3. Plasma accelerator with open gas wall and closed electron drift

The accelerator with closed electron drift is one of the kinds of the electric rocket engines and devices for ion plasma treatment of the surface material. However, the accelerators with closed electron drift and open (gas) walls were not researched for now in contrast to the well-known and widely used plasma accelerators with anode layer and accelerators with closed electron drift and dielectric walls [15, 16]. Therefore, this type accelerator could be interested in manipulating high-current flow of charged particle as well as can be attractive for many different high-tech applications for potential devices of low-cost and compact thrusters. More than, it has some advantages since the wall absence leads to exclusion of the wall material inclusions into the ion beam and exclusion of the secondary electrons formation due to emission and thus to conservation of the plasma electrons dynamics.

on the edges of system. As follow from experiment [17, 18], the accelerator has two operating modes: low-current with narrow anode layer and clear-cut plasma flow and high-current when plasma fills the entire volume of the accelerator. The transfer to the high-current mode occurs under influence of two parameters: worked gas pressure and applied voltage. In high-current quasi-neutral plasma mode of accelerator operation, plasma jet is observed (see, Figure 9 right). The preliminary results show that along the jet axis potential drop arise, which can be used for ion beam accelerating. The radial studies of plasma flow along system axis showed the significant increasing current density on the axis. It may indicate on plasma acceleration in that

Figure 10. (a) Model of discharge gap: 1-anode, 2-cathode, 3-permanent magnets system; (b) potential distribution in the

Figure 9. (a) Experimental sample: 1- cathode, 2- anode, and 3- permanent magnets system; (b)plasma jet in high-current

Modeling of Novel Plasma-Optical Systems http://dx.doi.org/10.5772/intechopen.77512 277

We will consider discharge gap, where ions production occurs due to ionization of the working gas by electrons (see Figure 10a). Electrons are magnetized, move along magnetic strength lines, and drift slowly to anode due to collisions. Ions are free and accelerated by electric field

3.1. One-dimensional hydrodynamic and hybrid model

direction.

gap for different parameters a value.

operation mode.

The sample of plasma accelerator with closed electron drift and open walls is shown in Figure 9 (left). This sample of cylindrical Hall-type plasma ion source that produced ion plasma flow converging toward the axis system was created for the properties exploration [17]. The discharge in the system burns due to ionization of the working gas by the electrons. Electrons are magnetized and formed stable negative space charge. The created ions accelerated from ionization zone to the cathode. As follow from discharge geometry an accumulation of ion space charge occurs as it is in the positive space charge lens (see above). The main part of generated ions leaves the system across radius, along with system axis, due to jets can appear

space charge. The lens will be used for focusing and manipulating beams of negatively charged particles. The value of positive charge potential formed at the axis and the steady state of the space charge depending on plasma dynamical parameters of the system are determined experimentally [12, 13]. Electric field value reaching 600–1000 V/cm realized under experimental condition is determined. Such electric field strength is sufficient for creation of short-focus elements to be used in systems for manipulating intense beams of negative ions and electrons. Experimental results [14] demonstrate an attractive possibilities application positive space charged plasma lens with magnetic electron insulation for focusing and manipulating wide-aperture high-current no relativistic electron beams. For relatively low-current mode for which electron beam space charged less than positive space charged plasma lens, it realizes electrostatic focusing is passing electron beam. In case of high-current mode, when electron beam space charge much more than space charge plasma lens the lens operates in plasma mode to create transparent plasma accelerating electrode and compensate space charge propagating electron beam. The lens magnetic field in this case uses for effective

In experiment was demonstrated a perspective applications of positive space charged PL with magnetic electron insulation for focusing and manipulating wide aperture high-current no relativistic electron beams (Eb = 16 keV; Ib = 100 A) [14]. Particularly, it was shown experimentally that under focusing these beams maximal compression factor was up to 30x, and beam

.

