*3.1.1 Hydrodynamic model*

To clarify the obtained experimental data, primarily a one-dimensional hydrodynamical theoretical model was developed (see **Figure 8a**). Here x-axis is directed along the radius of the system. We have considered the discharge gap, where ions production appear due to working gas ionization by electrons. 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.

The closed system of equations that describes such system in hydrodynamic approximation in stationary state has the form:

$$j\_e + j\_i = j\_d \tag{1}$$

$$j\_i(\mathbf{x}) = e\nu\_i \int\_0^\infty n\_\epsilon(\mathbf{x})d\mathbf{x} \tag{2}$$

$$j\_{\epsilon}(\mathbf{x}) = \epsilon \mu\_{\perp} \left( n\_{\epsilon} E(\mathbf{x}) - \frac{d}{d\mathbf{x}} (n\_{\epsilon} T\_{\epsilon}) \right) \tag{3}$$

$$T\_{\epsilon}(\mathbf{x}) = \frac{\beta}{j\_{\epsilon}(\mathbf{x})} \int\_{0}^{\mathbf{x}} j\_{\epsilon} \frac{d\phi}{ds} ds \tag{4}$$

$$n\_i(\mathbf{x}) = \sqrt{\frac{M}{2\varepsilon}} \Big|\_{0}^{\mathbf{x}} \frac{n\_\varepsilon(\mathbf{s})\nu\_i d\mathbf{s}}{\sqrt{\phi(\mathbf{x}) - \phi(\mathbf{s})}} \tag{5}$$

$$n\_{\epsilon} - n\_i = \frac{1}{4\pi\epsilon} \phi''\tag{6}$$

here *ji*, *je*, *ni*, *ne* - are ion and electron current density and density consequently, <sup>ν</sup><sup>i</sup> - 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 - electric field: *E x*ð Þ¼� *<sup>d</sup><sup>ϕ</sup> dx*, *ϕ* - potential, *ν<sup>e</sup>* is the frequency of elastic collisions with neutrals and ions, *ωeH* – the electron cyclotron frequency, Te – electron temperature.

### **Figure 8.**

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

### *Plasma Science and Technology*

In the framework of this model, both exact analytical and numerical solutions were obtained [7, 10]. Based on the idea of continuity of current transferring in the system and assumption that the discharge current density in gap volume is the sum of the ion and electron components are found exact analytical solutions describing electric potential distribution along acceleration gap, if Te = const. It has the next form in low-current mode:

$$
\phi = a \left( (\mathbf{x} - \mathbf{1})^2 - \mathbf{1} \right) + \mathbf{1}, \tag{7}
$$

where *<sup>a</sup>* is parameter equals *<sup>a</sup>* <sup>¼</sup> *<sup>ν</sup>id*<sup>2</sup> 2*μ*⊥*ϕ<sup>a</sup>* , d – gap length. The potential distribution in the gap for different value of the parameter a is present in **Figure 8b**. It is found under conditions when all electrons originated within the gap by impact ionization only and go out at the anode due to mobility in transverse magnetic field, the condition of full potential drop in the accelerating gap corresponds to equality gap length to the anode layer thickness. In case when the gap length is less than anode layer thickness, potential drop is not completed, and positive space charge dominated. For case when the gap length is more than anode layer, potential drop exceeds applied potential. This can be due to electron space charge dominated at the accelerator exit. It was shown that potential distribution is parabolic for different operation modes as in low-current mode well as in high current quasi neutral plasma mode and weakly depends on electron temperature. For high-current mode solution has form like expression (7):

$$\phi(\mathbf{x}) = \mathbf{1} + \frac{a^2}{2f^2} \left( (\mathbf{x} - \mathbf{1})^2 - \mathbf{1} \right) \tag{8}$$

where *f* ¼ *νid* ffiffiffiffiffiffiffi *M* 2*eϕ<sup>a</sup>* q – describes impact of ion density. Note, that if *a* = 2*f* <sup>2</sup> we obtain (7). This condition can be rewritten in form:

$$\frac{\tau\_{ed}}{\tau\_{id}^2} = 2\nu\_i \text{ or } \tau\_{ed}\nu\_i = 2\tau\_{id}^2 \nu\_i^2 = 1 \tag{9}$$

here *<sup>τ</sup>ed* <sup>¼</sup> *<sup>d</sup> <sup>μ</sup>*⊥*<sup>E</sup>* – electron lifetime, *<sup>τ</sup>id* <sup>¼</sup> *<sup>d</sup> vid* – ion lifetime. Indeed (9), is some generalization condition of self-sustained discharge in crossed E � H fields taking into consideration both electron and ion dynamic peculiarities.

