakabe*

A microscopic description for the partial DC conductivities in molten salts has been discussed by using a Langevin equation for the constituent ions. The memory function γ(t) can be written as in the form of a decaying function with time. In order to solve the mutual relation between the combined-velocity correlation

(t) and the memory function γ(t) in a short time region, a new recursion method is proposed. Practical application is carried out for molten NaCl by using MD simulation. The fitted function is described by three kinds of Gaussian functions and their physical backgrounds are discussed. Also the electrical conductivity in aqueous solution of electrolyte has been obtained, based on a generalized Langevin equation for cation and anion in it. This treatment can connect and compare with the work of computer simulation. The obtained results for concentration dependence of electrical conductivity are given by a function of the square root of concentration. The electrophoretic effect and the relaxation one are also

**Keywords:** conductivity of molten salts, conductivity of electrolytic solution,

The phenomena of transport properties in ionic liquids are of great important in the industrial science and technology, as well as in physics and chemistry. In connection with these, a number of experimental and theoretical studies have been published until the present time [1–3]. Ionic liquids are mainly classified into two categories; one is a group of molten salts and the other is a large number of electrolytic solutions, in particular, aqueous solutions of electrolytes.

In the case of molten salts, Sundheim discovered that the ratio of the partial conductivities of cation and anion were always equal to their inverse mass ratio,

Later on, this golden rule or a unified rule was theoretically explained by our

Paralleling to above discovery, a number of scientific studies in molten salts have

Salts and Ionic Conduction in

Electrolyte Solutions

[10] Zhang Q, Liao B, Lan Y, Lukas K, Liu W, Esfarjani K, et al. High thermoelectric performance by resonant dopant indium in nanostructured SnTe. Applied Physical Sciences. 2013;**110**(33): 13261-13266

[11] Bali A, Wang H, Snyder GJ, Mallik RC. Thermoelectric properties of indium doped PbTe1-ySey alloys. Journal of Applied Physics. 2014;**116**: 033707

[12] Kuropatwa BA, Guo Q, Assoud A, Kleinke H. Optimization of the telluride Tl10–x–ySnxBiyTe6 for the thermoelectric energy conversion. Journal of Inorganic and General Chemistry. 2014;**640**:774-780

[13] Dresselhaus MS, Chen G, Tang MY, Yang RG, Lee H, Wang DZ, et al. New directions for low-dimensional thermoelectric materials. Advanced Materials. 2007;**19**(8):1043-1053

#### **Chapter 5**

**References**

**179**:2707-2713

[1] Kurosaki K, Uneda H, Muta H, Yamanaka S. Thermoelectric properties of thallium antimony telluride. Journal of Alloys and Compounds. 2004:43-48

*Electromagnetic Field Radiation in Matter*

[9] Kim KT, Ha GH. Fabrication and enhanced thermoelectric properties of alumina nanoparticle-dispersed Bi0.5Sb1.5Te3 matrix composites. Journal of Nanomaterials. 2013;**2013**:1-6

[10] Zhang Q, Liao B, Lan Y, Lukas K,

thermoelectric performance by resonant dopant indium in nanostructured SnTe. Applied Physical Sciences. 2013;**110**(33):

Mallik RC. Thermoelectric properties of indium doped PbTe1-ySey alloys. Journal of Applied Physics. 2014;**116**:

[12] Kuropatwa BA, Guo Q, Assoud A, Kleinke H. Optimization of the telluride

[13] Dresselhaus MS, Chen G, Tang MY, Yang RG, Lee H, Wang DZ, et al. New

Tl10–x–ySnxBiyTe6 for the thermoelectric energy conversion. Journal of Inorganic and General Chemistry. 2014;**640**:774-780

directions for low-dimensional thermoelectric materials. Advanced Materials. 2007;**19**(8):1043-1053

Liu W, Esfarjani K, et al. High

[11] Bali A, Wang H, Snyder GJ,

13261-13266

033707

[2] Assoud A, Soheilnia N, Kleinke H. Crystal structure, electronic structure and physical properties of the new low-

Tl6Si2Te6 in comparison to Tl6Ge2Te6. Journal of Solid State Chemistry. 2006;

thermoelectric properties of Ag9TlTeX (X=5:06:0). Materials Transactions.

[4] Heremans JP, Jovovic V, Toberer ES,

Charoenphakdee A, et al. Enhancement of thermoelectric efficiency in PbTe by distortion of the electronic density of states. Science. 2008;**321**:554-557

[5] Sankar CR, Sanasy SB, Assoud A, Kleinke H. Syntheses, crystal structures and thermoelectric properties of two new thallium tellurides: Tl4ZrTe4 and Tl4HfTe4. Journal of Materials Chemistry. 2010;**20**:7485-7490

[6] Sankar CR, Guch M, Assoud A, Kleinke H. Structural, thermal, and physical properties of the thallium zirconium telluride Tl2ZrTe3. Chemistry of Materials. 2011;**23**:

[7] Raj C, Savitree BS, Holger K.

Thermoelectric properties of TlGdQ2(Q = Se, Te) and Tl9GdTe6. Journal of Electronic Materials. 2012;**41**(6):

[8] Sanasy SB. Thermoelectric properties of Tl10-xLnxTe6, with Ln=Ce, Pr, Nd, Sm, Gd, Tb, Dy, and Er, and 0.25<x< 1.32. Journal of Alloys and Compounds.

3886-3891

1662-1666

2013:126-134

**68**

valent thallium silicon telluride

[3] Kurosaki K, Goto K, Muta H, Yamanaka S. Fabrication and

2007;**48**(8):2083-2087

Saramat A, Kurosaki K,

## Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions

*Shigeru Tamaki, Shigeki Matsunaga and Masanobu Kusakabe*

#### **Abstract**

A microscopic description for the partial DC conductivities in molten salts has been discussed by using a Langevin equation for the constituent ions. The memory function γ(t) can be written as in the form of a decaying function with time. In order to solve the mutual relation between the combined-velocity correlation functions Z<sup>σ</sup> (t) and the memory function γ(t) in a short time region, a new recursion method is proposed. Practical application is carried out for molten NaCl by using MD simulation. The fitted function is described by three kinds of Gaussian functions and their physical backgrounds are discussed. Also the electrical conductivity in aqueous solution of electrolyte has been obtained, based on a generalized Langevin equation for cation and anion in it. This treatment can connect and compare with the work of computer simulation. The obtained results for concentration dependence of electrical conductivity are given by a function of the square root of concentration. The electrophoretic effect and the relaxation one are also discussed.

**Keywords:** conductivity of molten salts, conductivity of electrolytic solution, Langevin equation, MD simulation

#### **1. Introduction**

The phenomena of transport properties in ionic liquids are of great important in the industrial science and technology, as well as in physics and chemistry. In connection with these, a number of experimental and theoretical studies have been published until the present time [1–3]. Ionic liquids are mainly classified into two categories; one is a group of molten salts and the other is a large number of electrolytic solutions, in particular, aqueous solutions of electrolytes.

In the case of molten salts, Sundheim discovered that the ratio of the partial conductivities of cation and anion were always equal to their inverse mass ratio, namely, σ<sup>+</sup> (DC)/σ(DC) = m/m<sup>+</sup> [4].

Later on, this golden rule or a unified rule was theoretically explained by our group [5–9]. Detailed procedure will be shown in what follows.

Paralleling to above discovery, a number of scientific studies in molten salts have been developed from 1960s by several researchers [10, 11].

In order to study the structural and transport properties in molten salts, experimental investigations and molecular dynamics simulations have also been carried out from mid-70s of the last century [12–16].

We will apply the linear response theory for the electrolytic solution and to obtain Λ<sup>0</sup> and the concentration dependence of the conductivity in terms of pair-wise potentials and pair distribution functions among ions and water molecules, which can compare parallel with dynamical properties of MD simulation [28]. In addition, we will also clarify how the electrophoretic and relaxation effects

From these, we will see what is similar and what is different for the case of

Let us consider a molten salt composed of the density n<sup>+</sup> = n� = n0 (= N/V0), of the constituent ion's masses m<sup>+</sup> and m�, and of the charge z+ <sup>=</sup> � <sup>z</sup>� = z = 1, where N

As an extension, the generalized Langevin equation for an arbitrary cation or

ξ� *t* � *t* <sup>0</sup> ð Þ**v**<sup>i</sup>

After taking the ensemble average, equations of time evolution based on Eq. (1)

<ξ� *t* � *t* <sup>0</sup> ð Þ**v**<sup>i</sup>

And the equation of time evolution in relation to the diffusion constants of

ð*t*

independent for the averaging procedure and we have to define new memory

�ð Þ*t* < **v**<sup>i</sup>

�∞

� *t* <sup>0</sup> ð Þ**v**<sup>j</sup>

<ξ� *t* � *t* <sup>0</sup> ð Þ**v**<sup>i</sup>

friction force and the random fluctuating force, acting on the cation or anion *i*,

<ξ� *t* � *t* <sup>0</sup> ð Þ**v**<sup>i</sup>

� *t* <sup>0</sup> ð Þd*t*

�(*t*) are the retarded friction function in relation to the

� *t* <sup>0</sup> ð Þ**v**<sup>j</sup>

> � *t* <sup>0</sup> ð Þ**v**<sup>k</sup>

<sup>0</sup> þ **R**<sup>i</sup>

�ð Þ 0 >d*t*

� *t* <sup>0</sup> ð Þ**v**<sup>i</sup>

, can be obtainable from a generalized

�ð Þþ*t* z�e**E** (1)

<sup>0</sup> ð Þ for i ¼ j and i 6¼ j

<sup>∓</sup> ð Þ <sup>0</sup> <sup>&</sup>gt; ð Þ for i 6¼ <sup>k</sup> (3)

�ð Þ 0 >d *t*

�ð Þ 0 > ð Þ for i ¼ j and i 6¼ j (5)

(2)

<sup>0</sup> (4)

�(*t*) cannot be

treated by many researchers are explained in a microscopic view point.

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

**2. Generalized Langevin equations for the cation and anion in a**

being the total number of cation and/or anion in the volume V0.

(DC)/σ�(DC) = m�/m<sup>+</sup>

anion in the system under an external field **E** is written as follows:

ð

�∞

in respect to the partial ionic conductivities are then written as follows:

ð*t* �∞

As was previously illustrated [9], the retarded friction function ξ

ð*t* �∞ *t*

Drude theory, as a law of motion under an electric field [5].

�ð Þ*t =*d*t* ¼ �m�

�ð Þ 0 >*=*d*t* ¼ �m�

<sup>∓</sup>ð Þ <sup>0</sup> <sup>&</sup>gt;*=*d*<sup>t</sup>* ¼ �*m*�

�ð Þ 0 > *=*d*t* ¼ �m�

�ð Þ 0 > ¼ γσ

molten salts and that of electrolytic solutions.

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

**molten salt**

A golden rule, σ<sup>+</sup>

m�d**v**<sup>i</sup>

�(*t*) and **R**<sup>i</sup>

where ξ

respectively.

m�d<**v**<sup>i</sup>

and

m�d <**v**<sup>i</sup>

�ð Þ*t* **v**<sup>j</sup>

�ð Þ*t* **v**<sup>k</sup>

m�d< **v**<sup>i</sup>

functions as follows:

< ξ� *t* � *t* <sup>0</sup> ð Þ**v**<sup>i</sup>

and

**71**

constituent ions is written as follows:

�ð Þ*t* **v**<sup>i</sup>

� *t* <sup>0</sup> ð Þ**v**<sup>j</sup>

Following to these, we have been engaged in the study of transport properties in molten salts [6–9, 17]. We have carried out a theoretical study on the electrical conductivity of molten salts, starting from the Langevin equation and the velocity correlation functions for the constituent ions. Subsequently this treatment was successful to obtain the golden rule σ<sup>+</sup> /σ = m/m<sup>+</sup> in a microscopic view point.

It remains, however, unclear how the adopted Langevin equation can be effectively solved within a short time region, under an appropriate memory function, because our former theory was only successful to get the partial conductivities.

We like to discuss more generally the correlation between the velocity correlation functions incorporated with the partial DC conductivities and some of useful memory functions which are closely related to the friction constants acting on cations and anions in molten salts.

Preceding the investigation for molten salts, on the other hand, there have been a number of studies for ionic solutions since the discovery of Faraday, in which a typical example is electrolytic solution. During such long-termed history of electrochemistry, it was well established by Kohlrausch that the experimental results on the ionic conductivities in dilute electrolytic solutions indicated the law of independent migration of ions, *Λ<sup>c</sup>* = *Λ<sup>0</sup>* k*c* 1/2, where Λ<sup>0</sup> being the conductivity in the dilute limit and c the concentration and k the constant specified by the electrolyte dissolved in water.

The beginning of the modern aspect, in particular, on the thermodynamic and transport properties in electrolytic solutions might be originated from Debye-Hückel theory [18].

In order to explain the ionic conductivity in electrolytic solution, successful works following to Debye-Hückel theory have been reported by Onsager [19], Prigogine [20], and Fuoss and his co-worker [21]. In these theories, Λ<sup>0</sup> is treated by the Stokes law and the concentration dependence is mainly explained by the electrophoretic effect and relaxation one. Therefore, these treatments are based on a kind of mixing of the microscopic and partially macroscopic view point.

Starting from the Liouville equation, statistical mechanics of irreversible process for the ionic conductivity in electrolytic solution have been developed by Davis and Résibois [22] and Friedman [23], although they did not derive any explicit expressions for the friction constant in terms of inter-particle interactions.

It has been required to investigate the static and dynamic properties of dissolved ions in aqueous solutions from the microscopic view point. Along this requirement, the technique of molecular dynamic simulation has been applied, using some qualified inter-particle potentials. Various theoretical attempts have been recently tried to establish the dynamical behaviors of dissolved ions in these solutions, which is able to discuss parallel with results obtained by MD simulation [24–26].

Chandra and Bagchi [27] have developed a new theoretical approach to study the ionic conduction in electrolytic solutions, based on the combination of the mode coupling theory and the generalized Langevin equation, and they were successful to obtain the Onsager equation. However, there still remains the task to obtain how to derive the theoretical formula for Λ<sup>0</sup> in terms of inter-particle potentials and corresponding pair distribution functions.

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

We will apply the linear response theory for the electrolytic solution and to obtain Λ<sup>0</sup> and the concentration dependence of the conductivity in terms of pair-wise potentials and pair distribution functions among ions and water molecules, which can compare parallel with dynamical properties of MD simulation [28].

In addition, we will also clarify how the electrophoretic and relaxation effects treated by many researchers are explained in a microscopic view point.

From these, we will see what is similar and what is different for the case of molten salts and that of electrolytic solutions.

#### **2. Generalized Langevin equations for the cation and anion in a molten salt**

Let us consider a molten salt composed of the density n<sup>+</sup> = n� = n0 (= N/V0), of the constituent ion's masses m<sup>+</sup> and m�, and of the charge z+ <sup>=</sup> � <sup>z</sup>� = z = 1, where N being the total number of cation and/or anion in the volume V0.

A golden rule, σ<sup>+</sup> (DC)/σ�(DC) = m�/m<sup>+</sup> , can be obtainable from a generalized Drude theory, as a law of motion under an electric field [5].

As an extension, the generalized Langevin equation for an arbitrary cation or anion in the system under an external field **E** is written as follows:

$$\mathbf{m}^{\pm} \mathbf{d} \mathbf{v}\_{\mathbf{i}}^{\pm}(t) / \mathbf{d}t = -\mathbf{m}^{\pm} \int\_{-\infty}^{t} \xi^{\pm}(t - t') \mathbf{v}\_{\mathbf{i}}^{\pm}(t') \mathbf{d}t' + \mathbf{R}\_{\mathbf{i}}{}^{\pm}(t) + \mathbf{z}^{\pm} \mathbf{e} \mathbf{E} \tag{1}$$

where ξ �(*t*) and **R**<sup>i</sup> �(*t*) are the retarded friction function in relation to the friction force and the random fluctuating force, acting on the cation or anion *i*, respectively.

After taking the ensemble average, equations of time evolution based on Eq. (1) in respect to the partial ionic conductivities are then written as follows:

$$\mathbf{m}^{\pm}\mathbf{d} < \mathbf{v}\_{\mathrm{i}}^{\pm}(t)\mathbf{v}\_{\mathrm{j}}^{\pm}(\mathbf{0})>/\mathbf{d} \\ \mathbf{t} = -\mathbf{m}^{\pm} \int\_{-\mathrm{so}}^{t} \\ < \xi^{\pm}(t-t')\mathbf{v}\_{\mathrm{i}}^{\pm}(t')\mathbf{v}\_{\mathrm{j}}^{\pm}(\mathbf{0})>\mathbf{d}t' \quad (\text{for } \mathbf{i} = \text{j and } \mathbf{i} \neq \text{j}) \tag{2}$$

and

In order to study the structural and transport properties in molten salts, experimental investigations and molecular dynamics simulations have also been carried

Following to these, we have been engaged in the study of transport properties

It remains, however, unclear how the adopted Langevin equation can be effectively solved within a short time region, under an appropriate memory function, because our former theory was only successful to get the partial conductivities. We like to discuss more generally the correlation between the velocity correlation functions incorporated with the partial DC conductivities and some of useful memory functions which are closely related to the friction constants acting on

Preceding the investigation for molten salts, on the other hand, there have been a number of studies for ionic solutions since the discovery of Faraday, in which a typical example is electrolytic solution. During such long-termed history of electrochemistry, it was well established by Kohlrausch that the experimental results on the ionic conductivities in dilute electrolytic solutions indicated the law of indepen-

The beginning of the modern aspect, in particular, on the thermodynamic and transport properties in electrolytic solutions might be originated from Debye-H-

In order to explain the ionic conductivity in electrolytic solution, successful works following to Debye-Hückel theory have been reported by Onsager [19], Prigogine [20], and Fuoss and his co-worker [21]. In these theories, Λ<sup>0</sup> is treated by the Stokes law and the concentration dependence is mainly explained by the electrophoretic effect and relaxation one. Therefore, these treatments are based on a kind of mixing of the microscopic and partially macroscopic

Starting from the Liouville equation, statistical mechanics of irreversible process for the ionic conductivity in electrolytic solution have been developed by

It has been required to investigate the static and dynamic properties of dissolved ions in aqueous solutions from the microscopic view point. Along this requirement, the technique of molecular dynamic simulation has been applied, using some qualified inter-particle potentials. Various theoretical attempts have been recently tried to establish the dynamical behaviors of dissolved ions in these solutions, which

Chandra and Bagchi [27] have developed a new theoretical approach to study the ionic conduction in electrolytic solutions, based on the combination of the mode coupling theory and the generalized Langevin equation, and they were successful to obtain the Onsager equation. However, there still remains the task to obtain how to derive the theoretical formula for Λ<sup>0</sup> in terms of inter-particle potentials and

Davis and Résibois [22] and Friedman [23], although they did not derive any explicit expressions for the friction constant in terms of inter-particle

is able to discuss parallel with results obtained by MD simulation [24–26].

corresponding pair distribution functions.

limit and c the concentration and k the constant specified by the electrolyte

/σ = m/m<sup>+</sup> in a microscopic

1/2, where Λ<sup>0</sup> being the conductivity in the dilute

electrical conductivity of molten salts, starting from the Langevin equation and the velocity correlation functions for the constituent ions. Subsequently this

in molten salts [6–9, 17]. We have carried out a theoretical study on the

out from mid-70s of the last century [12–16].

*Electromagnetic Field Radiation in Matter*

cations and anions in molten salts.

dent migration of ions, *Λ<sup>c</sup>* = *Λ<sup>0</sup>* k*c*

dissolved in water.

ückel theory [18].

view point.

interactions.

**70**

view point.

treatment was successful to obtain the golden rule σ<sup>+</sup>

$$\mathbf{m}^{\pm}\mathbf{d} < \mathbf{v}\_{\mathbf{i}}^{\pm}(t)\,\mathbf{v}\_{\mathbf{k}}^{\mp}(\mathbf{0})>/\mathbf{d} \\ \mathbf{t} = -m^{\pm} \int\_{-\infty}^{t} \\ < \boldsymbol{\xi}^{\pm}(t-t')\mathbf{v}\_{\mathbf{i}}^{\pm}(t')\mathbf{v}\_{\mathbf{k}}^{\mp}(\mathbf{0})> \quad (\text{for } \mathbf{i} \neq \mathbf{k}) \tag{3}$$

And the equation of time evolution in relation to the diffusion constants of constituent ions is written as follows:

$$\mathbf{m}^{\pm}\mathbf{d} < \mathbf{v}\_{\mathbf{i}}^{\pm}(t)\mathbf{v}\_{\mathbf{i}}^{\pm}(\mathbf{0}) > /\mathbf{dt} = -\mathbf{m}^{\pm}\int\_{-\infty}^{t} < \mathbf{\xi}^{\pm}(t-t')\mathbf{v}\_{\mathbf{i}}^{\pm}(t')\mathbf{v}\_{\mathbf{i}}^{\pm}(\mathbf{0}) > \mathbf{d}t' \tag{4}$$

As was previously illustrated [9], the retarded friction function ξ �(*t*) cannot be independent for the averaging procedure and we have to define new memory functions as follows:

$$<\xi^{\pm}(t-t')\mathbf{v}\_{\mathrm{i}}^{\pm}(t')\mathbf{v}\_{\mathrm{j}}^{\pm}(\mathbf{0})> = \boldsymbol{\chi}\_{\sigma}^{\pm}(t) < \mathbf{v}\_{\mathrm{i}}^{\pm}(t')\mathbf{v}\_{\mathrm{j}}^{\pm}(\mathbf{0})> \quad (\text{for } \mathrm{i}=\mathrm{j} \text{and} \,\mathrm{i}\neq \mathrm{j})\tag{5}$$

and

$$<\xi^{\pm}(t - t')\mathbf{v}\_{\mathbf{i}}^{\pm}(t')\mathbf{v}\_{\mathbf{k}}^{\mp}(\mathbf{0})> = \boldsymbol{\eta}\_{\sigma}^{\pm}(t) < \mathbf{v}\_{\mathbf{i}}^{\pm}(t')\mathbf{v}\_{\mathbf{k}}^{\mp}(\mathbf{0})> \quad (\text{for } \mathbf{i} \neq \mathbf{k})\tag{6}$$

While, in the case of diffusion constants of constituent ions, that is, **E** = 0, we can define

$$<\xi^{\pm}(t-t')\mathbf{v}\_{\mathbf{i}}{}^{\pm}(t')\mathbf{v}\_{\mathbf{i}}{}^{\pm}(\mathbf{0})> = \gamma\_{\mathbf{D}}{}^{\pm}(t)<\mathbf{v}\_{\mathbf{i}}{}^{\pm}(t')\mathbf{v}\_{\mathbf{i}}{}^{\pm}(\mathbf{0})>\tag{7}$$

It is emphasized that the memory functions γσ �(*t*) is not equal to γ<sup>D</sup> �(*t*) as shown in previous paper [9]. In other words, the retarded friction function, ξ �(*t* � *t* 0 ), is a kind of vector function and is varied with the environment such as the existence of electric field **E**. Therefore, the memory function is varied in accordance with what sort of evolution is considered in the time-dependent correlation function [29].

Assuming that the ensemble average for the fluctuating force is zero and if we apply the following electric field,

$$\mathbf{E}(t) = \text{Re } \mathbf{E}\_0 \text{ exp}\left(\text{i}\alpha t\right) \tag{8}$$

<sup>~</sup>γð Þ¼ <sup>0</sup> <sup>α</sup>0*=*3<sup>μ</sup> � �1*=*<sup>2</sup>

<sup>ϕ</sup>þ�ð Þ*<sup>r</sup> <sup>=</sup>∂<sup>r</sup>*

experimental results obtained by Edwards et al. [31].

arbitrary ions located at the distance r, which describe as nij = 4πʃ<sup>0</sup>

*Pair distribution functions,* g*ij(*r*), for molten NaCl at 1148 K, obtained by MD simulation.*

<sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup>*=<sup>r</sup>* <sup>∂</sup>ϕþ� f g ð Þ*<sup>r</sup> <sup>=</sup>∂<sup>r</sup>* � � <sup>ɡ</sup>þ�ð Þ� *<sup>r</sup>* <sup>4</sup>π*<sup>r</sup>*

ϕ+�(*r*) and ɡ+�(*r*) in this equation are the inter-ionic potential between cation

Therefore, we have a golden rule for the partial conductivities in a microscopic

In the following sections, as a numerical example, the MD simulation on molten

Using these ɡij(r), we have also estimated the total neighboring numbers around

The nearest neighbor number is defined as nij(r1), where r1 is the position of the

, Cl�) as shown in **Figure 1**, which agree with those of

and anion and the corresponding pair distribution function, respectively.

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

NaCl at 1100 K is often utilized, for which the interionic potential functions suggested by Tosi and Fumi [30] for a study of solid alkali halides are applied. In order to make sure that the Tosi-Fumi potential for NaCl can be valid in the liquid state, we have estimated the partial pair distribution functions of molten NaCl

and

<sup>α</sup><sup>0</sup> <sup>¼</sup> *<sup>n</sup>* ð 0

scale as follows:

liquid, ɡij(r) (i,j = Na+

first minimum of ɡij(r).

**Figure 2a**–**c**.

**Figure 1.**

**73**

∞ *∂*2

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

, 1ð Þ¼ *=*μ 1*=*m<sup>þ</sup> ð Þþ 1*=*m� ð Þ (16)

σþð Þ DC *=*σ�ð Þ¼ DC m�*=*m<sup>þ</sup> (18)

2

r r 2

dr, as shown in

d*r* (17)

where Re means the real part and ω is the angular frequency, then the averaged ion's velocity induced by this external filed is equal to

$$<\mathbf{v}\_{i}^{\pm}(t)> = \text{Re } \mu^{\pm}(\omega) \mathbf{z}^{\pm} \mathbf{e} \mathbf{E}(t) \tag{9}$$

where μ�(ω) is the mobility of cation or anion.

Putting (9) into the equation of motion (1) after taking the ensemble average, we have

$$\boldsymbol{\mu}^{\pm}(\boldsymbol{\alpha}) = \left(\mathbf{1}/\boldsymbol{m}^{\pm}\right) \left[\mathbf{1}/\left\{\mathbf{i}\boldsymbol{\alpha} + \hat{\boldsymbol{\gamma}}^{\pm}(\boldsymbol{\alpha})\right\}\right] \tag{10}$$

where

$$\hat{\boldsymbol{\chi}}^{\pm}(\boldsymbol{\alpha}) = \int\_{0}^{\infty} \boldsymbol{\chi}^{\pm}(t) \, \exp\left(-i\boldsymbol{\alpha}t\right) \, \mathrm{d}t \tag{11}$$

Therefore, the current density is written as follows:

$$\mathbf{j}^{\pm}(t) = \mathbf{n} \mathbf{z}^{\pm 2} \mathbf{e}^2 < \mathbf{v}\_i^{\pm}(t) > = \text{Re } \mathbf{n} \mathbf{z}^{\pm 2} \mathbf{e}^2 \mu^{\pm}(\mathbf{o}) \mathbf{E}(t) \tag{12}$$

The partial conductivity is, then, equal to

$$\boldsymbol{\sigma}^{\pm}(\boldsymbol{\mathfrak{o}}) = \mathbf{n} \mathbf{z}^{\pm 2} \mathbf{e}^{2} \boldsymbol{\mu}^{\pm}(\boldsymbol{\mathfrak{o}}) = \left( \mathbf{n} \mathbf{z}^{\pm 2} \mathbf{e}^{2} / \mathbf{m}^{\pm} \right) \left\{ \mathbf{1} / \left( \mathbf{i} \boldsymbol{\mathfrak{o}} + \boldsymbol{\tilde{\boldsymbol{\eta}}}^{\pm}(\boldsymbol{\mathfrak{o}}) \right) \right\} \tag{13}$$

and in the limit of ω = 0,

$$\boldsymbol{\sigma}^{\pm}(\mathbf{D}\mathbf{C}) = \mathbf{n}\mathbf{z}^{\pm 2}\mathbf{e}^{2}\boldsymbol{\mu}^{\pm}(\mathbf{0}) = \left\{\mathbf{n}\mathbf{z}^{\pm 2}\mathbf{e}^{2}/\mathbf{m}^{\pm}\tilde{\boldsymbol{\gamma}}^{\pm}(\mathbf{0})\right\} \tag{14}$$

Therefore, ~γ�ð Þ 0 is equal to the effective friction constant acting on each ion. According to our previous studies [7–9], the following relation was recognized:

$$
\ddot{\mathsf{Y}}^{+}(\mathsf{O}) = \ddot{\mathsf{Y}}^{-}(\mathsf{O}) \equiv \ddot{\mathsf{y}}(\mathsf{O}) \tag{15}
$$

where ~γð Þ 0 is expressed as follows:

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

$$\ddot{\gamma}(\mathbf{0}) = \left(\mathbf{a}^0/3\mu\right)^{1/2}, \quad \left(\mathbf{1}/\mu\right) = \left(\mathbf{1}/\mathbf{m}^+\right) + \left(\mathbf{1}/\mathbf{m}^-\right) \tag{16}$$

and

<ξ� *t* � *t* <sup>0</sup> ð Þ**v**<sup>i</sup>

correlation function [29].

apply the following electric field,

can define

ξ �(*t* � *t* 0

we have

where

**72**

� *t* <sup>0</sup> ð Þ**v**<sup>k</sup>

*Electromagnetic Field Radiation in Matter*

<ξ� *t* � *t* <sup>0</sup> ð Þ**v**<sup>i</sup>

<sup>∓</sup> ð Þ <sup>0</sup> <sup>&</sup>gt; <sup>¼</sup> γσ

� *t* <sup>0</sup> ð Þ**v**<sup>i</sup>

It is emphasized that the memory functions γσ

ion's velocity induced by this external filed is equal to

where μ�(ω) is the mobility of cation or anion.

<**v**<sup>i</sup>

~γ�ð Þ¼ ω

Therefore, the current density is written as follows:

e2

<sup>σ</sup>�ð Þ¼ DC nz�<sup>2</sup>

e2 <**v**<sup>i</sup>

<sup>μ</sup>�ð Þ¼ <sup>ω</sup> nz�<sup>2</sup>

e2

�ðÞ¼ *<sup>t</sup>* nz�<sup>2</sup>

The partial conductivity is, then, equal to

<sup>σ</sup>�ð Þ¼ <sup>ω</sup> nz�<sup>2</sup>

where ~γð Þ 0 is expressed as follows:

j

and in the limit of ω = 0,

ð 0 ∞

�ð Þ*t* <**v**<sup>i</sup>

While, in the case of diffusion constants of constituent ions, that is, **E** = 0, we

�ð Þ 0 > ¼ γ<sup>D</sup>

), is a kind of vector function and is varied with the environment such

Assuming that the ensemble average for the fluctuating force is zero and if we

where Re means the real part and ω is the angular frequency, then the averaged

Putting (9) into the equation of motion (1) after taking the ensemble average,

�ð Þ*<sup>t</sup>* <sup>&</sup>gt; <sup>¼</sup> Re nz�<sup>2</sup>

e2

<sup>μ</sup>�ð Þ¼ <sup>0</sup> nz�<sup>2</sup>

Therefore, ~γ�ð Þ 0 is equal to the effective friction constant acting on each ion. According to our previous studies [7–9], the following relation was recognized:

shown in previous paper [9]. In other words, the retarded friction function,

as the existence of electric field **E**. Therefore, the memory function is varied in accordance with what sort of evolution is considered in the time-dependent

� *t* <sup>0</sup> ð Þ**v**<sup>k</sup>

�ð Þ*t* < **v**<sup>i</sup>

� *t* <sup>0</sup> ð Þ**v**<sup>i</sup>

**E**ðÞ¼ *t* Re **E**<sup>0</sup> exp ið Þ ω*t* (8)

�ð Þ*t* > ¼ Re μ�ð Þ ω z�e**E**ð Þ*t* (9)

<sup>μ</sup>�ð Þ¼ <sup>ω</sup> <sup>1</sup>*=m*� � � <sup>1</sup>*<sup>=</sup>* <sup>i</sup><sup>ω</sup> <sup>þ</sup> <sup>~</sup>γ�ð Þ <sup>ω</sup> � � � � (10)

e2

e2

~γþð Þ¼ 0 ~γ�ð Þ� 0 ~γð Þ 0 (15)

γ�ð Þ*t* exp ð Þ �iω*t* d*t* (11)

*<sup>=</sup>*m� � � <sup>1</sup>*<sup>=</sup>* <sup>i</sup><sup>ω</sup> <sup>þ</sup> <sup>~</sup>γ�ð Þ <sup>ω</sup> � � � � (13)

*<sup>=</sup>*m�~γ�ð Þ <sup>0</sup> � � (14)

μ�ð Þ ω **E**ð Þ*t* (12)

�(*t*) is not equal to γ<sup>D</sup>

<sup>∓</sup> ð Þ <sup>0</sup> <sup>&</sup>gt; ð Þ for i 6¼ <sup>k</sup> (6)

�ð Þ 0 > (7)

�(*t*) as

$$\mathbf{a}^{0} = n \int\_{0}^{\infty} \left[ \partial^{2} \phi^{+-}(r) / \partial r^{2} + (2/r) \{ \partial \phi^{+-}(r) / \partial r \} \right] \mathbf{g}^{+-}(r) \cdot 4\pi r^{2} \mathbf{d}r \tag{17}$$

ϕ+�(*r*) and ɡ+�(*r*) in this equation are the inter-ionic potential between cation and anion and the corresponding pair distribution function, respectively.

Therefore, we have a golden rule for the partial conductivities in a microscopic scale as follows:

$$
\sigma^+(\mathbf{DC})/\sigma^-(\mathbf{DC}) = \mathbf{m}^-/\mathbf{m}^+\tag{18}
$$

In the following sections, as a numerical example, the MD simulation on molten NaCl at 1100 K is often utilized, for which the interionic potential functions suggested by Tosi and Fumi [30] for a study of solid alkali halides are applied. In order to make sure that the Tosi-Fumi potential for NaCl can be valid in the liquid state, we have estimated the partial pair distribution functions of molten NaCl liquid, ɡij(r) (i,j = Na+ , Cl�) as shown in **Figure 1**, which agree with those of experimental results obtained by Edwards et al. [31].

Using these ɡij(r), we have also estimated the total neighboring numbers around arbitrary ions located at the distance r, which describe as nij = 4πʃ<sup>0</sup> r r 2 dr, as shown in **Figure 2a**–**c**.

The nearest neighbor number is defined as nij(r1), where r1 is the position of the first minimum of ɡij(r).

**Figure 1.** *Pair distribution functions,* g*ij(*r*), for molten NaCl at 1148 K, obtained by MD simulation.*

Zσ

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

Zσ

respectively, as follows:

and

and/or

expressions:

L �<sup>ð</sup> 0

∞

Therefore we have,

þð Þ¼ ω Z<sup>σ</sup>

In a similar way, we have,

<sup>σ</sup> ð Þ¼ ω Z<sup>σ</sup>

Z~σ

Z~�

**75**

γþð Þ*s* Z<sup>σ</sup>

� �

þðÞ� *t* <**v**<sup>i</sup>

�ðÞ� *t* <**v**<sup>k</sup>

*∂* Z<sup>σ</sup>

*∂* Z<sup>σ</sup>

Taking the Laplace transformation of ∂{Z<sup>σ</sup>

¼ exp ð Þ �iω*t* Z<sup>σ</sup> <sup>þ</sup> ½ � ð Þ*t* <sup>0</sup>

Here, we have used an evident condition Z<sup>σ</sup>

þð Þ *t* � *s* d*s*

�Z<sup>σ</sup>

<sup>þ</sup> <sup>½</sup> f g ð Þ*<sup>t</sup> <sup>=</sup>∂t*� � <sup>ð</sup>

<sup>L</sup> *<sup>∂</sup>* <sup>Z</sup><sup>σ</sup>

where < > means the ensemble average.

<sup>þ</sup> ð Þ*t* **v**<sup>j</sup>

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

� ð Þ*t* **v**<sup>l</sup>

<sup>σ</sup>þð Þ¼ DC nz2

<sup>σ</sup>�ð Þ¼ DC nz2

<sup>þ</sup> f g ð Þ*<sup>t</sup> <sup>=</sup>∂<sup>t</sup>* ¼ �<sup>ð</sup>

� f g ð Þ*<sup>t</sup> <sup>=</sup>∂<sup>t</sup>* ¼ �<sup>ð</sup>

<sup>þ</sup> ð Þ 0 > � < **v**<sup>i</sup>

�ð Þ 0 > � < **v**<sup>i</sup>

0

0

γþð Þ *t* � *s* Z<sup>σ</sup>

γ�ð Þ *t* � *s* Z<sup>σ</sup>

+

exp ð Þ �iω*<sup>t</sup> <sup>∂</sup>* <sup>Z</sup><sup>σ</sup>

∞

+

þð Þ¼� <sup>ω</sup> <sup>~</sup>γþð Þ <sup>ω</sup> <sup>Z</sup>~<sup>σ</sup>

þð Þ 0 *=* iω þ ~γ<sup>þ</sup> f g ð Þ ω ¼ 3kBT*=*m<sup>þ</sup> ð Þ*=* iω þ ~γ<sup>þ</sup> f g ð Þ ω (36)

�ð Þ 0 *=* iω þ ~γ� f g ð Þ ω ¼ 3kBT*=*m� ð Þ*=* iω þ ~γ� f g ð Þ ω (37)

∞ Zσ

∞ Zσ

Using (25) and (26), the partial DC conductivities (19) and (20) are written,

e2 *<sup>=</sup>*3kBT � �<sup>ð</sup>

e2 *<sup>=</sup>*3kBT � �<sup>ð</sup>

0

0

0

¼ �Z<sup>σ</sup>

¼ �<sup>ð</sup> 0

∞

ð 0

þð Þþ <sup>0</sup> <sup>i</sup>ωZ~<sup>σ</sup>

∞

<sup>∞</sup> <sup>þ</sup> <sup>i</sup><sup>ω</sup> ð 0

On the other hand, the right hand side of (29) is given by the following

∞

exp ð Þ �iω*s* Z<sup>σ</sup>

þð Þþ <sup>0</sup> <sup>i</sup>ωZ~<sup>σ</sup>

*t*

*t*

On the other hand, combining Eqs. (25) or (26) and (1), we have

<sup>þ</sup> ð Þ*t* **v**<sup>k</sup>

<sup>þ</sup> ð Þ*t* **v**<sup>k</sup>

� ð Þ 0 > (25)

� ð Þ 0 > (26)

þð Þ*t* d*t* (27)

�ð Þ*t* d*t* (28)

þð Þ*s* d*s* (29)

�ð Þ*s* d*s* (30)

<sup>þ</sup> ½ � f g ð Þ*<sup>t</sup> <sup>=</sup>∂<sup>t</sup>* <sup>d</sup>*<sup>t</sup>* (31)

þð Þ ω (33)

þð Þ*t* d*t* (32)

þð Þ ω

þð Þ ω (35)

(34)

(*t*)}/ ∂*t* in (29) as follows,

exp ð Þ �iω*t* Z<sup>σ</sup>

(*t* = ∞) = 0.

exp f g �iωð Þ *t* � *s* γþð Þ *t* � *s* dð Þ *t* � *s*

þð Þ*<sup>s</sup>* <sup>d</sup>*<sup>s</sup>* ¼ �~γþð Þ <sup>ω</sup> <sup>Z</sup>~<sup>σ</sup>

**Figure 2.**

*(a)* g*Na-Cl(*r*) and* n*Na-Cl(*r*) for molten NaCl at 1148 K, obtained by MD simulation. (b)* g*Na-Na(*r*) and* n*Na-Na(*r*) for molten NaCl at 1148 K, obtained by MD simulation. (c)* g*Cl-Cl(*r*) and* n*Cl-Cl(*r*) for molten NaCl at 1148 K, obtained by MD simulation.*

Then, the nearest neighbors around a Na<sup>+</sup> are nearly equal to 5.0, since the distance r1 is taken at the minimum position of ɡNa-Cl(r) as shown in **Figure 2a**.

The application of Tosi-Fumi potentials in the MD simulations for viscosity and electrical conductivity is also valid to reproduce their experimental results [5].

Therefore, the following MD simulations for molten NaCl must be reliable to see their microscopic view.

