*3.2.1 Introduction*

By stochastic heating, we mean a process in which, as a result of nonlinear dynamics, plasma particles move chaotically in the fields of regular electromagnetic waves. Their dynamics differ little from the dynamics of particles in random fields. Below, such regimes we will call regimes with dynamic chaos. The conditions for the occurrence of such modes will be the condition of overlapping nonlinear resonances (Chirikov's criterion). It is known that in the vicinity of resonances the dynamics of particles are described by equations of nonlinear oscillators, in particular, by the equation of a mathematical pendulum. Therefore, the algorithm for finding the conditions for the emergence of regimes with dynamic chaos (conditions of stochastic heating) can be described as follows:


These conditions will be the conditions of stochastic heating. Below, this algorithm is used for the case of Cherenkov resonances, as well as for cyclotron resonances.

#### *3.2.2 The case of Cherenkov resonances*

#### *3.2.2.1 Heating of particles in the field of several transverse electromagnetic waves*

Consider the dynamics of motion of charged particles in the field of several electromagnetic waves. Expressions for the electric and magnetic fields of these waves can be represented in this form

$$\begin{aligned} \overrightarrow{\dot{E}} &= \sum\_{n} \overrightarrow{E}\_{n}, \\ \overrightarrow{H} &= \sum\_{n} \overrightarrow{H}\_{n}, \\ \overrightarrow{E}\_{n} &= \text{Re}\left(\mathcal{E}\_{n}e^{i\varphi\_{n}}\right), \\ \overrightarrow{H}\_{n} &= \frac{c}{\alpha\_{n}}\left[\overrightarrow{k}\_{n}\overrightarrow{E}\_{n}\right], \end{aligned} \tag{8}$$

where *ψ<sup>n</sup>* ¼ *k* ! *n r* ! � *<sup>ω</sup>nt*. range on the microwave pump power, the temperature distribution inside the bulb,

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

By stochastic heating, we mean a process in which, as a result of nonlinear dynamics, plasma particles move chaotically in the fields of regular electromagnetic waves. Their dynamics differ little from the dynamics of particles in random fields. Below, such regimes we will call regimes with dynamic chaos. The conditions for the occurrence of such modes will be the condition of overlapping nonlinear resonances (Chirikov's criterion). It is known that in the vicinity of resonances the dynamics of particles are described by equations of nonlinear oscillators, in particular, by the equation of a mathematical pendulum. Therefore, the algorithm for finding the conditions for the emergence of regimes with dynamic chaos (condi-

1.The conditions for resonant interaction of waves with particles are found.

3.The conditions for the overlap of nonlinear resonances of these oscillators are

Below, this algorithm is used for the case of Cherenkov resonances, as well as for

2.Equations of nonlinear oscillators are determined, which describe the

*3.2.2.1 Heating of particles in the field of several transverse electromagnetic waves*

Consider the dynamics of motion of charged particles in the field of several electromagnetic waves. Expressions for the electric and magnetic fields of these

*<sup>n</sup>* ¼ Re E*ne*

*iψ<sup>n</sup>* � �, (8)

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

plasma electrical conductivity, etc.

**3.2 Stochastic plasma heating**

*3.2.1 Introduction*

found.

cyclotron resonances.

where *ψ<sup>n</sup>* ¼ *k*

**132**

! *n r* ! � *<sup>ω</sup>nt*.

*3.2.2 The case of Cherenkov resonances*

waves can be represented in this form

of changes in pressure in the flask on temperature.

tions of stochastic heating) can be described as follows:

dynamics of particles in the vicinity of resonances.

These conditions will be the conditions of stochastic heating.

*E* ! <sup>¼</sup> <sup>X</sup> *n E* ! *n*,

*H* ! <sup>¼</sup> <sup>X</sup> *n H* ! *n*,

*E* !

*H* ! *<sup>n</sup>* <sup>¼</sup> *<sup>c</sup> ωn k* ! *nE* ! *n* h i ,

There should be several such resonances.

These fields satisfy Maxwell's equations. The equations of motion of a charged particle in fields (8) have the traditional form

$$\frac{d\overrightarrow{P}}{dt} = \epsilon \overrightarrow{E} + \frac{e}{c} \left[\overrightarrow{v}\overrightarrow{H}\right].\tag{9}$$

It is convenient to write these equations in the following dimensionless variables for both dependent and independent variables: *<sup>ω</sup><sup>n</sup>* <sup>¼</sup> *<sup>ω</sup>n=ω*0, \_ *P* ! � *dP* ! *=dτ*, *τ* � *ω*0*t* , *P* ! � *P* ! *<sup>=</sup>mc*, \_ *r* ! ¼ *v* !*=c*, *k* ! *<sup>n</sup>* � *k* ! *nc=ω*0, *r* ! � *ω*<sup>0</sup> *r* !*=c*, *E* ! *<sup>n</sup>* � *eE* ! *<sup>n</sup>=mcω<sup>n</sup>* – is the wave force parameter.

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

$$
\dot{\chi} = \frac{\overrightarrow{P}}{\chi} \frac{e\overrightarrow{E}}{mco\_0}.\tag{10}
$$

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

$$\begin{aligned} \stackrel{\leftrightarrow}{\vec{P}} &= \sum\_{n} E\_{n} \left( \boldsymbol{\omega}\_{n} - \vec{\boldsymbol{k}}\_{n} \stackrel{\leftrightarrow}{\vec{r}} \right) + \sum\_{n} \vec{k}\_{n} \left( \stackrel{\leftrightarrow}{\vec{r}} \stackrel{\leftrightarrow}{\vec{E}}\_{n} \right), \\ \dot{\boldsymbol{\gamma}} &= \frac{\vec{P}}{\gamma} \sum\_{n} \boldsymbol{\omega}\_{n} \vec{E}\_{n}, \end{aligned} \tag{11}$$

where *E* ! *<sup>n</sup>* ¼ Re E ! *ne<sup>i</sup>ψ<sup>n</sup>* � �; *<sup>ψ</sup><sup>n</sup>* � *<sup>k</sup>* ! *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

$$
\dot{\gamma}\_n = a \iota\_n \left( \dot{\vec{r}} \, \vec{E}\_n \right). \tag{12}
$$

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

$$\overrightarrow{P} - \sum\_{n} \text{Re}\left(i\overrightarrow{\mathcal{E}}\_{n}e^{i\varphi\_{n}}\right) - \sum\_{n} \frac{\overrightarrow{k}\_{n}}{\alpha\_{n}}\gamma\_{n} = \overrightarrow{\mathcal{C}}.\tag{13}$$

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

$$\begin{aligned} \overrightarrow{P} &= \overline{\overrightarrow{P}} + \overline{\overrightarrow{P}} \end{aligned} \tag{14}$$
 
$$\begin{aligned} \gamma\_n &= \overline{\gamma}\_n + \tilde{\gamma}\_n. \end{aligned} \tag{14}$$

and

*<sup>γ</sup>*\_ <sup>¼</sup> <sup>1</sup> *γ* X *m*, *n*

*3.2.2.2 Resonances*

differential equation

! � *k* ! <sup>1</sup> � *k* !

where *χ*

rewritten as

**135**

where E¼E!

1E ! 2 .

*3.2.2.3 Particle dynamics near resonance*

<sup>¼</sup> <sup>X</sup> *m*, *n*

1 2*γ ω<sup>n</sup>* E ! *n* E !

Re *i*E ! *me iψ<sup>m</sup>* � �*ω<sup>n</sup>* Re <sup>E</sup>

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

some physical systems that are of considerable interest.

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

*dθ dt* <sup>¼</sup> *<sup>χ</sup>* ! *v*

> *dγ <sup>d</sup><sup>τ</sup>* <sup>¼</sup> <sup>1</sup> *γ*

resonance detuning Δð Þ*γ* can be expanded into a Taylor series:

Δð Þ¼ *γ* Δ *γ*<sup>0</sup> ð Þþ *δγ*

2, Ω � *ω*<sup>1</sup> � *ω*<sup>2</sup>

! *ne iψ<sup>n</sup>* � � <sup>¼</sup>

*<sup>m</sup>*½ � cosð*ψ<sup>m</sup>* þ *ψ<sup>n</sup>* þ *π=*2Þ þ cosð Þ *ψ <sup>m</sup>* � *ψ<sup>n</sup>* þ *π=*2 *:*

! � <sup>Ω</sup> <sup>¼</sup> <sup>Δ</sup>ð Þ*<sup>γ</sup>* , (19)

E � Ω � cos *θ*, (20)

Below, we use the obtained equations and integrals to analyze the dynamics of

In accordance with the algorithm described above, we will find resonances. In addition, we find equations that describe the dynamics of particles in the vicinity of resonances. All waves (1) are transverse and fast. In the original formulation of the problem, there is no mechanism for the resonant interaction of such waves separately with plasma particles. However, plasma particles can have a Cherenkov resonance with a beating wave (with a virtual wave; a combination wave). Indeed, let there be only two fast transverse waves (numbered 1 and 2) among those waves that act on a particle. The beats of these waves form a slow combination wave, the phase velocity of which can be close to the average particle velocity. In this case, the dynamics of particles can be described by the dynamics of a nonlinear pendulum (mathematical pendulum). Let's show it. Let us denote the phase difference of these waves through *θ*, i.e. *θ* � *ψ*<sup>1</sup> � *ψ*2*.* For this phase difference, we obtain the following

In this case, we assume that the parameters are close to the conditions of the Cherenkov resonance with the combination wave (Ω*=χ* ffi *v*). The second equation of system (11), taking into account the dynamics of slow and fast variables, can be

We will assume that the initial energy of a particle exactly corresponds to the Cherenkov resonance of a particle with a combination wave. It means that Δ *γ*<sup>0</sup> ð Þ¼ 0. In addition, we will take into account that as a result of the interaction of waves with particles, the energy of the particle has not changed much. In this case, the

