2. Ohm's law

Let us consider an axially symmetric system in which the magnetic field is directed along the axis (see figure 1). Let concentric electrodes be placed at the ends of such a system, while the interelectrode gap is filled with plasma.

Although the transverse conductivity across the magnetic field can be influenced by a large number of parameters, which include various types of instabilities, we will consider classical conductivity. Thus, this calculation can be considered as a lower estimate.

In the hydrodynamic plasma model, one can obtain the generalized Ohm's law from the momentum balance equations. Neglecting the motion of neutrals in the laboratory coordinate system (Paschmann et al. Reference Paschmann, Haaland and Rudolf2003), one can obtain the radial component of current density

(2.1)\begin{equation}{j_r} = \sigma _P^n{E_r},\end{equation}

where ${E_r}$ is the radial electric field and $\sigma _P^n$ is the transverse conductivity in the rest frame of neutrals.

Let us show that in a certain range of parameters, the main contribution to the transverse conductivity is made by collisions of ions with neutrals. In the model of three liquids (ions, electrons and neutral atoms), the transverse conductivity in the rest system of neutrals has the form (Song et al. Reference Song, Gombosi and Ridley2001)

(2.2)\begin{equation}\sigma _P^n = \frac{{e{n_e}}}{B}\frac{\kappa }{{{{(\kappa )}^2} + 1}},\quad \textrm{where}\;\kappa = \frac{{{\varOmega _i}}}{{{\nu _{in}}}} + \frac{{{\nu _{en}}}}{{{\varOmega _e}}} + \frac{{{\nu _{ei}}}}{{{\varOmega _e}}}.\end{equation}

Here, ${\varOmega _i},{\varOmega _e}$ are the Larmor frequencies of rotation of ions and electrons, while ${\nu _{in}},{\nu _{en}},{\nu _{ei}}$ are the corresponding collision frequencies. In obtaining this equation, the relation $M{\nu _{in}} \gg m{\nu _{en}}$ was assumed to be correct. As shown in figure 2, it is valid in a wide range of plasma parameters, in particular, for argon in the range of ${T_e} \in [2,35]\;\textrm{eV}$, ${T_i} \in [0.01,100]\;\textrm{eV}$. When obtaining figure 2, the values of the elastic collisions cross-section of ions with neutrals in the energy range of 4–400 eV were used from the study by Cramer (1959). At lower energies, we used (2.26) from Raizer et al. (Reference Raizer, Kisin and Allen2011), which determines the polarization cross-section. For the elastic collisions cross-section of electrons with neutrals we used the data from Ramsauer (Reference Ramsauer1922).

Figure 2. Relation between $M{\nu _{in}}$ and $m{\nu _{en}}$ in argon.

As follows from (2.2) the relationship between the magnetization parameters of electrons and ions determines which of the particle species makes the largest contribution to the transverse transport. Figure 3 shows the components $\kappa$ in their dependence on the magnetic field, neutral gas pressure and plasma density for argon. When analysing this graph, it should be considered that the upper and lower horizontal axes can be changed independently, since they are defined by different components of $\kappa$. So the lower axis determines the components ${\varOmega _i}/{\nu _{in}}$ and ${\nu _{en}}/{\varOmega _e}$, and the upper one ${\nu _{ei}}/{\varOmega _e}$.

Figure 3. Comparison of the contribution of electrons and ions to the transverse conductivity of plasma. The dashed line indicates the excess of ionic conductivity over the electronic one by 10 times in the considered temperature range of electrons and ions.

