**3.1 Mathematical description of the state of the plasma and its model**

Due to the variety of processes taking place in a spatially inhomogeneous plasma, an analytical description of a real plasma in the general case is very difficult. Therefore, simplified plasma models are usually considered, stipulating the conditions under which a real plasma can be close to its accepted model.

The state of a real plasma at an arbitrary pressure is determined by a) the concentration of particles of all kinds *N* (the number of particles per unit volume);

second group of such structures can be presented by the single- and double-ridged waveguides. Thus, an application of the above-mentioned electrodynamic structures enables improving the process of exciting plasma and enhancing the efficiency

*Image of the curve L*<sup>0</sup> *L*<sup>4</sup> *along which the electric field value is calculated.*

*Schematic image of the waveguide structure with a metal insert having a height h.*

*Microwave Heating - Electromagnetic Fields Causing Thermal and Non-Thermal Effects*

of transformation of energy.

**Figure 4.**

**Figure 5.**

**128**

b) their speed distribution functions *Ni*ð Þ *n* ; c) the population of the excited levels *Nk* (the number of particles per unit volume, excited in the state *k*); d) the spatial distribution of these quantities.

*NeN*<sup>i</sup> *Na*

*DOI: http://dx.doi.org/10.5772/intechopen.97167*

action with the same temperature *T*.

Maxwell function

unsuitable.

separately.

range.

**131**

may be written as

¼ 2

*Microwave Heating of Low-Temperature Plasma and Its Application*

*Ni*ð Þ¼ *n* 4*pNi*

ð Þ 2*p me* 3 2 *<sup>h</sup>*<sup>3</sup> ð Þ *kT* <sup>3</sup>

<sup>2</sup> *U*

where *me* – the electron mass; *E*<sup>i</sup> – ionization energy; *U*ið Þ *T* and *Ua*ð Þ *T* are the sums over the states of ions and atoms; g = 2 – the statistical weight of electrons. In plasma of a complex chemical composition, equation (3) is valid for the ions of each chemical element; chemical reactions can occur there, dissociation and recombination of molecules can occur. All these reactions obey the law of mass

The distribution of particles of any kind *i* by velocity *ν* is expressed by the

*Mi* <sup>2</sup>*p kT* � �<sup>3</sup> 2

∞ð

0

*<sup>p</sup>* <sup>¼</sup> <sup>X</sup> *i*

Local thermodynamic equilibrium is a state of a plasma in which all distribution functions are in equilibrium, except for one concerning radiation: there is no equilibrium of optical processes, as a result of which the Planck formula turns out to be

LTE is typical for most stationary plasmas obtained under laboratory conditions. Under conditions of LTE plasma, the detailed equilibrium with respect to optical transitions is violated; therefore, it is advisable to consider radiation and absorption

Plasma, in which radiation of a given wavelength is practically not absorbed, is

optically thin for this radiation. The radiation intensity *Jki* of an optically thin plasma in the LTE state within the spectral line with the following frequency *nki*

*Jki* <sup>¼</sup> *NkAkihnki* <sup>¼</sup> *<sup>N</sup>*0*Akihnki* exp � *Ek*

LTE plasma, described by a single parameter *T*, can exist in a limited pressure

The numerical model of the optical radiation source can be a spherical or cylin-

drical flask made of transparent anhydrous quartz glass filled with a metered amount of sulfur � 1 ... 3 mg (and it is also possible to introduce impurities, for example, CaBr2 or indium iodide InI, etc.) and a buffer gas (argon, neon, krypton) under the pressure of � 45 … 170 torrs. By changing the composition of the lamp bulb filling, it is possible to carry out theoretical studies of the output spectral characteristics of optical radiation and their dependence in the optical wavelength

tion) with velocities ranging from n to n + dn; *Ni* – concentration equal to

*Ni* ¼

The pressure *p* in the plasma is found from the equation of state

where *Mi* – the mass of particles; *Ni*ð Þ *n* – the number of particles (concentra-

*Ua*ð Þ *<sup>T</sup>* exp �*E*<sup>i</sup>

exp � *Mi <sup>n</sup>*<sup>2</sup>

*kT* � �, (3)

<sup>2</sup>*kT* � �, (4)

*Ni*ð Þ*v dv:* (5)

*NikT:* (6)

*kT* � �*:* (7)

It is extremely difficult to obtain information about all the listed characteristics since theoretical studies of the state of plasma-like media require the compilation and solution of a system of equations connecting the indicated quantities with external conditions.