> ∂Δ *∂γ* � �

*γ*0

*:* (21)

(18)

**Figure 8.** *Dispersion diagram of interacting waves.*

In this case, we can get the following expressions and equations that relate fast and slow variables:

$$\begin{aligned} \overrightarrow{\overline{P}} &= \sum\_{n} \overrightarrow{\overline{k}\_{n}} \overleftarrow{\gamma}\_{n} + C, \\\\ \overrightarrow{\overline{P}} &= \sum\_{n} \text{Re} \left( i \overrightarrow{\overline{\mathcal{E}}}\_{n} e^{i \boldsymbol{\wp}\_{n}} \right) + \sum\_{n} \overrightarrow{\overline{k}}\_{n} \overrightarrow{\overline{\gamma}}\_{n} / \boldsymbol{\wp}\_{n}, \\\\ \dot{\overline{\overline{\gamma}}}\_{n} &= \boldsymbol{\wp}\_{n} \overrightarrow{\overline{\overline{\boldsymbol{v}}}} \overrightarrow{\overline{E}}\_{n} = \boldsymbol{\wp}\_{n} \overrightarrow{\overline{\boldsymbol{v}}} \text{Re} \left( \overrightarrow{\overline{\boldsymbol{\mathcal{E}}}}\_{n} e^{i \boldsymbol{\wp}\_{n}} \right), \\\\ \dot{\overline{\overline{\gamma}}}\_{n} &= \boldsymbol{\wp}\_{n} \overrightarrow{\overline{\overline{\boldsymbol{v}}}} \overrightarrow{\overline{\boldsymbol{E}}}\_{n}, \\\\ \dot{\overline{\gamma}}\_{n} &= \text{Re} \left( \Gamma\_{n} e^{i \boldsymbol{\wp}\_{n}} \right), \end{aligned} \tag{15}$$

where Γ*<sup>n</sup>* ¼ �*iω<sup>n</sup> v* !E ! *<sup>n</sup>=ψ*\_ *<sup>n</sup>*. The equations for fast variables can be integrated

$$\begin{aligned} \check{\gamma}\_n &= \operatorname{Re} \left[ i o\_n \left( \overrightarrow{\overline{v}} \, \overrightarrow{\mathcal{E}}\_n \right) e^{i \nu\_n} / o\_n - \overrightarrow{k}\_n \, \overrightarrow{\overline{v}} \right], \\\\ \overset{\circ}{\overline{P}} &= \sum\_n \operatorname{Re} \left\{ i e^{i \nu\_n} \left[ \overrightarrow{\mathcal{E}}\_n + \overrightarrow{k}\_n \left( \overrightarrow{\overline{v}} \, \overrightarrow{\mathcal{E}}\_n \right) / o\_n \right] \right\}. \end{aligned} \tag{16}$$

The equations for the slow variables take the following form:

$$\dot{\overrightarrow{\vec{P}}} = \sum\_{m,n} \overrightarrow{k}\_n \frac{1}{\gamma} \left[ \text{Re} \left( i \overrightarrow{\mathcal{E}}\_m e^{i\wp\_m} \right) \right] \left[ \text{Re} \left( \overrightarrow{\mathcal{E}}\_n e^{i\wp\_n} \right) \right] \tag{17}$$

*Microwave Heating of Low-Temperature Plasma and Its Application DOI: http://dx.doi.org/10.5772/intechopen.97167*

and

$$\begin{split} \dot{\vec{\gamma}} &= \frac{1}{\gamma} \sum\_{m,n} \text{Re} \left( i \vec{\mathcal{E}}\_m e^{i\varphi\_m} \right) \omega\_n \text{Re} \left( \vec{\mathcal{E}}\_n e^{i\varphi\_n} \right) = \\\\ &= \sum\_{m,n} \frac{1}{2\gamma} \alpha\_n \vec{\mathcal{E}}\_n \vec{\mathcal{E}}\_m [\cos \left( \psi\_m + \psi\_n + \pi/2 \right) + \cos \left( \psi\_m - \psi\_n + \pi/2 \right)]. \end{split} \tag{18}$$

Below, we use the obtained equations and integrals to analyze the dynamics of some physical systems that are of considerable interest.

### *3.2.2.2 Resonances*

In this case, we can get the following expressions and equations that relate fast

þ<sup>X</sup> *n k* !

*<sup>i</sup>ψ<sup>n</sup> <sup>=</sup>ω<sup>n</sup>* � *<sup>k</sup>*

n o h i

! h i

! *<sup>n</sup>* þ *k* ! *<sup>n</sup> v* !E ! *n* � �

! *n v*

Re E ! *ne iψ<sup>n</sup>* h i � �

,

*=ω<sup>n</sup>*

*:*

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

*<sup>n</sup>*~*γn=ωn*,

(15)

(16)

(17)

,

and slow variables:

*Dispersion diagram of interacting waves.*

**Figure 8.**

where Γ*<sup>n</sup>* ¼ �*iω<sup>n</sup> v*

**134**

*P* ! <sup>¼</sup> <sup>X</sup> *n*

~ *P* !

~\_ *γ<sup>n</sup>* ¼ *ω<sup>n</sup> v*

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

!E ! *<sup>n</sup>=ψ*\_ *<sup>n</sup>*. The equations for fast variables can be integrated

> ~ *P* !

\_ *P* ! <sup>¼</sup> <sup>X</sup> *m*, *n k* ! *n* 1 *γ*

<sup>¼</sup> <sup>X</sup> *n*

~*γ<sup>n</sup>* ¼ Re Γ*ne*

~*γ<sup>n</sup>* ¼ Re *iω<sup>n</sup> v*

<sup>¼</sup> <sup>X</sup> *n*

*k* ! *n ωn*

!*E* ! *γ<sup>n</sup>* þ *C*,

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

*<sup>n</sup>* ¼ *ω<sup>n</sup> v*

*iψ<sup>n</sup>* � �,

> !E ! *n* � � *e*

Re *ie<sup>i</sup>ψ<sup>n</sup>* <sup>E</sup>

Re *i*E ! *me iψ <sup>m</sup>* h i � �

The equations for the slow variables take the following form:

Re *i*E ! *ne iψ<sup>n</sup>* � �

In accordance with the algorithm described above, we will find resonances. In addition, we find equations that describe the dynamics of particles in the vicinity of resonances. All waves (1) are transverse and fast. In the original formulation of the problem, there is no mechanism for the resonant interaction of such waves separately with plasma particles. However, plasma particles can have a Cherenkov resonance with a beating wave (with a virtual wave; a combination wave). Indeed, let there be only two fast transverse waves (numbered 1 and 2) among those waves that act on a particle. The beats of these waves form a slow combination wave, the phase velocity of which can be close to the average particle velocity. In this case, the dynamics of particles can be described by the dynamics of a nonlinear pendulum (mathematical pendulum). Let's show it. Let us denote the phase difference of these waves through *θ*, i.e. *θ* � *ψ*<sup>1</sup> � *ψ*2*.* For this phase difference, we obtain the following differential equation

$$\frac{d\theta}{dt} = \overrightarrow{\chi}\,\overrightarrow{v} - \Omega = \Delta(\chi),\tag{19}$$

where *χ* ! � *k* ! <sup>1</sup> � *k* ! 2, Ω � *ω*<sup>1</sup> � *ω*<sup>2</sup>

In this case, we assume that the parameters are close to the conditions of the Cherenkov resonance with the combination wave (Ω*=χ* ffi *v*). The second equation of system (11), taking into account the dynamics of slow and fast variables, can be rewritten as

$$\frac{d\boldsymbol{\eta}}{d\boldsymbol{\pi}} = \frac{1}{\boldsymbol{\chi}} \boldsymbol{\mathcal{E}} \cdot \boldsymbol{\Omega} \cdot \cos \theta,\tag{20}$$

where E¼E! 1E ! 2 .

#### *3.2.2.3 Particle dynamics near resonance*

We will assume that the initial energy of a particle exactly corresponds to the Cherenkov resonance of a particle with a combination wave. It means that Δ *γ*<sup>0</sup> ð Þ¼ 0. In addition, we will take into account that as a result of the interaction of waves with particles, the energy of the particle has not changed much. In this case, the resonance detuning Δð Þ*γ* can be expanded into a Taylor series:

$$
\Delta(\chi) = \Delta(\chi\_0) + \delta\chi \left(\frac{\partial \Delta}{\partial \chi}\right)\_{\chi\_0}.\tag{21}
$$

Then Eqs. (19) and (20) will be completely closed and take the following form

$$\begin{split} \frac{d\theta}{d\tau} &= \delta\gamma \left(\frac{\partial\Delta}{\partial\gamma}\right)\_{\gamma\_0}, \\ \frac{d\delta\chi}{d\tau} &= \frac{\mathcal{E}\Omega}{\chi\_0} \cos\theta. \end{split} \tag{22}$$

The system of equations (22) is equivalent to the equation of the mathematical pendulum

$$
\ddot{\theta} = \left(\frac{\partial \Delta}{\partial \dot{\gamma}}\right)\_{\gamma\_0} \frac{\mathcal{E} = \Omega}{\gamma\_0} \cos \theta. \tag{23}
$$

The half-width of the nonlinear resonance of the pendulum (23) is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>∂</sup>Δ*=∂<sup>γ</sup> <sup>γ</sup>*<sup>0</sup> E ¼ <sup>Ω</sup>*=γ*<sup>0</sup> ð Þ <sup>q</sup> . If there are many waves (three or more), then each pair of waves can organize a combination wave. The phase velocities of these waves can be easily selected in the required way for efficient particle heating. So if the distance between the phase velocities of the nearest combination waves turns out to be less than the sum of the half-widths of nonlinear resonances, then the dynamics of particles in the field of these waves will be chaotic.

