Renovations! Please excuse the mess, but we are in the process of overhauling the site.

THE LOW-VOLTAGE ARC IN A TIC, say, "diffusion" Edit, click, copy, click.

Another Gary Search Page, say, "HotBot", click,

Then, "HotBot", Search, click, paste for "diffusion" , click, then Find

Searching Entries for Find (Edit, then Find)

Seven types of string searches are available, 1) search for an author, (your name) 2) search for a title (or part of a title), 3) search for a journal, 5) search by year, or 6) search by keyword.

293

The diffusion mode is not the only possible TIC operating mode at high cesium pressures. If the interelectrode applied voltage in the diffusion mode is increased to relatively high voltages, ionization begins in the plasma volume, and the converter changes to arc mode operation.

The new ion source in the plasma volume leads to a marked increase in the current of the converter. At the same time, the potential distribution in the gap changes appreciably, the charged particle density increases sharply and the electron temperature increases. The potential distribution, typical for an arc mode, is a potential well for electrons, bounded at the edges by pre-electrode potential barriers (Fig. 9.1). In this case, the main potential drop occurs at the precathode potential electron barrier. This rather large potential drop sharply reduces the back electron current from the plasma to the cathode. As a result, a current close to the cathode emission current can be passed in the arc made at a rather high output voltage. High outputs and comparatively high efficiencies make the arc mode of the TIC very promising for practical use.

Calculation of the state of an arc plasma reduces to finding the distribution across the gap for the following parameters: charged particle density n, neutral atom density N_a, potential V, electron temperature T_e, and atom temperature T. As a result of the rapid energy exchange in collisions between ions and atoms, the ion temperature T_i may be assumed to be equal to the atom temperature T.

In the gap of a weakly ionized plasma, the density N_a of neutral atoms and their temperature T are calculated from the condition of constant pressure of the neutral gas:

With this weakly ionized plasma, the pressure of the neutral gas in the gap is essentially equal to the cesium pressure P_Cs in the surrounding space. The right side of (1.2) includes the heat transferred to the ions and atoms from the electron component (see (4.6.15) and (4.6.16)) and also the energy received by the ions in the electric field. Here k _a is the heat conduction of atoms, j_i is ion current and dV/dx is the electric field in the plasma. The atom temperature at the plasma boundaries is equal to the electrode temperature. Therefore, the boundary conditions for equation (1.2) are written in the form

where T_C and T_A are the cathode and anode temperatures, respectively.

In a weakly ionized arc plasma of a TIC, as in the diffusion mode, the terms on the right side of equation (1.2) are usually small, so that the heat flux transported by the atoms can usually be assumed constant across the gap. Since the heat conduction of atoms is rather weakly dependent on their temperature (k _a ~ T_1/2), the atom temperature distribution across the gap can be assumed to be linear.

Transport equations for electrons and ions. To find n, V, and T_e, a system of transport equations for electron current j_e, ion current j_i, and electron energy flux S_e must be used. According to (4.3.18), (4.3.19), and (4.7.2), these equations have the form:

where m _e and D_e and m _i and D_i are the electron and ion mobility's and diffusion coefficients, R_ie is the force acting on the ions as a result of electron-ion collisions, and a _e = k _e/D_enk and b _e,i = 5/2 + K^(T)_e,i are the dimensionless coefficients which are dependent on the charged particle scattering mechanism in the plasma. Unlike the diffusion mode of TIC operation, the values of j_e, j_i and S_e are now no longer constant.

*Formula (1.9) is obtained from (5.3.7) if the ground state population N₀ of the ground state is equal to the equilibrium population N(T_e) This will always be true under the conditions being considered here.

where –j_edV/dx is the work of the electric field, E_ion are ionization energy losses, D S_rad are radiation energy losses, and D S_ei and D S_ea are energy losses due to electron-atom and electron-ion collisions. The value of G is the ionization-recombination rate. According to (5.3.7)*

where t _i is the effective ionization time, calculated by expression (5.3.14):

Here s ₀(T_e) is the effective ionization cross section of a cesium atom (see (5.3.13)). The energy losses D S_rad, D S_ei, and D S_ea are calculated by expressions (5.4.21), (4.6.16), and (4.6.15), respectively.

