Electron Positron Plasma

Chapter 5

Abstract

Ian Joseph Lazarus

laboratory experiments.

1. Introduction

59

Electrostatic Waves in Magnetized

The behavior of arbitrary amplitude linear and nonlinear electrostatic waves that propagate in a magnetized four component, two-temperature, electron-positron plasma is presented. The characteristics of the dispersive properties of the associated linear modes using both fluid and kinetic theory are examined. The fluid theory analysis of the electrostatic linear waves shows the existence of electron acoustic, upper hybrid, electron plasma and electron cyclotron branches. A kinetic theory analysis is then used to study the acoustic mode, in particular the effect of Landau damping, which for the parameter regime considered is due to the cooler species. Consequently, it is found that a large enough drift velocity is required to produce wave growth. Nonlinear electrostatic solitary waves (ESWs), similar to those found in the broadband electrostatic noise observed in various regions of the earth's magnetosphere is further investigated. A set of nonlinear differential equations for the ESWs, which propagate obliquely to an external magnetic field is derived and numerically solved. The effect of various plasma parameters on the waves is explored and shows that as the electric driving force is increased, the electric field structure evolves from a sinusoidal wave to a spiky bipolar form. The results are relevant to both astrophysical environments and related laser-induced

Electron-Positron Plasmas

Keywords: electrons, positrons, electrostatic waves, nonlinear waves

Electron-positron plasmas play a significant role in the understanding of the early universe [1, 2], active galactic nuclei [3], gamma ray bursts (GRBs) [4], pulsar magnetospheres [5, 6] and the solar atmosphere [7]. These plasmas are also important in understanding extremely dense stars such as white dwarfs and pulsars, which are thought to be rotating neutron stars. The existence of these plasmas in neutron stars and in the pulsar magnetosphere is well documented [8]. The possibility for the co-existence of two types of cold and hot electron-positron populations in the pulsar magnetosphere has been suggested by [9] which was inspired by the pulsar model [10]. In their model, accelerated primary electrons moving on curved magnetic field lines emit curvature photons which produce electron-positron pairs. The secondary particles then produce curvature radiation, hence producing new electron-positron pairs, and so on. Therefore, both the electron and positron populations can be subdivided in two groups of distinct temperatures, one modeling the original plasma, and the second the higher-energy cascade-bred pairs. It is also

#### Chapter 5

## Electrostatic Waves in Magnetized Electron-Positron Plasmas

Ian Joseph Lazarus

#### Abstract

The behavior of arbitrary amplitude linear and nonlinear electrostatic waves that propagate in a magnetized four component, two-temperature, electron-positron plasma is presented. The characteristics of the dispersive properties of the associated linear modes using both fluid and kinetic theory are examined. The fluid theory analysis of the electrostatic linear waves shows the existence of electron acoustic, upper hybrid, electron plasma and electron cyclotron branches. A kinetic theory analysis is then used to study the acoustic mode, in particular the effect of Landau damping, which for the parameter regime considered is due to the cooler species. Consequently, it is found that a large enough drift velocity is required to produce wave growth. Nonlinear electrostatic solitary waves (ESWs), similar to those found in the broadband electrostatic noise observed in various regions of the earth's magnetosphere is further investigated. A set of nonlinear differential equations for the ESWs, which propagate obliquely to an external magnetic field is derived and numerically solved. The effect of various plasma parameters on the waves is explored and shows that as the electric driving force is increased, the electric field structure evolves from a sinusoidal wave to a spiky bipolar form. The results are relevant to both astrophysical environments and related laser-induced laboratory experiments.

Keywords: electrons, positrons, electrostatic waves, nonlinear waves

#### 1. Introduction

Electron-positron plasmas play a significant role in the understanding of the early universe [1, 2], active galactic nuclei [3], gamma ray bursts (GRBs) [4], pulsar magnetospheres [5, 6] and the solar atmosphere [7]. These plasmas are also important in understanding extremely dense stars such as white dwarfs and pulsars, which are thought to be rotating neutron stars. The existence of these plasmas in neutron stars and in the pulsar magnetosphere is well documented [8]. The possibility for the co-existence of two types of cold and hot electron-positron populations in the pulsar magnetosphere has been suggested by [9] which was inspired by the pulsar model [10]. In their model, accelerated primary electrons moving on curved magnetic field lines emit curvature photons which produce electron-positron pairs. The secondary particles then produce curvature radiation, hence producing new electron-positron pairs, and so on. Therefore, both the electron and positron populations can be subdivided in two groups of distinct temperatures, one modeling the original plasma, and the second the higher-energy cascade-bred pairs. It is also

known that in astrophysical and cosmic plasmas, a minority of cold electrons and heavy ions exist along with hot electron-positron pairs [11]. Hence, the formation of two temperature multispecies plasmas is possible due to the outflow of the electronpositron plasma from pulsars entering into an interstellar cold, low-density electron-ion plasma [12].

2. Linear waves in electron-positron plasmas: fluid theory approach

assumed to be in the x–z plane.

and

direction.

electrostatic potential.

namely the continuity equations,

the equations of motion,

∂vjc ∂t

electrons and positrons become

61

bution, their densities are, respectively

Electrostatic Waves in Magnetized Electron-Positron Plasmas

DOI: http://dx.doi.org/10.5772/intechopen.80958

Let us consider a homogeneous magnetized, four component electron-positron plasma, consisting of cool electrons and cool positrons with equal temperatures and equilibrium densities denoted by Tc and n0c, respectively, and hot electrons and hot positrons with equal temperatures and equilibrium densities denoted by Th and n0h, respectively. The temperatures are expressed in energy units and wave propagation is taken in the x-direction at an angle θ to the ambient magnetic field B0, which is

Assuming that the hot isothermal species are described by the Boltzmann distri-

eϕ Th

Th

(1)

, (2)

<sup>þ</sup> <sup>∇</sup>: njcvjc <sup>¼</sup> <sup>0</sup>, (3)

njcm

∇njc, (4)

neh ¼ n0<sup>h</sup> exp

nph <sup>¼</sup> <sup>n</sup>0<sup>h</sup> exp �e<sup>ϕ</sup>

where neh (nph) is the density of the hot electrons (positrons) and ϕ is the

Using Boltzmann distribution of hot electrons and positrons is justified provided they have sufficiently high temperatures, much greater than that of cooler species such that their thermal velocities parallel to the magnetic field exceed the phase velocity of the modes so that they are able to establish the Boltzmann distribution. The magnetic field effects on hot species are not felt since the perturbation wavelengths are shorter than their gyroradii such that both hot electrons and positrons follow essentially straight line orbits across the magnetic field

The dynamics of cooler isothermal species are governed by fluid equations,

∇ϕ þ ε<sup>j</sup>

where ε<sup>j</sup> = + 1(�1) for positrons (electrons), j ¼ e pð Þ for the electrons (posi-

In the above equations, nj and vj are the number densities and fluid velocities respectively of the jth species. In order to derive the linear dispersion relation, equations (3)–(5) are linearized. For perturbations varying as exp ð Þ i kx ð Þ � <sup>ω</sup><sup>t</sup> , <sup>∂</sup>=∂<sup>t</sup> is replaced with �i<sup>ω</sup> and <sup>∂</sup>=∂<sup>x</sup> with ik. Hence the perturbed densities for the

e m

vjc � B<sup>0</sup> � <sup>γ</sup>Tc

<sup>∂</sup>x<sup>2</sup> ¼ �e npc � nec <sup>þ</sup> nph � neh : (5)

∂njc ∂t

> e m

þ vjc:∇vjc ¼ �ε<sup>j</sup>

trons). The system is closed by the Poisson equation

ε0 ∂2 ϕ

Investigations into electron-positron plasma behavior have focused primarily on the relativistic regime. It is however plausible that nonrelativistic astrophysical electron-positron plasmas may exist, given the effect of cooling by cyclotron emission [13]. The study of nonrelativistic astrophysical electron-positron plasmas therefore plays an important role in understanding wave fluctuations. Due to the equal charge to mass ratio for these oppositely charged species, only one frequency scale exists and due to this symmetry, there exists different physical phenomena to the conventional electron-ion plasmas. Further, the frequent instabilities that arise in space plasma and astrophysical environments (e.g., solar flames and auroras), involve the growth of electrostatic and electromagnetic waves which gives rise to a growing wave mode. In particular, the linear behavior of the electrostatic modes using fluid and kinetic theory approaches allows one to understand the effect of plasma parameters such as the propagation angle, cool to hot temperature ratios, density ratios and the magnetic field strength on the waves.

Investigations conducted have focused on modulational instabilities and wave localization [14], envelope solitons [15], multidimensional effects [16]. Large amplitude solitons and electrostatic nonlinear potential structures in electronpositron plasmas having equal hot and cold components of both species have been studied by a number of authors [17–19]. In one such study [20], using the two-fluid model with a single temperature they investigated linear and nonlinear longitudinal and transverse electrostatic and electromagnetic waves in a nonrelativistic electronpositron plasma in the absence and presence of an external magnetic field. They found that several of the modes present in electron-ion plasmas also existed in electron-positron plasmas, but in a modified form. Collective modes in nonrelativistic electron-positron plasmas using the kinetic approach was studied by [21]. The author found that the dispersion relations for the longitudinal modes in the electron-positron plasma for both unmagnetized and magnetized electron-positron plasmas were similar to the modes in one-component electron or electron-ion plasmas. Moreover, the hybrid resonances present in the former are not found in an electron-positron plasma.

The understanding of nonlinear wave structures which gives rise to electrostatic solitary wave (ESWs) in space is important since it is known that satellite measurements using high-time resolution equipment aboard spacecraft S3-3 [22], Viking [23], Geotail [24], Polar [25], and Fast [26] have indicated the presence of Broadband Electrostatic Noise (BEN) in the auroral magnetosphere at altitudes between 3000 km to 8000 km and beyond. These observations have shown the presence of electrostatic solitary waves (ESWs), which are characterized by their spiky bipolar pulses. Hence, the study of nonlinear wave behavior in electron-positron plasmas propagating at oblique angles to an ambient magnetic field is explored to understand electrostatic solitary waves in space. Specifically, the spiky nature of the electrostatic potential structures and the effects of the propagation angle, cold and hot drift velocities, cool to hot density and temperature ratios and Mach number on the ESWs are examined.

In this chapter a two-temperature magnetized four component electron-positron plasma model is used to study linear wave modes using both the fluid and kinetic approaches as well as the behavior of the nonlinear structures of these electrostatic solitary waves (ESWs) which plays an important role in space and astrophysical environments.

#### 2. Linear waves in electron-positron plasmas: fluid theory approach

Let us consider a homogeneous magnetized, four component electron-positron plasma, consisting of cool electrons and cool positrons with equal temperatures and equilibrium densities denoted by Tc and n0c, respectively, and hot electrons and hot positrons with equal temperatures and equilibrium densities denoted by Th and n0h, respectively. The temperatures are expressed in energy units and wave propagation is taken in the x-direction at an angle θ to the ambient magnetic field B0, which is assumed to be in the x–z plane.

Assuming that the hot isothermal species are described by the Boltzmann distribution, their densities are, respectively

$$n\_{ch} = n\_{0h} \exp\left(\frac{e\phi}{T\_h}\right) \tag{1}$$

and

known that in astrophysical and cosmic plasmas, a minority of cold electrons and heavy ions exist along with hot electron-positron pairs [11]. Hence, the formation of two temperature multispecies plasmas is possible due to the outflow of the electron-

Investigations into electron-positron plasma behavior have focused primarily on

Investigations conducted have focused on modulational instabilities and wave localization [14], envelope solitons [15], multidimensional effects [16]. Large amplitude solitons and electrostatic nonlinear potential structures in electronpositron plasmas having equal hot and cold components of both species have been studied by a number of authors [17–19]. In one such study [20], using the two-fluid model with a single temperature they investigated linear and nonlinear longitudinal and transverse electrostatic and electromagnetic waves in a nonrelativistic electronpositron plasma in the absence and presence of an external magnetic field. They found that several of the modes present in electron-ion plasmas also existed in electron-positron plasmas, but in a modified form. Collective modes in nonrelativistic electron-positron plasmas using the kinetic approach was studied by [21]. The

author found that the dispersion relations for the longitudinal modes in the

electron-positron plasma for both unmagnetized and magnetized electron-positron plasmas were similar to the modes in one-component electron or electron-ion plasmas. Moreover, the hybrid resonances present in the former are not found in an

The understanding of nonlinear wave structures which gives rise to electrostatic solitary wave (ESWs) in space is important since it is known that satellite measurements using high-time resolution equipment aboard spacecraft S3-3 [22], Viking [23], Geotail [24], Polar [25], and Fast [26] have indicated the presence of Broadband Electrostatic Noise (BEN) in the auroral magnetosphere at altitudes between 3000 km to 8000 km and beyond. These observations have shown the presence of electrostatic solitary waves (ESWs), which are characterized by their spiky bipolar pulses. Hence, the study of nonlinear wave behavior in electron-positron plasmas propagating at oblique angles to an ambient magnetic field is explored to understand electrostatic solitary waves in space. Specifically, the spiky nature of the electrostatic potential structures and the effects of the propagation angle, cold and hot drift velocities, cool to hot density and temperature ratios and Mach number on

In this chapter a two-temperature magnetized four component electron-positron plasma model is used to study linear wave modes using both the fluid and kinetic approaches as well as the behavior of the nonlinear structures of these electrostatic solitary waves (ESWs) which plays an important role in space and astrophysical

positron plasma from pulsars entering into an interstellar cold, low-density

density ratios and the magnetic field strength on the waves.

the relativistic regime. It is however plausible that nonrelativistic astrophysical electron-positron plasmas may exist, given the effect of cooling by cyclotron emission [13]. The study of nonrelativistic astrophysical electron-positron plasmas therefore plays an important role in understanding wave fluctuations. Due to the equal charge to mass ratio for these oppositely charged species, only one frequency scale exists and due to this symmetry, there exists different physical phenomena to the conventional electron-ion plasmas. Further, the frequent instabilities that arise in space plasma and astrophysical environments (e.g., solar flames and auroras), involve the growth of electrostatic and electromagnetic waves which gives rise to a growing wave mode. In particular, the linear behavior of the electrostatic modes using fluid and kinetic theory approaches allows one to understand the effect of plasma parameters such as the propagation angle, cool to hot temperature ratios,

electron-ion plasma [12].

Charged Particles

electron-positron plasma.

the ESWs are examined.

environments.

60

$$n\_{ph} = n\_{0h} \exp\left(\frac{-e\phi}{T\_h}\right),\tag{2}$$

where neh (nph) is the density of the hot electrons (positrons) and ϕ is the electrostatic potential.

Using Boltzmann distribution of hot electrons and positrons is justified provided they have sufficiently high temperatures, much greater than that of cooler species such that their thermal velocities parallel to the magnetic field exceed the phase velocity of the modes so that they are able to establish the Boltzmann distribution. The magnetic field effects on hot species are not felt since the perturbation wavelengths are shorter than their gyroradii such that both hot electrons and positrons follow essentially straight line orbits across the magnetic field direction.

The dynamics of cooler isothermal species are governed by fluid equations, namely the continuity equations,

$$\frac{\partial n\_{j\epsilon}}{\partial t} + \nabla. \left( n\_{j\epsilon} \mathbf{v}\_{j\epsilon} \right) = \mathbf{0},\tag{3}$$

the equations of motion,

$$\frac{\partial \mathbf{v}\_{j\varepsilon}}{\partial t} + \mathbf{v}\_{j\varepsilon} \cdot \nabla \mathbf{v}\_{j\varepsilon} = -\varepsilon\_{j} \frac{\varepsilon}{m} \nabla \phi + \varepsilon\_{j} \frac{\varepsilon}{m} \left(\mathbf{v}\_{j\varepsilon} \times B\_{0}\right) - \frac{\chi T\_{\varepsilon}}{n\_{j\varepsilon} m} \nabla n\_{j\varepsilon} \tag{4}$$

where ε<sup>j</sup> = + 1(�1) for positrons (electrons), j ¼ e pð Þ for the electrons (positrons). The system is closed by the Poisson equation

$$
\varepsilon\_0 \frac{\partial^2 \phi}{\partial \mathbf{x}^2} = -e \left( \mathfrak{n}\_{\rm pc} - \mathfrak{n}\_{\rm ct} + \mathfrak{n}\_{\rm ph} - \mathfrak{n}\_{\rm eh} \right). \tag{5}
$$

In the above equations, nj and vj are the number densities and fluid velocities respectively of the jth species. In order to derive the linear dispersion relation, equations (3)–(5) are linearized. For perturbations varying as exp ð Þ i kx ð Þ � <sup>ω</sup><sup>t</sup> , <sup>∂</sup>=∂<sup>t</sup> is replaced with �i<sup>ω</sup> and <sup>∂</sup>=∂<sup>x</sup> with ik. Hence the perturbed densities for the electrons and positrons become

Charged Particles

$$n\_{\rm ce} = -\left(\frac{n\_{0\varepsilon}e\hbar^2\phi}{m}\right)\left(\frac{\omega^2 - \Omega^2\cos^2\theta}{\alpha^4 - \alpha^2(3k^2v\_{\rm tc}^2 + \Omega^2) + 3k^2v\_{\rm tc}^2\Omega^2\cos^2\theta}\right). \tag{6}$$

and

$$m\_{\rm pc} = \left(\frac{n\_{\rm 0c} e k^2 \phi}{m}\right) \left(\frac{\alpha^2 - \Omega^2 \cos^2 \theta}{\alpha^4 - \alpha^2 \left(3k^2 v\_{\rm tc}^2 + \Omega^2\right) + 3k^2 v\_{\rm tc}^2 \Omega^2 \cos^2 \theta}\right). \tag{7}$$

From equations (1) and (2), the perturbed densities for the hot species are given by,

$$n\_{eh} = n\_{oh} \frac{e\phi}{T\_h} \tag{8}$$

Taking short wavelength limit (k<sup>2</sup>

DOI: http://dx.doi.org/10.5772/intechopen.80958

<sup>ω</sup><sup>4</sup> � <sup>ω</sup><sup>2</sup> <sup>3</sup>k<sup>2</sup>

v2 tc <sup>þ</sup> <sup>ω</sup><sup>2</sup> UH � � <sup>þ</sup> <sup>3</sup>k<sup>2</sup>

Electrostatic Waves in Magnetized Electron-Positron Plasmas

reduces to,

where

<sup>ω</sup>pc <sup>¼</sup> noce<sup>2</sup> ð Þ <sup>=</sup>ε0<sup>m</sup>

equation (13) in the limit 3k<sup>2</sup>

obtains for the upper hybrid mode,

ω2

dicular and pure parallel propagations.

tion (10), reduces to:

mode, and

obtains,

63

2.1 Case I: pure perpendicular propagation

Taking the short wavelength limit (k<sup>2</sup>

model is due to the cooler species, where ω<sup>2</sup>

Considering the pure perpendicular (<sup>θ</sup> <sup>¼</sup> <sup>90</sup><sup>o</sup>

<sup>ω</sup><sup>4</sup> � <sup>ω</sup><sup>2</sup> <sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>3</sup>k<sup>2</sup>

<sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>3</sup>k<sup>2</sup>

<sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

<sup>þ</sup> <sup>¼</sup> <sup>3</sup>k<sup>2</sup>

λ2

ω2

v2 tc <sup>þ</sup> <sup>ω</sup><sup>2</sup> UH � �<sup>2</sup>

v2 tc <sup>þ</sup> <sup>ω</sup><sup>2</sup> UH � � �

> ω2 � ¼

Taking the negative square root of equation (13) yields

3k<sup>2</sup> v2 tc <sup>þ</sup> <sup>2</sup>ω<sup>2</sup> pc � �Ω<sup>2</sup> cos <sup>2</sup><sup>θ</sup>

> 3k<sup>2</sup> v2 tc <sup>þ</sup> <sup>ω</sup><sup>2</sup> UH

> > v2 tc þ

Hence the normal mode frequencies are, ω ¼ 0, which is a nonpropagating

v2 tc þ

λ2

UH <sup>þ</sup> <sup>3</sup>k<sup>2</sup>

showing that the behavior of the upper hybrid mode for the two temperature

k2 v2 ea

> k2 v2 ea

<sup>1</sup> <sup>þ</sup> <sup>1</sup> 2 k2 λ2 dh

v2

<sup>p</sup> <sup>þ</sup> <sup>2</sup>ω<sup>2</sup> pc.

UH <sup>¼</sup> <sup>Ω</sup><sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>1</sup> 2 k2 λ2 dh ! <sup>¼</sup> <sup>0</sup>: (17)

In order to gain physical insight into the solution space of the dispersion relation, the two extreme limits of equation (10) will now be considered, viz. pure perpen-

v2 tc <sup>þ</sup> <sup>2</sup>ω<sup>2</sup> pc � �Ω<sup>2</sup> cos <sup>2</sup>

<sup>1</sup>=<sup>2</sup> as the plasma frequency of the cooler species. If one solves

3k<sup>2</sup> v2 tc þ ω<sup>2</sup> UH

v2

≫ 4 3k<sup>2</sup>

3k<sup>2</sup> v2 tc <sup>þ</sup> <sup>2</sup>ω<sup>2</sup> pc � �Ω<sup>2</sup> cos <sup>2</sup><sup>θ</sup>

UH <sup>¼</sup> <sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>ω<sup>2</sup>

is the upper hybrid frequency associated with the cooler species [20], with

Dh ≫ 1), the dispersion relation equation (10)

pc (14)

� �, one

tcΩ<sup>2</sup> cos <sup>2</sup><sup>θ</sup> <sup>þ</sup> <sup>2</sup>ω<sup>2</sup>

θ ¼ 0, (13)

pcΩ<sup>2</sup> cos <sup>2</sup>θ

, (15)

, (16)

) limit, the general dispersion rela-

: (18)

dh ≫ 1) of the above relationship, one

tc: (19)

and

$$m\_{ph} = -n\_{oh} \frac{e\phi}{T\_h}.\tag{9}$$

Substituting equations (6)–(9), into Poisson's equation (5), the general dispersion relation for the two temperature electron-positron plasma is found to be

$$\log^2\left(\boldsymbol{\mu}^2-\boldsymbol{\Omega}^2\right)-3k^2v\_{\rm tr}^2\left(\boldsymbol{\mu}^2-\boldsymbol{\Omega}^2\cos^2\theta\right)-\frac{k^2v\_{\rm ea}^2}{1+\frac{1}{2}k^2\lambda\_{\rm Dh}^2}\left(\boldsymbol{\mu}^2-\boldsymbol{\Omega}^2\cos^2\theta\right)=0\tag{10}$$

where vea ¼ ð Þ n0<sup>c</sup>=n0<sup>h</sup> 1=2 vth is the acoustic speed of the electron-positron plasma, analogous in form to the electron acoustic speed in an electron-ion plasma [27]. The thermal velocity of the cool species is vtc ¼ ð Þ Tc=m 1=2 , Ω<sup>j</sup> ¼ Ω ¼ qj Bo=m is the gyrofrequency of the electrons and positrons and <sup>λ</sup>dh <sup>¼</sup> <sup>ε</sup>0Th=n0he<sup>2</sup> ð Þ<sup>1</sup>=<sup>2</sup> is the Debye length of the hot species.