#### **3. Linear response theory for the partial conductivities**

On the other hand, according to our previous investigations [6–9, 17, 29], the partial DC conductivities σ<sup>+</sup> (DC) and σ<sup>+</sup> (DC) are expressed as follows,

$$
\sigma^+(\mathbf{DC}) = \sigma^{++} + \sigma^{+-} = (\mathbf{1}/\mathbf{3k\_BT}) \int\_0^\infty < \mathbf{j}^+(t) \, \mathbf{j}(0) > \mathbf{d}t \tag{19}
$$

$$\left[\sigma^{-}(\mathbf{DC}) = \sigma^{--} + \sigma^{+-} = (\mathbf{1}/3\mathbf{k\_{B}T})\right]\_{0}^{\prime\prime} < \mathbf{j^{-}(t)}\,\mathbf{j(0)} > \mathbf{d}t \tag{20}$$

where

$$\boldsymbol{\sigma}^{\pm\pm} = (\mathbf{1}/3\mathbf{k}\_{\rm B}\mathbf{T})\Big|\_{0}^{\prime\prime} < \mathbf{j}^{\pm}(t)\mathbf{j}^{\pm}(\mathbf{0}) > \mathbf{d}t \tag{21}$$

$$\boldsymbol{\sigma}^{+-} = (\mathbf{1}/3\mathbf{k}\_{\rm B}\mathbf{T})\Big|\_{\rm 0}^{\rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \$$

and

$$\mathbf{j}(t) = \mathbf{j}^+(t) + \mathbf{j}^-(t) \tag{23}$$

where

$$\mathbf{j}^+(t) = \sum\_{i=1}^n \mathbf{z}^+ \mathbf{e} \mathbf{v}\_i^+(t), \quad \mathbf{j}^-(t) = \sum\_{\mathbf{k}=1}^n \mathbf{z}^- \mathbf{e} \mathbf{v}\_{\mathbf{k}}^-(t) \tag{24}$$

Considering the ensemble averages of (19) and (20), it is convenient to define the velocity correlation functions Z<sup>σ</sup> + (*t*) and Z<sup>σ</sup> �(*t*) as follows:

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

$$\mathbf{Z\_{v}}^{+}(t) \equiv <\mathbf{v\_{i}}^{+}(t)\,\mathbf{v\_{j}}^{+}(0)> - <\mathbf{v\_{i}}^{+}(t)\,\mathbf{v\_{k}}^{-}(0)>\tag{25}$$

and

$$\mathbf{Z\_{o}}^{-}(t) \equiv <\mathbf{v\_{k}}^{-}(t)\,\mathbf{v\_{l}}^{-}(0)> - <\mathbf{v\_{l}}^{+}(t)\,\mathbf{v\_{k}}^{-}(0)>\tag{26}$$

where < > means the ensemble average.

Using (25) and (26), the partial DC conductivities (19) and (20) are written, respectively, as follows:

$$\boldsymbol{\sigma}^{+}(\mathbf{D}\mathbf{C}) = \left(\mathbf{n}\mathbf{z}^{2}\mathbf{e}^{2}/3\mathbf{k}\_{\mathbb{B}}\mathbf{T}\right)\int\_{0}^{\infty}\mathbf{Z}\_{\sigma}{}^{+}(t)\mathbf{d}t\tag{27}$$

$$\boldsymbol{\sigma}^{-}(\mathbf{D}\mathbf{C}) = \left(\mathbf{n}\mathbf{z}^{2}\mathbf{e}^{2}/3\mathbf{k}\_{\mathbb{B}}\mathbf{T}\right)\int\_{0}^{\infty}\mathbf{Z}\_{\sigma}{}^{-}(t)\mathbf{d}t\tag{28}$$

On the other hand, combining Eqs. (25) or (26) and (1), we have

$$\left\{\partial\{\mathbf{Z}\_{\sigma}^{+}(t)\}\right\}/\partial\mathbf{t}=-\int\_{0}^{t} \boldsymbol{\eta}^{+}(t-s)\mathbf{Z}\_{\sigma}^{+}(s)\mathbf{ds}\tag{29}$$

and/or

Then, the nearest neighbors around a Na<sup>+</sup> are nearly equal to 5.0, since the distance r1 is taken at the minimum position of ɡNa-Cl(r) as shown in **Figure 2a**. The application of Tosi-Fumi potentials in the MD simulations for viscosity and

*(a)* g*Na-Cl(*r*) and* n*Na-Cl(*r*) for molten NaCl at 1148 K, obtained by MD simulation. (b)* g*Na-Na(*r*) and* n*Na-Na(*r*) for molten NaCl at 1148 K, obtained by MD simulation. (c)* g*Cl-Cl(*r*) and* n*Cl-Cl(*r*) for molten*

electrical conductivity is also valid to reproduce their experimental results [5].

**3. Linear response theory for the partial conductivities**

(DC) and σ<sup>+</sup>

σþð Þ¼ DC σþþ þ σþ� ¼ ð Þ 1*=*3kBT

σ�ð Þ¼ DC σ�� þ σþ� ¼ ð Þ 1*=*3kBT

σ�� ¼ ð Þ 1*=*3kBT

σþ� ¼ ð Þ 1*=*3kBT

their microscopic view.

*NaCl at 1148 K, obtained by MD simulation.*

*Electromagnetic Field Radiation in Matter*

**Figure 2.**

partial DC conductivities σ<sup>+</sup>

where

and

where

**74**

**j**

the velocity correlation functions Z<sup>σ</sup>

þðÞ¼ *<sup>t</sup>* <sup>X</sup><sup>n</sup>

i¼1

Therefore, the following MD simulations for molten NaCl must be reliable to see

On the other hand, according to our previous investigations [6–9, 17, 29], the

ð 0

ð 0

þð Þ*t* , **j**

Considering the ensemble averages of (19) and (20), it is convenient to define

(*t*) and Z<sup>σ</sup>

**j**ðÞ¼ *t* **j**

+

zþe**v**<sup>i</sup>

∞ <**j** �ð Þ*t* **j**

∞ <**j** þð Þ*t* **j**

þðÞþ*t* **j**

�ðÞ¼ *<sup>t</sup>* <sup>X</sup><sup>n</sup>

k¼1

�(*t*) as follows:

z�e**v**<sup>k</sup>

(DC) are expressed as follows,

þð Þ*t* **j**ð Þ 0 >d*t* (19)

�ð Þ*t* **j**ð Þ 0 >d*t* (20)

�ð Þ 0 >d*t* (21)

�ð Þ 0 >d*t* (22)

�ð Þ*t* (23)

�ð Þ*t* (24)

ð 0

ð 0 ∞ <**j**

∞ <**j**

$$\left\{\partial\{\mathbf{Z}\_{\sigma}^{-}(t)\}\right\}/\partial\mathbf{t}=-\int\_{0}^{t}\boldsymbol{\chi}^{-}(t-s)\mathbf{Z}\_{\sigma}^{-}(s)\mathbf{ds}\tag{30}$$

Taking the Laplace transformation of ∂{Z<sup>σ</sup> + (*t*)}/ ∂*t* in (29) as follows,

$$\mathcal{L}\left[\partial\{\mathbf{Z}\_{\sigma}^{+}(t)\}\rangle\langle\mathbf{t}\mathbf{t}\right] \equiv \int\_{0}^{\infty} \exp\left(-\text{i}\text{ot}\right) [\partial\{\mathbf{Z}\_{\sigma}^{+}(t)/\partial\mathbf{t}\}] \text{d}\mathbf{t} \tag{31}$$

$$= \left[ \exp\left(-\text{i}\text{ot}\right) \mathbf{Z}\_{\sigma}^{+}(t) \right]\_{0}^{\text{os}} + \text{i}\text{o} \int\_{0}^{\text{os}} \exp\left(-\text{i}\text{ot}\right) \mathbf{Z}\_{\sigma}^{+}(t) \text{d}t \tag{32}$$

$$=-\mathcal{Z}\_{\sigma}{}^{+}(\mathbf{0}) + \mathrm{i}\boldsymbol{\alpha}\tilde{\mathcal{Z}}\_{\sigma}{}^{+}(\mathbf{0})\tag{33}$$

Here, we have used an evident condition Z<sup>σ</sup> + (*t* = ∞) = 0.

On the other hand, the right hand side of (29) is given by the following expressions:

$$\begin{split} \mathcal{L}\left\{ -\int\_{0}^{\infty} \boldsymbol{\eta}^{+}(\boldsymbol{s}) \mathbf{Z}\_{\sigma}^{+}(t-\boldsymbol{s}) \mathbf{d} \mathbf{s} \right\} &= -\int\_{0}^{\infty} \exp\left\{-\mathrm{i}\boldsymbol{o}(t-\boldsymbol{s})\right\} \boldsymbol{\eta}^{+}(t-\boldsymbol{s}) \, \mathrm{d}(t-\boldsymbol{s}) \\ &\int\_{0}^{\infty} \exp\left(-\mathrm{i}\boldsymbol{o}\boldsymbol{s}\right) \mathbf{Z}\_{\sigma}^{+}(\boldsymbol{s}) \mathrm{d}\mathbf{s} = -\tilde{\boldsymbol{\eta}}^{+}(\boldsymbol{o}) \, \tilde{\mathbf{Z}}\_{\sigma}^{+}(\boldsymbol{o}) \end{split} \tag{34}$$

Therefore we have,

$$-\mathcal{Z}\_{\sigma}^{+}(\mathbf{0}) + i\alpha \tilde{\mathcal{Z}}\_{\sigma}^{+}(\mathbf{0}) = -\tilde{\chi}^{+}(\mathbf{0})\,\tilde{\mathcal{Z}}\_{\sigma}^{+}(\mathbf{0})\tag{35}$$

$$\tilde{\mathbf{Z}}\_{\sigma}^{+}(\mathbf{o}) = \mathbf{Z}\_{\sigma}^{+}(\mathbf{0}) / \{ \mathbf{i}\mathbf{o} + \tilde{\mathbf{j}}^{+}(\mathbf{o}) \} = (\mathbf{3k\_{B}T} / \mathbf{m}^{+}) / \{ \mathbf{i}\mathbf{o} + \tilde{\mathbf{j}}^{+}(\mathbf{o}) \} \tag{36}$$

In a similar way, we have,

$$\tilde{\mathbf{Z}}\_{\sigma}^{-}(\boldsymbol{\alpha}) = \mathbf{Z}\_{\sigma}^{-}(\mathbf{0}) / \{ \mathbf{i}\boldsymbol{\alpha} + \tilde{\boldsymbol{\gamma}}^{-}(\boldsymbol{\alpha}) \} = (\mathbf{\mathcal{Z}}\_{\mathbf{B}} \mathbf{T} / \mathbf{m}^{-}) / \{ \mathbf{i}\boldsymbol{\alpha} + \tilde{\boldsymbol{\gamma}}^{-}(\boldsymbol{\alpha}) \} \tag{37}$$

If an appropriate memory function γ(*t*), which is valid for both cation and anion in the system, is considered and its Laplace transformation is inserted into either (36) or (37), then we can get the partial AC conductivities.

concluded that the memory function in the generalized Langevin equation could be

where γn(*t*) is the n-th stage memory function and the first stage memory function is equal to γ(*t*) in Eqs. (29) and (30). The Fourier-Laplace transform of the

The method of Copley and Lovesey [36] was able to express the short time expansion for the velocity correlation function Z(*t*) (= < **v**i(*t*) **v**j(0)>) described as

> <sup>4</sup>*=*4! � �Z4 � *<sup>t</sup>* <sup>6</sup>*=*6! � �Z6 <sup>þ</sup> … � � (45)

above equation provides the following continued-fraction representation,

<sup>γ</sup>nþ1ð Þ *<sup>t</sup>* � *<sup>s</sup>* <sup>γ</sup>nð Þ*<sup>s</sup>* <sup>d</sup>*<sup>s</sup>* <sup>n</sup> <sup>¼</sup> 1, 2, 3, … (43)

~γnð Þ¼� ω δn*=*½ � ω þ ~γnþ<sup>1</sup> ð Þ ω (44)

�(*t*) and γ(*t*) in a short time region, in the

qð Þ *t* � *s* yð Þ*s* d*s* (47)

� � (48)

� � (49)

(46)

expressed as follows:

in the following form:

following section.

**77**

*<sup>∂</sup>* <sup>γ</sup><sup>n</sup> f g ð Þ*<sup>t</sup> <sup>=</sup>∂<sup>t</sup>* ¼ �<sup>ð</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

where the Mori coefficient δ<sup>n</sup> is equal to γn(0).

2

*<sup>=</sup>*2! � �Z2 <sup>þ</sup> *<sup>t</sup>*

Thus, they provided the following relations if several δn's are known:

Z0 ¼ ð Þ 3kB*T=*m , Z2 ¼ Z0δ1, Z4 ¼ Z0δ1ð Þ δ<sup>1</sup> þ δ<sup>2</sup> , Z6 ¼ Z0f g ð Þþ δ<sup>1</sup> þ δ<sup>2</sup> δ2δ<sup>3</sup> , …

Therefore, the problem is ascribed to the derivation of δn's. Because of a hard task in such repeating calculations, it is difficult to obtain a number of δn's. However, several applications along these procedures have been carried out [37, 38]. Instead of the method of continued-fraction described in the above, we will provide a simple but new method to obtain the mutual relation between the com-

�**(***t***) and γ(***t***)**

Here, we provide a new and useful method to solve the Langevin equation based

Let us consider a Langevin equation for an evolution function being equivalent

ð 0

qn*=*n! � �*<sup>t</sup>*

and the corresponding expansion formula for y(*t*) is written as follows:

ym*=*m! � �*<sup>t</sup>*

*t*

<sup>n</sup> qn <sup>¼</sup> <sup>q</sup>ð Þ <sup>n</sup> ð Þ <sup>0</sup>

<sup>m</sup> ym <sup>¼</sup> <sup>y</sup>ð Þ <sup>m</sup> ð Þ <sup>0</sup>

ZðÞ¼ *t* Z0 1 � *t*

bined velocity correlation function Z<sup>σ</sup>

**6. Recursion formulae for Z<sup>σ</sup>**

to (29) and (30), as follows:

on recursion process [29]. Its detail is shown below.

dyð Þ*t =*d*t* ¼

The power expansion for q(*t*) is defined as follows:

<sup>q</sup>ðÞ¼ *<sup>t</sup>* <sup>X</sup><sup>∞</sup>

<sup>y</sup>ðÞ¼ *<sup>t</sup>* <sup>X</sup><sup>∞</sup>

n¼0

m¼0

0

*t*

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

#### **4. Microscopic representation for the Z<sup>σ</sup> + (***t***) and Z<sup>σ</sup>** �**(***t***) in a molten salt**

We have already shown the microscopic expressions for Z<sup>σ</sup> + (*t*) and Z<sup>σ</sup> �(*t*) as Taylor expansion forms in a molten salt in which the inter-ionic potential between cation and anion and the corresponding pair distribution function are concerned by Koishi et al. [7]. In these combined velocity correlation functions, it can be shown that the odd power terms of the time *t* have vanishing coefficients which, it turns out, is related to the fact that any positions and their differentiations with time are uncorrelated in an ensemble average. In facts, the velocity auto-correlation function can be expressed in terms of even powers of the time *t* [32, 33].

The short-time expansion forms of Z<sup>σ</sup> + (t) and Z<sup>σ</sup> �(t) are actually shown in the following forms:

$$\mathbf{Z\_{\sigma}}^{+}(t) = \left(\mathfrak{Bk\_{B}T}/\mathfrak{m}^{+}\right) \left[\mathbf{1} - \left(t^{2}/2\right)\left(\mathfrak{a}^{0}/\mathfrak{A}\mu\right) + \left(\mathfrak{over}t^{4}\right)\right] \tag{38}$$

and

$$\mathbf{Z\_{\sigma}}^{-}(t) = \left(\mathbf{3k\_{B}T/m^{-}}\right) \left[\mathbf{1} - \left(t^{2}/2\right)\left(\mathbf{a}^{0}/3\mu\right) + \left(\mathbf{over}\,t^{4}\right)\right] \tag{39}$$

Thus, the partial conductivities for cation and anion in a molten salt are written as in the following Kubo-formulae:

$$\boldsymbol{\sigma}^{+}(\mathbf{D}\mathbf{C}) = \left(\mathbf{n}\_{0}\mathbf{e}^{2}/\mathbf{m}^{+}\right)\int\_{0}^{\infty}\left\{\mathbf{1} - \left(\mathbf{t}^{2}/2\right)\left(\mathbf{a}^{0}/3\boldsymbol{\mu}\right) + \left(\mathbf{over}\mathbf{t}^{4}\right)\right\}\mathbf{d}\mathbf{t} \tag{40}$$

and

$$\mathbf{o}^-\left(\mathbf{D}\mathbf{C}\right) = \left(\mathbf{n}\_0 \mathbf{e}^2/\mathbf{m}^-\right) \int\_0^\infty \left\{ 1 - \left(t^2/2\right) \left(\mathbf{a}^0/3\mu\right) + \left(\mathbf{over}\mathbf{t}^4\right) \right\} \mathbf{d}t \tag{41}$$

Using (14), (16), (40) and (41), we have a very interesting relation written in the following form:

$$\mathbf{1}/\ddot{\boldsymbol{\gamma}}(\mathbf{0}) = \int\_0^\infty \left\{ \mathbf{1} - \left( t^2/2 \right) (\ddot{\boldsymbol{\gamma}}(\mathbf{0}))^2 + \left( \mathbf{over} \, t^4 \right) \right\} \mathbf{d} \,\tag{42}$$

However, it is generally difficult to obtain Z<sup>σ</sup> �(t) from appropriate memory functions, by using the well-known method in statistical mechanics [33].

Under these circumstances, we explore a new method to solve Langevin Eqs. (29) and (30), in order to clarify a detailed correlation between γ(*t*) and Z<sup>σ</sup> �(*t*) within the short time region, which will be shown in later section.

#### **5. Method of continued-fraction based on Mori formulae**

Many years ago, Mori [34, 35] had generalized the Langevin equation starting from the Hamilton's canonical equation of motion in a system of a monatomic liquid with the component's mass as *m*. Along his theory, Copley and Lovesey [36] have

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

concluded that the memory function in the generalized Langevin equation could be expressed as follows:

$$\left\{\partial\{\boldsymbol{\gamma}\_{\mathbf{n}}(t)\}\right\}/\partial\!t = -\int\_{0}^{t} \boldsymbol{\gamma}\_{\mathbf{n}+1}(t-s)\boldsymbol{\gamma}\_{\mathbf{n}}(s)\,\mathrm{d}s \quad \mathbf{n} = \mathbf{1}, \mathbf{2}, \mathbf{3}, \dots \tag{43}$$

where γn(*t*) is the n-th stage memory function and the first stage memory function is equal to γ(*t*) in Eqs. (29) and (30). The Fourier-Laplace transform of the above equation provides the following continued-fraction representation,

$$
\tilde{\mathsf{Y}}\_{\mathfrak{n}}(\mathsf{o}) = -\mathsf{S}\_{\mathfrak{n}} / [\mathsf{o} + \tilde{\mathsf{Y}}\_{\mathfrak{n}+1}(\mathsf{o})] \tag{44}
$$

where the Mori coefficient δ<sup>n</sup> is equal to γn(0).

The method of Copley and Lovesey [36] was able to express the short time expansion for the velocity correlation function Z(*t*) (= < **v**i(*t*) **v**j(0)>) described as in the following form:

$$\mathbf{Z(t)} = \mathbf{Z\_0} \{ \mathbf{1} - \left( t^2 / 2! \right) \mathbf{Z\_2} + \left( t^4 / 4! \right) \mathbf{Z\_4} - \left( t^6 / 6! \right) \mathbf{Z\_6} + \dots \} \tag{45}$$

Thus, they provided the following relations if several δn's are known:

$$\mathbf{Z}\_0 = (\mathbf{3k}\_\mathsf{B} T/\mathbf{m}), \quad \mathbf{Z}\_2 = \mathbf{Z}\_0 \boldsymbol{\delta}\_1, \quad \mathbf{Z}\_4 = \mathbf{Z}\_0 \boldsymbol{\delta}\_1 (\boldsymbol{\delta}\_1 + \boldsymbol{\delta}\_2), \quad \mathbf{Z}\_6 = \mathbf{Z}\_0 \{ (\boldsymbol{\delta}\_1 + \boldsymbol{\delta}\_2) + \boldsymbol{\delta}\_2 \boldsymbol{\delta}\_3 \}, \dots \tag{46}$$

Therefore, the problem is ascribed to the derivation of δn's. Because of a hard task in such repeating calculations, it is difficult to obtain a number of δn's. However, several applications along these procedures have been carried out [37, 38].

Instead of the method of continued-fraction described in the above, we will provide a simple but new method to obtain the mutual relation between the combined velocity correlation function Z<sup>σ</sup> �(*t*) and γ(*t*) in a short time region, in the following section.

#### **6. Recursion formulae for Z<sup>σ</sup>** �**(***t***) and γ(***t***)**

Here, we provide a new and useful method to solve the Langevin equation based on recursion process [29]. Its detail is shown below.

Let us consider a Langevin equation for an evolution function being equivalent to (29) and (30), as follows:

$$\mathbf{d}\mathbf{y}(t)/\mathbf{d}t = \int\_{0}^{t} \mathbf{q}(t-s)\mathbf{y}(s)\,\mathrm{d}s \tag{47}$$

The power expansion for q(*t*) is defined as follows:

$$\mathbf{q}(t) = \sum\_{\mathbf{n}=0}^{\infty} \left( \mathbf{q}\_{\mathbf{n}} / \mathbf{n}! \right) t^{\mathbf{n}} \quad \left( \mathbf{q}\_{\mathbf{n}} = \mathbf{q}^{\left( \mathbf{n} \right)} \left( \mathbf{0} \right) \right) \tag{48}$$

and the corresponding expansion formula for y(*t*) is written as follows:

$$\mathbf{y}(t) = \sum\_{\mathbf{m}=0}^{\infty} (\mathbf{y}\_{\mathbf{m}}/\mathbf{m}!)t^{\mathbf{m}} \quad \left(\mathbf{y}\_{\mathbf{m}} = \mathbf{y}^{(\mathbf{m})}(\mathbf{0})\right) \tag{49}$$

If an appropriate memory function γ(*t*), which is valid for both cation and anion in the system, is considered and its Laplace transformation is inserted into either

Taylor expansion forms in a molten salt in which the inter-ionic potential between cation and anion and the corresponding pair distribution function are concerned by Koishi et al. [7]. In these combined velocity correlation functions, it can be shown that the odd power terms of the time *t* have vanishing coefficients which, it turns out, is related to the fact that any positions and their differentiations with time are uncorrelated in an ensemble average. In facts, the velocity auto-correlation function

+

2

2

Thus, the partial conductivities for cation and anion in a molten salt are written

(t) and Z<sup>σ</sup>

*<sup>=</sup>*<sup>2</sup> � � <sup>α</sup><sup>0</sup>*=*3<sup>μ</sup> � � <sup>þ</sup> over*<sup>t</sup>*

*<sup>=</sup>*<sup>2</sup> � � <sup>α</sup><sup>0</sup>*=*3<sup>μ</sup> � � <sup>þ</sup> over*<sup>t</sup>*

*<sup>=</sup>*<sup>2</sup> � � <sup>α</sup><sup>0</sup>*=*3<sup>μ</sup> � � <sup>þ</sup> over*<sup>t</sup>*

*<sup>=</sup>*<sup>2</sup> � � <sup>α</sup><sup>0</sup>*=*3<sup>μ</sup> � � <sup>þ</sup> over*<sup>t</sup>*

*<sup>=</sup>*<sup>2</sup> � �ð Þ <sup>~</sup>γð Þ <sup>0</sup> <sup>2</sup> <sup>þ</sup> over*<sup>t</sup>*

<sup>4</sup> � � � � (38)

*<sup>4</sup>* � � � � (39)

<sup>4</sup> � � � � d*t* (40)

<sup>4</sup> � � � � d*t* (41)

<sup>4</sup> n o � � <sup>d</sup>*<sup>t</sup>* (42)

�(t) from appropriate memory

�(*t*)

**+**

**(***t***) and Z<sup>σ</sup>**

�**(***t***) in a molten salt**

�(*t*) as

(*t*) and Z<sup>σ</sup>

�(t) are actually shown in the

+

(36) or (37), then we can get the partial AC conductivities.

We have already shown the microscopic expressions for Z<sup>σ</sup>

can be expressed in terms of even powers of the time *t* [32, 33].

þðÞ¼ *t* 3kBT*=*m<sup>þ</sup> ð Þ 1 � *t*

�ðÞ¼ *t* 3kBT*=*m� ð Þ 1 � *t*

0

0

ð 0 ∞

∞

∞

1 � *t* 2

1 � *t* 2

1 � *t* 2

functions, by using the well-known method in statistical mechanics [33]. Under these circumstances, we explore a new method to solve Langevin Eqs. (29) and (30), in order to clarify a detailed correlation between γ(*t*) and Z<sup>σ</sup>

within the short time region, which will be shown in later section.

**5. Method of continued-fraction based on Mori formulae**

Using (14), (16), (40) and (41), we have a very interesting relation written in

Many years ago, Mori [34, 35] had generalized the Langevin equation starting from the Hamilton's canonical equation of motion in a system of a monatomic liquid with the component's mass as *m*. Along his theory, Copley and Lovesey [36] have

*<sup>=</sup>*m<sup>þ</sup> � �ð

*<sup>=</sup>*m� � �<sup>ð</sup>

1*=*~γð Þ¼ 0

However, it is generally difficult to obtain Z<sup>σ</sup>

**4. Microscopic representation for the Z<sup>σ</sup>**

*Electromagnetic Field Radiation in Matter*

The short-time expansion forms of Z<sup>σ</sup>

Zσ

Zσ

as in the following Kubo-formulae:

<sup>σ</sup>þð Þ¼ DC n0e2

<sup>σ</sup>�ð Þ¼ DC n0e2

following forms:

and

and

**76**

the following form:

Putting (48) and (49) into the right hand side of Eq. (47), we have

$$\begin{aligned} \int\_{0}^{t} \mathbf{q}(t-s)\mathbf{y}(s)\mathbf{d}s &= \sum\_{\substack{\mathbf{n},\mathbf{m}=0\\\mathbf{n},\mathbf{m}=0}}^{\infty} \left(\mathbf{q}\_{\mathbf{n}}/n!\right) \left(\mathbf{y}\_{\mathbf{m}}/\mathbf{m}!\right) \int\_{0}^{t} (t-s)^{\mathbf{n}} s^{\mathbf{m}} \mathbf{d}s\\ &= \sum\_{\substack{\mathbf{n},\mathbf{m}=0\\\mathbf{n},\mathbf{m}=0}}^{\infty} \left(\mathbf{q}\_{\mathbf{n}}/n!\right) \left(\mathbf{y}\_{\mathbf{m}}/m!\right) t^{(n+m+1)} \int\_{0}^{1} (1-p)^{\mathbf{n}} p^{\mathbf{m}} \mathbf{d}p\\ &= \sum\_{\substack{\mathbf{n},\mathbf{m}=0\\\mathbf{n},\mathbf{m}=0}}^{\infty} \left(\mathbf{q}\_{\mathbf{n}}/n!\right) \left(\mathbf{y}\_{\mathbf{m}}/m!\right) t^{(n+m+1)} \mathbf{B} \left(\mathbf{n}+1,\mathbf{m}+1\right)\\ &= \sum\_{\mathbf{n},\mathbf{m}=0}^{\infty} \left(\mathbf{q}\_{\mathbf{n}}/n!\right) \left(\mathbf{y}\_{\mathbf{m}}/m!\right) t^{(n+m+1)} \left(\mathbf{n}!m!\right) / \left(\mathbf{n}+\mathbf{m}+1\right)! \\ &= \sum\_{\mathbf{n},\mathbf{m}=0}^{\infty} \left(\left\{ \left(\mathbf{q}\_{\mathbf{n}}\mathbf{y}\_{\mathbf{m}}\right) t^{(n+m+1)} / (\mathbf{n}\cdot\mathbf{m}+1) !\right\} \right. \end{aligned} \tag{50}$$

where B(n + 1, m + 1) and Γ(n + 1) mean the beta-function and the gammafunction, respectively, and

$$z\_{\mathbf{k}} = \sum\_{\mathbf{k} = \mathbf{n} + \mathbf{m} + \mathbf{1}} \mathbf{q}\_{\mathbf{n}} \mathbf{y}\_{\mathbf{m}} \tag{51}$$

q0 ¼ �~γð Þ <sup>0</sup> <sup>2</sup> (56)

h i (57)

<sup>γ</sup>ðÞ¼ *<sup>t</sup>* <sup>~</sup>γð Þ <sup>0</sup> <sup>2</sup> <sup>f</sup>ð Þ*<sup>t</sup>* (59)

fð Þ*t* d*t* ¼ 1*=*~γð Þ 0 (61)

exp ð Þ �iω*t* < **R**ið Þ*t* **R**jð Þ 0 >dt (63)

(0) > can be taken as in the follow-

<sup>2</sup> <sup>&</sup>gt; <sup>h</sup>ð Þ*<sup>t</sup>* (64)

<sup>2</sup> <sup>&</sup>gt;hðÞ¼ *<sup>t</sup>* <sup>~</sup>γð Þ <sup>0</sup> <sup>2</sup> <sup>f</sup>ð Þ*<sup>t</sup>* (65)

<sup>2</sup> <sup>&</sup>gt; <sup>¼</sup> ð Þ <sup>1</sup>*=*3μkB*<sup>T</sup>* <sup>~</sup>γð Þ <sup>0</sup> <sup>2</sup> (66)

γðÞ¼ *t* ð Þ 1*=*3μkBT < **R**ið Þ*t* **R**jð Þ 0 > (62)

fð Þ*t* d*t* (60)

� � � (58)

yðÞ¼ *t* y0 1 � *t*

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

y0 <sup>¼</sup> 3kBT*=*m� � � <sup>¼</sup> <sup>Z</sup><sup>σ</sup>

where

following form:

as follows:

[6–9],

and

ing form:

**79**

where < **R**ij

where f(0) = 1.

Therefore, we have immediately,

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

2

**7. Fluctuation dissipation theorem on the Laplace transformation of γ(***t***)**

Considering Eqs. (56) and (57), the memory function γ(*t*) can be taken as the

The Laplace transformation of (59) in the long wavelength limit is then written

ð 0

On the other hand, the memory function and its Laplace transformation are described as in the following forms, by using the fluctuation dissipation theorem

∞

<sup>~</sup>γð Þ¼ <sup>0</sup> <sup>~</sup>γð Þ <sup>0</sup> <sup>2</sup>

ð 0

~γ ωð Þ¼ ð Þ 1*=*3μkBT

The most simplest expression for < **R**i(*t*) **R**<sup>j</sup>

<sup>2</sup> > = < **R**i(0) **R**<sup>j</sup>

Putting (64) into (62) and using (59), we have

∞

ð 0 ∞

< **R**ið Þ*t* **R**jð Þ 0 > ¼ < **R**ij

This equation gives h(*t*) ∝ f(*t*), and if we take both functions are identical, then

(0)>.

γðÞ¼ *t* ð Þ 1*=*3μkBT < **R**ij

< **R**ij

*<sup>=</sup>*2! � �~γð Þ <sup>0</sup> <sup>2</sup> <sup>þ</sup> …

�ð Þ� 0 Z0

On the other hand, the left hand side of Eq. (47) is equal to the following formulae:

$$\mathbf{y}'(t) = \left\{ \sum\_{\mathbf{k}=0}^{\infty} (\mathbf{y}\_{\mathbf{k}}/\mathbf{k}!)t^{\mathbf{k}} \right\}' = \sum\_{\mathbf{k}=0}^{\infty} (\mathbf{y}\_{\mathbf{k}+1}/\mathbf{k}!)t^{\mathbf{k}} \tag{52}$$

Compare both expressions (50) and (52), we can get the recursion formulae as follows,

$$\mathbf{y}\_1 = \mathbf{0}; \quad \mathbf{y}\_{k+1} = \left\{ \sum\_{\mathbf{m}=0}^{k-1} \left( \mathbf{q}\_{k-\mathbf{m}-1} \mathbf{y}\_\mathbf{m} \right) \right\} \quad (\mathbf{k} = 1, 2, \dots) \tag{53}$$

Therefore, Eq. (49) is practically expressed in the following series:

$$\begin{aligned} \mathbf{y}\_1 &= \mathbf{0}; \quad \mathbf{y}\_2 = \mathbf{y}\_0 \mathbf{q}\_0; \quad \mathbf{y}\_3 = \mathbf{y}\_0 \mathbf{q}\_1 + \mathbf{y}\_1 \mathbf{q}\_0 = \mathbf{y}\_0 \mathbf{q}\_1; \\ \mathbf{y}\_4 &= \mathbf{q}\_2 \mathbf{y}\_0 + \mathbf{q}\_1 \mathbf{y}\_1 + \mathbf{q}\_0 \mathbf{y}\_2 = \mathbf{y}\_0 \left(\mathbf{q}\_0^2 + \mathbf{q}\_2\right); \\ \mathbf{y}\_5 &= \mathbf{y}\_0 \mathbf{q}\_3 + \mathbf{y}\_1 \mathbf{q}\_2 + \mathbf{y}\_2 \mathbf{q}\_1 + \mathbf{y}\_3 \mathbf{q}\_0 = \mathbf{y}\_0 \left(2 \mathbf{q}\_0 \mathbf{q}\_1 + \mathbf{q}\_3\right) \end{aligned} \tag{54}$$

and so on.

And vice versa, qn's are expressed as follows:

$$\begin{aligned} \mathbf{q}\_1 &= \left(\mathbf{1}/\mathbf{y}\_0\right)\mathbf{y}\_3; \quad \mathbf{q}\_2 = \left(\mathbf{y}\_4/\mathbf{y}\_0\right) - \mathbf{q}\_0\left(\mathbf{y}\_2/\mathbf{y}\_0\right) = \left(\mathbf{1}/\mathbf{y}\_0\right)\left\{\mathbf{y}\_4 - \mathbf{q}\_0\mathbf{y}\_2\right\}; \\ \mathbf{q}\_3 &= \left(\mathbf{1}/\mathbf{y}\_0\right)\left\{\mathbf{y}\_5 - \left(\mathbf{y}\_2\mathbf{y}\_3/\mathbf{y}\_0\right) - \mathbf{q}\_0\mathbf{y}\_3\right\} \end{aligned} \tag{55}$$

and so on.

This method can be immediately applicable in the following way, comparing with Eqs. (38) and (39).

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

$$\mathbf{q}\_0 = -\ddot{\mathbf{y}}(\mathbf{0})^2 \tag{56}$$

$$\mathbf{y}(t) = \mathbf{y}\_0 \left[ \mathbf{1} - \left( t^2 / 2! \right) \hat{\mathbf{y}}(\mathbf{0})^2 + \dots \right] \tag{57}$$

where

Putting (48) and (49) into the right hand side of Eq. (47), we have

0

ð Þ nþmþ1 ð 0

ð Þ <sup>n</sup>þmþ<sup>1</sup> *<sup>=</sup>*ð Þ <sup>n</sup> <sup>þ</sup> <sup>m</sup> <sup>þ</sup> <sup>1</sup> ! n o

> <sup>0</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> k¼0

ykþ<sup>1</sup>*=*k! � �*<sup>t</sup>*

<sup>2</sup> <sup>þ</sup> q2 � �;

� �

� � y4 � q0y2

� �g;

where B(n + 1, m + 1) and Γ(n + 1) mean the beta-function and the gamma-

k¼nþmþ1

Compare both expressions (50) and (52), we can get the recursion formulae as

qk�m�<sup>1</sup>ym � � ( )

y1 ¼ 0; y2 ¼ y0q0; y3 ¼ y0q1 þ y1q0 ¼ y0 q1;

y5 ¼ y0q3 þ y1q2 þ y2q1 þ y3q0 ¼ y0 2q0 q1 þ q3

This method can be immediately applicable in the following way, comparing

� � <sup>¼</sup> <sup>1</sup>*=*y0

On the other hand, the left hand side of Eq. (47) is equal to the following

*<sup>z</sup>*<sup>k</sup> <sup>¼</sup> <sup>X</sup>

yk*=*k! � �*<sup>t</sup>* k

k�1

m¼0

Therefore, Eq. (49) is practically expressed in the following series:

y4 ¼ q2 y0 þ q1 y1 þ q0 y2 ¼ y0 q0

� � � q0 y2*=*y0

� � � q0 y3 � �

( )

*t*

ð Þ *<sup>t</sup>* � *<sup>s</sup>* <sup>n</sup> *s* md*s*

1

ð Þ <sup>n</sup>þmþ<sup>1</sup> B nð Þ <sup>þ</sup> 1, m <sup>þ</sup> <sup>1</sup>

ð Þ <sup>n</sup>þmþ<sup>1</sup> ð Þ <sup>n</sup>!m! *<sup>=</sup>*ð Þ <sup>n</sup> <sup>þ</sup> <sup>m</sup> <sup>þ</sup> <sup>1</sup> !

ð Þ <sup>1</sup> � *<sup>p</sup>* <sup>n</sup> *<sup>p</sup>*<sup>m</sup> <sup>d</sup>*<sup>p</sup>*

ð Þ <sup>n</sup>þmþ<sup>1</sup> f g <sup>Γ</sup>ð Þ <sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> *<sup>=</sup>*f g <sup>Γ</sup>ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>m</sup> <sup>þ</sup> <sup>2</sup>

<sup>¼</sup> <sup>X</sup><sup>∞</sup> k¼1

qnym (51)

ð Þ *z*k*=*k! *t* k

<sup>k</sup> (52)

ð Þ k ¼ 1, 2, … (53)

(50)

(54)

(55)

qn*=*n! � � ym*=*m! � �<sup>ð</sup>

qn*=*n! � � ym*=*m! � �*<sup>t</sup>*

qn*=n*! � � ym*=m*! � �*<sup>t</sup>*

qn*=n*! � � ym*=m*! � �*<sup>t</sup>*

qn*=*n! � � ym*=*m! � �*<sup>t</sup>*

qnym � � *t*

ð 0 *t*

<sup>q</sup>ð Þ *<sup>t</sup>* � *<sup>s</sup>* <sup>y</sup>ð Þ*<sup>s</sup>* <sup>d</sup>*<sup>s</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

n, m¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> n, m¼0

*Electromagnetic Field Radiation in Matter*

<sup>¼</sup> <sup>X</sup><sup>∞</sup> n, m¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> n, m¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> n, m¼0

<sup>¼</sup> <sup>X</sup><sup>∞</sup> n, m¼0

y0

ðÞ¼ *<sup>t</sup>* <sup>X</sup><sup>∞</sup>

y1 <sup>¼</sup> 0; ykþ<sup>1</sup> <sup>¼</sup> <sup>X</sup>

And vice versa, qn's are expressed as follows:

� �y3; q2 <sup>¼</sup> y4*=*y0

� � y5 � y2y3*=y*<sup>0</sup>

k¼0

function, respectively, and

formulae:

follows,

and so on.

q1 ¼ 1*=*y0

q3 ¼ 1*=*y0

with Eqs. (38) and (39).

and so on.

**78**

$$\mathbf{y}\_0 = \left(\mathbf{3k\_BT/m^\pm}\right) \left(= \mathbf{Z\_{\sigma}}^\pm(\mathbf{0}) \equiv \mathbf{Z\_0}^\pm\right) \tag{58}$$

#### **7. Fluctuation dissipation theorem on the Laplace transformation of γ(***t***)**

Considering Eqs. (56) and (57), the memory function γ(*t*) can be taken as the following form:

$$\boldsymbol{\gamma}(t) = \boldsymbol{\tilde{\gamma}}(\mathbf{0})^2 \,\mathrm{f}(t) \tag{59}$$

where f(0) = 1.