The accelerator with closed electron drift is one of the kinds of the electric rocket engines and devices for ion plasma treatment of the surface material. However, the accelerators with closed electron drift and open (gas) walls were not researched for now in contrast to the well-known and widely used plasma accelerators with anode layer and accelerators with closed electron drift and dielectric walls [15, 16]. Therefore, this type accelerator could be interested in manipulating high-current flow of charged particle as well as can be attractive for many different high-tech applications for potential devices of low-cost and compact thrusters. More than, it has some advantages since the wall absence leads to exclusion of the wall material inclusions into the ion beam and exclusion of the secondary electrons formation due to emission and thus

The sample of plasma accelerator with closed electron drift and open walls is shown in Figure 9 (left). This sample of cylindrical Hall-type plasma ion source that produced ion plasma flow converging toward the axis system was created for the properties exploration [17]. The discharge in the system burns due to ionization of the working gas by the electrons. Electrons are magnetized and formed stable negative space charge. The created ions accelerated from ionization zone to the cathode. As follow from discharge geometry an accumulation of ion space charge occurs as it is in the positive space charge lens (see above). The main part of generated ions leaves the system across radius, along with system axis, due to jets can appear

3. Plasma accelerator with open gas wall and closed electron drift

focusing beam.

current density at the focus was about 100 A/cm<sup>2</sup>

276 Plasma Science and Technology - Basic Fundamentals and Modern Applications

to conservation of the plasma electrons dynamics.

Figure 9. (a) Experimental sample: 1- cathode, 2- anode, and 3- permanent magnets system; (b)plasma jet in high-current operation mode.

Figure 10. (a) Model of discharge gap: 1-anode, 2-cathode, 3-permanent magnets system; (b) potential distribution in the gap for different parameters a value.

on the edges of system. As follow from experiment [17, 18], the accelerator has two operating modes: low-current with narrow anode layer and clear-cut plasma flow and high-current when plasma fills the entire volume of the accelerator. The transfer to the high-current mode occurs under influence of two parameters: worked gas pressure and applied voltage. In high-current quasi-neutral plasma mode of accelerator operation, plasma jet is observed (see, Figure 9 right). The preliminary results show that along the jet axis potential drop arise, which can be used for ion beam accelerating. The radial studies of plasma flow along system axis showed the significant increasing current density on the axis. It may indicate on plasma acceleration in that direction.

#### 3.1. One-dimensional hydrodynamic and hybrid model

We will consider discharge gap, where ions production occurs due to ionization of the working gas by electrons (see Figure 10a). Electrons are magnetized, move along magnetic strength lines, and drift slowly to anode due to collisions. Ions are free and accelerated by electric field move to the system axis. Will assume that the discharge current density in gap volume is the sum of the ion and electron components:

$$j\_e + j\_i = j\_d \tag{8}$$

δ ¼ r<sup>e</sup> φ<sup>A</sup>

This can be due to the electron space charge dominated at the accelerator exit.

come back to potential distribution (14) [18, 19].

∂f o <sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>v</sup><sup>0</sup>

f <sup>0</sup>ð Þ¼ 0; v; t

– potential of ionization.

Right part (Eq. (16)) we can write:

conditions:

wherein right parts take into account only single ionizations:

<sup>∂</sup><sup>x</sup> ¼ �h i <sup>σ</sup>ieve nef <sup>0</sup>,

ð Þ <sup>2</sup>πMT <sup>3</sup>=<sup>2</sup> exp � Mv<sup>2</sup>

∂f 0

1

where for ion density and current density is valid:

This expression coincides with one for classical anode layer [20] accurate within √2. Note in case when parameter a<1(the gap length less than δ) potential drop is not completed (see Figure 1b). For case a > 1, when the gap length d > δ potential drop exceed applied potential.

Another case is quasi-neutral plasma: ni ≈ ne, that correspond to high-current mode [19]. In this case from the solution follows one interesting conclusion: when electron density does not change along the gap we could get generalization condition of self-sustained discharge in crossed electrical and magnetic fields (EXH), taking into consideration both electron and ion dynamics peculiarity. More than in this case solution for high-current mode is reduced to form (Eq. (14)) also. Even if we consider more general model description, assuming that electron heating losing occurs mostly by different kind of collisions it can show that potential distribution along gap weakly depend on electron temperature saving close to parabolic distribution. And under assumption that time of energetic losses equal electron lifetime in the gap we again

For a more detailed study of the influence of ion dynamics on the system process, we applied a one-dimension mode hybrid model used for calculations of a span mode with neutral-particle ionization. In this model dynamics of ions and neutrals is described by kinetic equations,