To the right of the dashed line in figure 3, there is a domain where the ionic conductivity is more than 10 times greater than the electronic one. Most experiments with end electrodes are performed using the parameters from this domain (Gueroult et al. Reference Gueroult, Rax and Fisch2019a; Liziakin et al. Reference Liziakin, Gavrikov and Smirnov2020). Therefore, it will be further assumed that the plasma conductivity across the magnetic field is determined by the transport of ions. However, it is worth remembering that such a consideration is true only for the indicated B/p and B/n_e. In this case, the conductivity takes the form

(2.3)\begin{equation}\sigma _P^n = \frac{{e{n_e}}}{B}\frac{\chi }{{{{(\chi )}^2} + 1}} = \frac{{{e^2}{n_e}}}{{M{\nu _{in}}}}\frac{1}{{{{(\chi )}^2} + 1}},\quad \textrm{where}\;\chi = \frac{{{\varOmega _i}}}{{{\nu _{in}}}} = \frac{{eB}}{M}\frac{1}{{{n_n}{\sigma _{in}}{u_i}}}.\end{equation}

Here ${\sigma _{in}}$ is the elastic collisions cross-section of ions with neutrals.

3. Radial profile of the potential

It was found in the previous section that in the considered parameter range, the main current across the magnetic field was transported by ions.

Consider the relation between the plasma potential and the potential of the end electrodes in the case when the potential of electrodes monotonically increases with r $({U_{e1}} < {U_{e2}})$. Figure 4 shows an equivalent electrical circuit. The outer cylindrical surface and electrode 3 are grounded, and their potentials are equal to zero $({U_{e3}} = 0)$. On the two inner electrodes $({U_{e1}},{U_{e2}})$, a negative potential relative to electrode 3 is maintained by the external power source. The symbols ${V_{\textrm{sh1}}},{V_{\textrm{sh2}}}$ indicate the near-electrode potential drop. We should point out that the discrete current loops schematically indicated in figure 4 are in fact continuous, and the magnitude of the near-electrode potential drop is radially varying.

Figure 4. Equivalent electrical circuit of an axisymmetric system with an axial magnetic field and end electrodes.

The elementary increment of the plasma potential along the radius can be written in the form

(3.1)\begin{equation}\textrm{d}U(r) = {I_r}(r)\,\textrm{d}R(r),\end{equation}

where

(3.2)\begin{equation}\textrm{d}R(r) = \frac{1}{{\sigma _P^n(r)}}\frac{{\textrm{d}r}}{{2{\rm \pi} rL}},\end{equation}

whereas ${I_r}$ is the total current along the radial coordinate and L is the length of the plasma cylinder. Taking into account the stationarity of the flow of currents and the law of the charge conservation, one can find that the radial current in an arbitrary coordinate r should be equal to the total current of all the electrodes located in the region $\tilde{r} < r$

(3.3)\begin{equation}{I_r}(r) = \int_0^r {j(\tilde{r})\,\textrm{d}S(\tilde{r})} ,\end{equation}

where j is the density of current to the end electrodes and $\textrm{d}S = 2{\rm \pi}\tilde{r}\,\textrm{d}\tilde{r}$. Therefore, the plasma potential at an arbitrary point should be equal to

(3.4)\begin{equation}U(r) = \int_{{r_g}}^r {\int_0^{\hat{r}} {j(\tilde{r})\,\textrm{d}S(\tilde{r})} } \frac{1}{{\sigma _P^n(\hat{r})}}\frac{{\textrm{d}\hat{r}}}{{2{\rm \pi}\hat{r}L}},\end{equation}

where $\tilde{r},\hat{r}$ are integration variables and ${r_g}$ is the inner edge of the grounded electrode.

The plasma conductivity under the integral is a function of the ion velocity, which in turn is a function of the radial electric field.