The Laplace transformation of (59) in the long wavelength limit is then written as follows:

$$
\tilde{\boldsymbol{\eta}}(\mathbf{0}) = \tilde{\boldsymbol{\eta}}(\mathbf{0})^2 \int\_0^\infty \mathbf{f}(t) \,\mathrm{d}t \tag{60}
$$

Therefore, we have immediately,

$$\int\_{0}^{\infty} \mathbf{f}(t) \, \mathbf{d}t = \mathbf{1}/\tilde{\gamma}(\mathbf{0})\tag{61}$$

On the other hand, the memory function and its Laplace transformation are described as in the following forms, by using the fluctuation dissipation theorem [6–9],

$$\gamma(t) = (\mathbf{1}/\Im \mu \mathbf{k}\_\mathbf{B} \mathbf{T}) < \mathbf{R}\_i(t) \, \mathbf{R}\_j(\mathbf{0}) > \tag{62}$$

and

$$\ddot{\boldsymbol{\gamma}}(\boldsymbol{\alpha}) = (\mathbf{1}/3\mu\mathbf{k}\_{\rm B}\mathbf{T})\int\_{0}^{\infty} \exp\left(-\text{i}\boldsymbol{\alpha}t\right) < \mathbf{R}\_{\rm i}(t)\,\mathbf{R}\_{\rm j}(\mathbf{0}) > \mathbf{dt} \tag{63}$$

The most simplest expression for < **R**i(*t*) **R**<sup>j</sup> (0) > can be taken as in the following form:

$$<\mathbf{R}\_{\mathbf{i}}(t)\,\mathbf{R}\_{\mathbf{j}}(0)> = <\mathbf{R}\_{\mathbf{i}}\,^2 > \mathbf{h}(t) \tag{64}$$

where < **R**ij <sup>2</sup> > = < **R**i(0) **R**<sup>j</sup> (0)>. Putting (64) into (62) and using (59), we have

$$\boldsymbol{\gamma}(t) = (\mathbf{1}/3\mu \mathbf{k}\_{\rm B} \mathbf{T}) < \mathbf{R}\_{\rm i}^{2} > \mathbf{h}(t) = \boldsymbol{\bar{\gamma}}(\mathbf{0})^{2} \mathbf{f}(t) \tag{65}$$

This equation gives h(*t*) ∝ f(*t*), and if we take both functions are identical, then

$$<\mathbf{R}\_{\rm i}\textsubur} \, ^{2}\text{>} \, = \, (\mathbf{1}/\text{3\mu k}\_{\rm B}T)\ddot{\gamma}(\mathbf{0})^{2} \tag{66}$$

Putting this relation into (62), we obtain again the relation (59), which indicates that the assumption, h(*t*) = f(*t*), is exactly justified.

On the other hand, we can get an agreement if we use even and odd serial

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

2 . Therefore, the method utilizing the odd and even power series has a more rapid

least mean square method as so-called Levenberg-Marquart method [44].

The fitting parameters, which are equal to ym's, are obtained by the non-linear

The primary value in this non-linear least mean square method is inferred by

It is inevitable that the coefficients of ym's (m = 3, 4, … ) are slightly variable because of the termination effect in the expansion form. But we have no difficulty

By using these obtained ym's, it is immediately possible to obtain qn's. And thereafter we can get a fitted curve indicating the curve of γ(*t*) within a short time region. In this figure, the fitting curve of γ(*t*) is obtained for the time range of

It is therefore emphasized that the utilization of odd terms within the short time region is necessary for the derivation of qn's from the ym's obtained by MD

For references, several analytic functional forms describing γ(*t*) can also be given. The following two-types of functional forms are known as model functions

However, an inevitable fact is that the theoretical memory function must be an expansion form of only even powers of the time, even though it is numerically close

ai exp �ð Þ <sup>π</sup>*=*<sup>4</sup> bi~γð Þ <sup>0</sup> <sup>2</sup>

a2 þ ð Þ b1b2

1*=*2 a3

Is it possible to get a model function to fit the obtained curve of γ(*t*) by MD simulation? To answer this question, we have carried out the fitting procedure by using a combination of poly-Gaussian functions [29]. Practically, the following form composed of three kinds of Gaussian functions is good enough to reproduce the obtained curve of γ(*t*) under the condition of Eq. (61) for molten NaCl at

ð Þ **<sup>a</sup>** � **<sup>1</sup>** <sup>γ</sup>ðÞ¼ *<sup>t</sup>* <sup>~</sup>γð Þ <sup>0</sup> <sup>2</sup> sechf g ð Þ <sup>π</sup>*=*<sup>2</sup> <sup>~</sup>γð Þ <sup>0</sup> *<sup>t</sup>* (68)

*t* <sup>2</sup> <sup>n</sup> �o (69)

*t* <sup>2</sup> <sup>n</sup> �o (70)

<sup>1</sup>*=*<sup>2</sup> <sup>¼</sup> 1 (71)

exp{�~γ(0)t} agrees, at least within the

exp �ð Þ <sup>π</sup>*=*<sup>4</sup> <sup>~</sup>γð Þ <sup>0</sup> <sup>2</sup>

. This fact encourages us that the combined velocity correla-

�(*t*), in comparison with the method utilizing only

�(*t*) up to *t*

15.

�(*t*) in molten systems must be practically analyzed in terms of

powers over *t*

tion functions Z<sup>σ</sup>

even power series.

simulation.

1100 K,

where

ai ¼ 1, and bð Þ 2b3

X 3

i¼1

**81**

<sup>2</sup> up to *t* 9

convergence for obtaining Z<sup>σ</sup>

utilization of simplex method.

even and odd powers of the time over *t*

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

to elucidate γ(*t*) in an appropriate short time range.

<sup>0</sup> <sup>&</sup>lt; <sup>t</sup> <sup>&</sup>lt; 5.0 � <sup>10</sup>�<sup>14</sup> seconds, from the expansion form of Z<sup>σ</sup>

being suitable for the auto-velocity correlation functions in liquids.

to the exponentially decaying function which includes the odd powers.

3

i¼1

a1 þ ð Þ b3b1

1*=*2

Using (70) and (71), we could reproduce the obtained curve of γ(*t*) by MD simulation in molten NaCl at 1100 K. And these are approximated to as {a1 = 0.2, a2 = 0.3 and a3 = 0.5}, which values correspond to the existing fractions of each short

n o*=*ð Þ b1b2b3

ð Þ **<sup>a</sup>** � **<sup>2</sup>** <sup>γ</sup>ðÞ¼ *<sup>t</sup>* <sup>~</sup>γð Þ <sup>0</sup> <sup>2</sup>

<sup>γ</sup>ðÞ¼ *<sup>t</sup>* <sup>~</sup>γð Þ <sup>0</sup> <sup>2</sup><sup>X</sup>

1*=*2

The γ(*t*) is expressed by the form of ~γ(0)<sup>2</sup>

short time region, with that of MD simulation.

Therefore, the general form for the memory function γ(*t*) is always written in the form of Eq. (59).

#### **8. Former theories of velocity correlation functions in molten salts**

Various analytic forms for memory functions were proposed [7, 8, 12, 39–43] and all these functions are qualitatively useful to obtain the combined velocity correlation functions, although some of these theories cannot predict the result obtained by MD simulation.

For example, if we use an approximate form for the memory function as

$$\begin{aligned} \mathbf{Z}\_{\sigma}^{\pm}(\mathbf{t}) &= \left(\mathbf{3k}\_{\mathbf{B}} \mathbf{T}/\mathbf{m}^{\pm}\right) \exp\left\{-\underset{\mathbf{y}}{\uparrow}(\mathbf{0})t/2\right\} \left[\cos\left(\sqrt{\mathbf{3}}\,\widetilde{\chi}\,(\mathbf{0})t/2\right)\right. \\ &\left. + \left\{\left(\widetilde{\chi}\,(\mathbf{0})/2\right)/\left(\sqrt{\mathbf{3}}\,\widetilde{\chi}\,(\mathbf{0})/2\right)\right\} \sin\left(\sqrt{\mathbf{3}}\,\widetilde{\chi}\,(\mathbf{0})t/2\right)\right] \\ &= \left(\mathbf{3k}\_{\mathbf{B}} \mathbf{T}/\mathbf{m}^{\pm}\right) \left[\mathbf{1} - \left(t^{2}/2!\right)\widetilde{\chi}\,(\mathbf{0})^{2} - \left(t^{3}/3!\right)\mathbf{3}\,\widetilde{\chi}\,(\mathbf{0})^{3}/8 + \left(\mathbf{over}\,t^{4}\right)\right] \tag{67} \end{aligned}$$

As shown in our previous results [29], the calculated Z<sup>σ</sup> + (*t*) for cation by using Eq. (67) agrees with that of MD simulation [7] qualitatively and semiquantitatively.

However, the time expansion forms of Z<sup>σ</sup> �(*t*) are essentially equal to the even powers expansion forms, which contradicts to the expression of (67). It is, therefore, necessary to seek an appropriate memory function which can be expanded as the even powers of the time t, even though the obtained result is numerically very close to the expression of <sup>γ</sup>(*t*) = <sup>~</sup>γ(0)<sup>2</sup> exp{�~γ(0)*t*}.

#### **9. Application of recursion method for the derivation of γ(***t***) from Z<sup>σ</sup>** �**(***t***)**

So far, we are successful to obtain the mutual relation between γ(*t*) and Z<sup>σ</sup> �(*t*) within a short time region to satisfy the Langevin equations in molten salts.

There are several works to obtain the auto-velocity correlation functions in monatomic liquids from appropriate memory functions γ(t) [39, 41, 42].

However, it is not known what sorts of model functions are suitable for the combined velocity correlation function Z<sup>σ</sup> �(*t*) until the present time. In order to elucidate this question, we will try to calculate the coefficients *y*m's of simulated Zσ �(*t*) of molten NaCl in a short time region, and from these the corresponding γ(*t*) will be obtained.

Previously we have already carried out the MD simulation for the combined velocity correlation functions Z<sup>σ</sup> �(*t*) [7].

We try two types of power expansion forms in order to fit the combined correlation functions Z<sup>σ</sup> �(*t*) by MD simulation. One is an arbitrary expansion form given by the even power series of the time *t*, which is theoretically exact for the combined correlation function. Another one is the series of even and odd powers for higher order terms over *t* <sup>2</sup> one. Practical reason for the use of latter case will be given below.

In the case of the utilization of only even powers, it was quite difficult to get to the simulated Z<sup>σ</sup> �(*t*) even if the power's number is taken up to 36th order of time *t*.

#### *Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

On the other hand, we can get an agreement if we use even and odd serial powers over *t* <sup>2</sup> up to *t* 9 . This fact encourages us that the combined velocity correlation functions Z<sup>σ</sup> �(*t*) in molten systems must be practically analyzed in terms of even and odd powers of the time over *t* 2 .

Therefore, the method utilizing the odd and even power series has a more rapid convergence for obtaining Z<sup>σ</sup> �(*t*), in comparison with the method utilizing only even power series.

The fitting parameters, which are equal to ym's, are obtained by the non-linear least mean square method as so-called Levenberg-Marquart method [44].

The primary value in this non-linear least mean square method is inferred by utilization of simplex method.

It is inevitable that the coefficients of ym's (m = 3, 4, … ) are slightly variable because of the termination effect in the expansion form. But we have no difficulty to elucidate γ(*t*) in an appropriate short time range.

By using these obtained ym's, it is immediately possible to obtain qn's. And thereafter we can get a fitted curve indicating the curve of γ(*t*) within a short time region. In this figure, the fitting curve of γ(*t*) is obtained for the time range of <sup>0</sup> <sup>&</sup>lt; <sup>t</sup> <sup>&</sup>lt; 5.0 � <sup>10</sup>�<sup>14</sup> seconds, from the expansion form of Z<sup>σ</sup> �(*t*) up to *t* 15.

It is therefore emphasized that the utilization of odd terms within the short time region is necessary for the derivation of qn's from the ym's obtained by MD simulation.

For references, several analytic functional forms describing γ(*t*) can also be given. The following two-types of functional forms are known as model functions being suitable for the auto-velocity correlation functions in liquids.

$$\mathbf{r}(\mathbf{a} - \mathbf{1}) \quad \mathbf{\tilde{y}}(t) = \mathbf{\tilde{y}}(\mathbf{0})^2 \operatorname{sech}\{ (\mathfrak{m}/2)\mathbf{\tilde{y}}(\mathbf{0})t \} \tag{68}$$

$$\gamma(\mathbf{a} - \mathbf{2}) \quad \gamma(t) = \tilde{\gamma}(\mathbf{0})^2 \exp\left\{ - \left( \pi/4 \right) \tilde{\gamma}(\mathbf{0})^2 t^2 \right\} \tag{69}$$

The γ(*t*) is expressed by the form of ~γ(0)<sup>2</sup> exp{�~γ(0)t} agrees, at least within the short time region, with that of MD simulation.

However, an inevitable fact is that the theoretical memory function must be an expansion form of only even powers of the time, even though it is numerically close to the exponentially decaying function which includes the odd powers.

Is it possible to get a model function to fit the obtained curve of γ(*t*) by MD simulation? To answer this question, we have carried out the fitting procedure by using a combination of poly-Gaussian functions [29]. Practically, the following form composed of three kinds of Gaussian functions is good enough to reproduce the obtained curve of γ(*t*) under the condition of Eq. (61) for molten NaCl at 1100 K,

$$\gamma(t) = \tilde{\gamma}(\mathbf{0})^2 \sum\_{i=1}^3 \mathbf{a\_i} \exp\left\{- (\pi/4) \mathbf{b\_i} \tilde{\gamma}(\mathbf{0})^2 t^2\right\} \tag{70}$$

where

Putting this relation into (62), we obtain again the relation (59), which indicates

Therefore, the general form for the memory function γ(*t*) is always written in

Various analytic forms for memory functions were proposed [7, 8, 12, 39–43] and all these functions are qualitatively useful to obtain the combined velocity correlation functions, although some of these theories cannot predict the result

� ð Þ <sup>0</sup> *<sup>t</sup>=*<sup>2</sup> � � cos <sup>√</sup><sup>3</sup> <sup>γ</sup>

sin √3 γ

<sup>4</sup> � � h i

3 *=*3! � �3 γ � ð Þ 0 *t=*2

*=*8 þ over*t*

(*t*) for cation by using

(67)

�(*t*)

h � �

� ð Þ 0 *t=*2 � �i

+

�(*t*) are essentially equal to the even

�(*t*) until the present time. In order to

� ð Þ <sup>0</sup> <sup>3</sup>

**8. Former theories of velocity correlation functions in molten salts**

For example, if we use an approximate form for the memory function as

� ð Þ 0 *=*2

powers expansion forms, which contradicts to the expression of (67). It is, therefore, necessary to seek an appropriate memory function which can be expanded as the even powers of the time t, even though the obtained result is numerically very

So far, we are successful to obtain the mutual relation between γ(*t*) and Z<sup>σ</sup>

However, it is not known what sorts of model functions are suitable for the

�(*t*) of molten NaCl in a short time region, and from these the corresponding γ(*t*)

We try two types of power expansion forms in order to fit the combined correla-

In the case of the utilization of only even powers, it was quite difficult to get to

�(*t*) by MD simulation. One is an arbitrary expansion form given by

<sup>2</sup> one. Practical reason for the use of latter case will be given below.

�(*t*) even if the power's number is taken up to 36th order of time *t*.

Previously we have already carried out the MD simulation for the combined

the even power series of the time *t*, which is theoretically exact for the combined correlation function. Another one is the series of even and odd powers for higher

elucidate this question, we will try to calculate the coefficients *y*m's of simulated

within a short time region to satisfy the Langevin equations in molten salts. There are several works to obtain the auto-velocity correlation functions in

monatomic liquids from appropriate memory functions γ(t) [39, 41, 42].

�(*t*) [7].

� ð Þ <sup>0</sup> <sup>2</sup> � *<sup>t</sup>*

that the assumption, h(*t*) = f(*t*), is exactly justified.

*Electromagnetic Field Radiation in Matter*

�ðÞ¼ <sup>t</sup> 3kBT*=*m� � � exp � <sup>γ</sup>

� ð Þ <sup>0</sup> *<sup>=</sup>*<sup>2</sup> � �*<sup>=</sup>* <sup>√</sup><sup>3</sup> <sup>γ</sup>

n o � �

2 *=*2! � � γ

Eq. (67) agrees with that of MD simulation [7] qualitatively and semi-

**9. Application of recursion method for the derivation of γ(***t***)**

As shown in our previous results [29], the calculated Z<sup>σ</sup>

þ γ

<sup>¼</sup> 3kBT*=*m� � � <sup>1</sup> � *<sup>t</sup>*

However, the time expansion forms of Z<sup>σ</sup>

close to the expression of <sup>γ</sup>(*t*) = <sup>~</sup>γ(0)<sup>2</sup> exp{�~γ(0)*t*}.

the form of Eq. (59).

obtained by MD simulation.

Zσ

quantitatively.

**from Z<sup>σ</sup>**

will be obtained.

tion functions Z<sup>σ</sup>

order terms over *t*

the simulated Z<sup>σ</sup>

Zσ

**80**

�**(***t***)**

combined velocity correlation function Z<sup>σ</sup>

velocity correlation functions Z<sup>σ</sup>

$$\sum\_{i=1}^{3} \mathbf{a}\_{i} = \mathbf{1}, \quad \text{and} \left\{ \left( \mathbf{b}\_{2} \mathbf{b}\_{3} \right)^{1/2} \mathbf{a}\_{1} + \left( \mathbf{b}\_{3} \mathbf{b}\_{1} \right)^{1/2} \mathbf{a}\_{2} + \left( \mathbf{b}\_{1} \mathbf{b}\_{2} \right)^{1/2} \mathbf{a}\_{3} \right\} / \left( \mathbf{b}\_{1} \mathbf{b}\_{2} \mathbf{b}\_{3} \right)^{1/2} = \mathbf{1} \tag{71}$$

Using (70) and (71), we could reproduce the obtained curve of γ(*t*) by MD simulation in molten NaCl at 1100 K. And these are approximated to as {a1 = 0.2, a2 = 0.3 and a3 = 0.5}, which values correspond to the existing fractions of each short range configuration i = 1, i = 2, and i = 3, respectively. And values of {b1 = 97.50, b2 = 6.52, and b3 = 0.38} correspond to their structural decaying speeds, respectively.

According to **Figure 2a**, the averaged nearest neighbor's number around the Na<sup>+</sup> ion is equal to 5.0. Any local coordination numbers around a Na<sup>+</sup> are possible to be

It is possible to consider that stable short range configurations seem to be two types. One is the case of cubic structure-type configuration having with the coordination of 6 chlorine ions around the centered sodium ion as shown in **Figure 3a**, which is similar to the solid type configuration with a sort of lengthen fluctuation of

The other is close to a tetrahedral coordination of chlorine ions around the

For simplicity, here we assume that the decaying or releasing of these two types of rather stable short range configurations is nearly the same, then the combined

On the other hand, there exist two types of rather unstable short range configurations as shown in **Figure 4a** and **b**, respectively, in which the surrounded Cl� ions

Totally, the local configuration types of Cl� ions around a centered Na<sup>+</sup> ion are

As shown in the previous section, the combined velocity correlation functions

forms and the corresponding memory function includes the terms of odd and even

tions for cation and anion are identical and Eq. (15) is automatically justified by the

In conclusion, we have newly obtained the mutual relation between the memory

In addition, it is emphasized that the γ(*t*) obtained from the simulated Z<sup>σ</sup>

recursion method to solve the Langevin equation and it may be applicable for

Hereafter, we will consider the strong electrolytic solution composed of N+ cations, N� anions and X water molecules in a volume VM. For simplicity, we take that N+ = N� = N and ions charges are equal to z+ <sup>=</sup> � <sup>z</sup>� = z. Then the number densities of ions and water molecules are equal to n<sup>+</sup> = n� = n = N/VM and *x* = X/*V*M, respectively. And furthermore we assume that the dissociation of electrolyte is

In the present system, a generalized Langevin equation for the cation (or anion)

�ð Þ t' d*t*

where γ�(*t*) is the memory function incorporating with the friction force acting on its cation (or anion). **F** is the induced internal field yielded by the change of ion's

<sup>2</sup> in their expansion

�(*t*), by using a

<sup>0</sup> þ z�eð Þþ **E** þ **F** z�e**ε**ið Þ*t* (72)

�(*t*). This fact means that the memory func-

+ (*t*)

**10. Discussion and conclusions in the case of molten salts**

�(*t*) can be analyzed in terms of odd and even powers over *t*

present new type of experiment such as computer simulation.

function γ(*t*) and the combined velocity correlation function Z<sup>σ</sup>

**11. Generalized Langevin equation in electrolytic solution**

finding a suitable memory function in all liquid matters.

4, 5, and 6 under the condition of density fluctuation in the liquid state.

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

the interionic distance.

listed in **Table 1**.

powers in its expansion form.

agrees completely with that from Z<sup>σ</sup>

complete under the condition of N ≪ *X*.

�ð Þ*t =*d*t* ¼ �m�

m�d**v**<sup>i</sup>

**83**

i under an external field **E** is written as follows:

ð 0 *t*

γ� *t* � *t* <sup>0</sup> ð Þ**v**<sup>i</sup>

Zσ

centered sodium ion as shown in **Figure 3b**.

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

around a Na+ ion are spatially asymmetric.

configurational decaying is given by i = 3 and b3.

#### **Figure 3.**

*(a) A stable short range configuration of 6 Cl ions around a Na+ ion. (b) Another stable short range configuration of 4 Cl ions around a Na<sup>+</sup> ion.*

#### **Figure 4.**

*(a) A rather unstable short range configuration of 5 Cl ions around a Na+ ion. (b) Another unstable short range configuration of 4 Cl ions around a Na<sup>+</sup> ion.*


#### **Table 1.**

*Local configuration types of Cl ions around a centered Na+ ion.*

#### *Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

According to **Figure 2a**, the averaged nearest neighbor's number around the Na<sup>+</sup> ion is equal to 5.0. Any local coordination numbers around a Na<sup>+</sup> are possible to be 4, 5, and 6 under the condition of density fluctuation in the liquid state.

It is possible to consider that stable short range configurations seem to be two types. One is the case of cubic structure-type configuration having with the coordination of 6 chlorine ions around the centered sodium ion as shown in **Figure 3a**, which is similar to the solid type configuration with a sort of lengthen fluctuation of the interionic distance.

The other is close to a tetrahedral coordination of chlorine ions around the centered sodium ion as shown in **Figure 3b**.

For simplicity, here we assume that the decaying or releasing of these two types of rather stable short range configurations is nearly the same, then the combined configurational decaying is given by i = 3 and b3.

On the other hand, there exist two types of rather unstable short range configurations as shown in **Figure 4a** and **b**, respectively, in which the surrounded Cl� ions around a Na+ ion are spatially asymmetric.

Totally, the local configuration types of Cl� ions around a centered Na<sup>+</sup> ion are listed in **Table 1**.

#### **10. Discussion and conclusions in the case of molten salts**

As shown in the previous section, the combined velocity correlation functions Zσ �(*t*) can be analyzed in terms of odd and even powers over *t* <sup>2</sup> in their expansion forms and the corresponding memory function includes the terms of odd and even powers in its expansion form.

In addition, it is emphasized that the γ(*t*) obtained from the simulated Z<sup>σ</sup> + (*t*) agrees completely with that from Z<sup>σ</sup> �(*t*). This fact means that the memory functions for cation and anion are identical and Eq. (15) is automatically justified by the present new type of experiment such as computer simulation.

In conclusion, we have newly obtained the mutual relation between the memory function γ(*t*) and the combined velocity correlation function Z<sup>σ</sup> �(*t*), by using a recursion method to solve the Langevin equation and it may be applicable for finding a suitable memory function in all liquid matters.

#### **11. Generalized Langevin equation in electrolytic solution**

Hereafter, we will consider the strong electrolytic solution composed of N+ cations, N� anions and X water molecules in a volume VM. For simplicity, we take that N+ = N� = N and ions charges are equal to z+ <sup>=</sup> � <sup>z</sup>� = z. Then the number densities of ions and water molecules are equal to n<sup>+</sup> = n� = n = N/VM and *x* = X/*V*M, respectively. And furthermore we assume that the dissociation of electrolyte is complete under the condition of N ≪ *X*.

In the present system, a generalized Langevin equation for the cation (or anion) i under an external field **E** is written as follows:

$$\mathbf{m}^{\pm} \mathbf{d} \mathbf{v}\_{i}^{\pm}(t)/\mathbf{d}t = -\mathbf{m}^{\pm} \int\_{0}^{t} \boldsymbol{\gamma}^{\pm}(t - t') \mathbf{v}\_{i}^{\pm}(\mathbf{t'}) \mathbf{d}t' + \mathbf{z}^{\pm} \mathbf{e}(\mathbf{E} + \mathbf{F}) + \mathbf{z}^{\pm} \mathbf{e} \mathbf{e}\_{i}(t) \tag{72}$$

where γ�(*t*) is the memory function incorporating with the friction force acting on its cation (or anion). **F** is the induced internal field yielded by the change of ion's

range configuration i = 1, i = 2, and i = 3, respectively. And values of {b1 = 97.50,

*(a) A stable short range configuration of 6 Cl ions around a Na+ ion. (b) Another stable short range*

**Configuration type**

**Coordination of 6 Cl ions**

**Existing probability, ai**

**Coordination of 5 Cl ions**

*(a) A rather unstable short range configuration of 5 Cl ions around a Na+ ion. (b) Another unstable short*

i = 1 0.2 0.2 i = 2 0.3 0.3 i = 3 0.15 0.35 0.5

b2 = 6.52, and b3 = 0.38} correspond to their structural decaying speeds,

respectively.

**Figure 3.**

**Degree of stability**

**Figure 4.**

**Table 1.**

**82**

**Coordination of 4 Cl ions**

*range configuration of 4 Cl ions around a Na<sup>+</sup> ion.*

*Local configuration types of Cl ions around a centered Na+ ion.*

*configuration of 4 Cl ions around a Na<sup>+</sup> ion.*

*Electromagnetic Field Radiation in Matter*

distribution which is resulted from the applying external field **E**, and **ε**i(*t*) is the random fluctuating force acting on the ion i.

According to Berne and Rice [16], the internal field **F** induced by the asymmetric ion's distribution in an ionic melt is expressed as follows:

$$\mathbf{F} = -\boldsymbol{\mathfrak{G}} \cdot \mathbf{E} = -4\pi \mathbf{n} / 3 \mathbf{k}\_{\mathbf{B}} \mathbf{T} \int\_{\rm d}^{\rm \infty} \left\{ \mathbf{d} \boldsymbol{\Phi}^{+ -}(r) / \mathbf{dr} \right\} \mathbf{g}^{+ -}(r) \, r^{3} \mathbf{dr} \cdot \mathbf{E} \tag{73}$$

where ɡ+�(*r*) is the pair distribution function between cation and anion, and d is the hard-core contact distance between cation and anion. Hereafter, we will use this result.

If we take **E** = **E**<sup>0</sup> e�iω*<sup>t</sup>* , then the ensemble average for **v**<sup>i</sup> �(*t*) is written in the following form:

$$<\mathbf{v}\_{l}^{\pm}(t)> = \text{Re } \mu^{\pm}(\mathbf{o}) \mathbf{z}^{\pm} \mathbf{e} \to \mathbf{e}^{-\text{i}\alpha t} \tag{74}$$

Starting from Eq. (80) with an infinitesimal external field **E**, it is also easily obtainable the following Kubo-Green formulae for the partial conductivities σ<sup>+</sup> and

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

ð 0

ð 0 ∞ <**j**

�ð Þ*t* and **j**ðÞ¼ *t* **j**

In order to obtain the partial conductivities based on Eqs. (81) and (82), it is

In the next section, we will discuss velocity correlation functions described in terms of inter-molecular (or ionic) potentials and pair distribution functions in

**13. Short time expansion of velocity correlation functions in electrolytic**

þð Þ 0 > þ *t*

2 *=*2!<**v**<sup>i</sup>

*<sup>=</sup>*2*m*� <sup>þ</sup><sup>X</sup>

� � **r**<sup>q</sup> <sup>w</sup><sup>j</sup> � � <sup>þ</sup><sup>X</sup>

— � **<sup>r</sup>**<sup>l</sup> � ð Þþj

<sup>ϕ</sup>�<sup>w</sup> <sup>j</sup>**r**<sup>k</sup>

Since the water molecule is not spherical in its molecular configuration, it is

X

q¼1 **p**q

> N X þ, N�

> > i, k

w. However, we tentatively assume that its

ϕþ� j**r**<sup>i</sup>

X

q, s

The short time expansion of velocity correlation function, < **v**<sup>i</sup>

þð Þ 0 **v**<sup>j</sup>

*<sup>=</sup>*2m<sup>þ</sup> <sup>þ</sup><sup>X</sup>

N�

k6¼l

N X�, X k, q

position is located at the center of oxygen atom in the H2O molecule.

þ higher order over*t*

N�

k¼1 **p**i �2

ϕ�� j**r**<sup>k</sup>

In the present aqueous solution of electrolyte, the total Hamiltonian of the

∞ <**j**

þð Þ� *t* **j**ð Þ 0 > d*t* (81)

�ð Þ� *t* **j**ð Þ 0 > d*t* (82)

�ð Þ*t* (83)

(0)>, < **v**<sup>k</sup>

+ (*t*) **v**<sup>j</sup> +

w2*=*2mw <sup>þ</sup> V (85)

<sup>þ</sup> � **r**<sup>k</sup> � ð Þj

> <sup>w</sup> � **<sup>r</sup>**<sup>s</sup> <sup>w</sup><sup>j</sup> � �

> > (86)

<sup>ϕ</sup>w w <sup>j</sup>**r**<sup>q</sup>

�(*t*)

(0) > for

�(*t*) and **j**(*t*) are defined by the following

þðÞþ*t* **j**

+ (*t*) **v**<sup>j</sup> +

þð Þ 0 **v** € j þ ð Þ 0 >

<sup>4</sup> � � (84)

σ<sup>þ</sup> ¼ ð Þ 1*=*3kBT

σ� ¼ ð Þ 1*=*3kBT

X n **v**i

necessary to study the velocity correlation functions, < **v**<sup>i</sup>

�(0)>.

σ� [6, 7, 28]:

and

expressions:

�(0) > and < **v**<sup>i</sup>

**solutions**

cation is written as

<**v**<sup>i</sup>

where

<sup>V</sup> <sup>¼</sup> <sup>X</sup> Nþ

i6¼j

þ N X þ, X

**85**

i, q

þð Þ*t* **v**<sup>j</sup>

system is written as follows:

<sup>H</sup> <sup>¼</sup> <sup>X</sup> Nþ

ϕþþ j**r**<sup>i</sup>

i¼1 **p**i þ2

<sup>þ</sup> � **r**<sup>j</sup> þj � � <sup>þ</sup><sup>X</sup>

> <sup>þ</sup> � **r**<sup>q</sup> <sup>w</sup><sup>j</sup> � � <sup>þ</sup>

<sup>ϕ</sup><sup>þ</sup><sup>w</sup> <sup>j</sup>**r**<sup>i</sup>

difficult to define the position of **r**<sup>q</sup>

order to obtain the ~γ�(0).

**v**l

where the current densities **j**

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

**j**

+ (*t*) **v**<sup>k</sup>

�ðÞ¼ *t* z�e

þð Þ 0 > ¼ <**v**<sup>i</sup>

Inserting (74) into (72) and taking ensemble average under the assumption of <**ε**i(*t*) > = 0, we have

$$\mathbf{m}^{\pm} < \mathbf{d} \mathbf{v}\_{\mathbf{i}}^{\pm}(t) > / \mathbf{d}t = -\mathbf{m}^{\pm} \int\_{0}^{t} \boldsymbol{\gamma}^{\pm}(t - t') \, \mathbf{v}\_{\mathbf{i}}^{\pm}(t') \mathbf{d}t' + \mathbf{z}^{\pm} \mathbf{e}(\mathbf{1} - \mathbf{\delta}) \mathbf{E} \tag{75}$$

Therefore,

$$\mu^{\pm}(\mathfrak{o}) = (\mathfrak{1} \text{-} \mathfrak{d}) / \mathfrak{m}^{\pm} \left\{ -\text{i}\mathfrak{o} + \tilde{\chi}^{\pm}(\mathfrak{o}) \right\} \tag{76}$$

where

$$\tilde{\boldsymbol{\chi}}^{\pm}(\boldsymbol{\alpha}) = \int\_{0}^{\infty} \boldsymbol{\chi}^{\pm}(t) \, \dot{\mathbf{e}}^{\text{int}} \, \mathbf{d}t \tag{77}$$

The dc current density **j** � is then written as follows:

$$\mathbf{j}^{\pm} = \mathbf{n} \mathbf{z}^{\pm} \mathbf{e} < \mathbf{v}\_{\mathbf{i}}^{\pm}(t) >\_{\omega=0} = \mathbf{n} \mathbf{z}^{2} \mathbf{e}^{2} \boldsymbol{\mu}^{\pm}(\mathbf{0}) \, \mathbf{E}\_{0} = \mathbf{n} \mathbf{z}^{2} \mathbf{e}^{2} (\mathbf{1} \text{-} \boldsymbol{\delta}) \, \mathbf{E}\_{0} / \mathbf{m}^{\pm} \hat{\boldsymbol{\gamma}}^{\pm}(\mathbf{0}) \tag{78}$$

On the other hand, **j** � is expressed as **j** � = σ�**E**0, where σ� being equal to the partial conductivity for cation or anion. Therefore, σ� is written as follows:

$$\mathbf{e}^{\pm} = \mathbf{n} \mathbf{z}^{2} \mathbf{e}^{2} (\mathbf{1} \mathbf{-} \mathbf{\delta}) / \mathbf{m}^{\pm} \mathbf{\tilde{\gamma}}^{\pm} (\mathbf{0}) \tag{79}$$

The Laplace transformation of the memory function in the long wavelength limit ~γ�(0) in Eq. (79) will be obtained in later section.

In the next section, we will discuss velocity correlation functions.

#### **12. Linear response theory for electrolytic solutions**

Eq. (79) is also obtainable from the following simplified Langevin equation:

$$\mathbf{m}^{\pm} \mathbf{d} \mathbf{v}\_{\mathbf{i}}^{\pm}(t) / \mathbf{d}t = -\mathbf{m}^{\pm} \tilde{\mathbf{y}}^{\pm}(\mathbf{0}) \mathbf{v}\_{\mathbf{i}}^{\pm}(t) + \mathbf{z}^{\pm} \mathbf{e}(\mathbf{E} + \mathbf{F}) + \mathbf{z}^{\pm} \mathbf{e} \mathbf{e}\_{\mathbf{i}}(t) \tag{80}$$

Its derivation can be easily seen in a standard book of statistical physics.

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

Starting from Eq. (80) with an infinitesimal external field **E**, it is also easily obtainable the following Kubo-Green formulae for the partial conductivities σ<sup>+</sup> and σ� [6, 7, 28]:

$$\boldsymbol{\sigma}^{+} = (\mathbf{1}/3\mathbf{k}\_{\rm B}\mathbf{T})\Big|\_{0}^{\prime\prime} \prec \mathbf{j}^{+}(t) \cdot \mathbf{j}(0) > \mathbf{d}t \tag{81}$$

and

distribution which is resulted from the applying external field **E**, and **ε**i(*t*) is the

ð d ∞

According to Berne and Rice [16], the internal field **F** induced by the asymmetric

where ɡ+�(*r*) is the pair distribution function between cation and anion, and d is the hard-core contact distance between cation and anion. Hereafter, we will use this

, then the ensemble average for **v**<sup>i</sup>

Inserting (74) into (72) and taking ensemble average under the assumption of

γ� *t* � *t* <sup>0</sup> ð Þ **v**<sup>i</sup>

ð 0

~γ�ð Þ¼ ω

e2

partial conductivity for cation or anion. Therefore, σ� is written as follows:

e2

In the next section, we will discuss velocity correlation functions.

The Laplace transformation of the memory function in the long wavelength

Eq. (79) is also obtainable from the following simplified Langevin equation:

Its derivation can be easily seen in a standard book of statistical physics.

� is expressed as **j**

<sup>σ</sup>� <sup>¼</sup> nz<sup>2</sup>

ð 0 ∞

� is then written as follows:

<sup>μ</sup>�ð Þ <sup>0</sup> **<sup>E</sup>**<sup>0</sup> <sup>¼</sup> nz<sup>2</sup>

t

<sup>d</sup>ϕþ�ð Þ*<sup>r</sup> <sup>=</sup>*d*<sup>r</sup>* � �ɡþ�ð Þ*<sup>r</sup> <sup>r</sup>*

�ð Þ*<sup>t</sup>* <sup>&</sup>gt; <sup>¼</sup> Re <sup>μ</sup>�ð Þ <sup>ω</sup> <sup>z</sup>�e**E**<sup>0</sup> <sup>e</sup>�iω*<sup>t</sup>* (74)

� *t* <sup>0</sup> ð Þd*t*

<sup>μ</sup>�ð Þ¼ <sup>ω</sup> ð Þ <sup>1</sup>–<sup>δ</sup> *<sup>=</sup>*m� �i<sup>ω</sup> <sup>þ</sup> <sup>~</sup>γ�ð Þ <sup>ω</sup> � � (76)

e2

<sup>γ</sup>�ð Þ*<sup>t</sup>* <sup>e</sup><sup>i</sup>ω*<sup>t</sup>* <sup>d</sup>*<sup>t</sup>* (77)

� = σ�**E**0, where σ� being equal to the

ð Þ 1–δ *=*m�~γ�ð Þ 0 (79)

�ðÞþ*t* z�eð Þþ **E** þ **F** z�e**ε**ið Þ*t* (80)

ð Þ 1–δ **E**0*=*m�~γ�ð Þ 0 (78)

3

d*r* � **E** (73)

�(*t*) is written in the

<sup>0</sup> þ z�e 1ð Þ –δ **E** (75)

random fluctuating force acting on the ion i.

*Electromagnetic Field Radiation in Matter*

result.

following form:

<**ε**i(*t*) > = 0, we have

Therefore,

where

**j**

m� <d**v**<sup>i</sup>

The dc current density **j**

� ¼ nz�e<**v**<sup>i</sup>

On the other hand, **j**

m�d**v**<sup>i</sup>

**84**

If we take **E** = **E**<sup>0</sup> e�iω*<sup>t</sup>*

ion's distribution in an ionic melt is expressed as follows:

**F** ¼ �δ � **E** ¼ �4πn*=*3kBT

< **v**<sup>i</sup>

�ð Þ*t* >*=*d*t* ¼ �m�

�ð Þ*<sup>t</sup>* <sup>&</sup>gt;ω¼<sup>0</sup> <sup>¼</sup> nz<sup>2</sup>

limit ~γ�(0) in Eq. (79) will be obtained in later section.

�ð Þ*t =*d*t* ¼ �m�~γ�ð Þ 0 **v**<sup>i</sup>

**12. Linear response theory for electrolytic solutions**

$$\boldsymbol{\sigma}^{-} = (\mathbf{1}/3\mathbf{k}\_{\text{B}}\mathbf{T})\Big|\_{0}^{\prime\prime} \prec \mathbf{j}^{-}(t) \cdot \mathbf{j}(\mathbf{0}) > \mathbf{d}t \tag{82}$$

where the current densities **j** �(*t*) and **j**(*t*) are defined by the following expressions:

$$\mathbf{j}^{\pm}(t) = \mathbf{z}^{\pm} \mathbf{e} \sum\_{\mathbf{n}} \mathbf{v}\_{\mathbf{i}}^{\pm}(t) \quad \text{and} \quad \mathbf{j}(t) = \mathbf{j}^{+}(t) + \mathbf{j}^{-}(t) \tag{83}$$

In order to obtain the partial conductivities based on Eqs. (81) and (82), it is necessary to study the velocity correlation functions, < **v**<sup>i</sup> + (*t*) **v**<sup>j</sup> + (0)>, < **v**<sup>k</sup> �(*t*) **v**l �(0) > and < **v**<sup>i</sup> + (*t*) **v**<sup>k</sup> �(0)>.

In the next section, we will discuss velocity correlation functions described in terms of inter-molecular (or ionic) potentials and pair distribution functions in order to obtain the ~γ�(0).