> ∂f i <sup>∂</sup><sup>t</sup> <sup>þ</sup> vi

where f0, fi distribution function of neutrals and ions consequently that satisfies boundary

where σ max – maximal ionization cross-section, ve(Te) – average electron thermal velocity, Ui

For electron part we will use hydrodynamic description and solve (Eqs. (8), (10), and(13)),

The result of simulation hybrid model after achieving dynamic equilibrium by system in comparisons with hydrodynamic model is shown in Figure 11. One can see that taking into

ð vif <sup>i</sup>

2T

ni ¼ ð f i dv, ji ¼ ∂f i ∂x þ e <sup>M</sup> <sup>E</sup> ∂f i

� �, <sup>v</sup> <sup>&</sup>gt; <sup>0</sup>; f <sup>0</sup>ð Þ¼ <sup>0</sup>; <sup>v</sup>; <sup>t</sup> <sup>0</sup>, <sup>v</sup> <sup>&</sup>lt; <sup>0</sup>; f <sup>i</sup>

h i¼ σieve σmaxveð Þ Te exp ð Þ �Ui=Te , (18)

<sup>∂</sup><sup>v</sup> <sup>¼</sup> h i <sup>σ</sup>ieve nef <sup>0</sup> (16)

Modeling of Novel Plasma-Optical Systems http://dx.doi.org/10.5772/intechopen.77512

dv (19)

ð Þ¼ 0; v; t 0 (17)

� � ffiffiffiffiffiffi 2ν<sup>e</sup> νi

s

(15)

279

where ji, je are ion and electron current density consequently:

$$j\_i(\mathbf{x}) = e\nu\_i \int\_0^\mathbf{x} n\_c(\mathbf{x})d\mathbf{x} \tag{9}$$

$$j\_\epsilon(\mathbf{x}) = e\mu\_\perp \left( n\_\epsilon E(\mathbf{x}) - \frac{d}{d\mathbf{x}} (n\_\epsilon T\_\epsilon) \right) \tag{10}$$

<sup>ν</sup>i-is the ionization frequency, <sup>μ</sup><sup>⊥</sup> <sup>¼</sup> <sup>e</sup>ν<sup>e</sup> mω<sup>2</sup> eH electron transverse mobility, E xð Þ¼� <sup>d</sup><sup>φ</sup> dx-electric field, φ - potential, ν<sup>e</sup> is the frequency of elastic collisions with neutrals and ions, ωeH is the electron cyclotron frequency, Te – electron temperature that could write in form [15]:

$$T\_{\epsilon}(\mathbf{x}) = \frac{\beta}{j\_{\epsilon}(\mathbf{x})} \int\_{0}^{\mathbf{x}} j\_{\epsilon} \frac{d\rho}{d\mathbf{s}} ds \tag{11}$$

for ion density we could write:

$$m\_i(\mathbf{x}) = \sqrt{\frac{M}{2\varepsilon}} \int\_0^\mathbf{x} \frac{n\_e(\mathbf{s})\nu\_i d\mathbf{s}}{\sqrt{\varrho(\mathbf{x}) - \varrho(\mathbf{s})}} \tag{12}$$

and Poisson equation closed this system of equations:

$$n\_{\epsilon} - n\_i = \frac{1}{4\pi\epsilon} \varphi''\tag{13}$$

In some cases equations system Eqs.(9) and (13) allows exact analytical solutions. Dimensionless these equations and assume that position x = 0 correspond to anode and initial potential on this boundary is equal applied voltage, position x = 1 correspond to cathode, and ion current on the cathode is equal to discharge current. If we neglect diffusion for case ne>> ni, that corresponds to a low-current mode, we can get exact solution in form [18, 19]:

$$\varphi = a \left( \left( \mathbf{x} - \mathbf{1} \right)^2 - 1 \right) + 1,\\ a = \frac{\nu\_i d^2}{2 \mu\_\perp \varphi\_a} \tag{14}$$

where d is gap length. One can see that at the parameter a = 1, the total applied potential falls inside the gap (see Figure 1b). For this case, gap length is equal: d ¼ ffiffiffiffiffiffiffiffiffiffi 2μ⊥φ<sup>A</sup> νi q . Under assumption that electrons originated from the gap only by impact ionization, and go to the anode due to classical transverse mobility, last expression can rewritten in the form:

Modeling of Novel Plasma-Optical Systems http://dx.doi.org/10.5772/intechopen.77512 279

$$
\delta = \rho\_\epsilon(\varphi\_A) \sqrt{\frac{2\nu\_\epsilon}{\nu\_i}} \tag{15}
$$

This expression coincides with one for classical anode layer [20] accurate within √2. Note in case when parameter a<1(the gap length less than δ) potential drop is not completed (see Figure 1b). For case a > 1, when the gap length d > δ potential drop exceed applied potential. This can be due to the electron space charge dominated at the accelerator exit.