It is enough for us to consider the dynamics of particles in the field of three waves. Two of these waves propagate in the same direction, the third propagates towards them. The condition for overlapping nonlinear resonances can be written as

$$\left(v\_{ph\_{i+1}} - v\_{ph\_i}\right) \le \frac{\sqrt{\mathcal{E}\_0}}{\chi\_0^2 \sqrt{k\_0 v\_0}} \left[\sqrt{\mathcal{E}\_i \Delta a v\_{0i}} + \sqrt{\mathcal{E}\_{i+1} \Delta a v\_{0(i+1)}}\right],\tag{24}$$

where *vphi* ¼ Δ*ω*0*<sup>i</sup>=*ð Þ *k*<sup>0</sup> þ *ki* , *i* ¼ f g 1, 2, *::n* , Δ*ω*0*<sup>i</sup>* � 1 � *ωi*.

The left side of inequality (24) describes the distance between nonlinear resonances. The right side represents itself the sum of half–widths of two adjacent nonlinear resonances. If inequality (24) is satisfied, then the dynamics of particles become chaotic. This fact is confirmed by both analytical and numerical studies.

Let us briefly describe the results of a numerical study of the original system of Eq. (11) for the case of interaction of particles with three waves (see **Figure 8**). The dynamics of particles was investigated in a field of small and identical field strengths E*<sup>i</sup>* ¼ 0*:*03 and with large –E*<sup>i</sup>* ¼ 0*:*3 . In **Figure 9** shows the dependence of the change in energy on time for particles with an initial velocity equal to zero. The wave vectors of the waves were equal: *k*<sup>1</sup> ¼ �0*:*8, *k*<sup>2</sup> ¼ �1, *k*<sup>0</sup> ¼ 1*:*2. In **Figure 10** shows the temporal dynamics of particle energy with large field strength E*<sup>i</sup>* ¼ 0*:*3.

**Figures 9** and **10** it is seen that at low strengths of the electromagnetic field of the waves, the particle performs regular oscillations, being in one nonlinear Cherenkov resonance with one of the combination waves. With an increase in the field strength under the action of the fields, the particle transitions from resonance to resonance, the dynamics of particle motion is irregular with significant changes in the particle energy.

obtained above can be used in the first approximation. This means that we can assume that condition (24) of overlap of nonlinear Cherenkov resonances is satisfied in the field of laser radiation. If these conditions are met, we can assume that the dynamics of particles is chaotic. Then, by averaging over random phases and random positions of particles, we can find the following expression for the mean

<sup>2</sup> D E <sup>≈</sup>E<sup>4</sup>ð Þ <sup>Δ</sup>*<sup>ω</sup>* <sup>2</sup> � *<sup>τ</sup><sup>=</sup>* <sup>4</sup>*γ*<sup>2</sup>

0

� �*:* (25)

square of the change in the energy of particles

*The energy of one particle at* E*<sup>i</sup>* ¼ 0*:*3 *k*<sup>1</sup> ¼ �0*:*8, *k*<sup>2</sup> ¼ �1, *k*<sup>3</sup> ¼ 1*:*2*.*

*The energy of one particle at* E*<sup>i</sup>* ¼ 0*:*03 *and k*<sup>1</sup> ¼ �0*:*8, *k*<sup>2</sup> ¼ �1, *k*<sup>3</sup> ¼ 1*:*2*.*

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

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

**Figure 10.**

**137**

**Figure 9.**

ð Þ Δ*γ*

In this section, it will be shown that using regimes with dynamic chaos, it is possible to propose rather simple and efficient schemes for heating solid-state plasma up to temperatures required for nuclear fusion. Moreover, the heating process proceeds extremely quickly, so that all known plasma instabilities do not have time to develop. To prove the possibility of such heating, we will use all the results obtained above. We will assume that the frequency of the laser radiation that acts on a solid target is much higher than the plasma frequency. Then the results

*Microwave Heating of Low-Temperature Plasma and Its Application DOI: http://dx.doi.org/10.5772/intechopen.97167*

Then Eqs. (19) and (20) will be completely closed and take the following form

The system of equations (22) is equivalent to the equation of the mathematical

E ¼ Ω *γ*0

. If there are many waves (three or more), then each pair of

h i q

*γ*0

waves can organize a combination wave. The phase velocities of these waves can be easily selected in the required way for efficient particle heating. So if the distance between the phase velocities of the nearest combination waves turns out to be less than the sum of the half-widths of nonlinear resonances, then the dynamics of

It is enough for us to consider the dynamics of particles in the field of three waves. Two of these waves propagate in the same direction, the third propagates towards them. The condition for overlapping nonlinear resonances can be written as

<sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

The left side of inequality (24) describes the distance between nonlinear resonances. The right side represents itself the sum of half–widths of two adjacent nonlinear resonances. If inequality (24) is satisfied, then the dynamics of particles become chaotic. This fact is confirmed by both analytical and numerical studies. Let us briefly describe the results of a numerical study of the original system of Eq. (11) for the case of interaction of particles with three waves (see **Figure 8**). The

strengths E*<sup>i</sup>* ¼ 0*:*03 and with large –E*<sup>i</sup>* ¼ 0*:*3 . In **Figure 9** shows the dependence of the change in energy on time for particles with an initial velocity equal to zero. The wave vectors of the waves were equal: *k*<sup>1</sup> ¼ �0*:*8, *k*<sup>2</sup> ¼ �1, *k*<sup>0</sup> ¼ 1*:*2. In **Figure 10** shows the temporal dynamics of particle energy with large field strength E*<sup>i</sup>* ¼ 0*:*3. **Figures 9** and **10** it is seen that at low strengths of the electromagnetic field of

In this section, it will be shown that using regimes with dynamic chaos, it is possible to propose rather simple and efficient schemes for heating solid-state plasma up to temperatures required for nuclear fusion. Moreover, the heating process proceeds extremely quickly, so that all known plasma instabilities do not have time to develop. To prove the possibility of such heating, we will use all the results obtained above. We will assume that the frequency of the laser radiation that acts on a solid target is much higher than the plasma frequency. Then the results

E*i*Δ*ω*0*<sup>i</sup>* <sup>p</sup> <sup>þ</sup>

∂Δ *∂γ* � �

*γ*0 ,

(22)

cos *θ:* (23)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E*<sup>i</sup>*þ<sup>1</sup>Δ*ω*<sup>0</sup>ð Þ *<sup>i</sup>*þ<sup>1</sup>

, (24)

cos *θ:*

*dθ <sup>d</sup><sup>τ</sup>* <sup>¼</sup> *δγ*

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

*dδγ <sup>d</sup><sup>τ</sup>* <sup>¼</sup> <sup>E</sup><sup>Ω</sup> *γ*0

€*<sup>θ</sup>* <sup>¼</sup> <sup>∂</sup><sup>Δ</sup> *∂γ* � �

particles in the field of these waves will be chaotic.

≤

*γ*2 0

where *vphi* ¼ Δ*ω*0*<sup>i</sup>=*ð Þ *k*<sup>0</sup> þ *ki* , *i* ¼ f g 1, 2, *::n* , Δ*ω*0*<sup>i</sup>* � 1 � *ωi*.

*vphi*þ<sup>1</sup> � *vphi* � �

The half-width of the nonlinear resonance of the pendulum (23) is

ffiffiffiffiffi E0 p

ffiffiffiffiffiffiffiffiffi *k*0*v*<sup>0</sup>

dynamics of particles was investigated in a field of small and identical field

the waves, the particle performs regular oscillations, being in one nonlinear Cherenkov resonance with one of the combination waves. With an increase in the field strength under the action of the fields, the particle transitions from resonance to resonance, the dynamics of particle motion is irregular with significant changes

pendulum

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>∂</sup>Δ*=∂<sup>γ</sup> <sup>γ</sup>*<sup>0</sup> E ¼ <sup>Ω</sup>*=γ*<sup>0</sup> ð Þ

in the particle energy.

**136**

**Figure 9.** *The energy of one particle at* E*<sup>i</sup>* ¼ 0*:*03 *and k*<sup>1</sup> ¼ �0*:*8, *k*<sup>2</sup> ¼ �1, *k*<sup>3</sup> ¼ 1*:*2*.*

**Figure 10.** *The energy of one particle at* E*<sup>i</sup>* ¼ 0*:*3 *k*<sup>1</sup> ¼ �0*:*8, *k*<sup>2</sup> ¼ �1, *k*<sup>3</sup> ¼ 1*:*2*.*

obtained above can be used in the first approximation. This means that we can assume that condition (24) of overlap of nonlinear Cherenkov resonances is satisfied in the field of laser radiation. If these conditions are met, we can assume that the dynamics of particles is chaotic. Then, by averaging over random phases and random positions of particles, we can find the following expression for the mean square of the change in the energy of particles

$$
\left< \left( (\Delta \boldsymbol{\gamma})^2 \right) \approx \mathcal{E}^4 (\Delta \boldsymbol{\omega})^2 \cdot \boldsymbol{\tau} / \left( 4 \boldsymbol{\gamma}\_0^2 \right). \tag{25}
$$

Here, the angle brackets denote averaging over phases and positions of particles

$$
\begin{split}
\langle L \rangle &\equiv \frac{1}{2\pi} \int\_0^{2\pi} d \left( \stackrel{\rightarrow}{k} \stackrel{\rightarrow}{r} \right) \cdot \lim \frac{1}{T} \int\_{-T}^T L \cdot dt, \\
\stackrel{\rightarrow}{\mathcal{E}}\_n &= \stackrel{\rightarrow}{E}\_n e^{-i\nu\_n}.
\end{split}
\tag{26}
$$

It should be noted that many other heating mechanisms are also less efficient than heating in the dynamic chaos regime. In particular, one can point to the well–known turbulent heating. In turbulent heating schemes, radiation incident on plasma as a result of nonlinear processes excites random fluctuations of fields in the plasma. It is these random fluctuations that heat the plasma particles. As we saw above, this mechanism is less efficient than dynamic heating. In addition, the transformation of regular fields incident on the plasma into random fields requires a significant time. The closest to the one considered is the scenario of plasma heating, which is associated with collisions of particles of dense (solid-state) plasma. The collision frequency, as is known, is proportional to the plasma density *<sup>n</sup>* <sup>¼</sup> <sup>10</sup>22*cm*�<sup>3</sup> and at a

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

�<sup>1</sup> and the amplitude of the laser wave E ¼ <sup>0</sup>*:*1, then the heating of the plasma to a temperature of 7 keV occurs in a time <sup>Δ</sup>*tH* <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�14*s*, i.e. in a time significantly shorter than the time of collision between particles. Thus, there is a range of laser radiation and plasma parameters at which dynamic heating is much

Let us estimate the possibility of using dynamic heating of solid–state plasma to thermonuclear temperatures. In this case, we need to heat the plasma ions. In this case, direct dynamic heating of ions is ineffective. Indeed, as follows from formula

�<sup>1</sup> heats plasma electrons *<sup>n</sup>* <sup>¼</sup> 1022*cm*�<sup>3</sup> to a temperature of 7 keV.