#### **13. Short time expansion of velocity correlation functions in electrolytic solutions**

The short time expansion of velocity correlation function, < **v**<sup>i</sup> + (*t*) **v**<sup>j</sup> + (0) > for cation is written as

$$\begin{array}{rcl}<\mathbf{v}\_{\mathbf{i}}^{+}(t)\,\mathbf{v}\_{\mathbf{j}}^{+}(\mathbf{0})> &=&<\mathbf{v}\_{\mathbf{i}}^{+}(\mathbf{0})\,\mathbf{v}\_{\mathbf{j}}^{+}(\mathbf{0})>+t^{2}/2!<\mathbf{v}\_{\mathbf{i}}^{+}(\mathbf{0})\,\mathbf{v}\_{\mathbf{j}}^{+}(\mathbf{0})>\\&\quad+\text{(higher order over}\,t^{4}\text{)}\end{array} \tag{84}$$

In the present aqueous solution of electrolyte, the total Hamiltonian of the system is written as follows:

$$\mathbf{H} = \sum\_{i=1}^{N+} \mathbf{p}\_i^{+2} / 2\mathbf{m}^+ + \sum\_{\mathbf{k}=1}^{N-} \mathbf{p}\_i^{-2} / 2\mathbf{m}^- + \sum\_{\mathbf{q}=1}^{X} \mathbf{p}\_{\mathbf{q}}^{\text{ w2}} / 2\mathbf{m}^{\text{w}} + \mathbf{V} \tag{85}$$

where

$$\begin{split} \mathbf{V} &= \sum\_{i \neq j}^{\mathbf{N} + \mathbf{v}} \Phi^{++} \left( |\mathbf{r}\_{\mathbf{i}}^{+} - \mathbf{r}\_{\mathbf{j}}^{+}| \right) + \sum\_{\mathbf{k} \neq \mathbf{l}}^{\mathbf{N} - \mathbf{r}} \Phi^{--} \left( |\mathbf{r}\_{\mathbf{k}}^{-} - \mathbf{r}\_{\mathbf{l}}^{-}| \right) + \sum\_{\mathbf{i}, \mathbf{k}}^{\mathbf{N} + \mathbf{N} - \mathbf{r}} \Phi^{+-} \left( |\mathbf{r}\_{\mathbf{i}}^{+} - \mathbf{r}\_{\mathbf{k}}^{-}| \right) \\ &+ \sum\_{\mathbf{i}, \mathbf{q}}^{\mathbf{N} + \mathbf{X}} \Phi^{+ \mathbf{w}} \left( |\mathbf{r}\_{\mathbf{i}}^{+} - \mathbf{r}\_{\mathbf{q}}^{\mathbf{w}}| \right) + \sum\_{\mathbf{k}, \mathbf{q}}^{\mathbf{N} - \mathbf{X}} \Phi^{- \mathbf{w}} \left( |\mathbf{r}\_{\mathbf{k}}^{-} - \mathbf{r}\_{\mathbf{q}}^{\mathbf{w}}| \right) + \sum\_{\mathbf{q}, \mathbf{s}}^{\mathbf{X}} \Phi^{\mathbf{w} \mathbf{w}} \left( |\mathbf{r}\_{\mathbf{q}}^{\mathbf{w}} - \mathbf{r}\_{\mathbf{s}}^{\mathbf{w}}| \right) \end{split} \tag{86}$$

Since the water molecule is not spherical in its molecular configuration, it is difficult to define the position of **r**<sup>q</sup> w. However, we tentatively assume that its position is located at the center of oxygen atom in the H2O molecule.

From the Poisson's equation of motion,

$$\mathbf{p}\_i^+ \cdot \ddot{\mathbf{p}}\_i^+ = -\sum\_{j=1}^{N+} \left( \mathbf{p}\_i^+ \mathbf{p}\_j^+ / \mathbf{m}^+ \right) \partial^2 \mathbf{V} / \partial \mathbf{r}\_i^+ \partial \mathbf{r}\_j^+ - \sum\_{k=1}^{N-} \left( \mathbf{p}\_i^+ \mathbf{p}\_k^- / \mathbf{m}^- \right) \partial^2 \mathbf{V} / \partial \mathbf{r}\_k^- \partial \mathbf{r}\_i^+$$

$$-\sum\_{q=1}^{X} \left( \mathbf{p}\_i^+ \mathbf{p}\_q^w / \mathbf{m}^w \right) \partial^2 \mathbf{V} / \partial \mathbf{r}\_q^w \partial \mathbf{r}\_i^+ \tag{87}$$

Using this relation, the distinct velocity correlation function is written as

�ð Þ 0 > þ *t*

�ð Þ 0 > þ *t*

2 *=*2!<**v**<sup>i</sup>

2

) incorporation with the partial conductivity σ<sup>+</sup> is therefore expressed

þð Þ*t* **v**<sup>k</sup>

where μ is equal to the reduced mass of m+ and m�. In deriving (95), we have

These initial conditions are confirmed by our own molecular dynamic simula-

þð Þ 0 > ¼ 3kBT*=*m<sup>þ</sup> ð Þ and <**v**<sup>i</sup>

þð Þ 0 **v**<sup>j</sup>

<sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup>*=<sup>r</sup> <sup>∂</sup>*ϕ�<sup>w</sup>ð Þ*<sup>r</sup> <sup>∂</sup><sup>r</sup>* � �ɡ�<sup>w</sup>ð Þ*<sup>r</sup>* <sup>4</sup>π*<sup>r</sup>*

�ð Þ 0 >

*<sup>=</sup>*2! � � <sup>n</sup><sup>&</sup>lt; <sup>ϕ</sup>þ� <sup>&</sup>gt; *<sup>=</sup>*3<sup>μ</sup> <sup>þ</sup> <sup>x</sup><sup>&</sup>lt; <sup>ϕ</sup>�<sup>w</sup> <sup>&</sup>gt;*=*3m� ð Þ �

<sup>4</sup> � �<sup>g</sup> (97)

þð Þ 0 **v** € k � ð Þ 0 >

<sup>4</sup> � � (94)

�ð Þ 0 >

*<sup>=</sup>*2! � � <sup>n</sup><sup>&</sup>lt; <sup>ϕ</sup>þ� <sup>&</sup>gt;*=*3<sup>μ</sup> <sup>þ</sup> <sup>x</sup><ϕþ<sup>w</sup> <sup>&</sup>gt; *<sup>=</sup>*3m<sup>þ</sup> ð Þ �

<sup>4</sup> � �<sup>g</sup> (95)

*=*2! � �kBT n< ϕþ� > *=*mþm�

þð Þ 0 **v**<sup>k</sup>

2

+

(*t*)(= < **j**

�ð Þ 0 > ¼ 0 (96)

d*r* (98)

+ (*t*)

þð Þ 0 **v**<sup>k</sup>

þð Þ 0 **v**<sup>k</sup>

þ higher order over*t* 4 � �

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

þ higher order over*t*

Using (92) and (94), the combined velocity correlation function Z<sup>σ</sup>

þð Þ 0 > � <**v**<sup>i</sup>

2

tion, which will be shown in the later section. In a similar way, we have

�ð Þ 0 > � < **v**<sup>i</sup>

2

<sup>ϕ</sup>�<sup>w</sup>ð Þ*<sup>r</sup> <sup>∂</sup><sup>r</sup>*

�(0) > is also vanished to be zero.

**14. Derivation of γ**~�**(***0***) in electrolytic solutions**

þ higher order over*t*

ɡ�w(*r*) of this equation means the pair distribution function between anion and water molecule. And it is also emphasized that the contribution from ϕ��(*r*)

It is impossible to obtain the partial conductivities by the insertion of (95) and (97) into (81) and (82), because we knew only the terms up to t2 in their explicit forms. However, these equations can be utilized for the derivation of ~γ�(0) as

According to the fluctuation dissipation theorem applied for the present system with the condition of no external field or of infinitesimal external field, the Laplace transformation of the memory function γ�(*t*) and that of the ensemble average of time correlation function for the fluctuating random force <**ε**i(*t*)**ε**i(0) > have the

follows:

< **v**<sup>i</sup>

**j**(0)>/n<sup>2</sup>

<**v**<sup>i</sup>

þð Þ 0 **v**<sup>j</sup>

Zσ

<sup>&</sup>lt;ϕ�<sup>w</sup> <sup>&</sup>gt; <sup>¼</sup>

�(*t*) **v**<sup>j</sup>

shown in the next section.

following relation [25, 28]:

where

to < **v**<sup>i</sup>

**87**

as follows:

þð Þ*t* **v**<sup>k</sup>

z2 e2

Zσ

þðÞ� *t* <**v**<sup>i</sup>

assumed the initial conditions as follows:

þð Þ 0 > ¼ < **v**<sup>i</sup>

�ðÞ� *t* <**v**<sup>i</sup>

ð 0 ∞ *∂*2

�ð Þ 0 > ¼ < **v**<sup>i</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

¼ < **v**<sup>i</sup>

þð Þ*t* **v**<sup>j</sup>

¼ 3kBT*=*m<sup>þ</sup> ð Þ 1– *t*

þð Þ 0 **v**<sup>i</sup>

�ð Þ*t* **v**<sup>j</sup>

¼ 3kBT*=*m� ð Þ 1– *t*

þ higher order over*t*

and

$$\begin{aligned} \mathbf{p}\_i^+ \cdot \ddot{\mathbf{p}}\_{i \neq i^\*}^+ &= -\sum\_{\mathbf{j}}^{\mathbf{N}+} \left( \mathbf{p}\_{\mathbf{i}^\*} \, ^+ \mathbf{p}\_{\mathbf{j}} \, ^+ / \mathbf{m}^+ \right) \partial^2 \mathbf{V} / \partial \mathbf{r}\_i^+ \partial \mathbf{r}\_{\mathbf{j}}^+ - \sum\_{\mathbf{k}=1}^{\mathbf{N}-} \left( \mathbf{p}\_{\mathbf{i}} \, ^+ \mathbf{p}\_{\mathbf{k}} \, ^- / \mathbf{m}^- \right) \partial^2 \mathbf{V} / \partial \mathbf{r}\_{\mathbf{k}} \, ^- \partial \mathbf{r}\_{\mathbf{i}}^+ \\ &- \sum\_{\mathbf{q}=1}^{\mathbf{X}} \left( \mathbf{p}\_{\mathbf{i}} \, ^+ \mathbf{p}\_{\mathbf{q}} \, ^\mathbf{w} / \mathbf{m}^\mathbf{w} \right) \partial^2 \mathbf{V} / \partial \mathbf{r}\_{\mathbf{q}} \, ^\mathbf{w} \partial \mathbf{r}\_{\mathbf{i}}^+ \end{aligned} \tag{88}$$

Since the second derivative of the potential term V is independent for the product of momenta, all other terms other than i = j in (87) must vanish on averaging. And in a similar way, the meaningful terms of (88) for averaging must be also equal to the case i 6¼ i' = j. Therefore, taking the ensemble averages for (87) and (88), we have

$$<\mathbf{p}\_i^+ \ddot{\mathbf{p}}\_j^+ > = <\mathbf{p}\_i^+ \ddot{\mathbf{p}}\_i^+ > \mathbf{ <} \mathbf{p}\_i^+ \ddot{\mathbf{p}}\_{j \ne i}^+ > = -\mathbf{k}\_\mathbb{B} \mathbf{T} \{ \mathbf{n} < \phi^{+-} > +\mathbf{x} < \phi^{+\mathbf{w}} > \} \tag{89}$$

where

$$<\phi^{+-}> = \int\_0^\infty \left\{ \partial^2 \phi^{+-}(r) \partial r^2 + (2/r) \partial \phi^{+-}(r) \partial r \right\} \mathbf{g}^{+-}(\mathbf{r}) \, 4\pi r^2 \mathbf{d}r \tag{90}$$

and

$$<\phi^{+\text{w}}> = \int\_0^\infty \left\{ \partial^2 \phi^{+\text{w}}(r) \partial r^2 + (2/\text{r}) \partial \phi^{+\text{w}}(r) \partial r \right\} \mathbf{g}^{+\text{w}}(r) \, 4\pi r^2 \text{d}r \tag{91}$$

In this equation, ɡ+w(*r*) is the pair distribution function between cation and water molecule.

It is emphasized that there is no contribution from ϕ++(*r*) in Eq. (89) because of the cancelation by the terms of i = j and i 6¼ j in < **p**<sup>þ</sup> *<sup>i</sup>* **p**€<sup>þ</sup> *<sup>j</sup>* > [7].

Insertion of (89) into (84) gives us the following form:

$$\begin{aligned} \mathbf{Z}^+(\mathbf{t}) &\equiv <\mathbf{v}\_{\mathbf{i}}^+(\mathbf{t}) \, \mathbf{v}\_{\mathbf{j}}^+(\mathbf{0}) > \\ &= <\mathbf{v}\_{\mathbf{i}}^+(\mathbf{0}) \, \mathbf{v}\_{\mathbf{j}}^+(\mathbf{0}) > - \, \left(\mathbf{t}^2/2\right) \mathbf{k}\_{\mathbb{B}} \mathbf{T} \{ (\mathbf{n} < \boldsymbol{\phi}^{+ \mathrm{-}} > +\mathbf{x} < \boldsymbol{\phi}^{+ \mathrm{w}} >) / \mathbf{m}^+ \mathbf{m}^+\} \\ &+ \, \left( \text{higher order over} \, t^4 \right) \end{aligned} \tag{92}$$

In a similar way, the term < **p**<sup>þ</sup> *<sup>i</sup>* **p**€� *<sup>k</sup>* > can be described as follows:

$$<\mathbf{p}\_i^+ \ddot{\mathbf{p}}\_k^-> = \mathbf{k}\_\mathsf{B} \mathbf{T} \,\mathbf{n} < \boldsymbol{\phi}^{+-}>\tag{93}$$

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

Using this relation, the distinct velocity correlation function is written as follows:

$$<\mathbf{v}\_{\mathbf{i}}^{+}(t)\,\mathbf{v}\_{\mathbf{k}}^{-}(0)> = <\mathbf{v}\_{\mathbf{i}}^{+}(0)\,\mathbf{v}\_{\mathbf{k}}^{-}(0)> + t^{2}/2! < \mathbf{v}\_{\mathbf{i}}^{+}(0)\,\mathbf{v}\_{\mathbf{k}}^{-}(0)>$$

$$+ \left(\text{higher order over}\,t^{4}\right)$$

$$= <\mathbf{v}\_{\mathbf{i}}^{+}(0)\,\mathbf{v}\_{\mathbf{k}}^{-}(0)> + \left(t^{2}/2!\right)\mathbf{k}\_{\mathbf{k}}\mathbf{T}\,\mathbf{n} < \boldsymbol{\phi}^{+-} > /\mathbf{m}^{+}\mathbf{m}^{-}$$

$$+ \left(\text{higher order over}\,t^{4}\right) \tag{94}$$

Using (92) and (94), the combined velocity correlation function Z<sup>σ</sup> + (*t*)(= < **j** + (*t*) **j**(0)>/n<sup>2</sup> z2 e2 ) incorporation with the partial conductivity σ<sup>+</sup> is therefore expressed as follows:

$$\begin{split} \mathbf{Z\_{\sigma}}^{+}(t) & \equiv & < \mathbf{v\_{i}}^{+}(t)\,\mathbf{v\_{j}}^{+}(0)> - \, < \mathbf{v\_{i}}^{+}(t)\,\mathbf{v\_{k}}^{-}(0)> \\ & = & (3\mathbf{k\_{B}}\,\mathbf{T}/\mathbf{m}^{+})\left\{\mathbf{1}\text{--}(t^{2}/2!)\left(\mathbf{n} < \phi^{+-} > /3\mu + \mathbf{x} < \phi^{+\mathbf{w}} > /3\mathbf{m}^{+}\right) \\ & \quad + \text{(higher order overt}^{4}\text{)}\right\} \end{split} \tag{95}$$

where μ is equal to the reduced mass of m+ and m�. In deriving (95), we have assumed the initial conditions as follows:

$$<\mathbf{v}\_{\mathrm{i}}^{+}(\mathbf{0})\mathbf{v}\_{\mathrm{j}}^{+}(\mathbf{0})> = <\mathbf{v}\_{\mathrm{i}}^{+}(\mathbf{0})\mathbf{v}\_{\mathrm{i}}^{+}(\mathbf{0})> = (\mathbf{3}\mathbf{k}\_{\mathrm{B}}\mathbf{T}/\mathbf{m}^{+}) \quad \text{and} \quad <\mathbf{v}\_{\mathrm{i}}^{+}(\mathbf{0})\mathbf{v}\_{\mathrm{k}}^{-}(\mathbf{0})> = \mathbf{0} \tag{96}$$

These initial conditions are confirmed by our own molecular dynamic simulation, which will be shown in the later section. In a similar way, we have

$$\begin{split} \mathbf{Z}\_{\sigma}^{-}(t) \equiv &< \mathbf{v}\_{\text{i}}^{-}(t) \, \mathbf{v}\_{\text{j}}^{-}(0) > \, - \, < \mathbf{v}\_{\text{i}}^{+}(0) \, \mathbf{v}\_{\text{j}}^{-}(0) > \\ &= \left( 3 \mathbf{k}\_{\text{B}} \mathbf{T} / \mathbf{m}^{-} \right) \left\{ \mathbf{1} \left( t^{2} / 2! \right) \left( \mathbf{n} < \phi^{+-} > / 3 \mu + \mathbf{x} < \phi^{- \text{w}} > / 3 \mathbf{m}^{-} \right) \\ &\quad + \left( \text{higher order over} \, t^{4} \right) \right\} \end{split} \tag{97}$$

where

From the Poisson's equation of motion,

*∂*2 V*=∂***r**<sup>i</sup>

*∂*2 V*=∂***r**<sup>q</sup>

> *∂*2 V*=∂***r**<sup>i</sup>

*∂*2 V*=∂***r**<sup>q</sup>

*<sup>i</sup>* **p**€<sup>þ</sup>

<sup>þ</sup>*∂***r**<sup>j</sup>

<sup>w</sup>*∂***r**<sup>i</sup>

<sup>þ</sup>*∂***r**<sup>j</sup>

<sup>w</sup>*∂***r**<sup>i</sup> þ

<sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup>*=<sup>r</sup> <sup>∂</sup>*ϕþ�ð Þ*<sup>r</sup> <sup>∂</sup><sup>r</sup>* � �ɡþ�ð Þ<sup>r</sup> <sup>4</sup>π*<sup>r</sup>*

<sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup>*=*<sup>r</sup> *<sup>∂</sup>*ϕþ<sup>w</sup>ð Þ*<sup>r</sup> <sup>∂</sup><sup>r</sup>* � �ɡþ<sup>w</sup>ð Þ*<sup>r</sup>* <sup>4</sup>π*<sup>r</sup>*

In this equation, ɡ+w(*r*) is the pair distribution function between cation and

It is emphasized that there is no contribution from ϕ++(*r*) in Eq. (89) because of

*<sup>i</sup>* **p**€<sup>þ</sup> *<sup>j</sup>* > [7].

*<sup>=</sup>*2! � �kBT n<ϕþ� <sup>&</sup>gt; <sup>þ</sup> <sup>x</sup><sup>&</sup>lt; <sup>ϕ</sup><sup>þ</sup><sup>w</sup> ð Þ <sup>&</sup>gt; *<sup>=</sup>*mþm<sup>þ</sup> f g

*<sup>k</sup>* > can be described as follows:

*<sup>k</sup>* > ¼ kBT n<ϕþ� > (93)

<sup>4</sup> � � (92)

Since the second derivative of the potential term V is independent for the product of momenta, all other terms other than i = j in (87) must vanish on averaging. And in a similar way, the meaningful terms of (88) for averaging must be also equal to the case i 6¼ i' = j. Therefore, taking the ensemble averages for

<sup>þ</sup> �<sup>X</sup> N�

k¼1 **p**i <sup>þ</sup>**p**<sup>k</sup> �*=*m� � �*∂*<sup>2</sup>

<sup>þ</sup> �<sup>X</sup> N�

k¼1 **p**i <sup>þ</sup>**p**<sup>k</sup> �*=*m� � �*∂*<sup>2</sup>

V*=∂***r**<sup>k</sup>

þ (87)

<sup>j</sup>6¼<sup>i</sup> <sup>&</sup>gt; ¼ �kBT n<ϕþ� <sup>&</sup>gt; <sup>þ</sup> <sup>x</sup><ϕþ<sup>w</sup> f g <sup>&</sup>gt; (89)

2

2

d*r* (90)

d*r* (91)

�*∂***r**<sup>i</sup> þ

V*=∂***r**<sup>k</sup>

�*∂***r**<sup>i</sup> þ

(88)

**p**<sup>þ</sup> *<sup>i</sup>* **p**€<sup>þ</sup>

and

**p**<sup>þ</sup> *<sup>i</sup>* **p**€<sup>þ</sup>

< **p**<sup>þ</sup> *<sup>i</sup>* **p**€<sup>þ</sup>

where

and

water molecule.

ZþðÞ� *t* <**v**<sup>i</sup>

**86**

¼ <**v**<sup>i</sup>

*<sup>i</sup>* ¼ �<sup>X</sup> Nþ

j¼1

�<sup>X</sup> X

q¼1

<sup>i</sup>6¼i' ¼ �<sup>X</sup> Nþ

(87) and (88), we have

*<sup>j</sup>* > ¼ <**p**<sup>þ</sup>

<ϕþ� > ¼

<sup>&</sup>lt;ϕþ<sup>w</sup> <sup>&</sup>gt; <sup>¼</sup>

þð Þ*t* **v**<sup>j</sup>

þð Þ 0 **v**<sup>j</sup>

**p**i <sup>þ</sup>**p**<sup>j</sup> <sup>þ</sup>*=*m<sup>þ</sup> � �

*Electromagnetic Field Radiation in Matter*

**p**i <sup>þ</sup>**p**<sup>q</sup> <sup>w</sup>*=*mw � �

j

�<sup>X</sup> X

q¼1

**p**i' <sup>þ</sup>**p**<sup>j</sup> <sup>þ</sup>*=*m<sup>þ</sup> � �

**p**i <sup>þ</sup>**p**<sup>q</sup> <sup>w</sup>*=*m<sup>w</sup> � �

*<sup>i</sup>* **p**€<sup>þ</sup>

ð 0

ð 0 ∞ *∂*2

the cancelation by the terms of i = j and i 6¼ j in < **p**<sup>þ</sup>

þð Þ 0 > � *t*

þð Þ 0 >

þ higher order over*t*

In a similar way, the term < **p**<sup>þ</sup>

∞ *∂*2

*<sup>i</sup>* > þ <**p**<sup>þ</sup>

<sup>ϕ</sup>þ�ð Þ*<sup>r</sup> <sup>∂</sup><sup>r</sup>*

<sup>ϕ</sup>þ<sup>w</sup>ð Þ*<sup>r</sup> <sup>∂</sup><sup>r</sup>*

Insertion of (89) into (84) gives us the following form:

2

< **p**<sup>þ</sup> *<sup>i</sup>* **p**€�

*<sup>i</sup>* **p**€�

$$<\phi^{-\mathrm{w}}> = \int\_0^\infty \left\{ \partial^2 \phi^{-\mathrm{w}}(r) \partial r^2 + (2/r) \partial \phi^{-\mathrm{w}}(r) \partial r \right\} \mathrm{g}^{-\mathrm{w}}(r) \, 4\pi r^2 \mathrm{d}r \tag{98}$$

ɡ�w(*r*) of this equation means the pair distribution function between anion and water molecule. And it is also emphasized that the contribution from ϕ��(*r*) to < **v**<sup>i</sup> �(*t*) **v**<sup>j</sup> �(0) > is also vanished to be zero.

It is impossible to obtain the partial conductivities by the insertion of (95) and (97) into (81) and (82), because we knew only the terms up to t2 in their explicit forms. However, these equations can be utilized for the derivation of ~γ�(0) as shown in the next section.

#### **14. Derivation of γ**~�**(***0***) in electrolytic solutions**

According to the fluctuation dissipation theorem applied for the present system with the condition of no external field or of infinitesimal external field, the Laplace transformation of the memory function γ�(*t*) and that of the ensemble average of time correlation function for the fluctuating random force <**ε**i(*t*)**ε**i(0) > have the following relation [25, 28]:

$$\mathbf{m}^{\pm 2} < \mathbf{v}\_{\mathbf{i}}^{\pm}(\mathbf{0}) \cdot \mathbf{v}\_{\mathbf{j}}^{\pm}(\mathbf{0}) > \tilde{\mathbf{\gamma}}^{\pm}(\mathbf{0}) = \mathbf{m}^{\pm 2} < \mathbf{v}\_{\mathbf{i}}^{\pm}(\mathbf{0}) \cdot \mathbf{v}\_{\mathbf{i}}^{\pm}(\mathbf{0}) > \tilde{\mathbf{\gamma}}^{\pm}(\mathbf{0})$$

$$= \mathbf{z}^2 \mathbf{e}^2 \int\_0^\infty < \mathbf{e}\_{\mathbf{i}}(t) \mathbf{e}\_{\mathbf{i}}(0) > \mathbf{e}^{\text{int}} \,\mathbf{d}t \tag{99}$$

The fluctuation dissipation theorem tells us the following relation:

$$\dot{\mathbf{q}}^{\pm}(\mathbf{o}) = \left(\mathbf{1}/\mathbf{3}\,\mathrm{m}^{\pm}\mathrm{k}\_{\mathrm{B}}\mathbf{T}\right) \left(\mathbf{z}^{2}\,\mathrm{e}^{2}\right) \int\_{0}^{\infty} < \mathbf{e}\_{i}(t)\mathbf{e}\_{i}(0) > \mathbf{e}^{\mathrm{int}}\,\mathrm{d}t \tag{100}$$

As seen in Eq. (79), the Laplace transformation of memory function in the long

anion, which means that the auto-correlation function of random fluctuating force. <**ε**i0(*t*) **ε**i0(0) > may be represented by an exponential decaying function with

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

<**ε**i0ð Þ*t* **ε**i0ð Þ 0 > ¼ <**ε**i0ð Þ 0 **ε**i0ð Þ 0 > exp �~γ�

<**ε**i0ð Þ 0 **ε**i0ð Þ 0 > ¼ 3 m�kBT ~γ�

<sup>0</sup> ð Þ <sup>0</sup> � �<sup>2</sup>

By the analogy with this relation, we can infer the following relation:

<sup>¼</sup> <sup>x</sup><ϕ�<sup>w</sup> <sup>&</sup>gt;*=*3m� � � exp � <sup>x</sup><ϕ�<sup>w</sup> <sup>&</sup>gt;*=*3m� � �<sup>1</sup>*=*<sup>2</sup>

And the Laplace transformation of this equation in the long wavelength limit is

Putting Eq. (113) into (79), we obtain the formulae of the partial conductivities, σ<sup>+</sup> and σ�, which are expressed in terms of the pair distribution functions and pair

ð Þ <sup>1</sup>–<sup>δ</sup> *<sup>=</sup>*m<sup>þ</sup> <sup>x</sup><ϕ<sup>þ</sup><sup>w</sup> <sup>&</sup>gt; *<sup>=</sup>*3m<sup>þ</sup> ð Þ<sup>1</sup>*=*<sup>2</sup> <sup>þ</sup> <sup>n</sup><ϕþ� ð Þ <sup>&</sup>gt;*=*3<sup>μ</sup>

ð Þ <sup>1</sup>–<sup>δ</sup> *<sup>=</sup>*m� <sup>x</sup><sup>&</sup>lt; <sup>ϕ</sup>�<sup>w</sup> <sup>&</sup>gt;*=*3m� ð Þ<sup>1</sup>*=*<sup>2</sup> <sup>þ</sup> <sup>n</sup><ϕþ� ð Þ <sup>&</sup>gt;*=*3<sup>μ</sup>

þ n<ϕþ� ð Þ >*=*3μ exp � n<ϕþ� ð Þ >*=*3μ

<sup>~</sup>γ�ð Þ¼ <sup>0</sup> <sup>x</sup><ϕ�<sup>w</sup> <sup>&</sup>gt;*=*3m� � �<sup>1</sup>*=*<sup>2</sup> h i <sup>þ</sup> <sup>n</sup><ϕþ� ð Þ <sup>&</sup>gt;*=*3<sup>μ</sup>

<sup>0</sup> (0) as follows:

γ�

<sup>0</sup> ðÞ¼ *t* ~γ�

<sup>0</sup> ð Þ¼ 0 γ<sup>00</sup>

<sup>1</sup> ð Þ¼ 0 γ<sup>01</sup>

Therefore, Eq. (102) is explicitly written as follows:

�ð Þ*t*

�ðÞþ*t* γ<sup>1</sup>

Insertion of (107) into (106) gives us

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

Compare (106) and (109) we have

β0 � ¼ ~γ�

> β1 � ¼ ~γ�

**15. Partial conductivities σ<sup>+</sup> and σ**�

ð Þ 1–δ *=*mþ~γþð Þ 0

ð Þ 1–δ *=*m�~γ�ð Þ 0

potentials as follows [28],

e2

e2

<sup>σ</sup><sup>þ</sup> <sup>¼</sup> nz2

<sup>σ</sup>� <sup>¼</sup> nz<sup>2</sup>

<sup>¼</sup> nz<sup>2</sup> e2

and

**89**

<sup>¼</sup> nz2 e2

γ�ðÞ¼ *t* γ<sup>0</sup>

equal to

<sup>0</sup> (0), corresponds to effective friction constants for cation and

exp �~γ�

<sup>0</sup> ð Þ <sup>0</sup> *<sup>t</sup>* � � (107)

<sup>0</sup> ð Þ <sup>0</sup> � �<sup>2</sup> (108)

<sup>0</sup> ð Þ <sup>0</sup> *<sup>t</sup>* � � (109)

*t*

<sup>1</sup>*=*<sup>2</sup> h i (113)

1*=*2 *t* h i (112)

� � �<sup>1</sup>*=*<sup>2</sup> <sup>¼</sup> <sup>x</sup><sup>&</sup>lt; <sup>ϕ</sup>�<sup>w</sup> <sup>&</sup>gt;*=*3m� � �<sup>1</sup>*=*<sup>2</sup> (110)

� � �<sup>1</sup>*=*<sup>2</sup> <sup>¼</sup> <sup>n</sup><sup>&</sup>lt; <sup>ϕ</sup>�<sup>w</sup> <sup>&</sup>gt;*=*3<sup>μ</sup> � �<sup>1</sup>*=*<sup>2</sup> (111)

h i

<sup>1</sup>*=*<sup>2</sup> h i (114)

<sup>1</sup>*=*<sup>2</sup> h i (115)

wavelength limit, ~γ�

the time constant of ~γ�

Therefore, we obtain

In the long wavelength limit, this relation is expressed by

$$\dot{\mathbf{y}}^{\pm}(\mathbf{0}) = \left(\mathbf{1}/3\,\mathrm{m}^{\pm}\mathrm{k}\_{\mathrm{B}}\mathbf{T}\right) \left(\mathbf{z}^{2}\,\mathrm{e}^{2}\right) \int\_{0}^{\infty} < \mathbf{e}\_{i}(t)\mathbf{e}\_{i}(0) > \mathrm{d}t \tag{101}$$

Let us go back to the memory function γ�(t) and assume a combined exponential decay functions for it as follows, although this assumption is not exactly consistent with Eq. (84), but technically acceptable one as discussed in the case of molten salt [29],

$$\gamma^{\pm}(t) = \gamma\_0^{\pm}(t)\left(\left\{=\gamma\_{00}{}^{\pm}\exp\left(-\mathfrak{J}\_0{}^{\pm}t\right)\right\} + \gamma\_1^{\pm}(t)\left\{=\gamma\_{01}{}^{\pm}\exp\left(-\mathfrak{J}\_1{}^{\pm}t\right)\right\} \tag{102}$$

In this equation, the pre-exponential factor γ00� is subject to the interactions between the central ion and surrounding water molecules. The decaying constants are related to the time dependence of its configuration change. The pre-exponential factor, γ01�, is equal to the magnitude of memory function at t = 0 in respect to the friction force acting on the central cation or anion caused by interactions between its central ion and neighboring ions. In other words, the first term on the right hand side of this equation means the case of dilute limit of electrolytic solution and the second one is equal to the effective friction caused by the addition of a moderate amount of electrolyte. Therefore, the first term is related to either <ϕ+w > or < ϕ�<sup>w</sup> >, while the second one is related to the term <ϕ+� > .

Using (94) and (96), γ00� and γ01� are expressed as follows:

$$\gamma\_{00}{}^{\pm} + \gamma\_{01}{}^{\pm} = -\ddot{\mathbf{Z}}\_{\sigma}^{\pm}(\mathbf{0})/\mathbf{Z}\_{\sigma}^{\pm}(\mathbf{0}) = \mathbf{x} < \boldsymbol{\Phi}^{\pm \mathbf{w}} > /3\mathbf{m}^{\pm} + \mathbf{n} < \boldsymbol{\Phi}^{+} > /3\boldsymbol{\mu} \tag{103}$$

In the dilute limit of n ≪ x, we have

$$\left(\chi\_{00}^{\ \pm}\right) = \mathbf{x} < \boldsymbol{\Phi}^{\pm \mathbf{w}} > / \mathbf{3} \mathbf{m}^{\pm} \tag{104}$$

And then we have

$$\left(\chi\_{01}^{\ \pm}\right) = \mathbf{n} < \Phi^{+-} > / 3\mu \tag{105}$$

At the dilution limit of electrolyte where the contribution of γ<sup>1</sup> �(*t*) can be neglected, the Laplace transformation of γ<sup>0</sup> �(*t*) in the long wavelength limit is then described as follows:

$$\begin{split} \dot{\chi}\_{0}^{\pm}(\mathbf{0}) &= \chi\_{00}{}^{\pm} / \mathfrak{h}\_{0}{}^{\pm} = \mathbf{x} < \boldsymbol{\Phi}^{\pm \mathbf{w}} > \boldsymbol{\beta} \mathbf{m}^{\pm} \mathfrak{h}\_{0}{}^{\pm} \\ &= \left( \mathbf{1} / \mathbf{3} \, \mathbf{m}^{\pm} \mathbf{k}\_{\mathbb{B}} \mathbf{T} \right) \left( \mathbf{z}^{2} \mathbf{e}^{2} \right) \int\_{0}^{\infty} < \mathbf{e}\_{i0}(t) \mathbf{e}\_{i0}(0) > \mathbf{d}t \end{split} \tag{106}$$

where the auto-correlation function of random fluctuating force <**ε**i0(*t*)**ε**i0(0) > is only related to either <ϕ+w > or < ϕ�<sup>w</sup> >.

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

As seen in Eq. (79), the Laplace transformation of memory function in the long wavelength limit, ~γ� <sup>0</sup> (0), corresponds to effective friction constants for cation and anion, which means that the auto-correlation function of random fluctuating force.

<**ε**i0(*t*) **ε**i0(0) > may be represented by an exponential decaying function with the time constant of ~γ� <sup>0</sup> (0) as follows:

$$<\mathfrak{e}\_{i0}(t)\mathfrak{e}\_{i0}(\mathbf{0})> = <\mathfrak{e}\_{i0}(\mathbf{0})\mathfrak{e}\_{i0}(\mathbf{0})> \exp\left\{-\tilde{\eta}\_{0}^{\pm}(\mathbf{0})t\right\}\tag{107}$$

Insertion of (107) into (106) gives us

$$<\mathbf{e}\_{i0}(\mathbf{0})\mathbf{e}\_{i0}(\mathbf{0})> = \mathbf{3}\,\mathbf{m}^{\pm}\mathbf{k}\_{\mathrm{B}}\mathbf{T}\big(\tilde{\boldsymbol{\gamma}}\_{0}^{\pm}(\mathbf{0})\big)^{2}\tag{108}$$

Therefore, we obtain

m�<sup>2</sup> <**v**<sup>i</sup>

salt [29],

γ�ðÞ¼ *t* γ<sup>0</sup>

γ<sup>00</sup>

� þ γ<sup>01</sup>

And then we have

described as follows:

**88**

�ð Þ� 0 **v**<sup>j</sup>

*Electromagnetic Field Radiation in Matter*

�ðÞ ¼ *t* γ<sup>00</sup>

while the second one is related to the term <ϕ+� > .

<sup>σ</sup> ð Þ 0 *=*Z�

� ¼ �Z€�

In the dilute limit of n ≪ x, we have

neglected, the Laplace transformation of γ<sup>0</sup>

<sup>0</sup> ð Þ¼ 0 γ<sup>00</sup>

~γ�

Using (94) and (96), γ00� and γ01� are expressed as follows:

γ<sup>00</sup>

γ<sup>01</sup>

�*=*β<sup>0</sup>

<**ε**i0(*t*)**ε**i0(0) > is only related to either <ϕ+w > or < ϕ�<sup>w</sup> >.

<sup>¼</sup> <sup>1</sup>*=*3 m�kBT � � z2

where the auto-correlation function of random fluctuating force

At the dilution limit of electrolyte where the contribution of γ<sup>1</sup>

� <sup>¼</sup> <sup>x</sup><ϕ�<sup>w</sup> <sup>&</sup>gt;*=*3m�β<sup>0</sup>

0

∞

e2 � �<sup>ð</sup>

�ð Þ <sup>0</sup> <sup>&</sup>gt;~γ�ð Þ¼ <sup>ω</sup> <sup>m</sup>�<sup>2</sup> <sup>&</sup>lt; **<sup>v</sup>**<sup>i</sup>

The fluctuation dissipation theorem tells us the following relation:

<sup>~</sup>γ�ð Þ¼ <sup>ω</sup> <sup>1</sup>*=*3 m�kBT � � z2

In the long wavelength limit, this relation is expressed by

<sup>~</sup>γ�ð Þ¼ <sup>0</sup> <sup>1</sup>*=*3 m�kBT � � z2

� exp �β<sup>0</sup> �*<sup>t</sup>* � � � � <sup>þ</sup> <sup>γ</sup><sup>1</sup>

<sup>¼</sup> <sup>z</sup><sup>2</sup> e2 ð 0

e2 � �<sup>ð</sup>

�ð Þ� 0 **v**<sup>i</sup>

∞

0

e2 � �<sup>ð</sup>

Let us go back to the memory function γ�(t) and assume a combined exponential decay functions for it as follows, although this assumption is not exactly consistent with Eq. (84), but technically acceptable one as discussed in the case of molten

In this equation, the pre-exponential factor γ00� is subject to the interactions between the central ion and surrounding water molecules. The decaying constants are related to the time dependence of its configuration change. The pre-exponential factor, γ01�, is equal to the magnitude of memory function at t = 0 in respect to the friction force acting on the central cation or anion caused by interactions between its central ion and neighboring ions. In other words, the first term on the right hand side of this equation means the case of dilute limit of electrolytic solution and the second one is equal to the effective friction caused by the addition of a moderate amount of electrolyte. Therefore, the first term is related to either <ϕ+w > or < ϕ�<sup>w</sup> >,

∞

0

∞

�ðÞ ¼ *t* γ<sup>01</sup>

<sup>σ</sup> ð Þ¼ <sup>0</sup> <sup>x</sup><sup>&</sup>lt; <sup>ϕ</sup>�<sup>w</sup> <sup>&</sup>gt;*=*3m� <sup>þ</sup> <sup>n</sup><ϕþ� <sup>&</sup>gt;*=*3<sup>μ</sup> (103)

� � � <sup>¼</sup> <sup>x</sup><ϕ�<sup>w</sup> <sup>&</sup>gt;*=*3m� (104)

� � � <sup>¼</sup> <sup>n</sup><ϕþ� <sup>&</sup>gt;*=*3<sup>μ</sup> (105)

�

�(*t*) can be

�(*t*) in the long wavelength limit is then

<**ε**i0ð Þ*t* **ε**i0ð Þ 0 >d*t* (106)

�*t* � � � � � (102)

�ð Þ 0 >~γ�ð Þ ω

<sup>&</sup>lt; **<sup>ε</sup>**ið Þ*<sup>t</sup>* **<sup>ε</sup>**ið Þ <sup>0</sup> <sup>&</sup>gt;eiω*<sup>t</sup>* <sup>d</sup>*<sup>t</sup>* (99)

<sup>&</sup>lt;**ε**ið Þ*<sup>t</sup>* **<sup>ε</sup>**ið Þ <sup>0</sup> <sup>&</sup>gt;eiω*<sup>t</sup>* <sup>d</sup>*<sup>t</sup>* (100)

<**ε**ið Þ*t* **ε**ið Þ 0 >d*t* (101)

� exp �β<sup>1</sup>

$$\gamma\_0^{\pm}(t) = \left(\tilde{\gamma}\_0^{\pm}(\mathbf{0})\right)^2 \exp\left(-\tilde{\gamma}\_0^{\pm}(\mathbf{0})t\right) \tag{109}$$

Compare (106) and (109) we have

$$\mathfrak{H}\_0^{\pm} = \tilde{\eta}\_0^{\pm}(\mathbf{0}) = \left(\chi\_{00}^{\pm}\right)^{1/2} = \left(\mathbf{x} < \boldsymbol{\Phi}^{\pm \mathbf{w}} > / \mathbf{3} \mathbf{m}^{\pm}\right)^{1/2} \tag{110}$$

By the analogy with this relation, we can infer the following relation:

$$\mathfrak{H}\_1^{\pm} = \tilde{\eta}\_1^{\pm}(\mathbf{0}) = \left(\chi\_{01}^{\pm}\right)^{1/2} = \left(\mathbf{n} < \boldsymbol{\Phi}^{\pm \mathbf{w}} > / \mathbf{3}\boldsymbol{\mu}\right)^{1/2} \tag{111}$$

Therefore, Eq. (102) is explicitly written as follows:

$$\begin{split} \boldsymbol{\eta}^{\pm}(t) &= \boldsymbol{\eta}\_{0}^{\pm}(t) + \boldsymbol{\eta}\_{1}^{\pm}(t) \\ &= \left( \mathbf{x} < \boldsymbol{\Phi}^{\pm \mathbf{w}} > \boldsymbol{\beta} \mathbf{m}^{\pm} \right) \exp\left[ - \left( \mathbf{x} < \boldsymbol{\Phi}^{\pm \mathbf{w}} > \boldsymbol{\beta} \mathbf{m}^{\pm} \right)^{1/2} t \right] \\ &+ \left( \mathbf{n} < \boldsymbol{\Phi}^{+ -} > \boldsymbol{\beta} \mathbf{\mu} \right) \exp\left[ - \left( \mathbf{n} < \boldsymbol{\Phi}^{+ -} > \boldsymbol{\beta} \mathbf{\mu} \right)^{1/2} t \right] \end{split} \tag{112}$$

And the Laplace transformation of this equation in the long wavelength limit is equal to

$$\bar{\mathbf{y}}^{\pm}(\mathbf{0}) = \left[ \left( \mathbf{x} < \boldsymbol{\Phi}^{\pm \mathbf{w}} > \left/ 3\mathbf{m}^{\pm} \right)^{1/2} \right] + \left[ \left( \mathbf{n} < \boldsymbol{\Phi}^{+ -} > \left< 3\boldsymbol{\mu} \right)^{1/2} \right] \right. \tag{113}$$

### **15. Partial conductivities σ<sup>+</sup> and σ**�

Putting Eq. (113) into (79), we obtain the formulae of the partial conductivities, σ<sup>+</sup> and σ�, which are expressed in terms of the pair distribution functions and pair potentials as follows [28],

$$\begin{aligned} \mathbf{\sigma}^+ &= \mathbf{n} \mathbf{z}^2 \mathbf{e}^2 (\mathbf{1} - \mathbf{\delta}) / \mathbf{m}^+ \mathbf{\bar{\gamma}}^+(\mathbf{0}) \\ &= \mathbf{n} \mathbf{z}^2 \mathbf{e}^2 (\mathbf{1} - \mathbf{\delta}) / \mathbf{m}^+ \left[ \left( \mathbf{x} < \mathbf{\phi}^{+\mathbf{w}} > \left/ 3 \mathbf{m}^+ \right)^{1/2} + \left( \mathbf{n} < \mathbf{\phi}^{+-} > \left< 3 \mathbf{\mu} \right)^{1/2} \right) \right] \end{aligned} \tag{114}$$

and

$$\begin{aligned} \mathbf{\bar{\sigma}}^- &= \mathbf{n} \mathbf{z}^2 \mathbf{e}^2 (\mathbf{1} - \mathbf{\delta}) / \mathbf{m}^- \mathbf{\bar{\gamma}}^-(\mathbf{0}) \\ &= \mathbf{n} \mathbf{z}^2 \mathbf{e}^2 (\mathbf{1} - \mathbf{\delta}) / \mathbf{m}^- \left[ \left( \mathbf{x} < \boldsymbol{\phi}^{-\mathbf{w}} > \left/ \mathbf{3} \mathbf{m}^- \right)^{1/2} + \left( \mathbf{n} < \boldsymbol{\phi}^{+-} > \left/ \mathbf{3} \boldsymbol{\mu} \right)^{1/2} \right) \right] \end{aligned} \tag{115}$$

If the concentration c is defined as the number of moles of electrolyte in the unit volume (actually taken as 1 cc), then the number density n is equal to cNA, where NA being the Avogadro's number. Then, the partial conductivities, σ<sup>+</sup> and σ�, are written as follows:

**16. Pair potentials in electrolytic solution**

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

numerically equal to r0 = 5 A and κ = 3.44 A�<sup>1</sup>

neglected, may be given as follows:

method [46–48].

water-molecule.

behavior.

necessary or not.

is written as follows:

direction of cation.