Another case is quasi-neutral plasma: ni ≈ ne, that correspond to high-current mode [19]. In this case from the solution follows one interesting conclusion: when electron density does not change along the gap we could get generalization condition of self-sustained discharge in crossed electrical and magnetic fields (EXH), taking into consideration both electron and ion dynamics peculiarity. More than in this case solution for high-current mode is reduced to form (Eq. (14)) also. Even if we consider more general model description, assuming that electron heating losing occurs mostly by different kind of collisions it can show that potential distribution along gap weakly depend on electron temperature saving close to parabolic distribution. And under assumption that time of energetic losses equal electron lifetime in the gap we again come back to potential distribution (14) [18, 19].

For a more detailed study of the influence of ion dynamics on the system process, we applied a one-dimension mode hybrid model used for calculations of a span mode with neutral-particle ionization. In this model dynamics of ions and neutrals is described by kinetic equations, wherein right parts take into account only single ionizations:

$$\frac{\partial f\_o}{\partial t} + \upsilon\_0 \frac{\partial f\_0}{\partial \mathbf{x}} = - \langle \sigma\_{it} \upsilon\_\epsilon \rangle n\_\epsilon f\_{0'} \frac{\partial f\_i}{\partial t} + \upsilon\_i \frac{\partial f\_i}{\partial \mathbf{x}} + \frac{e}{M} E \frac{\partial f\_i}{\partial \mathbf{v}} = \langle \sigma\_{it} \upsilon\_\epsilon \rangle n\_\epsilon f\_0 \tag{16}$$

where f0, fi distribution function of neutrals and ions consequently that satisfies boundary conditions:

$$f\_0(0, \mathbf{v}, t) = \frac{1}{\left(2\pi M T\right)^{3/2}} \exp\left(-\frac{M\mathbf{v}^2}{2T}\right), \mathbf{v} > 0; f\_0(0, \mathbf{v}, \mathbf{t}) = 0, \mathbf{v} < 0; f\_i(0, \mathbf{v}, \mathbf{t}) = 0 \tag{17}$$

Right part (Eq. (16)) we can write:

move to the system axis. Will assume that the discharge current density in gap volume is the

ð x

0

φ - potential, ν<sup>e</sup> is the frequency of elastic collisions with neutrals and ions, ωeH is the electron

0 j e dφ

neð Þ<sup>s</sup> <sup>ν</sup>ids ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>þ</sup> <sup>1</sup>, a <sup>¼</sup> <sup>ν</sup>id<sup>2</sup>

2μ⊥φ<sup>a</sup>

� �

dx ð Þ neTe

electron transverse mobility, E xð Þ¼� <sup>d</sup><sup>φ</sup>

<sup>d</sup> (8)

neð Þx dx (9)

ds ds (11)

<sup>φ</sup>ð Þ� <sup>x</sup> <sup>φ</sup>ð Þ<sup>s</sup> <sup>p</sup> (12)

φ<sup>00</sup> (13)

ffiffiffiffiffiffiffiffiffiffi 2μ⊥φ<sup>A</sup> νi q

(10)

(14)

. Under assump-

dx-electric field,

j <sup>e</sup> þ j <sup>i</sup> ¼ j

<sup>e</sup>ð Þ¼ <sup>x</sup> <sup>e</sup>μ<sup>⊥</sup> neE xð Þ� <sup>d</sup>

j i ð Þ¼ x eν<sup>i</sup>

mω<sup>2</sup> eH

cyclotron frequency, Te – electron temperature that could write in form [15]:

nið Þ¼ x

corresponds to a low-current mode, we can get exact solution in form [18, 19]:

<sup>φ</sup> <sup>¼</sup> a xð Þ � <sup>1</sup> <sup>2</sup> � <sup>1</sup> � �

inside the gap (see Figure 1b). For this case, gap length is equal: d ¼

to classical transverse mobility, last expression can rewritten in the form:

and Poisson equation closed this system of equations:

Teð Þ¼ <sup>x</sup> <sup>β</sup> j <sup>e</sup>ð Þx ð x

> ffiffiffiffiffi M 2e r ð x

> > 0

ne � ni <sup>¼</sup> <sup>1</sup>

In some cases equations system Eqs.(9) and (13) allows exact analytical solutions. Dimensionless these equations and assume that position x = 0 correspond to anode and initial potential on this boundary is equal applied voltage, position x = 1 correspond to cathode, and ion current on the cathode is equal to discharge current. If we neglect diffusion for case ne>> ni, that

where d is gap length. One can see that at the parameter a = 1, the total applied potential falls

tion that electrons originated from the gap only by impact ionization, and go to the anode due

4πe

sum of the ion and electron components:

<sup>ν</sup>i-is the ionization frequency, <sup>μ</sup><sup>⊥</sup> <sup>¼</sup> <sup>e</sup>ν<sup>e</sup>

for ion density we could write:

where ji, je are ion and electron current density consequently:

278 Plasma Science and Technology - Basic Fundamentals and Modern Applications

j

$$
\langle \sigma\_{\dot{\varkappa}} \upsilon\_{\varepsilon} \rangle = \sigma\_{\max} \upsilon\_{\varepsilon}(T\_{\varepsilon}) \exp \left( -\mathcal{U}\_{i}/T\_{\varepsilon} \right), \tag{18}
$$

where σ max – maximal ionization cross-section, ve(Te) – average electron thermal velocity, Ui – potential of ionization.

For electron part we will use hydrodynamic description and solve (Eqs. (8), (10), and(13)), where for ion density and current density is valid:

$$m\_i = \begin{cases} f\_i d\upsilon\_i \dot{\jmath}\_i = \int \upsilon\_i f\_i d\upsilon \end{cases} \tag{19}$$

The result of simulation hybrid model after achieving dynamic equilibrium by system in comparisons with hydrodynamic model is shown in Figure 11. One can see that taking into

4.5 cm, spaced by some distance from each other. For ions and neutrals description we use

We solved this equation by splitting on the Vlasov equation for finding ions and neutrals

and to correct the found trajectories taking into account the collision integral in which we took

Dt <sup>¼</sup> St f i,n

To solve these equations, the PIC method [10] with Boris scheme [11] was used to avoid singularities at the axis. For initial electric field distribution was taken electric field in the

in this field. The probability of a collision of a particle with energy ε<sup>j</sup> during time Δt was found

where σ(ε) – collision cross-section (elastic, ionization or excitation), nj – density of similar

collision has occurred. It is determine the ratio of the cross-sections with the random number generator, which collision has occurred – elastic, excitation, or ionization. Independent of this particle parameters change or new ion adds in computational box. The emerging ions begin to move toward the system axis. The evolution of all particles that are in the modeling region is traced at each time step. For this motion, equations were solved, and new velocities and positions of the particles were found. Particles that move out the modeling box boundaries are excluded from consideration. After sufficiently long-time particle density distribution was found. The ion charge density and current density are calculated from coordinates and veloc-

� �,j rð Þ¼ ; <sup>t</sup> <sup>X</sup>

Pj ¼ 1 � exp �vjΔtσ ε<sup>j</sup>

from interval [0,1] with the help of a random number generator. If β < Pj

into account the processes of ionization and elastic and inelastic collisions:

The Vlasov equations were solved by the method of characteristics:

dvk ! dt <sup>¼</sup> qk <sup>M</sup> <sup>E</sup> ! þ 1 c ½ � vk � B � �,

Df i,n

1 с ½ � v � B � � ∂f <sup>i</sup>

1 с ½ � v � B � � ∂f <sup>i</sup>

∂vi

¼ St f i,n

∂vi

drk ! dt <sup>¼</sup> vk

<sup>r</sup>ln ð Þ rc=ra . The Monte-Carlo method was used for modeling of ionization