3.1. Strongly negative equipotential end

The simplest case of connecting the end electrodes is by short-circuiting all of the electrodes to each other except for the outermost one (${U_{e1}} = {U_{e2}}$, while ${U_{e3}} = 0$, figure 4). A large negative potential $e{U_e} \gg {T_e}$ is applied to electrodes 1 and 2. This configuration looks most like a classical reflex discharge (Liziakin et al. Reference Liziakin, Gavrikov, Murzaev, Usmanov and Smirnov2016). In this case, j from (3.4) can be substituted by the density of the ion saturation current (Lieberman & Lichtenberg Reference Lieberman and Lichtenberg2005) ${j_{is}} = 0.52en{(k{T_e})^{0.5}}{M^{ - 0.5}}$ times the number of ends with electrodes ${N_{\textrm{surf}}}$ (1 or 2). If one assumes that the neutral density, plasma density, electron temperature and the cross-section of elastic collisions of ions with neutrals do not depend on the radial coordinate, then the potential increment along the radius can be represented in the form

(3.5)\begin{align}- \textrm{d}U(r) = \frac{{0.52}}{e}\frac{{{M^{0.5}}{{(k{T_e})}^{0.5}}{n_n}{\sigma _{in}}{N_{\textrm{surf}}}}}{{BL}}\int_0^{\hat{r}} {(\tilde{r}\,\textrm{d}\tilde{r})\left( {U^{\prime}(\hat{r})\left[ {1 + {{\left( {\frac{{e{B^2}}}{{M{n_n}{\sigma_{in}}}}\frac{1}{{U^{\prime}(\hat{r})}}} \right)}^2}} \right]} \right)\frac{{\textrm{d}\hat{r}}}{{\hat{r}}}} .\end{align}

Here, for simplification, ${\sigma _{in}}$ is taken to be constant and is equal to $2.5 \times {10^{ - 15}}\;\textrm{c}{\textrm{m}^2}$, whereas the velocity of ions equals ${u_i} = E/B = U^{\prime}/B$. Then, performing integration with respect to the variable $\tilde{r}$ and denoting

(3.6a,b)\begin{equation}{C_1} = \frac{{0.52}}{{2e}}\frac{{{M^{0.5}}{{(k{T_e})}^{0.5}}{n_n}{\sigma _{in}}{N_{\textrm{surf}}}}}{{BL}},\quad {C_2} = {\left( {\frac{{e{B^2}}}{{M{n_n}{\sigma_{in}}}}} \right)^2},\end{equation}

one can get

(3.7)\begin{equation}U(r) = {C_1}\int_{{r_g}}^r {\hat{r}U^{\prime}(\hat{r})\left( {1 + \frac{{{C_2}}}{{{{(U^{\prime}(\hat{r}))}^2}}}} \right)\,\textrm{d}\hat{r}} + U({r_g}).\end{equation}

Differentiating both sides of (3.7), expressing $U^{\prime}$ and integrating from the inner edge of the grounded electrode ${r_g}$, one can obtain

(3.8)\begin{align}U(r) & = \sqrt {{C_2}} \int_{{r_g}}^r {\sqrt {\dfrac{{\hat{r}}}{{1/{C_1} - \hat{r}}}} \,\textrm{d}\hat{r}} \notag\\ & = \sqrt {\dfrac{{{C_2}{C_1}r}}{{1 - {C_1}r}}} \dfrac{{(\sqrt {{C_1}r} ({C_1}r - 1) + \sqrt {1 - {C_1}r} {{\sin }^{ - 1}}(\sqrt {{C_1}r} ))}}{{C_1^{3/2}\sqrt r }} - F({r_g}) + U({r_g}). \end{align}

Here and after, $F(r)$ is an antiderivative of the function. For ${C_1}r \ll 1$, one can expand this in a Taylor series to the first non-zero term

(3.9)\begin{equation}U(r) = {\textstyle{2 \over 3}}\sqrt {{C_1}{C_2}} ({r^{3/2}} - r_g^{3/2}) + U({r_g}).\end{equation}

Figure 5 demonstrates potential profiles, calculated by (3.9) with $U({r_g}) = 0$, under fixed magnetic field, electron temperature, length and radius of installation in argon gas. In this example, ${r_g} = 0.25\;\textrm{m}$. The ion-saturation current is the maximal current which can flow through a layer (sheath) of a cold negative electrode. Therefore, the presented profiles are an estimate from above for the possible potential values. For the conditions shown in figure 5, the dimensionless parameter s ⁻¹ is in the range (2–55) × 10⁻³.