Equations (1.7) and (1.8) are the particle and energy conservation laws for a unit volume of plasma. By using (1.6) and (1.7), from (1.8) we can obtain the continuity equation for the total electron energy flux S_e:

As already noted, the energy D S_ei + D S_ea lost from the electrons to the heavy components is usually small in a weakly ionized TIC plasma, and in most cases, it can be disregarded.

The ion current in a TIC plasma of a practical converter is considerably less than the electron current. At the same time, the variation of ion current D j_i is equal in absolute value to variation of electron current D j_e, because the total current is constant across the gap:

This means that the relative variation of electron current across the gap is small even if there is intensive ionization in the TIC plasma and considerable variation in the ion current. * Therefore , when solving the system of transport equations for not very large voltage drops eV_d « E_ion, the electron current may be assumed constant:

As a result, the continuity equation for electron current dj_e/dx = eG , can often be replaced by condition (1.13) and thus eliminated from the system of transport equations.

*Since an energy equal to E_ion is expended on the formation of each ion, variation of the electron and ion currents across the gap may not exceed the value j_eV_d/E_ion. In converter modes, V_d usually does not exceed 1 V, and E_ion ~ 4 eV, where V_d is the total potential drop in the gap. Therefore, D j_e/j_e < 1/4. Actually, D j_e/j_e is much less than 1/4, because part of the energy obtained by the electrons from the field is dissipated at the electrodes, radiated, etc.

The contribution of ions to the total current j can now be calculated approximately by setting the current of the device after calculation equal to j = j_e + j_iC, where j_iC is the ion current at the pre-cathode boundary of the plasma. The degree of accuracy of this calculation is illustrated by Table 9.1 in which the plasma parameters are compared which were obtained as a result of solving the complete system of equations (1.4)-(1.7) and (1.11) (the first line) and provided that j_e(x) = const (the second line). It is obvious that the results of the two calculations are similar.

Consideration of electron-ion and electron-atom scattering. The system of differential transport equations presented above was written on the assumption that the plasma is weakly ionized (n « N_a). However, electron-ion scattering and electron-electron interaction can be important as current through the converter increases, even with a weakly ionized plasma.

The approximate values of the kinetic coefficients with simultaneous electron-ion and electron-atom scattering, as shown in Chapter 4, can be obtained by interpolating the values of m _e, D_e, a _e and b _e between the two limiting cases, where the electrons are scattered only by atoms or only by ions:

where m _ea = (e/3kT_e)l_eav_e@ is the electron mobility for electron-atom scattering with a constant mean free path (see (4.3.14)) and m _ei = 0.582 mv_e³@ /4e³lnL is the electron mobility for a completely ionized plasma (see (4.5.1)).

The values of the coefficients a _e = k _e/D_enk and b _e = 5/2 + K^(T)_e in the limiting cases of scattering from atoms and ions are obtained from formulas (4.3.14)-(4.3.16) and (4.5.1), respectively, using the Einstein relation (4.3.20). In this case, a _e = b _e = 2 for electron-atom scattering with a constant mean free path; for Coulomb scattering a _e = 1.6 and b _e = 3.2. The force R_ie = -R_ei for mixed scattering is given by expression (4.7.1), but one may assume that R_ie » j_e/m _ei (see (4.6.3)) in this approximation where the mobility m _e is expressed by formula (1.14). The term in (1.5) containing R_ie makes a contribution on the order of (m _i/m _ei) j_e to the value of j_i. In most cases in the arc mode of a TIC, we have (m _i/m _ei)( j_e/j_i) « 1. The value of R_ie in equation (1.5) may then be disregarded.