� �nj rj !Þ

. To determine the probability of collision a random number β is chosen

j qj vjð Þ<sup>t</sup> R r!; rj

! ð Þ<sup>t</sup>

� � (25)

n o (22)

� � � (24)

n o (20)

Modeling of Novel Plasma-Optical Systems http://dx.doi.org/10.5772/intechopen.77512 281

¼ 0 (21)

! (23)

, then assumed that

Boltzmann kinetic equation:

plasma absence: E rð Þ¼ Ua

from expression [21]:

particles at the point rj

ities of particles according to formulas:

rð Þ¼ r; t

1 V X j qj R r!; rj ! ð Þ<sup>t</sup>

trajectories:

∂f i,n <sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>v</sup> ! i,n ∂f i,n ∂ r ! þ e <sup>M</sup> <sup>E</sup> <sup>þ</sup>

> ∂f i,n <sup>∂</sup><sup>t</sup> <sup>þ</sup> <sup>v</sup> ! i,n ∂f i,n ∂ r ! þ e <sup>M</sup> <sup>E</sup> <sup>þ</sup>

Figure 11. Comparison simulation results for hybrid and hydrodynamic models:(a) potential distribution in gap, (b) electron density.

account span mode with single ionization does not have a big impact on the result potential distribution (see Figure 11a). Electron density distribution (Figure 11b) looks more consistent along gap in hybrid model as compared to a hydrodynamic model. It is the result of influence on the system processes of neutrals dynamics and ionization, which are not taken into account in the hydrodynamic model.

Figure 12 shows ion density changing in the gap during the time in high-current mode (under applied anode potential equal 1200 V). It can be seen that at first ions number increases in nearanode region and remains almost constant in gap, then it increases almost linearly throughout the gap, and finally increases sharply at the cathode region.

Note that for correct description (especially high-current mode), it is necessary to model ionization and plasma creation, as well as motion of neutrals and formed ions in the whole volume of accelerator, thus need consider two-dimensional model.

#### 3.2. Two-dimensional model

We will consider cylindrical geometry, where the anode is a cylinder with diameter 6.7 cm and applied potential about 1–2.5 kV, and cathode consist from two cylinders with diameter

Figure 12. Ion density in the gap in high-current mode depending on time.

4.5 cm, spaced by some distance from each other. For ions and neutrals description we use Boltzmann kinetic equation:

$$\frac{\partial f\_{i,n}}{\partial t} + \overrightarrow{\overline{\sigma}}\_{i,n} \frac{\partial f\_{i,n}}{\partial \cdot \overrightarrow{r}} + \frac{e}{M} \left( E + \frac{1}{c} [\boldsymbol{\upsilon} \times \boldsymbol{B}] \right) \frac{\partial f\_i}{\partial \boldsymbol{\upsilon}\_i} = \text{St} \left\{ f\_{i,n} \right\} \tag{20}$$

We solved this equation by splitting on the Vlasov equation for finding ions and neutrals trajectories:

$$\frac{\partial f\_{i,n}}{\partial t} + \overrightarrow{\overline{\upsilon}}\_{i,n} \frac{\partial f\_{i,n}}{\partial \overrightarrow{r}} + \frac{e}{M} \left( E + \frac{1}{c} [\upsilon \times B] \right) \frac{\partial f\_i}{\partial \upsilon\_i} = 0 \tag{21}$$

and to correct the found trajectories taking into account the collision integral in which we took into account the processes of ionization and elastic and inelastic collisions:

$$\frac{Df\_{i,n}}{Dt} = St \begin{Bmatrix} f\_{i,n} \end{Bmatrix} \tag{22}$$

The Vlasov equations were solved by the method of characteristics:

account span mode with single ionization does not have a big impact on the result potential distribution (see Figure 11a). Electron density distribution (Figure 11b) looks more consistent along gap in hybrid model as compared to a hydrodynamic model. It is the result of influence on the system processes of neutrals dynamics and ionization, which are not taken into account

Figure 11. Comparison simulation results for hybrid and hydrodynamic models:(a) potential distribution in gap, (b)

Figure 12 shows ion density changing in the gap during the time in high-current mode (under applied anode potential equal 1200 V). It can be seen that at first ions number increases in nearanode region and remains almost constant in gap, then it increases almost linearly throughout

Note that for correct description (especially high-current mode), it is necessary to model ionization and plasma creation, as well as motion of neutrals and formed ions in the whole

We will consider cylindrical geometry, where the anode is a cylinder with diameter 6.7 cm and applied potential about 1–2.5 kV, and cathode consist from two cylinders with diameter

in the hydrodynamic model.

electron density.