The basic equations describing the nonlinear states of plasma have limitations, primarily related to the possibility of obtaining their solutions. Too simple mathematical plasma models also have limited capabilities due to their inability to adequately reflect the behavior of real plasma. The strongest difference between the real state of the plasma and its mathematical description is observed in the so-called boundary zones, where the plasma passes from one physical state to another (for example, from a state with a low degree of ionization to a state with a high degree of ionization). In this case, the plasma cannot be described using simple smooth functions and a probabilistic approach is required to describe it. Effects such as a spontaneous change in the state of plasma are a consequence of the complex nonlinear interaction of charged particles that make up it. Therefore, to describe plasma, models of the state of the plasma are used, which relate the values of its main parameters, and, therefore, determine its basic properties and behavior.

#### *3.1.1 Local thermodynamic equilibrium (LTE) model*

To describe the low–temperature plasma in the bulb of an electrodeless sulfur lamp, which is formed under the action of an electromagnetic field, one can use the LTE model. This model makes it possible to qualitatively and quantitatively describe the continuous emission spectrum of the lamp, as well as the distribution of the main physical quantities of sulfuric plasma.

According to the LTE model, the temperature in different elements of the volume of the medium is different, there is a radiation flux outward (the radiation field is anisotropic), but for each element of the volume of the medium, the Boltzmann and Maxwell distributions, as well as the Saha formula, are valid. Moreover, all of them for a given volume include the same local temperature value, which is the same for all types of particles.

Basic equations describing the LTE model:

– the number of atoms or ions in an arbitrary excited state *k* (population of the state *k*) is determined by the Boltzmann formula

$$N\_k = N\_0 \frac{\mathcal{g}\_k}{\mathcal{g}\_0} \exp\left(\frac{-E\_k}{kT}\right) = N \frac{\mathcal{g}\_k}{U} \exp\left(\frac{-E\_k}{kT}\right),\tag{1}$$

where *N*<sup>0</sup> – the population of the ground state; *g*<sup>0</sup> – the statistical weight of this state; *gk* – statistical weight of the excited state; *Ek* – the energy of the excited state, measured from the ground level.

Statistical sums over all energy levels *En* of the corresponding ions (atoms) is equal

$$U = \sum\_{n} \mathbf{g}\_{n} \exp\left(-\frac{E\_{n}}{kT}\right). \tag{2}$$

In the case of a single ionization of a gas, the concentrations of atoms, ions, and electrons are related to each other by the Saha formula

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b) their speed distribution functions *Ni*ð Þ *n* ; c) the population of the excited levels *Nk* (the number of particles per unit volume, excited in the state *k*); d) the spatial

*Microwave Heating - Electromagnetic Fields Causing Thermal and Non-Thermal Effects*

It is extremely difficult to obtain information about all the listed characteristics since theoretical studies of the state of plasma-like media require the compilation and solution of a system of equations connecting the indicated quantities with

The basic equations describing the nonlinear states of plasma have limitations, primarily related to the possibility of obtaining their solutions. Too simple mathematical plasma models also have limited capabilities due to their inability to adequately reflect the behavior of real plasma. The strongest difference between the real state of the plasma and its mathematical description is observed in the so-called boundary zones, where the plasma passes from one physical state to another (for example, from a state with a low degree of ionization to a state with a high degree of ionization). In this case, the plasma cannot be described using simple smooth functions and a probabilistic approach is required to describe it. Effects such as a spontaneous change in the state of plasma are a consequence of the complex nonlinear interaction of charged particles that make up it. Therefore, to describe plasma, models of the state of the plasma are used, which relate the values of its main parameters, and, therefore, determine its basic properties and behavior.