It is noted that the study of linear electrostatic waves using a simple fluid model cannot handle the possible Landau damping of the modes. Hence, Landau damping is not significant since phase velocities are far away from the thermal velocities of either the hot or cooler species, i.e., vth ≫ v<sup>ϕ</sup> ≫ vtc with Th ≫ Tc. The effects of the temperature variation on the acoustic mode in terms of Landau damping using kinetic theory are discussed in the next section.

For a single species electron-positron plasma, with temperature Tc, equation (10) reduces to,

$$
\alpha^4 - \alpha^2 \left(\Omega^2 + 3k^2 v\_{tc}^2\right) + 3k^2 v\_{tc}^2 \Omega^2 \cos^2 \theta = 0. \tag{11}
$$

This is identical to the dispersion relation of [20] for their single temperature electron-positron model.

For wave frequencies much lower than the gyrofrequency and satisfying ω ≪ Ω cos θ, the associated electron-acoustic (or positron-acoustic) mode is found to be,

$$
\omega^2 = \frac{k^2 v\_{ea}^2 \cos^2 \theta}{1 + \frac{1}{2} k^2 \lambda\_{Dh}^2} + 3k^2 v\_{tc}^2 \cos^2 \theta. \tag{12}
$$

Taking short wavelength limit (k<sup>2</sup> λ2 Dh ≫ 1), the dispersion relation equation (10) reduces to,

$$
\alpha^4 - \alpha^2 \left( 3k^2 v\_{\rm tc}^2 + \alpha\_{\rm UH}^2 \right) + \left( 3k^2 v\_{\rm tc}^2 + 2\alpha\_{\rm pc}^2 \right) \Omega^2 \cos^2 \theta = 0,\tag{13}
$$

where

nec ¼ � <sup>n</sup>0cek<sup>2</sup>

npc <sup>¼</sup> <sup>n</sup>0cek<sup>2</sup>

and

Charged Particles

and

<sup>ω</sup><sup>2</sup> <sup>ω</sup><sup>2</sup> � <sup>Ω</sup><sup>2</sup> � � � <sup>3</sup>k<sup>2</sup>

where vea ¼ ð Þ n0<sup>c</sup>=n0<sup>h</sup>

length of the hot species.

(10) reduces to,

to be,

62

electron-positron model.

v2

1=2

thermal velocity of the cool species is vtc ¼ ð Þ Tc=m

kinetic theory are discussed in the next section.

<sup>ω</sup><sup>4</sup> � <sup>ω</sup><sup>2</sup> <sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>3</sup>k<sup>2</sup>

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

v2 ea cos <sup>2</sup>θ

<sup>1</sup> <sup>þ</sup> <sup>1</sup> 2 k2 λ2 Dh

tc <sup>ω</sup><sup>2</sup> � <sup>Ω</sup><sup>2</sup> cos <sup>2</sup>

by,

ϕ m !

ϕ m ! <sup>ω</sup><sup>4</sup> � <sup>ω</sup><sup>2</sup> <sup>3</sup>k<sup>2</sup>

<sup>ω</sup><sup>4</sup> � <sup>ω</sup><sup>2</sup> <sup>3</sup>k<sup>2</sup>

<sup>ω</sup><sup>2</sup> � <sup>Ω</sup><sup>2</sup> cos <sup>2</sup><sup>θ</sup>

<sup>ω</sup><sup>2</sup> � <sup>Ω</sup><sup>2</sup> cos <sup>2</sup><sup>θ</sup>

!

!

v2

v2

tcΩ<sup>2</sup> cos <sup>2</sup>θ

tcΩ<sup>2</sup> cos <sup>2</sup>θ

: (9)

<sup>θ</sup> � � <sup>¼</sup> 0 (10)

θ ¼ 0: (11)

θ: (12)

Bo=m is the gyro-

<sup>ω</sup><sup>2</sup> � <sup>Ω</sup><sup>2</sup> cos <sup>2</sup>

, Ω<sup>j</sup> ¼ Ω ¼ qj

vth is the acoustic speed of the electron-positron plasma,

1=2

: (6)

: (7)

(8)

v2 tc <sup>þ</sup> <sup>Ω</sup><sup>2</sup> � � <sup>þ</sup> <sup>3</sup>k<sup>2</sup>

v2 tc <sup>þ</sup> <sup>Ω</sup><sup>2</sup> � � <sup>þ</sup> <sup>3</sup>k<sup>2</sup>

From equations (1) and (2), the perturbed densities for the hot species are given

eϕ Th

> eϕ Th

> > ea

neh ¼ noh

nph ¼ �noh

sion relation for the two temperature electron-positron plasma is found to be

<sup>θ</sup> � � � <sup>k</sup><sup>2</sup> <sup>v</sup><sup>2</sup>

Substituting equations (6)–(9), into Poisson's equation (5), the general disper-

<sup>1</sup> <sup>þ</sup> <sup>1</sup> 2 k2 λ2 Dh

analogous in form to the electron acoustic speed in an electron-ion plasma [27]. The

It is noted that the study of linear electrostatic waves using a simple fluid model cannot handle the possible Landau damping of the modes. Hence, Landau damping is not significant since phase velocities are far away from the thermal velocities of either the hot or cooler species, i.e., vth ≫ v<sup>ϕ</sup> ≫ vtc with Th ≫ Tc. The effects of the temperature variation on the acoustic mode in terms of Landau damping using

For a single species electron-positron plasma, with temperature Tc, equation

This is identical to the dispersion relation of [20] for their single temperature

v2 tcΩ<sup>2</sup> cos <sup>2</sup>

<sup>þ</sup> <sup>3</sup>k<sup>2</sup> v2 tc cos <sup>2</sup>

v2 tc � � <sup>þ</sup> <sup>3</sup>k<sup>2</sup>

For wave frequencies much lower than the gyrofrequency and satisfying ω ≪ Ω cos θ, the associated electron-acoustic (or positron-acoustic) mode is found

frequency of the electrons and positrons and <sup>λ</sup>dh <sup>¼</sup> <sup>ε</sup>0Th=n0he<sup>2</sup> ð Þ<sup>1</sup>=<sup>2</sup> is the Debye

$$
\alpha\_{UH}^2 = \Omega^2 + 2\alpha\_{pc}^2 \tag{14}
$$

is the upper hybrid frequency associated with the cooler species [20], with <sup>ω</sup>pc <sup>¼</sup> noce<sup>2</sup> ð Þ <sup>=</sup>ε0<sup>m</sup> <sup>1</sup>=<sup>2</sup> as the plasma frequency of the cooler species. If one solves equation (13) in the limit 3k<sup>2</sup> v2 tc <sup>þ</sup> <sup>ω</sup><sup>2</sup> UH � �<sup>2</sup> ≫ 4 3k<sup>2</sup> v2 tcΩ<sup>2</sup> cos <sup>2</sup><sup>θ</sup> <sup>þ</sup> <sup>2</sup>ω<sup>2</sup> pcΩ<sup>2</sup> cos <sup>2</sup>θ � �, one obtains for the upper hybrid mode,

$$
\rho\_+^2 = \left(\mathfrak{B}k^2 v\_{\mathfrak{t}\mathfrak{c}}^2 + \alpha\_{\text{UH}}^2\right) - \frac{\left(\mathfrak{B}k^2 v\_{\mathfrak{t}\mathfrak{c}}^2 + 2\alpha\_{\text{pc}}^2\right)\mathfrak{Q}^2 \cos^2\theta}{\mathfrak{B}^2 v\_{\mathfrak{t}\mathfrak{c}}^2 + \alpha\_{\text{UH}}^2},
\tag{15}
$$

Taking the negative square root of equation (13) yields

$$
\rho\_-^2 = \frac{\left(3k^2v\_{tc}^2 + 2\alpha\_{pc}^2\right)\Omega^2\cos^2\theta}{3k^2v\_{tc}^2 + \alpha\_{UH}^2},\tag{16}
$$

In order to gain physical insight into the solution space of the dispersion relation, the two extreme limits of equation (10) will now be considered, viz. pure perpendicular and pure parallel propagations.

#### 2.1 Case I: pure perpendicular propagation

Considering the pure perpendicular (<sup>θ</sup> <sup>¼</sup> <sup>90</sup><sup>o</sup> ) limit, the general dispersion relation (10), reduces to:

$$
\alpha^4 - \alpha^2 \left( \Omega^2 + 3k^2 v\_{tc}^2 + \frac{k^2 v\_{\alpha t}^2}{1 + \frac{1}{2}k^2 \lambda\_{dh}^2} \right) = 0. \tag{17}
$$

Hence the normal mode frequencies are, ω ¼ 0, which is a nonpropagating mode, and

$$
\omega \alpha^2 = \Omega^2 + 3k^2 v\_{tc}^2 + \frac{k^2 v\_{ca}^2}{\mathbf{1} + \frac{1}{2}k^2 \lambda\_{dh}^2}. \tag{18}
$$

Taking the short wavelength limit (k<sup>2</sup> λ2 dh ≫ 1) of the above relationship, one obtains,

$$
\alpha^2 = \alpha\_{UH}^2 + 3k^2 v\_{tc}^2. \tag{19}
$$

showing that the behavior of the upper hybrid mode for the two temperature model is due to the cooler species, where ω<sup>2</sup> UH <sup>¼</sup> <sup>Ω</sup><sup>2</sup> <sup>p</sup> <sup>þ</sup> <sup>2</sup>ω<sup>2</sup> pc.

Now taking the long wavelength limit (k<sup>2</sup> λ2 dh ≪ 1) of the dispersion relation for perpendicular propagation, equation (18) reduces to

$$
\alpha^2 = \Omega^2 + k^2 \left( \mathfrak{J} v\_{\rm tc}^2 + v\_{\rm eat}^2 \right). \tag{20}
$$

This is the cyclotron mode for the electron-positron plasma with contributions from both the thermal motion of the adiabatic cooler species and the acoustic motion due to the two species of different temperatures. To try and understand the physical implications, the above expression for the dispersion relation can be written as,

$$
\omega \alpha^2 = \Omega^2 + k^2 v\_{\text{ea}}^2 \left( \mathbf{1} + \mathfrak{F} \frac{T\_c}{T\_h} \frac{n\_{0h}}{n\_{0c}} \right). \tag{21}
$$

ω2 � <sup>¼</sup> <sup>3</sup>k<sup>2</sup>

(longbroken). The fixed plasma parameters are R ¼ 0:333, Tc=Th ¼ 0:01 and n0<sup>c</sup>=n0<sup>h</sup> ¼ 0:11.

3Tc

n0<sup>c</sup> n0 � �<sup>1</sup>=<sup>2</sup>

, the particle density by the total equilibrium plasma density <sup>n</sup><sup>0</sup> <sup>¼</sup> <sup>n</sup>0<sup>c</sup> <sup>þ</sup> <sup>n</sup>0<sup>h</sup>, the temperatures by Th, the spatial length by <sup>λ</sup><sup>D</sup> <sup>¼</sup> <sup>ε</sup>0Th=n0e<sup>2</sup> ð Þ<sup>1</sup>=<sup>2</sup>

þ

<sup>0</sup><sup>h</sup> ¼ n0<sup>h</sup>=n0, n

sure of the plasma densities and the strength of the magnetic field. A typical result can be seen in Figure 1 [28] for the normalized real frequency as a function of the normalized wavenumber showing the acoustic and cyclotron branches for a range

3. Linear waves in electron-positron plasmas: kinetic theory approach

In this section the kinetic theory approach is used to study the acoustic mode that was investigated in the previous section using fluid theory. The focus is on this mode since it is a micro-instability arising from resonances in velocity space. This instability is kinetic in nature and the growth rate of the wave is a function of the slope of the velocity distribution function. When the wave phase velocity along B0 sees a negative slope of the velocity distribution <sup>∂</sup><sup>f</sup> <sup>0</sup>=∂V<sup>∥</sup> < 0 � �, the particles on

cos <sup>2</sup>θ <sup>R</sup><sup>2</sup> <sup>3</sup><sup>k</sup>

0

A numerical analysis of the general dispersion relation can be performed focusing on the effects of the density and temperature ratios of the hot and cool electrons and positrons. If one normalizes the fluid speeds by the thermal velocity vth =

Equating equations (24) and (27) in the limit k<sup>2</sup>

Electrostatic Waves in Magnetized Electron-Positron Plasmas

DOI: http://dx.doi.org/10.5772/intechopen.80958

which the two modes may couple is determined to be,

ð Þ Th=m

Figure 1.

ω0 <sup>4</sup> � <sup>ω</sup><sup>0</sup>

65

1=2

and the time by ω�<sup>1</sup>

general dispersion relation,

<sup>R</sup><sup>2</sup> <sup>þ</sup> <sup>3</sup><sup>k</sup> 0 <sup>2</sup> Tc Th þ

¼ ω=ωp, k

<sup>2</sup> 1

of propagation angles.

where ω<sup>0</sup>

ð Þ <sup>k</sup>λ<sup>d</sup> crit <sup>¼</sup> Th

k 0 2 n 0 0c

¼ kλD, n 0

n 0 <sup>0</sup><sup>h</sup> <sup>þ</sup> <sup>1</sup> 2 k 0 2

!

0

v2 tc <sup>þ</sup> <sup>2</sup>ω<sup>2</sup>

Normalized real frequency as a function of the normalized wavenumber showing the acoustic and cyclotron branches for various angles of propagation θ = 0<sup>o</sup> (solid), 9<sup>o</sup> (dotted), 22:5<sup>o</sup> (broken), 45<sup>o</sup> (dashddot) and 90<sup>o</sup>

λ2

n0 <sup>n</sup>0cR<sup>2</sup> � <sup>2</sup> � �<sup>1</sup>=<sup>2</sup>

<sup>p</sup> <sup>¼</sup> <sup>n</sup>0e<sup>2</sup> ð Þ <sup>=</sup>ε0<sup>m</sup> �1=<sup>2</sup> in equation (10), you get the normalized

0 <sup>2</sup> Tc Th þ k 0 2 n 0 0c

<sup>0</sup><sup>c</sup> ¼ n0<sup>c</sup>=n<sup>0</sup> and R ¼ ωp=Ω is a mea-

n0 <sup>0</sup><sup>h</sup> <sup>þ</sup> <sup>1</sup> 2 k 0 2

!

pc (27)

dh ≫ 1, the critical k value for

: (28)

,

¼ 0, (29)

For Tc=Th ≪ 1, one requires n0<sup>h</sup> ≫ n0c, i.e., a plasma dominated by the hot species, in order for the second term in brackets to affect the dispersive properties of the wave.

#### 2.2 Case II: pure parallel propagation

Considering the limit of parallel propagation (<sup>θ</sup> <sup>¼</sup> <sup>0</sup><sup>o</sup> ), the general dispersion relation (10) reduces to,

$$
\alpha^4 - \alpha^2 \left( \Omega^2 + 3k^2 v\_{tc}^2 + \frac{k^2 v\_{ea}^2}{1 + \frac{1}{2}k^2 \lambda\_{dh}^2} \right) + \Omega^2 \left( 3k^2 v\_{tc}^2 + \frac{k^2 v\_{ea}^2}{1 + \frac{1}{2}k^2 \lambda\_{dh}^2} \right) = 0,\tag{22}
$$

from which it can be shown

$$
\omega^2 = \frac{1}{2} \left[ \Omega^2 + 3k^2 v\_{\text{tc}}^2 + \frac{k^2 v\_{\text{ea}}^2}{1 + \frac{1}{2}k^2 \lambda\_{dh}^2} \pm \left( \Omega^2 - 3k^2 v\_{\text{tc}}^2 - \frac{k^2 v\_{\text{ea}}^2}{1 + \frac{1}{2}k^2 \lambda\_{dh}^2} \right) \right]. \tag{23}
$$

There exist two possible solutions. Taking the positive sign of the relevant term in equation (23) as the first option yields,

$$
\alpha\_+^2 = \Omega^2,\tag{24}
$$

which is a constant frequency, nonpropagating cyclotron mode.

Now taking the negative sign of the term in equation (23) yields the normal mode frequency

$$
\omega\_-^2 = \mathfrak{B}^2 v\_{tc}^2 + \frac{k^2 v\_{ea}^2}{1 + \frac{1}{2}k^2 \lambda\_{dh}^2},
\tag{25}
$$

which may be written for k<sup>2</sup> λ2 dh ≪ 1 as

$$
\rho\_-^2 = k^2 \nu\_{ea}^2 \left( \mathbf{1} + \mathbf{3} \frac{T\_c}{T\_h} \frac{n\_{0h}}{n\_{0c}} \right),
\tag{26}
$$

which is identified fundamentally, as the electron-acoustic mode, with a correction term to its phase velocity due to the thermal motion of the cooler species.

In the limit k<sup>2</sup> λ2 dh ≫ 1, one obtains Electrostatic Waves in Magnetized Electron-Positron Plasmas DOI: http://dx.doi.org/10.5772/intechopen.80958

Figure 1.

Now taking the long wavelength limit (k<sup>2</sup>

2.2 Case II: pure parallel propagation

v2 tc þ

> v2 tc þ

!

relation (10) reduces to,

<sup>ω</sup><sup>4</sup> � <sup>ω</sup><sup>2</sup> <sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>3</sup>k<sup>2</sup>

<sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> 2

mode frequency

In the limit k<sup>2</sup>

64

from which it can be shown

<sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>3</sup>k<sup>2</sup>

in equation (23) as the first option yields,

which may be written for k<sup>2</sup>

λ2

ten as,

Charged Particles

the wave.

perpendicular propagation, equation (18) reduces to

λ2

tc <sup>þ</sup> <sup>v</sup><sup>2</sup> ea

<sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>3</sup>v<sup>2</sup>

from both the thermal motion of the adiabatic cooler species and the acoustic motion due to the two species of different temperatures. To try and understand the physical implications, the above expression for the dispersion relation can be writ-

v2

For Tc=Th ≪ 1, one requires n0<sup>h</sup> ≫ n0c, i.e., a plasma dominated by the hot species, in order for the second term in brackets to affect the dispersive properties of

ea <sup>1</sup> <sup>þ</sup> <sup>3</sup> Tc

<sup>þ</sup> <sup>Ω</sup><sup>2</sup> <sup>3</sup>k<sup>2</sup>

� <sup>Ω</sup><sup>2</sup> � <sup>3</sup>k<sup>2</sup>

" # !

There exist two possible solutions. Taking the positive sign of the relevant term

ω2 <sup>þ</sup> <sup>¼</sup> <sup>Ω</sup><sup>2</sup>

Now taking the negative sign of the term in equation (23) yields the normal

k2 v2 ea

Tc Th n0<sup>h</sup> n0<sup>c</sup>

� �

which is identified fundamentally, as the electron-acoustic mode, with a correc-

tion term to its phase velocity due to the thermal motion of the cooler species.

<sup>1</sup> <sup>þ</sup> <sup>1</sup> 2 k2 λ2 dh

v2 tc þ

which is a constant frequency, nonpropagating cyclotron mode.

v2 tc þ

> v2 tc � <sup>k</sup><sup>2</sup>

k2 v2 ea

> v2 ea

, (24)

, (25)

, (26)

<sup>1</sup> <sup>þ</sup> <sup>1</sup> 2 k2 λ2 dh

<sup>1</sup> <sup>þ</sup> <sup>1</sup> 2 k2 λ2 dh

!

Th n0<sup>h</sup> n0<sup>c</sup>

� �

<sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup>

Considering the limit of parallel propagation (<sup>θ</sup> <sup>¼</sup> <sup>0</sup><sup>o</sup>

k2 v2 ea

> k2 v2 ea

<sup>1</sup> <sup>þ</sup> <sup>1</sup> 2 k2 λ2 dh

ω2 � <sup>¼</sup> <sup>3</sup>k<sup>2</sup>

ω2 � <sup>¼</sup> <sup>k</sup><sup>2</sup> v2 ea 1 þ 3

dh ≫ 1, one obtains

λ2 dh ≪ 1 as

<sup>1</sup> <sup>þ</sup> <sup>1</sup> 2 k2 λ2 dh

This is the cyclotron mode for the electron-positron plasma with contributions

dh ≪ 1) of the dispersion relation for

: (21)

), the general dispersion

¼ 0, (22)

: (23)

� �: (20)

Normalized real frequency as a function of the normalized wavenumber showing the acoustic and cyclotron branches for various angles of propagation θ = 0<sup>o</sup> (solid), 9<sup>o</sup> (dotted), 22:5<sup>o</sup> (broken), 45<sup>o</sup> (dashddot) and 90<sup>o</sup> (longbroken). The fixed plasma parameters are R ¼ 0:333, Tc=Th ¼ 0:01 and n0<sup>c</sup>=n0<sup>h</sup> ¼ 0:11.

$$
\rho\_-^2 = \mathfrak{Z}k^2 \nu\_{tc}^2 + \mathfrak{Z}\rho\_{pc}^2 \tag{27}
$$

Equating equations (24) and (27) in the limit k<sup>2</sup> λ2 dh ≫ 1, the critical k value for which the two modes may couple is determined to be,

$$(\mathbb{k}\lambda\_d)\_{crit} = \left(\frac{T\_h}{3T\_c}\frac{n\_{0c}}{n\_0}\right)^{1/2} \left(\frac{n\_0}{n\_{0c}R^2} - 2\right)^{1/2}.\tag{28}$$

A numerical analysis of the general dispersion relation can be performed focusing on the effects of the density and temperature ratios of the hot and cool electrons and positrons. If one normalizes the fluid speeds by the thermal velocity vth = ð Þ Th=m 1=2 , the particle density by the total equilibrium plasma density <sup>n</sup><sup>0</sup> <sup>¼</sup> <sup>n</sup>0<sup>c</sup> <sup>þ</sup> <sup>n</sup>0<sup>h</sup>, the temperatures by Th, the spatial length by <sup>λ</sup><sup>D</sup> <sup>¼</sup> <sup>ε</sup>0Th=n0e<sup>2</sup> ð Þ<sup>1</sup>=<sup>2</sup> , and the time by ω�<sup>1</sup> <sup>p</sup> <sup>¼</sup> <sup>n</sup>0e<sup>2</sup> ð Þ <sup>=</sup>ε0<sup>m</sup> �1=<sup>2</sup> in equation (10), you get the normalized general dispersion relation,

$$
\alpha^{'4} - \alpha^{'2} \left( \frac{1}{R^2} + 3k^{'2} \frac{T\_c}{T\_h} + \frac{k^{'2} n\_{0c}^{'}}{n\_{0h}^{'} + \frac{1}{2}k^{'2}} \right) + \frac{\cos^2 \theta}{R^2} \left( 3k^{'2} \frac{T\_c}{T\_h} + \frac{k^{'2} n\_{0c}^{'}}{n\_{0h}^{'} + \frac{1}{2}k^{'2}} \right) = 0,\tag{29}
$$

where ω<sup>0</sup> ¼ ω=ωp, k 0 ¼ kλD, n 0 <sup>0</sup><sup>h</sup> ¼ n0<sup>h</sup>=n0, n 0 <sup>0</sup><sup>c</sup> ¼ n0<sup>c</sup>=n<sup>0</sup> and R ¼ ωp=Ω is a measure of the plasma densities and the strength of the magnetic field. A typical result can be seen in Figure 1 [28] for the normalized real frequency as a function of the normalized wavenumber showing the acoustic and cyclotron branches for a range of propagation angles.