Equations (1.4)-(1.7) and (1.11) are a system of six first-order differential equations in the desired functions: n(x), V(x), T_e(x), j_e(x), j_i(x), and S_e(x). Six boundary conditions (three conditions for each plasma boundary) are needed to solve this system. The boundary conditions relate the particle and energy fluxes of the electrons to the boundary values of the plasma parameters.

Here j_sC(E_C) and j_is are the electron and ion emission currents from the cathode and n_C and T_eC are the electron density and temperature near the cathode.

The values of j_eC, j_iC, and S_eC should be set equal to the values of j_e(0), j_i(0), and S_e(0) - the particle and energy fluxes in the pre-cathode boundary of the plasma calculated from (1.4)-(1.6).

The anode surface is taken as the point of zero potential, so that the plasma potential near the cathode is V(0) = V_d - V_C.

The boundary conditions (2.1)-(2.3) correspond to the case of eV_C » kT_eC. The boundary conditions near the anode, where usually eVA » kT_eC according to (6.7.17), (6.7.18), and (6.7.22), can be written in the following form:

Here j_sA is the electron emission from the anode, T_A is the anode temperature and n_A and T_eA are the electron density and temperature near the anode.

There are not yet any reliable boundary conditions for intermediate barriers (on the order of one or several kT_e). Therefore, during numerical calculations with intermediate barriers, the coefficients should be interpolated, for example, from (1 - r) in (2.1) to (1 - f₀)^-1 in (2.4). Calculations show that the results are insensitive to the specific type of interpolation.

where j_s⁰ is emission with zero field at the cathode surface and E_C is the field intensity at the surface. The field intensity can be calculated by formula (6.2.17). However, when deriving expression (6.2.17), the space charge of electrons and ions emitted from the cathode were not taken into account. In most cases, the charged particles emitted from the cathode in the arc mode have a negligible effect on the state of a quasi-neutral plasma near the boundary to the space charge zone. But to include this contribution, the space charge of electrons and ions from the cathode can simply be added to the space charge of the electrons and ions emerging from the plasma by calculating the latter as if there were no emission from the electrode. In this way, the following expression is obtained for the field intensity at the cathode:

where v_e = (2kT_C/p m)^1/2, v_i = (2kT_C/p m)^1/2, and t _e = T_eC/T_C, and function h⁺(z) is given by formula (10.1.16). The third and fourth terms in the brackets take into account the contributions to the space charge in the pre-cathode layer of the cathode ion and electron emission, respectively. These terms are small in a developed discharge, and their contribution does not affect the calculated value of E_C.

If the voltage V_d across the TIC gap is assumed to be known, the six transcendental equations (2.1)-(2.6) can be used to calculate the integration constants of the six differential equations (1.4)-(1.7) and (1.11).

The virtual cathode. When the current through the converter decreases, near the point of extinction of the discharge, a negative space charge may be formed at the cathode that retards the electrons emitted from the cathode. This occurs for small values of the compensation parameter a = (j_is/j_sC)Ö M/m@, where in the diffusion mode f _C < m so that a transition from the arc to the diffusion mode involves a change in the polarity of the pre-cathode potential barrier. In this case, the latter term in the brackets in (2.8) may be comparable in magnitude to the sum of the first two terms (with sufficiently small currents j and, accordingly, small values of n_C). The field intensity at the cathode may approach zero at sufficiently small currents. With a further decrease of the current, the potential distribution in the pre-cathode sheath becomes non-monotonic and a potential barrier D V_C develops which retards part of the electrons emitted from the cathode. Thus, the virtual cathode is introduced. With such a barrier, the current from the plasma is determined by the total height of the retarding barrier eV_C (Fig. 9.2a). According to (2.1)-(2.3), the boundary conditions with a virtual cathode are written in the following form:

When calculating the reflection coefficients by the formulas of §3, Chapter 6, E₀ should be understood as the total pre-cathode potential barrier (i.e., E₀ = eV_C). The value of D V_C is calculated separately by Poisson’s equation for the space-charge region.