(25), this time is proportional to the fourth power of the mass (*τ<sup>H</sup>* � ð Þ *mi*

In this case, the ion heating scheme may look as follows: the laser field E ¼

This heating takes place over time *<sup>t</sup>*<10�<sup>13</sup>*s:* During the time *<sup>t</sup>* � <sup>10</sup>�<sup>9</sup>*s*, the heated electrons transfer their energy to the ions. This time is rather short. During this time, a solid-state target of radius r = 0.1 will not increase its size too much. Note that the rapid heating of electrons and the rapid transfer of energy from electrons to ions make it possible to avoid the development of plasma instabilities.

We saw above that the dynamics of charged particles in the field of a combination wave in the vicinity of the Cherenkov resonance of particles with a combination wave is described by the equation of a mathematical pendulum. If there are several combination waves (we saw above that three transverse electromagnetic waves can generate two combination waves), then to describe the dynamics of particles, it is necessary to analyze a model that contains two equations of a mathematical pendulum. As we saw above, stochastic instability developed when the nonlinear resonances of these mathematical pendulums crossed (see [5, 6]). The particle dynamics became random. Thus, in this model of the interaction of charged particles with electromagnetic waves, the result turned out to be analogous to the motion of particles in a random field. Above, using the example of plasma heating by three laser waves, an expression was obtained that characterizes the efficiency of plasma heating in the field of three regular laser waves. Another common scheme for realizing plasma heating is that the plasma is placed in an external constant magnetic field. To analyze the appearance of conditions for effective plasma heating in such installations based on regimes with dynamic chaos, we note that the presence of an external magnetic field leads to the fact that regimes with dynamic chaos can be realized even when the plasma is exposed to only one external electromagnetic wave It turns out that the role of a large number of waves, in this case, is played by resonances (cyclotron resonances), and also that the dynamics of particles in the vicinity of cyclotron resonances is described by the model of a mathematical pendulum. Overlapping of nonlinear cyclotron resonances leads, as above,

�<sup>1</sup> . If the frequency of the laser radiation *<sup>ω</sup>* <sup>¼</sup>

4 ).

temperature T = 7 keV is *<sup>v</sup>* <sup>¼</sup> <sup>10</sup>12*<sup>s</sup>*

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

more efficient than other heating mechanisms.

*3.2.3 Plasma heating in an external magnetic field*

<sup>5</sup> � <sup>10</sup>15*<sup>s</sup>*

<sup>0</sup>*:*1,*<sup>ω</sup>* <sup>¼</sup> <sup>5</sup> � <sup>10</sup><sup>15</sup>*<sup>s</sup>*

**139**

In deriving (25), we assumed that Δ*ω*<sup>01</sup> ≈ Δ*ω*<sup>02</sup> � Δ*ω*0, E<sup>0</sup> ≈E<sup>1</sup> ≈E<sup>2</sup> � E and that the averaging time is much longer than the decoupling time of correlations of particle motion (*τ* > >*τk*). The decoupling time of the correlation can be estimated by the value *τ<sup>k</sup>* � 1*=ω* � ln*K*. Here *K* is the ratio of the width of nonlinear resonances to the distance between them. At *K* >1, the decoupling time of the correlation is commensurate with the period of the HF field.

A similar analysis of the particle dynamics can be carried out for the case of a large number of waves interacting with particles. The analytical analysis practically does not differ from the one carried out above. Numerical calculations were carried out as well. Let us note the most important results of these studies. The growth rate of the average energy of an ensemble of particles and its maximum energy depends both on the strength of the electromagnetic waves, on the number of combination waves participating in the interaction, as well as on the distance between their nonlinear resonances. Thus, the maximum energy that particles can accumulate in the case of overlap of all Cherenkov resonances from *N*combination waves is determined by the sum of the distances between these resonances

$$\sum\_{i=0}^{N-1} \left( \upsilon\_{p h\_{i+1}} - \upsilon\_{p h\_i} \right) = \upsilon\_{p h\_N} - \upsilon\_{p h\_0}.\tag{27}$$

#### *3.2.2.4 Comparison of heating efficiency*

It is of interest to compare the efficiency of plasma heating by fields of regular electromagnetic waves (in a regime with dynamic chaos) with plasma heating by random fields. In random fields, we can write the following equation for the particle energy

$$\frac{d\chi}{d\tau} = \left(\overrightarrow{v}\,\overrightarrow{\mathcal{E}}\_n\right). \tag{28}$$

Here E ! *<sup>n</sup>* – the field strength of the random wave. Under the same assumptions under which formula (25) was obtained, we find

$$
\left\langle \left( \Delta \boldsymbol{\gamma} \right)^{2} \right\rangle = \boldsymbol{\nu}^{2} \mathcal{E}\_{n}^{2} \boldsymbol{\pi}. \tag{29}
$$

Let us assume that the energy in the field of the noise wave is equal to the energy of the field of coherent radiation. In this case E*<sup>n</sup>* <sup>2</sup> � <sup>Δ</sup>*ω<sup>n</sup>* ¼ E<sup>2</sup> � <sup>Δ</sup>*ω:* Here <sup>Δ</sup>*<sup>ω</sup>* < < <sup>Δ</sup>*ω<sup>n</sup>* is the width of the spectrum of the noise field, Δ*ω* ¼ *ω=Q* is the width of the spectrum of coherent radiation, *<sup>Q</sup>*is the Q–factor of the optical resonator (*<sup>Q</sup>* � <sup>10</sup><sup>6</sup> � 107 ).

$$K \equiv \frac{\left\langle \left(\Delta \boldsymbol{\gamma}\right)^{2} \right\rangle}{\left\langle \left(\Delta \boldsymbol{\gamma}\right)^{2} \right\rangle\_{n}} > \frac{\mathcal{E}^{2} (\Delta \boldsymbol{\alpha}\_{0})^{2} Q}{4 \gamma\_{0}^{2} v\_{0}^{2}}. \tag{30}$$

In the vast majority of cases *K* > >1 .

*Microwave Heating of Low-Temperature Plasma and Its Application DOI: http://dx.doi.org/10.5772/intechopen.97167*

Here, the angle brackets denote averaging over phases and positions of particles

In deriving (25), we assumed that Δ*ω*<sup>01</sup> ≈ Δ*ω*<sup>02</sup> � Δ*ω*0, E<sup>0</sup> ≈E<sup>1</sup> ≈E<sup>2</sup> � E and that

A similar analysis of the particle dynamics can be carried out for the case of a large number of waves interacting with particles. The analytical analysis practically does not differ from the one carried out above. Numerical calculations were carried out as well. Let us note the most important results of these studies. The growth rate of the average energy of an ensemble of particles and its maximum energy depends both on the strength of the electromagnetic waves, on the number of combination waves participating in the interaction, as well as on the distance between their nonlinear resonances. Thus, the maximum energy that particles can accumulate in the case of overlap of all Cherenkov resonances from *N*combination waves is

� lim <sup>1</sup> *T* ð *T*

�*T*

*L* � *dt*,

¼ *vphN* � *vph*<sup>0</sup> *:* (27)

*:* (28)

*<sup>n</sup>τ:* (29)

<sup>2</sup> � <sup>Δ</sup>*ω<sup>n</sup>* ¼ E<sup>2</sup> � <sup>Δ</sup>*ω:* Here <sup>Δ</sup>*<sup>ω</sup>* < < <sup>Δ</sup>*ω<sup>n</sup>* is

).

*:* (30)

(26)

h i *<sup>L</sup>* � <sup>1</sup> 2*π*

E ! *<sup>n</sup>* ¼ *E* ! *ne* �*iψ<sup>n</sup> :*

commensurate with the period of the HF field.

2 ð*π*

0

determined by the sum of the distances between these resonances

*vphi*þ<sup>1</sup> � *vphi* � �

> *dγ <sup>d</sup><sup>τ</sup>* <sup>¼</sup> *<sup>v</sup>* !E ! *n* � �

ð Þ Δ*γ* <sup>2</sup> D E

coherent radiation, *<sup>Q</sup>*is the Q–factor of the optical resonator (*<sup>Q</sup>* � <sup>10</sup><sup>6</sup> � 107

ð Þ Δ*γ* <sup>2</sup> D E

ð Þ Δ*γ* <sup>2</sup> D E

*<sup>n</sup>* – the field strength of the random wave.

*K* �

It is of interest to compare the efficiency of plasma heating by fields of regular electromagnetic waves (in a regime with dynamic chaos) with plasma heating by random fields. In random fields, we can write the following equation for the particle energy

Under the same assumptions under which formula (25) was obtained, we find

<sup>¼</sup> *<sup>v</sup>*<sup>2</sup> E2

Let us assume that the energy in the field of the noise wave is equal to the energy of

the width of the spectrum of the noise field, Δ*ω* ¼ *ω=Q* is the width of the spectrum of

*n*

> E2 ð Þ Δ*ω*<sup>0</sup> 2 *Q*

4*γ*<sup>2</sup> 0*v*2 0

X *N*�1

*i*¼0

*3.2.2.4 Comparison of heating efficiency*

the field of coherent radiation. In this case E*<sup>n</sup>*

In the vast majority of cases *K* > >1 .