#### 3. Linear waves in electron-positron plasmas: kinetic theory approach

In this section the kinetic theory approach is used to study the acoustic mode that was investigated in the previous section using fluid theory. The focus is on this mode since it is a micro-instability arising from resonances in velocity space. This instability is kinetic in nature and the growth rate of the wave is a function of the slope of the velocity distribution function. When the wave phase velocity along B0 sees a negative slope of the velocity distribution <sup>∂</sup><sup>f</sup> <sup>0</sup>=∂V<sup>∥</sup> < 0 � �, the particles on

average will gain energy from the wave, consequently the wave losses energy and becomes damped, an effect known as Landau damping. The wave mode is hence subjected to Landau damping and wave enhancement. Therefore the focus in this section is primarily on the effect of the temperatures of the plasma species.

The same plasma model as in the previous section is considered, i.e., a four component magnetized electron-positron plasma, consisting of cool electrons and cool positrons with equal temperatures and equilibrium densities denoted by Tc and n0<sup>c</sup> respectively, and hot electrons and hot positrons with equal temperatures and equilibrium densities denoted by Th and n0h, respectively.

We begin by deriving the general dispersion relation where each species j has an isotropic, drifting Maxwellian velocity distribution with temperatures Tj drifting parallel to the magnetic field B0 ¼ B0^z, with drift velocities Voj.

Hence, the equilibrium velocity distribution for the electron and positron species is chosen to be,

$$f\_{a0} = \frac{n\_{a0}}{\left(2\pi v\_{\text{ij}}^2\right)^{\frac{3}{2}}} \exp\left\{\frac{-\left[V\_x^2 + V\_y^2 + \left(V\_x - V\_{o\text{j}}\right)^2\right]}{2v\_{\text{ij}}^2}\right\},\tag{30}$$

The Vlasov equations are,

$$\frac{\partial f\_a}{\partial t} + \mathbf{V}.\nabla f\_a + \frac{q\_a}{m}(\mathbf{E} + \mathbf{V} \times \mathbf{B}).\frac{\partial f\_a}{\partial \mathbf{V}} = \mathbf{0},\tag{31}$$

and the equations of motion for the electrons and positrons is given by,

$$m\frac{d\mathbf{V}}{dt} = q\_a \{ \mathbf{E} + \mathbf{V} \times \mathbf{B} \},\tag{32}$$

Γpj ¼ e

drift velocities of the cool (hot) species, respectively.

Electrostatic Waves in Magnetized Electron-Positron Plasmas

DOI: http://dx.doi.org/10.5772/intechopen.80958

3.1 Approximate solutions of the kinetic dispersion relation

sions for the frequency and growth rate of the acoustic mode.

� <sup>2</sup><sup>z</sup> <sup>1</sup> � <sup>2</sup>z<sup>2</sup>

0, Imð Þz >0 1, Imð Þ¼ z 0 2, Imð Þz < 0

and

Z-function [30] is given by

where for ∣z∣ ≫ 1, δ ¼

Then for the cool species,

Z zpc � �Γpc <sup>≈</sup><sup>Z</sup> <sup>ω</sup> � <sup>k</sup>:Voc ffiffi

VohÞ [31] and ∣ω∣ ≪ Ω,

and

∑ ∞ p¼�∞

67

Z zð Þ¼ <sup>i</sup> ffiffiffi <sup>π</sup> <sup>p</sup> <sup>e</sup> �z<sup>2</sup>

Z zð Þ¼ <sup>i</sup> ffiffiffi

<sup>π</sup> <sup>p</sup> <sup>δ</sup><sup>e</sup> �z<sup>2</sup> � 1 z 1 þ 1 2z<sup>2</sup> þ

> 8 ><

> >:

zpc <sup>¼</sup> <sup>ω</sup> � <sup>k</sup>:Voc � <sup>p</sup><sup>Ω</sup> ffiffi 2 <sup>p</sup> <sup>k</sup>∥vtc

zph <sup>¼</sup> <sup>ω</sup> � <sup>k</sup>:Voh � <sup>p</sup><sup>Ω</sup> ffiffi 2 <sup>p</sup> <sup>k</sup>∥vth

<sup>p</sup> <sup>k</sup>∥vtc !Γoc <sup>þ</sup> <sup>∑</sup>

∑ ∞ p¼�∞

From the definition of the Z-function, Zð Þþ ξ Zð Þ¼ �ξ 0, hence

2

�α<sup>j</sup> Ip α<sup>j</sup>

<sup>α</sup><sup>j</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> ⊥v2 tj Ω2 j

where Ip is the modified Bessel function of order p. The components of k parallel (perpendicular) to B0 are given by k<sup>∥</sup> (k⊥) respectively, while Voc and Voh are the

The general dispersion relation (33) can be numerically solved without any approximations. However, to get some insight into the solutions, here, approximate expansions of the plasma dispersion function are used to obtain analytical expres-

In proceeding, for the temperatures it is assumed that Th ≫ Tcð Þ � 0 . In addition low frequency modes satisfying ∣ω∣ ≪ Ω are considered. The series expansion of the

3 þ

Assuming the drift of the electrons and positrons to be weak (i.e., small Voc and

<sup>≈</sup> �p<sup>Ω</sup> ffiffi 2 <sup>p</sup> <sup>k</sup>∥vtc

<sup>≈</sup> �p<sup>Ω</sup> ffiffi 2 <sup>p</sup> <sup>k</sup>∥vth

> <sup>Z</sup> <sup>p</sup>Ωffiffi 2

<sup>p</sup> <sup>k</sup>∥vtc ! <sup>þ</sup> <sup>Z</sup> �p<sup>Ω</sup> ffiffi

Z zpc � �Γpc <sup>≈</sup>Z zð Þ oc <sup>Γ</sup>oc: (42)

<sup>p</sup> <sup>k</sup>∥vtc ( ) ! <sup>Γ</sup>pc:

∞ p¼1 4z<sup>4</sup> <sup>15</sup> � …

3 <sup>4</sup>z<sup>4</sup> <sup>þ</sup> …

� � for <sup>∣</sup>z∣ ≪ 1 and (37)

� � for <sup>∣</sup>z∣ ≫ <sup>1</sup>: (38)

for p 6¼ 0 (39)

for p 6¼ 0: (40)

2

(41)

� �, (35)

, (36)

where j ¼ c hð Þ for the cool (hot) species and α ¼ ec, pc, eh and ph for the cool electrons, cool positrons, hot electrons and hot positrons respectively, and vtj <sup>¼</sup> Tj=<sup>m</sup> � �<sup>1</sup>=<sup>2</sup> is the thermal velocity of the <sup>j</sup> th species.

Following standard techniques for electron-ion plasmas [29], the general kinetic dispersion relation for the four component, two temperature electron-positron plasma is given by

$$\begin{split} &k^2 + \frac{2}{\lambda\_{Dc}^2} \left[ \mathbf{1} + \frac{\boldsymbol{\alpha} - \mathbf{k} \cdot \mathbf{V}\_{\mathbf{oc}}}{\sqrt{2} k\_{\parallel} v\_{tc}} \sum\_{p = -\infty}^{\infty} Z(\mathbf{z}\_{pc}) \boldsymbol{\Gamma}\_{pc} \right] \\ &+ \frac{2}{\lambda\_{Dh}^2} \left[ \mathbf{1} + \frac{\boldsymbol{\alpha} - \mathbf{k} \cdot \mathbf{V}\_{\mathbf{oh}}}{\sqrt{2} k\_{\parallel} v\_{th}} \sum\_{p = -\infty}^{\infty} Z(\mathbf{z}\_{ph}) \boldsymbol{\Gamma}\_{ph} \right] = \mathbf{0}, \end{split} \tag{33}$$

where <sup>λ</sup>Dc,h <sup>¼</sup> <sup>ε</sup>0Th=n0c,he<sup>2</sup> ð Þ<sup>1</sup>=<sup>2</sup> is the Debye length for the cool (hot) species and zpj is the argument of the plasma dispersion function or Z-function [30] and is given by,

$$z\_{p\circ} = \frac{o - \mathbf{k} . \mathbf{V}\_{\bullet \circ} - p\Omega\_{\circ}}{\sqrt{2}k\_{\parallel}v\_{t\circ}},\tag{34}$$

where,

Electrostatic Waves in Magnetized Electron-Positron Plasmas DOI: http://dx.doi.org/10.5772/intechopen.80958

$$
\Gamma\_{p\circ} = e^{-a\_{\circ}} I\_p(a\_{\circ}), \tag{35}
$$

and

average will gain energy from the wave, consequently the wave losses energy and becomes damped, an effect known as Landau damping. The wave mode is hence subjected to Landau damping and wave enhancement. Therefore the focus in this section is primarily on the effect of the temperatures of the plasma species. The same plasma model as in the previous section is considered, i.e., a four component magnetized electron-positron plasma, consisting of cool electrons and cool positrons with equal temperatures and equilibrium densities denoted by Tc and n0<sup>c</sup> respectively, and hot electrons and hot positrons with equal temperatures and

We begin by deriving the general dispersion relation where each species j has an isotropic, drifting Maxwellian velocity distribution with temperatures Tj drifting

Hence, the equilibrium velocity distribution for the electron and positron species

<sup>x</sup> <sup>þ</sup> <sup>V</sup><sup>2</sup>

<sup>m</sup> ð Þ <sup>E</sup> <sup>þ</sup> <sup>V</sup> � <sup>B</sup> :

<sup>y</sup> þ Vz � Voj � �<sup>2</sup> h i

∂f α

dt <sup>¼</sup> <sup>q</sup>αf g <sup>E</sup> <sup>þ</sup> <sup>V</sup> � <sup>B</sup> , (32)

9 =

<sup>∂</sup><sup>V</sup> <sup>¼</sup> <sup>0</sup>, (31)

;, (30)

2v<sup>2</sup> tj

th species.

Z zpc � �Γpc

Z zph � �Γph

¼ 0,

, (34)

(33)

� <sup>V</sup><sup>2</sup>

and the equations of motion for the electrons and positrons is given by,

where j ¼ c hð Þ for the cool (hot) species and α ¼ ec, pc, eh and ph for the cool

Following standard techniques for electron-ion plasmas [29], the general kinetic

" #

" #

where <sup>λ</sup>Dc,h <sup>¼</sup> <sup>ε</sup>0Th=n0c,he<sup>2</sup> ð Þ<sup>1</sup>=<sup>2</sup> is the Debye length for the cool (hot) species and zpj is the argument of the plasma dispersion function or Z-function [30] and is given

> zpj <sup>¼</sup> <sup>ω</sup> � <sup>k</sup>:Voj � <sup>p</sup>Ω<sup>j</sup> ffiffi 2 <sup>p</sup> <sup>k</sup>∥vtj

∑ ∞ p¼�∞

∑ ∞ p¼�∞

electrons, cool positrons, hot electrons and hot positrons respectively, and

<sup>1</sup> <sup>þ</sup> <sup>ω</sup> � <sup>k</sup>:Voc ffiffi 2 <sup>p</sup> <sup>k</sup>∥vtc

<sup>1</sup> <sup>þ</sup> <sup>ω</sup> � <sup>k</sup>:Voh ffiffi 2 <sup>p</sup> <sup>k</sup>∥vth

dispersion relation for the four component, two temperature electron-positron

8 < :

equilibrium densities denoted by Th and n0h, respectively.

<sup>f</sup> <sup>α</sup><sup>0</sup> <sup>¼</sup> <sup>n</sup>α<sup>0</sup>

∂f α ∂t

vtj <sup>¼</sup> Tj=<sup>m</sup> � �<sup>1</sup>=<sup>2</sup> is the thermal velocity of the <sup>j</sup>

<sup>k</sup><sup>2</sup> <sup>þ</sup> 2 λ2 Dc

> þ 2 λ2 Dh

The Vlasov equations are,

2πv<sup>2</sup> tj � �<sup>3</sup> 2 exp

is chosen to be,

Charged Particles

plasma is given by

by,

66

where,

parallel to the magnetic field B0 ¼ B0^z, with drift velocities Voj.

<sup>þ</sup> <sup>V</sup>:∇<sup>f</sup> <sup>α</sup> <sup>þ</sup> <sup>q</sup><sup>α</sup>

m dV

$$a\_{\circ} = \frac{k\_{\perp}^{2} v\_{\circ}^{2}}{\Omega\_{\circ}^{2}},\tag{36}$$

where Ip is the modified Bessel function of order p. The components of k parallel (perpendicular) to B0 are given by k<sup>∥</sup> (k⊥) respectively, while Voc and Voh are the drift velocities of the cool (hot) species, respectively.

#### 3.1 Approximate solutions of the kinetic dispersion relation

The general dispersion relation (33) can be numerically solved without any approximations. However, to get some insight into the solutions, here, approximate expansions of the plasma dispersion function are used to obtain analytical expressions for the frequency and growth rate of the acoustic mode.

In proceeding, for the temperatures it is assumed that Th ≫ Tcð Þ � 0 . In addition low frequency modes satisfying ∣ω∣ ≪ Ω are considered. The series expansion of the Z-function [30] is given by

$$Z(\mathbf{z}) = i\sqrt{\pi}e^{-\mathbf{z}^2} - 2\mathbf{z}\left[\mathbf{1} - \frac{2\mathbf{z}^2}{3} + \frac{4\mathbf{z}^4}{15} - \dots\right] \text{ for } |\mathbf{z}| \ll \mathbf{1} \text{ and }\tag{37}$$

$$Z(\mathbf{z}) = i\sqrt{\pi}\delta\mathbf{e}^{-\mathbf{z}^2} - \frac{\mathbf{1}}{\mathbf{z}} \left[\mathbf{1} + \frac{\mathbf{1}}{2\mathbf{z}^2} + \frac{\mathbf{3}}{4\mathbf{z}^4} + \dots \right] \text{ for } |\mathbf{z}| \gg \mathbf{1}. \tag{38}$$

where for ∣z∣ ≫ 1, δ ¼ 0, Imð Þz >0 1, Imð Þ¼ z 0 2, Imð Þz < 0 8 >< >:

Assuming the drift of the electrons and positrons to be weak (i.e., small Voc and VohÞ [31] and ∣ω∣ ≪ Ω,

$$z\_{pc} = \frac{\alpha - \mathbf{k}.\mathbf{V}\_{\mathbf{oc}} - p\Omega}{\sqrt{2}k\_{\parallel}v\_{tc}} \approx \frac{-p\Omega}{\sqrt{2}k\_{\parallel}v\_{tc}} \quad \text{for } p \neq 0 \tag{39}$$

and

$$z\_{ph} = \frac{\boldsymbol{\alpha} - \mathbf{k} \cdot \mathbf{V}\_{\mathbf{o}h} - p\boldsymbol{\Omega}}{\sqrt{2}k\_{\parallel}v\_{th}} \approx \frac{-p\boldsymbol{\Omega}}{\sqrt{2}k\_{\parallel}v\_{th}} \text{ for } p \neq \mathbf{0}. \tag{40}$$

Then for the cool species,

$$\sum\_{p=-\infty}^{\infty} Z(z\_{p\epsilon})\Gamma\_{p\epsilon} \approx Z\left(\frac{\omega - \mathbf{k}.\mathbf{V\_{oc}}}{\sqrt{2}k\_{\parallel}v\_{lc}}\right)\Gamma\_{\alpha\epsilon} + \sum\_{p=1}^{\infty} \left\{ Z\left(\frac{p\Omega}{\sqrt{2}k\_{\parallel}v\_{lc}}\right) + Z\left(\frac{-p\Omega}{\sqrt{2}k\_{\parallel}v\_{lc}}\right) \right\}\Gamma\_{p\epsilon}.\tag{41}$$

From the definition of the Z-function, Zð Þþ ξ Zð Þ¼ �ξ 0, hence

$$\sum\_{p=-\infty}^{\infty} Z(\mathbf{z}\_{p\mathbf{c}}) \Gamma\_{p\mathbf{c}} \approx Z(\mathbf{z}\_{\mathbf{c}\mathbf{c}}) \Gamma\_{\mathbf{c}\mathbf{c}}.\tag{42}$$

Taking the cooler species to be stationary, Voc is therefore set to zero, allowing only the hot species to drift. Then,

$$\mathbf{z}\_{\alpha^\*} = \frac{\alpha}{\sqrt{2}k\_{\parallel}v\_{tt}}.\tag{43}$$

where cos θ ¼ k∥=k and vea ¼ ð Þ n0c=n0<sup>h</sup>

Electrostatic Waves in Magnetized Electron-Positron Plasmas

expression (12) obtained from fluid theory.

DOI: http://dx.doi.org/10.5772/intechopen.80958

γ ¼

ω4 r k3 ∥ π 8 � �1=<sup>2</sup> <sup>m</sup> Th � �3=<sup>2</sup>

Th, the spatial length by <sup>λ</sup>dj <sup>¼</sup> <sup>ε</sup>0Tj

write the normalized real frequency as,

ω2

γ<sup>r</sup> ¼

Figure 2.

69

Voh <sup>¼</sup> <sup>0</sup>:5, <sup>n</sup>0<sup>c</sup> <sup>¼</sup> <sup>0</sup>:<sup>1</sup> and <sup>θ</sup> <sup>¼</sup> <sup>45</sup><sup>o</sup>

Tc=Th ¼ 0:005 (solid), 0.01 (dotted), and 0.02 (broken).

and the approximate normalized growth rate as,

ω4 r k3 ∥λ3 d π 8 1=2

The approximate solution of the growth rate is determined by taking the imag-

We note that in equation (50), it is the cooler species that provides the Landau damping, i.e., the velocity distribution function sees a negative slope <sup>∂</sup><sup>f</sup> <sup>0</sup>=∂V<sup>∥</sup> < 0 � �. It is also seen from equation (50) that for an unstable mode (γ>0), it is necessary that V0<sup>h</sup>>ωr=k∥, i.e., the drift velocity parallel to B0 of the hot species has to be

density by the total equilibrium plasma density n<sup>0</sup> ¼ n0<sup>c</sup> þ n0<sup>h</sup>, the temperatures by

∥λ2 d

1 þ 6k<sup>2</sup> ∥ Tc Th ω2 r

Normalized growth rate as a function of the normalized wavenumber. The fixed parameters are R ¼ 0:333,

. The parameter labeling the curve is the cool to hot temperature ratio

For a fixed value of kλd, the real frequency increases with an increase in the cool to hot temperature ratio. This can be seen from the approximate analytical expression (51). Figure 2 displays the normalized growth rate as a function of the normalized wavenumber for varying cool to hot species temperature ratios Tc=Th. It is noted that as the Tc=Th decreases, the growth rate increases, implying that the

λ2 d <sup>þ</sup> <sup>3</sup>k<sup>2</sup> ∥λ2 d Tc Th

2 1ð Þþ � <sup>n</sup>0<sup>c</sup> <sup>k</sup><sup>2</sup>

� �<sup>1</sup>=<sup>2</sup> <sup>1</sup>�n0<sup>c</sup> n0<sup>c</sup> � � <sup>k</sup> ! : Voh ! ωr � 1

, and the time by ω�<sup>1</sup>

 !<sup>Γ</sup>oh " #

oc <sup>þ</sup> <sup>n</sup>0<sup>h</sup> n0<sup>c</sup> � � <sup>k</sup>:Voh

� �Γoh � �

e�z<sup>2</sup>

1 þ 6k<sup>2</sup> ∥ Tc m ω<sup>2</sup> r

electron-positron plasma. It is noted that equation (49) is consistent with the

inary part of equation (48), and hence solving for γ, one finds

� Th Tc � �3=<sup>2</sup>

larger than the phase velocity to overcome the damping terms.

n0je<sup>2</sup> � �<sup>1</sup>=<sup>2</sup>

<sup>r</sup> <sup>¼</sup> <sup>2</sup>n0ck<sup>2</sup>

Normalizing the fluid speeds by the thermal velocity vth = ð Þ Th=m

vth is the acoustic speed of the

<sup>ω</sup><sup>r</sup> � 1

h i : (50)

1=2

, (51)

<sup>p</sup> <sup>¼</sup> <sup>n</sup>0e<sup>2</sup> ε0m � ��1=<sup>2</sup>

� � , (52)

, the particle

, one may

For modes satisfying ω=k<sup>∥</sup> ≫ vtc, one may assume ∣zoc∣ ≫ 1, i.e., the wave phase speed along Bo is much larger than the cool electron thermal speed. For instability (i.e., a growing wave with Imð Þz >0), δ is set equal to zero in equation (38). Hence using the series expansion equation (38), equation (41) becomes

$$\sum\_{p=-\infty}^{\infty} Z(\mathbf{z}\_{pc}) \Gamma\_{pc} \approx \left[ -\frac{\mathbf{1}}{\mathbf{z}\_{oc}} - \frac{\mathbf{1}}{2\mathbf{z}\_{oc}^3} - \frac{\mathbf{3}}{4\mathbf{z}\_{oc}^5} \right] \Gamma\_{oc}.\tag{44}$$

Similarly, using the series expansion equation (37) (where e�z<sup>2</sup> oh ≈1 for ∣zoh∣ ≪ 1), we have for the hot species,

$$\sum\_{p=-\infty}^{\infty} Z(z\_{ph})\Gamma\_{ph} \approx \left(i\sqrt{\pi} - 2z\_{oh} + \frac{4z\_{oh}^3}{3}\right)\Gamma\_{oh}.\tag{45}$$

It is noted that for relatively high temperature Th, the thermal velocity of the hot species is much larger than the wave phase velocity. Hence, for large Th, we have assumed that ∣zoh∣ ≪ 1.

Substituting (44) and (45), λD, λDc and λDh, into the dispersion relation (33), whereas before <sup>λ</sup><sup>D</sup> <sup>¼</sup> <sup>ε</sup>0Th=n0e<sup>2</sup> ð Þ<sup>1</sup>=<sup>2</sup> , gives

$$k^2 \lambda\_D^2 + 2\frac{\frac{n\_0}{n\_0}}{\frac{T\_c}{T\_h}} \left[ i\sqrt{\pi} x\_{oc} e^{-x\_{oc}^2} - \frac{1}{2x\_{oc}^2} - \frac{3}{4x\_{oc}^4} \right] + 2\frac{n\_{0h}}{n\_0} \left[ 1 + i\sqrt{\pi} x\_{oh} \Gamma\_{oh} \right] = 0. \tag{46}$$

For the cool species we have assumed <sup>∣</sup>αc<sup>∣</sup> <sup>¼</sup> <sup>∣</sup>k<sup>2</sup> ⊥v2 tc=Ω<sup>2</sup> <sup>∣</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> ρ2 <sup>c</sup> ≪ 1 (where ρ<sup>c</sup> is the gyroradius of the cool species), i.e., long wavelength fluctuations in comparison to <sup>ρ</sup>c. Since in general for <sup>∣</sup>x∣ ≪ 1 we can write <sup>Γ</sup>pð Þ¼ <sup>x</sup> <sup>e</sup>�xIpð Þ <sup>x</sup> <sup>≈</sup>ð Þ <sup>x</sup>=<sup>2</sup> <sup>p</sup> ð Þ 1=p! ð Þ 1 � x , hence we have Γoc ≈ 1.