To clarify the effect of electron temperature on the characteristics of accelerating layer, it was assumed that the electrons received energy from the electric field, thus *Te* ¼ *β* � *φ*, where 0<β≤1. In this assumption, the solution has the form:

$$\phi(\mathbf{x}) = \frac{a}{\mathbf{1} + \boldsymbol{\beta}} \mathbf{x}(\mathbf{x} - \mathbf{2}) + \mathbf{1} \tag{10}$$

Consider a more complete description, assuming that heat loss occurs due to different types of collision. Entering the characteristic time τ<sup>0</sup> – energy loss by collision, expression for electron temperature (4) can be represented as:

$$T\_{\epsilon} = \frac{j\_d E \tau\_0}{e n\_{\epsilon}} \left( \mathbf{1} - e^{-t\_{\hat{\gamma}\_0}} \right) \tag{11}$$

If we assume that τ<sup>0</sup> is equal to electron lifetime τed, we obtain:

*The Emerging Field Trends Erosion-Free Electric Hall Thrusters Systems DOI: http://dx.doi.org/10.5772/intechopen.99096*

$$T\_{\epsilon} = \frac{j\_d d}{\mu\_\perp en\_\epsilon} \tag{12}$$

Thus, the second term in expression (3) disappears and we come back to the solution (7). The numerical solution of system Eq. (1)–(6) showed that the electron density changes extraordinarily little along the gap with typical operating parameters, so our assumption about ne = const is justified.

So, even in such a simplified model, we obtain a result explaining the appearance a space charge and finding the optimal length of accelerating gap. Nevertheless, although the hydrodynamic model can well describe the dynamics of the electron and ionic components, it does not make allowance for ionization state processes, as well as the influence of neutral atoms in the working gas. A purely kinetic description cannot also be used because of a significant difference between the velocities of electrons and ions. Therefore, a description using the hybrid model may be an optimal decision.

## *3.1.2 Hybrid model*

In the framework of this model [18, 19], the hydrodynamic description is used for the electron component, and the kinetic description for the ionic and neutral ones. This approach also allows the limited stay time of ions in the system to be considered. A one-dimensional model was considered with regard for only the single ionization. In this case, we can write the following equations for neutrals and ions, respectively.

$$\frac{\partial f\_o}{\partial t} + v\_0 \frac{\partial f\_0}{\partial \mathbf{x}} = -\langle \sigma\_{i\epsilon} v\_{\epsilon} \rangle n\_{\epsilon} f\_0 \tag{13}$$

$$\frac{\partial f\_i}{\partial t} + v\_i \frac{\partial f\_i}{\partial \mathbf{x}} + \frac{e}{M} E \frac{\partial f\_i}{\partial v} = \langle \sigma\_{i\epsilon} \nu\_{\epsilon} \rangle n\_{\epsilon} f\_{\ 0} \tag{14}$$

here f0, fi – distribution function of neutrals and ions consequently, that satisfy boundary conditions:

$$f\_0(0, v, t) = \frac{1}{\left(2\pi MT\right)^{\frac{3}{2}}} \exp\left(-\frac{Mv^2}{2T}\right), v > 0$$

$$f\_0(0, v, t) = 0, v < 0; \tag{15}$$

$$f\_i(0, v, t) = 0$$

In (13) and (14) right part expression is:

$$
\langle \sigma\_{i\epsilon} v\_{\epsilon} \rangle = \sigma\_{\max} v\_{\epsilon}(T\_{\epsilon}) \exp\left(\frac{-U\_i}{T\_{\epsilon}}\right) \tag{16}
$$

here σmax – maximal ionization cross-section, v*e*(*Te*) – average electron thermal velocity, *Ui* – ionization potential.