Figure 5. Potential radial profile in the configuration when all the electrodes (except for the external grounded electrode) are at a large negative potential $e{U_e} \gg {T_e}$.

3.2. Floating-gap configuration

In the experimental implementation of the electrode geometry described in the previous section, an arc discharge may occur between negative and grounded electrodes across the magnetic field. To avoid this, one can install another electrode between these electrodes with an intermediate voltage or under a floating potential (${I_2} = 0$ in figure 4). Thus, a buffer zone appears that prevents disruption of the discharge. In this case, the plasma potential will be determined by the system of equations

(3.10)\begin{equation}U(r) = \left\{ {\begin{array}{*{20}{@{}l}} {\tfrac{2}{3}\sqrt {{C_1}{C_2}} ({r^{3/2}} - r_e^{3/2}) + U({r_e}),}&{r \in [0,{r_e}]}\\ {\displaystyle\int_0^{{r_e}} {{j_{is}}\,\textrm{d}S} \displaystyle \int_{{r_g}}^r {\textrm{d}R} ,}&{r \in [{r_e},{r_g}]} \end{array}} \right.,\end{equation}

where ${r_e}$ is the outer radius of the negative electrode. Substituting ${j_{is}}$ into the second equation of system (3.10) and taking into account (3.2), one can get

(3.11)\begin{equation}U(r) = r_e^2{C_1}\int_{{r_g}}^r {U^{\prime}(\hat{r})\left( {1 + \frac{{{C_2}}}{{{{U^{\prime}}^2}(\hat{r})}}} \right)} \frac{{\textrm{d}\hat{r}}}{{\hat{r}}} + U({r_g}).\end{equation}

Then, similar to (3.7), differentiating both sides of (3.11), expressing $U^{\prime}$ and performing integration, one can get

(3.12)\begin{equation}U(r) = 2\sqrt {r_e^2{C_1}{C_2}} (\sqrt {r - r_e^2{C_1}} - \sqrt {{r_g} - r_e^2{C_1}} ) + U({r_g}),\quad r \in [{r_e},{r_g}].\end{equation}

3.3. Thermionic element on the axis of the system

Equation (3.1) shows that an increase in the potential of the plasma volume can be achieved by increasing the radial current. In an experimental implementation, this can be done by using a thermionic cathode.

When electrons are injected into a vacuum, their current is limited by the Child–Langmuir law (Langmuir & Compton Reference Langmuir and Compton1931). When electrons are injected into neutral plasma, the space charge of electrons is compensated by the space charge of ions in the cathode layer. Although the current of thermoelectrons can significantly exceed the ion-saturation current, its magnitude is bounded just the same. At a low-ion-saturation current, a virtual cathode is formed. The critical thermionic current as a function of the potential of the cathode layer was found by Poulos (Reference Poulos2019).

Consider the case when a thermal cathode with a radius ${r_{\textrm{th}}}$ is situated on the axis. Then the potential increment along the radius will be a piecewise-defined function

(3.13)\begin{equation}\textrm{d}U = \left\{ {\begin{array}{*{20}{@{}l}} {\left( {\displaystyle\int_0^r {{j_{\textrm{th}}}(\tilde{r})\,\textrm{d}S} + \displaystyle\int_0^r {{j_{is}}(\tilde{r})\,\textrm{d}S} } \right)\,\textrm{d}R,}&{r \in [0,{r_{\textrm{th}}}]}\\ {\left( {\displaystyle\int_0^{{r_{\textrm{th}}}} {{j_{\textrm{th}}}\,\textrm{d}S} + \displaystyle\int_0^r {{j_{is}}(\tilde{r})\,\textrm{d}S} } \right)\,\textrm{d}R,}&{r \in [{r_{\textrm{th}}},{r_e}]}\\ {\left( {\displaystyle\int_0^{{r_{\textrm{th}}}} {{j_{\textrm{th}}}\,\textrm{d}S} + \displaystyle\int_0^{{r_e}} {{j_{is}}\,\textrm{d}S} } \right)\,\textrm{d}R,}&{r \in [{r_e},{r_g}]} \end{array}} \right..\end{equation}