**91**

according to Boltzmann law,

A number of research works to obtain the model potentials in electrolytic solutions have been presented since the Debye-Hückel theory [18]. In particular, various qualified model potentials, which satisfy the experimental data such as the hydration free energy and the enthalpies in condensed and gas phases, have recently been proposed in order to carry out the molecular dynamic simulation. It is not our intension to compare or evaluate these potentials and therefore we like to refer only some of these for our interests [24–27, 45]. It may be possible to estimate these potentials by using wave mechanical approach. In fact the ion-water molecule interactions were obtained by such an elaborating

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

The essential point for these model potentials in electrolytic solutions is that the dielectric character should be concerned. According to Sack [49], the watermolecules around the constituent ion are strongly oriented and the ion's orientating ability to neighboring water-molecules decreases with increasing of the distance between the ion and those water-molecules. Oka [50] also estimated the change of effective dielectric constant as a function of distance between the ion and

We propose the following model function to satisfy these results as follows:

where ε<sup>0</sup> (=78.35) is the dielectric constant of water. Other parameters are

The insertion of this dielectric function ε(*r*) for the long range part of interparticle potential is not necessary in molecular dynamic simulations. The dynamics produces automatically the configuration of constituents to satisfy the dielectric

On the analogy of the inter-ionic potentials in molten salts, ϕ+�(*r*) in aqueous solution, where the dipole-dipole and dipole-quadrupole dispersion forces are

where A+� is a constant in relation to the magnitude of repulsive force between

contact between cation i and anion j. A+� and B+� are also given in the literature [27]. The difference between this expression and that of ionic crystal or of molten salt is only ascribed to whether the introduction of the dielectric function ε(*r*) is

For simplicity, the pair potentials ϕ+ w(*r*) and ϕ� w(*r*) are assumed to be a combined form of the repulsive potential and the charge-dipole potential. Then the pair potential between cation and water molecule centered at the oxygen atom, ϕ+ w(*r*),

<sup>þ</sup><sup>w</sup>ð Þ� *<sup>r</sup>* <sup>z</sup>þe<sup>μ</sup> cos <sup>θ</sup> <sup>1</sup>–3l<sup>2</sup>

μ is the dipole of water molecule and l its length. θ is the dipole's angle in the

where ϕrep+w(*r*) is repulsive potential and its explicit form will be given be later.

It is well-known that the above expression is converted to the following form

<sup>ϕ</sup>þ�ð Þ¼ *<sup>r</sup>* <sup>z</sup>þz�e2 *<sup>=</sup>*f g *<sup>r</sup>* <sup>ε</sup>ð Þ*<sup>r</sup>* <sup>þ</sup> <sup>A</sup>þ� exp Bþ� di

cation i and anion j. B+� the softness parameter and (di

<sup>ϕ</sup><sup>þ</sup><sup>w</sup>ð Þ¼ *<sup>r</sup>* <sup>ϕ</sup>rep

εð Þ¼ *r* 1 þ ð Þ ε<sup>0</sup> � 1 ½ � 1 � exp f g �ð Þ *r* � r0 *=*κ (124)

, respectively.

<sup>þ</sup> þ dj

<sup>+</sup> + dj

*=*8*r* <sup>2</sup> *= r*

2

� � *<sup>r</sup>* (125)

�) is the hard core

<sup>ε</sup>ð Þ*<sup>r</sup>* (126)

$$\mathbf{e}^{+} = \mathbf{n} \mathbf{e} \mu^{+} = \mathbf{c} \mathbf{N}\_{\mathbf{A}} \mathbf{z}^{2} \mathbf{e}^{2} (\mathbf{1} - \mathbf{\delta}) / \mathbf{m}^{+} \left[ \left( \mathbf{x} < \boldsymbol{\Phi}^{+ \mathbf{w}} > \left/ \mathbf{3} \mathbf{m}^{+} \right)^{1/2} + \left( \mathbf{c} \mathbf{N}\_{\mathbf{A}} < \boldsymbol{\Phi}^{+ -} > \left/ \mathbf{3} \boldsymbol{\mu} \right)^{1/2} \right) \right] \tag{116}$$

and

$$\mathbf{\sigma}^- = \mathbf{n} \mathbf{e} \mu^- = \mathbf{c} \mathbf{N\_A} \mathbf{z}^2 \mathbf{e}^2 (\mathbf{1} \text{-6}) / \mathbf{m}^- \left[ \left( \mathbf{x} < \boldsymbol{\Phi}^{-\mathbf{w}} > \left/ 3 \mathbf{m}^- \right)^{1/2} + \left( \mathbf{c} \mathbf{N\_A} < \boldsymbol{\Phi}^{+-} > \left/ 3 \boldsymbol{\mu} \right)^{1/2} \right) \right] \tag{117}$$

In these equations, μ<sup>+</sup> and μ� are called as the mobility of cation and anion. The partial molar conductance Λ<sup>+</sup> and Λ� are defined as Λ� = σ�/c. Then the total conductance Λ<sup>c</sup> is written as follows:

$$\begin{split} \Lambda\_{\text{c}} &= \Lambda^{+} + \Lambda^{-} = \text{N}\_{\text{A}} \text{z}^{2} \text{e}^{2} \left( \text{1-\delta} \right) \times \left[ \text{1/m}^{+} \left\{ \left( \text{x} < \text{\phi}^{+\text{w}} > \text{\textdegree /3m}^{+} \right)^{1/2} + \left( \text{cN}\_{\text{A}} < \text{\phi}^{+-} > \text{\textdegree /3\mu} \right)^{1/2} \right\} \right] \\ &+ \text{1/m}^{-} \left\{ \left( \text{x} < \text{\phi}^{-\text{w}} > \text{\textdegree /3m}^{-} \right)^{1/2} + \left( \text{cN}\_{\text{A}} < \text{\phi}^{+-} > \text{\textdegree /3\mu} \right)^{1/2} \right\} \end{split} \tag{118}$$

Under the condition of n(=cNA) ≪ x, they are approximated to as follows:

$$\boldsymbol{\Lambda}^{+} = \mathbf{N}\_{\mathbf{A}} \mathbf{z}^{2} \mathbf{e}^{2} (\mathbf{1} - \mathbf{\hat{s}}) (\mathbf{3}/\mathbf{x} \mathbf{m}^{+} < \boldsymbol{\phi}^{+ \mathbf{w}} >)^{1/2} \left[ \mathbf{1} \mathbf{-} (\mathbf{c} \mathbf{N}\_{\mathbf{A}} \mathbf{m}^{+} < \boldsymbol{\phi}^{+ -} > / \mathbf{\mu} \mathbf{x} < \boldsymbol{\phi}^{+ \mathbf{w}} >)^{1/2} \right] \tag{119}$$

and

$$\mathbf{A}^- = \mathbf{N}\_\mathbf{A} \mathbf{z}^2 \mathbf{e}^2 (\mathbf{1} - \mathbf{\hat{s}}) (\mathbf{3}/\mathbf{x} \mathbf{m}^- < \boldsymbol{\phi}^{-\mathbf{w}} > )^{1/2} \left[ \mathbf{1} \mathbf{-} (\mathbf{c} \mathbf{N}\_\mathbf{A} \mathbf{m}^- < \boldsymbol{\phi}^{+-} > / \boldsymbol{\mu} \mathbf{x} < \boldsymbol{\phi}^{-\mathbf{w}} >)^{1/2} \right] \tag{120}$$

From Eqs. (119) and (120), we have a form of Λ<sup>c</sup> = (Λ<sup>+</sup> + Λ�) ≃ Λ<sup>0</sup> + Λ1– kc1/2. Λ<sup>0</sup> and k are written as follows:

$$\Lambda\_0 = \text{N}\_\text{A} \text{z}^2 \text{e}^2 \left\{ \left( 3/\text{xm}^+ < \phi^{+\text{w}} > \right)^{1/2} + \left( 3/\text{xm}^- < \phi^{-\text{w}} > \right)^{1/2} \right\} \tag{121}$$

$$\Lambda\_1 = \text{N}\_\text{A} \text{\&}^2 \text{e}^2 \left\{ \left( \text{3/xm}^+ < \text{\phi}^{+\text{w}} > \right)^{1/2} + \left( \text{3/xm}^- < \text{\phi}^{-\text{w}} > \right)^{1/2} \right\} \tag{122}$$

and

$$\mathbf{k} = \mathbf{N\_A} \mathbf{z}^2 \mathbf{e}^2 \left(\mathbf{1}\text{-\'6}\right) \left(\mathbf{3N\_A} \mathbf{<} \phi^{+-} > \left/\mu\right)^{1/2} \left\{ \left(\mathbf{1}\left(\mathbf{x} < \phi^{+\mathbf{w}} > + \left(\mathbf{1}\left(\mathbf{x} < \phi^{-\mathbf{w}} >\right)\right)\right) - \left(\mathbf{1}\mathbf{x} > \phi^{-\mathbf{w}} >\right)\right) \right\} \right.\tag{123}$$

As seen in these expressions, Λ<sup>0</sup> means the conductance in the dilution limit of electrolyte and Λ<sup>1</sup> is the correction term appeared by the so-called relaxation effect. The last term kc1/2 is composed of the so-called electrophoretic effect and the combined term of both effects.

In the case of a moderate concentration of electrolyte, in particular, of relatively weak electrolyte, we have to take account of the degree of association between the opposite ions into the expression for the partial conductivities.

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

#### **16. Pair potentials in electrolytic solution**

If the concentration c is defined as the number of moles of electrolyte in the unit volume (actually taken as 1 cc), then the number density n is equal to cNA, where NA being the Avogadro's number. Then, the partial conductivities, σ<sup>+</sup> and σ�, are

In these equations, μ<sup>+</sup> and μ� are called as the mobility of cation and anion. The partial molar conductance Λ<sup>+</sup> and Λ� are defined as Λ� = σ�/c. Then the

Under the condition of n(=cNA) ≪ x, they are approximated to as follows:

ð Þ <sup>1</sup>–<sup>δ</sup> <sup>3</sup>*=*xm� <sup>&</sup>lt;ϕ�<sup>w</sup> ð Þ <sup>&</sup>gt; <sup>1</sup>*=*<sup>2</sup> <sup>1</sup>– cNA <sup>m</sup>� <sup>&</sup>lt;ϕþ� <sup>&</sup>gt;*=*μx<ϕ�<sup>w</sup> ð Þ <sup>&</sup>gt;

From Eqs. (119) and (120), we have a form of Λ<sup>c</sup> = (Λ<sup>+</sup> + Λ�) ≃ Λ<sup>0</sup> + Λ1– kc1/2.

<sup>1</sup>*=*<sup>2</sup> <sup>þ</sup> <sup>3</sup>*=*xm� <sup>&</sup>lt;ϕ�<sup>w</sup> ð Þ <sup>&</sup>gt; <sup>1</sup>*=*<sup>2</sup> n o

<sup>1</sup>*=*<sup>2</sup> <sup>þ</sup> <sup>3</sup>*=*xm� <sup>&</sup>lt;ϕ�<sup>w</sup> ð Þ <sup>&</sup>gt; <sup>1</sup>*=*<sup>2</sup> n o

1*=*2

As seen in these expressions, Λ<sup>0</sup> means the conductance in the dilution limit of electrolyte and Λ<sup>1</sup> is the correction term appeared by the so-called relaxation effect. The last term kc1/2 is composed of the so-called electrophoretic effect and the

In the case of a moderate concentration of electrolyte, in particular, of relatively weak electrolyte, we have to take account of the degree of association between the

<sup>þ</sup>1*=*m� <sup>x</sup><sup>&</sup>lt; <sup>ϕ</sup>�<sup>w</sup> <sup>&</sup>gt; *<sup>=</sup>*3m� ð Þ<sup>1</sup>*=*<sup>2</sup> <sup>þ</sup> cNA <sup>&</sup>lt;ϕþ� ð Þ <sup>&</sup>gt;*=*3<sup>μ</sup> <sup>1</sup>*=*<sup>2</sup> <sup>n</sup> oi

e2 ð Þ <sup>1</sup>–<sup>δ</sup> <sup>3</sup>*=*xm<sup>þ</sup> <sup>&</sup>lt;ϕþ<sup>w</sup> ð Þ <sup>&</sup>gt;

<sup>e</sup><sup>2</sup> <sup>3</sup>*=*xm<sup>þ</sup> <sup>&</sup>lt;ϕ<sup>þ</sup><sup>w</sup> ð Þ <sup>&</sup>gt;

e2 <sup>3</sup>*=*xm<sup>þ</sup> <sup>&</sup>lt;ϕ<sup>þ</sup><sup>w</sup> ð Þ <sup>&</sup>gt;

opposite ions into the expression for the partial conductivities.

e2 ð Þ <sup>1</sup>–<sup>δ</sup> 3NA <sup>&</sup>lt;ϕþ� ð Þ <sup>&</sup>gt;*=*<sup>μ</sup>

ð Þ <sup>1</sup>–<sup>δ</sup> *<sup>=</sup>*m<sup>þ</sup> <sup>x</sup><ϕþ<sup>w</sup> <sup>&</sup>gt;*=*3m<sup>þ</sup> ð Þ1*=*<sup>2</sup> <sup>þ</sup> cNA <sup>&</sup>lt;ϕþ� ð Þ <sup>&</sup>gt;*=*3<sup>μ</sup>

ð Þ <sup>1</sup>–<sup>δ</sup> *<sup>=</sup>*m� <sup>x</sup><ϕ�<sup>w</sup> <sup>&</sup>gt;*=*3m� ð Þ1*=*<sup>2</sup> <sup>þ</sup> cNA <sup>&</sup>lt;ϕþ� ð Þ <sup>&</sup>gt;*=*3<sup>μ</sup>

e2 ð Þ� <sup>1</sup>–<sup>δ</sup> <sup>1</sup>*=*m<sup>þ</sup> <sup>x</sup> <sup>&</sup>lt;ϕ<sup>þ</sup><sup>w</sup> <sup>&</sup>gt;*=*3m<sup>þ</sup> ð Þ<sup>1</sup>*=*<sup>2</sup> <sup>þ</sup> cNA <sup>&</sup>lt; <sup>ϕ</sup>þ� ð Þ <sup>&</sup>gt; *<sup>=</sup>*3<sup>μ</sup> <sup>1</sup>*=*<sup>2</sup> <sup>h</sup> n o

1*=*2 h i

1*=*2 h i

<sup>1</sup>*=*<sup>2</sup> <sup>1</sup>– cNA <sup>m</sup><sup>þ</sup> <sup>&</sup>lt;ϕþ� <sup>&</sup>gt;*=*μx<sup>&</sup>lt; <sup>ϕ</sup>þ<sup>w</sup> ð Þ <sup>&</sup>gt; <sup>1</sup>*=*<sup>2</sup> h i

<sup>1</sup>*=*<sup>2</sup> h i

<sup>f</sup> <sup>1</sup>*=*x<ϕ<sup>þ</sup><sup>w</sup> <sup>&</sup>gt; <sup>þ</sup> <sup>1</sup>*=*x<ϕ�<sup>w</sup> <sup>ð</sup> <sup>ð</sup> <sup>&</sup>gt;Þg (123)

(116)

(117)

(118)

(119)

(120)

(121)

(122)

written as follows:

and

<sup>σ</sup><sup>þ</sup> <sup>¼</sup> neμ<sup>þ</sup> <sup>¼</sup> c NA z2

*Electromagnetic Field Radiation in Matter*

<sup>σ</sup>� <sup>¼</sup> neμ� <sup>¼</sup> c NA z2

<sup>Λ</sup><sup>c</sup> <sup>¼</sup> <sup>Λ</sup><sup>þ</sup> <sup>þ</sup> <sup>Λ</sup>� <sup>¼</sup> NA z2

<sup>Λ</sup><sup>þ</sup> <sup>¼</sup> NA z2

<sup>Λ</sup>� <sup>¼</sup> NAz2

e2

Λ<sup>0</sup> and k are written as follows:

<sup>Λ</sup><sup>0</sup> <sup>¼</sup> NAz<sup>2</sup>

<sup>Λ</sup><sup>1</sup> <sup>¼</sup> NAδz2

combined term of both effects.

and

and

**90**

<sup>k</sup> <sup>¼</sup> NA z2

e2

e2

total conductance Λ<sup>c</sup> is written as follows:

A number of research works to obtain the model potentials in electrolytic solutions have been presented since the Debye-Hückel theory [18]. In particular, various qualified model potentials, which satisfy the experimental data such as the hydration free energy and the enthalpies in condensed and gas phases, have recently been proposed in order to carry out the molecular dynamic simulation. It is not our intension to compare or evaluate these potentials and therefore we like to refer only some of these for our interests [24–27, 45]. It may be possible to estimate these potentials by using wave mechanical approach. In fact the ion-water molecule interactions were obtained by such an elaborating method [46–48].

The essential point for these model potentials in electrolytic solutions is that the dielectric character should be concerned. According to Sack [49], the watermolecules around the constituent ion are strongly oriented and the ion's orientating ability to neighboring water-molecules decreases with increasing of the distance between the ion and those water-molecules. Oka [50] also estimated the change of effective dielectric constant as a function of distance between the ion and water-molecule.

We propose the following model function to satisfy these results as follows:

$$e(r) = \mathbf{1} + (\mathbf{e}\_0 - \mathbf{1})[\mathbf{1} - \exp\left\{-(r - \mathbf{r}\_0)/\mathbf{x}\right\}] \tag{124}$$

where ε<sup>0</sup> (=78.35) is the dielectric constant of water. Other parameters are numerically equal to r0 = 5 A and κ = 3.44 A�<sup>1</sup> , respectively.

The insertion of this dielectric function ε(*r*) for the long range part of interparticle potential is not necessary in molecular dynamic simulations. The dynamics produces automatically the configuration of constituents to satisfy the dielectric behavior.

On the analogy of the inter-ionic potentials in molten salts, ϕ+�(*r*) in aqueous solution, where the dipole-dipole and dipole-quadrupole dispersion forces are neglected, may be given as follows:

$$\Phi^{+-}(r) = \left(\mathbf{z}^{+}\mathbf{z}^{-}\mathbf{e}^{2}\right) / \{re(r)\} + \mathbf{A}^{+-}\exp\left[\mathbf{B}^{+-}\left(\mathbf{d}\_{\mathrm{i}}^{+} + \mathbf{d}\_{\mathrm{j}}^{-} - r\right)\right] \tag{125}$$

where A+� is a constant in relation to the magnitude of repulsive force between cation i and anion j. B+� the softness parameter and (di <sup>+</sup> + dj �) is the hard core contact between cation i and anion j. A+� and B+� are also given in the literature [27]. The difference between this expression and that of ionic crystal or of molten salt is only ascribed to whether the introduction of the dielectric function ε(*r*) is necessary or not.

For simplicity, the pair potentials ϕ+ w(*r*) and ϕ� w(*r*) are assumed to be a combined form of the repulsive potential and the charge-dipole potential. Then the pair potential between cation and water molecule centered at the oxygen atom, ϕ+ w(*r*), is written as follows:

$$\phi^{+\mathbf{w}}(r) = \phi\_{\rm rep}^{\rm +\mathbf{w}}(r) - \mathbf{z}^{+} \mathbf{e}\mu \cos\theta (\mathbf{1} \text{-} \mathfrak{J}^{2} / 8r^{2}) / \left\{ r^{2} \varepsilon(r) \right\} \tag{126}$$

where ϕrep+w(*r*) is repulsive potential and its explicit form will be given be later. μ is the dipole of water molecule and l its length. θ is the dipole's angle in the direction of cation.

It is well-known that the above expression is converted to the following form according to Boltzmann law,

$$\boldsymbol{\Phi}^{+\mathrm{w}}(\boldsymbol{r}) = \boldsymbol{\phi}\_{\mathrm{rep}}\,^{+\mathrm{w}}(\boldsymbol{r}) - \mathrm{z}^{+2}\mathrm{e}^{2}\mu^{2}\left(\mathbf{1}\mathrm{-3l}^{2}/8\boldsymbol{r}^{2}\right) / \left\{\mathbf{3}\mathrm{k}\_{\mathrm{B}}\mathrm{Tr}\,\boldsymbol{r}^{2}\boldsymbol{e}(\boldsymbol{r})\right\}\tag{127}$$

On the other hand, a modified Lennard-Jones potential for water molecule, ϕww(*r*), is useful and is written as follows [45]:

$$\Phi^{\rm ww}(r) = 4\mathbf{C} \left[ \left( \mathbf{d}^{\rm w}/r \right)^{12} - \left( \mathbf{d}^{\rm w}/r \right)^{6} \right] - 2\mu^{2}/r^{3} \tag{128}$$

mþd**v**<sup>i</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

molecule q.

mþ**v**<sup>i</sup>

m�**v**�

and

n< mþ**v**<sup>i</sup>

¼ 1*=*VM

þ 1*=*VM

þð Þ¼ τ

<sup>k</sup> ð Þ¼ τ

ð 0

In a similar way, we have

ð 0

momenta is written as follows:

ð 0

<sup>þ</sup> nw <sup>&</sup>lt; <sup>m</sup><sup>w</sup>**v**<sup>q</sup>

and charge neutrality condition. Therefore, we have

different from the case of molten salts.

n< mþ**v**<sup>i</sup>

< **v**<sup>k</sup>

**93**

ð 0 τ

τ Nþ X and N�

<sup>τ</sup> <sup>N</sup>�<sup>X</sup> and Nþ

m<sup>w</sup>**v**<sup>q</sup>

l6¼k

þð Þτ > þ n< m�**v**<sup>k</sup>

<sup>w</sup>ð Þ¼ <sup>τ</sup>

j6¼i

þð Þ*t =*d*t* ¼

**f**ij d*t* þ

**f**kl d*t* þ

ð 0

<sup>τ</sup><sup>X</sup> **<sup>f</sup>**ij <sup>þ</sup> **<sup>f</sup>**ji � �d*<sup>t</sup>* <sup>þ</sup> <sup>1</sup>*=*VM

where nw is the number density of water molecules.

þð Þτ > þ n< m�**v**<sup>k</sup>

This equation indicates that the partial conductivity ratio < **v**<sup>i</sup>

�(τ) > is not equal to the inverse mass ratio m�/m<sup>+</sup>

water-molecules by each cation and anion, as x<sup>+</sup> and x�, we have

z<sup>þ</sup> þ z� ð Þe**E**d*t* þ n< mþ**v**<sup>i</sup>

ð 0

ð 0

<sup>τ</sup> <sup>X</sup> X

q6¼ð Þ k and i

In a unit volume, the total summation of the ensemble averages of these

�ð Þ<sup>τ</sup> <sup>&</sup>gt; <sup>þ</sup> nw <sup>&</sup>lt; <sup>m</sup><sup>w</sup>**v**<sup>q</sup>

ð 0

The summation of last three terms on the right hand side of this equation is equal to zero, because there is no external force at *t* = 0. All other terms on the right hand side of this equation are equal to zero by considering the law of action and reaction

�ð Þ<sup>τ</sup> <sup>&</sup>gt; <sup>þ</sup> nw <sup>&</sup>lt; mw**v**<sup>q</sup>

Some of water molecules may be simultaneously pulled by the dissolved ions under an external field **E**. Here, we neglect the relative time-relaxation for velocities of particles undergoing the co-operative motion. Taking the numbers of pulled

Here, xr is equal to the number density of un-pulled water molecules.

Nþ X and N�

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

j6¼i

At the time of steady state, τ, after applying the external field **E**, we have

<sup>τ</sup><sup>X</sup> X

q6¼i

<sup>τ</sup><sup>X</sup> X

q6¼k

**F**iq d*t* þ

**F**kq d*t* þ

**f**ij is the force acting on the ion i from the ion j and **F**iq is that from the water

**<sup>f</sup>**ij <sup>þ</sup><sup>X</sup> X

q6¼i

ð 0 τ

ð 0 τ

**<sup>F</sup>**iq <sup>þ</sup> **<sup>F</sup>**kq � �d*<sup>t</sup>* <sup>þ</sup> <sup>m</sup><sup>w</sup>**v**<sup>q</sup>

<sup>w</sup>ð Þ<sup>τ</sup> <sup>&</sup>gt;

þð Þ 0 > þ n< m�**v**<sup>k</sup>

nx<sup>þ</sup> <sup>þ</sup> nx� <sup>þ</sup> xr <sup>¼</sup> nw (138)

<sup>w</sup>ð Þ <sup>0</sup> <sup>&</sup>gt; (136)

<sup>τ</sup><sup>X</sup> **<sup>F</sup>**iq <sup>þ</sup> **<sup>F</sup>**kq <sup>þ</sup> **<sup>F</sup>**qi <sup>þ</sup> **<sup>F</sup>**qk � �d*<sup>t</sup>*

zþe**E**d*t* þ mþ**v**<sup>i</sup>

z�e**E**d*t* þ m�**v**<sup>k</sup>

**F**iq þ zþe**E** (132)

þð Þ 0 (133)

�ð Þ 0 (134)

<sup>w</sup>ð Þ <sup>0</sup> (135)

�ð Þ 0 >

<sup>w</sup>ð Þ<sup>τ</sup> <sup>&</sup>gt; <sup>¼</sup> 0 (137)

+ (τ)>/

, which is essentially

In this equation, the term 4C(dw/*r*) <sup>12</sup> is equal to the repulsive part and the parameters C and d<sup>w</sup> for water molecule in the gaseous state are equal to 230.9 kB and 2.824 Å, respectively.

The repulsive part of inter-ionic potential for ϕ++(*r*) may be approximately described as the form of A++exp[B++(d+ + d+ � *<sup>r</sup>*)] similar to the repulsive one in Eq. (125), since its interaction occurs around the distance of close contact where the dielectric behavior of neighboring water molecules must be neglected.

Now let us assume that the repulsive potential ϕrep+w(*r*) is represented by the root mean square of 4C(dw/*r*) <sup>12</sup> and A++exp[B++ (d+ + d+ - *r*)] as follows:

$$\boldsymbol{\Phi}\_{\rm rep}^{\rm +w}(\mathbf{r}) = \left\{ \mathbf{4C} (\mathbf{d}^{\rm w}/r)^{\rm 2} \, \mathbf{A}^{++} \exp\left[ \mathbf{B}^{++} \left( \mathbf{d}^{+} + \mathbf{d}^{+} - r \right) \right] \right\}^{1/2} \tag{129}$$

Insertion of (129) into Eq. (127) gives us the following expression,

$$\begin{aligned} \boldsymbol{\Phi}^{+\mathbf{w}}(r) &= \left\{ \mathbf{4C} (\mathbf{d}^{\mathbf{w}}/r)^{12} \mathbf{A}^{++} \exp\left[ \mathbf{B}^{++} (\mathbf{d}^{+} + \mathbf{d}^{+} - r) \right] \right\}^{1/2} \\ &- \mathbf{z}^{2} \mathbf{e}^{2} \mu^{2} (\mathbf{1} \mathbf{-3} \mathbf{l}^{2} / 8 \mathbf{r}^{2}) / \left\{ \mathbf{3k}\_{\mathrm{B}} \mathrm{Tr}(r) r^{2} \right\} \end{aligned} \tag{130}$$

In a similar way, the inter-particle potential between anion and water molecule is expressed as follows:

$$\Phi^{-\mathbf{w}}(r) = \left\{ 4\mathbf{C} (\mathbf{d}^{\mathbf{w}}/r)^{12} \mathbf{A}^{--} \exp\left[\mathbf{B}^{--} (\mathbf{d}^{-} + \mathbf{d}^{-} - r)\right] \right\}^{1/2} $$
 
$$ -\mathbf{z}^{2} \mathbf{e}^{2} \mu^{2} (\mathbf{1} \mathbf{-} \mathbf{J}^{2} / 8r^{2}) / \left\{ 3 \mathbf{k}\_{\mathsf{B}} \mathrm{Tr} \left(r^{2}\right) \right\} \tag{131} $$

The dipole moment of water molecule is known to be μ = 0.38 (in the unit of e times 1 Å = 1.6 � <sup>10</sup>�<sup>29</sup> <sup>C</sup>�m) and l <sup>≒</sup>0.5 Å. Therefore, all parameters in (130) and (131) are known. According to Bopp et al. [51], the repulsive parts in (130) and (131) are converted to the exponential decaying functions similar to the repulsive part in (125) [46, 47].

Under these circumstances, it is possible to use either our empirical expressions (130) and (131), or to apply the inter-particle potentials derived by Bopp et al. [51]. It is also possible to estimate the repulsion terms in (130) and (131) by using wave mechanical approach. In fact, the ion-water molecule interactions were obtained by such an elaborating method [33, 52]. However, we will use the above empirical potentials for numerical application, for simplicity.

#### **17. Momentum conservation and the tag of water molecules by ion's movement**

We will investigate the tag of water molecules by ion's moving in the electrolytic solutions from the view point of equation of motion under an applied field **E** [28].

Under this situation, the second law of motion for the cation i can be written as follows:

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

$$\mathbf{m}^+ \mathbf{d} \mathbf{v}\_{\mathbf{i}}^+(t) / \mathbf{d}t = \sum\_{\mathbf{j} \neq \mathbf{i}}^{N+\text{and }N-} \mathbf{f}\_{\mathbf{i}\mathbf{j}} + \sum\_{\mathbf{q} \neq \mathbf{i}}^{X} \mathbf{F}\_{\mathbf{i}\mathbf{q}} + \mathbf{z}^+ \mathbf{e} \mathbf{E} \tag{132}$$

**f**ij is the force acting on the ion i from the ion j and **F**iq is that from the water molecule q.

At the time of steady state, τ, after applying the external field **E**, we have

$$\mathbf{m}^+\mathbf{v}\_{\mathbf{i}}^+(\tau) = \int\_0^{\tau\,\mathbf{N}+\textrm{and}} \sum\_{\mathbf{j}\neq\mathbf{i}}^{\tau\,\mathbf{N}+\textrm{and}} \mathbf{f}\_{\mathbf{i}\mathbf{j}}\,\mathbf{d}\mathbf{t} + \int\_0^{\tau} \sum\_{\mathbf{q}\neq\mathbf{i}}^{\mathbf{X}} \mathbf{F}\_{\mathbf{i}\mathbf{q}}\,\mathbf{d}\mathbf{t} + \int\_0^{\tau} \mathbf{z}^+\mathbf{e}\mathbf{E}\,\mathbf{d}\mathbf{t} + \mathbf{m}^+\mathbf{v}\_{\mathbf{i}}^+(\mathbf{0})\tag{133}$$

In a similar way, we have

$$\mathbf{m}^-\mathbf{v}\_{\mathbf{k}}^-(\tau) = \int\_0^{\tau\,\mathrm{N}-\mathrm{and}} \sum\_{\mathbf{l}\neq\mathbf{k}}^{\tau\,\mathrm{N}-\mathrm{and}} \mathbf{f}\_{\mathrm{kl}}\,\mathrm{d}t + \int\_0^{\tau} \sum\_{\mathbf{q}\neq\mathbf{k}}^{\mathrm{X}} \mathbf{F}\_{\mathrm{k}\mathbf{q}}\,\mathrm{d}t + \int\_0^{\tau} \mathbf{z}^-\,\mathrm{e}\mathbf{E}\,\mathrm{d}t + \,\mathrm{m}^-\mathbf{v}\_{\mathrm{k}}^-(\mathbf{0})\tag{134}$$

and

<sup>ϕ</sup>þwð Þ¼ *<sup>r</sup>* <sup>ϕ</sup>rep

*Electromagnetic Field Radiation in Matter*

ϕww(*r*), is useful and is written as follows [45]:

In this equation, the term 4C(dw/*r*)

<sup>þ</sup><sup>w</sup>ð Þ¼ <sup>r</sup> 4C d<sup>w</sup> ð Þ *<sup>=</sup><sup>r</sup>*

<sup>ϕ</sup>þ<sup>w</sup>ð Þ¼ *<sup>r</sup>* 4C d<sup>w</sup> ð Þ *<sup>=</sup><sup>r</sup>*

� z2 e2

<sup>ϕ</sup>�<sup>w</sup>ð Þ¼ *<sup>r</sup>* 4C d<sup>w</sup> ð Þ *<sup>=</sup><sup>r</sup>*

� <sup>z</sup><sup>2</sup> e2

potentials for numerical application, for simplicity.

μ<sup>2</sup> 1–3l<sup>2</sup>

μ<sup>2</sup> 1–3l2

*=*8*r* <sup>2</sup> � �*=* 3kBTε *r*

and 2.824 Å, respectively.

root mean square of 4C(dw/*r*)

ϕrep

is expressed as follows:

part in (125) [46, 47].

**movement**

as follows:

**92**

<sup>þ</sup>wð Þ� *<sup>r</sup>* <sup>z</sup>þ<sup>2</sup>

<sup>ϕ</sup>wwð Þ¼ *<sup>r</sup>* 4C d<sup>w</sup> ð Þ *<sup>=</sup><sup>r</sup>*

e2

μ<sup>2</sup> 1–3l<sup>2</sup>

On the other hand, a modified Lennard-Jones potential for water molecule,

parameters C and d<sup>w</sup> for water molecule in the gaseous state are equal to 230.9 kB

The repulsive part of inter-ionic potential for ϕ++(*r*) may be approximately described as the form of A++exp[B++(d+ + d+ � *<sup>r</sup>*)] similar to the repulsive one in Eq. (125), since its interaction occurs around the distance of close contact where the

Now let us assume that the repulsive potential ϕrep+w(*r*) is represented by the

<sup>12</sup> and A++exp[B++ (d+ + d+

<sup>12</sup> <sup>A</sup>þþ exp Bþþ <sup>d</sup><sup>þ</sup> <sup>þ</sup> <sup>d</sup><sup>þ</sup> � *<sup>r</sup>* n o � � � �

<sup>12</sup> <sup>A</sup>þþ exp Bþþ <sup>d</sup><sup>þ</sup> <sup>þ</sup> <sup>d</sup><sup>þ</sup> � *<sup>r</sup>* n o � � � �

<sup>12</sup> <sup>A</sup>�� exp B�� <sup>d</sup>� <sup>þ</sup> <sup>d</sup>� ½ � ð Þ � *<sup>r</sup>* n o<sup>1</sup>*=*<sup>2</sup>

dielectric behavior of neighboring water molecules must be neglected.

Insertion of (129) into Eq. (127) gives us the following expression,

*<sup>=</sup>*8r<sup>2</sup> � �*<sup>=</sup>* 3kBTεð Þ*<sup>r</sup> <sup>r</sup>*

In a similar way, the inter-particle potential between anion and water molecule

The dipole moment of water molecule is known to be μ = 0.38 (in the unit of e times 1 Å = 1.6 � <sup>10</sup>�<sup>29</sup> <sup>C</sup>�m) and l <sup>≒</sup>0.5 Å. Therefore, all parameters in (130) and (131) are known. According to Bopp et al. [51], the repulsive parts in (130) and (131) are converted to the exponential decaying functions similar to the repulsive

Under these circumstances, it is possible to use either our empirical expressions (130) and (131), or to apply the inter-particle potentials derived by Bopp et al. [51]. It is also possible to estimate the repulsion terms in (130) and (131) by using wave mechanical approach. In fact, the ion-water molecule interactions were obtained by such an elaborating method [33, 52]. However, we will use the above empirical

**17. Momentum conservation and the tag of water molecules by ion's**

We will investigate the tag of water molecules by ion's moving in the electrolytic solutions from the view point of equation of motion under an applied field **E** [28]. Under this situation, the second law of motion for the cation i can be written

*=*8*r* <sup>2</sup> � �*=* 3kBT*r*

<sup>12</sup> � <sup>d</sup><sup>w</sup> ð Þ *<sup>=</sup>*<sup>r</sup> 6 h i 2

� <sup>2</sup>μ<sup>2</sup> *=r*

<sup>12</sup> is equal to the repulsive part and the


1*=*2

<sup>2</sup> � � (130)

<sup>2</sup> � � � � (131)

1*=*2

(129)

<sup>ε</sup>ð Þ*<sup>r</sup>* � � (127)

<sup>3</sup> (128)

$$\mathbf{m}^{\mathbf{w}}\mathbf{v}\_{\mathbf{q}}^{\mathbf{w}}(\tau) = \int\_{0}^{\tau} \sum\_{\mathbf{q} \neq (\mathbf{k} \text{ and } \mathbf{i})}^{\mathbf{X}} \left(\mathbf{F}\_{\mathbf{i}\mathbf{q}} + \mathbf{F}\_{\mathbf{k}\mathbf{q}}\right) \mathbf{d}t + \mathbf{m}^{\mathbf{w}}\mathbf{v}\_{\mathbf{q}}^{\mathbf{w}}(\mathbf{0})\tag{135}$$

In a unit volume, the total summation of the ensemble averages of these momenta is written as follows:

$$\begin{split} \mathbf{n} &< \mathbf{m}^+ \mathbf{v}\_{\mathrm{i}}^+(\tau) > + \mathbf{n} < \mathbf{m}^- \mathbf{v}\_{\mathrm{k}}^-(\tau) > + \mathbf{n}\_{\mathrm{w}} < \mathbf{m}^\mathbf{w} \mathbf{v}\_{\mathrm{q}}^\mathbf{w}(\tau) \\ &= 1/\mathcal{V}\_{\mathrm{M}} \Big[ \int\_0^\tau \sum (\mathbf{f}\_{\mathrm{ij}} + \mathbf{f}\_{\mathrm{ji}}) \mathrm{d}t + \mathbf{1}/\mathcal{V}\_{\mathrm{M}} \Big]\_0^\tau \sum (\mathcal{F}\_{\mathrm{iq}} + \mathcal{F}\_{\mathrm{k}q} + \mathcal{F}\_{\mathrm{qi}} + \mathcal{F}\_{\mathrm{q}k}) \mathrm{d}t \\ &+ \mathbf{1}/\mathcal{V}\_{\mathrm{M}} \Big[ \int\_0^\tau (\mathbf{z}^+ + \mathbf{z}^-) \mathrm{e} \mathrm{E} \mathrm{d}t + \mathbf{n} < \mathbf{m}^+ \mathbf{v}\_{\mathrm{i}}^+(\mathbf{0}) > + \mathbf{n} < \mathbf{m}^- \mathbf{v}\_{\mathrm{k}}^-(\mathbf{0}) > \\ &+ \mathbf{n}\_{\mathrm{w}} < \mathbf{m}^\mathbf{w} \mathbf{v}\_{\mathrm{q}}^\mathbf{w}(\mathbf{0}) > \end{split} \tag{136}$$

where nw is the number density of water molecules.