For electrons we can use hydrodynamic model and to solve system (1)–(6), where instead Eq. (2) and (5), the equations are used:

$$j\_i = \int v\_i f\_i dv \text{ and } n\_i = \int f\_i dv \tag{17}$$

The numerical solution of the system of Eqs. (13)–(17) showed that in the stationary case the difference between two models is insignificant (see **Figure 9a**). A comparison between the results of both models obtained in model experiments testifies to an insignificant influence of the neutral component of a working gas on the formation of the potential drop across the discharge gap for the examined initial conditions.

To solve the Eqs. (13) and (14), we chose a time step that satisfies the condition Δ*t* < <sup>Δ</sup>*<sup>x</sup>=vimax*, where Δx – spatial step, *vimax* – maximal ion velocity. The **Figure 10** shows ion density changing in the gap during the time in high-current mode (under applied anode potential equal 1200 V). At first ions number increases in near anode 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, collisions, and plasma creation, as well as motion of neutrals and formed ions in the whole volume of accelerator, thus need consider two-dimensional hybrid model.

### *3.1.3 2D-hybrid model and results*

In the framework of this model the kinetic description is used in cylindrical geometry for the ionic and neutral components and the hydrodynamic

**Figure 9.**

*a) Comparison between the results of numerical calculations for the potential distribution in the discharge gap in the hydrodynamic and hybrid models. b) Ions number dependence on r and z (r = 0, z = 0 – center of the system).*

**Figure 10.** *Ion distribution in the gap on different time steps.*

*The Emerging Field Trends Erosion-Free Electric Hall Thrusters Systems DOI: http://dx.doi.org/10.5772/intechopen.99096*

one-dimension on each special layer zi<z<zi+1 description for the electron ones. Thus, for ions and neutrals description we use Boltzmann kinetic equation:

$$\frac{\partial f\_{i,n}}{\partial t} + \overrightarrow{\nu}\_{i,n} \frac{\partial f\_{i,n}}{\partial \overrightarrow{r}} + \frac{e}{M} \left( E + \frac{1}{c} [\nu \times B] \right) \frac{\partial f\_i}{\partial v\_i} = \text{St} \left\{ \left. f\_{i,n} \right\} \right\} \tag{18}$$

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

$$\frac{\partial f\_{i,n}}{\partial t} + \overrightarrow{\nu}\_{i,n} \frac{\partial f\_{i,n}}{\partial \overrightarrow{r}} + \frac{e}{M} \left( E + \frac{1}{c} [\nu \times B] \right) \frac{\partial f\_i}{\partial v\_i} = \mathbf{0} \tag{19}$$

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

$$\frac{Df\_{i,n}}{Dt} = \text{St}\left\{ \left. f\_{i,n} \right\} \right. \tag{20}$$

The Vlasov equations were solved by the method of characteristics [20]:

$$\frac{d\overrightarrow{\boldsymbol{v}}\_{k}}{dt} = \frac{q\_{k}}{M} \left( \overrightarrow{\boldsymbol{E}} + \frac{\mathbf{1}}{c} [\boldsymbol{v}\_{k} \times \boldsymbol{B}] \right), \frac{d\overrightarrow{\boldsymbol{r}}\_{k}}{dt} = \overrightarrow{\boldsymbol{v}}\_{k} \tag{21}$$

To solve these equations the PIC method [21] with Boris scheme [22] was used to avoid singularities at the axis. For initial electric field distribution was taken electric field in the plasma absence: *E r*ð Þ¼ *Ua rln r*ð Þ *<sup>c</sup>=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 [23]:

$$P\_j = \mathbf{1} - \exp\left(-\nu\_j \Delta t \sigma(\varepsilon\_j) n\_j \left(\overrightarrow{r}\_j\right)\right) \tag{22}$$