Second and higher order terms in zoh are also neglected since we have assumed ∣zoh∣ ≪ 1. Setting ω ¼ ω<sup>r</sup> þ iγ and assuming γ=ω<sup>r</sup> ≪ 1 one may write

$$\frac{1}{a\nu^2} \approx \frac{1}{a\nu\_r^2} \left(1 - \frac{2i\gamma}{a\nu\_r}\right). \tag{47}$$

Using the above manipulation the dispersion relation equation (46) becomes

$$\begin{split} &k^{2}\lambda\_{\rm D}^{2} + 2\frac{\frac{n\_{0\rm c}}{n\_{\rm D}}}{\frac{\cdot}{T\_{\rm h}}} \Biggl[ i\sqrt{\pi} \left( \frac{\alpha\_{r} + i\gamma}{\sqrt{2}k\_{\parallel}v\_{\rm tr}} \right) e^{-\frac{2}{\alpha\_{\rm v}}} - \frac{k\_{\parallel}^{2}v\_{\rm tr}^{2}}{\alpha\_{r}^{2}} \left( 1 - \frac{2i\gamma}{\alpha\_{r}} \right) - \frac{3k\_{\parallel}^{4}v\_{\rm tr}^{4}}{\alpha\_{r}^{4}} \left( 1 - \frac{2i\gamma}{\alpha\_{r}} \right)^{2} \right] \\ &+ 2\frac{n\_{0\rm h}}{n\_{\rm D}} \left[ 1 + i\sqrt{\pi} \left( \frac{\alpha\_{r} + i\gamma - \mathbf{k} \cdot \mathbf{V}\_{\rm ob}}{\sqrt{2}k\_{\parallel}v\_{\rm th}} \right) \Gamma\_{\rm ob} \right] = \mathbf{0}. \end{split} \tag{48}$$

Taking the real part of equation (48) with the charge neutrality condition noc þ noh ¼ 1, gives

$$
\alpha\_r^2 = \frac{k^2 v\_{ea}^2 \cos^2 \theta}{1 + \frac{1}{2}k^2 \lambda\_{Dh}^2} + 3k^2 v\_{tc}^2 \cos^2 \theta,\tag{49}
$$

Electrostatic Waves in Magnetized Electron-Positron Plasmas DOI: http://dx.doi.org/10.5772/intechopen.80958

Taking the cooler species to be stationary, Voc is therefore set to zero, allowing

For modes satisfying ω=k<sup>∥</sup> ≫ vtc, one may assume ∣zoc∣ ≫ 1, i.e., the wave phase speed along Bo is much larger than the cool electron thermal speed. For instability (i.e., a growing wave with Imð Þz >0), δ is set equal to zero in equation (38). Hence

zoc

� <sup>1</sup> 2z<sup>3</sup> oc

<sup>π</sup> <sup>p</sup> � <sup>2</sup>zoh <sup>þ</sup>

It is noted that for relatively high temperature Th, the thermal velocity of the hot species is much larger than the wave phase velocity. Hence, for large Th, we have

> þ 2 n0<sup>h</sup> n0

Substituting (44) and (45), λD, λDc and λDh, into the dispersion relation (33),

the gyroradius of the cool species), i.e., long wavelength fluctuations in comparison

Second and higher order terms in zoh are also neglected since we have assumed

Using the above manipulation the dispersion relation equation (46) becomes

� �<sup>2</sup> " #

Γoh

<sup>þ</sup> <sup>3</sup>k<sup>2</sup> v2 tc cos <sup>2</sup>

Taking the real part of equation (48) with the charge neutrality condition

<sup>1</sup> � <sup>2</sup>i<sup>γ</sup> ωr � �

> <sup>1</sup> � <sup>2</sup>i<sup>γ</sup> ωr � �

> > ¼ 0:

� <sup>3</sup>k<sup>4</sup> ∥v4 tc ω<sup>4</sup> r

, gives

� <sup>3</sup> 4z<sup>4</sup> oc

to <sup>ρ</sup>c. Since in general for <sup>∣</sup>x∣ ≪ 1 we can write <sup>Γ</sup>pð Þ¼ <sup>x</sup> <sup>e</sup>�xIpð Þ <sup>x</sup> <sup>≈</sup>ð Þ <sup>x</sup>=<sup>2</sup> <sup>p</sup>

∣zoh∣ ≪ 1. Setting ω ¼ ω<sup>r</sup> þ iγ and assuming γ=ω<sup>r</sup> ≪ 1 one may write

1 <sup>ω</sup><sup>2</sup> <sup>≈</sup> <sup>1</sup> ω2 r

e �z<sup>2</sup> oc � <sup>k</sup><sup>2</sup> ∥v2 tc ω2 r

<sup>π</sup> <sup>p</sup> <sup>ω</sup><sup>r</sup> <sup>þ</sup> <sup>i</sup><sup>γ</sup> � <sup>k</sup>:Voh ffiffi 2 <sup>p</sup> <sup>k</sup>∥vth !

> <sup>1</sup> <sup>þ</sup> <sup>1</sup> 2 k2 λ2 Dh

ω2 <sup>r</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> v2 ea cos <sup>2</sup>θ

" #

� �

� �

� <sup>3</sup> 4z<sup>5</sup> oc

> 4z<sup>3</sup> oh 3

<sup>1</sup> <sup>þ</sup> <sup>i</sup> ffiffiffi

⊥v2 tc=Ω<sup>2</sup>

<sup>π</sup> <sup>p</sup> zohΓoh

<sup>∣</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> ρ2

� � <sup>¼</sup> <sup>0</sup>: (46)

: (47)

<sup>1</sup> � <sup>2</sup>i<sup>γ</sup> ωr

θ, (49)

<sup>c</sup> ≪ 1 (where ρ<sup>c</sup> is

ð Þ 1=p!

(48)

: (43)

Γoc: (44)

Γoh: (45)

oh ≈1 for ∣zoh∣ ≪ 1),

zoc <sup>¼</sup> <sup>ω</sup> ffiffi 2 <sup>p</sup> <sup>k</sup>∥vtc

using the series expansion equation (38), equation (41) becomes

� �Γpc <sup>≈</sup> � <sup>1</sup>

Similarly, using the series expansion equation (37) (where e�z<sup>2</sup>

� �Γph ≈ i ffiffiffi

Z zpc

Z zph

� �

For the cool species we have assumed <sup>∣</sup>αc<sup>∣</sup> <sup>¼</sup> <sup>∣</sup>k<sup>2</sup>

∑ ∞ p¼�∞

∑ ∞ p¼�∞

only the hot species to drift. Then,

Charged Particles

we have for the hot species,

assumed that ∣zoh∣ ≪ 1.

k2 λ2 <sup>D</sup> þ 2

k2 λ2 <sup>D</sup> þ 2

68

whereas before <sup>λ</sup><sup>D</sup> <sup>¼</sup> <sup>ε</sup>0Th=n0e<sup>2</sup> ð Þ<sup>1</sup>=<sup>2</sup>

i ffiffiffi <sup>π</sup> <sup>p</sup> zoce �z<sup>2</sup> oc � <sup>1</sup> 2z<sup>2</sup> oc

n0<sup>c</sup> n0 Tc Th

ð Þ 1 � x , hence we have Γoc ≈ 1.

n0<sup>c</sup> n0 Tc Th

þ 2 n0<sup>h</sup> n0

noc þ noh ¼ 1, gives

i ffiffiffi

<sup>1</sup> <sup>þ</sup> <sup>i</sup> ffiffiffi

<sup>π</sup> <sup>p</sup> <sup>ω</sup><sup>r</sup> <sup>þ</sup> <sup>i</sup><sup>γ</sup> ffiffi 2 <sup>p</sup> <sup>k</sup>∥vtc !

where cos θ ¼ k∥=k and vea ¼ ð Þ n0c=n0<sup>h</sup> 1=2 vth is the acoustic speed of the electron-positron plasma. It is noted that equation (49) is consistent with the expression (12) obtained from fluid theory.

The approximate solution of the growth rate is determined by taking the imaginary part of equation (48), and hence solving for γ, one finds

$$\gamma = \frac{\frac{\alpha\_r^4}{k\_\parallel^3} \left(\frac{\pi}{8}\right)^{1/2} \left(\frac{m}{T\_h}\right)^{3/2} \left[ -\left(\frac{T\_h}{T\_\epsilon}\right)^{3/2} e^{-x\_{\alpha r}^2} + \left(\frac{n\_{0h}}{n\_{0r}}\right) \left(\frac{\mathbf{k} \cdot \mathbf{V}\_{\alpha h}}{\alpha\_r} - \mathbf{1}\right) \Gamma\_{oh}\right]}{\left[\mathbf{1} + \frac{6k\_{1\mu}^2}{\alpha\_r^2}\right]}. \tag{50}$$

We note that in equation (50), it is the cooler species that provides the Landau damping, i.e., the velocity distribution function sees a negative slope <sup>∂</sup><sup>f</sup> <sup>0</sup>=∂V<sup>∥</sup> < 0 � �. It is also seen from equation (50) that for an unstable mode (γ>0), it is necessary that V0<sup>h</sup>>ωr=k∥, i.e., the drift velocity parallel to B0 of the hot species has to be larger than the phase velocity to overcome the damping terms.

Normalizing the fluid speeds by the thermal velocity vth = ð Þ Th=m 1=2 , the particle density by the total equilibrium plasma density n<sup>0</sup> ¼ n0<sup>c</sup> þ n0<sup>h</sup>, the temperatures by Th, the spatial length by <sup>λ</sup>dj <sup>¼</sup> <sup>ε</sup>0Tj n0je<sup>2</sup> � �<sup>1</sup>=<sup>2</sup> , and the time by ω�<sup>1</sup> <sup>p</sup> <sup>¼</sup> <sup>n</sup>0e<sup>2</sup> ε0m � ��1=<sup>2</sup> , one may write the normalized real frequency as,

$$
\alpha\_r^2 = \frac{2n\_{0c}k\_{\parallel}^2\lambda\_d^2}{2(1-n\_{0c}) + k^2\lambda\_d^2} + 3k\_{\parallel}^2\lambda\_d^2 \frac{T\_c}{T\_h},\tag{51}
$$

and the approximate normalized growth rate as,

$$\gamma\_r = \frac{\frac{\alpha\_r^4}{k\_\parallel^3 \lambda\_d^3} \left(\frac{\pi}{8}\right)^{1/2} \left[ \left(\frac{1-n\_{0c}}{n\_{0c}}\right) \left(\stackrel{\rightarrow}{k} \cdot \frac{\overrightarrow{V}\_{oh}}{\alpha\_r} - \mathbf{1}\right) \Gamma\_{oh} \right]}{\left[\mathbf{1} + \frac{6k\_\parallel^2 \Gamma\_h}{\alpha\_r^2}\right]},\tag{52}$$

For a fixed value of kλd, the real frequency increases with an increase in the cool to hot temperature ratio. This can be seen from the approximate analytical expression (51). Figure 2 displays the normalized growth rate as a function of the normalized wavenumber for varying cool to hot species temperature ratios Tc=Th. It is noted that as the Tc=Th decreases, the growth rate increases, implying that the

#### Figure 2.

Normalized growth rate as a function of the normalized wavenumber. The fixed parameters are R ¼ 0:333, Voh <sup>¼</sup> <sup>0</sup>:5, <sup>n</sup>0<sup>c</sup> <sup>¼</sup> <sup>0</sup>:<sup>1</sup> and <sup>θ</sup> <sup>¼</sup> <sup>45</sup><sup>o</sup> . The parameter labeling the curve is the cool to hot temperature ratio Tc=Th ¼ 0:005 (solid), 0.01 (dotted), and 0.02 (broken).

instability is more easily excited with lower temperature ratios. This may be explained as follows. As the temperature of the cooler species is increased, the associated Landau damping increases, resulting in a reduction of the overall growth rate. It is noted that a cutoff kλ<sup>d</sup> value is reached beyond which the mode is damped. The general equation of state for the four species is given by

∂pj ∂x þ 3pj

In the above equations, nj, v<sup>j</sup> and pj are the densities, fluid velocities and pres-

quency. Here m = me = mp is the common mass of the electrons and the positrons. Adiabatic compression, γ ¼ ð Þ 2 þ N =N =3, is assumed, where N =1 implies one

Upon linearizing and combining equations (53)–(58) and taking the limit

of the hot (cool) species, the dispersion relation equation for a magnetized twotemperature four component electron-positron plasma, where all species are

> <sup>s</sup> <sup>þ</sup> <sup>3</sup>k<sup>2</sup> v2 tc <sup>þ</sup> <sup>2</sup>ω<sup>2</sup>

Solving the above dispersion relation gives the cyclotron mode,

<sup>s</sup> <sup>þ</sup> <sup>3</sup>k<sup>2</sup> v2 tc � <sup>2</sup>ω<sup>2</sup>

� <sup>¼</sup> <sup>2</sup>ω<sup>2</sup>

In the nonlinear regime, a transformation to a stationary frame

vecx ¼ � neco

<sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>ω<sup>2</sup>

s ¼ ð Þ x � Vt ð Þ Ω=V is performed, and v, t, x and ϕ are normalized with respect to vth,

Integrating equation (53) and using the initial conditions nec<sup>0</sup> ¼ n<sup>0</sup> and vecx ¼ v<sup>0</sup> at s ¼ 0, yields the normalized velocity for the cool electrons in the x-direction.

Similarly the cool positrons, hot electrons and hot positrons velocities are determined. Substituting these into the normalized form of equations (53)–(57), gives

, ρ ¼ vth=Ω, and Th=e, respectively. V is the phase velocity of the wave. In equations (53)–(57), <sup>∂</sup>=∂<sup>t</sup> is replaced by �Ωð Þ <sup>∂</sup>=∂<sup>s</sup> and <sup>∂</sup>=∂<sup>x</sup> by ð Þ <sup>Ω</sup>=<sup>V</sup> ð Þ <sup>∂</sup>=∂<sup>s</sup> , and the diving electric field amplitude is defined as <sup>E</sup> ¼ �ð Þ <sup>∂</sup>ψ=∂<sup>s</sup> , where <sup>ψ</sup> <sup>¼</sup> <sup>e</sup>ϕ=Th.

<sup>1</sup>=<sup>2</sup> and vtc <sup>¼</sup> ð Þ Tc=<sup>m</sup>

λ2 Dh <sup>1</sup>=<sup>2</sup>

<sup>s</sup> Ω<sup>2</sup> cos <sup>2</sup>θ

<sup>s</sup> <sup>þ</sup> <sup>3</sup>k<sup>2</sup> v2 tc

∂vjx

<sup>∂</sup>x<sup>2</sup> ¼ �e npc � nec <sup>þ</sup> nph � neh : (58)

th species. <sup>Ω</sup> <sup>¼</sup> <sup>Ω</sup><sup>e</sup> <sup>¼</sup> <sup>Ω</sup><sup>p</sup> <sup>¼</sup> eB0=<sup>m</sup> is the cyclotron fre-

<sup>s</sup> Ω<sup>2</sup> cos <sup>2</sup>

<sup>1</sup>=<sup>2</sup> are the plasma frequencies of the cool and hot

<sup>s</sup> Ω<sup>2</sup> cos <sup>2</sup>θ

<sup>s</sup> <sup>þ</sup> <sup>3</sup>k<sup>2</sup> v2 tc

nec ð Þþ <sup>V</sup> � <sup>v</sup><sup>0</sup> <sup>V</sup> (62)

<sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>ω<sup>2</sup>

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>0</sup>, (57)

<sup>1</sup>=<sup>2</sup> are the thermal velocities

θ ¼ 0: (59)

.

(60)

and <sup>λ</sup>Dh <sup>¼</sup> <sup>ε</sup>0Th=nohe<sup>2</sup> ð Þ<sup>1</sup>=<sup>2</sup>

, (61)

∂pj ∂t þ vjx

The system is closed by the Poisson equation

Electrostatic Waves in Magnetized Electron-Positron Plasmas

DOI: http://dx.doi.org/10.5772/intechopen.80958

ε0 ∂2 ϕ

<sup>ω</sup><sup>4</sup> � <sup>ω</sup><sup>2</sup> <sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>ω<sup>2</sup>

<sup>þ</sup> <sup>¼</sup> <sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>ω<sup>2</sup>

ω2

sures, respectively, of the j

vtc ≪ ω=k ≪ vth, where vth ¼ ð Þ Th=m

governed by the fluid equations is,

where <sup>ω</sup>pc,ph <sup>¼</sup> <sup>n</sup>0c,he<sup>2</sup> ð Þ <sup>=</sup>ε0<sup>m</sup>

species respectively and <sup>ω</sup><sup>s</sup> <sup>¼</sup> <sup>ω</sup>pc<sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>=3k<sup>2</sup>

ω2

and the acoustic mode,

4.2 Nonlinear analysis

Ω�<sup>1</sup>

71

degree of freedom.

#### 4. Nonlinear electrostatic solitary waves in electron-positron plasmas

The study of nonlinear effects in electron-positron plasmas is important since these plasmas exhibit different wave phenomena as compared to electron-ion plasmas. It is therefore important to understand the nonlinear structures, especially the solitary waves that exist in electron-positron plasmas. Satellite observations in the Earth's magnetosphere have shown the existence of electrostatic solitary waves which forms part of broadband electrostatic noise (BEN) and electrostatic solitary waves (ESWs) in various regions of the Earth's magnetosphere. The characteristic features of these ESWs are solitary bipolar pulses and consist of small scale, large amplitude parallel electric fields. These large amplitude spiky structures have been interpreted in terms of either solitons [32] or isolated electron holes in the phase space corresponding to positive electrostatic potential [33]. Given that electronpositron plasmas are increasingly observed in astrophysical environments, as well as in laboratory experiments [34], the above mentioned satellite observations also lead one to explore if such nonlinear structures are also possible in electron-positron plasmas. There is a distinct possibility that a pulsar magnetosphere can support coexistence of two types of cold and hot electron-positron populations [10, 35, 28]. In this section we investigate nonlinear electrostatic spiky structures in a magnetized four component two-temperature electron-positron plasma.

#### 4.1 Basic equations

The model considered, as in the previous section is a homogeneous magnetized, four component, collisionless, electron-positron plasma, consisting of cool electrons (ec) and cool positrons (pc) with equal temperatures Tc and initial densities (nec<sup>0</sup> ¼ npc0), and hot electrons (eh) and hot positrons (ph) with equal temperatures Th and densities (neh<sup>0</sup> ¼ nph0). Wave propagation is taken in the x-direction at an angle θ to the magnetic field B0, which is assumed to be in the x-z plane.

The continuity and momentum equations for the four species are given by

∂nj ∂t þ ∂ njvjx <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>0</sup> (53)

$$\frac{\partial v\_{j\mathbf{x}}}{\partial t} + v\_{j\mathbf{x}} \frac{\partial v\_{j\mathbf{x}}}{\partial \mathbf{x}} + \frac{\mathbf{1}}{n\_j m} \frac{\partial p\_j}{\partial \mathbf{x}} = -\frac{\varepsilon\_j \varepsilon}{m} \frac{\partial \phi}{\partial \mathbf{x}} + \varepsilon\_j \Delta v\_{j\mathbf{y}} \sin \theta \tag{54}$$

$$\frac{\partial v\_{\dot{\mathcal{Y}}}}{\partial t} + v\_{\dot{\mathcal{X}}} \frac{\partial v\_{\dot{\mathcal{Y}}}}{\partial \boldsymbol{\infty}} = \varepsilon\_{\dot{\mathcal{Y}}} \boldsymbol{\Omega} v\_{\dot{\mathcal{X}}} \cos \theta - \varepsilon\_{\dot{\mathcal{Y}}} \boldsymbol{\Omega} v\_{\dot{\mathcal{X}}} \sin \theta \tag{55}$$

$$\frac{\partial v\_{j\pm}}{\partial t} + v\_{j\varepsilon} \frac{\partial v\_{j\varepsilon}}{\partial \mathbf{x}} = -\varepsilon\_j \boldsymbol{\Omega} v\_{j\nu} \cos \theta,\tag{56}$$

where ε<sup>j</sup> = + 1(�1) for positrons (electrons) and j ¼ ec, pc, eh, ph for the cool electrons, cool positrons, hot electrons, and the hot positrons, respectively.

The density of the cool electrons (positrons) is nec (npc), and that of the hot electrons (positrons) is neh (nph).

Electrostatic Waves in Magnetized Electron-Positron Plasmas DOI: http://dx.doi.org/10.5772/intechopen.80958

instability is more easily excited with lower temperature ratios. This may be explained as follows. As the temperature of the cooler species is increased, the associated Landau damping increases, resulting in a reduction of the overall growth rate. It is noted that a cutoff kλ<sup>d</sup> value is reached beyond which the mode is damped.

4. Nonlinear electrostatic solitary waves in electron-positron plasmas

tized four component two-temperature electron-positron plasma.

The model considered, as in the previous section is a homogeneous magnetized, four component, collisionless, electron-positron plasma, consisting of cool electrons

(nec<sup>0</sup> ¼ npc0), and hot electrons (eh) and hot positrons (ph) with equal temperatures Th and densities (neh<sup>0</sup> ¼ nph0). Wave propagation is taken in the x-direction at an

The continuity and momentum equations for the four species are given by

∂pj <sup>∂</sup><sup>x</sup> ¼ � <sup>ε</sup>je m ∂ϕ ∂x

∂vjz

electrons, cool positrons, hot electrons, and the hot positrons, respectively. The density of the cool electrons (positrons) is nec (npc), and that of the hot

where ε<sup>j</sup> = + 1(�1) for positrons (electrons) and j ¼ ec, pc, eh, ph for the cool

∂ njvjx 

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>0</sup> (53)

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>ε</sup>jΩvjz cos <sup>θ</sup> � <sup>ε</sup>jΩvjx sin <sup>θ</sup> (55)

<sup>∂</sup><sup>x</sup> ¼ �εjΩvjy cos <sup>θ</sup>, (56)

þ εjΩvjy sin θ (54)

(ec) and cool positrons (pc) with equal temperatures Tc and initial densities

angle θ to the magnetic field B0, which is assumed to be in the x-z plane.

∂nj ∂t þ

∂vjx ∂t

electrons (positrons) is neh (nph).

70

þ vjx

∂vjy ∂t

∂vjx ∂x þ 1 njm

∂vjy

þ vjx

þ vjx

∂vjz ∂t

4.1 Basic equations

Charged Particles

The study of nonlinear effects in electron-positron plasmas is important since these plasmas exhibit different wave phenomena as compared to electron-ion plasmas. It is therefore important to understand the nonlinear structures, especially the solitary waves that exist in electron-positron plasmas. Satellite observations in the Earth's magnetosphere have shown the existence of electrostatic solitary waves which forms part of broadband electrostatic noise (BEN) and electrostatic solitary waves (ESWs) in various regions of the Earth's magnetosphere. The characteristic features of these ESWs are solitary bipolar pulses and consist of small scale, large amplitude parallel electric fields. These large amplitude spiky structures have been interpreted in terms of either solitons [32] or isolated electron holes in the phase space corresponding to positive electrostatic potential [33]. Given that electronpositron plasmas are increasingly observed in astrophysical environments, as well as in laboratory experiments [34], the above mentioned satellite observations also lead one to explore if such nonlinear structures are also possible in electron-positron plasmas. There is a distinct possibility that a pulsar magnetosphere can support coexistence of two types of cold and hot electron-positron populations [10, 35, 28]. In this section we investigate nonlinear electrostatic spiky structures in a magneThe general equation of state for the four species is given by

$$\frac{\partial p\_j}{\partial t} + v\_{j\mathbf{x}} \frac{\partial p\_j}{\partial \mathbf{x}} + 3p\_j \frac{\partial v\_{j\mathbf{x}}}{\partial \mathbf{x}} = \mathbf{0},\tag{57}$$

The system is closed by the Poisson equation

$$
\varepsilon\_0 \frac{\partial^2 \phi}{\partial \mathbf{x}^2} = -e \left( n\_{\rm pc} - n\_{\rm ce} + n\_{\rm ph} - n\_{\rm ch} \right). \tag{58}
$$

In the above equations, nj, v<sup>j</sup> and pj are the densities, fluid velocities and pressures, respectively, of the j th species. <sup>Ω</sup> <sup>¼</sup> <sup>Ω</sup><sup>e</sup> <sup>¼</sup> <sup>Ω</sup><sup>p</sup> <sup>¼</sup> eB0=<sup>m</sup> is the cyclotron frequency. Here m = me = mp is the common mass of the electrons and the positrons. Adiabatic compression, γ ¼ ð Þ 2 þ N =N =3, is assumed, where N =1 implies one degree of freedom.