The summation of last three terms on the right hand side of this equation is equal to zero, because there is no external force at *t* = 0. All other terms on the right hand side of this equation are equal to zero by considering the law of action and reaction and charge neutrality condition.

Therefore, we have

$$\mathbf{n} \mathbf{<} \mathbf{m}^+ \mathbf{v}\_{\mathbf{i}}^+(\mathbf{r}) > \mathbf{+} \mathbf{n} \mathbf{<} \mathbf{m}^- \mathbf{v}\_{\mathbf{k}}^-(\mathbf{r}) > \mathbf{+} \mathbf{n}\_{\mathbf{w}} \mathbf{<} \mathbf{m}^w \mathbf{v}\_{\mathbf{q}}^w(\mathbf{r}) > \mathbf{=} \mathbf{0} \tag{137}$$

This equation indicates that the partial conductivity ratio < **v**<sup>i</sup> + (τ)>/ < **v**<sup>k</sup> �(τ) > is not equal to the inverse mass ratio m�/m<sup>+</sup> , which is essentially different from the case of molten salts.

Some of water molecules may be simultaneously pulled by the dissolved ions under an external field **E**. Here, we neglect the relative time-relaxation for velocities of particles undergoing the co-operative motion. Taking the numbers of pulled water-molecules by each cation and anion, as x<sup>+</sup> and x�, we have

$$\mathbf{n}\mathbf{x}^+ + \mathbf{n}\mathbf{x}^- + \mathbf{x}^\mathbf{r} = \mathbf{n}\_\mathbf{w} \tag{138}$$

Here, xr is equal to the number density of un-pulled water molecules.

Since, the movements of remainder water molecules under the external field must be isotropic, we have xr < mw**v**w(τ) > = 0. Then nw < mw**v**<sup>q</sup> w(τ) > is expressed as follows:

$$\mathbf{m}\_{\rm w} < \mathbf{m}^{\rm w} \mathbf{v}\_{\rm q}^{\rm w}(\boldsymbol{\pi}) > \\ = \mathbf{x}^{+} < \mathbf{m}^{\rm w} \mathbf{v}\_{\rm i}^{+}(\boldsymbol{\pi}) > + \mathbf{x}^{-} < \mathbf{m}^{\rm w} \mathbf{v}\_{\rm k}^{-}(\boldsymbol{\pi}) > \tag{139}$$

procedure of MD simulation of electrolyte aqueous solution will be briefly described as follows for reader's benefit. In MD for the electrolyte aqueous solution, the rigid body models (TIP4P) [57] are used for water molecules. The interactions between constituent TIP4P water molecules are expressed as the charged L-J type potentials, as,

> *zizje*<sup>2</sup> *r* þ *A <sup>r</sup>*<sup>12</sup> � *<sup>B</sup>*

The interactions between alkali metal cation and halide anion, TIP4P- alkali

*zizje*<sup>2</sup> *r* þ *C <sup>r</sup>*<sup>9</sup> � *<sup>D</sup>*

In (144) and (145), i and j stand for the constituent atoms; *e* is the elementary charge. The used charges for the constituent species zi and the interaction parameters are taken from the literature; TIP4P – TIP4P [57]; TIP4P – alkali metal cation, TIP4P – halide anon, between alkali metal cation, between halide anion, and between alkali metal cation and halide anion [58]. The Ewald method is used for the calculation of the Coulomb interaction. For the structure calculation, MD is performed in NTP constant condition [59–61] under the pressure of 1 atm at 283 K. MD is performed for 50,000 steps with 0.1 fs one time step in 1.1% NaCl aqueous solution. MD cell contains about 10,000 molecules (i.e. 30,000 atoms) for the calculation of the

**Solute Water (TIP4P) Cation Anion** Li<sup>+</sup> Cl� 10,000 112 112 Na+ Cl� 10,000 112 112 K<sup>+</sup> Cl� 10,000 112 112

*Pair distribution function of water molecules around a Li + ion,* g*Li-w(*r*) in the electrolyte solution of LiCl. And numbers of locating water molecules around a Li + ion within the sphere of the length* r *centered at its*

*Li + ion,* n*Li-w(*r*), in its solution obtained by MD simulation.*

*<sup>r</sup>*<sup>6</sup> (144)

*<sup>r</sup>*<sup>6</sup> (145)

*ij*ð Þ¼ *r*

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

*ij*ð Þ¼ *r*

metal anion, and TIP4P – halide anion are expressed as [58]:

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

**Table 2.**

**Figure 5.**

**95**

*The numbers of ions in MD cell.*

Insertion of this equation into (66) gives us the following relation:

$$<\langle \mathbf{m}^+ + \mathbf{x}^+ \mathbf{m}^\mathbf{w} \rangle \mathbf{v}^+(\mathbf{r}) > \langle \mathbf{m}^- + \mathbf{x}^- \mathbf{m}^\mathbf{w} \rangle \mathbf{v}^-(\mathbf{r}) > = \mathbf{0} \tag{140}$$

Hereafter, we omit the suffix of ion i or k. Therefore, we have

$$|<\mathbf{v}^+(\tau)>|/|<\mathbf{v}^-(\tau)>|=(\mathbf{m}^-+\mathbf{x}^-\mathbf{m}^\mathbf{w})/(\mathbf{m}^++\mathbf{x}^+\mathbf{m}^\mathbf{w})\tag{141}$$

We cannot apply the above treatment for H+ and OH� ions, because their conduction mechanisms differ from that of all other dissolved ions. Their mechanisms are known as the Grotthus-type conduction which is a kind of hopping conduction of electrons or holes [3].

It is, however, straightforward to obtain the following relation for all dissolved ions in their dilute limits except for H+ and OH� ones,

$$|<\mathbf{v}^+(\mathbf{r})>|(\mathbf{m}^+ + \mathbf{x}^+\mathbf{m}^\mathbf{w})=|<\mathbf{v}^-(\mathbf{r})>|(\mathbf{m}^- + \mathbf{x}^-\mathbf{m}^\mathbf{w})\tag{142}$$

This relation seems to be valid for all aqueous solutions of equivalent electrolytes in the dilution limit.

Using Eqs. (114) and (115), Eq. (142) for the dilution limit of electrolytic solution is expressed as follows:

<sup>σ</sup>þ*=*σ� <sup>¼</sup> mass of an anion plus masses of water molecules pulled by its anion *<sup>=</sup>* mass ofa cation plus masses of water molecules pulled by its cation

$$= (\mathbf{m}^- + \mathbf{x}^- \mathbf{m}^\mathbf{w})/(\mathbf{m}^+ + \mathbf{x}^+ \mathbf{m}^\mathbf{w}) = (\mathbf{m}^- < \boldsymbol{\Phi}^{-\mathbf{w}} > )^{1/2}/(\mathbf{m}^+ < \boldsymbol{\Phi}^{+\mathbf{w}} > )^{1/2} \tag{143}$$

This equation may correspond to the inverse mass ratio for the partial conductivities of molten salt [6].

#### **18. Numerical results in electrolytic solutions**

According to the theoretical results we have discussed so far, the pair distribution functions appear in the essential equations [28]. Therefore, how to obtain the pair distribution functions is one of the matters of vital importance.

There are several standard theoretical methods to obtain the pair distribution functions in molecular liquids from the knowledge of inter-particle potentials [33]. In the calculation of site-site distribution function for such a molecular liquid, the reference interaction-site model (RISM) approximation proposed by Chandler and Anderson [52] seems to be useful. Until the present time, the extension of RISM approximation, in order to obtain the potentials of mean force and also the site-site pair distribution functions ɡμν(r)'s in electrolytic solutions, has been carried out by several authors [53–55]. These attempts cover the insufficient experimental knowledge for pair distribution functions ɡ+�(*r*), ɡ+ w(*r*) and ɡ� <sup>w</sup> (*r*).

However, we will use the ɡμν(*r*)'s in aqueous solution of sodium chloride obtained by our own MD simulation. The essential numerical procedure of MD simulation in this study is same as our previous works of molten salts [56]. The

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

procedure of MD simulation of electrolyte aqueous solution will be briefly described as follows for reader's benefit. In MD for the electrolyte aqueous solution, the rigid body models (TIP4P) [57] are used for water molecules. The interactions between constituent TIP4P water molecules are expressed as the charged L-J type potentials, as,

$$
\omega\_{\vec{\eta}}(r) = \frac{\mathbf{z}\_i \mathbf{z}\_j e^2}{r} + \frac{A}{r^{12}} - \frac{B}{r^6} \tag{144}
$$

The interactions between alkali metal cation and halide anion, TIP4P- alkali metal anion, and TIP4P – halide anion are expressed as [58]:

$$
\omega\_{\vec{\eta}}(r) = \frac{\mathbf{z}\_i \mathbf{z}\_j e^2}{r} + \frac{\mathbf{C}}{r^9} - \frac{D}{r^6} \tag{145}
$$

In (144) and (145), i and j stand for the constituent atoms; *e* is the elementary charge. The used charges for the constituent species zi and the interaction parameters are taken from the literature; TIP4P – TIP4P [57]; TIP4P – alkali metal cation, TIP4P – halide anon, between alkali metal cation, between halide anion, and between alkali metal cation and halide anion [58]. The Ewald method is used for the calculation of the Coulomb interaction. For the structure calculation, MD is performed in NTP constant condition [59–61] under the pressure of 1 atm at 283 K. MD is performed for 50,000 steps with 0.1 fs one time step in 1.1% NaCl aqueous solution. MD cell contains about 10,000 molecules (i.e. 30,000 atoms) for the calculation of the


**Table 2.**

Since, the movements of remainder water molecules under the external field must be

<sup>&</sup>lt; <sup>m</sup><sup>þ</sup> <sup>þ</sup> <sup>x</sup>þm<sup>w</sup> ð Þ**v**þð Þ<sup>τ</sup> <sup>&</sup>gt; <sup>þ</sup> <sup>&</sup>lt; <sup>m</sup>� <sup>þ</sup> <sup>x</sup>�m<sup>w</sup> ð Þ**v**�ð Þ<sup>τ</sup> <sup>&</sup>gt; <sup>¼</sup> 0 (140)

<sup>∣</sup><**v**þð Þ<sup>τ</sup> <sup>&</sup>gt;∣*=*∣<**v**�ð Þ<sup>τ</sup> <sup>&</sup>gt; <sup>∣</sup> <sup>¼</sup> <sup>m</sup>� <sup>þ</sup> <sup>x</sup>�m<sup>w</sup> ð Þ*<sup>=</sup>* <sup>m</sup><sup>þ</sup> <sup>þ</sup> <sup>x</sup>þm<sup>w</sup> ð Þ (141)

We cannot apply the above treatment for H+ and OH� ions, because their conduction mechanisms differ from that of all other dissolved ions. Their mechanisms are known as the Grotthus-type conduction which is a kind of hopping

It is, however, straightforward to obtain the following relation for all dissolved

This relation seems to be valid for all aqueous solutions of equivalent electrolytes

Using Eqs. (114) and (115), Eq. (142) for the dilution limit of electrolytic solu-

This equation may correspond to the inverse mass ratio for the partial conduc-

According to the theoretical results we have discussed so far, the pair distribution functions appear in the essential equations [28]. Therefore, how to obtain the

There are several standard theoretical methods to obtain the pair distribution functions in molecular liquids from the knowledge of inter-particle potentials [33]. In the calculation of site-site distribution function for such a molecular liquid, the reference interaction-site model (RISM) approximation proposed by Chandler and Anderson [52] seems to be useful. Until the present time, the extension of RISM approximation, in order to obtain the potentials of mean force and also the site-site pair distribution functions ɡμν(r)'s in electrolytic solutions, has been carried out by several authors [53–55]. These attempts cover the insufficient experimental knowl-

However, we will use the ɡμν(*r*)'s in aqueous solution of sodium chloride obtained by our own MD simulation. The essential numerical procedure of MD simulation in this study is same as our previous works of molten salts [56]. The

σþ*=*σ� ¼ mass of an anion plus masses of water molecules pulled by its anion

<sup>∣</sup><**v**þð Þ<sup>τ</sup> <sup>&</sup>gt;<sup>∣</sup> <sup>m</sup><sup>þ</sup> <sup>þ</sup> <sup>x</sup>þmw ð Þ¼ <sup>∣</sup><**v**�ð Þ<sup>τ</sup> <sup>&</sup>gt;<sup>∣</sup> <sup>m</sup>� <sup>þ</sup> <sup>x</sup>�m<sup>w</sup> ð Þ (142)

 *=* mass ofa cation plus masses of water molecules pulled by its cation 

*<sup>=</sup>* <sup>m</sup><sup>þ</sup> <sup>&</sup>lt;ϕþ<sup>w</sup> ð Þ <sup>&</sup>gt;

<sup>1</sup>*=*<sup>2</sup> (143)

þð Þ<sup>τ</sup> <sup>&</sup>gt; <sup>þ</sup> <sup>x</sup>� <sup>&</sup>lt; <sup>m</sup><sup>w</sup>**v**<sup>k</sup>

w(τ) > is expressed as follows:

�ð Þτ > (139)

isotropic, we have xr < mw**v**w(τ) > = 0. Then nw < mw**v**<sup>q</sup>

Hereafter, we omit the suffix of ion i or k.

ions in their dilute limits except for H+ and OH� ones,

<sup>¼</sup> <sup>m</sup>� <sup>þ</sup> <sup>x</sup>�m<sup>w</sup> ð Þ*<sup>=</sup>* <sup>m</sup><sup>þ</sup> <sup>þ</sup> <sup>x</sup>þm<sup>w</sup> ð Þ¼ <sup>m</sup>� <sup>&</sup>lt;ϕ�<sup>w</sup> ð Þ <sup>&</sup>gt; <sup>1</sup>*=*<sup>2</sup>

**18. Numerical results in electrolytic solutions**

pair distribution functions is one of the matters of vital importance.

edge for pair distribution functions ɡ+�(*r*), ɡ+ w(*r*) and ɡ� <sup>w</sup> (*r*).

<sup>w</sup>ð Þ<sup>τ</sup> <sup>&</sup>gt; <sup>¼</sup> <sup>x</sup><sup>þ</sup> <sup>&</sup>lt; <sup>m</sup><sup>w</sup>**v**<sup>i</sup>

Insertion of this equation into (66) gives us the following relation:

nw < m<sup>w</sup>**v**<sup>q</sup>

*Electromagnetic Field Radiation in Matter*

Therefore, we have

in the dilution limit.

tion is expressed as follows:

tivities of molten salt [6].

**94**

conduction of electrons or holes [3].

*The numbers of ions in MD cell.*

#### **Figure 5.**

*Pair distribution function of water molecules around a Li + ion,* g*Li-w(*r*) in the electrolyte solution of LiCl. And numbers of locating water molecules around a Li + ion within the sphere of the length* r *centered at its Li + ion,* n*Li-w(*r*), in its solution obtained by MD simulation.*

structure and the velocity autocorrelation function for alkali halide aqueous solution. The numbers of the constituent ions in the MD cell are listed in **Table 2**.

The main part of MD is performed using SIGRESS ME package (Fujitsu) at the supercomputing facilities in Kyushu University.

The obtained figures of ɡij(*r*) are shown in **Figures 5**–**8**. And using these data, we have estimated the numbers of water molecules involved within a sphere of radius r from the centered ion, nij(r) (i = Li<sup>+</sup> , Na<sup>+</sup> , K<sup>+</sup> and Cl; j = oxygen of water molecule) = 4πʃ<sup>0</sup> <sup>r</sup> ɡij(*r*)r2 dr, which are also figured in them.

Using Eq. (143), that is, σ<sup>+</sup> /σ = (m + xmw)/(m+ + x<sup>+</sup> mw), and taking an assumption that the pulling water molecules for Na<sup>+</sup> ion is equal to 6.0 although its

#### **Figure 6.**

*Pair distribution function of water molecules around a Na + ion,* g*Na-w(*r*) in the electrolyte solution of NaCl. And numbers of locating water molecules around a Na + ion within the sphere of the length* r *centered at its Na + ion,* n*Na-w(*r*), in its solution obtained by MD simulation.*

plausible justification seems to be difficult, then we obtain the pulling water molecules for other ions as shown in **Table 3**, in which the hydration numbers seen in a

*Assumption that the pulling number x+ of Na+ ion is equal to be 6.0 and also that the pulling numbers of water*

*Pair distribution function of water molecules around a Cl ion,* g*Cl-w(*r*) in the electrolyte solution of MCl (M = Li, Na and K). And numbers of locating water molecules around a Cl ion within the sphere of the length*

> **Hydration numbers in the text book [36]**

**Li<sup>+</sup> 7.6 4.3 0.6 4.1** Na+ 6.0\* 5.6 1.7 5.7 <sup>K</sup><sup>+</sup> 2.8 5.5 1.3 6.4 Cl 2.8 6.0 0.7 6.5

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

r *centered at its Cl ion,* n*Cl-w(*r*), in its solution obtained by MD simulation.*

*molecules for Cl are not changed even for that the pairing positive ions are different.*

*Numbers of pulling water molecules, x<sup>+</sup> or x and hydration numbers.*

Using these pulling numbers for the constituent ions, we can estimate the term,

both terms are satisfactory, which is a kind of proof for the assumption x+ is equal to 6.0. It is emphasized that the pulling number of water molecules by moving ion has no relation to the hydration number of water molecules as seen in **Table 3**. The hydration of water molecules around electrolytic ions is originated essentially by the thermodynamic stability which is related not only to the interaction energies among ions and water molecules but also to the configuration entropy terms. This is because that the pulling number is not always related to the hydration one.

**19. Discussion on the electrical conductivities in electrolytic solutions**

The present theory seems essentially comparable to the treatments developed by Onsager [19], Fuoss et al. [21], Prigogine [20], Friedman [23], Chandra and Bagchi

mw) as shown in **Table 4**. As seen in this table, agreements for

**Hydration numbers obtained from MD simulations**

text book [62] and our results obtained by MD simulation, for reference.

(m + xmw)/(m+ + x+

**Ions Pulling water molecules, x<sup>+</sup> or x**

**Figure 8.**

*\**

**97**

**Table 3.**

[27], and Matsunaga and Tamaki [28].

#### **Figure 7.**

*Pair distribution function of water molecules around a K+ ion,* g*K-w(*r*) in the electrolyte solution of KCl. And numbers of locating water molecules around a K+ ion within the sphere of the length* r *centered at its K+ ion,* n*K-w(*r*), in its solution obtained by MD simulation.*

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

#### **Figure 8.**

structure and the velocity autocorrelation function for alkali halide aqueous solution.

The main part of MD is performed using SIGRESS ME package (Fujitsu) at the

The obtained figures of ɡij(*r*) are shown in **Figures 5**–**8**. And using these data, we have estimated the numbers of water molecules involved within a sphere of radius

/σ = (m + xmw)/(m+ + x<sup>+</sup>

, K<sup>+</sup> and Cl; j = oxygen of water

mw), and taking an

, Na<sup>+</sup>

assumption that the pulling water molecules for Na<sup>+</sup> ion is equal to 6.0 although its

*Pair distribution function of water molecules around a Na + ion,* g*Na-w(*r*) in the electrolyte solution of NaCl. And numbers of locating water molecules around a Na + ion within the sphere of the length* r *centered at its*

*Pair distribution function of water molecules around a K+ ion,* g*K-w(*r*) in the electrolyte solution of KCl. And numbers of locating water molecules around a K+ ion within the sphere of the length* r *centered at its K+ ion,*

<sup>r</sup> ɡij(*r*)r2 dr, which are also figured in them.

The numbers of the constituent ions in the MD cell are listed in **Table 2**.

supercomputing facilities in Kyushu University.

*Na + ion,* n*Na-w(*r*), in its solution obtained by MD simulation.*

n*K-w(*r*), in its solution obtained by MD simulation.*

r from the centered ion, nij(r) (i = Li<sup>+</sup>

*Electromagnetic Field Radiation in Matter*

Using Eq. (143), that is, σ<sup>+</sup>

molecule) = 4πʃ<sup>0</sup>

**Figure 6.**

**Figure 7.**

**96**

*Pair distribution function of water molecules around a Cl ion,* g*Cl-w(*r*) in the electrolyte solution of MCl (M = Li, Na and K). And numbers of locating water molecules around a Cl ion within the sphere of the length* r *centered at its Cl ion,* n*Cl-w(*r*), in its solution obtained by MD simulation.*


*\* Assumption that the pulling number x+ of Na+ ion is equal to be 6.0 and also that the pulling numbers of water molecules for Cl are not changed even for that the pairing positive ions are different.*

#### **Table 3.**

*Numbers of pulling water molecules, x<sup>+</sup> or x and hydration numbers.*

plausible justification seems to be difficult, then we obtain the pulling water molecules for other ions as shown in **Table 3**, in which the hydration numbers seen in a text book [62] and our results obtained by MD simulation, for reference.

Using these pulling numbers for the constituent ions, we can estimate the term, (m + xmw)/(m+ + x+ mw) as shown in **Table 4**. As seen in this table, agreements for both terms are satisfactory, which is a kind of proof for the assumption x+ is equal to 6.0.

It is emphasized that the pulling number of water molecules by moving ion has no relation to the hydration number of water molecules as seen in **Table 3**. The hydration of water molecules around electrolytic ions is originated essentially by the thermodynamic stability which is related not only to the interaction energies among ions and water molecules but also to the configuration entropy terms. This is because that the pulling number is not always related to the hydration one.

#### **19. Discussion on the electrical conductivities in electrolytic solutions**

The present theory seems essentially comparable to the treatments developed by Onsager [19], Fuoss et al. [21], Prigogine [20], Friedman [23], Chandra and Bagchi [27], and Matsunaga and Tamaki [28].


**References**

[1] Janz GJ. Molten Salts Handbook. New

*DOI: http://dx.doi.org/10.5772/intechopen.91369*

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions*

Restriction on the Distribution of Ions in

[12] Hansen JP, McDonald IR. Statistical mechanics of dense ionized matter. IV. Density and charge fluctuations in a simple molten salt. Physics Review.

Electrolytes. 49, 1991; 1968

[13] Giaquinta PV, Parrinello M, Tosi MP. Hydrodynamic correlation functions for molten salts. Physics and Chemistry of Liquids. 1976;**5**:305

[14] Cicotti G, Jaccuci G, McDonald IR. Transport properties of molten alkali halides. Physics Review. 1976;**A13**:426

[16] Biggin S, Enderby JE. The structure of molten calcium chloride. Journal of

[17] Matsunaga S, Koishi T, Tamaki S. Velocity correlation functions and partial conductivities of molten AgI-

[18] Debye P, Hückel E. The theory of electrolytes. I. Lowering of freezing point and related phenomena. Physikalishce Zeitschrift. 1923;**24**:185

[19] Onsager L. On the theory of

[20] Prigogine I. Non-Equilibrium Statistical Mechanics. New York: Wiley;

1927;**28**:277

1962

1959

electrolytes. II. Physikalische Zeitschrift.

[21] Fuoss RM, Accascina F. Electrolytic Conductance. New York: Interscience;

[15] Trullàs J, Padró JA. Diffusive transport properties in monovalent and divalent metal-ion halide melts: A computer simulation study. Physics

Review. 1996;**B55**:12210

Physics C. 1981;**14**:3577

AgBr by molecular dynamics simulation. Materials Science and Engineering A. 2007;**449-451**:693

1975;**A11**:2111

[2] Smedley SI. The Interpretation of Ionic Conductivity in Liquids. New

[3] Conway BE. In: Eyring H, editor. Physical Chemistry (an Advanced Treatise) Vol. IX A; Electrochemistry. New York: Academic Press; 1970

[4] Sundheim BR. Transference numbers in molten Salta. The Journal of Physical

[5] Koishi T, Arai Y, Shirakawa Y, Tamaki S. Transport coefficients in molten NaCl by computer simulation. Journal of the Physical Society of Japan.

[6] Koishi T, Tamaki S. Partial

Physics. 2002;**116**:3018

**121**:333

**99**

**123**(19):194501

Physics. 1964;**40**:1347

conductivities of a molten salt based on Langevin equation. Journal of the Physical Society of Japan. 1999;**68**:964

[7] Koishi T, Kawase S, Tamaki S. A theory of electrical conductivity of molten salt. The Journal of Chemical

[8] Koishi T, Tamaki S. A theory of electrical conductivity of molten salt II. The Journal of Chemical Physics. 2004;

[9] Koishi T, Tamaki S. A theory of transport properties of molten salts. The Journal of Chemical Physics. 2005;

[10] Berne BJ, Rice SA. On the kinetic theory of dense fluids, XVI; the ideal ionic melt. The Journal of Chemical

[11] Stillinger FH, Lovett R. Ion-pair theory of concentrated electrolytes. I. Basic concept. The Journal of Chemical

Physics. 1968;**48**:3858. General

York: Academic Press; 1967

York: Plenum Press; 1980

Chemistry. 1956;**60**:1381

1997;**66**:3188

**Table 4.**

*The ratio of ionic conductivity and the calculation results by using Table 3.*

Friedman [23] used a technique of diagram expansion starting from Kubo-Green formula for the conductivity of electrolytic solution and the obtained expression was also written in the form of Λ<sup>c</sup> = (Λ<sup>+</sup> + Λ) = Λ<sup>0</sup> + Λ<sup>1</sup> – kc1/2. However, his theory is very much sophisticated and too mathematical to understand with a physical insight.

Recent theoretical work carried out by Chandra and Bagchi [27] is basically started from a Kubo-Green type theory, that is, the partial conductivities are derived from velocity correlation functions. Their treatment seems to be a modernized and beautiful and therefore it is very much appreciable. However, the friction force of their theory involves various terms which make it difficult to calculate practically the partial conductivities. In fact, there still remains the task to represent a microscopic formula for Λ0.

The present treatment is easily to understand in view of physical insight and is successful for deriving the formula of Λ0.

The short-time expansion forms for < **v**<sup>i</sup> + (*t*) **v**<sup>i</sup> + (0)>, < **v**<sup>k</sup> (*t*) **v**<sup>k</sup> (0) > and < **v**<sup>i</sup> + (*t*) **v**<sup>k</sup> (0) > are expressed in terms inter-particle potentials and corresponding pair distribution functions as seen in (95) and (97). In the case of molten salts, all these velocity correlation functions yield some physical quantities in relation to a part of partial conductivities [6]. In the present case, however, Z<sup>σ</sup> + (*t*) and Z<sup>σ</sup> (*t*) play its role. Such an essential difference between the case of molten salt and the electrolytic solution may be ascribed to the difference in the momentum conservation of the system.

#### **Author details**

Shigeru Tamaki<sup>1</sup> , Shigeki Matsunaga<sup>2</sup> \* and Masanobu Kusakabe<sup>3</sup>

1 Niigata University, Niigata, Japan

2 National Institute of Technology, Nagaoka College, Nagaoka, Japan

3 Niigata Institute of Technology, Kashiwazaki, Japan

\*Address all correspondence to: matsu@nagaoka-ct.ac.jp

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

#### **References**

Friedman [23] used a technique of diagram expansion starting from Kubo-Green formula for the conductivity of electrolytic solution and the obtained expression was also written in the form of Λ<sup>c</sup> = (Λ<sup>+</sup> + Λ) = Λ<sup>0</sup> + Λ<sup>1</sup> – kc1/2. However, his theory is very much sophisticated and too mathematical to understand with a physical

Li<sup>+</sup> Cl 0.595 0.598 Na+ Cl 0.659 0.655 K<sup>+</sup> Cl 0.963 0.960

*The ratio of ionic conductivity and the calculation results by using Table 3.*

**/σ (m + xmw)/(m+**

Recent theoretical work carried out by Chandra and Bagchi [27] is basically started from a Kubo-Green type theory, that is, the partial conductivities are derived from velocity correlation functions. Their treatment seems to be a modernized and beautiful and therefore it is very much appreciable. However, the friction force of their theory involves various terms which make it difficult to calculate practically the partial conductivities. In fact, there still remains the task to represent

The present treatment is easily to understand in view of physical insight and is

pair distribution functions as seen in (95) and (97). In the case of molten salts, all these velocity correlation functions yield some physical quantities in relation to a

play its role. Such an essential difference between the case of molten salt and the electrolytic solution may be ascribed to the difference in the momentum conserva-

part of partial conductivities [6]. In the present case, however, Z<sup>σ</sup>

2 National Institute of Technology, Nagaoka College, Nagaoka, Japan

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

, Shigeki Matsunaga<sup>2</sup>

3 Niigata Institute of Technology, Kashiwazaki, Japan

\*Address all correspondence to: matsu@nagaoka-ct.ac.jp

1 Niigata University, Niigata, Japan

provided the original work is properly cited.

+ (*t*) **v**<sup>i</sup> +

(0) > are expressed in terms inter-particle potentials and corresponding

\* and Masanobu Kusakabe<sup>3</sup>

(0)>, < **v**<sup>k</sup>

(*t*) **v**<sup>k</sup>

+

(0) > and

**+ x+ mw)**

(*t*)

(*t*) and Z<sup>σ</sup>

insight.

**Table 4.**

< **v**<sup>i</sup> + (*t*) **v**<sup>k</sup>

tion of the system.

**Author details**

Shigeru Tamaki<sup>1</sup>

**98**

a microscopic formula for Λ0.

successful for deriving the formula of Λ0. The short-time expansion forms for < **v**<sup>i</sup>

**Electrolyte σ<sup>+</sup>**

*Electromagnetic Field Radiation in Matter*

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[7] Koishi T, Kawase S, Tamaki S. A theory of electrical conductivity of molten salt. The Journal of Chemical Physics. 2002;**116**:3018

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[9] Koishi T, Tamaki S. A theory of transport properties of molten salts. The Journal of Chemical Physics. 2005; **123**(19):194501

[10] Berne BJ, Rice SA. On the kinetic theory of dense fluids, XVI; the ideal ionic melt. The Journal of Chemical Physics. 1964;**40**:1347

[11] Stillinger FH, Lovett R. Ion-pair theory of concentrated electrolytes. I. Basic concept. The Journal of Chemical Physics. 1968;**48**:3858. General

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[12] Hansen JP, McDonald IR. Statistical mechanics of dense ionized matter. IV. Density and charge fluctuations in a simple molten salt. Physics Review. 1975;**A11**:2111

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[14] Cicotti G, Jaccuci G, McDonald IR. Transport properties of molten alkali halides. Physics Review. 1976;**A13**:426

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#### *Electrical Conductivity of Molten Salts and Ionic Conduction in Electrolyte Solutions DOI: http://dx.doi.org/10.5772/intechopen.91369*

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*Electromagnetic Field Radiation in Matter*

Physics. 1964;**40**:3276

in ionic Solutions, 537; 1964

electrolytes. The Journal of Chemical

[32] March NH, Tosi MP. Atomic Dynamics in Liquids. London: The

[34] Mori H. Transport, collective motion, and Brownian motion. Progress in Theoretical Physics. 1965;**33**:423-455

[35] Mori H. A continued-fraction representation of the time-correlation functions. Progress of Theoretical

[36] Copley JRD, Lovesey SW. The dynamic properties of monatomic liquids. Reports on Progress in Physics.

[37] Tankeshwar K, Pathak KN,

Ranganathan S. Self-diffusion coefficients of Lennard-Jones fluids. Journal of Physics C: Solid State Physics. 1987;**20**:5749

[38] Joslin CG, Gray CG. Calculation of transport coefficients using a modified Mori formalism. Molecular Physics.

[39] Douglass DC. Self Difusion and velocity correlation. The Journal of Chemical Physics. 1961;**35**(1):81

[40] Levesque D, Verlet L. Computer "experiment" on classical fluids III. Physics Review. 1970;**A2**(1970):2514

[42] Balucani U, Zoppi M. Dynamics of the Liquid State. Clarendon: Oxford; 1994

[43] Hoshino K, Shimojo F, Munejiri S. Mode-coupling analysis of atomic dynamics for liquid Ge, Sn and Na. Journal of the Physical Society of Japan.

[41] Tankeshwar K, Pathak K, Ranganathan S. Theory of transport coefficients of simple liquids. Journal of Physics: Condensed Matter. 1990;

**2**(1990):5891-5905

2002;**71**(1):119-124

Physics. 1965;**34**:399-416

1975;**38**:461-563

1986;**58**:789

[33] Hansen JP, McDonald IR. Theory of Simple Liquids. 2nd ed. New York:

MacMillan Press Ltd; 1976

Academic; 1986

[23] Friedman HL. A cluster expansion for the electrical conductance of solutions. Physica. 1964;**30**:509. On the limiting law for electrical conductance

[24] Chandra A, Wei D, Patey GN. The frequency dependent conductivity of electrolyte solutions. The Journal of Chemical Physics. 1993;**99**:2083

[25] Smith DE, Dang LX. Computer simulations of NaCl association in polarizable water. The Journal of Chemical Physics. 1994;**100**:3757

[26] Koneshan S, Rasaiaha JC. Computer simulation studies of aqueous sodium chloride solutions of 298 K and 683 K. The Journal of Chemical Physics. 2000;

[27] Chandra A, Bagchi B. Ion conductance in electrolyte solutions. The Journal of Chemical Physics. 1999;**110**:10024

conduction in electrolyte solution. Journal of Solution Chemistry. 2014;**43**:1771

[29] Kusakabe M, Takeno S, Koishi T, Matsunga S, Tamaki S. A theoretical

conductivities of molten salts. Molecular

[30] Tosi MP, Fumi FG. Ionic sizes and born repulsive parameters in the NaCltype alkali halides—II: The generalized Huggins-Mayer form. Journal of Physics and Chemistry of Solids. 1964;**25**:45

[31] Edwards FG, Enderby JE, How RA, Page DI. The structure of molten sodium chloride. Journal of Physics C.

[28] Matsunaga S, Tamaki S. Ionic

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[45] Berendsen HJC, Griegera JR, Straatsma TP. The missing term in effective pair potentials. The Journal of Chemical Physics. 1987;**9**:6269

[46] Clementi E, Popkie H. Study of the structure of molecular complexes. I. Energy surface of a water molecule in the field of a lithium positive ion. The Journal of Chemical Physics. 1972;**57**:1077

[47] McGlynn SP et al. Introduction to Applied Quantum Chemistry. New York: Rinehart and Winston Inc.; 1972, 1972

[48] Kistenmacher H, Popkie H, Clemnti E. Study of the structure of molecular complexes. V., heat of formation for the Li<sup>+</sup> , Na<sup>+</sup> , K+ , F, and Cl ion complexes with a single water molecule. The Journal of Chemical Physics. 1974;**59**:5842

[49] Sack H. The dielectric constants of solutions of electrolytes at small concentrations. Physikalishce Zeitschrift. 1927;**28**:199

[50] Oka S. Über den Sattigungszustand einen Dipolflussigkeit in der Umgebung eines Ions. Proceedings of the Physico-Mathematical Society of Japan. 1932;**14**: 441

[51] Bopp P, Dietz W, Heinzinger K. A molecular dynamics study of aqueous solutions X.: First results for a NaCl solution with a central force model for water. Zeitschrift für Naturforschung. 1979;**34a**:1424-1435

[52] Chandler D, Anderson HC. Optimized expansions for classical fluids. II. Theory of molecular liquids. The Journal of Chemical Physics. 1972;**57**:1930

[53] Hirata F, Rossky PJ, Pettitt BM. The interionic potential of mean force in a molecular polar solvent from an

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[54] Pettitt BM, Rossky PJ. Alkali halides in water: Ion-solvent correlations and ion-ion potentials of mean force at infinite dilution. The Journal of Chemical Physics. 1986;**84**:5836

[55] Kovalenko A, Hirata F. Potentials of mean force of simple ions in anbient aqueous solution. I: Three dimensional reference interaction site model approach. The Journal of Chemical Physics. 2000; **112**:10391. II: Solvation structure from the three-dimensional reference interaction site model approach, and comparison with simulation. 10403(2000)

[56] Matsunaga S. Structural features in molten RbAg4I5 by molecular dynamics simulation. Molecular Simulation. 2013; **39**:119

[57] Jorgensen WL, Chandrasekhar J, Madura JD, Impey RW, Klein ML. Comparison of simple partial functions for simulating liquid water. The Journal of Chemical Physics. 1983;**79**:926

[58] Zhengwei P, Ewig CS, Hwang M-J, Waldman M, Hagler AT. Derivation of class II force field. 4. Van der Waals parameters of alkali metal Cations and halide anions. The Journal of Physical Chemistry. A. 1997;**101**:7243

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[60] Nosé S. A molecular dynamics method for simulations in the canonical ensemble. Molecular Physics. 1984;**52**:255

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[62] Conway BE. Ionic Hydration in Chemistry and Biophysics. Amsterdam: Elsevier; 1981

**103**

**Chapter 6**

**Abstract**

**1. Introduction**

surface of its body.

*Leonid Chervinsky*

Study of the Electromagnetic

Radiation on the Animal Body

The rapid technical development of human society on Earth leads to the pollution of its atmosphere and an increase in the electromagnetic radiation of the Sun and its main part—light and ultraviolet radiation. In order to properly protect and control the effects of electromagnetic radiation on the human body, it is necessary to know and understand the process of absorption and conversion of electromagnetic radiation falling on the surface of the body. The material contains the original results of experimental studies on electromagnetic radiation transmission through a sample of quasi-vital skin with pigs of different ages. The reliable results of the percentage ratio of the amount of electromagnetic radiation of the optical spectrum that passes under the skin through the skin layer and the individual wool depending on the species and age of the animal are obtained. The results of the experiment showed that the electromagnetic radiation of the Sun affects the body of the animal through the skin, as well as inside the cylinders of separate wool. This new knowledge is important for biologists and applied engineers to monitor and control electromagnetic radiation for young and old animals with different wools.

**Keywords:** electromagnetic radiation, transmission, reflection, scattering and

In life, when a person or an animal is under the influence of electromagnetic radiation of the Sun or a special source of electromagnetic radiation of different spectral composition, the biological effect causes radiation energy to enter the

The human body is covered with clothes. The body of the animal is protected by different hairs or wools. It is known that the electromagnetic radiation of the optical spectrum of the short wavelength range (ultraviolet radiation) is very active in the body. Therefore, it is of scientific interest to study the penetration of optical radiation of different wavelengths (spectrum) into the human and animal body. Experimental studies were carried out quickly on the skin with the hair of recently dead animals.

The results of such studies are presented below in the described material.

**2. Methods and installation for studying the paths of penetration of optical and, in particular, ultraviolet radiation into the animal's body**

According to the law of geometric optics, the interaction of the optical radiation with the irradiated biological object is characterized by the spectral optical

absorption, skin, animal body surface, conductivity, separate wool

#### **Chapter 6**

## Study of the Electromagnetic Radiation on the Animal Body

*Leonid Chervinsky*

#### **Abstract**

The rapid technical development of human society on Earth leads to the pollution of its atmosphere and an increase in the electromagnetic radiation of the Sun and its main part—light and ultraviolet radiation. In order to properly protect and control the effects of electromagnetic radiation on the human body, it is necessary to know and understand the process of absorption and conversion of electromagnetic radiation falling on the surface of the body. The material contains the original results of experimental studies on electromagnetic radiation transmission through a sample of quasi-vital skin with pigs of different ages. The reliable results of the percentage ratio of the amount of electromagnetic radiation of the optical spectrum that passes under the skin through the skin layer and the individual wool depending on the species and age of the animal are obtained. The results of the experiment showed that the electromagnetic radiation of the Sun affects the body of the animal through the skin, as well as inside the cylinders of separate wool. This new knowledge is important for biologists and applied engineers to monitor and control electromagnetic radiation for young and old animals with different wools.