To describe the low–temperature plasma in the bulb of an electrodeless sulfur lamp, which is formed under the action of an electromagnetic field, one can use the

LTE model. This model makes it possible to qualitatively and quantitatively describe the continuous emission spectrum of the lamp, as well as the distribution

According to the LTE model, the temperature in different elements of the volume of the medium is different, there is a radiation flux outward (the radiation field is anisotropic), but for each element of the volume of the medium, the

Boltzmann and Maxwell distributions, as well as the Saha formula, are valid. Moreover, all of them for a given volume include the same local temperature value,

– the number of atoms or ions in an arbitrary excited state *k* (population of the

where *N*<sup>0</sup> – the population of the ground state; *g*<sup>0</sup> – the statistical weight of this state; *gk* – statistical weight of the excited state; *Ek* – the energy of the excited state,

Statistical sums over all energy levels *En* of the corresponding ions (atoms) is

*gn* exp � *En*

In the case of a single ionization of a gas, the concentrations of atoms, ions, and

*kT* � �

<sup>¼</sup> *<sup>N</sup> gk*

*<sup>U</sup>* exp �*Ek kT* � �

, (1)

*:* (2)

distribution of these quantities.

*3.1.1 Local thermodynamic equilibrium (LTE) model*

of the main physical quantities of sulfuric plasma.

which is the same for all types of particles. Basic equations describing the LTE model:

state *k*) is determined by the Boltzmann formula

*Nk* ¼ *N*<sup>0</sup>

measured from the ground level.

equal

**130**

*gk g*0 exp �*Ek kT* � �

*<sup>U</sup>* <sup>¼</sup> <sup>X</sup> *n*

electrons are related to each other by the Saha formula

external conditions.

$$\frac{N\_c N\_i}{N\_a} = 2 \frac{(2p \ m\_e)^{\frac{3}{2}}}{h^3} (kT)^{\frac{3}{2}} \frac{U}{U\_a(T)} \exp\left(\frac{-E\_i}{kT}\right),\tag{3}$$

where *me* – the electron mass; *E*<sup>i</sup> – ionization energy; *U*ið Þ *T* and *Ua*ð Þ *T* are the sums over the states of ions and atoms; g = 2 – the statistical weight of electrons.

In plasma of a complex chemical composition, equation (3) is valid for the ions of each chemical element; chemical reactions can occur there, dissociation and recombination of molecules can occur. All these reactions obey the law of mass action with the same temperature *T*.

The distribution of particles of any kind *i* by velocity *ν* is expressed by the Maxwell function

$$N\_i(n) = 4pN\_i \left(\frac{M\_i}{2p\,kT}\right)^{\frac{3}{2}} \exp\left(-\frac{M\_i n^2}{2kT}\right),\tag{4}$$

where *Mi* – the mass of particles; *Ni*ð Þ *n* – the number of particles (concentration) with velocities ranging from n to n + dn; *Ni* – concentration equal to

$$N\_i = \bigcap\_{0}^{\infty} N\_i(v) \, dv. \tag{5}$$

The pressure *p* in the plasma is found from the equation of state

$$p = \sum\_{i} \mathbf{N}\_{i} \mathbf{k} \mathbf{T}. \tag{6}$$

Local thermodynamic equilibrium is a state of a plasma in which all distribution functions are in equilibrium, except for one concerning radiation: there is no equilibrium of optical processes, as a result of which the Planck formula turns out to be unsuitable.

LTE is typical for most stationary plasmas obtained under laboratory conditions. Under conditions of LTE plasma, the detailed equilibrium with respect to optical transitions is violated; therefore, it is advisable to consider radiation and absorption separately.

Plasma, in which radiation of a given wavelength is practically not absorbed, is optically thin for this radiation. The radiation intensity *Jki* of an optically thin plasma in the LTE state within the spectral line with the following frequency *nki* may be written as

$$J\_{ki} = N\_k A\_{ki} h n\_{ki} = N\_0 A\_{ki} h n\_{ki} \exp\left(-\frac{E\_k}{kT}\right). \tag{7}$$

LTE plasma, described by a single parameter *T*, can exist in a limited pressure range.

The numerical model of the optical radiation source can be a spherical or cylindrical flask made of transparent anhydrous quartz glass filled with a metered amount of sulfur � 1 ... 3 mg (and it is also possible to introduce impurities, for example, CaBr2 or indium iodide InI, etc.) and a buffer gas (argon, neon, krypton) under the pressure of � 45 … 170 torrs. By changing the composition of the lamp bulb filling, it is possible to carry out theoretical studies of the output spectral characteristics of optical radiation and their dependence in the optical wavelength

range on the microwave pump power, the temperature distribution inside the bulb, plasma electrical conductivity, etc.