Upon linearizing and combining equations (53)–(58) and taking the limit vtc ≪ ω=k ≪ vth, where vth ¼ ð Þ Th=m <sup>1</sup>=<sup>2</sup> and vtc <sup>¼</sup> ð Þ Tc=<sup>m</sup> <sup>1</sup>=<sup>2</sup> are the thermal velocities of the hot (cool) species, the dispersion relation equation for a magnetized twotemperature four component electron-positron plasma, where all species are governed by the fluid equations is,

$$
\alpha \alpha^4 - \alpha^2 \left(\Omega^2 + 2\alpha\_s^2 + 3k^2 v\_{tc}^2\right) + 2\alpha\_s^2 \Omega^2 \cos^2 \theta = 0. \tag{59}
$$

where <sup>ω</sup>pc,ph <sup>¼</sup> <sup>n</sup>0c,he<sup>2</sup> ð Þ <sup>=</sup>ε0<sup>m</sup> <sup>1</sup>=<sup>2</sup> are the plasma frequencies of the cool and hot species respectively and <sup>ω</sup><sup>s</sup> <sup>¼</sup> <sup>ω</sup>pc<sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>=3k<sup>2</sup> λ2 Dh <sup>1</sup>=<sup>2</sup> and <sup>λ</sup>Dh <sup>¼</sup> <sup>ε</sup>0Th=nohe<sup>2</sup> ð Þ<sup>1</sup>=<sup>2</sup> . Solving the above dispersion relation gives the cyclotron mode,

$$\rho\_+^2 = \left(\Omega^2 + 2a\_s^2 + 3k^2v\_{tc}^2\right) - \frac{2a\_s^2\Omega^2\cos^2\theta}{\Omega^2 + 2a\_s^2 + 3k^2v\_{tc}^2} \tag{60}$$

and the acoustic mode,

$$\rho\_{-}^{2} = \frac{2\rho\_{s}^{2}\Omega^{2}\cos^{2}\theta}{\Omega^{2} + 2\rho\_{s}^{2} + 3k^{2}v\_{tc}^{2}},\tag{61}$$

#### 4.2 Nonlinear analysis

In the nonlinear regime, a transformation to a stationary frame s ¼ ð Þ x � Vt ð Þ Ω=V is performed, and v, t, x and ϕ are normalized with respect to vth, Ω�<sup>1</sup> , ρ ¼ vth=Ω, and Th=e, respectively. V is the phase velocity of the wave. In equations (53)–(57), <sup>∂</sup>=∂<sup>t</sup> is replaced by �Ωð Þ <sup>∂</sup>=∂<sup>s</sup> and <sup>∂</sup>=∂<sup>x</sup> by ð Þ <sup>Ω</sup>=<sup>V</sup> ð Þ <sup>∂</sup>=∂<sup>s</sup> , and the diving electric field amplitude is defined as <sup>E</sup> ¼ �ð Þ <sup>∂</sup>ψ=∂<sup>s</sup> , where <sup>ψ</sup> <sup>¼</sup> <sup>e</sup>ϕ=Th.

Integrating equation (53) and using the initial conditions nec<sup>0</sup> ¼ n<sup>0</sup> and vecx ¼ v<sup>0</sup> at s ¼ 0, yields the normalized velocity for the cool electrons in the x-direction.

$$
\upsilon\_{\rm exx} = -\left(\frac{n\_{\rm eco}}{n\_{\rm ec}}\right)(V - \upsilon\_0) + V \tag{62}
$$

Similarly the cool positrons, hot electrons and hot positrons velocities are determined. Substituting these into the normalized form of equations (53)–(57), gives

the following set of nonlinear first-order differential equations for the cool electron species in the stationary frame.

$$\frac{\partial \varphi}{\partial t} = -E \tag{63}$$

$$\frac{\partial E}{\partial \mathbf{s}} = R^2 \mathbf{M}^2 \left( n\_{pcn} - n\_{ecn} + n\_{plm} - n\_{ehn} \right) \tag{64}$$

$$\frac{\partial n\_{ecn}}{\partial \mathbf{\dot{s}}} = \frac{n\_{ecn}^3 \left[ E + M \sin \theta v\_{eym} \right]}{\left( \frac{n\_{ec0}}{n\_0} \right)^2 \left( M - \delta\_c \right)^2 - \mathfrak{Z}\_{\frac{T\_c}{T\_h}}^{T\_c} p\_{ecn} n\_{ecn}} \tag{65}$$

$$\frac{\partial v\_{\text{evyn}}}{\partial \boldsymbol{\delta}} = \frac{M n\_{\text{ecn}}}{(M - \delta\_{\text{c}})} \left( \frac{n\_0}{n\_{\text{ec0}}} \right) \left[ - \left( M - \frac{(M - \delta\_{\text{c}})}{n\_{\text{ecn}}} \left( \frac{n\_{\text{ec0}}}{n\_0} \right) \right) \sin \theta + v\_{\text{ecm}} \cos \theta \right] \tag{66}$$

$$\frac{\partial v\_{ecm}}{\partial \mathbf{s}} = -\left(\frac{n\_0}{n\_{ec0}}\right) \frac{n\_{ecn}v\_{eyn}M\cos\theta}{(M-\delta\_c)}\tag{67}$$

$$\frac{\partial p\_{ecn}}{\partial \mathbf{\hat{s}}} = \frac{3p\_{ecn}n\_{ecn}^2 \left[E + M \sin \theta \nu\_{eyn}\right]}{\left(\frac{n\_{ec0}}{n\_0}\right)^2 \left(M - \delta\_c\right)^2 - 3\frac{T\_c}{T\_h}p\_{ecn}n\_{ecn}}\tag{68}$$

The set of differential equations for the cool positrons are given by,

$$\frac{\partial n\_{p\text{cn}}}{\partial \mathbf{s}} = \frac{n\_{p\text{cn}}^3}{\left(M - \delta\_c\right)^2} \left(\frac{n\_0}{n\_{p\text{c0}}}\right)^2 \left[-E - M\sin\theta v\_{p\text{cmathcal}}\right] \tag{69}$$

$$\frac{\partial v\_{\rm pcyn}}{\partial \boldsymbol{\sigma}} = \frac{M n\_{\rm pcn}}{(\boldsymbol{M} - \boldsymbol{\delta\_c})} \left(\frac{n\_0}{n\_{\rm pc0}}\right) \left[ \left(\boldsymbol{M} - \frac{(\boldsymbol{M} - \boldsymbol{\delta\_c})}{n\_{\rm pcn}} \left(\frac{n\_{\rm pc0}}{n\_0}\right)\right) \sin \theta - v\_{\rm pcx} \cos \theta \right] \tag{70}$$

$$\frac{\partial v\_{pcn}}{\partial s} = \left(\frac{n\_0}{n\_{pc0}}\right) \frac{n\_{pcn}v\_{pcn}M\cos\theta}{(M-\delta\_c)}\tag{71}$$

higher E<sup>0</sup> value of 3.5, the potential structure has a spiky bipolar form showing that as the period of the wave increases and the frequency of the wave decreases.

nec0=n<sup>0</sup> = npc0=n<sup>0</sup> =0.5, Tc=Th = 0.0, and E<sup>0</sup> = (a) 0.05 [linear waveform], (b) 0.5 [sinusoidal waveform],

, R =10.0, δ<sup>c</sup> = δ<sup>h</sup> = 0.0,

Numerical solution of the normalized electric field for the parameters M = 3.5, θ = 2<sup>o</sup>

(c) 1.5 [sawtooth waveform] and (d) 3.5 [bipolar waveform].

Electrostatic Waves in Magnetized Electron-Positron Plasmas

DOI: http://dx.doi.org/10.5772/intechopen.80958

Linear and nonlinear electrostatic waves in a magnetized four component twotemperature electron-positron plasma have been investigated. In the linear analysis fluid and kinetic theory approaches are employed to describe the wave motion. The fluid theory approach focused on the wave dynamics of both the acoustic and cyclotron branches. Solutions of the dispersion relation from fluid theory yielded electron-acoustic, upper hybrid, electron plasma and electron cyclotron branches. Perpendicular and parallel wave propagation was examined showing its influence on the dispersive properties of the wave. The kinetic theory approach further examined Landau damping effects on the acoustic mode, analyzing the frequency and growth rate of the wave. The analysis shows that a large enough drift velocity (Voh) is required to produce wave growth. Both fluid and kinetic theory show excellent agreement for the real frequencies of the acoustic mode and solutions of the corresponding dispersion relation can be explored as a function of several

5. Conclusion

73

Figure 3.

$$\frac{\partial p\_{pcn}}{\partial \mathbf{s}} = \frac{3p\_{pcn}n\_{pcn}^2 \left[ -E - \mathcal{M}\sin\theta v\_{pcpn} \right]}{\left(\frac{n\_{p0}}{n\_0}\right)^2 (\mathcal{M} - \delta\_c)^2 - \mathfrak{Z}\frac{T\_c}{T\_h}p\_{pcn}n\_{pcn}}\tag{72}$$

Similar sets of differential equations can be derived for the hot electrons and hot positron species. The velocities are normalized with respect to the thermal velocity of the hot species vth ¼ ð Þ Th=m <sup>1</sup>=<sup>2</sup> and the densities with respect to the total density n0. The equilibrium density of the cool (hot) electrons is nec<sup>0</sup> ð Þ neh<sup>0</sup> , and that of the cool (hot) positrons npc<sup>0</sup> nph<sup>0</sup> , with nec<sup>0</sup> <sup>þ</sup> neh<sup>0</sup> <sup>¼</sup> npc<sup>0</sup> <sup>þ</sup> nph<sup>0</sup> <sup>¼</sup> <sup>n</sup>0. <sup>R</sup> <sup>¼</sup> <sup>ω</sup>p=Ω, where <sup>ω</sup><sup>p</sup> <sup>¼</sup> <sup>n</sup>0e<sup>2</sup> ð Þ <sup>=</sup>ε0<sup>m</sup> <sup>1</sup>=<sup>2</sup> is the total plasma frequency, <sup>M</sup> <sup>¼</sup> <sup>V</sup>=vth is the Mach number and δc,h ¼ v0c,0<sup>h</sup>=vth is the normalized drift velocity of cool (hot) species at s =0. The system of nonlinear first-order differential equations can now be solved numerically using the Runge-Kutta (RK4) technique [36]. The initial values can be determined self consistently where the actual normalized electric fields are given by Enorm ¼ �ð Þ <sup>1</sup>=<sup>M</sup> ð Þ <sup>∂</sup>ψ=∂<sup>s</sup> and wave propagation is taken almost parallel to the ambient magnetic field B0.

Numerical results to investigate the effect of parameters such as the electric driving force E0, densities nec<sup>0</sup> and nph0, temperature ratio Tc=Th, Mach number M, drift velocities δc,h and propagation angle θ on the wave can be explored. A typical numerical result is seen in Figure 3a–d [37] showing the evolution of the system for various driving electric field amplitudes E0. It is seen that as E<sup>0</sup> increases, the electric field structure evolves from a sinusoidal wave to a sawtooth structure. For a Electrostatic Waves in Magnetized Electron-Positron Plasmas DOI: http://dx.doi.org/10.5772/intechopen.80958

#### Figure 3.

the following set of nonlinear first-order differential equations for the cool electron

<sup>M</sup><sup>2</sup> npcn � necn <sup>þ</sup> nphn � nehn

ecn E þ M sin θvecyn 

ð Þ <sup>M</sup> � <sup>δ</sup><sup>c</sup> <sup>2</sup> � <sup>3</sup> Tc

necnvecynM cos θ

ecn E þ M sin θvecyn 

> npc<sup>0</sup> n0

ð Þ M � δ<sup>c</sup>

pcn �E � M sin θvpcyn 

npcnvpcynM cos θ

ð Þ <sup>M</sup> � <sup>δ</sup><sup>c</sup> <sup>2</sup> � <sup>3</sup> Tc

Similar sets of differential equations can be derived for the hot electrons and hot positron species. The velocities are normalized with respect to the thermal velocity

n0. The equilibrium density of the cool (hot) electrons is nec<sup>0</sup> ð Þ neh<sup>0</sup> , and that of the

number and δc,h ¼ v0c,0<sup>h</sup>=vth is the normalized drift velocity of cool (hot) species at s =0. The system of nonlinear first-order differential equations can now be solved numerically using the Runge-Kutta (RK4) technique [36]. The initial values can be determined self consistently where the actual normalized electric fields are given by Enorm ¼ �ð Þ <sup>1</sup>=<sup>M</sup> ð Þ <sup>∂</sup>ψ=∂<sup>s</sup> and wave propagation is taken almost parallel to the ambi-

Numerical results to investigate the effect of parameters such as the electric driving force E0, densities nec<sup>0</sup> and nph0, temperature ratio Tc=Th, Mach number M, drift velocities δc,h and propagation angle θ on the wave can be explored. A typical numerical result is seen in Figure 3a–d [37] showing the evolution of the system for various driving electric field amplitudes E0. It is seen that as E<sup>0</sup> increases, the electric field structure evolves from a sinusoidal wave to a sawtooth structure. For a

ð Þ <sup>M</sup> � <sup>δ</sup><sup>c</sup> <sup>2</sup> � <sup>3</sup> Tc

nec<sup>0</sup> n0

ð Þ M � δ<sup>c</sup>

<sup>∂</sup><sup>s</sup> ¼ �<sup>E</sup> (63)

sin θ þ veczn cos θ

(69)

sin θ � vpczn cos θ

(65)

(66)

(67)

(68)

(70)

(71)

(72)

(64)

Th pecnnecn

Th pecnnecn

�E � M sin θvpcyn

Th ppcnnpcn

<sup>1</sup>=<sup>2</sup> and the densities with respect to the total density

, with nec<sup>0</sup> <sup>þ</sup> neh<sup>0</sup> <sup>¼</sup> npc<sup>0</sup> <sup>þ</sup> nph<sup>0</sup> <sup>¼</sup> <sup>n</sup>0. <sup>R</sup> <sup>¼</sup> <sup>ω</sup>p=Ω,

<sup>1</sup>=<sup>2</sup> is the total plasma frequency, <sup>M</sup> <sup>¼</sup> <sup>V</sup>=vth is the Mach

∂ψ

species in the stationary frame.

∂vecyn

Charged Particles

∂vpcyn

<sup>∂</sup><sup>s</sup> <sup>¼</sup> Mnpcn ð Þ M � δ<sup>c</sup>

of the hot species vth ¼ ð Þ Th=m

cool (hot) positrons npc<sup>0</sup> nph<sup>0</sup>

where <sup>ω</sup><sup>p</sup> <sup>¼</sup> <sup>n</sup>0e<sup>2</sup> ð Þ <sup>=</sup>ε0<sup>m</sup>

ent magnetic field B0.

72

<sup>∂</sup><sup>s</sup> <sup>¼</sup> Mnecn ð Þ M � δ<sup>c</sup> ∂E <sup>∂</sup><sup>s</sup> <sup>¼</sup> <sup>R</sup><sup>2</sup>

∂necn

n0 nec<sup>0</sup> 

∂veczn

<sup>∂</sup>pecn

<sup>∂</sup><sup>s</sup> <sup>¼</sup> <sup>n</sup><sup>3</sup>

∂vpczn

n0 npc<sup>0</sup> 

<sup>∂</sup>ppcn

∂npcn

<sup>∂</sup><sup>s</sup> <sup>¼</sup> <sup>n</sup><sup>3</sup>

nec<sup>0</sup> n0 <sup>2</sup>

<sup>∂</sup><sup>s</sup> ¼ � <sup>n</sup><sup>0</sup>

<sup>∂</sup><sup>s</sup> <sup>¼</sup> <sup>3</sup>pecnn<sup>2</sup>

pcn ð Þ <sup>M</sup> � <sup>δ</sup><sup>c</sup> <sup>2</sup>

<sup>∂</sup><sup>s</sup> <sup>¼</sup> <sup>n</sup><sup>0</sup>

npc<sup>0</sup> n0 <sup>2</sup>

<sup>∂</sup><sup>s</sup> <sup>¼</sup> <sup>3</sup>ppcnn<sup>2</sup>

nec<sup>0</sup> n0 <sup>2</sup>

The set of differential equations for the cool positrons are given by,

n0 npc<sup>0</sup> <sup>2</sup>

<sup>M</sup> � ð Þ <sup>M</sup> � <sup>δ</sup><sup>c</sup> npcn

npc<sup>0</sup>

� <sup>M</sup> � ð Þ <sup>M</sup> � <sup>δ</sup><sup>c</sup>

nec<sup>0</sup>

necn

Numerical solution of the normalized electric field for the parameters M = 3.5, θ = 2<sup>o</sup> , R =10.0, δ<sup>c</sup> = δ<sup>h</sup> = 0.0, nec0=n<sup>0</sup> = npc0=n<sup>0</sup> =0.5, Tc=Th = 0.0, and E<sup>0</sup> = (a) 0.05 [linear waveform], (b) 0.5 [sinusoidal waveform], (c) 1.5 [sawtooth waveform] and (d) 3.5 [bipolar waveform].

higher E<sup>0</sup> value of 3.5, the potential structure has a spiky bipolar form showing that as the period of the wave increases and the frequency of the wave decreases.

#### 5. Conclusion

Linear and nonlinear electrostatic waves in a magnetized four component twotemperature electron-positron plasma have been investigated. In the linear analysis fluid and kinetic theory approaches are employed to describe the wave motion. The fluid theory approach focused on the wave dynamics of both the acoustic and cyclotron branches. Solutions of the dispersion relation from fluid theory yielded electron-acoustic, upper hybrid, electron plasma and electron cyclotron branches. Perpendicular and parallel wave propagation was examined showing its influence on the dispersive properties of the wave. The kinetic theory approach further examined Landau damping effects on the acoustic mode, analyzing the frequency and growth rate of the wave. The analysis shows that a large enough drift velocity (Voh) is required to produce wave growth. Both fluid and kinetic theory show excellent agreement for the real frequencies of the acoustic mode and solutions of the corresponding dispersion relation can be explored as a function of several

plasma parameters. In the nonlinear analysis, the two-fluid model is used to derive a set of differential equations for the electrostatic solitary waves in a magnetized twotemperature electron-positron plasma. In particular, electrostatic solitary waves and their electric fields, similar to those found in the Broadband Electrostatic Noise are explored. For the onset of spiky ESWs, it is noted that as the wave speed increases, a larger driving electric field is required.

References

p. 202

2005;76:1143

Journal. 1969;157:869

Physics. 1982;54:1

p. 124

1983;58:235

E. 1995;bf52:1968

75

[1] Weinberg S. Gravitation and Cosmology. New York: Wiley; 1972

University Press; 1983

[2] Rees MJ. In: Gibbons GW, Hawking SW, Siklas S, editors. The Very Early Universe. Cambridge: Cambridge

DOI: http://dx.doi.org/10.5772/intechopen.80958

Electrostatic Waves in Magnetized Electron-Positron Plasmas

Ion Plasma. Astrophysics and Space

[13] Bhattacharyya R, Janaki MS, Dasgupta B. Relaxation in electronpositron plasma: a possibility. Physics

[14] Stenflo L, Shukla PK, Yu MY.

and Space Science. 1985;117:303

positron plasmas and pulsar microstructure. Physical Review A.

[16] Yu MY, Shukla PK, Stenflo L. Alfven Vortices in a Strongly

[17] Pillay R, Bharuthram R. Large Amplitude Solitons in a Multi-Species Electron-Positron Plasma. Astrophysics and Space Science. 1992;198:85-93

[18] Verheest F, Lakhina GS. Oblique Solitary Alfv e′ n Modes in Relativistic Electron-Positron Plasmas. Astrophysics

[19] Lazarus IJ, Bharuthram R, Hellberg MA. Modified Korteweg-de Vries-Zakharov-Kuznetsov solitons in symmetric two-temperature electronpositron plasmas. Journal of Plasma

[20] Zank GP, Greaves RG. Linear and nonlinear modes in nonrelativistic electron-positron plasmas. Physics

[21] Iwamoto N. Collective modes in nonrelativistic electron-positron

and Space Science. 1996;240:215

Physics. 2008;74:519

Review. 1995;51:6079

Magnetized Electron-Positron Plasma. The Astrophysical Journal. 1986;309:

1985;31:951

L63

Electromagnetic Waves in Magnetized Electron-Positron Plasmas. Astrophysics

[15] Mofiz UA, de Angelis U, Forlani A. Solitons in weakly nonlinear electron-

Science. 1997;250:109

Letters A. 2003;315:120

Nonlinear Propagation of

[3] Miller HR, Witta PJ. Active Galactic Nuclei. Berlin: Springer-Verlag; 1987.

[4] Piran T. The physics of gamma-ray bursts. Reviews of Modern Physics.

[5] Goldreich P, Julian WH. Pulsar Electrodynamics. The Astrophysical

[6] Michel FC. Theory of pulsar magnetospheres. Reviews of Modern

[7] Tandberg EH, Emslie AG. The Physics of Solar Flares. Cambridge: Cambridge University Press; 1988.

[8] Beskin VS, Gurevich AV, Istomin YN. Electrodynamics of pulsar

magnetospheres. Soviet Physics—JETP.

[9] Bharuthram R. Arbitrary Amplitude Double Layers in a Multi-Species Electron-Positron Plasma. Astrophysics and Space Science. 1992;189:213

[10] Sturrock PA. A Model of Pulsars. The Astrophysical Journal. 1971;164:529

[11] Berezhiani VI, Mahajan SM. Large relativistic density pulses in electronpositron-ion plasmas. Physical Review

[12] Shatashvili NL, Javakhishvili JI, Kaya H. Nonlinear Wave Dynamics in Two Temperature Electron-Positron-

### Acknowledgements

Thank you to Prof. Ramesh Bharuthram (University of the Western Cape, South Africa), Prof. Gurbax Lakhina and Prof. Satyavir Singh (Indian Institute of Geomagnetism, Navi Mumbai, India and Dr. Suleman Moolla (University of KwaZulu-Natal, Durban, South Africa) for your valuable contributions.

### Author details

Ian Joseph Lazarus Department of Physics, Durban University of Technology, Durban, South Africa

\*Address all correspondence to: lazarusi@dut.ac.za

© 2018 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.

Electrostatic Waves in Magnetized Electron-Positron Plasmas DOI: http://dx.doi.org/10.5772/intechopen.80958

#### References

plasma parameters. In the nonlinear analysis, the two-fluid model is used to derive a set of differential equations for the electrostatic solitary waves in a magnetized twotemperature electron-positron plasma. In particular, electrostatic solitary waves and their electric fields, similar to those found in the Broadband Electrostatic Noise are explored. For the onset of spiky ESWs, it is noted that as the wave speed increases, a

Thank you to Prof. Ramesh Bharuthram (University of the Western Cape, South Africa), Prof. Gurbax Lakhina and Prof. Satyavir Singh (Indian Institute of Geomagnetism, Navi Mumbai, India and Dr. Suleman Moolla (University of KwaZulu-

Department of Physics, Durban University of Technology, Durban, South Africa

© 2018 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: lazarusi@dut.ac.za

provided the original work is properly cited.