**Keywords:** electromagnetic radiation, transmission, reflection, scattering and absorption, skin, animal body surface, conductivity, separate wool

#### **1. Introduction**

In life, when a person or an animal is under the influence of electromagnetic radiation of the Sun or a special source of electromagnetic radiation of different spectral composition, the biological effect causes radiation energy to enter the surface of its body.

The human body is covered with clothes. The body of the animal is protected by different hairs or wools. It is known that the electromagnetic radiation of the optical spectrum of the short wavelength range (ultraviolet radiation) is very active in the body. Therefore, it is of scientific interest to study the penetration of optical radiation of different wavelengths (spectrum) into the human and animal body. Experimental studies were carried out quickly on the skin with the hair of recently dead animals. The results of such studies are presented below in the described material.

#### **2. Methods and installation for studying the paths of penetration of optical and, in particular, ultraviolet radiation into the animal's body**

According to the law of geometric optics, the interaction of the optical radiation with the irradiated biological object is characterized by the spectral optical

absorption coefficients *α*(*λ*), the reflection *ρ*(*λ*), and the transmission *τ*(*λ*), which are interconnected by the dependence (1):

$$a(\lambda) + \pi(\lambda) + \rho(\lambda) = \mathbf{1} \tag{1}$$

Measurement of spectral optical characteristics (absorption coefficients *α*(*λ*), reflection *ρ*(*λ*), and transmission *τ*(*λ*)) in the cover of animals is associated with the difficulties of the methodological and technical capabilities in the formation of the experiment.

Firstly, there is a large discrepancy in the magnitude of the coefficients. Based on the analysis of the literature [1, 2] and the results of our own research [3–6], it was found that the magnitude of the reflection coefficients of optical radiation for the shelter of animals is within 0.10–0.60 incident radiation and the transmittance is less than 0.05–0.00001 incident radiation (depending on the thickness of the coating and body pigmentation). Using one method and device measures is impossible without large errors. Therefore, for their measurements, use different methods and different technical devices.

Secondly, the measurement of the optical coefficients of individual skin and wool is associated with similar difficulties due to the large difference in the geometric dimensions of the samples studied: the cross-sectional area of an individual wool has an order of magnitude 0.01 mm<sup>2</sup> , the area of the skin between the individual hairs is 1 mm2 , and the area of irradiation of the skin and wool is regulated by the size of the animal (about 1 m<sup>2</sup> ), and therefore various technical means are used to measure them.

Thirdly, in the system research of the spectrum of optical radiation, there is a need for variable optical systems. In order to increase the flux of the intensity of incident or measured radiation (thus increasing the accuracy of the measurement), the focusing optical systems are used in the studies, which, unfortunately, have different degrees of transparency for different sections of the optical radiation spectrum. So, for the measurement of ultraviolet radiation, optics are used from crystal glass and for visible radiation, from "ordinary" glass, and for infrared radiation, from a special high-temperature quartz glass.

#### **Figure 1.**

*The block diagram of the installation for the investigation of spectral optical characteristics of the cover of animals.*

**105**

*Study of the Electromagnetic Radiation on the Animal Body*

of the universal luminescent microscope "LUMAM-I3."

of the investigated object micro- and macrosize of the area: from 1 cm2

• Study of optical characteristics of large areas of skin with a wool

Our long-term research in this area has allowed us to develop a method for determining all the named spectral coefficients of the skin and animal wool and the creation of a universal installation for measuring spectral optical coefficients based on a universal fluorescent microscope LUMAM-I3 (Russia) (http://www.laborato-

characteristics of individual wool, areas of clear skin, and areas of the skin with

Determination of the spectral coefficients of absorption, reflection, and transmission of the skin and wool cover and its components in animals of different species, age, and breed was summarized on this research facility, made on the basis

The structural-optical scheme of the universal device for the study of the optical

The versatility of the proposed installation lies in the fact that the shift of focusing lenses allows you to focus on the irradiation beam (cut from the flow of the optical radiation from the source by a system of quartz and interference filter system) on the plane

this case, the sensitivity of the photodetector by the corrective optical system is regu-

sensitivity to the radiation spectrum. Due to this, it is possible to conduct structural investigations of optical characteristics in the same direction at the same installation:

to 1 W/mm2

• Study of optical characteristics of skin micro areas between individual wools

• Study of optical characteristics and light conductivity of individual wool in ani-

It is important that as a result of the study of the optical characteristics and their analysis, it is possible to determine the quantitative scheme of the distribution of electromagnetic radiation parts of the optical spectrum penetrating the skin separately through the skin, through the cylinder separate wool, and as a whole through

In order to increase the reliability of the results obtained in the experiments of measuring the radiation flux from the source was carried out with the help of a photoelectron multiplier, the signal from which was perceived through an amplifier with a sensitive galvanometer, or the corresponding interface through an analog-

Meanwhile, the measurements of the measuring galvanometer have always been

This condition allowed for the calculation of optical coefficients to accurately determine the relative radiation flux through the indicators of the galvanometer,

In particular, the transmission coefficients of the optical radiation in the investi-

τλ = \_ *I*λ *IO*

where *I*λ is the photocurrent in measuring the radiation flux with wavelength *λ* passing through the sample (μΑ) and *I*ολ is the photocurrent in measuring the

directly proportional to the flux emitted by the photomultiplier (the condition was satisfied so that the intensity of the radiation flux coming into the perimeter window of the photomultiplier was within the linear section of its sensitivity

digital converter was fed for mathematical processing on a computer.

gated samples were determined by the formula (2):

to 0.01 mm2

and takes into account its selective

(2)

. In

*DOI: http://dx.doi.org/10.5772/intechopen.89430*

rium.dp.ua/item/53).

wool is shown in **Figure 1**.

lated in the range from 0.01 mW/mm2

mals of all ages and species

leather and wool.

characteristic).

and not in energy units.

*Study of the Electromagnetic Radiation on the Animal Body DOI: http://dx.doi.org/10.5772/intechopen.89430*

*Electromagnetic Field Radiation in Matter*

experiment.

hairs is 1 mm2

measure them.

different technical devices.

has an order of magnitude 0.01 mm<sup>2</sup>

from a special high-temperature quartz glass.

size of the animal (about 1 m<sup>2</sup>

are interconnected by the dependence (1):

absorption coefficients *α*(*λ*), the reflection *ρ*(*λ*), and the transmission *τ*(*λ*), which

Measurement of spectral optical characteristics (absorption coefficients *α*(*λ*), reflection *ρ*(*λ*), and transmission *τ*(*λ*)) in the cover of animals is associated with the difficulties of the methodological and technical capabilities in the formation of the

Firstly, there is a large discrepancy in the magnitude of the coefficients. Based on the analysis of the literature [1, 2] and the results of our own research [3–6], it was found that the magnitude of the reflection coefficients of optical radiation for the shelter of animals is within 0.10–0.60 incident radiation and the transmittance is less than 0.05–0.00001 incident radiation (depending on the thickness of the coating and body pigmentation). Using one method and device measures is impossible without large errors. Therefore, for their measurements, use different methods and

Secondly, the measurement of the optical coefficients of individual skin and wool is associated with similar difficulties due to the large difference in the geometric dimensions of the samples studied: the cross-sectional area of an individual wool

Thirdly, in the system research of the spectrum of optical radiation, there is a need for variable optical systems. In order to increase the flux of the intensity of incident or measured radiation (thus increasing the accuracy of the measurement), the focusing optical systems are used in the studies, which, unfortunately, have different degrees of transparency for different sections of the optical radiation spectrum. So, for the measurement of ultraviolet radiation, optics are used from crystal glass and for visible radiation, from "ordinary" glass, and for infrared radiation,

, and the area of irradiation of the skin and wool is regulated by the

*α(λ) + τ(λ) + ρ(λ) = 1* (1)

, the area of the skin between the individual

), and therefore various technical means are used to

**104**

**Figure 1.**

*animals.*

*The block diagram of the installation for the investigation of spectral optical characteristics of the cover of* 

Our long-term research in this area has allowed us to develop a method for determining all the named spectral coefficients of the skin and animal wool and the creation of a universal installation for measuring spectral optical coefficients based on a universal fluorescent microscope LUMAM-I3 (Russia) (http://www.laboratorium.dp.ua/item/53).

The structural-optical scheme of the universal device for the study of the optical characteristics of individual wool, areas of clear skin, and areas of the skin with wool is shown in **Figure 1**.

Determination of the spectral coefficients of absorption, reflection, and transmission of the skin and wool cover and its components in animals of different species, age, and breed was summarized on this research facility, made on the basis of the universal luminescent microscope "LUMAM-I3."

The versatility of the proposed installation lies in the fact that the shift of focusing lenses allows you to focus on the irradiation beam (cut from the flow of the optical radiation from the source by a system of quartz and interference filter system) on the plane of the investigated object micro- and macrosize of the area: from 1 cm2 to 0.01 mm2 . In this case, the sensitivity of the photodetector by the corrective optical system is regulated in the range from 0.01 mW/mm2 to 1 W/mm2 and takes into account its selective sensitivity to the radiation spectrum. Due to this, it is possible to conduct structural investigations of optical characteristics in the same direction at the same installation:


It is important that as a result of the study of the optical characteristics and their analysis, it is possible to determine the quantitative scheme of the distribution of electromagnetic radiation parts of the optical spectrum penetrating the skin separately through the skin, through the cylinder separate wool, and as a whole through leather and wool.

In order to increase the reliability of the results obtained in the experiments of measuring the radiation flux from the source was carried out with the help of a photoelectron multiplier, the signal from which was perceived through an amplifier with a sensitive galvanometer, or the corresponding interface through an analogdigital converter was fed for mathematical processing on a computer.

Meanwhile, the measurements of the measuring galvanometer have always been directly proportional to the flux emitted by the photomultiplier (the condition was satisfied so that the intensity of the radiation flux coming into the perimeter window of the photomultiplier was within the linear section of its sensitivity characteristic).

This condition allowed for the calculation of optical coefficients to accurately determine the relative radiation flux through the indicators of the galvanometer, and not in energy units.

In particular, the transmission coefficients of the optical radiation in the investigated samples were determined by the formula (2):

$$
\pi\_{\lambda} = \frac{I\_{\lambda}}{I\_{O\lambda}} \tag{2}
$$

where *I*λ is the photocurrent in measuring the radiation flux with wavelength *λ* passing through the sample (μΑ) and *I*ολ is the photocurrent in measuring the

radiation flux with wavelength *λ* from the source supplied to the photomultiplier without a sample (μΑ).

If, in studies, the value of the photocurrent after passing through the sample is much smaller than the photocurrent from the source, *I*<sup>λ</sup> ≪ *I*ολ, observed in studies with short-wave ultraviolet radiation at wavelengths less than 365 nm from powerful sources of ultraviolet radiation (due to strong absorption in the irradiation), in the measurement, an error is introduced due to the over-excitation of the photodetector by powerful radiation, and therefore, we have a decrease in its sensitivity (due to the outflow of the linear portion of its h sensitivity characteristics).

To eliminate such an effect, the magnitude of the excitatory radiation from the source was measured by the use of radiation absorbents (nonselective dimming filters of the type NS) with a known attenuation coefficient.

In the described studies, the expression for determining the spectral coefficients of optical transmission has the formula (3):

\*\*Sessonn nor weiterinning une spectrume coënnica:

 $\mathsf{T}\_{\lambda} = \frac{I\_{\lambda} \cdot \mathsf{T}\_{\lambda \phi}}{I\_{O\lambda \phi}}$ 

where *I*λ is the photocurrent in measuring the radiation flux with wavelength *λ* passing through the sample (μΑ); *τλφ* is the transmission coefficient of a neutral filter; and *I*ολφ is the photocurrent in measuring radiation emitted with wavelength *λ* by the photodetector through a dimming filter (μΑ).

An important optical parameter is also the reflection coefficient of optical radiation from the formula (4):

$$
\text{It's also the representation coefficient or } \mathfrak{glrica} \text{ rand}
$$

$$
\rho\_{\lambda} = \frac{I\_{\lambda} - I\_{\lambda}^{\ell}}{I\_{\lambda \mathfrak{c}} - \mathbf{r}\_{\lambda}^{\prime}} \cdot \rho\_{\lambda \mathfrak{c}} \tag{4}
$$

where *I*λ is the photocurrent in measuring radiation reflected from the sample (μΑ); *I*λ is the photocurrent in the measurement of radiation, with a wavelength *λ* reflected from the standard (μΑ); *I*′λ is the extraneous current that shows the galvanometer in the absence of the sample and reference (μΑ); and *ρ*λе is the reflection coefficient of radiation with wavelength *λ*, for the standard.

A standard used that is commonly accepted in optical studies is the metal plate with a precipitated layer of sulfuric acid barium. The sulfuric acid barium has a comparatively identical reflection coefficient for optical radiation in the range of 640–250 nm equal to 0.96–0.98 [1].

The last important absorption coefficient of optical radiation by the sample was determined by the known formula (1):

$$\alpha\_{\bar{k}} = \mathfrak{I} - \rho\_{\bar{k}} - \mathfrak{r}\_{\bar{k}} \tag{5}$$

**107**

**Figure 2.**

*spectrum by a separate cylinder of wool under the skin.*

*Study of the Electromagnetic Radiation on the Animal Body*

Methods of mathematical statistics are used to process experimental results with given accuracy and reliability. In the processing of experimental data, gross errors were removed from the series. Measurement of optical and electrical quantities was performed at least three times, and the average value of the measured value was analyzed. The variance, coefficient of variation, and mean square deviation of the

**3. Investigation of the light conductivity of individual wool from pigs**

In studies devoted to the study of the laws of penetration of optical radiation through the wool coating of animals, the possibility of penetration of radiation under the skin by individual wool, as in fibers, was studied. The main objects were the wool (bristles) of pigs. The choice is based on the fact that pigs' wool has the largest cross-sectional area among other farm animals, and the radiation that passes through the wool cylinder to the photomultiplier tube makes it a valid signal in the

That is, the value of the photocurrent is greater than the sensitivity threshold of the photoelectron multiplier. This fact increases the reliability of the measurement

To determine the spectral coefficients of optical conductivity, a system of interference filters with a bandwidth of 10 nm in the range from 300 to 760 nm was used. The corrective lens system is used to ensure uniform irradiation of the sensing area of the sensitive surface of the photo-measuring sensor. To ensure the purity of the study, a rubber seal was applied to the skin from above, in which a separate wool was passed through the hole. The radiation flux was focused on the wool surface above the seal, and the radiation that passed through its cylinder under the skin was recorded. Then the wool was cut off over the compaction (the radiation was directed perpendicular to the surface of the trimmed hair) and, finally, cut off the follicular

*The structure of the installation for the study of the penetration of electromagnetic radiation of the optical* 

*DOI: http://dx.doi.org/10.5772/intechopen.89430*

obtained data were determined.

linear part of its sensitivity profile.

results. The study scheme is shown in **Figure 2**.

It is important to recognize that when measuring the spectral reflection coefficients in the UV region, along with reflected radiation, the radiation of longwave luminescence of the objects under investigation, which originated from this ultraviolet radiation, was recorded. This radiation made an additional error in the magnitude of the reflection coefficients, but the error introduced did not have a significant effect (given that the intensity of the luminescence was 10n times less than the intensity of the reflected radiation).

The results obtained by this method, the results of the optical characteristics of the skin samples with the wool of different species of animals, and their analysis allow us to determine the quantitative scheme of the distribution of optical radiation penetrating under the skin separately through the skin itself, by individual wool, and are generalized through the leather and wool cover [7–9].

*Electromagnetic Field Radiation in Matter*

without a sample (μΑ).

characteristics).

tion from the formula (4):

640–250 nm equal to 0.96–0.98 [1].

determined by the known formula (1):

than the intensity of the reflected radiation).

radiation flux with wavelength *λ* from the source supplied to the photomultiplier

If, in studies, the value of the photocurrent after passing through the sample is much smaller than the photocurrent from the source, *I*<sup>λ</sup> ≪ *I*ολ, observed in studies with short-wave ultraviolet radiation at wavelengths less than 365 nm from powerful sources of ultraviolet radiation (due to strong absorption in the irradiation), in the measurement, an error is introduced due to the over-excitation of the photodetector by powerful radiation, and therefore, we have a decrease in its sensitivity (due to the outflow of the linear portion of its h sensitivity

To eliminate such an effect, the magnitude of the excitatory radiation from the source was measured by the use of radiation absorbents (nonselective dimming

In the described studies, the expression for determining the spectral coefficients

(3)

⋅ *ρ* (4)

*αλ = 1 − ρλ − τλ* (5)

τλ = \_ *I*<sup>λ</sup> ⋅ τ *IO*

where *I*λ is the photocurrent in measuring the radiation flux with wavelength *λ* passing through the sample (μΑ); *τλφ* is the transmission coefficient of a neutral filter; and *I*ολφ is the photocurrent in measuring radiation emitted with wavelength *λ*

An important optical parameter is also the reflection coefficient of optical radia-

where *I*λ is the photocurrent in measuring radiation reflected from the sample (μΑ); *I*λ is the photocurrent in the measurement of radiation, with a wavelength *λ* reflected from the standard (μΑ); *I*′λ is the extraneous current that shows the galvanometer in the absence of the sample and reference (μΑ); and *ρ*λе is the reflection

A standard used that is commonly accepted in optical studies is the metal plate with a precipitated layer of sulfuric acid barium. The sulfuric acid barium has a comparatively identical reflection coefficient for optical radiation in the range of

The last important absorption coefficient of optical radiation by the sample was

It is important to recognize that when measuring the spectral reflection coefficients in the UV region, along with reflected radiation, the radiation of longwave luminescence of the objects under investigation, which originated from this ultraviolet radiation, was recorded. This radiation made an additional error in the magnitude of the reflection coefficients, but the error introduced did not have a significant effect (given that the intensity of the luminescence was 10n times less

The results obtained by this method, the results of the optical characteristics of the skin samples with the wool of different species of animals, and their analysis allow us to determine the quantitative scheme of the distribution of optical radiation penetrating under the skin separately through the skin itself, by individual

wool, and are generalized through the leather and wool cover [7–9].

*ρλ* = \_ *I*<sup>λ</sup> − *I*λ′ *I* − ґλ′

filters of the type NS) with a known attenuation coefficient.

of optical transmission has the formula (3):

by the photodetector through a dimming filter (μΑ).

coefficient of radiation with wavelength *λ*, for the standard.

**106**

Methods of mathematical statistics are used to process experimental results with given accuracy and reliability. In the processing of experimental data, gross errors were removed from the series. Measurement of optical and electrical quantities was performed at least three times, and the average value of the measured value was analyzed. The variance, coefficient of variation, and mean square deviation of the obtained data were determined.

#### **3. Investigation of the light conductivity of individual wool from pigs**

In studies devoted to the study of the laws of penetration of optical radiation through the wool coating of animals, the possibility of penetration of radiation under the skin by individual wool, as in fibers, was studied. The main objects were the wool (bristles) of pigs. The choice is based on the fact that pigs' wool has the largest cross-sectional area among other farm animals, and the radiation that passes through the wool cylinder to the photomultiplier tube makes it a valid signal in the linear part of its sensitivity profile.

That is, the value of the photocurrent is greater than the sensitivity threshold of the photoelectron multiplier. This fact increases the reliability of the measurement results. The study scheme is shown in **Figure 2**.

To determine the spectral coefficients of optical conductivity, a system of interference filters with a bandwidth of 10 nm in the range from 300 to 760 nm was used. The corrective lens system is used to ensure uniform irradiation of the sensing area of the sensitive surface of the photo-measuring sensor. To ensure the purity of the study, a rubber seal was applied to the skin from above, in which a separate wool was passed through the hole. The radiation flux was focused on the wool surface above the seal, and the radiation that passed through its cylinder under the skin was recorded.

Then the wool was cut off over the compaction (the radiation was directed perpendicular to the surface of the trimmed hair) and, finally, cut off the follicular

#### **Figure 2.**

*The structure of the installation for the study of the penetration of electromagnetic radiation of the optical spectrum by a separate cylinder of wool under the skin.*

sphere from the lower part of the skin. Registration of the radiation flux under the skin was carried out in stages, both visually and using a photo-measuring device. **Figure 3** shows the correct results of studying the optical conductivity of individual wool in light pigs (Landrace, Large White).

Analysis of the data shows that white wool conducts up to 10% of the visible electromagnetic radiation falling from above on its surface, depending on the diameter and structure of the wool. When the hair is trimmed over the skin, the radiation emitted into the subcutaneous structure increases. The results of experimental studies of the passage of optical radiation in the cylinder of an individual hair under the skin, on samples of the skin with hair from Large White pigs of different ages, are shown in **Table 1**.

From the tabular data, it can be seen that clipping the sphere of the follicle of an individual wool several times increases the value of the measured radiation. This indicates that the follicle sphere dissipates and absorbs part of the radiation energy (turns into another), that is, the follicle is the particular basis of the photobiological reaction. Practical confirmation of this fact is important for further understanding of the mechanism of penetration of optical radiation into the animal's body and its place of interaction with the biological structures of the skin.

A visual observation of an experiment on the transmission of optical radiation under the skin for different animals (when irradiating a part of the skin surface with wool) also revealed that the follicles have a brighter glow than the inner surface of the skin. Therefore, it is permissible to assert that follicles can be considered as light bulbs in the skin structure and, consequently, primary cells of active photobiological reactions. And there is a realistic explanation: the energy of optical radiation, which reaches the follicles inside the wool cylinder, directly affects the biologically active structures (nerve and blood principles that supply the necessary bioproducts for the formation and growth of wool) and causes a greater photobiological effect than the radiation that passes at the same depth under the skin between individual wool and reaches less active biological entities in the structure of the skin itself [2, 5]. This path is also important because of the fact that the energy of optical radiation enters the wool under the skin directly to the follicle unchanged. In the immediate proximity of the follicle are salogen and sweat glands filled with reactive cellular organic components.

#### **Figure 3.**

*Spectral coefficients of transmission of a wool and its segments, at a cross section of a wool: \_\_ S = 0.12 mm<sup>2</sup> , \_ \_ S = 0.08 mm2 , \_ \_ S = 0.062 mm2 .*

**109**

**Figure 4.**

is shown in **Figure 4**.

*Study of the Electromagnetic Radiation on the Animal Body*

**Cross section of the wool, mm2**

3 years 3.5 0.11–0.13 Whole wool Is 12–14 3 years 3.5 0.11–0.13 2.6–2.8\* Is 24–28 3 years 3.5 0.11–0.13 2.6–2.8\* The cut 74–90 12 months 2.3 0.085 × 0.09 Whole wool Is 9–12 12 months 2.3 0.085 × 0.09 2.6–2.8\* Is 20–25 12 months 2.3 0.085 × 0.09 2.6–2.8\* The cut 52–76 8 months 2.0 0.07–0.074 Whole wool Is 4–7 8 months 2.0 0.07–0.074 2.6–2.8\* Is 10–12 8 months 2.0 0.07–0.074 2.6–2.8\* The cut 26–34

**Wool length (to the cut), mm**

**Follicles Light** 

**conductivity, μΑ**

The suitability of this assumption is confirmed by the results of biochemical studies of the effect of optical radiation on the skin, carried out by other authors [3, 5]. To obtain a three-coordinate model describing the spectral dependence of the transmission of electromagnetic radiation on the optical spectrum of a singlecylinder hair on its length, a multiple regression analysis of experimental data was carried out in the Mathcad 2001 Pro software environment. The result of the study

*The length of the wool segment was measured from the cut above the rubber seal to the surface of the skin.*

*The results of experimental studies of the light conductivity of a single wool under the skin of an animal.*

It can be seen from the above that, with the decrease in the wavelength of the optical radiation falling on the surface of a separate one wool, its light conductivity

*The dependence of the coefficient transmission of a separate cotton wool τ from the spectral composition of the* 

*electromagnetic radiation λ and the length of its light-conducting part L (distance to its follicle).*

decreases. This corresponds to the results obtained by other authors [2, 4]. The established fact of the propagation of optical radiation energy inside a single-haired cylinder allows determining the intensity of the electromagnetic radiation of the optical spectrum at a specific location of wool (e.g., at the place where it appears above the skin, or at the growth site from the follicle), depending on the distance to the fall site radiation to the surface of a single layer, which is important to know in biological studies when studying the mechanism of the action

*DOI: http://dx.doi.org/10.5772/intechopen.89430*

**Thickness skin, mm**

**Age of animals**

*\**

**Table 1.**


*Study of the Electromagnetic Radiation on the Animal Body DOI: http://dx.doi.org/10.5772/intechopen.89430*

**Table 1.**

*Electromagnetic Field Radiation in Matter*

are shown in **Table 1**.

wool in light pigs (Landrace, Large White).

sphere from the lower part of the skin. Registration of the radiation flux under the skin was carried out in stages, both visually and using a photo-measuring device. **Figure 3** shows the correct results of studying the optical conductivity of individual

Analysis of the data shows that white wool conducts up to 10% of the visible electromagnetic radiation falling from above on its surface, depending on the diameter and structure of the wool. When the hair is trimmed over the skin, the radiation emitted into the subcutaneous structure increases. The results of experimental studies of the passage of optical radiation in the cylinder of an individual hair under the skin, on samples of the skin with hair from Large White pigs of different ages,

From the tabular data, it can be seen that clipping the sphere of the follicle of an individual wool several times increases the value of the measured radiation. This indicates that the follicle sphere dissipates and absorbs part of the radiation energy (turns into another), that is, the follicle is the particular basis of the photobiological reaction. Practical confirmation of this fact is important for further understanding of the mechanism of penetration of optical radiation into the animal's body and its

A visual observation of an experiment on the transmission of optical radiation under the skin for different animals (when irradiating a part of the skin surface with wool) also revealed that the follicles have a brighter glow than the inner surface of the skin. Therefore, it is permissible to assert that follicles can be considered as light bulbs in the skin structure and, consequently, primary cells of active photobiological reactions. And there is a realistic explanation: the energy of optical radiation, which reaches the follicles inside the wool cylinder, directly affects the biologically active structures (nerve and blood principles that supply the necessary bioproducts for the formation and growth of wool) and causes a greater photobiological effect than the radiation that passes at the same depth under the skin between individual wool and reaches less active biological entities in the structure of the skin itself [2, 5]. This path is also important because of the fact that the energy of optical radiation enters the wool under the skin directly to the follicle unchanged. In the immediate proximity of the follicle are salogen and sweat glands filled with reactive cellular organic components.

*Spectral coefficients of transmission of a wool and its segments, at a cross section of a wool: \_\_ S = 0.12 mm<sup>2</sup>*

place of interaction with the biological structures of the skin.

**108**

**Figure 3.**

*\_ \_ S = 0.08 mm2*

*, \_ \_ S = 0.062 mm2*

*.*

*The results of experimental studies of the light conductivity of a single wool under the skin of an animal.*

The suitability of this assumption is confirmed by the results of biochemical studies of the effect of optical radiation on the skin, carried out by other authors [3, 5].

To obtain a three-coordinate model describing the spectral dependence of the transmission of electromagnetic radiation on the optical spectrum of a singlecylinder hair on its length, a multiple regression analysis of experimental data was carried out in the Mathcad 2001 Pro software environment. The result of the study is shown in **Figure 4**.

It can be seen from the above that, with the decrease in the wavelength of the optical radiation falling on the surface of a separate one wool, its light conductivity decreases. This corresponds to the results obtained by other authors [2, 4].

The established fact of the propagation of optical radiation energy inside a single-haired cylinder allows determining the intensity of the electromagnetic radiation of the optical spectrum at a specific location of wool (e.g., at the place where it appears above the skin, or at the growth site from the follicle), depending on the distance to the fall site radiation to the surface of a single layer, which is important to know in biological studies when studying the mechanism of the action

#### **Figure 4.**

*The dependence of the coefficient transmission of a separate cotton wool τ from the spectral composition of the electromagnetic radiation λ and the length of its light-conducting part L (distance to its follicle).*

*,* 

of optical radiation through the skin and wools on an animal's body in order to make the desired effective dose and control its effects.

For the visual confirmation of the light conductivity of the wool of different species of animals, the study of the light conductivity of individual wools was carried out by the method of photographing.

The photographs were taken according to the above scheme on **Figure 5**.

**Figure 6** shows an example of a photo of the light transmission of the wool of different animals.

Photo analysis shows that pet hair is mostly transparent to visible radiation. The coat of natural animals is darker with respect to visible radiation. This is natural because wildlife is much more exposed to the open environment and longer exposed

**Figure 5.**

*Scheme for photographing the light conductivity of a single wool. 1, wool; 2, seals; 3, camera; OP, optical radiation.*

**Figure 6.**

*Photos of the light conductivity of single wools from different animals: (a) horse tail, (b) wool of pigs, and (c) of the one wools' follicle.*

**111**

**Figure 7.**

*Study of the Electromagnetic Radiation on the Animal Body*

to sunlight. In summer, when there are long days and intense sunlight, most wild animals have a dark coat color to protect against excessive radiation; in winter, most animals "melt," turning their wool into lighter and thicker ones, which protects from cold and contributes to a wider use of heat when there is less solar radiation in

The calculations presented further confirm that the wool coating plays a significant role in the transfer of the energy of optical radiation of an animal's organism (especially in domestic animals that genetically experience the need for solar radiation under cultivation in a closed space) due to better conductivity and also due to the transfer of energy of this radiation directly to biologically active

**4. Investigation of optical characteristics of skin and wool cover animals**

Given the biological significance of radiation that reaches subcutaneous body structures directly on individual wool, it should be noted that under natural conditions biological action results in radiation that passes under the skin in any way and reaches active structures. Therefore, it is also important to know what proportion of the energy of optical radiation falling on the surface of the animal passes directly through the skin. For this purpose, complex studies were conducted on samples of the skin with wool, in which the radiation was focused on the areas of the skin

Based on the results of experimental studies, the generalized curves of the spectral transmittance of optical radiation versus skin thickness for animals of light

For animals listed in **Table 1**, the coefficients of complete transmission of electromagnetic radiation of the visible spectrum of the skin with an area of 1 cm2 were investigated. The three-coordinate **Figure 8** shows, after mathematical processing and generalization of the experimental data, the results of studies of the dependence of the total transmittance of skin samples with wool for the radiation of

**Figure 8** shows that in the spectral range of the ultraviolet and visible areas of optical radiation with a decrease in the wavelength of radiation, the intensity and

*The transmission coefficient of electromagnetic optical radiation τ through only the skin of agricultural animals* 

*of breeds depending on the wavelength λ. Note: the skin thickness is indicated on the curves.*

between the hairs in accordance with the scheme shown in **Figure 1**.

optical spectral radiation and the thickness of the animal skin.

*DOI: http://dx.doi.org/10.5772/intechopen.89430*

summer.

centers in the skin.

breeds are shown in **Figure 7**.

#### *Study of the Electromagnetic Radiation on the Animal Body DOI: http://dx.doi.org/10.5772/intechopen.89430*

*Electromagnetic Field Radiation in Matter*

out by the method of photographing.

different animals.

**Figure 5.**

*radiation.*

the desired effective dose and control its effects.

of optical radiation through the skin and wools on an animal's body in order to make

The photographs were taken according to the above scheme on **Figure 5**. **Figure 6** shows an example of a photo of the light transmission of the wool of

*Scheme for photographing the light conductivity of a single wool. 1, wool; 2, seals; 3, camera; OP, optical* 

*Photos of the light conductivity of single wools from different animals: (a) horse tail, (b) wool of pigs, and* 

For the visual confirmation of the light conductivity of the wool of different species of animals, the study of the light conductivity of individual wools was carried

Photo analysis shows that pet hair is mostly transparent to visible radiation. The coat of natural animals is darker with respect to visible radiation. This is natural because wildlife is much more exposed to the open environment and longer exposed

**110**

**Figure 6.**

*(c) of the one wools' follicle.*

to sunlight. In summer, when there are long days and intense sunlight, most wild animals have a dark coat color to protect against excessive radiation; in winter, most animals "melt," turning their wool into lighter and thicker ones, which protects from cold and contributes to a wider use of heat when there is less solar radiation in summer.

The calculations presented further confirm that the wool coating plays a significant role in the transfer of the energy of optical radiation of an animal's organism (especially in domestic animals that genetically experience the need for solar radiation under cultivation in a closed space) due to better conductivity and also due to the transfer of energy of this radiation directly to biologically active centers in the skin.

#### **4. Investigation of optical characteristics of skin and wool cover animals**

Given the biological significance of radiation that reaches subcutaneous body structures directly on individual wool, it should be noted that under natural conditions biological action results in radiation that passes under the skin in any way and reaches active structures. Therefore, it is also important to know what proportion of the energy of optical radiation falling on the surface of the animal passes directly through the skin. For this purpose, complex studies were conducted on samples of the skin with wool, in which the radiation was focused on the areas of the skin between the hairs in accordance with the scheme shown in **Figure 1**.

Based on the results of experimental studies, the generalized curves of the spectral transmittance of optical radiation versus skin thickness for animals of light breeds are shown in **Figure 7**.

For animals listed in **Table 1**, the coefficients of complete transmission of electromagnetic radiation of the visible spectrum of the skin with an area of 1 cm2 were investigated. The three-coordinate **Figure 8** shows, after mathematical processing and generalization of the experimental data, the results of studies of the dependence of the total transmittance of skin samples with wool for the radiation of optical spectral radiation and the thickness of the animal skin.

**Figure 8** shows that in the spectral range of the ultraviolet and visible areas of optical radiation with a decrease in the wavelength of radiation, the intensity and

#### **Figure 7.**

*The transmission coefficient of electromagnetic optical radiation τ through only the skin of agricultural animals of breeds depending on the wavelength λ. Note: the skin thickness is indicated on the curves.*

#### **Figure 8.**

*Three-coordinate picture of the transmission τ of the spectrum of optical radiation to the depth of the skin and wool of the animal, depending on the wavelength of optical radiation λ and the thickness of the cover δ.*

depth of penetration of its energy into the thickness of the leather and wools of farm animals decrease, which corresponds to the results of studies of other scientist experimenters.

#### **5. Investigation of the percentage efficiency of radiation penetration through the structure of the coating (through the skin and the wool cylinders) into the animal's body**

Establishing the fact of transmission of the electromagnetic radiation of the optical spectrum of an individual wool into the depths of the animal's body, together with the transmission of radiation through the entire skin and coat, is important for understanding and developing research on the mechanism of the action of electromagnetic radiation on the animal and possibly humans. As shown above, electromagnetic radiation on a separate cotton wool with its initial growth (follicle) more effectively affects subcutaneous biological structures than radiation penetrating the skin through the thickness, since the follicle has a well-developed network for the life of the blood vessels and nerve endings.

In the immediate vicinity of the follicle are sebaceous and sweat glands, filled with reactive organic secretions, and it is easy for them to consume the energy of absorbed radiation from the follicles themselves for the development of the skin and body [5].

Although the individual wool in the body of the animal comes a smaller amount of electromagnetic radiation of the optical spectrum than through the skin between the one wool, it comes directly to the biologically active components of the body cells of the animal and can have a greater effect on the photobiological reactions of its development.

In accordance with the foregoing, it is important to know how much of the optical radiation affects the body of an animal through the surface of the skin and how much it passes through individual wools (one wool). To do this, it is necessary to conduct quite complex experimental studies. The difficulty is that it is necessary to irradiate only a certain micro part of the skin surface (or individual wool) and measure the scattered radiation of low intensity that has penetrated deep into the skin.

That is, it is necessary to have a powerful source of optical radiation with a focusing system, and to measure the penetration of the radiation, a photodetector

**113**

**Table 2.**

*White breed of pigs.*

*Study of the Electromagnetic Radiation on the Animal Body*

conductivity is carried out for a skin area of 1 mm<sup>2</sup>

with a perceptual window should be used with dimensions corresponding to the dimensions of the irradiated surface (to reduce the error of measurements from scattering of radiation into skin structures), or to use a complex optical integrated system that further reduces the sensitivity of the instrument (accuracy of measurements). Below is a specific example of the method of comparative analysis of the intensity of the penetration of optical radiation through the skin and on individual

The criterion for the radiation flux passed into the subcutaneous layers of the animal is taken by the photocurrent of the measuring device, since the value of the photocurrent is directly proportional to the intensity of the radiation coming to the receiving window of the photo registration element. Comparison of light

> *Isk* = \_ *I*<sup>0</sup>*sk Ssk*

. The relative photocurrent values for comparison are deter-

 , *Iw* = \_ *I*<sup>0</sup>*<sup>w</sup> Sw*

where *I*<sup>0</sup>*sk* is the value of the photocurrent from the radiation that passed through the skin to the photodiode's receiving window (impressions of the experiment), *Ssk* is the area of the skin from which the penetrating radiation was recorded, *I0w* is the magnitude of the photocurrent from the radiation that passed inside the cylinder of a single wool on the photodiode receiving window (impressions of the experiment), and *Sw* is the area of the cross section of a single wool from which the penetrating

The results of research and calculations for comparing the pure transmission of electromagnetic radiation of wool and skin thickness are presented in

wool during the growth of an animal according to formula (6):

The relative transmittance of electromagnetic radiation of the solar spectrum is determined to understand the change in the quantitative ratio between the penetration of electromagnetic radiation through the skin and through a separate layer of

> *kc* = \_ *Iw Isk*

Based on the data in **Table 2**, a curve was constructed for seeing the change in the relative transmission coefficient of electromagnetic radiation into the animal's

 4 ± 0.2 5.0 3.8 ± 0.4 0.76 ± 0.08 0.03 ± 0.006 48 ± 3 1.6 ± 0.2 3 ± 0.4 5.0 4 ± 0.4 0.8 ± 0.08 0.025 ± 0.005 36 ± 3 1.44 ± 0.2 2.8 ± 0.3 5.0 5 ± 0.4 1.0 ± 0.08 0.016 ± 0.004 22 ± 3 1.35 ± 0.2 1.8 ± 0.2 5.0 5.8 ± 0.4 1.16 ± 0.08 0.01 ± 0.003 12 ± 3 1.2 ± 0.2

*Comparison table of skin transmittance and a separate wool with increasing age in an animal of a Large* 

**Light conductance and geometric parameters of pigs Skin Single wool**

*І0sk***, μΑ** *Іsk***, μΑ/mm2** *Sw***, mm2** *І0w***, 10<sup>−</sup><sup>3</sup> μΑ** *Іw***, μΑ/mm2**

, and the conditional wool of the

(6)

(7)

*DOI: http://dx.doi.org/10.5772/intechopen.89430*

wool.

same section, 1 mm2

mined from formula (6):

radiation was recorded.

body with age, given in **Figure 9**.

*δ***, mm** *Ssk***,** 

**mm2**

*Note: δ is the thickness of the skin where the radiation was measured.*

**Table 2**.

**Age, month**

#### *Study of the Electromagnetic Radiation on the Animal Body DOI: http://dx.doi.org/10.5772/intechopen.89430*

*Electromagnetic Field Radiation in Matter*

experimenters.

**Figure 8.**

and body [5].

its development.

**cylinders) into the animal's body**

network for the life of the blood vessels and nerve endings.

depth of penetration of its energy into the thickness of the leather and wools of farm animals decrease, which corresponds to the results of studies of other scientist

*Three-coordinate picture of the transmission τ of the spectrum of optical radiation to the depth of the skin and wool of the animal, depending on the wavelength of optical radiation λ and the thickness of the cover δ.*

**5. Investigation of the percentage efficiency of radiation penetration through the structure of the coating (through the skin and the wool** 

Establishing the fact of transmission of the electromagnetic radiation of the optical spectrum of an individual wool into the depths of the animal's body, together with the transmission of radiation through the entire skin and coat, is important for understanding and developing research on the mechanism of the action of electromagnetic radiation on the animal and possibly humans. As shown above, electromagnetic radiation on a separate cotton wool with its initial growth (follicle) more effectively affects subcutaneous biological structures than radiation penetrating the skin through the thickness, since the follicle has a well-developed

In the immediate vicinity of the follicle are sebaceous and sweat glands, filled with reactive organic secretions, and it is easy for them to consume the energy of absorbed radiation from the follicles themselves for the development of the skin

Although the individual wool in the body of the animal comes a smaller amount of electromagnetic radiation of the optical spectrum than through the skin between the one wool, it comes directly to the biologically active components of the body cells of the animal and can have a greater effect on the photobiological reactions of

In accordance with the foregoing, it is important to know how much of the optical radiation affects the body of an animal through the surface of the skin and how much it passes through individual wools (one wool). To do this, it is necessary to conduct quite complex experimental studies. The difficulty is that it is necessary to irradiate only a certain micro part of the skin surface (or individual wool) and measure the scattered radiation of low intensity that has penetrated deep into

That is, it is necessary to have a powerful source of optical radiation with a focusing system, and to measure the penetration of the radiation, a photodetector

**112**

the skin.

with a perceptual window should be used with dimensions corresponding to the dimensions of the irradiated surface (to reduce the error of measurements from scattering of radiation into skin structures), or to use a complex optical integrated system that further reduces the sensitivity of the instrument (accuracy of measurements). Below is a specific example of the method of comparative analysis of the intensity of the penetration of optical radiation through the skin and on individual wool.