Denoting

(3.14)\begin{equation}{C_3} = \frac{{M{n_n}{\sigma _{in}}{N_{\textrm{surf}}}}}{{2{e^2}{n_e}BL}},\end{equation}

and considering that

(3.15)\begin{equation}{C_1} = {j_{is}}{C_3},\end{equation}

one can obtain, similar to the previous sections,

(3.16)\begin{align}U(r) = \left\{ {\begin{array}{*{20}{@{}l}} {\tfrac{2}{3}\sqrt {({j_{\textrm{th}}} + {j_{is}}){C_3}{C_2}} ({r^{3/2}} - r_{\textrm{th}}^{3/2}) + U({r_{\textrm{th}}}),}&{r \in [0,{r_{\textrm{th}}}]}\\ {\displaystyle\int_{{r_e}}^r {\sqrt {\dfrac{{({j_{\textrm{th}}}r_{\textrm{th}}^2 + {j_{is}}{r^2}){C_3}{C_2}}}{{r - ({j_{\textrm{th}}}r_{\textrm{th}}^2 - {j_{is}}{r^2}){C_3}}}} \,\textrm{d}r} + U({r_e}),}&{r \in [{r_{\textrm{th}}},{r_e}]}\\ {2\sqrt {({j_{\textrm{th}}}r_{\textrm{th}}^2 + {j_{is}}r_e^2){C_3}{C_2}} \sqrt {r - ({j_{\textrm{th}}}r_{\textrm{th}}^2 + {j_{is}}r_e^2){C_3}} - F({r_g}) + U({r_g}),}&{r \in [{r_e},{r_g}]} \end{array}} \right..\end{align}

The constants in the expression are chosen so the function is continuous.

3.4. Slightly negative electrodes

In §§3.1–3.3, particular cases of connecting end electrodes to an electrical circuit were considered. The main simplification that allowed obtaining analytical equations for $U(r)$ is that a large negative potential $e{U_e} \gg {T_e}$ is applied to the negative electrode. In these cases, as seen from (3.9), (3.12) and (3.16), the plasma potential does not depend on the electrode potential. For such a voltage, the electrons from the plasma cannot get onto the negative electrode, and therefore their current can be neglected. In this section, we consider the case when the electrodes are slightly negative biased $(e{U_e} \sim {T_e})$. In this case ${I_r}$ from (3.1) should be written in the form

(3.17)\begin{equation}{I_r}(r) = 2{\rm \pi}\int_0^r {\left( {{j_{es}}(\tilde{r})exp \left( {\frac{{e({U_e}(\tilde{r}) - U(\tilde{r}))}}{{k{T_e}}}} \right) - {j_{is}}(\tilde{r})} \right)} \tilde{r}\,\textrm{d}\tilde{r}.\end{equation}

Since now the potential of the electrode may not differ much from the potential of the plasma $(e{U_e}/{T_e}\sim 1)$, regions may appear where $U^{\prime}(r) \to 0$, in this case, in (2.3) for the collision frequency, the ion thermal velocity ${u_T}$ should be added

(3.18)\begin{equation}{u_i} = \frac{{U^{\prime}(\hat{r})}}{B} + {u_T}.\end{equation}

For ${u_T}$ we use room temperature of 300 K.