Here E !

**138**

*d k*! *r* ! � �

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

the averaging time is much longer than the decoupling time of correlations of particle motion (*τ* > >*τk*). The decoupling time of the correlation can be estimated by the value *τ<sup>k</sup>* � 1*=ω* � ln*K*. Here *K* is the ratio of the width of nonlinear resonances to the distance between them. At *K* >1, the decoupling time of the correlation is

It should be noted that many other heating mechanisms are also less efficient than heating in the dynamic chaos regime. In particular, one can point to the well–known turbulent heating. In turbulent heating schemes, radiation incident on plasma as a result of nonlinear processes excites random fluctuations of fields in the plasma. It is these random fluctuations that heat the plasma particles. As we saw above, this mechanism is less efficient than dynamic heating. In addition, the transformation of regular fields incident on the plasma into random fields requires a significant time.

The closest to the one considered is the scenario of plasma heating, which is associated with collisions of particles of dense (solid-state) plasma. The collision frequency, as is known, is proportional to the plasma density *<sup>n</sup>* <sup>¼</sup> <sup>10</sup>22*cm*�<sup>3</sup> and at a temperature T = 7 keV is *<sup>v</sup>* <sup>¼</sup> <sup>10</sup>12*<sup>s</sup>* �<sup>1</sup> . If the frequency of the laser radiation *<sup>ω</sup>* <sup>¼</sup> <sup>5</sup> � <sup>10</sup>15*<sup>s</sup>* �<sup>1</sup> and the amplitude of the laser wave E ¼ <sup>0</sup>*:*1, then the heating of the plasma to a temperature of 7 keV occurs in a time <sup>Δ</sup>*tH* <sup>¼</sup> <sup>2</sup> � <sup>10</sup>�14*s*, i.e. in a time significantly shorter than the time of collision between particles. Thus, there is a range of laser radiation and plasma parameters at which dynamic heating is much more efficient than other heating mechanisms.

Let us estimate the possibility of using dynamic heating of solid–state plasma to thermonuclear temperatures. In this case, we need to heat the plasma ions. In this case, direct dynamic heating of ions is ineffective. Indeed, as follows from formula (25), this time is proportional to the fourth power of the mass (*τ<sup>H</sup>* � ð Þ *mi* 4 ).

In this case, the ion heating scheme may look as follows: the laser field E ¼ <sup>0</sup>*:*1,*<sup>ω</sup>* <sup>¼</sup> <sup>5</sup> � <sup>10</sup><sup>15</sup>*<sup>s</sup>* �<sup>1</sup> heats plasma electrons *<sup>n</sup>* <sup>¼</sup> 1022*cm*�<sup>3</sup> to a temperature of 7 keV.

This heating takes place over time *<sup>t</sup>*<10�<sup>13</sup>*s:* During the time *<sup>t</sup>* � <sup>10</sup>�<sup>9</sup>*s*, the heated electrons transfer their energy to the ions. This time is rather short. During this time, a solid-state target of radius r = 0.1 will not increase its size too much. Note that the rapid heating of electrons and the rapid transfer of energy from electrons to ions make it possible to avoid the development of plasma instabilities.

#### *3.2.3 Plasma heating in an external magnetic field*

We saw above that the dynamics of charged particles in the field of a combination wave in the vicinity of the Cherenkov resonance of particles with a combination wave is described by the equation of a mathematical pendulum. If there are several combination waves (we saw above that three transverse electromagnetic waves can generate two combination waves), then to describe the dynamics of particles, it is necessary to analyze a model that contains two equations of a mathematical pendulum. As we saw above, stochastic instability developed when the nonlinear resonances of these mathematical pendulums crossed (see [5, 6]). The particle dynamics became random. Thus, in this model of the interaction of charged particles with electromagnetic waves, the result turned out to be analogous to the motion of particles in a random field. Above, using the example of plasma heating by three laser waves, an expression was obtained that characterizes the efficiency of plasma heating in the field of three regular laser waves. Another common scheme for realizing plasma heating is that the plasma is placed in an external constant magnetic field. To analyze the appearance of conditions for effective plasma heating in such installations based on regimes with dynamic chaos, we note that the presence of an external magnetic field leads to the fact that regimes with dynamic chaos can be realized even when the plasma is exposed to only one external electromagnetic wave It turns out that the role of a large number of waves, in this case, is played by resonances (cyclotron resonances), and also that the dynamics of particles in the vicinity of cyclotron resonances is described by the model of a mathematical pendulum. Overlapping of nonlinear cyclotron resonances leads, as above,

to the development of local instability (stochastic instability). As a result, we get method for effective plasma heating [7–11].

Consider a charged particle (electron) that moves in an external constant magnetic field *H*<sup>0</sup> of magnitude directed along the axis *z* and in the field of an electromagnetic wave of arbitrary polarization. The components of the electric and magnetic fields of such a wave can be represented as

$$\begin{aligned} \overrightarrow{E} &= \text{Re}\left(\overrightarrow{\mathbf{E}}\_{0}\mathbf{e}^{i\varphi}\right),\\ \overrightarrow{H} &= \text{Re}\left(\frac{\mathbf{1}}{k\_{0}}\left[\overrightarrow{k}\overrightarrow{E}\right]\right), \end{aligned} \tag{31}$$

In these variables, Eqs. (32) taking into account the integral (34) take the form

*Jn* � *kzv*⊥*εyJ*

*n*¼�∞

At (*ε*<sup>0</sup> < <1) the effective interaction of the particle with the wave occurs when

Assuming condition (38) had performed and introducing the resonant phase *θ<sup>s</sup>* ¼ *sθ* � *τ* from the system of equations (36), after averaging, we obtain the

*kzWsε*<sup>0</sup> cos *θs*,

*Ws* � cos *θs*,

<sup>0</sup> þ *αzpzJs*.

*3.2.4 The condition for the appearance of dynamic chaos (the condition of stochastic*

We will assume that the particle energy changes little as a result of the interaction with the electromagnetic wave *γ* ¼ *γ* ð Þ <sup>0</sup> þ ~*γ* , ~*γ* < <*γ*0, and the resonance condition (38) is exactly satisfied for a particle with energy *γo*. Then, doing decomposition Δ*s*ð Þ*γ* near *γ<sup>o</sup>* the last two equations of the system (39), we obtain a closed

*θ<sup>s</sup>* ¼ Δ*<sup>s</sup>* � *kzvz* þ *s*

*ω<sup>H</sup> γ*0

ð Þ 1 � *kzvz Ws* � *ε*<sup>0</sup> cos *θs*,

*ω<sup>H</sup> <sup>γ</sup>* � 1,

*n μ*

� *v*⊥*εyJn* 0

cosð Þ¼ *<sup>x</sup>* � *<sup>μ</sup>* sin *<sup>θ</sup>* <sup>X</sup><sup>∞</sup>

Δ*<sup>s</sup> γ<sup>o</sup>* ð Þ� *kzvz*<sup>0</sup> þ *s*

*<sup>p</sup>*\_ <sup>⊥</sup> <sup>¼</sup> <sup>1</sup> *p*⊥

*<sup>p</sup>*\_ *<sup>z</sup>* <sup>¼</sup> <sup>1</sup> *γ*

*<sup>γ</sup>*\_ <sup>¼</sup> *<sup>ε</sup>*<sup>0</sup> *γ*

*<sup>μ</sup> Js* � *αyp*⊥*Js*

system of two equations for determining ~*γ* and *θ<sup>s</sup>*

\_

X *n n μ*

> *kxv*<sup>⊥</sup> *p*⊥ *εy*

> > 0 *n*,

*Jn* cos *θn*,

<sup>X</sup>*Jn* sin *<sup>θ</sup><sup>n</sup>* <sup>þ</sup>

*kxvz p*⊥ *εz* X *n J* 0 *<sup>n</sup>* � sin *θn*,

*Jn*ð Þ *μ* cosð Þ *x* � *nθ* , (37)

� 1 ¼ 0*:* (38)

(36)

(39)

*p*\_ <sup>⊥</sup> ¼ ð Þ 1 � *kzvz*

\_ *<sup>θ</sup>* ¼ � *<sup>ω</sup><sup>H</sup> γ*

*<sup>p</sup>*\_ <sup>k</sup> <sup>¼</sup> <sup>X</sup> *n*

*<sup>γ</sup>*\_ <sup>¼</sup> <sup>X</sup> *n*

*z*\_ ¼ *vz*,

X *n εx n μ Jn* � *εyJ* 0 *n* � � cos *<sup>θ</sup><sup>n</sup>* <sup>þ</sup> *kxvzε<sup>z</sup>*

<sup>þ</sup> ð Þ <sup>1</sup> � *kzvz p*⊥

cos *θ<sup>n</sup> Jn v*⊥*ε<sup>x</sup>*

where *μ* ¼ *kxp*⊥*=ω<sup>H</sup>* .

following equations of motion

where *Ws* � *<sup>α</sup>xp*<sup>⊥</sup> *<sup>s</sup>*

*heating)*

**141**

*θ<sup>n</sup>* ¼ *kzz* þ *kxξ* � *nθ* � *τ:*

X *n*

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

cos *θ<sup>n</sup> εzJ* ½ � *<sup>n</sup>* þ ð Þ *kzv*⊥*ε<sup>x</sup>* � *kxv*⊥*ε<sup>z</sup>*

*n μ* þ *vzε<sup>z</sup>* � �

In obtaining (36), we used the expansion

one of the resonance conditions is satisfied

*εxJ* 0 *<sup>n</sup>* � *ε<sup>y</sup> n μ Jn* � � sin *<sup>θ</sup><sup>n</sup>* <sup>þ</sup>

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

� �,

where *ψ* � *ωt* � k ! r ! , E ! <sup>0</sup> ¼ *α* !*E*0; *α* ! ¼ *αx*, *iαy*, *α<sup>z</sup>* � � – wave polarization vector; *k*<sup>0</sup> ¼ *ω=c*; *ω*, *k* ! – frequency and wave vector of the wave. We introduce the following dimensionless variables: *p* ! <sup>1</sup> ¼ *p* !*=mc* , *k* ! <sup>1</sup> ¼ *k* ! *=k*0, *τ* ¼ *ωt*, *r* ! <sup>1</sup> ¼ *k*<sup>0</sup> *r* !, *ε* ! ¼ *eE* ! <sup>0</sup>*=mcω*, *v* ! <sup>1</sup> ¼ *v* !*=c*, *<sup>υ</sup>ph*<sup>1</sup> <sup>¼</sup> *<sup>υ</sup>ph=<sup>c</sup>* <sup>¼</sup> *<sup>ω</sup>=kc*.