The criterion for the radiation flux passed into the subcutaneous layers of the animal is taken by the photocurrent of the measuring device, since the value of the photocurrent is directly proportional to the intensity of the radiation coming to the receiving window of the photo registration element. Comparison of light conductivity is carried out for a skin area of 1 mm<sup>2</sup> , and the conditional wool of the same section, 1 mm2 . The relative photocurrent values for comparison are determined from formula (6):

$$I\_{sk} = \frac{I\_{\text{Ook}}}{S\_{sk}}, \ I\_{\text{uv}} = \frac{I\_{\text{Ouv}}}{S\_{\text{uv}}} \tag{6}$$

where *I*<sup>0</sup>*sk* is the value of the photocurrent from the radiation that passed through the skin to the photodiode's receiving window (impressions of the experiment), *Ssk* is the area of the skin from which the penetrating radiation was recorded, *I0w* is the magnitude of the photocurrent from the radiation that passed inside the cylinder of a single wool on the photodiode receiving window (impressions of the experiment), and *Sw* is the area of the cross section of a single wool from which the penetrating radiation was recorded.

The results of research and calculations for comparing the pure transmission of electromagnetic radiation of wool and skin thickness are presented in **Table 2**.

The relative transmittance of electromagnetic radiation of the solar spectrum is determined to understand the change in the quantitative ratio between the penetration of electromagnetic radiation through the skin and through a separate layer of wool during the growth of an animal according to formula (6):

$$
\lambda \mathbf{k}\_c = \frac{I\_{\alpha \nu}}{I\_{s k}} \tag{7}
$$

Based on the data in **Table 2**, a curve was constructed for seeing the change in the relative transmission coefficient of electromagnetic radiation into the animal's body with age, given in **Figure 9**.


#### **Table 2.**

*Comparison table of skin transmittance and a separate wool with increasing age in an animal of a Large White breed of pigs.*

#### **Figure 9.**

*Change of the relative efficiency of penetration of electromagnetic radiation of the optical spectrum under the skin of an animal by individual wool and through the skin during the growth of the animal (on the example of a White pig).*

From the figure, it is clear that if the animal grows, then the skin becomes coarse and wool cylinders become thicker. This means that older animals have more efficient energy that passes through individual wools.

The analysis of the results of experimental studies shows that the working hypothesis in the process of animal development, the role of wool and skin changes in the interaction of an animal with the environment is real:


**115**

**Author details**

Leonid Chervinsky

National University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: lchervinsky@gmail.com

provided the original work is properly cited.

*Study of the Electromagnetic Radiation on the Animal Body*

*DOI: http://dx.doi.org/10.5772/intechopen.89430*

*Study of the Electromagnetic Radiation on the Animal Body DOI: http://dx.doi.org/10.5772/intechopen.89430*

*Electromagnetic Field Radiation in Matter*

*Change of the relative efficiency of penetration of electromagnetic radiation of the optical spectrum under the skin of an animal by individual wool and through the skin during the growth of the animal (on the example of* 

From the figure, it is clear that if the animal grows, then the skin becomes coarse and wool cylinders become thicker. This means that older animals have more

The analysis of the results of experimental studies shows that the working hypothesis in the process of animal development, the role of wool and skin changes

1.In young animals, the skin is thin and easily permeable to optical radiation. Therefore, it is natural that the role of the protective screen is woolen, which consists of a thick layer of thin worsted. This layer reduces the flow of optical radiation to the skin by absorbing or dispersing it into the environment and

2.With the development of the animal, its skin thickens, its surface and thickness increase, and the penetrating ability of the skin for optical radiation decreases; therefore, the role of the wool as a network of light conductors increases: the distance between them increases on the skin (they are less overlapping each other), their value per unit area of the animal's body decreases, the thickness of each wool increases, and the inner structure of the wool cylinder

3.Shirting cover with animal growth gradually loses protective functions from optical radiation, acquiring functions, possibly the main conductor of electromagnetic radiation of the optical spectrum in an animal's organism. With the growth of animal's wool become more transparent for optical radiation, which on them as light transmissions with less losses enters the skin to active

4.In real conditions, when an animal is under the radiation of the Sun or a special source of optical radiation of different spectral composition, the biological effect results in all the energy of radiation that has reached the subcutaneous structures of the body by any of these paths and through the skin and indi-

efficient energy that passes through individual wools.

in the interaction of an animal with the environment is real:

protects the young animal from excessive light energy.

**Figure 9.**

*a White pig).*

becomes lighter.

structures.

vidual wools as light conductors.

**114**

#### **Author details**

Leonid Chervinsky National University of Life and Environmental Sciences of Ukraine, Kyiv, Ukraine

\*Address all correspondence to: lchervinsky@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References Chapter 7**

[1] Smith KC, editor. The Science of Photobiology. NY: Pl. Press; 1977. 430 p

[2] Radha Rani G, Raju GSN. Transmission and reflection characteristics of electromagnetic energy in biological tissues. International Journal of Electronics and Communication Engineering. 2013;**6**(1):119-129. ISSN 0974-2166

[3] Cardona-Hernández Leonel Fierro-Arias MA, Cabrera-Pérez AL, Vidal-Flores AA. Effects of electromagnetic radiation on skin. Dermatologia Revista Mexicana. 2017;**61**(4):292-302

[4] Bashkatov AN, Genina EA, Tuchin VV. Optical properties of skin, subcutaneous, and muscle tissues: A review. Journal of Innovative Optical Health Sciences. 2011;**04**:9-38

[5] Bahar L, Eralp A, Rumevleklioglu Y, Erturk SE, Yuncu M. The effect of electromagnetic radiation on the development of skin ultrastructural and immunohistochemically evaluation with P63. PSP Fresenius Environmental Bulletin. 2018;**27**(3):1764-1771

[6] Chervinsky LS. The action lights on the derma animal's. In: Extra on International Conf. 1st Congress of the World Association for Laser Therapy «WALT»; 1996 May 5-9; Jerusalem, Israel. 1996. p. 2922

[7] Chervinsky LS. Investigation of the light-conductivity of the separate one wool and skins translucence. In: PITTCON'98; 1998 March 1-5; New Orleans, Louisiana, USA. p. 652

[8] Chervinsky LS. About the mechanism of photo reactivation of the biological objects. In: The European Biomedical Optics Week, BIOS Europe'97; 1997 September 4-8; San Remo, Italy. 1997. p. 3198

[9] Chervinsky LS. Primary mechanism of action of optical radiation on living organisms. International Journal of Biosensors & Bioelectronics. 2018;**4**(4):204. DOI: 10.15406/ ijbsbe.2018.04.00126

Square-Wave Electric Impulses of

Impulse Generator Device, Affect

The influence of the medium-strength electric forces (MSE) on the proliferation of adherent chicken embryo fibroblasts (CEF), VERO, MDBK, MRC-5, and HeLa; lymphoblast cells, FB1 and K562; and cell multiplications were analyzed by growth index (GI). Impulse generator device PGen-1 provided 100 V/cm square-wave impulses of 10 ms. Treatment: Samples were subjected to one or three MSE. GIs were compared with controls after 72 hours and one or three treatments: Monolayers: CEF: GI in the control is 16.76, and after one and three MSE, it is 15.81 and 7.09. Vero cells: GI in the controls is 8.39, and after one and three MSE, it is 5.39 and 5.69. MDBK cells: GI in controls is 8.39, and after one and three MSE, it is 5.39 and 5.69. MRC-5 cells: GI in controls is 5.58, and after one and three MSE, it is 4.18 and 2.60. HeLa cells: GI in controls is 13.69, and after one and three MSE, it is 10.16 and 5.37. Suspension cells: Lymphoblast FB1: GI in controls is 6.55, and after one and three MSE, it is 13.48 and 12.25. Lymphoblast K562: GI in controls is 9.07, and after one or three MSE, it is 12.37 and 13.55. To conclude: MSE in monolayer cells inhibits the GI, depending on the nature of cells. MSE enhances the multiplication of

**Keywords:** square-wave electric impulses, monolayer cells, lymphoblast cells, growth pattern, growth index decrease, growth index increase, Caspase-3

Different electric and magnetic field forces can interact with the living systems at enzymatic, cellular, or organism levels [1, 2]. Despite a numerous experimental

10 ms and 100 V/cm of Field

Force, Produced by PGen-1

the Proliferation Patterns of

*Bratko Filipič, Lidija Gradišnik, Kristine Kovacs,*

*Ferenc Somogyvari, Hrvoje Mazija and Toth Sandor*

Different Animal Cells

**Abstract**

lymphoblast FB1 or K562.

**1. Introduction**

**117**

containing cells, percent of dead cells

### **References Chapter 7**

## Square-Wave Electric Impulses of 10 ms and 100 V/cm of Field Force, Produced by PGen-1 Impulse Generator Device, Affect the Proliferation Patterns of Different Animal Cells

*Bratko Filipič, Lidija Gradišnik, Kristine Kovacs, Ferenc Somogyvari, Hrvoje Mazija and Toth Sandor*

#### **Abstract**

The influence of the medium-strength electric forces (MSE) on the proliferation of adherent chicken embryo fibroblasts (CEF), VERO, MDBK, MRC-5, and HeLa; lymphoblast cells, FB1 and K562; and cell multiplications were analyzed by growth index (GI). Impulse generator device PGen-1 provided 100 V/cm square-wave impulses of 10 ms. Treatment: Samples were subjected to one or three MSE. GIs were compared with controls after 72 hours and one or three treatments: Monolayers: CEF: GI in the control is 16.76, and after one and three MSE, it is 15.81 and 7.09. Vero cells: GI in the controls is 8.39, and after one and three MSE, it is 5.39 and 5.69. MDBK cells: GI in controls is 8.39, and after one and three MSE, it is 5.39 and 5.69. MRC-5 cells: GI in controls is 5.58, and after one and three MSE, it is 4.18 and 2.60. HeLa cells: GI in controls is 13.69, and after one and three MSE, it is 10.16 and 5.37. Suspension cells: Lymphoblast FB1: GI in controls is 6.55, and after one and three MSE, it is 13.48 and 12.25. Lymphoblast K562: GI in controls is 9.07, and after one or three MSE, it is 12.37 and 13.55. To conclude: MSE in monolayer cells inhibits the GI, depending on the nature of cells. MSE enhances the multiplication of lymphoblast FB1 or K562.

**Keywords:** square-wave electric impulses, monolayer cells, lymphoblast cells, growth pattern, growth index decrease, growth index increase, Caspase-3 containing cells, percent of dead cells

#### **1. Introduction**

Different electric and magnetic field forces can interact with the living systems at enzymatic, cellular, or organism levels [1, 2]. Despite a numerous experimental

**116**

*Electromagnetic Field Radiation in Matter*

[1] Smith KC, editor. The Science of Photobiology. NY: Pl. Press; 1977. 430 p [9] Chervinsky LS. Primary mechanism of action of optical radiation on living organisms. International Journal of Biosensors & Bioelectronics. 2018;**4**(4):204. DOI: 10.15406/

ijbsbe.2018.04.00126

International Journal of Electronics and Communication Engineering. 2013;**6**(1):119-129. ISSN 0974-2166

[3] Cardona-Hernández Leonel Fierro-Arias MA, Cabrera-Pérez AL,

[4] Bashkatov AN, Genina EA,

Bulletin. 2018;**27**(3):1764-1771

Israel. 1996. p. 2922

[6] Chervinsky LS. The action lights on the derma animal's. In: Extra on International Conf. 1st Congress of the World Association for Laser Therapy «WALT»; 1996 May 5-9; Jerusalem,

[7] Chervinsky LS. Investigation of the light-conductivity of the separate one wool and skins translucence. In: PITTCON'98; 1998 March 1-5; New Orleans, Louisiana, USA. p. 652

[8] Chervinsky LS. About the

Remo, Italy. 1997. p. 3198

mechanism of photo reactivation of the biological objects. In: The European Biomedical Optics Week, BIOS Europe'97; 1997 September 4-8; San

Tuchin VV. Optical properties of skin, subcutaneous, and muscle tissues: A review. Journal of Innovative Optical Health Sciences. 2011;**04**:9-38

[5] Bahar L, Eralp A, Rumevleklioglu Y, Erturk SE, Yuncu M. The effect of electromagnetic radiation on the development of skin ultrastructural and immunohistochemically evaluation with P63. PSP Fresenius Environmental

Vidal-Flores AA. Effects of electromagnetic radiation on skin. Dermatologia Revista Mexicana.

2017;**61**(4):292-302

[2] Radha Rani G, Raju GSN. Transmission and reflection characteristics of electromagnetic energy in biological tissues.

approaches were performed about this subject, most of the obtained data are completely different and unfortunately very often incomparable. Different basic facts are the reason for this discrepancy. Basically, the living systems *per se* can generate electric or magnetic fields and impulses. These are in the field force range from 10 to 500 mV/cm. The external electric and magnetic range can cause physiological responses at a cellular level. They are of very short forces that can be compared to this duration and are in the range of milliseconds to seconds. Usually they do not result in any important alteration in the system [3, 4]. On the contrary, the biological effects that an electric field with a field force below 1–10 mV/cm causes to a cell are likely due to interaction mechanisms occurring in the cell membrane. For the field forces that are above 1–10 mV/cm, the effects are due to interaction mechanisms occurring in the intracellular compartments. Electric field forces of 10–500 V/cm act as environmental stress factors and result only in a transitory defensive response [5]. In addition, many reports show that these interactions are field force dependent and they can provoke both the enhancement [6] and the inhibition [7] of different cellular functions. It is important that the studied types of responses show the "sensitivity windows" within the above field force range [8]. In the frame of a defined window, the system provides a definite response, while when it is outside the frame, the response disappears or suddenly turns in the opposite direction. The electric field forces that are above 500 V/cm can provoke the sustained response like in the case of electroporation. The most interesting, even the least studied, are the biological effects of medium-strength electric forces of 100 V/cm. The complication, in fact, is that these electric fields can be applied as impulses, long-lasting DC fluxes, or AC waves, and each of them can induce very different types of responses. The applied effects can be a direct ionic current or the electric field generated between the two plates of a condenser, or as a magnetic field, that can provide different interacting doses and different kinds of energies. The named forces can affect the different immune responses of animals [9, 10] or in humans [11] through the induction [12] or augmenting [13] of different immune response elements.

**2. Material and methods**

*DOI: http://dx.doi.org/10.5772/intechopen.90506*

**2.1 The electric impulse generator device PGen-1**

tion of the PGen-1 device [20] (**Figure 1)**.

*Electric circuits of the electric impulse generator device PGen-1.*

**2.2 The dosimetry**

**Figure 1.**

**119**

The electric impulse generator device PGen-1 was developed and physically realized by Dr. Sandor Toth and Dr. Ferenc Somogyvari. The PGen-1 device provides 1–300 V/cm square-wave impulses of 1–10 ms duration, with a repetition option of 1–9. It has also a continuous work option. Its repetition intervals can be set between 1 and 10 s. The device consists of two separate circuits: (1) a low-voltage circuit, running on transformed and rectified net current, being stabilized by monolithic integrated stabilizers. They work as a power source for the regulator. (2) A high-voltage circuit serves for the impulse generator itself. The outgoing voltage is regulated by a phase-splitting dimmer and is rectified and stabilized. The low-voltage settings secure the filter condenser. The analogous regulator system is composed from a stable multivibrator, governing the counter, and two synchronized mono-stable multi vibrators, generating the outgoing impulses and the visual control of the signals. The outgoing square-wave impulses are characterized in the **Appendix.** The sample chamber has a 50 ml capacity and is a polypropylene tube with a 2.5 cm in diameter, with a platinum wire electrode. It is detachable from the basic device and is autoclavable. Further details are available in the patent descrip-

*Square-Wave Electric Impulses of 10 ms and 100 V/cm of Field Force, Produced…*

The adsorbed doses (AD in J/g) of the electric impulses in the samples were calculated according to Pakhomov et al. [21] by the use of their formula as follows:

*AD* <sup>¼</sup> *<sup>E</sup>*<sup>2</sup> � *<sup>d</sup>*<sup>2</sup> � *<sup>W</sup>* � *<sup>n</sup>*

ð Þ *<sup>R</sup>* � *<sup>M</sup>* (1)

Exogenously added electric impulses can induce the synthesis of antiviral substances, are not interferon, interleukins, or tumor necrosis factors. Therefore, we named them "interferon-like molecules." In our experiments, we also found [14] that medium-strength square-wave impulses of direct ionic current (DC) can result in a short-term direct antiviral resistance to virus infections and in a consequence to alter the membrane properties of the target cells [15]. We therefore decided to study the changes in the expression of membrane-bound surface marker molecules that are on the surface of the immune competent cells in the human blood [16–18]. Our experiments were aimed to establish whether such exogenous electric stimulation of human leukocytes could be utilized as an immune enhancer and an antiviral protector ex vivo, preferably coupled to the dialysis process. In order to detect the potential hazards of such an application, we have to study further the effects of electrostimulation on some other parameters of the human blood.

Least but not the last, the nature and biological conditions of the target system can determine the type and extent of the response. The given conditions can induce an enhancement of cell proliferation in a suspension cell culture and the inhibition of cell proliferation in an adherent growing in monolayer cell culture [19]. Therefore, the herein presented experiments are aimed to investigate the influence and some mechanisms of the medium-strength square-wave electric impulses of the field forces of 100 V/cm, on the proliferation pattern of different animal cells growing in a monolayer or growing as the suspension culture.

*Square-Wave Electric Impulses of 10 ms and 100 V/cm of Field Force, Produced… DOI: http://dx.doi.org/10.5772/intechopen.90506*

#### **2. Material and methods**

approaches were performed about this subject, most of the obtained data are completely different and unfortunately very often incomparable. Different basic facts are the reason for this discrepancy. Basically, the living systems *per se* can generate electric or magnetic fields and impulses. These are in the field force range from 10 to 500 mV/cm. The external electric and magnetic range can cause physiological responses at a cellular level. They are of very short forces that can be compared to this duration and are in the range of milliseconds to seconds. Usually they do not result in any important alteration in the system [3, 4]. On the contrary, the biological effects that an electric field with a field force below 1–10 mV/cm causes to a cell are likely due to interaction mechanisms occurring in the cell membrane. For the field forces that are above 1–10 mV/cm, the effects are due to interaction mechanisms occurring in the intracellular compartments. Electric field forces of 10–500 V/cm act as environmental stress factors and result only in a transitory defensive response [5]. In addition, many reports show that these interactions are field force dependent and they can provoke both the enhancement [6] and the inhibition [7] of different cellular functions. It is important that the studied types of responses show the "sensitivity windows" within the above field force range [8]. In the frame of a defined window, the system provides a definite response, while when it is outside the frame, the response disappears or suddenly turns in the opposite direction. The electric field forces that are above 500 V/cm can provoke the sustained response like in the case of electroporation. The most interesting, even the least studied, are the biological effects of medium-strength electric forces of 100 V/cm. The complication, in fact, is that these electric fields can be applied as impulses, long-lasting DC fluxes, or AC waves, and each of them can induce very different types of responses. The applied effects can be a direct ionic current or the electric field generated between the two plates of a condenser, or as a magnetic field, that can provide different interacting doses and different kinds of energies. The named forces can affect the different immune responses of animals [9, 10] or in humans [11] through the induction [12] or augmenting [13] of different

Exogenously added electric impulses can induce the synthesis of antiviral substances, are not interferon, interleukins, or tumor necrosis factors. Therefore, we named them "interferon-like molecules." In our experiments, we also found [14] that medium-strength square-wave impulses of direct ionic current (DC) can result in a short-term direct antiviral resistance to virus infections and in a consequence to alter the membrane properties of the target cells [15]. We therefore

decided to study the changes in the expression of membrane-bound surface marker molecules that are on the surface of the immune competent cells in the human blood [16–18]. Our experiments were aimed to establish whether such exogenous electric stimulation of human leukocytes could be utilized as an immune enhancer and an antiviral protector ex vivo, preferably coupled to the dialysis process. In order to detect the potential hazards of such an application, we have to study further the effects of electrostimulation on some other parameters of the

growing in a monolayer or growing as the suspension culture.

Least but not the last, the nature and biological conditions of the target system can determine the type and extent of the response. The given conditions can induce an enhancement of cell proliferation in a suspension cell culture and the inhibition of cell proliferation in an adherent growing in monolayer cell culture [19]. Therefore, the herein presented experiments are aimed to investigate the influence and some mechanisms of the medium-strength square-wave electric impulses of the field forces of 100 V/cm, on the proliferation pattern of different animal cells

immune response elements.

*Electromagnetic Field Radiation in Matter*

human blood.

**118**

#### **2.1 The electric impulse generator device PGen-1**

The electric impulse generator device PGen-1 was developed and physically realized by Dr. Sandor Toth and Dr. Ferenc Somogyvari. The PGen-1 device provides 1–300 V/cm square-wave impulses of 1–10 ms duration, with a repetition option of 1–9. It has also a continuous work option. Its repetition intervals can be set between 1 and 10 s. The device consists of two separate circuits: (1) a low-voltage circuit, running on transformed and rectified net current, being stabilized by monolithic integrated stabilizers. They work as a power source for the regulator. (2) A high-voltage circuit serves for the impulse generator itself. The outgoing voltage is regulated by a phase-splitting dimmer and is rectified and stabilized. The low-voltage settings secure the filter condenser. The analogous regulator system is composed from a stable multivibrator, governing the counter, and two synchronized mono-stable multi vibrators, generating the outgoing impulses and the visual control of the signals. The outgoing square-wave impulses are characterized in the **Appendix.** The sample chamber has a 50 ml capacity and is a polypropylene tube with a 2.5 cm in diameter, with a platinum wire electrode. It is detachable from the basic device and is autoclavable. Further details are available in the patent description of the PGen-1 device [20] (**Figure 1)**.

#### **2.2 The dosimetry**

The adsorbed doses (AD in J/g) of the electric impulses in the samples were calculated according to Pakhomov et al. [21] by the use of their formula as follows:

$$AD = \frac{\left(E^2 \times d^2 \times W \times n\right)}{\left(R \times M\right)}\tag{1}$$

#### **Figure 1.**

*Electric circuits of the electric impulse generator device PGen-1.*


#### **Table 1.**

*The adsorbed dose (AD in J/g) of the electric impulses in the samples.*

In it E is the E is field in the sample (V/m), W is the pulse width (10<sup>8</sup> s), n is the number of electric impulses delivered to the sample, d is the gap in the cuvette, R is the resistance of the cuvette with the sample (8–9 Ù), and M is the mass of the medium in g. The AD values in J/g of one and three impulses are shown in **Table 1**.

#### **2.3 The cells used in the experiments**

The following cells were used: (a) monolayer cells (chicken embryonic fibroblasts, VERO, MDBK, MRC-5, and HeLa cells) and (b) cells growing in suspension (lymphoblast cells like FB1 and K562).

#### **2.4 Cell multiplication**

The monolayer cell cultures were resuspended in 3 50 ml of medium EMEM complemented with surplus Ca2+ and 2% of fetal calf serum (FCS). The cell lines growing in suspension were multiplied in the medium RPMI 1640 + 10% FCS. For the experiments they were resuspended in 3 50 ml of the medium RPMI 1640 with surplus Ca2+ and 2% FCS.

#### **2.5 The cell treatments with the electric impulses**

The aliquots of 3 50 ml of cells suspensions with the 105 cells/mL were put in the three electro-induction chambers with built-in platinum wire electrodes, for treatment. Samples were subjected to one or three square-wave impulses of 10 ms with a field force of 100 V/cm **(Figure 2)**. The untreated cells in the electro-induction chamber with built-in platinum wire electrodes represented the cell control.

sample was measured three times and the averages were calculated. The GI was calculated by the following formula: GI = absorbance after 4 days/absorbance of the

*Scheme of the experiments. (1) chicken embryonic fibroblasts; (2) cercopithecus arthiopis kidney; (3) Madin-Darby bovine kidney epithelial cells; (4) Medical Research Council cell strain five fibroblasts derived from lung tissue; (5) Henrietta lacks cervical tumor immortal cell line; (6) Homo sapiens blood lymphoblast; (7) Homo*

*Square-Wave Electric Impulses of 10 ms and 100 V/cm of Field Force, Produced…*

*DOI: http://dx.doi.org/10.5772/intechopen.90506*

From the GI of the pulsed cells by the one or three impulses and untreated control cells, the percentage of GI inhibition were calculated by the following formula: percent of GI inhibition = 100 GI of pulsed cells with one or three

After the cell's treatment, the monolayer cells were seeded into a 6-well plates containing 3.3 <sup>10</sup><sup>5</sup> cells/well. Twenty-four hours later and 72 h in the parallels, monolayer cells were detached and Trypan blue positive (=blue) cells were counted.

a. Bring adherent cells into suspension by the trypsin/EDTA as described

previously, and resuspend them in a volume of fresh medium EMEM at least equivalent to the volume of trypsin. Centrifuge and resuspend the cells that grow in suspension in a small volume of medium. In addition, gently pipette

impulses/GI untreated control 100 were calculated [24, 25].

The cell numbers were normalized to control cells as 100% [26, 27].

initial sample [22, 23].

*sapiens bone marrow lymphoblast.*

**Figure 2.**

**2.7 Viability assay**

*2.7.1 Procedure*

**121**

*2.6.2 Percentage of GI inhibition*

to break up clumps.

#### **2.6 The growth parameters determination**

#### *2.6.1 Growth index (GI)*

The GI was determined by the use of spectrophotometer measurements. For the experiments, there were three samples for (1) untreated cell control, (2) one impulse, and (3) three impulses. Cells were cultured in microtiter plates with EMEM medium supplemented with 10% FCS. On the next day, the medium was replaced by new medium containing 5% of FCS. The initial number of cells was determined separately. After 3 h of 5% CO2 incubation at 37°C, the cells were fixed with a 0.25% solution of Glutar aldehyde, and the plates were cooled to 4°C. The cells from untreated control, cells pulsed with one impulse, and cells pulsed with three impulses were incubated for 3 days at 37°C and 5% CO2. After the incubation microtiter plates with cells were fixed with a 0.25% solution of Glutar aldehyde for 20 minutes, washed with phosphate buffer saline (PBS, pH = 7.4), and stained with 4% solution of Methyl blue for 45 minutes at 37°C. Finally, the plates were thoroughly washed with tap water, air-dried, and the colour was extracted by adding of 100 μl of 0.1 M/HCl. The optical density (OD) was measured in AUTOEIA (Lab system) automatic spectrophotometer at 570/650 nm. Every

*Square-Wave Electric Impulses of 10 ms and 100 V/cm of Field Force, Produced… DOI: http://dx.doi.org/10.5772/intechopen.90506*

#### **Figure 2.**

In it E is the E is field in the sample (V/m), W is the pulse width (10<sup>8</sup> s), n is the number of electric impulses delivered to the sample, d is the gap in the cuvette, R is the resistance of the cuvette with the sample (8–9 Ù), and M is the mass of the medium in g. The AD values in J/g of one and three impulses are shown in **Table 1**.

**Number of impulses AD in J/g** 1 0.163 3 0.490

The following cells were used: (a) monolayer cells (chicken embryonic fibroblasts, VERO, MDBK, MRC-5, and HeLa cells) and (b) cells growing in suspension

The monolayer cell cultures were resuspended in 3 50 ml of medium EMEM complemented with surplus Ca2+ and 2% of fetal calf serum (FCS). The cell lines growing in suspension were multiplied in the medium RPMI 1640 + 10% FCS. For the experiments they were resuspended in 3 50 ml of the medium RPMI 1640

The aliquots of 3 50 ml of cells suspensions with the 105 cells/mL were put in the three electro-induction chambers with built-in platinum wire electrodes, for treatment. Samples were subjected to one or three square-wave impulses of 10 ms with a field force of 100 V/cm **(Figure 2)**. The untreated cells in the electro-induction chamber with built-in platinum wire electrodes represented the cell control.

The GI was determined by the use of spectrophotometer measurements. For the experiments, there were three samples for (1) untreated cell control, (2) one impulse, and (3) three impulses. Cells were cultured in microtiter plates with EMEM medium supplemented with 10% FCS. On the next day, the medium was replaced by new medium containing 5% of FCS. The initial number of cells was determined separately. After 3 h of 5% CO2 incubation at 37°C, the cells were fixed with a 0.25% solution of Glutar aldehyde, and the plates were cooled to 4°C. The cells from untreated control, cells pulsed with one impulse, and cells pulsed with three impulses were incubated for 3 days at 37°C and 5% CO2. After the incubation microtiter plates with cells were fixed with a 0.25% solution of Glutar aldehyde for 20 minutes, washed with phosphate buffer saline (PBS, pH = 7.4), and stained with 4% solution of Methyl blue for 45 minutes at 37°C. Finally, the plates were thoroughly washed with tap water, air-dried, and the colour was extracted by

adding of 100 μl of 0.1 M/HCl. The optical density (OD) was measured in AUTOEIA (Lab system) automatic spectrophotometer at 570/650 nm. Every

**2.3 The cells used in the experiments**

*Electromagnetic Field Radiation in Matter*

*The adsorbed dose (AD in J/g) of the electric impulses in the samples.*

(lymphoblast cells like FB1 and K562).

**2.4 Cell multiplication**

**Table 1.**

*2.6.1 Growth index (GI)*

**120**

with surplus Ca2+ and 2% FCS.

**2.5 The cell treatments with the electric impulses**

**2.6 The growth parameters determination**

*Scheme of the experiments. (1) chicken embryonic fibroblasts; (2) cercopithecus arthiopis kidney; (3) Madin-Darby bovine kidney epithelial cells; (4) Medical Research Council cell strain five fibroblasts derived from lung tissue; (5) Henrietta lacks cervical tumor immortal cell line; (6) Homo sapiens blood lymphoblast; (7) Homo sapiens bone marrow lymphoblast.*

sample was measured three times and the averages were calculated. The GI was calculated by the following formula: GI = absorbance after 4 days/absorbance of the initial sample [22, 23].

#### *2.6.2 Percentage of GI inhibition*

From the GI of the pulsed cells by the one or three impulses and untreated control cells, the percentage of GI inhibition were calculated by the following formula: percent of GI inhibition = 100 GI of pulsed cells with one or three impulses/GI untreated control 100 were calculated [24, 25].

#### **2.7 Viability assay**

After the cell's treatment, the monolayer cells were seeded into a 6-well plates containing 3.3 <sup>10</sup><sup>5</sup> cells/well. Twenty-four hours later and 72 h in the parallels, monolayer cells were detached and Trypan blue positive (=blue) cells were counted. The cell numbers were normalized to control cells as 100% [26, 27].

#### *2.7.1 Procedure*

a. Bring adherent cells into suspension by the trypsin/EDTA as described previously, and resuspend them in a volume of fresh medium EMEM at least equivalent to the volume of trypsin. Centrifuge and resuspend the cells that grow in suspension in a small volume of medium. In addition, gently pipette to break up clumps.


g. Approximately hundreds of cells were acquired for analyses and expressed as

h. As electric fields were increased, greater numbers of cells became Caspase-3 positive with a homogeneous shift of cells into the Caspase-3-positive gate, making cell percentages with active Caspase-3 the most accurate and meaningful quantification of active Caspase-3 as an apoptosis marker.

For the level of statistical significance determination (\**p* < 0.1, \*\**p* < 0.05), the T-test was used. All the data are shown as mean value standard deviation. The tests were performed in triplicate and each was repeated three to four times.

**3.1 The growth parameters, percentages of dead cells, and percentages of**

*3.1.1 The growth parameters and percentage of dead cells after 24 h of incubation*

The main effect of the cell's treatment with one or three impulse with the adsorbed dose (AD) of one impulse 0.163 J/g and of three impulses 0.490 J/g was the GI index inhibition expressed in percentage. The detailed results **(Table 2,**

Chicken fibroblasts: after one impulse 74.41% and after three impulses 57.2% MDBK cells: after one impulse 31.35% and after three impulses 30.4% Vero cells: after one impulse 37.2% and after three impulses 24.7% MRC-5 cells: after one impulse 38.8% and after three impulses 41.9% HeLa cells: after one impulse 78% and after three impulses 27%

When the percentages of dead cells after one impulse or three impulses were

Chicken fibroblasts: after one impulse 9.2% and after three impulses 19.6%

MDBK cells: after one impulse 11.2% and after three impulses 12.7%. Vero cells: after one impulse 8.6% and after three impulses 24.7%. MRC-5 cells: after one impulse 14.7% and after three impulses 32.6%. HeLa cells: after one impulse there were 7.4% of dead cells, and after three

*3.1.2 The growth parameters and percentage of dead cells after 72 h of incubation*

After the cell's treatment with one or three impulses with the adsorbed dose (AD) of one impulse was 0.163 J/g and of three impulses with 0.490 J/g, was the growth index inhibition expressed in percentage. The results presented in **Table 2**

**Caspase-3 positive cells of different monolayer cells**

percentage of cells showing positive fluorescence.

*DOI: http://dx.doi.org/10.5772/intechopen.90506*

*Square-Wave Electric Impulses of 10 ms and 100 V/cm of Field Force, Produced…*

**2.9 Statistical evaluation**

**Figure 3)** show the following:

of dead cells.

evaluated, the following data were obtained:

impulses, there were 11.3% of dead cells.

and **Figure 4** show the following:

**123**

**3. Results**


non‐viable cell count dead cells ð Þ *<sup>=</sup>*ml

<sup>¼</sup> No*:*of dead cells counted No*:*of large corner squares counted � dilution factor � <sup>10</sup>*:*<sup>0000</sup> (2)

#### **2.8 Caspase-3 assay**

#### *2.8.1 Procedure*


*Square-Wave Electric Impulses of 10 ms and 100 V/cm of Field Force, Produced… DOI: http://dx.doi.org/10.5772/intechopen.90506*


#### **2.9 Statistical evaluation**

For the level of statistical significance determination (\**p* < 0.1, \*\**p* < 0.05), the T-test was used. All the data are shown as mean value standard deviation. The tests were performed in triplicate and each was repeated three to four times.

#### **3. Results**

b. Under sterile conditions take 100–200 μl of cell suspension.

pipetting.

*Electromagnetic Field Radiation in Matter*

the cover slip).

magnification.

using the equation below:

non‐viable cell count dead cells ð Þ *<sup>=</sup>*ml

ImmunoChemistry Technologies LLC).

<sup>¼</sup> No*:*of dead cells counted

of >100.

**2.8 Caspase-3 assay**

*2.8.1 Procedure*

**122**

c. Add an equal volume of Trypan blue (dilution factor = 2) and mix by gentle

d. Clean the haemocytometer and the cover slips in 70% ethanol. Clean and dry them with two-site by rubbing with cotton sheets wrapped in cotton cloth.

e. Moisten the cover slip with water or exhaled breath. Slide the cover slip over the chamber back and forth using slight pressure until Newton's refraction rings appear (Newton's refraction rings are seen as rainbow-like rings under

f. Fill both sides of the chamber with cell suspension (approximately 5–10 μl), and view under an inverted phase contrast microscope using the 20�

g. Count the number of viable (seen as bright cells) and nonviable cells (stained blue). Ideally >100 cells should be counted in order to increase the accuracy of the cell count. Note the number of squares counted to obtain your count

h. Calculate the concentration of nonviable cells and the percentage of cells

No*:*of large corner squares counted � dilution factor � <sup>10</sup>*:*<sup>0000</sup>

a. Caspase-3 activities in pulsed cells were assayed using a commercial method

b. The monolayer or suspension cells were pulsed with one or three impulses.

d. According to the manufacturer's recommendations, cells were labelled with carboxyfluorescein caspase-3 inhibitors for 1 h at 37°C under 5% CO2 and

c. One hour after treatment, cells were gently removed from cuvettes and

e. Cells were washed with PBS buffer to remove the unbound reagent.

f. Cell fluorescence was determined by the use of fluorescent microscope.

based on fluorochrome-labeled inhibitors of caspases (FLICA,

resuspended in a medium EMEM without foetal serum.

protected from light as it was previously described [28].

(2)

#### **3.1 The growth parameters, percentages of dead cells, and percentages of Caspase-3 positive cells of different monolayer cells**

#### *3.1.1 The growth parameters and percentage of dead cells after 24 h of incubation*

The main effect of the cell's treatment with one or three impulse with the adsorbed dose (AD) of one impulse 0.163 J/g and of three impulses 0.490 J/g was the GI index inhibition expressed in percentage. The detailed results **(Table 2, Figure 3)** show the following:

Chicken fibroblasts: after one impulse 74.41% and after three impulses 57.2% MDBK cells: after one impulse 31.35% and after three impulses 30.4% Vero cells: after one impulse 37.2% and after three impulses 24.7% MRC-5 cells: after one impulse 38.8% and after three impulses 41.9% HeLa cells: after one impulse 78% and after three impulses 27%

When the percentages of dead cells after one impulse or three impulses were evaluated, the following data were obtained:

Chicken fibroblasts: after one impulse 9.2% and after three impulses 19.6% of dead cells.

MDBK cells: after one impulse 11.2% and after three impulses 12.7%. Vero cells: after one impulse 8.6% and after three impulses 24.7%. MRC-5 cells: after one impulse 14.7% and after three impulses 32.6%. HeLa cells: after one impulse there were 7.4% of dead cells, and after three impulses, there were 11.3% of dead cells.

*3.1.2 The growth parameters and percentage of dead cells after 72 h of incubation*

After the cell's treatment with one or three impulses with the adsorbed dose (AD) of one impulse was 0.163 J/g and of three impulses with 0.490 J/g, was the growth index inhibition expressed in percentage. The results presented in **Table 2** and **Figure 4** show the following:


**Monolayer**

**Cell treatment**

**125**

 **Time of treatment**

**Chicken fibroblasts**

> GI inh. %

% of dead cells

% of Casp.3 pos. cells

CPD

GI GI inh. %

% of dead cells

% of Casp.3 pos. cells

CPD

*1Cercopithecus*

*2Madin-Darby*

*3Medical Research Council cell strain five fibroblasts derived from lung tissue.*

*4Henrietta lacks cervical tumor immortal cell line.*

*5*Homo sapiens *blood lymphoblast.*

*6*Homo sapiens *bone marrow lymphoblast.*

*7GI, growth index.*

*8CPD, cumulative population doubling.*

**Table 2.** *The growth parameters,*

 *percentages*

 *of dead cells, and percentages*

 *of Caspase-3*

 *positive cells.*

*…*

 *bovine kidney epithelial cells.*

 *arthiopis kidney.*

0.02 0.003

 0.04 0.005

 0.53 0.004

0.35 0.03

1.03 0.11

 2.33 0.21

 2.38 0.21

0.98 0.08

57.7 63.5

 44.7

15 11.7

30.5

55.09

11.4

1.7

0.9

*Square-Wave Electric Impulses of 10 ms and 100 V/cm of Field Force, Produced*

38.9

70.1

14.5

2.2

1.2

48.3

31.9

37.7

46.9

10.1

2.3

 1.12 0.11

 1.88 0.17

 0.93 0.08

 1.05 0.09

 5.68 0.51

 5.33 0.48

72 h

1.78 0.16

0.92 0.08

0.55 0.05

0.20 0.01

2.95 0.26

 0.22 0.02

 0.94 0.08

*DOI: http://dx.doi.org/10.5772/intechopen.90506*

79.4 19.6

 14.5

12.7

9.9

19.3

25.5

8.2

1.02

1.3

24.7

32.6

11.3

1.3

1.7

30.4

41.9

13.6

27.1

13.4

3.7

 **MDBK1**

 **cells**

**Vero2**

**MRC-53**

**HeLa4**

**FB15**

**K 5626**

**Suspension**

 **cells**

#### *Electromagnetic Field Radiation in Matter*