The potential distribution in the pre-cathode sheath depicted in Fig. 9.2a, corresponds to the case where the potential difference between the plasma and the cathode surface accelerates electrons toward the plasma: V_C > D V_C. Another case is possible, where the potential difference between the plasma and cathode has the opposite polarity (Fig. 9.2b). In this case, the potential barrier D V_C, which decelerates the electrons emitted from the plasma to the cathode, has values only fractions of a kT for the modes in the TIC of interest. The potential barrier for the ions emitted from the plasma is equal to V_C > D V_C and is also small. This permits the use of expressions (6.7.17), (6.7.18), and (6.7.23) as the boundary conditions for the pre-cathode region of the plasma. In the notation of this chapter, the boundary conditions near the cathode are now described in the following form:

Since expressions (2.12)-(2.14) make a transition to the boundary conditions for the diffusion mode at D V_C = 0, the use of the above system of boundary conditions makes it possible to follow the transition from the arc mode with the virtual cathode to the diffusion mode, where the potential distribution in the pre-cathode sheath is again monotonic.

The point where the virtual cathode appears can be fixed by formula (2.8). The virtual cathode appears when E_C = 0, i.e., when the right side of (2.8) reaches 0. It should be recalled, however, that formula (2.8) is obtained on the assumption that the pre-cathode potential drop is sufficiently large (eV_C » kT_eC) and that the charged particles emitted from the cathode make a negligible contribution to the density at the boundary of a quasi-neutral plasma.* (Both these conditions are important when determining the form of the first and second terms in (2.8).) If the pre-cathode drop V_C is small when the virtual cathode occurs, so that the above assumptions are not fulfilled, then a special calculation of the pre-electrode sheath must be carried out to determine the point where the virtual cathode appears.

Boundary conditions for a cathode with a non-uniform work function. If the work function of the cathode is non-uniform, a virtual cathode may be formed on the spots with a small work function and not on the spots with a large work function (see Fig. 6.4a). Then, as the pre-cathode voltage drop increases, the current j_sC also increases because of increased exposure of the small work function spots. The increase of emission current with the applied field is more rapid in this case than in the case of the normal Schottky effect. This is the anomalous Schottky effect described in Chapter 2. As was pointed out there, the increase of current as the field increases may be described by a formula similar to (2.7) by introducing into the exponent the anomaly coefficient b (E):

*The first and second hypotheses are related to each other, because cathode emission electrons can make an appreciable contribution to the space charge near the cathode and a relatively small contribution to the density near the boundary of the quasi-neutral plasma only with a sufficiently large value of V_C (see §7, Chapter 6).

With large currents through the converter, as in a developed discharge, the field at the surface is very large and all of the surface is exposed (b = 1). Nevertheless, even in this case, the spottiness of the cathode can have an effect on the parameters of the gas-discharge plasma and the form of the current-voltage characteristic.

Accounting for cathode spottiness requires some change in the boundary conditions for the cathode. If, for example, there are spots of two types with work functions f _C1 and f _C2 and corresponding emission currents j⁽⁰⁾_s1 and j⁽⁰⁾_s2 at the cathode surface, then the electron current and energy flux for each of the spots are calculated individually by formulas (2.1)-(2.3) and (2.9)-(2.11) with their own values of j⁽⁰⁾_s, j_is and V_C.

To apply these formulas, it is necessary that the spot sizes significantly exceed the Debye length L_D, since only then can the contact of each spot with the plasma be considered independently. Moreover, the spot sizes should be small compared to the distance over which plasma parameters in the pre-cathode sheath vary significantly.

Boundary conditions in the presence of a cathode with a developed surface. The geometry of the cathode surface is sometimes altered to increase the cathode emitting area, and therefore, to increase the cathode emission current [1]. A cathode with a developed surface consisting of a series of rectangular slots is shown in Fig. 9.3 as an example. The lateral edges of the slots give an additional contribution to the emission, and the total cathode emission current increases in proportion to its area. The case where the slot dimensions d exceed not only the Debye length L_D, but also the length of the electron free path l_e, is of most interest. With these slot widths, the plasma fills the slot cavity, and electrons emitted by the lateral edge of the slot in a direction parallel to the frontal surface of the cathode (plane AB in Fig. 9.3) do not reach the opposite lateral edge, but are scattered in the slot cavity. The direction of electron velocity changes as a result of scattering, so that most of the scattered electrons are "captured" by the plasma.