Without loss of generality, we can assume that the vector *k* ! has only two nonzero components *kx* and *kz*. The equations of motion of a particle can be reduced to the form

$$\begin{aligned} \dot{\overrightarrow{P}} &= \left(\mathbf{1} - \frac{\overrightarrow{k}\,\overrightarrow{p}}{\chi}\right) \text{Re}\left(\overrightarrow{e}\,e^{i\mathbf{p}}\right) + \frac{\overrightarrow{k}}{\chi} \text{Re}\left(\overrightarrow{p}\,\overrightarrow{e}\right)e^{i\mathbf{p}} + \frac{o\mu}{\chi}\left[\overrightarrow{p}\,\overrightarrow{e}\right],\\ \dot{\overrightarrow{r}} &= \overrightarrow{p}/\chi,\\ \dot{\nu} &= \overrightarrow{k}\,\overrightarrow{p}/\chi - \mathbf{1}, \end{aligned} \tag{32}$$

where *τ* � *ωt*, *e* ! � *H* ! <sup>0</sup>*=H*0; *ω<sup>H</sup>* � *eH*0*=mcω*; *ψ* ¼ *k* ! *r* ! � *τ* [7].

The last term on the right–hand side of the first vector equation describes the Lorentz force that acts on a particle in a constant external field. Multiplying the first of equations (32) by *p* ! and taking into account that *<sup>p</sup>*<sup>2</sup> <sup>¼</sup> *<sup>γ</sup>*<sup>2</sup> � 1, we obtain the following equation for changing the particle energy

$$
\dot{\varphi} = \operatorname{Re} \left( \nu \overrightarrow{\varepsilon} \right) \varepsilon^{i\varphi}. \tag{33}
$$

Using this equation, from equations (32) we find the following integral of motion

$$
\overrightarrow{p} - \text{Re}\left(i\overrightarrow{e}e^{i\varphi}\right) + a\_H \left[\overrightarrow{r'}\overrightarrow{e}\right] - \overrightarrow{k}\gamma = \text{const.}\tag{34}
$$

For what follows, it is convenient to pass to new variables *<sup>p</sup>*⊥, *<sup>p</sup>*k, *<sup>θ</sup>*, *<sup>ξ</sup>* and *<sup>η</sup>*, which are related to the old following ratios

$$\begin{aligned} p\_x &= p\_\perp \cos \theta, \\ p\_y &= p\_\perp \sin \theta, \\ p\_x &= p\_\parallel, \\ \infty &= \xi - \frac{p\_\perp}{a\mu\_H} \sin \theta, \\ y &= \eta + \frac{p\_\perp}{a\mu\_H} \cos \theta. \end{aligned} \tag{35}$$

*Microwave Heating of Low-Temperature Plasma and Its Application DOI: http://dx.doi.org/10.5772/intechopen.97167*

In these variables, Eqs. (32) taking into account the integral (34) take the form

$$\begin{split} \dot{p}\_{\perp} &= (1 - k\_{x}v\_{z}) \sum\_{n} \left( \varepsilon\_{x} \frac{n}{\mu} J\_{n} - \varepsilon\_{y} l\_{n}' \right) \cos \theta\_{n} + k\_{x} v\_{x} \varepsilon\_{z} \sum\_{n} \frac{n}{\mu} J\_{n} \cos \theta\_{n}, \\ \dot{\theta} &= -\frac{\alpha \mu}{\gamma} + \frac{(1 - k\_{y}v\_{z})}{p\_{\perp}} \sum\_{n} \left( \varepsilon\_{y} l\_{n}' - \varepsilon\_{\gamma} \frac{n}{\mu} J\_{n} \right) \sin \theta\_{n} + \frac{k\_{x} v\_{\perp}}{p\_{\perp}} \varepsilon\_{\gamma} \sum\_{n} J\_{n} \sin \theta\_{n} + \frac{k\_{x} v\_{x}}{p\_{\perp}} \varepsilon\_{z} \sum\_{n} l\_{n}' \cdot \sin \theta\_{n}, \\ \dot{p}\_{\parallel} &= \sum\_{n} \cos \theta\_{n} [\varepsilon\_{y} l\_{n} + (k\_{x} v\_{\perp} \varepsilon\_{x} - k\_{x} v\_{\perp} \varepsilon\_{z})] \frac{n}{\mu} f\_{n} - k\_{x} v\_{\perp} \varepsilon\_{y} l\_{n}', \\ \dot{\gamma} &= \sum\_{n} \cos \theta\_{n} \left[ J\_{n} \left( v\_{\perp} \varepsilon\_{x} \frac{n}{\mu} + v\_{\perp} \varepsilon\_{x} \right) - v\_{\perp} \varepsilon\_{y} l\_{n}' \right], \\ \dot{\varepsilon} &= v\_{z}, \\ \theta\_{n} &= k\_{x} \varepsilon + k\_{x} \xi - n \theta - \tau. \end{split} \tag{36}$$

In obtaining (36), we used the expansion

$$\cos\left(\pi - \mu \sin \theta\right) = \sum\_{n=-\infty}^{\infty} J\_n(\mu) \cos\left(\pi - n\theta\right),\tag{37}$$

where *μ* ¼ *kxp*⊥*=ω<sup>H</sup>* .

to the development of local instability (stochastic instability). As a result, we get

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

Consider a charged particle (electron) that moves in an external constant magnetic field *H*<sup>0</sup> of magnitude directed along the axis *z* and in the field of an electromagnetic wave of arbitrary polarization. The components of the electric and

<sup>¼</sup> Re E!

<sup>¼</sup> Re <sup>1</sup> *k*0 *k* ! *E* h i ! � �

0*e <sup>i</sup><sup>ψ</sup>* � � ,

! ¼ *αx*, *iαy*, *α<sup>z</sup>*

– frequency and wave vector of the wave. We introduce the

!*=mc* , *k* ! <sup>1</sup> ¼ *k* !

nonzero components *kx* and *kz*. The equations of motion of a particle can be reduced

þ *k* ! *γ*

<sup>0</sup>*=H*0; *ω<sup>H</sup>* � *eH*0*=mcω*; *ψ* ¼ *k*

*γ*\_ ¼ Re *v ε*

The last term on the right–hand side of the first vector equation describes the Lorentz force that acts on a particle in a constant external field. Multiplying the first

> ! � � *e*

Using this equation, from equations (32) we find the following integral of motion

þ *ω<sup>H</sup> r* ! *e* ! h i

For what follows, it is convenient to pass to new variables *<sup>p</sup>*⊥, *<sup>p</sup>*k, *<sup>θ</sup>*, *<sup>ξ</sup>* and *<sup>η</sup>*,

*px* ¼ *p*<sup>⊥</sup> cos *θ*, *py* ¼ *p*<sup>⊥</sup> sin *θ*, *pz* <sup>¼</sup> *<sup>p</sup>*k, *<sup>x</sup>* <sup>¼</sup> *<sup>ξ</sup>* � *<sup>p</sup>*<sup>⊥</sup>

*<sup>y</sup>* <sup>¼</sup> *<sup>η</sup>* <sup>þ</sup> *<sup>p</sup>*<sup>⊥</sup> *ω<sup>H</sup>*

*ω<sup>H</sup>*

sin *θ*,

cos *θ:*

Re *p* ! *ε* ! � � *e <sup>i</sup><sup>ψ</sup>* <sup>þ</sup> *<sup>ω</sup><sup>H</sup> <sup>γ</sup> <sup>p</sup>* ! *e* ! h i ,

> ! *r* ! � *τ* [7].

! and taking into account that *<sup>p</sup>*<sup>2</sup> <sup>¼</sup> *<sup>γ</sup>*<sup>2</sup> � 1, we obtain the

� *k* !

,

� � – wave polarization vector;

*=k*0, *τ* ¼ *ωt*, *r*

!

*<sup>i</sup><sup>ψ</sup> :* (33)

*γ* ¼ *const:* (34)

! <sup>1</sup> ¼ *k*<sup>0</sup> *r* !, *ε* ! ¼

has only two

(31)

(32)

(35)

method for effective plasma heating [7–11].

! r ! , E ! <sup>0</sup> ¼ *α*

following dimensionless variables: *p*

where *ψ* � *ωt* � k

!

\_ *P* !

\_ *r* ! ¼ *p* !*=γ*,

where *τ* � *ωt*, *e*

of equations (32) by *p*

*ψ*\_ ¼ *k* ! *p* !*=γ* � 1,

<sup>¼</sup> <sup>1</sup> � *<sup>k</sup>*

! � *H* !

! *p* ! *γ* !

following equation for changing the particle energy

*p*

which are related to the old following ratios

! � Re *i ε*

!*e <sup>i</sup><sup>ψ</sup>* � �

*k*<sup>0</sup> ¼ *ω=c*; *ω*, *k*

<sup>0</sup>*=mcω*, *v* ! <sup>1</sup> ¼ *v*

to the form

*eE* !