3.2. Two-dimensional model

the gap, and finally increases sharply at the cathode region.

280 Plasma Science and Technology - Basic Fundamentals and Modern Applications

Figure 12. Ion density in the gap in high-current mode depending on time.

volume of accelerator, thus need consider two-dimensional model.

$$\frac{d\overrightarrow{v\_k}}{dt} = \frac{q\_k}{M} \left( \overrightarrow{E} + \frac{1}{c} [v\_k \times B] \right), \frac{d\overrightarrow{r\_k}}{dt} = \overrightarrow{v\_k} \tag{23}$$

To solve these equations, the PIC method [10] with Boris scheme [11] was used to avoid singularities at the axis. For initial electric field distribution was taken electric field in the plasma absence: E rð Þ¼ Ua <sup>r</sup>ln ð Þ rc=ra . The Monte-Carlo method was used for modeling of ionization in this field. The probability of a collision of a particle with energy ε<sup>j</sup> during time Δt was found from expression [21]:

$$P\_{\vec{\jmath}} = 1 - \exp\left(-\upsilon\_{\vec{\jmath}} \Delta t \sigma(\varepsilon\_{\vec{\jmath}}) n\_{\vec{\jmath}} \left(\overrightarrow{r\_{\vec{\jmath}}}\right)\right) \tag{24}$$

where σ(ε) – collision cross-section (elastic, ionization or excitation), nj – density of similar particles at the point rj . To determine the probability of collision a random number β is chosen from interval [0,1] with the help of a random number generator. If β < Pj , then assumed that collision has occurred. It is determine the ratio of the cross-sections with the random number generator, which collision has occurred – elastic, excitation, or ionization. Independent of this particle parameters change or new ion adds in computational box. The emerging ions begin to move toward the system axis. The evolution of all particles that are in the modeling region is traced at each time step. For this motion, equations were solved, and new velocities and positions of the particles were found. Particles that move out the modeling box boundaries are excluded from consideration. After sufficiently long-time particle density distribution was found. The ion charge density and current density are calculated from coordinates and velocities of particles according to formulas:

$$\rho(r,t) = \frac{1}{V} \sum\_{\vec{j}} q\_{\vec{j}} \mathbb{R}\left(\overrightarrow{r}, \overrightarrow{r\_{\vec{j}}}\,(t)\right),\\\dot{j}(r,t) = \sum\_{\vec{j}} q\_{\vec{j}} \overline{v}\_{\vec{j}}(t) \mathbb{R}\left(\overrightarrow{r}, \overrightarrow{r\_{\vec{j}}}\,(t)\right) \tag{25}$$

where R(r,rj ) – usual standard PIC – core, that characterizes particle size and shape and charges distribution in it. After that, the Poisson equation was solved, and new electric field distribution was found. Since electrons are magnetized we consider their movement in radial plane only, thus can solve for electrons one-dimensional hydrodynamic equations on each layer at z separately. Solve it we find electron density, calculate electric field on each layer and correct particle trajectories. After that, the procedure was repeated again. Modeling time is large enough for establish of ion multiplication process. The formation of the sufficient number of ions is possible due to magnetic field presence, which isolates anode from the cathode. Ions practically do not feel the magnetic field action and move from anode to the axis, where create a space charge, first in the center of the system. Electrons move along the magnetic field strength line, but due to collisions with neutrals, they start moving across the magnetic field. An internal electric field is formed, which slow down the ions and pushes out them from the volume along system axis. Figure 13 shows results of modeling high-current mode (Ua = 1.2 kV, pressure 0.15 Pa, and magnetic field at the axis is 0.03 T). Figure 13a shows how the ions number to axis increases when ionization process is steady-state. One can see that number of ions increase not only to axis but also along axis from center to edge too. Figure 13b shows ion space charge distribution for different time step. One can see that ions create space charge in center of the system first, but then under electric field action they leave center and move along z-axis in both direction.

layer potential drop exceed applied potential. The generalization condition of self-sustained discharge in crossed ExH fields taking into consideration both electron and ion dynamic peculiarity were obtained. The performed modeling showed that in high-current mode the ions moving to the system center and then along the axis in both directions are able to create space charge. Experimental model of accelerator that formed ion flow converging toward the axis system was created [17, 18]. In high-current mode of accelerator operation is observed plasma jet. It is shown at the jet axis forms potential drop that could be used for ion beam accelerating.