If the slot dimensions h and d greatly exceed the electron mean free path, and also the energy relaxation length of the fast electrons, then the reverse electron flow from the plasma to the cathode and the energy carried by the electrons from the plasma can be calculated with the same formulas as those used for a plane cathode surface. However, with a developed cathode, the increased area should appear in the calculation. In addition, we are concerned with the value of ion current. As will be shown below, ions incident to the cathode from the plasma are formed at a considerable distance from the cathode. If this distance, or ionization length, (see §4) is large compared to h, then the ion current to the cathode may also be calculated as for a plane cathode surface without regard to the increase in area. The variation of the density within the slot can also be disregarded.

Under these conditions, only the boundary conditions for j_eC and S_eC change form, namely the right sides of expressions (2.1) and (2.3) increase b _S times, where b _S is the factor of cathode area increase compared

to the area of the frontal surface AB.

If the depth h of the slots is comparable in value to the ion generation length in the pre-cathode sheath, then the boundary conditions should be changed in form to account for the real geometry of the cathode. The problem ceases to be one-dimensional.

We discuss initially the boundary conditions for ion current. As already noted in Chapter 6, when there is an accelerating pre-electrode potential barrier the value of ion current actually is not dependent on the specific type of boundary condition if the width of the ion diffusion region greatly exceeds the mean free path. The latter condition is usually fulfilled for ions, because the width of the ionization region in a TIC greatly exceeds l_i. Therefore, the value of the ion currents j_ic and j_iA in the arc mode are essentially independent of g ₀. This is illustrated by Table 9.2, in which are presented values of the ion current, the density, and the potential barrier near the cathode and anode, obtained from complete calculations of the arc mode with different selections of g ₀ (we recall that g _i = g ₀(p T_e/4T)^1/2).* It is obvious that the value of g _i is actually equivalent to a shift of the boundary for the quasi-neutral plasma nearer to the electrode (to a point with greater field intensity and with less charged particle density). Therefore, the portion of the potential drop in the pre-electrode sheath (that treated as a quasi-neutral plasma) increases and the voltage included in the space charge region (i.e., the potential barrier) decreases. (Compare the values of V_C and V_A for different values of g _i). The total potential drop in the pre-electrode sheaths (including the space charge region and the quasi-neutral region of the pre-electrode sheath), in this case, does not vary in the same way as the total voltage drop V_d across the gap. It is true that the field intensity at the cathode, E_C, varies as V_C varies, which affects the value of cathode emission current j_s. However, it is obvious from (2.8) that @ is very weakly dependent on V_C (roughly, E_C ~ V_C^1/4?).

*The data obtained in [21] where the boundary condition j_iA = -eg ₀(2kT_eA/M)^1/2n_A was used instead of (2.5), are presented in Table 9.2.

Correct selection of the value of g ₀ (g ₀ ~ 0.76) means a more accurate assignment of the ion drift velocity (v_di)₀ = g ₀(2kT_eA/M)^1/2, and together with it the density and potential at the boundary of the quasi-neutral plasma.

Boundary condition (2.1) for electron current at the cathode for large values of V_C, being an accurate expression for j_eC, matches well with the transport equation (1.4) for electron current in the plasma volume , because the condition j_e « (e/4)nv_e@ is usually fulfilled up to the plasma pre-cathode boundary. A greater error may be expected by using expression (2.4) for j_eA, because the current j_eA is comparable to the random current (e/4)n_av_e@ near the anode. As can be seen from (2.4) and (6.7.13), at j_eA = 0, the coefficient in the expression for j_eA varies from a value of 1/4 for eV_A/kT_eA » 1 to a value of 1/2 at V_A = 0. As indicated by the calculations, variation of the expression for j_eA within these limits weakly affects the distribution of the plasma parameters in the volume. Only the values of plasma parameters near the anode itself undergo significant variation. The nature of these variations is illustrated by Table 9.3, obtained from a solution of the complete system of equations for different boundary conditions near the anode.