**140**

magnetic fields of such a wave can be represented as

*E* !

*H* !

!*=c*, *<sup>υ</sup>ph*<sup>1</sup> <sup>¼</sup> *<sup>υ</sup>ph=<sup>c</sup>* <sup>¼</sup> *<sup>ω</sup>=kc*.

!*E*0; *α*

! <sup>1</sup> ¼ *p*

Without loss of generality, we can assume that the vector *k*

Re *ε* !*e <sup>i</sup>*<sup>Ψ</sup> � �

At (*ε*<sup>0</sup> < <1) the effective interaction of the particle with the wave occurs when one of the resonance conditions is satisfied

$$\Delta\_s(\chi\_o) \equiv k\_x v\_{x0} + s \frac{\alpha\_H}{\chi\_0} - \mathbf{1} = \mathbf{0}.\tag{38}$$

Assuming condition (38) had performed and introducing the resonant phase *θ<sup>s</sup>* ¼ *sθ* � *τ* from the system of equations (36), after averaging, we obtain the following equations of motion

$$\begin{aligned} \dot{p}\_{\perp} &= \frac{1}{p\_{\perp}} (\mathbf{1} - k\_{\bar{x}} \nu\_{\bar{x}}) W\_{\bar{s}} \cdot e\_{0} \cos \theta\_{\bar{s}}, \\\\ \dot{p}\_{x} &= \frac{1}{\chi} k\_{\bar{x}} W\_{\bar{s}} e\_{0} \cos \theta\_{\bar{s}}, \\\\ \dot{\theta}\_{\bar{s}} &= \Delta\_{\mathfrak{s}} \equiv k\_{\bar{x}} \nu\_{\mathfrak{z}} + s \frac{\alpha\_{H}}{\chi} - \mathbf{1}, \\\\ \dot{\mathcal{Y}} &= \frac{\varepsilon\_{0}}{\chi} W\_{\mathfrak{s}} \cdot \cos \theta\_{\mathfrak{s}}, \end{aligned} \tag{39}$$

where *Ws* � *<sup>α</sup>xp*<sup>⊥</sup> *<sup>s</sup> <sup>μ</sup> Js* � *αyp*⊥*Js* <sup>0</sup> þ *αzpzJs*.

### *3.2.4 The condition for the appearance of dynamic chaos (the condition of stochastic heating)*

We will assume that the particle energy changes little as a result of the interaction with the electromagnetic wave *γ* ¼ *γ* ð Þ <sup>0</sup> þ ~*γ* , ~*γ* < <*γ*0, and the resonance condition (38) is exactly satisfied for a particle with energy *γo*. Then, doing decomposition Δ*s*ð Þ*γ* near *γ<sup>o</sup>* the last two equations of the system (39), we obtain a closed system of two equations for determining ~*γ* and *θ<sup>s</sup>*

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

$$\begin{aligned} \frac{d\tilde{\boldsymbol{\gamma}}}{d\boldsymbol{\tau}} &= \frac{\boldsymbol{\varepsilon}\_0}{\boldsymbol{\gamma}\_0} \boldsymbol{W}\_s \cos \theta\_s, \\\\ \frac{d\theta\_s}{d\boldsymbol{\tau}} &= \frac{k\_x^2 - 1}{\boldsymbol{\gamma}\_0} \boldsymbol{\tilde{\boldsymbol{\gamma}}}. \end{aligned} \tag{40}$$

Equations (40) represent the equation of a mathematical pendulum. Of them, we find the width of the nonlinear resonance

$$\begin{aligned} \Delta \dot{\theta}\_s &= 4 \sqrt{\varepsilon\_0 \left(k\_x^2 - 1\right)} \cdot W\_s / \chi\_0^2, \\ \Delta \tilde{\chi}\_s &= 4 \sqrt{\varepsilon\_0 W\_s / \left(k\_x^2 - 1\right)}. \end{aligned} \tag{41}$$

To find the distance between resonances, we write the resonance conditions (38) and the averaged conservation law (34) for two adjacent resonances (see [6, 7, 11])

$$\begin{aligned} k\_x p\_{s+1} + (s+1)a\_H - \chi\_{s+1} &= \mathbf{0}, \\ \chi\_{s+1} - p\_{s+1}/k\_x &= \mathbf{C}, \\ k\_x p\_s + s a\_H - \chi\_s &= \mathbf{0}, \\ \chi\_s - p\_s/k\_x &= \mathbf{C}. \end{aligned} \tag{42}$$

From these conditions, we find the following value of the distance between resonances

$$
\delta\eta = a\nu\_H / \left(\mathbf{1} - k\_x^2\right). \tag{43}
$$

for the occurrence of conditions for stochastic heating, the field strengths signifi-

The main experiments were carried out on a setup, the scheme of which is described in detail in [9]. The main element of this setup is a cylindrical resonator, the general view of which is shown in **Figure 12**. The resonator is made of a copper tube with an inner diameter of 16 cm and a length of 66 cm. The modes *H*10*<sup>x</sup>* and *H*20*<sup>x</sup>* are excited in this resonator. The central axis of the resonator coincided with the direction of the external inhomogeneous magnetic field. This field formed a magnetic trap. The mirror ratio was chosen equal to 1.2 ... 2. The length of the uniform part of the magnetic field of the trap in the cavity was varied from 25 to 66 cm. A loop probe was located in the central part of the cavity. Plasma in the cavity was generated by an electron beam with an energy of 400–600 eV and a current of 60–100 mA due to a beam–plasma discharge [9]. The pressure in the resonator could be regulated in the range of 10–<sup>4</sup> … 10–<sup>6</sup> mm Hg. In the main series of

The sequence of working of the equipment in time is as follows. An electron beam is injected into the cavity. As a result of the beam–plasma discharge, a plasma

The experiments investigated the fluxes of microwave, optical and X–ray radiation. Simultaneously, using a set of foil plates (up to 15 layers of aluminum foil), electron fluxes with energies up to 1 MeV were recorded. The results are shown in **Figure 13**. Estimation of the electron energy at the maximums of the X–ray radiation intensity showed that at t = 2 μsec the electron energy reaches 100–150 keV,

while at t = 1 μsec the electron energy is 8–10 times higher ( 1 MeV).

frequency of 2.7 GHz, for which the cyclotron resonance was performed at a magnetic induction at the minimum of the trap equal to 0.1 T. The length of the uniform part of the magnetic field of the trap in the resonator was varied from 25 to 66 cm. The oscillation power of the magnetron could be varied in the range 0.1–1.0 MW in a pulse with a duration of 1.8 μs. and was fed into the resonator through a waveguide with a cross-section of 72x34 mm. By varying the delay time between the electron beam pulse and the high-frequency power pulse, the required plasma density was selected in the range 10<sup>7</sup> – 10<sup>9</sup> cm–3 at a pressure of 10–<sup>5</sup> – 10–<sup>4</sup> mm Hg.

.

. The resonator was excited at a

cantly decrease ( <sup>10</sup><sup>4</sup> V/cm) [10, 11].

**Figure 11.**

**143**

experiments, the plasma density was within <sup>10</sup><sup>9</sup> cm–<sup>3</sup>

*Location of resonances and one of the integrals (27) on the plane pz*, *<sup>γ</sup> .*

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

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

is formed with a density of up to <sup>10</sup><sup>11</sup> cm–<sup>3</sup>

Argon was used as a plasma-forming gas.

From expressions (42) and (43) it follows that in carrying out the inequality

$$
\varepsilon\_0 > \alpha\_H^2 / 4 \left[ \sqrt{W\_s} + \sqrt{W\_{s+1}} \right]^2 \left( 1 - k\_x^2 \right), \tag{44}
$$

nonlinear resonances overlap. A regime of stochastic instability sets in, and inequality (44) is the condition for stochastic heating of plasma particles. Note that the width of the nonlinear resonance as well as the distance between resonances must be calculated along with the integrals of motion (see **Figure 11**). In this figure, the dotted lines show the boundaries of nonlinear resonances (the position of the separatrices), the solid lines are the cyclotron resonances themselves, and the bold arrow denotes the integral. In all cases, the particle dynamics run according to integrals. When the integral line coincides with the resonance line, the autoresonance condition occurs. Under autoresonance conditions, particles can resonantly acquire unlimited energy.

#### *3.2.5 Experimental studies of stochastic heating of plasma in a constant magnetic field*

After theoretical work, a large series of experimental studies of stochastic plasma heating was carried out. At the same time, theoretical estimates showed that for stochastic heating of the plasma by the field of one external electromagnetic wave, the field strength of this wave should be sufficiently high (� <sup>10</sup><sup>6</sup> V/cm) [8]. This result, for example, follows from an analysis of conditions (44). Additional numerical studies have shown that if several waves are excited in the experimental setup, for example, two waves with the same frequencies but different wave vectors, then *Microwave Heating of Low-Temperature Plasma and Its Application DOI: http://dx.doi.org/10.5772/intechopen.97167*

*d*~*γ <sup>d</sup><sup>τ</sup>* <sup>¼</sup> *<sup>ε</sup>*<sup>0</sup> *γ*0

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

*dθ<sup>s</sup> <sup>d</sup><sup>τ</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup>

we find the width of the nonlinear resonance

Δ\_ *θ<sup>s</sup>* ¼ 4

Δ~*γ<sup>s</sup>* ¼ 4

*γ<sup>s</sup>* � *ps*

*ε*<sup>0</sup> >*ω*<sup>2</sup>

onantly acquire unlimited energy.