Natal, Durban, South Africa) for your valuable contributions.

larger driving electric field is required.

Acknowledgements

Charged Particles

Author details

Ian Joseph Lazarus

74

[1] Weinberg S. Gravitation and Cosmology. New York: Wiley; 1972

[2] Rees MJ. In: Gibbons GW, Hawking SW, Siklas S, editors. The Very Early Universe. Cambridge: Cambridge University Press; 1983

[3] Miller HR, Witta PJ. Active Galactic Nuclei. Berlin: Springer-Verlag; 1987. p. 202

[4] Piran T. The physics of gamma-ray bursts. Reviews of Modern Physics. 2005;76:1143

[5] Goldreich P, Julian WH. Pulsar Electrodynamics. The Astrophysical Journal. 1969;157:869

[6] Michel FC. Theory of pulsar magnetospheres. Reviews of Modern Physics. 1982;54:1

[7] Tandberg EH, Emslie AG. The Physics of Solar Flares. Cambridge: Cambridge University Press; 1988. p. 124

[8] Beskin VS, Gurevich AV, Istomin YN. Electrodynamics of pulsar magnetospheres. Soviet Physics—JETP. 1983;58:235

[9] Bharuthram R. Arbitrary Amplitude Double Layers in a Multi-Species Electron-Positron Plasma. Astrophysics and Space Science. 1992;189:213

[10] Sturrock PA. A Model of Pulsars. The Astrophysical Journal. 1971;164:529

[11] Berezhiani VI, Mahajan SM. Large relativistic density pulses in electronpositron-ion plasmas. Physical Review E. 1995;bf52:1968

[12] Shatashvili NL, Javakhishvili JI, Kaya H. Nonlinear Wave Dynamics in Two Temperature Electron-PositronIon Plasma. Astrophysics and Space Science. 1997;250:109

[13] Bhattacharyya R, Janaki MS, Dasgupta B. Relaxation in electronpositron plasma: a possibility. Physics Letters A. 2003;315:120

[14] Stenflo L, Shukla PK, Yu MY. Nonlinear Propagation of Electromagnetic Waves in Magnetized Electron-Positron Plasmas. Astrophysics and Space Science. 1985;117:303

[15] Mofiz UA, de Angelis U, Forlani A. Solitons in weakly nonlinear electronpositron plasmas and pulsar microstructure. Physical Review A. 1985;31:951

[16] Yu MY, Shukla PK, Stenflo L. Alfven Vortices in a Strongly Magnetized Electron-Positron Plasma. The Astrophysical Journal. 1986;309: L63

[17] Pillay R, Bharuthram R. Large Amplitude Solitons in a Multi-Species Electron-Positron Plasma. Astrophysics and Space Science. 1992;198:85-93

[18] Verheest F, Lakhina GS. Oblique Solitary Alfv e′ n Modes in Relativistic Electron-Positron Plasmas. Astrophysics and Space Science. 1996;240:215

[19] Lazarus IJ, Bharuthram R, Hellberg MA. Modified Korteweg-de Vries-Zakharov-Kuznetsov solitons in symmetric two-temperature electronpositron plasmas. Journal of Plasma Physics. 2008;74:519

[20] Zank GP, Greaves RG. Linear and nonlinear modes in nonrelativistic electron-positron plasmas. Physics Review. 1995;51:6079

[21] Iwamoto N. Collective modes in nonrelativistic electron-positron

plasmas. Physical Review E. 1993;47: 604

[22] Mozer FS, Ergun R, Temerin M, Cattell CA, Dombeck J, Wygant JR. New Features of Time Domain Electric-Field Structures in the Auroral Acceleration Region. Physical Review Letters. 1997; 79:1281

[23] Andre M, Koskinen H, Gustafsson G, Lundin R. Ion waves and upgoing ion beams observed by the Viking satellite. Geophysical Research Letters. 1987;14: 463

[24] Matsumoto H, Kojima H, Miyatake T, Omura Y, Okada M, Nagano I, et al. Electrostatic Solitary Waves (ESW) in the magnetotail: BEN wave forms observed by GEOTAIL. Geophysical Research Letters. 1994;21:2915

[25] Franz JR, Kintner PM, Pichett JS. POLAR Observations of Coherent Electric Field Structures. Geophysical Research Letters. 1998;25:1277

[26] Ergun RE, Carlson CW, McFadden JP, Mozer FS, Delroy GT, Peria W, et al. FAST satellite observations of largeamplitude solitary wave structures. Geophysical Research Letters. 1998;25: 2041

[27] Gary SP, Tokar RL. The electron acoustic Mode. Physics of Fluids. 1985; 28:2439

[28] Lazarus IJ, Bharuthram R, Singh SV, Pillay SR, Lakhina GS. Linear electrostatic waves in two-temperature electron-positron plasmas. Journal of Plasma Physics. 2012;78:621

[29] Bharuthram R, Pather T. The kinetic dust-acoustic instability in a magnetized dusty plasma. Planetary and Space Science. 1996;44:137

[30] Fried BD, Conte SD. The Plasma Dispersion Function. New York: Academic Press; 1961

[31] Rosenberg M. Ion- and dustacoustic instabilities in dusty plasmas. Planetary and Space Science. 1993;41: 229

[32] Temerin MA, Cerny K, Lotko W, Mozer FS. Observations of double layers and solitary waves in the auroral plasma. Physical Review Letters. 1982;48:1175

[33] Omura Y, Kojima H, Matsumoto H. Computer simulation of Electrostatic Solitary Waves: A nonlinear model of broadband electrostatic noise. Geophysical Research Letters. 1994;21: 2923

[34] Greaves RG, Surko CM. An Electron-Positron Beam-Plasma Experiment. Physical Review Letters. 1995;75:3846

[35] Bharuthram R. Arbitrary Amplitude Double Layers in a Multi-Species Electron-Positron Plasma. Astrophysics and Space Science. 1992;189:213

[36] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Receipes in Fortran 90-The Art of Parallel Scientific Computing. 1996;2:702, 731, 1297-704, 740, 1308

[37] Lazarus IJ, Bharuthram R, Moolla S, Singh SV, Lakhina GS. Nonlinear electrostatic solitary waves in electronpositron plasmas. Journal of Plasma Physics. 2016;82:1

**77**

Section 4

Application of Charge

Particles

### Section 4

## Application of Charge Particles

plasmas. Physical Review E. 1993;47:

[31] Rosenberg M. Ion- and dustacoustic instabilities in dusty plasmas. Planetary and Space Science. 1993;41:

[32] Temerin MA, Cerny K, Lotko W, Mozer FS. Observations of double layers and solitary waves in the auroral plasma. Physical Review Letters. 1982;48:1175

[33] Omura Y, Kojima H, Matsumoto H. Computer simulation of Electrostatic Solitary Waves: A nonlinear model of

Geophysical Research Letters. 1994;21:

[35] Bharuthram R. Arbitrary Amplitude Double Layers in a Multi-Species Electron-Positron Plasma. Astrophysics and Space Science. 1992;189:213

[36] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Receipes in Fortran 90-The Art of Parallel Scientific Computing. 1996;2:702, 731,

[37] Lazarus IJ, Bharuthram R, Moolla S, Singh SV, Lakhina GS. Nonlinear electrostatic solitary waves in electronpositron plasmas. Journal of Plasma

broadband electrostatic noise.

[34] Greaves RG, Surko CM. An Electron-Positron Beam-Plasma Experiment. Physical Review Letters.

229

2923

1995;75:3846

1297-704, 740, 1308

Physics. 2016;82:1

[22] Mozer FS, Ergun R, Temerin M, Cattell CA, Dombeck J, Wygant JR. New Features of Time Domain Electric-Field Structures in the Auroral Acceleration Region. Physical Review Letters. 1997;

[23] Andre M, Koskinen H, Gustafsson G, Lundin R. Ion waves and upgoing ion beams observed by the Viking satellite. Geophysical Research Letters. 1987;14:

[24] Matsumoto H, Kojima H, Miyatake T, Omura Y, Okada M, Nagano I, et al. Electrostatic Solitary Waves (ESW) in the magnetotail: BEN wave forms observed by GEOTAIL. Geophysical Research Letters. 1994;21:2915

[25] Franz JR, Kintner PM, Pichett JS. POLAR Observations of Coherent Electric Field Structures. Geophysical Research Letters. 1998;25:1277

[26] Ergun RE, Carlson CW, McFadden JP, Mozer FS, Delroy GT, Peria W, et al. FAST satellite observations of largeamplitude solitary wave structures. Geophysical Research Letters. 1998;25:

[27] Gary SP, Tokar RL. The electron acoustic Mode. Physics of Fluids. 1985;

[28] Lazarus IJ, Bharuthram R, Singh SV,

electrostatic waves in two-temperature electron-positron plasmas. Journal of

[29] Bharuthram R, Pather T. The kinetic dust-acoustic instability in a magnetized dusty plasma. Planetary and Space

[30] Fried BD, Conte SD. The Plasma Dispersion Function. New York:

Pillay SR, Lakhina GS. Linear

Plasma Physics. 2012;78:621

Science. 1996;44:137

Academic Press; 1961

76

604

Charged Particles

79:1281

463

2041

28:2439

**79**

**Chapter 6**

**Abstract**

infections.

**1. Introduction**

Biological Effects of Negatively

*Suni Lee, Yasumitsu Nishimura, Naoko Kumagai-Takei,* 

To identify health-promoting indoor air conditions, we developed negatively charged particle-dominant indoor air conditions (NCPDIAC). Experiments assessing the biological effects of NCPDIAC comprised (1) 2.5-h stays in NCPDIAC or control rooms, (2) 2-week nightly stays in control followed by NCPDIAC rooms, (3) 3-month OFF to ON and ON to OFF trials in individual living homes equipped with NPCDIAC in their sleeping or living rooms, and (4) in vitro assays comparing the immune effects between negatively charged particle-dominant and control cell culture incubators. The most significant difference examined between NCPDIAC and control rooms in the 2.5-h stays was an increase in interleukin (IL)-2 with occupancy of the NCPDIAC room. For the 2-week nightly stay experiments, natural killer (NK) cell activity increased with occupancy of the NCPDIAC room. The 3-month OFF to ON trial showed an increase in NK cell activity, while the ON to OFF trial yielded a decrease in NK cell activity. Additionally, the in vitro assays also showed an increase in NK cell activity. The use of NCPDIAC resulted in increased NK cell activity, which has the effect of enhancing immune surveillance for the occurrence of cancer and improving symptoms associated with viral

**Keywords:** indoor air, negatively charged particle, natural killer cell activity

Indoor air conditions can sometimes affect human health. For example, sick building syndrome (SBS) is one of the most well-known health impairments caused by indoor air conditions [1–3]. Volatile organic compounds (VOCs) are considered to be the cause of SBS [1–3]. SBS can induce a variety of signs or symptoms such as headache; eye, nose, and throat irritation; fatigue; dizziness; and nausea. The condition of patients with SBS may worsen following exposure to certain VOCs, with individual patients revealing specific hypersensitivity to particular chemicals. It is considered that some pathophysiological alterations in the psycho-neuroimmune-endocrine network at the level of genes, molecules, proteins, cells, and

*Hidenori Matsuzaki, Megumi Maeda, Nagisa Sada,* 

Charged Particle-Dominant

Indoor Air Conditions

*Kei Yoshitome and Takemi Otsuki*

#### **Chapter 6**

## Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions

*Suni Lee, Yasumitsu Nishimura, Naoko Kumagai-Takei, Hidenori Matsuzaki, Megumi Maeda, Nagisa Sada, Kei Yoshitome and Takemi Otsuki*

#### **Abstract**

To identify health-promoting indoor air conditions, we developed negatively charged particle-dominant indoor air conditions (NCPDIAC). Experiments assessing the biological effects of NCPDIAC comprised (1) 2.5-h stays in NCPDIAC or control rooms, (2) 2-week nightly stays in control followed by NCPDIAC rooms, (3) 3-month OFF to ON and ON to OFF trials in individual living homes equipped with NPCDIAC in their sleeping or living rooms, and (4) in vitro assays comparing the immune effects between negatively charged particle-dominant and control cell culture incubators. The most significant difference examined between NCPDIAC and control rooms in the 2.5-h stays was an increase in interleukin (IL)-2 with occupancy of the NCPDIAC room. For the 2-week nightly stay experiments, natural killer (NK) cell activity increased with occupancy of the NCPDIAC room. The 3-month OFF to ON trial showed an increase in NK cell activity, while the ON to OFF trial yielded a decrease in NK cell activity. Additionally, the in vitro assays also showed an increase in NK cell activity. The use of NCPDIAC resulted in increased NK cell activity, which has the effect of enhancing immune surveillance for the occurrence of cancer and improving symptoms associated with viral infections.

**Keywords:** indoor air, negatively charged particle, natural killer cell activity

#### **1. Introduction**

Indoor air conditions can sometimes affect human health. For example, sick building syndrome (SBS) is one of the most well-known health impairments caused by indoor air conditions [1–3]. Volatile organic compounds (VOCs) are considered to be the cause of SBS [1–3]. SBS can induce a variety of signs or symptoms such as headache; eye, nose, and throat irritation; fatigue; dizziness; and nausea. The condition of patients with SBS may worsen following exposure to certain VOCs, with individual patients revealing specific hypersensitivity to particular chemicals. It is considered that some pathophysiological alterations in the psycho-neuroimmune-endocrine network at the level of genes, molecules, proteins, cells, and

organs may occur in SBS patients, which then defines or determines their sensitivity to such low concentrations of VOCs [1–3]. However, the precise nature of these alterations is yet to be delineated. Consequently, the only advice presently available to SBS patients is to avoid exposure to VOCs for which the patient shows particular sensitivity [1–3].

Furthermore, most homes in Japan possess air-conditioning units in each room. During the winter season, the room adjacent to the bathroom which is utilized for changing clothes is narrow and particularly cold. Moreover, the lavatory is also cold. Consequently, there is a risk of changes in blood pressure and the onset of cardiovascular events caused by drastic changes in room temperature in these areas [4, 5].

Thus, decreasing the amount and use of chemicals and maintaining appropriate room temperatures at home are things that can be considered with respect to the task of establishing health-promoting indoor conditions. Additionally, although there are few reports detailing the use of indoor air under negatively charged conditions, a consideration of air electrical charges may assist with this task [6–9].

#### **2. Development of NCPDIAC**

The development of NCPDIAC has previously been reported [10].

As shown in **Figure 1**, NCPDIAC was established using extraporous charcoal paint and loading an electric voltage (approximately 72–100 V) behind the room walls [10]. The charcoal paints were mainly used for deodorization and dehumidification. As a result, the surface of the walls acquired a slightly negative charge, and small positively charged particles 20–30 nm in diameter collected on the surface of walls [10]. Thus, although negatively charged particles were not introduced into the

**Figure 1.** *NCPDIAC was established using extraporous charcoal paint and loading an electric voltage behind the walls.*

**81**

*Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions*

was such that negatively charged particles were predominant [10].

and 1564 m3

indoor conditions, the balance between positively and negatively charged particles

Results of the 2.5H (2.5-h) stay experiments have previously been reported [10]. Three control rooms and three NCPDIAC rooms were built in the wide subunderground laboratory. Both types of rooms were built in a large subunderground laboratory in the Comprehensive Housing R&D Institute, SEKISUI HOUSE, Ltd., at Kizu-town, Kyoto prefecture, Japan. The area and volume of the laboratory were

rooms is shown in **Figure 2A**. All of the healthy volunteers (HV) referred to in **Figure 2B** were unaware of the room type (control or NCPDIAC) they were to occupy during the experimental period. The following items were measured immediately prior to (prestay) and following (poststay) entry into the rooms as previously reported [10].

1.General conditions: blood chemistry including liver [alanine aminotransferase (ALT), aspartate aminotransferase (AST), and gamma-glutamyl transferase (γGT)] and kidney functions [creatinine, blood urea nitrogen (BUN) and uric acid], blood sugar and lactic acid levels, and peripheral blood counts (white blood cell, red blood cell, hemoglobin, hematocrit and platelet) were measured using peripheral venous blood. Blood pressure and pulse rate were also measured.

2.Stress markers: Levels of blood cortisol and salivary cortisol, chromogranin A, amylase, and secretory immunoglobulin A were measured as stress markers.

3.Parameters related to the autonomic nervous system: The autonomic nervous system was examined using the Flicker test, a stabilometer, and heart rate monitor for 3 min. The Flicker test and flicking frequencies of red, green, and yellow colors were monitored. A Gravicoder GS-7TM instrument (Anima Inc., Tokyo, Japan) was used as a stabilometer and the Romberg ratio was used as the parameter for body sway. The ratio was calculated from the whole trajectory of the body sway during a 30-s standing period with eyes closed divided by that with eyes open. The heart rate was monitored using a Heart Rate Monitor S810iTM instrument (Polar Electro, Kempele, Finland) for 3 min. During monitoring, HV sat on chairs and were kept at rest. The R wave intervals in the electrocardiogram were estimated and the standard deviation (SD) or R wave interval was considered as an index of heart rate fluctuation.

4.Immunological parameters: Serum levels of immunoglobulin (Ig) E and Ig A, and cytokines related to the Th1/Th2 balance [Interferon (IFN)-γ, tumor necrosis factor (TNF)-α, Interleukin (IL)-2, IL-4, IL-6 and IL-10] were evaluated. Individual samples for cytokine measurement were applied to the Cytometric Bead Array of Human Th1/Th2 cytokine kit II (CBA, BD Bioscience, San Jose, CA, USA) and measurements were made using FACSCalibur flow cytometry (BD Bioscience) according to the manufacturer's instructions. For samples that revealed less than the lower limit of the values from analytical methods for cytokines and immunoglobulins, the 1/10 value of the minimum values among the entire measurable samples was substituted

instead of 0 or left as "unmeasurable," as previously reported [10].

, respectively, and those of the experimental rooms

, respectively. The appearance of the control and NCPDIAC

*DOI: http://dx.doi.org/10.5772/intechopen.79934*

**3. 2.5H stay experiments**

approximately 539 m2

and 22.8 m3

were 9.1 m2

*Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions DOI: http://dx.doi.org/10.5772/intechopen.79934*

indoor conditions, the balance between positively and negatively charged particles was such that negatively charged particles were predominant [10].

#### **3. 2.5H stay experiments**

*Charged Particles*

sensitivity [1–3].

these areas [4, 5].

**2. Development of NCPDIAC**

task [6–9].

organs may occur in SBS patients, which then defines or determines their sensitivity to such low concentrations of VOCs [1–3]. However, the precise nature of these alterations is yet to be delineated. Consequently, the only advice presently available to SBS patients is to avoid exposure to VOCs for which the patient shows particular

Furthermore, most homes in Japan possess air-conditioning units in each room. During the winter season, the room adjacent to the bathroom which is utilized for changing clothes is narrow and particularly cold. Moreover, the lavatory is also cold. Consequently, there is a risk of changes in blood pressure and the onset of cardiovascular events caused by drastic changes in room temperature in

Thus, decreasing the amount and use of chemicals and maintaining appropriate room temperatures at home are things that can be considered with respect to the task of establishing health-promoting indoor conditions. Additionally, although there are few reports detailing the use of indoor air under negatively charged conditions, a consideration of air electrical charges may assist with this

The development of NCPDIAC has previously been reported [10].

As shown in **Figure 1**, NCPDIAC was established using extraporous charcoal paint and loading an electric voltage (approximately 72–100 V) behind the room walls [10]. The charcoal paints were mainly used for deodorization and dehumidification. As a result, the surface of the walls acquired a slightly negative charge, and small positively charged particles 20–30 nm in diameter collected on the surface of walls [10]. Thus, although negatively charged particles were not introduced into the

*NCPDIAC was established using extraporous charcoal paint and loading an electric voltage behind the walls.*

**80**

**Figure 1.**

Results of the 2.5H (2.5-h) stay experiments have previously been reported [10]. Three control rooms and three NCPDIAC rooms were built in the wide subunderground laboratory. Both types of rooms were built in a large subunderground laboratory in the Comprehensive Housing R&D Institute, SEKISUI HOUSE, Ltd., at Kizu-town, Kyoto prefecture, Japan. The area and volume of the laboratory were approximately 539 m2 and 1564 m3 , respectively, and those of the experimental rooms were 9.1 m2 and 22.8 m3 , respectively. The appearance of the control and NCPDIAC rooms is shown in **Figure 2A**. All of the healthy volunteers (HV) referred to in **Figure 2B** were unaware of the room type (control or NCPDIAC) they were to occupy during the experimental period. The following items were measured immediately prior to (prestay) and following (poststay) entry into the rooms as previously reported [10].


5.Blood viscosity: Blood viscosity was measured using a Micro-Channel Flow Analyzer MC-FAN (MC Laboratory Inc., Tokyo, Japan) according to the manufacturer's instructions. Briefly, peripheral heparinized blood sample (100 μl) was placed into the instrument and allowed to flow through the microchannel chips, which are a model for capillary vessels, and the flowing time was recorded. The flowing blood sample was visualized using a CCD camera equipped to the microscope.

These experiments were approved by the institutional ethical committee (#114). Samples were only taken from HV who provided written informed consent.

The electrical charge in these rooms was measured using an Ion Counter EB-1000TM instrument made by Eco Holistic Inc., Suita, Japan.

Differences in the positively and negatively charged particles in control and NCPDIAC rooms are shown in **Figure 2C**. The number of positively charged particles in the rooms with NCPDIAC was reduced. However, the number of negatively charged particles in control and NCPDIAC rooms did not differ. Negatively charged air conditions were therefore formed by reducing the number of positively charged particles in NCPDIAC rooms.

Differences between control and NCPDIAC room values for all items measured were determined by calculating [poststay]-[prestay]. As shown in **Figure 2D**, differences were found in IL-2 [10]. The increase in IL-2 levels (by approximately 1 pg/ml)

#### **Figure 2.**

*(A) Control and NCPDIAC rooms were constructed as shown. Healthy volunteers occupied rooms for 2.5 h, being unaware of the room type (control or NCPDIAC). Volunteers remained within the rooms in a stable state, without sleeping or excitement. (B) A total of 60 Japanese volunteers participated in experiments for each room type. The gender ratio and average age of the volunteers were almost identical in both groups (occupants of control and NCPDIAC rooms). (C) The box-and-whisker plots show the number of positively and negatively charged particles per 1 cm3 air in control and NCPDIAC rooms. Although there was no difference in the level of positively and negatively charged particles in control rooms, NCPDIAC rooms possessed significantly lower levels of positively charged particles compared with control rooms. Furthermore, there was no difference in the level of negatively charged particles between control and NCPDIAC rooms. Statistical differences were assayed using the student T test. (D) The box-and-whisker plots show a comparison of IL-2 levels ([Post-Stay]-[Pre-Stay], pg/ml) in volunteers who occupied the control and NCPDIAC rooms during the 2.5-h stay experiments. The most significant difference found in the examined values using [poststay]-[prestay] related to IL-2 levels, which increased significantly following NCPDIAC room stays compared with control room stays. This difference was analyzed using the Mann-Whitney U test.*

**83**

**Figure 3.**

*Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions*

was considered not to be caused by any pathophysiological conditions. Additionally, it was considered that levels could return to base values in individual HV. As a result, it appears that NCPDIAC affected the immune system without any adverse effects with respect to the signs, symptoms, or measured items in the present experiments [10].