here *σ*(*ε*) – collision cross-section (elastic, ionization or excitation), *nj* – density of similar particles at the point *r*<sup>j</sup> . To determine the probability of collision a random number *s* is chosen from interval [0, 1] with the help of a random number generator. If s<Pj, then assumed that collision has occurred. It is determined by the ratio of the cross-sections with the random number generator, which collision has occurred – elastic, excitation, or ionization. In dependence of this particle parameters change or new ion add in computational box. 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 quite a long time particle density distribution was found. The ion charge density and current density are calculated from coordinates and velocities particles according to formulas:

$$\rho(r,t) = \frac{1}{V} \sum\_{j} q\_{j} R\left(\overrightarrow{r}, \overrightarrow{r}\_{j}(t)\right),\\j(r,t) = \sum\_{j} q\_{j} v\_{j}(t) R\left(\overrightarrow{r}, \overrightarrow{r}\_{j}(t)\right) \tag{23}$$

where R(r, rj) – usual standard PIC – core, that characterizes particle size and shape and charge distribution in it. For a cylindrical coordinate system it has form [24]:

$$R\left(\left(r\_1, x\_k\right), \left(r\_j, x\_j\right)\right) = \begin{cases} \frac{1}{V\_i} \ast \frac{r\_{i+1}^2 - r\_j^2}{r\_{i+1}^2 - r\_i^2} \ast \frac{h\_x - \left|z\_k - x\_j\right|}{h\_x}, r\_i < r\_j < r\_{i+1}, \left|z\_k - x\_j\right| < h\_x\\ \frac{1}{V\_i} \ast \frac{r\_{i-1}^2 - r\_j^2}{r\_{i-1}^2 - r\_i^2} \ast \frac{h\_x - \left|z\_k - x\_j\right|}{h\_x}, r\_{i-1} < r\_i < r\_{i+1}, \left|z\_k - x\_j\right| < h\_x\\ 0, i \end{cases} \tag{24}$$

here *Vi* ¼ 2*πrihrhz* – volume of the cell, hr, hz – steps in the spatial coordinates.

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. 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

**Figure 11.** *Ion's trajectories (calculation for Ua = 1 kV, H = 0.03 T) for different time step a) Nt = 50, b) Nt = 200.*

*The Emerging Field Trends Erosion-Free Electric Hall Thrusters Systems DOI: http://dx.doi.org/10.5772/intechopen.99096*

line, but due to collisions with neutrals, they start move 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. The **Figure 9b** shows results of modeling highcurrent mode (Ua = 1.2 kV, pressure 0.15 Pa and magnetic field at the axis is 0.03 T). In is shown 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 along axis from center to edge too. In **Figure 11** the calculated ion trajectories for different time steps are shown. One can see that the ions that appear due to ionization move to center of the system. Coinciding on the system axis, they accumulate and create a positive space charge, and then diverge along the axis in both directions under the action of created own electric field. The ion space charge distribution for these case is shown in **Figure 12**.

The electrons trajectories for this case are shown in **Figure 13a**. One can see that the electrons are magnetized, moving along magnetic strength lines, and their trajectories are almost parallel to the surface of the anode. The **Figure 14** shows the potential distribution for different time steps. One can see that at the beginning of ionization, the potential drop is not complete in the gap and even has a negative sign in the center of the system. With ions number increasing, they coincide in the

**Figure 12.** *Ion space charge for time step 70 (a) and 340 (b).*

**Figure 13.** *Electron (a) and ions (b) trajectories in accelerator for H = 0.03 T.*

center of the system and form a positive space charge cloud, the potential of which even exceeds the applied potential Ua. This creates an electric field under which ions begin to move along the axis of symmetry in both directions, from the center to the edges, taking out with them part of space charge, reducing it in the center and creating bulks of space charge at the ends of the system (see **Figure 12**).

If we look at the potential distribution along the z-axis, we see that the maximum potential with distance from the center first declines, but then gradually begins to increase (see **Figure 15**). One can see that at a distance of 0.16 m from system center maximum potential is 222 V(0.22 Ua), while at a distance 0.21 m it is already 396 V, which is equal to 0.4 Ua, and current density of ion beam reached 0.7 mA/cm<sup>2</sup> . Thus, the formed ions, initially accumulating in the center of the system, under action of own created electric field can accelerated and create a powerful ion flux from both edges of the accelerator.