If the discharge is self-sustained, that is, the plasma is created only by electrodes, then the area occupied by the plasma is bounded by the condition $r < {r_g}$. If the plasma is maintained by an external ionization source (electron beam (Maggs et al. Reference Maggs, Carter and Taylor2007), radio-frequency (rf) discharge (Litvak et al. Reference Litvak, Agnew, Anderegg, Cluggish, Freeman, Gilleland, Isler, Lee, Miller, Ohkawa, Putvinski, Sevier, Umstadter and Winslow2003; Shinohara & Horii Reference Shinohara and Horii2007; Gilmore et al. Reference Gilmore, Lynn, Desjardins, Zhang, Watts, Hsu, Betts, Kelly and Schamiloglu2015; Gueroult et al. Reference Gueroult, Evans, Zweben, Fisch and Levinton2016; Liziakin et al. Reference Liziakin, Gavrikov, Usmanov, Timirkhanov and Smirnov2017; Vorona et al. Reference Vorona, Gavrikov, Kuzmichev, Liziakin, Melnikov, Murzaev, Smirnov, Timirkhanov and Usmanov2019), microwave plasma source (Amatucci et al. Reference Amatucci, Walker, Ganguli, Antoniades, Duncan, Bowles, Gavrishchaka and Koepke1996)), then the plasma density may not be equal to zero for $r > {r_g}$. In this case, when moving away along the radius from the inner edge of the grounded electrode, the plasma potential tends to the value

(3.19)\begin{equation}U(r > {r_g}) = {T_e}\ln \left( {\sqrt {\frac{\rm \pi}{2}\frac{m}{M}} } \right).\end{equation}

Then the distribution of the plasma potential is the solution of the equation

(3.20)\begin{align} U(r) & = \dfrac{{M{n_n}}}{{{e^2}L}}\int_{{r_{ex}}}^r {\int_0^{\hat{r}} {\left( {{j_{es}}(\tilde{r})exp \left( {\dfrac{{e({U_e}(\tilde{r}) - U(\tilde{r}))}}{{k{T_e}}}} \right) - {j_{is}}(\tilde{r})} \right)\tilde{r}\,\textrm{d}\tilde{r}} } {\sigma _{in}}\notag\\ &\quad \times \left( {1 + {{\left( {\dfrac{{eB}}{{M{n_a}{\sigma_{in}}}}} \right)}^2}\dfrac{1}{{{{\left( {\dfrac{{U^{\prime}(\hat{r})}}{B} + {v_T}} \right)}^2}}}} \right)\notag\\ &\quad \times\left( {\dfrac{{U^{\prime}(\hat{r})}}{B} + {v_T}} \right)\dfrac{1}{{n(\hat{r})}}\dfrac{{\textrm{d}\hat{r}}}{{\hat{r}}} + U({r_{ex}}). \end{align}

Here, integration is carried out from the outer boundary ${r_{ex}}$ of the electrode system. It is also considered in this equation that the plasma density and electron temperature depend on the radial coordinate. A methodology for numerically solving this equation is presented in Appendix A. The computer program LaPotential that solves this equation is presented by Oiler & Liziakin (Reference Oiler and Liziakin2021).

An example of the calculation results for a case when the voltage at the electrodes is not large $(e{U_e}/{T_e}\sim 1)$ is presented in figure 6. A positive current density corresponds to the electronic current to the electrodes, whereas a negative value corresponds to the ionic current.

Figure 6. An example of solving (3.20) with parameters $B = 1.4\;\textrm{kG}$, $P = 0.2\;\textrm{m}\;\textrm{Torr}$, ${n_e} = {10^{11}}\;\textrm{c}{\textrm{m}^{ - \textrm{3}}}$, ${T_e} = 5\;\textrm{eV}$.

4. Comparison with experimental data

To test the model described above, a series of experiments was carried out. The experimental set-up is shown in figure 7. The vacuum chamber was a cylinder with a diameter of 85 cm and a length of 220 cm. A longitudinal magnetic field was created by a four-coil solenoid. On the chamber axis, the magnetic field was 1.4 kG. The working gas was argon at a pressure of 0.3 mTorr.