In Table 9.3, n_m is the maximum value of electron density in the plasma. It is obvious that a variation in the boundary condition for j_eA has a relatively minor effect on the value of the output voltage and, consequently, on the calculated current-voltage characteristic. Refinement of boundary condition (2.4) is reflected only in the values of V_A and T_eA.*

In concluding this section, we consider the boundary conditions in the presence of a virtual cathode. In the case depicted in Fig. 9.2a, the non-monoticity in the potential distribution near the cathode is very significant, because it can lead to significant decrease of the electron emission from the cathode. In the case depicted in Fig. 9.2b, the value of D V_C, as indicated above, is only a fraction of a kT. Accurate calculation of the value of D V_C is difficult, but the boundary conditions

*The limits within which the expression for j_eA can vary are exaggerated In Table 9.3, because the pre-anode barrier in a TIC usually does not approach zero, and therefore, the first line is not realistic.

can be simplified. One of the possible simplifications is illustrated by Table 9.4. The calculated values of the main plasma parameters, obtained for D V_C = 0, are presented in the first line of the table. The results for a calculation for D V_C = 0.35kT_C/e, which corresponds approximately to the value of D V_C obtained in the Knudsen mode,* are presented in the second line. It is obvious that these results hardly differ.@ This permits the system of boundary conditions to be simplified for the theoretical calculations. For example, we can assume that D V_C = 0. This approach was used in [2, 3].

The linearity of the system of transport equations and boundary conditions. We turn our attention to the following property of the system of transport equations and boundary conditions [4]. In the absence of recombination, kinetic reflection, and energy losses due to radiation and Coulomb interaction, the system of equations and boundary conditions is linear in charged particle density n and fluxes j_e, j_i, S_e, j_sC, and j_sA. This means that, if all these values increase simultaneously by some factor, and if the values of V(x) and T_e(x) are unchanged, all the terms in the transport and continuity equations and in the boundary conditions increase by the same factor. Thus, the system of equations and boundary conditions remains unchanged. The property of linearity is most easily used if electron emission from the anode and ion emission from the cathode are negligible (the latter condition is almost always fulfilled in the TIC discharge mode). Then, if the transport equations and boundary conditions are divided by the value of cathode emission current j_sC, so that density n and fluxes j_e, j_i, and S_e are measured in relative units , then the equations and boundary conditions do not depend

*Compare §2, Chapter 10, where the results from calculating the value of D V_C for the Knudsen mode are presented.

@For both lines of Table 9.4, to simplify the numerical calculations, it was assumed that f₀ = f₁ = 0 in formulas (2.12)-(2.14).

We also turn our attention to the fact that, under conditions where emission from the plasma, kinetic reflection, recombination, and Coulomb collisions are not significant, solution of the system of equations may be expressed as a function of the parameter pd, where p is the pressure of a neutral gas and d is the interelectrode spacing. To prove this, let us denote the values of the mobility and the electron and ion diffusion coefficient at some pressure p₀ by m _e0, D_e0, m _i0, and D_i0. The values of the kinetic coefficients at any other pressure @ are related to the values of m _e0, D_e0, m _i0, and D_i0 by the relations

where x = x/d. Since N_ad in (3.4) and (3.5) are proportional to pd, and the boundary conditions are not dependent on p and d, then solutions of the system of equations (3.1)-(3.5), expressed as functions of the dimensionless distance x , are dependent only on parameter pd and are not dependent on p and d separately. This is also true of the current-voltage characteristic of the converter. With this linear approximation, the current-voltage characteristics, plotted in coordinates of j/j_sC and V_d, are calculated as functions of pd and are universal for all emission currents j_sC.