In the next step, 2W (2-week) nightly stay experiments were performed as

Approximately 1 year after obtaining the results of the 2.5H experiments and with subsequent discussions, new participants were recruited for our 2W nightly stay experiments. This study was approved by the institutional ethical committee (#176) and samples were only taken from HV who provided written informed consent. The dormitory belonging to SEKISUI HOUSE, Ltd., Kizugawa-City, Kyoto Prefecture, Japan, and ordinarily used for the training of employees was utilized for our 2W nightly stay experiments. These employees usually receive training for 3 months as shown in **Figure 3A**. Then, in the case of volunteers for our experiments, employees were moved at the second month from their original room into a control

*(A) For the 2W (2-week) nightly stay experiments the dormitory for training employees was used. (B) All volunteers initially occupied the control room every night for 2 weeks. Thereafter, volunteers occupied NCPDIAC rooms without being aware of which room type (control or NCPDIAC) they had initially occupied. Sample collections were performed at T1, T2, and T3 time points. T1 represented the time point prior to volunteers occupying the control room. T2 represented the time point immediately following the 2W nightly stay in control rooms, and just prior to occupancy of the NCPDIAC room, while T3 represented the time point following the 2W nightly stay in the NCPDIAC room. Although HV agreed to be recruited in this study, they were unaware of the room type (control or NCPDIAC) they had initially entered. (C) The box-and-whisker plots show the* 

*there was no difference between the number of positively and negatively charged particles in control rooms, NCPDIAC rooms possessed significantly lower levels of positively charged particles compared with control rooms. Furthermore, there was no difference in the level of negatively charged particles between control and NCPDIAC rooms. Statistical analyses were performed using the student T test. (D) The box-and-whisker plots show the actual NK cell activity (%) at T1, T2, and T3 time points. There were no differences among the three time points. Statistical analyses were performed using the ANOVA test. (E) The box-and-whisker plots show the relative NK activity after setting T1 of individual volunteers to 1.0. Statistical analyses comparing T1 and T2 or T3 were performed using the student T test. (F) Relative NK activity after setting T2 of individual volunteers to 1.0. There was a significant increase in relative NK activity after the 2W nightly stay in the NCPDIAC rooms.* 

 *air in control and NCPDIAC rooms. Although* 

*number of positively and negatively charged particles per 1 cm3*

*Statistical analyses were performed using the student T test.*

*DOI: http://dx.doi.org/10.5772/intechopen.79934*

**4. 2W nightly stay experiments**

previously reported [11].

was considered not to be caused by any pathophysiological conditions. Additionally, it was considered that levels could return to base values in individual HV. As a result, it appears that NCPDIAC affected the immune system without any adverse effects with respect to the signs, symptoms, or measured items in the present experiments [10].

#### **4. 2W nightly stay experiments**

*Charged Particles*

camera equipped to the microscope.

particles in NCPDIAC rooms.

5.Blood viscosity: Blood viscosity was measured using a Micro-Channel Flow Analyzer MC-FAN (MC Laboratory Inc., Tokyo, Japan) according to the manufacturer's instructions. Briefly, peripheral heparinized blood sample (100 μl) was placed into the instrument and allowed to flow through the microchannel chips, which are a model for capillary vessels, and the flowing time was recorded. The flowing blood sample was visualized using a CCD

These experiments were approved by the institutional ethical committee (#114).

Differences in the positively and negatively charged particles in control and NCPDIAC rooms are shown in **Figure 2C**. The number of positively charged particles in the rooms with NCPDIAC was reduced. However, the number of negatively charged particles in control and NCPDIAC rooms did not differ. Negatively charged air conditions were therefore formed by reducing the number of positively charged

Differences between control and NCPDIAC room values for all items measured were determined by calculating [poststay]-[prestay]. As shown in **Figure 2D**, differences were found in IL-2 [10]. The increase in IL-2 levels (by approximately 1 pg/ml)

*(A) Control and NCPDIAC rooms were constructed as shown. Healthy volunteers occupied rooms for 2.5 h, being unaware of the room type (control or NCPDIAC). Volunteers remained within the rooms in a stable state, without sleeping or excitement. (B) A total of 60 Japanese volunteers participated in experiments for each room type. The gender ratio and average age of the volunteers were almost identical in both groups (occupants of control and NCPDIAC rooms). (C) The box-and-whisker plots show the number of positively and negatively* 

*of positively and negatively charged particles in control rooms, NCPDIAC rooms possessed significantly lower levels of positively charged particles compared with control rooms. Furthermore, there was no difference in the level of negatively charged particles between control and NCPDIAC rooms. Statistical differences were assayed using the student T test. (D) The box-and-whisker plots show a comparison of IL-2 levels ([Post-Stay]-[Pre-Stay], pg/ml) in volunteers who occupied the control and NCPDIAC rooms during the 2.5-h stay experiments. The most significant difference found in the examined values using [poststay]-[prestay] related to IL-2 levels, which increased significantly following NCPDIAC room stays compared with control room stays. This difference* 

 *air in control and NCPDIAC rooms. Although there was no difference in the level* 

Samples were only taken from HV who provided written informed consent. The electrical charge in these rooms was measured using an Ion Counter

EB-1000TM instrument made by Eco Holistic Inc., Suita, Japan.

**82**

**Figure 2.**

*charged particles per 1 cm3*

*was analyzed using the Mann-Whitney U test.*

In the next step, 2W (2-week) nightly stay experiments were performed as previously reported [11].

Approximately 1 year after obtaining the results of the 2.5H experiments and with subsequent discussions, new participants were recruited for our 2W nightly stay experiments. This study was approved by the institutional ethical committee (#176) and samples were only taken from HV who provided written informed consent.

The dormitory belonging to SEKISUI HOUSE, Ltd., Kizugawa-City, Kyoto Prefecture, Japan, and ordinarily used for the training of employees was utilized for our 2W nightly stay experiments. These employees usually receive training for 3 months as shown in **Figure 3A**. Then, in the case of volunteers for our experiments, employees were moved at the second month from their original room into a control

#### **Figure 3.**

*(A) For the 2W (2-week) nightly stay experiments the dormitory for training employees was used. (B) All volunteers initially occupied the control room every night for 2 weeks. Thereafter, volunteers occupied NCPDIAC rooms without being aware of which room type (control or NCPDIAC) they had initially occupied. Sample collections were performed at T1, T2, and T3 time points. T1 represented the time point prior to volunteers occupying the control room. T2 represented the time point immediately following the 2W nightly stay in control rooms, and just prior to occupancy of the NCPDIAC room, while T3 represented the time point following the 2W nightly stay in the NCPDIAC room. Although HV agreed to be recruited in this study, they were unaware of the room type (control or NCPDIAC) they had initially entered. (C) The box-and-whisker plots show the number of positively and negatively charged particles per 1 cm3 air in control and NCPDIAC rooms. Although there was no difference between the number of positively and negatively charged particles in control rooms, NCPDIAC rooms possessed significantly lower levels of positively charged particles compared with control rooms. Furthermore, there was no difference in the level of negatively charged particles between control and NCPDIAC rooms. Statistical analyses were performed using the student T test. (D) The box-and-whisker plots show the actual NK cell activity (%) at T1, T2, and T3 time points. There were no differences among the three time points. Statistical analyses were performed using the ANOVA test. (E) The box-and-whisker plots show the relative NK activity after setting T1 of individual volunteers to 1.0. Statistical analyses comparing T1 and T2 or T3 were performed using the student T test. (F) Relative NK activity after setting T2 of individual volunteers to 1.0. There was a significant increase in relative NK activity after the 2W nightly stay in the NCPDIAC rooms. Statistical analyses were performed using the student T test.*

(both were the same in terms of NCPDIAC). After occupying the control room every night for 2 weeks, they were then moved into NCPDIAC rooms, without being aware of the room type (control or NCPDIAC) they had initially occupied. As shown in **Figure 3B**, sample collections were performed prior to volunteers occupying the control room (T1), after their 2W nightly stay in the control room (T2), and finally after occupying the NCPDIAC room (T3). The measured items remained unchanged during the 2.5H stay experiments. Additionally, several parameters that had not changed during the hour-based time period but were altered during the week-based time period when environmental factors or physiological conditions had changed were included as biological parameters. Those parameters comprised NK cell activity, along with urine 17 hydroxycorticosteroid (OHCS) and 8-oxo-2′-deoxyguanosine (OHdG) levels.

All volunteers initially occupied the control room every night for 2 weeks. Thereafter, volunteers occupied NCPDIAC rooms without being aware of which room type (control or NCPDIAC) they had initially occupied. Sample collection was performed at T1, T2, and T3 time points. T1 represented the time point prior to volunteers occupying the control room. T2 represented the time point immediately following the 2W nightly stay in control rooms, and just prior to occupancy of the NCPDIAC room, while T3 represented the time point following the 2W nightly stay in the NCPDIAC room.

Among all the items measured, significant change was only found in the NK activity. The NK cell activity was determined using a 51Cr-release assay according to a method outlined in previous reports [12, 13]. The effector cell (mononuclear cell) to-target cell (K562 cell) ratio was 10:1. However, there was a wide range of individual actual NK cell activity in the vicinity of approximately 10% to greater than 50%. In the 2W nightly stay experiments, the NK cell activity among HV ranged from less than 30% to near 60% at T1 (**Figure 3B**). Although actual NK activity measurements did not reveal any statistical significance (**Figure 3D**) due to the wide variation in individual volunteers, when the relative NK activity was set to 1.0 at T1, there was a tendency toward increased NK cell activity at T3 (**Figure 3E**) [11]. Since the room conditions prior to T1 and T2 collections were basically the same, the relative NK cell activity between T2 and T3 was compared, with T2 being set to 1.0 in individual HV [11]. As shown in **Figure 3F**, there was a significant increase in relative NK activity at T3, after the 2W nightly stay with NCPDIAC [11].

#### **5. In vitro experiments**

The in vitro experiments were performed to examine cellular alterations under negative particle dominant conditions. Results have previously been reported [14].

Freshly isolated peripheral blood mononuclear cells derived from seven healthy volunteers were cultured in a standard CO2 incubator at 37°C with 5% CO2 under humidified conditions (standard cell culture conditions comprised a humidity of 95%). For these experiments, it was impossible to use charcoal paint or to load an electric voltage. Thus, negatively charged particles were forced in and circulated. The difference between positively and negatively charged particles in 1 cm3 of incubator air was approximately 3000 (**Figure 4A**) [14].

This study was approved by the institutional ethical committee (#883) and samples were only taken from the HV who provided written informed consent. Peripheral blood mononuclear cells were derived from samples obtained from HV.

For the NK cell activity, K562 cells, a human immortalized myelogenous leukemia cell line, were stained with Vybrant™ Dio Cell-Labeling Solution by incubation for 20 min at room temperature [14]. Dio-stained cells were then washed with phosphate buffered slain (PBS), and peripheral blood mononuclear cells (PBMC)

**85**

**Figure 4.**

*Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions*

were incubated with 5000 Dio-labeled K562 cells in 96-well round bottom plates at an effector cell-to-target cell (E/T) ratio of 2.5:1, 5:1, or 10:1 for 5 h in experimental or standard incubators. Following incubation, cells were collected and stained with propidium iodide (PI) at 5 μg/ml and the percentage of PI+ Dio-labeled cells among the total Dio-labeled cells, representing the percentage of lysed cells, was examined using FACSCalibur flow cytometry. The percentage of specific lysis induced by effector cells was calculated after analyzing the substrate and spontaneous dead cell

*. (B) The box-and-whisker plots show the increase* 

*(A) Peripheral blood mononuclear cells from HV were incubated in standard (STD) or experimental (EXP) incubators. Negatively charged particles (20–30 nm in diameter) were forced into incubators and then circulated. The experimental incubator was set to generate negatively charged particles using a neutralizing instrument that created negatively charged particles (SJ-M200, Keyence Co. Ltd., Osaka, Japan). This instrument yielded negatively charged particles that were set to directly enter the inside of the incubator (by making a hole). Since the interior volume of the incubator was 49 l, the negatively charged particles entered* 

*in actual NK cell activity, represented by the E/T (effector vs. target cells) ratio, which, being 5:1, showed greater increase with experimental incubators and negatively charged particle-dominant culture conditions compared with standard incubators. (C) The LOG10 value of the "immune index" as [NC-activisty X IFN-*γ *concentration X IL-2 concentration]/[IL-10 concentration] was compared between standard and experimental* 

*incubator conditions. Experimental incubator conditions showed a significant increase in this index.*

Additionally, peripheral blood mononuclear cells were cultured in RPMI1640 culture medium with antibiotics just as for standard cell cultures for 1 or 2 weeks without any stimulants such as cytokines. The concentrations of IFN-γ, IL-2, and IL-10 (as well as IL-6, TNF-α, and IL-4) in supernatants were measured. Additionally, other items such as surface CD25, CD69, programmed death-1 (PD1), and CD44 expression in CD4+ T helper cells, CD8+ T cells, and NK cells were measured [14]. It is noteworthy that NK cell activity was significantly higher when incubations were performed in the experimental incubator compared with those performed in the standard incubator (**Figure 4B**). Additionally, if we calculate the "immune index" as [NC-activity X IFN-γ concentration X IL-2 concentration]/[IL-10 concentration] and compare this index from standard and experimental incubators using log10 titer (**Figure 4C**), there is a significantly greater increase in this "Log10 Immune Index" associated with the use of experimental incubators compared with the use of standard incubators. The higher value of this item was assumed to reflect

These results indicated that a predominance of negatively charged particles

induces immune stimulation at non-pathophysiological levels [14].

numbers expressed in wells without effector [14].

*and passed out at a rate of approximately 3000 particles/cm3*

stimulation of immune status [14].

*DOI: http://dx.doi.org/10.5772/intechopen.79934*

*Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions DOI: http://dx.doi.org/10.5772/intechopen.79934*

#### **Figure 4.**

*Charged Particles*

in the NCPDIAC room.

**5. In vitro experiments**

(both were the same in terms of NCPDIAC). After occupying the control room every night for 2 weeks, they were then moved into NCPDIAC rooms, without being aware of the room type (control or NCPDIAC) they had initially occupied. As shown in **Figure 3B**, sample collections were performed prior to volunteers occupying the control room (T1), after their 2W nightly stay in the control room (T2), and finally after occupying the NCPDIAC room (T3). The measured items remained unchanged during the 2.5H stay experiments. Additionally, several parameters that had not changed during the hour-based time period but were altered during the week-based time period when environmental factors or physiological conditions had changed were included as biological parameters. Those parameters comprised NK cell activity, along with urine 17 hydroxycorticosteroid (OHCS) and 8-oxo-2′-deoxyguanosine (OHdG) levels. All volunteers initially occupied the control room every night for 2 weeks. Thereafter, volunteers occupied NCPDIAC rooms without being aware of which room type (control or NCPDIAC) they had initially occupied. Sample collection was performed at T1, T2, and T3 time points. T1 represented the time point prior to volunteers occupying the control room. T2 represented the time point immediately following the 2W nightly stay in control rooms, and just prior to occupancy of the NCPDIAC room, while T3 represented the time point following the 2W nightly stay

Among all the items measured, significant change was only found in the NK activity. The NK cell activity was determined using a 51Cr-release assay according to a method outlined in previous reports [12, 13]. The effector cell (mononuclear cell) to-target cell (K562 cell) ratio was 10:1. However, there was a wide range of individual actual NK cell activity in the vicinity of approximately 10% to greater than 50%. In the 2W nightly stay experiments, the NK cell activity among HV ranged from less than 30% to near 60% at T1 (**Figure 3B**). Although actual NK activity measurements did not reveal any statistical significance (**Figure 3D**) due to the wide variation in individual volunteers, when the relative NK activity was set to 1.0 at T1, there was a tendency toward increased NK cell activity at T3 (**Figure 3E**) [11]. Since the room conditions prior to T1 and T2 collections were basically the same, the relative NK cell activity between T2 and T3 was compared, with T2 being set to 1.0 in individual HV [11]. As shown in **Figure 3F**, there was a significant increase in

relative NK activity at T3, after the 2W nightly stay with NCPDIAC [11].

incubator air was approximately 3000 (**Figure 4A**) [14].

The in vitro experiments were performed to examine cellular alterations under negative particle dominant conditions. Results have previously been reported [14]. Freshly isolated peripheral blood mononuclear cells derived from seven healthy volunteers were cultured in a standard CO2 incubator at 37°C with 5% CO2 under humidified conditions (standard cell culture conditions comprised a humidity of 95%). For these experiments, it was impossible to use charcoal paint or to load an electric voltage. Thus, negatively charged particles were forced in and circulated. The difference between positively and negatively charged particles in 1 cm3

This study was approved by the institutional ethical committee (#883) and samples were only taken from the HV who provided written informed consent. Peripheral blood mononuclear cells were derived from samples obtained from HV. For the NK cell activity, K562 cells, a human immortalized myelogenous leukemia cell line, were stained with Vybrant™ Dio Cell-Labeling Solution by incubation for 20 min at room temperature [14]. Dio-stained cells were then washed with phosphate buffered slain (PBS), and peripheral blood mononuclear cells (PBMC)

of

**84**

*(A) Peripheral blood mononuclear cells from HV were incubated in standard (STD) or experimental (EXP) incubators. Negatively charged particles (20–30 nm in diameter) were forced into incubators and then circulated. The experimental incubator was set to generate negatively charged particles using a neutralizing instrument that created negatively charged particles (SJ-M200, Keyence Co. Ltd., Osaka, Japan). This instrument yielded negatively charged particles that were set to directly enter the inside of the incubator (by making a hole). Since the interior volume of the incubator was 49 l, the negatively charged particles entered and passed out at a rate of approximately 3000 particles/cm3 . (B) The box-and-whisker plots show the increase in actual NK cell activity, represented by the E/T (effector vs. target cells) ratio, which, being 5:1, showed greater increase with experimental incubators and negatively charged particle-dominant culture conditions compared with standard incubators. (C) The LOG10 value of the "immune index" as [NC-activisty X IFN-*γ *concentration X IL-2 concentration]/[IL-10 concentration] was compared between standard and experimental incubator conditions. Experimental incubator conditions showed a significant increase in this index.*

were incubated with 5000 Dio-labeled K562 cells in 96-well round bottom plates at an effector cell-to-target cell (E/T) ratio of 2.5:1, 5:1, or 10:1 for 5 h in experimental or standard incubators. Following incubation, cells were collected and stained with propidium iodide (PI) at 5 μg/ml and the percentage of PI+ Dio-labeled cells among the total Dio-labeled cells, representing the percentage of lysed cells, was examined using FACSCalibur flow cytometry. The percentage of specific lysis induced by effector cells was calculated after analyzing the substrate and spontaneous dead cell numbers expressed in wells without effector [14].

Additionally, peripheral blood mononuclear cells were cultured in RPMI1640 culture medium with antibiotics just as for standard cell cultures for 1 or 2 weeks without any stimulants such as cytokines. The concentrations of IFN-γ, IL-2, and IL-10 (as well as IL-6, TNF-α, and IL-4) in supernatants were measured. Additionally, other items such as surface CD25, CD69, programmed death-1 (PD1), and CD44 expression in CD4+ T helper cells, CD8+ T cells, and NK cells were measured [14].

It is noteworthy that NK cell activity was significantly higher when incubations were performed in the experimental incubator compared with those performed in the standard incubator (**Figure 4B**). Additionally, if we calculate the "immune index" as [NC-activity X IFN-γ concentration X IL-2 concentration]/[IL-10 concentration] and compare this index from standard and experimental incubators using log10 titer (**Figure 4C**), there is a significantly greater increase in this "Log10 Immune Index" associated with the use of experimental incubators compared with the use of standard incubators. The higher value of this item was assumed to reflect stimulation of immune status [14].

These results indicated that a predominance of negatively charged particles induces immune stimulation at non-pathophysiological levels [14].

#### **6. 3M ON and OFF trials**

Finally, long-term (3-month, or 3M) stay experiments were performed and the results have previously been reported [15].

Following the aforementioned experiments (2.5H and 2W nightly stay experiments and evaluations), it seemed that NCPDIAC stimulated NK activity with no adverse effects on HV [15]. The increase in NK activity could be accounted for by the short-term, albeit slight, increase in IL-2, which may activate NK cells during the 2W period. The following experiments were then applied in living homes. The homes of seven volunteers were modified for NCPDIAC, targeting mainly sleeping rooms and living rooms. A switch panel approximately 4 × 22 × 29 cm in size had been fitted. Volunteers would then switch this panel ON and OFF every 3 months. Then, prior to and following every 3-month ON or OFF living period, clinical measurements including NK cell activity and others (as measured in the 2W nightly stay experiments) were performed. A total of 16 OFF to ON (3M ON) and 13 ON to OFF (3M OFF) trials were performed as shown in **Figure 5A** [15].

Blood samples were taken just before switching ON or OFF. Thus, during the OFF to ON (3M ON) period, HV stayed at home with NCPDIAC (sleeping room and living room). During the ON to OFF (3M OFF) period, HV occupied rooms in their homes without NCPDIAC [15].

All seven HV comprised Japanese living in Japan and were asked to join this project by first-class registered architects who are colleagues of the authors. The average age of the volunteers was 54.86 ± 9.15 years and included five males and two females.