Figure 7. Experimental scheme.

End electrodes in the form of truncated cones were placed at the ends of the chamber. Three internal electrodes $(r < 15\;\textrm{cm)}$ from each end were connected between one another and constant voltage was maintained on them. The fourth and fifth electrodes $(15 < r < 25\;\textrm{cm)}$ were under floating potential. The walls with $r > 25\;\textrm{cm}$ were at ground potential. A thermal cathode made of LaB₆ with a diameter of 2 cm was installed at the left end (figure 7) in the very centre of the electrode. The emissive electrode was electrically connected with three internal electrodes. The end electrodes, together with the grounded cylindrical surface of the vacuum chamber, formed the geometry of the reflex discharge (Hooper Reference Hooper1970). Two modes of discharge functioning were investigated. The first mode was when the thermal cathode is not heated, that is, the discharge operates as a usual reflex discharge with cold electrodes. The second mode was when the thermal cathode is heated. In the first case, the voltage at the electrodes was maintained at a level of −1 kV, while the discharge current was 50 mA; in the second case, the voltage was −0.5 kV, and the current was 10 A. The density profiles in both modes are presented in figure 8. Table 1 summarizes the parameters of both modes. The calculated values for s ⁻¹ correspond to 0.01 and 0.03 for the cold and hot modes, respectively.

Figure 8. Radial profile of the plasma density.

Table 1. Experimental parameters of the reflex discharge.

Plasma density and electron temperature were measured by the double probe, and the plasma potential was measured with an emission probe (Murzaev et al. Reference Murzaev, Liziakin, Gavrikov, Timirkhanov and Smirnov2019). The temperature of electrons weakly depended on the radial coordinate and was 4 eV in the electrode region and 2 eV in the floating electrode region for the cold cathode mode. For the hot cathode mode, the electron temperature was ${T_e} = 8.5 \pm 1.5\;\textrm{eV}$.

Figure 9 presents a comparison of the plasma potential profile calculated by the system of (3.10) and that obtained experimentally in the mode with cold electrodes. Two profiles are presented with different cross-sections. The first profile corresponds to elastic collisions ${\sigma _{in}} = 25 \times {10^{ - 20}}\;{\textrm{m}^2}$ and the second corresponds to the sum of elastic and charge exchange cross-section ${\sigma _{in}} = 50 \times {10^{ - 20}}\;{\textrm{m}^2}$. Thus, there is an input parameter in this model that could quantitatively affect predictions, and these parameters are not all well-constrained experimentally. While $U \sim \sigma _{in}^{ - 0.5}T_e^{0.25}$ doubling the cross-section gives the same result as decreasing T_e by 4 times. It can be reasonable to consider that the double probe method can overestimate T_e. Figure 9 also shows the numerical solution of (3.20) by LaPotential taking into account polarization cross-section, space variation of plasma density and electron temperature, and ion temperatures with T_i equal to 0.025 and 0.2 eV. Taking these features into account significantly improved the agreement with the experimental data. Figure 10 shows a comparison of the plasma potential profile calculated using the system of (3.16) with the experimental one under the operation of the thermal cathode with a current of 5.6 A. To exclude the influence of anode processes (not considered in this article), we set the calculated values of the plasma potential at ${r_g} = 25\;\textrm{cm}$, equal to the experimental value at this point. In both cases, good agreement was observed between the experimental and theoretical values. All parameters that are used in the theoretical model are summarized in table 2.

Figure 9. Comparison of the plasma potential profile calculated by the system of (3.10) with the experimental one without a thermal cathode.

Figure 10. Comparison of the plasma potential profile calculated by the system of (3.16) with the experimental one under the thermal cathode functioning with a current of 5.6 A.

Table 2. Input parameters of theoretical model for analytical solution.