*<sup>H</sup>=*<sup>4</sup> ffiffiffiffiffiffiffi *Ws* <sup>p</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffi

integrals. When the integral line coincides with the resonance line, the

resonances

**142**

*Ws* cos *θs*,

(40)

(41)

(42)

*<sup>z</sup>* � 1 *γ*0 ~*γ:*

Equations (40) represent the equation of a mathematical pendulum. Of them,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ε*<sup>0</sup> *k*<sup>2</sup> *<sup>z</sup>* � 1 <sup>q</sup> � � � *Ws=γ*<sup>2</sup>

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ε*0*Ws= k*<sup>2</sup>

<sup>q</sup> � �*:*

To find the distance between resonances, we write the resonance conditions (38) and the averaged conservation law (34) for two adjacent resonances (see [6, 7, 11])

*kzps*þ<sup>1</sup> <sup>þ</sup> ð Þ *<sup>s</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ω</sup><sup>H</sup>* � *<sup>γ</sup><sup>s</sup>*þ<sup>1</sup> <sup>¼</sup> 0,

From these conditions, we find the following value of the distance between

*δγ* <sup>¼</sup> *<sup>ω</sup>H<sup>=</sup>* <sup>1</sup> � *<sup>k</sup>*<sup>2</sup>

From expressions (42) and (43) it follows that in carrying out the inequality

h i p <sup>2</sup>

nonlinear resonances overlap. A regime of stochastic instability sets in, and inequality (44) is the condition for stochastic heating of plasma particles. Note that the width of the nonlinear resonance as well as the distance between resonances must be calculated along with the integrals of motion (see **Figure 11**). In this figure, the dotted lines show the boundaries of nonlinear resonances (the position of the separatrices), the solid lines are the cyclotron resonances themselves, and the bold arrow denotes the integral. In all cases, the particle dynamics run according to

autoresonance condition occurs. Under autoresonance conditions, particles can res-

*3.2.5 Experimental studies of stochastic heating of plasma in a constant magnetic field*

After theoretical work, a large series of experimental studies of stochastic plasma heating was carried out. At the same time, theoretical estimates showed that for stochastic heating of the plasma by the field of one external electromagnetic wave, the field strength of this wave should be sufficiently high (� <sup>10</sup><sup>6</sup> V/cm) [8]. This result, for example, follows from an analysis of conditions (44). Additional numerical studies have shown that if several waves are excited in the experimental setup, for example, two waves with the same frequencies but different wave vectors, then

*z*

*Ws*þ<sup>1</sup>

� �*:* (43)

� �, (44)

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

*<sup>γ</sup><sup>s</sup>*þ<sup>1</sup> � *ps*þ<sup>1</sup>*=kz* <sup>¼</sup> *<sup>C</sup>*, *kzps* þ *sω<sup>H</sup>* � *γ<sup>s</sup>* ¼ 0,

*=kz* ¼ *C:*

*<sup>z</sup>* � 1

0,

**Figure 11.** *Location of resonances and one of the integrals (27) on the plane pz*, *<sup>γ</sup> .*

for the occurrence of conditions for stochastic heating, the field strengths significantly decrease ( <sup>10</sup><sup>4</sup> V/cm) [10, 11].

The main experiments were carried out on a setup, the scheme of which is described in detail in [9]. The main element of this setup is a cylindrical resonator, the general view of which is shown in **Figure 12**. The resonator is made of a copper tube with an inner diameter of 16 cm and a length of 66 cm. The modes *H*10*<sup>x</sup>* and *H*20*<sup>x</sup>* are excited in this resonator. The central axis of the resonator coincided with the direction of the external inhomogeneous magnetic field. This field formed a magnetic trap. The mirror ratio was chosen equal to 1.2 ... 2. The length of the uniform part of the magnetic field of the trap in the cavity was varied from 25 to 66 cm. A loop probe was located in the central part of the cavity. Plasma in the cavity was generated by an electron beam with an energy of 400–600 eV and a current of 60–100 mA due to a beam–plasma discharge [9]. The pressure in the resonator could be regulated in the range of 10–<sup>4</sup> … 10–<sup>6</sup> mm Hg. In the main series of experiments, the plasma density was within <sup>10</sup><sup>9</sup> cm–<sup>3</sup> .

The sequence of working of the equipment in time is as follows. An electron beam is injected into the cavity. As a result of the beam–plasma discharge, a plasma is formed with a density of up to <sup>10</sup><sup>11</sup> cm–<sup>3</sup> . The resonator was excited at a frequency of 2.7 GHz, for which the cyclotron resonance was performed at a magnetic induction at the minimum of the trap equal to 0.1 T. The length of the uniform part of the magnetic field of the trap in the resonator was varied from 25 to 66 cm. The oscillation power of the magnetron could be varied in the range 0.1–1.0 MW in a pulse with a duration of 1.8 μs. and was fed into the resonator through a waveguide with a cross-section of 72x34 mm. By varying the delay time between the electron beam pulse and the high-frequency power pulse, the required plasma density was selected in the range 10<sup>7</sup> – 10<sup>9</sup> cm–3 at a pressure of 10–<sup>5</sup> – 10–<sup>4</sup> mm Hg. Argon was used as a plasma-forming gas.

The experiments investigated the fluxes of microwave, optical and X–ray radiation. Simultaneously, using a set of foil plates (up to 15 layers of aluminum foil), electron fluxes with energies up to 1 MeV were recorded. The results are shown in **Figure 13**. Estimation of the electron energy at the maximums of the X–ray radiation intensity showed that at t = 2 μsec the electron energy reaches 100–150 keV, while at t = 1 μsec the electron energy is 8–10 times higher ( 1 MeV).

plasma and enhancing efficiency in its heating it is necessary to optimize not only the shape and special distribution of the electric field in the cavity but also its intensity. In this regard, a preference is given to the resonant electrodynamic structures having a concentrated capacity as a parameter. Among such structures is the coaxial resonator loaded on a capacity as well as the toroidal resonator. In the case of an application of the interference method for forming an electromagnetic field and enhancing the effectiveness of plasma heating a great interest is using the single- and double-ridged waveguides instead of the regular waveguides. On the other hand, a comparison of the theoretical and experimental data showed that the most effective is heating of any plasma when the interaction of plasma particles with regular electromagnetic waves occurs in the dynamic chaos regime. Note that the described mechanisms relate to the interaction of the wave-particle type. Another fundamental interaction (the interaction of wave-wave type) can also be used to heat the plasma. But in that event, such a heating mechanism (turbulent heating) contains two stages. At the first stage, the energy of regular waves is

The experimental implementation of the conditions for stochastic heating of plasma by the field of regular electromagnetic waves with a high rate of energy transfer from electromagnetic waves to the energy of thermal motion of plasma electrons has been carried out. It is shown that the average energy of plasma

Also, it is necessary to note that in the experiment stochastic heating and, accordingly, X-ray radiation from the plasma was observed only when several spatial modes were excited in the resonator or when the resonator was excited by two frequencies. This fact is in full agreement with the results of the analysis of

1 Kharkiv National University of Radio Electronics, Kharkiv, Ukraine

4 Harbin Institute of Technology, Harbin, China

provided the original work is properly cited.

\*Address all correspondence to: tetyana.frolova@nure.ua

2 National Science Center 'Kharkiv Institute of Physics and Technology', Kharkiv,

3 Radio Astronomy Institute, National Academy of Sciences of Ukraine, Kharkiv,

© 2021 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,

\*, Vyacheslav Buts2,3, Gennadiy Churyumov1,4, Eugene Odarenko1

transformed into the energy of less efficient noise vibrations.

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

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

electrons reached values 1 MeV in times less than 1 μsec.

theoretical models.

**Author details**

Tetyana Frolova<sup>1</sup>

Ukraine

Ukraine

**145**

and Vladimir Gerasimov<sup>1</sup>

*Some elements of the resonator: 5 – below-cutoff waveguide; 6 – movable piston; 9 – gas supply system; 10 – loop sealed microwave probe; 11 – vacuum window made of Lavsan film with mesh.*

**Figure 13.** *High-frequency power oscillogram: 1 – incident wave; 2 – backward wave; 3 – absorbed wave.*

### **4. Conclusions**

In this chapter, there were considered different approaches to exciting plasma by a regular electromagnetic field. As a microwave source, there was used a magnetron generator as well as two types of electrodynamic structures: resonators (the cylindrical resonator) and waveguides (the rectangular waveguide). An application of the given electrodynamic structures allowed the formation of an electromagnetic field needed for effective exciting plasma in the area of location of an active medium (for example, a bulb with gases mixture). The carried out investigations have pointed to distinct aspects of forming a regular electromagnetic field and its excitation as well as the features of plasma heating. It is significant that for exciting

## *Microwave Heating of Low-Temperature Plasma and Its Application DOI: http://dx.doi.org/10.5772/intechopen.97167*

plasma and enhancing efficiency in its heating it is necessary to optimize not only the shape and special distribution of the electric field in the cavity but also its intensity. In this regard, a preference is given to the resonant electrodynamic structures having a concentrated capacity as a parameter. Among such structures is the coaxial resonator loaded on a capacity as well as the toroidal resonator. In the case of an application of the interference method for forming an electromagnetic field and enhancing the effectiveness of plasma heating a great interest is using the single- and double-ridged waveguides instead of the regular waveguides. On the other hand, a comparison of the theoretical and experimental data showed that the most effective is heating of any plasma when the interaction of plasma particles with regular electromagnetic waves occurs in the dynamic chaos regime. Note that the described mechanisms relate to the interaction of the wave-particle type. Another fundamental interaction (the interaction of wave-wave type) can also be used to heat the plasma. But in that event, such a heating mechanism (turbulent heating) contains two stages. At the first stage, the energy of regular waves is transformed into the energy of less efficient noise vibrations.

The experimental implementation of the conditions for stochastic heating of plasma by the field of regular electromagnetic waves with a high rate of energy transfer from electromagnetic waves to the energy of thermal motion of plasma electrons has been carried out. It is shown that the average energy of plasma electrons reached values 1 MeV in times less than 1 μsec.

Also, it is necessary to note that in the experiment stochastic heating and, accordingly, X-ray radiation from the plasma was observed only when several spatial modes were excited in the resonator or when the resonator was excited by two frequencies. This fact is in full agreement with the results of the analysis of theoretical models.