#### **Figure 5.**

*(A) Seven healthy volunteers utilized sleeping and living rooms with NCPDIAC in their homes. During this period, the occupants themselves would switch NCPDIAC ON and OFF. Sixteen trials of OFF to ON (3M ON) and thirteen trials of ON to OFF (3M OFF) were executed. (B) The levels of positively and negatively charged particles in the representative six rooms including the homes of HV where the NCPDIAC apparatus was set were measured using as ion counter (EM-1000, Eco Holistic Inc., Suita, Japan) during the OFF and ON periods. During the OFF period, there was no difference between positively and negatively charged particles; however, during the ON period, the level of positively charged particles was significantly reduced, thereby establishing a difference between the number of positively and negatively charged particles. Statistical analyses were performed using the student T test. (C) Changes in actual NK cell activity prior to and following the 3M ON and 3M OFF periods. Values tended to increase during the 3M ON and decrease during the 3M OFF periods. (D) Relative NK cell activities with NK cell activity previously set to 1.0. There was a significant increase in relative NK cell activity during the 3M ON period and a significant decrease during the 3M OFF period. Statistical analyses were performed using the student T test.*

**87**

**7. Discussion**

*Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions*

All volunteers had built or renovated their residential homes prior to being recruited to this project and agreed to set up an NCPDIAC device for the experiments [15]. This study was approved by the institutional ethical committee (#854) and samples were only taken from HV who provided written informed consent. The clinical parameters measured were similar to those determined for the 2W nightly stay experiments with additional cytokines being measured. Twentynine cytokines were measured using the Luminex 26 Cytokine Plex Kit Human Cytokine/Chemokine Panel (MPXHCYTO60KPMX26, Merck Millipore, Billerica, MA) [15]. Additionally, adipokines and cytokines related to oxidative stress in serum from 16 ON and 13 OFF trials were measured using the Human Adipokine Magnetic 14-Plex Panel with Luminex instruments (Bio-rad, Hercules, CA, USA). Fourteen of the cytokines examined comprised IL-1β, IL-10, IL-6, monocyte chemotactic protein (MCP)-1, leptin, SAA (serum amyloid A), hepatocyte growth factor (HGF), insulin, lipocalin-2, TNF-α, B cell activating factor (BAFF: belonging to the tumor necrosis factor family), resistin, plasminogen activator inhibitor

After analyzing all of the results, it was determined that all items measured except for NK activity revealed no significant difference between 3M ON and 3M OFF conditions. NK activity was measured as the E/T ratio and was calculated to be 10:1 and 20:1. With an actual NK activity of 20:1, there was a tendency of NK activity to increase in the 3M ON period and decrease in the 3M OFF period [15]. Additionally, the relative changes in NK activity set before as 1.0 for individual values of 3M ON and 3M OFF periods revealed a significant increase during the 3M ON periods and a decrease during the 3M OFF periods in the 20:1 E/T ratio as shown in **Figure 5C** [with an E/T ratio of 10:1, similar significant results were obtained in the

Taken together, it was shown that NCPDIAC cause enhancement of NK cell activity even in living homes. The apparatus utilized for establishing NCPDIAC may possess advantages in reducing the occurrence of cancers, as well as reducing signs

For adipokines and cytokines related to oxidative stress, there were no significant changes observed [16]. However, with the exception of one case, serum amyloid A (SAA) levels decreased significantly during the ON trials (data not shown) [16]. Considering that SAA is an acute phase-reactive protein like C-reactive protein (CRP), this observed decrease may indicate a prevention of cardiovascular and atherosclerotic changes, since an increase in high-sensitive CRP is associated with

Initial assessment of the biological effects of NCPDIAC began with evaluations

of 2.5H stay experiments since similar experiments had yet to be reported and investigations should involve collaboration with HV. Thus, our initial 2.5H stay experiments demonstrated a small but significant increase in IL-2. Additionally, it was assumed that there were no adverse effects during the 2.5H period. These experiments were then followed with 2W nightly stay experiments. However, to set up NCPDIAC in the homes of HV was very difficult since sleeping and/or living rooms required alterations. Thus, we used a dormitory that belonged to the collaborating house company where many trainees (company employees) would be staying during the 3M period. With these experiments, we found an enhancement of NK cell activity during the 2W nightly stay. Additionally, there were no adverse effects

or symptoms associated with virus-infected diseases such as influenza [15].

3M ON (p = 0.017) and 3M OFF (p = 0.012) periods] [15].

subsequent detection of these events [16].

as determined by the various parameters examined.

*DOI: http://dx.doi.org/10.5772/intechopen.79934*

(PAI)-1, and IL-8 [16].

#### *Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions DOI: http://dx.doi.org/10.5772/intechopen.79934*

All volunteers had built or renovated their residential homes prior to being recruited to this project and agreed to set up an NCPDIAC device for the experiments [15].

This study was approved by the institutional ethical committee (#854) and samples were only taken from HV who provided written informed consent.

The clinical parameters measured were similar to those determined for the 2W nightly stay experiments with additional cytokines being measured. Twentynine cytokines were measured using the Luminex 26 Cytokine Plex Kit Human Cytokine/Chemokine Panel (MPXHCYTO60KPMX26, Merck Millipore, Billerica, MA) [15]. Additionally, adipokines and cytokines related to oxidative stress in serum from 16 ON and 13 OFF trials were measured using the Human Adipokine Magnetic 14-Plex Panel with Luminex instruments (Bio-rad, Hercules, CA, USA). Fourteen of the cytokines examined comprised IL-1β, IL-10, IL-6, monocyte chemotactic protein (MCP)-1, leptin, SAA (serum amyloid A), hepatocyte growth factor (HGF), insulin, lipocalin-2, TNF-α, B cell activating factor (BAFF: belonging to the tumor necrosis factor family), resistin, plasminogen activator inhibitor (PAI)-1, and IL-8 [16].

After analyzing all of the results, it was determined that all items measured except for NK activity revealed no significant difference between 3M ON and 3M OFF conditions. NK activity was measured as the E/T ratio and was calculated to be 10:1 and 20:1. With an actual NK activity of 20:1, there was a tendency of NK activity to increase in the 3M ON period and decrease in the 3M OFF period [15]. Additionally, the relative changes in NK activity set before as 1.0 for individual values of 3M ON and 3M OFF periods revealed a significant increase during the 3M ON periods and a decrease during the 3M OFF periods in the 20:1 E/T ratio as shown in **Figure 5C** [with an E/T ratio of 10:1, similar significant results were obtained in the 3M ON (p = 0.017) and 3M OFF (p = 0.012) periods] [15].

Taken together, it was shown that NCPDIAC cause enhancement of NK cell activity even in living homes. The apparatus utilized for establishing NCPDIAC may possess advantages in reducing the occurrence of cancers, as well as reducing signs or symptoms associated with virus-infected diseases such as influenza [15].

For adipokines and cytokines related to oxidative stress, there were no significant changes observed [16]. However, with the exception of one case, serum amyloid A (SAA) levels decreased significantly during the ON trials (data not shown) [16]. Considering that SAA is an acute phase-reactive protein like C-reactive protein (CRP), this observed decrease may indicate a prevention of cardiovascular and atherosclerotic changes, since an increase in high-sensitive CRP is associated with subsequent detection of these events [16].

#### **7. Discussion**

*Charged Particles*

**6. 3M ON and OFF trials**

results have previously been reported [15].

their homes without NCPDIAC [15].

Finally, long-term (3-month, or 3M) stay experiments were performed and the

Following the aforementioned experiments (2.5H and 2W nightly stay experiments and evaluations), it seemed that NCPDIAC stimulated NK activity with no adverse effects on HV [15]. The increase in NK activity could be accounted for by the short-term, albeit slight, increase in IL-2, which may activate NK cells during the 2W period. The following experiments were then applied in living homes. The homes of seven volunteers were modified for NCPDIAC, targeting mainly sleeping rooms and living rooms. A switch panel approximately 4 × 22 × 29 cm in size had been fitted. Volunteers would then switch this panel ON and OFF every 3 months. Then, prior to and following every 3-month ON or OFF living period, clinical measurements including NK cell activity and others (as measured in the 2W nightly stay experiments) were performed. A total of 16 OFF to ON (3M ON) and 13 ON to

Blood samples were taken just before switching ON or OFF. Thus, during the OFF to ON (3M ON) period, HV stayed at home with NCPDIAC (sleeping room and living room). During the ON to OFF (3M OFF) period, HV occupied rooms in

All seven HV comprised Japanese living in Japan and were asked to join this project by first-class registered architects who are colleagues of the authors. The average age of the volunteers was 54.86 ± 9.15 years and included five males and two females.

*(A) Seven healthy volunteers utilized sleeping and living rooms with NCPDIAC in their homes. During this period, the occupants themselves would switch NCPDIAC ON and OFF. Sixteen trials of OFF to ON (3M ON) and thirteen trials of ON to OFF (3M OFF) were executed. (B) The levels of positively and negatively charged particles in the representative six rooms including the homes of HV where the NCPDIAC apparatus was set were measured using as ion counter (EM-1000, Eco Holistic Inc., Suita, Japan) during the OFF and ON periods. During the OFF period, there was no difference between positively and negatively charged particles; however, during the ON period, the level of positively charged particles was significantly reduced, thereby establishing a difference between the number of positively and negatively charged particles. Statistical analyses were performed using the student T test. (C) Changes in actual NK cell activity prior to and following the 3M ON and 3M OFF periods. Values tended to increase during the 3M ON and decrease during the 3M OFF periods. (D) Relative NK cell activities with NK cell activity previously set to 1.0. There was a significant increase in relative NK cell activity during the 3M ON period and a significant decrease during the 3M OFF* 

*period. Statistical analyses were performed using the student T test.*

OFF (3M OFF) trials were performed as shown in **Figure 5A** [15].

**86**

**Figure 5.**

Initial assessment of the biological effects of NCPDIAC began with evaluations of 2.5H stay experiments since similar experiments had yet to be reported and investigations should involve collaboration with HV. Thus, our initial 2.5H stay experiments demonstrated a small but significant increase in IL-2. Additionally, it was assumed that there were no adverse effects during the 2.5H period. These experiments were then followed with 2W nightly stay experiments. However, to set up NCPDIAC in the homes of HV was very difficult since sleeping and/or living rooms required alterations. Thus, we used a dormitory that belonged to the collaborating house company where many trainees (company employees) would be staying during the 3M period. With these experiments, we found an enhancement of NK cell activity during the 2W nightly stay. Additionally, there were no adverse effects as determined by the various parameters examined.

#### *Charged Particles*

During these studies employing HV, in vitro assays were performed. From the in vitro experiments, we confirmed the increased NK cell activity and slight immune-stimulatory effects of negatively charged particles. Again, there were no adverse effects to the human body as determined by the various parameters examined.

These results encouraged us to apply NCPDIAC in the actual homes (sleeping and living rooms) of volunteers. Following approval by the institutional ethical committee and obtaining written informed consent, the homes of 7 HV were set up with NCPDIAC. Results showed that the 3M ON period enhanced and the 3M OFF period reduced NK cell activity.

It was important to proceed with these experiments in a methodical, step-bystep fashion as careful consideration and confirmation of the results were required to preserve the health and well-being of the volunteers.

#### **8. Conclusions**

In this chapter, the establishment of NCPDIAC and results of the experiments for short-term (2.5H, 2.5-h), mid-term (2W, 2-week), and relatively long-term (3M, 3-month in actual living homes) stays were shown.

Long-term monitoring in actual living homes comprising 6-month, 12-month, and 3-year duration periods has commenced, and the results will be reported in due course.

It is extremely important that appropriate living environments are investigated and created that mitigate or prevent the onset of many diseases such as cancers and virus-infected diseases. In addition to NCPDIAC, other devices that maintain stable temperature as well as increase air tightness may prevent acute accidents related to cardiovascular events. Our recent long-term trial monitoring experiments have included these devices. We hope that these health-promoting indoor environments enable people to live healthier and happier lives.

#### **Notes and Acknowledgements**

All experiments described in this chapter were approved by the Ethics Committee of the Kawasaki Medical School, Kurashiki, Japan.

All authors thank Ms. Shoko Yamamoto and Tamayo Hatayama for their skillful technical assistance.

#### **Conflicts of interest**

For the short-term (2.5H) and mid-term (2W) stay experiments and the in vitro experiments, the Department of Hygiene, Kawasaki Medical School, obtained research funding from SEKISUI HOUSE Ltd., Osaka, Japan. For the long-term (3M) stay experiments, the Department of Hygiene obtained research funding from Yamada SXL Hone Co. Ltd., Takasaki, Japan. Additionally, the extraporous charcoal paints were provided by Artech Kohboh, Co. Ltd., Omura, Nagasaki, Japan.

**89**

**Author details**

Megumi Maeda3

Suni Lee1

provided the original work is properly cited.

, Yasumitsu Nishimura1

Prefectural University of Hiroshima, Shobara, Japan

Life Science, Okayama University, Okayama, Japan

, Nagisa Sada1,4, Kei Yoshitome1

1 Department of Hygiene, Kawasaki Medical School, Kurashiki, Japan

and Pharmaceutical Sciences, Okayama University, Okayama, Japan

\*Address all correspondence to: takemi@med.kawasaki-m.ac.jp

2 Department of Life Science, Faculty of Life and Environmental Science,

© 2018 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,

3 Department of Biofunctional Chemistry, Graduate School of Environmental and

4 Department of Biophysical Chemistry, Graduate School of Medicine, Dentistry

, Naoko Kumagai-Takei1

, Hidenori Matsuzaki<sup>2</sup>

\*

and Takemi Otsuki1

,

*Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions*

*DOI: http://dx.doi.org/10.5772/intechopen.79934*

*Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions DOI: http://dx.doi.org/10.5772/intechopen.79934*

### **Author details**

*Charged Particles*

examined.

**8. Conclusions**

course.

period reduced NK cell activity.

to preserve the health and well-being of the volunteers.

3-month in actual living homes) stays were shown.

enable people to live healthier and happier lives.

**Notes and Acknowledgements**

technical assistance.

**Conflicts of interest**

During these studies employing HV, in vitro assays were performed. From the in vitro experiments, we confirmed the increased NK cell activity and slight immune-stimulatory effects of negatively charged particles. Again, there were no adverse effects to the human body as determined by the various parameters

These results encouraged us to apply NCPDIAC in the actual homes (sleeping and living rooms) of volunteers. Following approval by the institutional ethical committee and obtaining written informed consent, the homes of 7 HV were set up with NCPDIAC. Results showed that the 3M ON period enhanced and the 3M OFF

It was important to proceed with these experiments in a methodical, step-bystep fashion as careful consideration and confirmation of the results were required

In this chapter, the establishment of NCPDIAC and results of the experiments for short-term (2.5H, 2.5-h), mid-term (2W, 2-week), and relatively long-term (3M,

Long-term monitoring in actual living homes comprising 6-month, 12-month, and 3-year duration periods has commenced, and the results will be reported in due

It is extremely important that appropriate living environments are investigated and created that mitigate or prevent the onset of many diseases such as cancers and virus-infected diseases. In addition to NCPDIAC, other devices that maintain stable temperature as well as increase air tightness may prevent acute accidents related to cardiovascular events. Our recent long-term trial monitoring experiments have included these devices. We hope that these health-promoting indoor environments

All experiments described in this chapter were approved by the Ethics

All authors thank Ms. Shoko Yamamoto and Tamayo Hatayama for their skillful

For the short-term (2.5H) and mid-term (2W) stay experiments and the in vitro

experiments, the Department of Hygiene, Kawasaki Medical School, obtained research funding from SEKISUI HOUSE Ltd., Osaka, Japan. For the long-term (3M) stay experiments, the Department of Hygiene obtained research funding from Yamada SXL Hone Co. Ltd., Takasaki, Japan. Additionally, the extraporous charcoal

paints were provided by Artech Kohboh, Co. Ltd., Omura, Nagasaki, Japan.

Committee of the Kawasaki Medical School, Kurashiki, Japan.

**88**

Suni Lee1 , Yasumitsu Nishimura1 , Naoko Kumagai-Takei1 , Hidenori Matsuzaki<sup>2</sup> , Megumi Maeda3 , Nagisa Sada1,4, Kei Yoshitome1 and Takemi Otsuki1 \*

1 Department of Hygiene, Kawasaki Medical School, Kurashiki, Japan

2 Department of Life Science, Faculty of Life and Environmental Science, Prefectural University of Hiroshima, Shobara, Japan

3 Department of Biofunctional Chemistry, Graduate School of Environmental and Life Science, Okayama University, Okayama, Japan

4 Department of Biophysical Chemistry, Graduate School of Medicine, Dentistry and Pharmaceutical Sciences, Okayama University, Okayama, Japan

\*Address all correspondence to: takemi@med.kawasaki-m.ac.jp

© 2018 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**

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[2] Redlich CA, Sparer J, Cullen MR. Sick-building syndrome. Lancet. 1997;**349**:1013-1016

[3] Menzies D, Bourbeau J. Buildingrelated illnesses. The New England Journal of Medicine. 1997;**337**:1524-1531

[4] Saeki K, Obayashi K, Tone N, Kurumatani N. A warmer indoor environment in the evening and shorter sleep onset latency in winter: The HEIJO-KYO study. Physiology & Behavior. 2015;**149**:29-34. DOI: 10.1016/j.physbeh.2015.05.022

[5] Saeki K, Obayashi K, Tone N, Kurumatani N. Daytime cold exposure and salt intake based on nocturnal urinary sodium excretion: A crosssectional analysis of the HEIJO-KYO study. Physiology & Behavior. 2015;**152**:300-306. DOI: 10.1016/j. physbeh.2015.10.015

[6] Krueger AP, Reed EJ. Biological impact of small air ions. Science. 1976;**193**:1209-1213

[7] Misiaszek J, Gray F, Yates A. The calming effects of negative air ions on manic patients: A pilot study. Biological Psychiatry. 1987;**22**:107-110

[8] Weber C, Henne B, Loth F, Schoenhofen M, Falkenhagen D. Development of cationically modified cellulose adsorbents for the removal of endotoxins. ASAIO Journal. 1995;**41**:M430-M434

[9] Yamada R, Yanoma S, Akaike M, Tsuburaya A, Sugimasa Y, Takemiya S, Motohashi H, Rino Y, Takanashi Y, Imada T. Water-generated negative

air ions activate NK cell and inhibit carcinogenesis in mice. Cancer Letters. 2006;**239**:190-197

[10] Takahashi K, Otsuki T, Mase A, Kawado T, Kotani M, Ami K, et al. Negatively-charged air conditions and responses of the human psychoneuro-endocrino-immune network. Environment International. 2008; **34**:765-772. DOI: 10.1016/j. envint.2008.01.003

[11] Takahashi K, Otsuki T, Mase A, Kawado T, Kotani M, Nishimura Y, et al. Two weeks of permanence in negatively-charged air conditions causes alteration of natural killer cell function. International Journal of Immunopathology and Pharmacology. 2009;**22**:333-342

[12] Domzig W, Stadler BM, Herberman RB. Interleukin 2 dependence of human natural killer (NK) cell activity. Journal of Immunology. 1983;**130**:1970-1973

[13] Ishikawa H, Saeki T, Otani T, Suzuki T, Shimozuma K, Nishino H, Fukuda S, Morimoto K. Aged garlic extract prevents a decline of NK cell number and activity in patients with advanced cancer. The Journal of Nutrition. 2006;**136**:816S-820S

[14] Nishimura Y, Takahashi K, Mase A, Kotani M, Ami K, Maeda M, Shirahama T, Lee S, Matsuzaki H, Kumagai-Takei N, Yoshitome K, Otsuki T. Exposure to negatively charged-particle dominant airconditions on human lymphocytes in vitro activates immunological responses. Immunobiology. 2015;**220**:1359-1368. DOI: 10.1016/j.imbio.2015.07.006

[15] Nishimura Y, Takahashi K, Mase A, Kotani M, Ami K, Maeda M, Shirahama T, Lee S, Matsuzaki H, Kumagai-Takei N, Yoshitome K, Otsuki T. Enhancement of NK cell

**91**

*Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions*

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[16] Lee S, Yamamoto S, Nishimura Y, Matsuzaki H, Yoshitome K, Hatayama T,

Ikeda M, Yu M, Sada N, Kumagai-Takei N, Otsuki T. Decrease in serum amyloid A (SAA) protein following threemonth stays under negatively charged particle-dominant indoor air conditions.

Biomedical and Environmental

Sciences. 2018;**31**:335-342. DOI: 10.3967/

pone.0132373

bes2018.044

*Biological Effects of Negatively Charged Particle-Dominant Indoor Air Conditions DOI: http://dx.doi.org/10.5772/intechopen.79934*

cytotoxicity induced by long-term living in negatively charged-particle dominant indoor air-conditions. PLoS One. 2015;**10**:e0132373. DOI: 10.1371/journal. pone.0132373

[16] Lee S, Yamamoto S, Nishimura Y, Matsuzaki H, Yoshitome K, Hatayama T, Ikeda M, Yu M, Sada N, Kumagai-Takei N, Otsuki T. Decrease in serum amyloid A (SAA) protein following threemonth stays under negatively charged particle-dominant indoor air conditions. Biomedical and Environmental Sciences. 2018;**31**:335-342. DOI: 10.3967/ bes2018.044

**90**

*Charged Particles*

**References**

[1] Hodgson MJ. Clinical diagnosis and management of building-related illness and the sick-building syndrome. Occupational Medicine. 1989;**4**:93-606

air ions activate NK cell and inhibit carcinogenesis in mice. Cancer Letters.

[10] Takahashi K, Otsuki T, Mase A, Kawado T, Kotani M, Ami K, et al. Negatively-charged air conditions and responses of the human psychoneuro-endocrino-immune network. Environment International. 2008; **34**:765-772. DOI: 10.1016/j. envint.2008.01.003

[11] Takahashi K, Otsuki T, Mase A, Kawado T, Kotani M, Nishimura Y, et al. Two weeks of permanence in negatively-charged air conditions causes alteration of natural killer cell function. International Journal of Immunopathology and Pharmacology.

[12] Domzig W, Stadler BM, Herberman RB. Interleukin 2 dependence of human natural killer (NK) cell activity. Journal of Immunology. 1983;**130**:1970-1973

[13] Ishikawa H, Saeki T, Otani T, Suzuki T, Shimozuma K, Nishino H, Fukuda S, Morimoto K. Aged garlic extract prevents a decline of NK cell number and activity in patients with advanced cancer. The Journal of Nutrition. 2006;**136**:816S-820S

[14] Nishimura Y, Takahashi K, Mase A, Kotani M, Ami K, Maeda M, Shirahama T, Lee S, Matsuzaki H, Kumagai-Takei N, Yoshitome K, Otsuki T. Exposure to negatively charged-particle dominant airconditions on human lymphocytes in vitro activates immunological responses. Immunobiology. 2015;**220**:1359-1368. DOI: 10.1016/j.imbio.2015.07.006

[15] Nishimura Y, Takahashi K, Mase A,

Kotani M, Ami K, Maeda M, Shirahama T, Lee S, Matsuzaki H, Kumagai-Takei N, Yoshitome K, Otsuki T. Enhancement of NK cell

2006;**239**:190-197

2009;**22**:333-342

[2] Redlich CA, Sparer J, Cullen MR. Sick-building syndrome. Lancet.

[3] Menzies D, Bourbeau J. Buildingrelated illnesses. The New England Journal of Medicine. 1997;**337**:1524-1531

[4] Saeki K, Obayashi K, Tone N, Kurumatani N. A warmer indoor environment in the evening and shorter sleep onset latency in winter: The HEIJO-KYO study. Physiology & Behavior. 2015;**149**:29-34. DOI: 10.1016/j.physbeh.2015.05.022

[5] Saeki K, Obayashi K, Tone N, Kurumatani N. Daytime cold exposure and salt intake based on nocturnal urinary sodium excretion: A crosssectional analysis of the HEIJO-KYO study. Physiology & Behavior. 2015;**152**:300-306. DOI: 10.1016/j.

[6] Krueger AP, Reed EJ. Biological impact of small air ions. Science.

[7] Misiaszek J, Gray F, Yates A. The calming effects of negative air ions on manic patients: A pilot study. Biological

Psychiatry. 1987;**22**:107-110

[8] Weber C, Henne B, Loth F, Schoenhofen M, Falkenhagen D. Development of cationically modified cellulose adsorbents for the removal of endotoxins. ASAIO Journal.

[9] Yamada R, Yanoma S, Akaike M, Tsuburaya A, Sugimasa Y, Takemiya S, Motohashi H, Rino Y, Takanashi Y, Imada T. Water-generated negative

physbeh.2015.10.015

1976;**193**:1209-1213

1995;**41**:M430-M434

1997;**349**:1013-1016

### *Edited by Malek Maaza and Mahmoud Izerrouken*

A charged particle is a particle that carries an electric charge and can be discussed in many aspects. This book focuses on cutting-edge and important research topics such as flavor physics to search for new physics via charged particles that appear in different extensions of the standard model, as well as the analysis of ultra-high energy muons using the pair-meter technique. Also included in this book are the idea of the Eloisatron to PeVatron, the important research field of electrostatic waves in magnetized electron/positron plasmas, and the application of charge bodies.

Published in London, UK © 2019 IntechOpen © underworld111 / iStock

Charged Particles

Charged Particles

*Edited by Malek Maaza and Mahmoud Izerrouken*