Modeling of Novel Plasma-Optical Systems http://dx.doi.org/10.5772/intechopen.77512 283

Note also that the presented plasma device is attractive for many different high-tech practical applications, for example, like PL with positive space cloud for focusing negative, intense charge particles beams (electrons and negative ions), and for potential devices small rocket engines.

Now the vacuum-plasma technologies are widely used in erosion plasma sources, such as cathodic arc and laser-produced plasma, for thin films synthesis, and coatings with control properties. However, the micro-droplets present in the formed ion plasma flow restrict the applicability of this method of film synthesis. The micro-droplets component of erosion for the majority of metals is an essential part of general losses of the cathode material in a vacuum-arc, comparable with ion component. Therefore, for prevention of effect of micro-droplets on a substrate, it is necessary to eliminate droplets. The formation and propagation of microdroplets, and also the mechanisms of decreasing of their effects on quality and rate deposition films and coatings were investigated in detail in works [22–26]. It is known from several approaches micro-droplets density reduction in ion plasma flows. Therefore, proposed methods do not provide the complete solution of the problem of micro-droplets elimination. In paper Anders [26], the conclusion has been formulated that micro-droplets cannot be evaporated in the arc plasma flow without additional energy source. A new approach to the elimination or reduction of micro-droplets from the dense metal plasma flow based on the use of a cylindrical electrostatic PL configuration to generate an energetic radial electron beam within the low-energy ion plasma flow has been proposed and described in [27, 28]. The pumping of energy into arc plasma flow by the self-consistently formed radial beam of highenergy electrons for evaporation of micro-droplets could serve as additional energy source. The beam is formed by double layer, appeared in a cylindrical channel of the novel plasmaoptical system in crossed radial electrical and longitudinal magnetic fields. Here we detailed

4. The model of the plasma-dynamical filter for micro-droplets

describe the model of device for filtering dense plasma flow from micro-droplets.

We will consider electrostatic PL configuration through which a low-energy arc ion plasma flow passes. Figure 14 shows experimental device (left) and simplified model (right). A dense arc plasma flow with micro-droplets is propagated from the cathodic arc source and passes

The experimental results are in good accordance with theory data.

eliminations

4.1. Model approach

#### 3.3. Conclusion remarks

First, the original approach to use plasma accelerators with closed electron drift and open walls for creation cost-effective low-maintenance plasma device for production converging toward axis accelerating ion beam was described. Based on the idea of continuity of current transferring in the system are found exact analytical solutions describing electric potential distribution along acceleration gap. It was shown that potential distribution is parabolic for different operation modes as in low-current mode as well as in high-current quasi-neutral plasma mode and cannot depend on electron temperature. It is found under conditions that everything electrons originated within the gap by impact ionization only, and go out at the anode due to mobility in transverse magnetic field, the condition full potential drop in the accelerating gap corresponds to equality gap length to the anode layer thickness. In case when the gap length less than anode layer thickness potential drop is not completed. For case when the gap length more than anode

Figure 13. (a) Ions number dependence on r and z (r = 0,z = 0–center of the system); (b) ion space charge for time step 70 (left) and 340(right).

layer potential drop exceed applied potential. The generalization condition of self-sustained discharge in crossed ExH fields taking into consideration both electron and ion dynamic peculiarity were obtained. The performed modeling showed that in high-current mode the ions moving to the system center and then along the axis in both directions are able to create space charge. Experimental model of accelerator that formed ion flow converging toward the axis system was created [17, 18]. In high-current mode of accelerator operation is observed plasma jet. It is shown at the jet axis forms potential drop that could be used for ion beam accelerating. The experimental results are in good accordance with theory data.

Note also that the presented plasma device is attractive for many different high-tech practical applications, for example, like PL with positive space cloud for focusing negative, intense charge particles beams (electrons and negative ions), and for potential devices small rocket engines.