These fields satisfy Maxwell's equations. The equations of motion of a charged

It is convenient to write these equations in the following dimensionless variables

!*=c*, *E* ! *<sup>n</sup>* � *eE* !

> *n k* ! *<sup>n</sup>* \_ *r* ! *E* ! *n* � �,

*eE* ! *mcω*<sup>0</sup>

! � *ω*<sup>0</sup> *r*

Eq. (9) can be conveniently supplemented with the energy equation

*<sup>γ</sup>*\_ <sup>¼</sup> *<sup>P</sup>* ! *γ*

*En ω<sup>n</sup>* � *k*

! *n r* ! � *ωnτ*. Let us introduce some auxiliary characteristic of the particle, which we will further call the partial energy of the particle, which satisfies the following equation

> Re *i*E ! *ne iψ<sup>n</sup>* � � �<sup>X</sup>

*<sup>γ</sup>*\_*<sup>n</sup>* <sup>¼</sup> *<sup>ω</sup><sup>n</sup>* \_ *r* ! *E* ! *n*

From the definition of this partial energy, it follows that it determines the value of the energy that a particle would have if it moved only in the field of one *n*-th electromagnetic wave. Using the definition of this partial energy, we obtain from

A possible dispersion diagram of three waves interacting with particles is shown

in **Figure 8**. This figure shows both the dispersion characteristics of the waves themselves (*ω*0,*ω*1, *ω*2; *k*0, *k*1, *k*2) and the dispersion characteristics of the combination waves with which the Cherenkov resonance of plasma particles (*vph*1; *vph*2) occurs. In the general case, Eqs. (11) and (12) together with the integral (13) can be studied only by numerical methods. To obtain analytical results, we will assume that the force parameter of each of the waves acting on the particle is small. In this case, all the characteristics of a particle (its energy, momentum, coordinate, velocity) can be represented as a sum of slowly varying and rapidly changing quantities

> *P* ! ¼ *P* ! þ ~ *P* ! ,

*γ<sup>n</sup>* ¼ *γ<sup>n</sup>* þ ~*γn:*

*n*

*k* ! *n ωn*

*γ<sup>n</sup>* ¼ *C* !

Substituting fields (8) into Eqs. (9) and (10) and using these dimensionless variables, we can obtain the following, convenient for further analysis, equations

> ! *n* \_ *r* ! � � þ<sup>X</sup>

*:* (9)

� *dP* !

*:* (10)

� �*:* (12)

*:* (13)

(14)

*<sup>n</sup>=mcω<sup>n</sup>* – is the wave force

*=dτ*, *τ* � *ω*0*t* ,

(11)

*P* !

*dP* ! *dt* <sup>¼</sup> *eE* ! þ *e c v* !*H* h i !

for both dependent and independent variables: *<sup>ω</sup><sup>n</sup>* <sup>¼</sup> *<sup>ω</sup>n=ω*0, \_

*Microwave Heating of Low-Temperature Plasma and Its Application*

*nc=ω*0, *r*

particle in fields (8) have the traditional form

*DOI: http://dx.doi.org/10.5772/intechopen.97167*

!*=c*, *k* ! *<sup>n</sup>* � *k* !

> \_ *P* !

*<sup>γ</sup>*\_ <sup>¼</sup> *<sup>P</sup>* ! *γ* X *n ωnE* ! *n*,

Eqs. (11) and (12) the following integral of motion

*P* ! �<sup>X</sup> *n*

! *ne<sup>i</sup>ψ<sup>n</sup>* � �; *<sup>ψ</sup><sup>n</sup>* � *<sup>k</sup>*

*<sup>n</sup>* ¼ Re E

<sup>¼</sup> <sup>X</sup> *n*

*P* ! � *P* ! *<sup>=</sup>mc*, \_ *r* ! ¼ *v*

parameter.

where *E* !

**133**

Buffer gas (argon) – serves for initial ionization and obtaining a glow discharge (gas pressure is set at the initial stage). We obtain the dependence of the dynamics of changes in pressure in the flask on temperature.

#### **3.2 Stochastic plasma heating**