The above properties of the system of equations permit certain conclusions about the relationships between different TIC parameters [4,5,6]. However, under TIC operating conditions, the range over which the linear approximation applies is very small, because recombination and Coulomb collisions are significant in almost all TIC modes.*

Calculation of the state of the plasma and of the current-voltage characteristic. Special algorithms and computer programs that permit TIC calculations over a wide range of external parameters - T_C, T_A, P_Cs, d, j_s⁽⁰⁾, and j_sA - were developed to solve the system of equations (2.1)-(2.6). The calculations carried out by these programs are a unique computer experiment. One of the positive aspects of this experiment is the possibility of varying independently all the parameters of the device (for example, cesium pressure and the electrode work function) and, at the same time, of following the dependence of the state of the plasma and of the current-voltage characteristic on each parameter separately. The system of differential equations is non-linear and, therefore, cannot be solved by analytical methods. The main difficulty in a numerical solution of the problem is that not all the values of the desired function are known at any one end of the integration space. Therefore, the missing initial values at one end of the space must be found in some manner to solve the problem. The difficulties in the numerical integration are aggravated by the fact that the right sides of equations (1.7) and (1.11) are exponentially dependent on electron temperature T_e, so that slight errors in the calculation of the initial conditions fundamentally alter the entire solution. As a result, it is either impossible to integrate the system of equations to the end of the space or the discrepancies in the boundary conditions at the other end are excessive and can no longer be minimized.

In various investigations in which the TIC plasma parameters and current-voltage characteristics have been calculated by numerical methods, different adjustment methods have been used to overcome this difficulty [2, 3, 7-9]. We outline here in more detail the method used in [9]. In this investigation, equations (1.4)-(1.6) were solved for the derivatives dn/dx, dV/dx, and dT_e/dx, after which the system of transport equations was written in the following form

where f_i are the unknown functions n, V, T_e, j_e, j_i, and S_e, and F_i are the known functions of f_i. Thus, a system of six nonlinear ordinary first-order differential equations is obtained.

As already noted, in most cases, the electron current j_e in the plasma may be assumed constant. Thus, the differential equation dj_e/dx = eG is omitted. When integrating the system of the remaining five differential equations (1.4)-(1.6), (1.7), and (1.11), it is convenient to assume that the value of j_e is given, while the voltage drop across the gap V_d is determined as a result of the calculation.

The system of differential equations may be integrated from x = d to x = 0, i.e., from the anode end of the plasma to the cathode end. Equations (2.4)-(2.6) link the five unknown values n_A, V_A, T_eA, j_iA, and S_eA at the anode end of the plasma. By selecting in some manner two values, for example, T_eA and n_A, the remaining unknowns V_A, j_iA, and S_eA can be calculated from equations (2.4)-(2.6). Afterwards, the system of differential equations may be integrated numerically, and the values of the plasma parameters near the cathode n_C, T_eC, j_iC, and S_eC and the plasma potential are calculated with respect to the anode

Expressions (2.1)-(2.3) and (3.7) form a system of four equations which should be satisfied by the values of n_C, T_eC, j_iC, S_eC, V_d and V_C. If, for example, V_d and V_C are calculated from (2.3) and (3.7), then the remaining values (n_C, T_eC, j_iC, and S_eC) should satisfy equations (2.1) and (2.2). Of course, equations (2.1) and (2.2) will in reality not be satisfied. Let us denote the differences between the left and right sides in (2.1) and (2.2) by d j_eC and d j_iC, respectively. The values of d j_eC and d j_iC are functions of the parameters T_eA and n_A, selected arbitrarily at the anode end of the plasma.

The boundary conditions near the cathode are fulfilled if the values of d j_eC and d j_iC are equal to zero. Thus, the problem reduces to the solution of two equations

Assume that the solution of the problem is known for some value of j_e. Now, if we change the current j_e by a small value D j_e, equations (3.8) will no longer be satisfied at the previous values of T_eA and n_A. The quantities d j_eC and d j_iC will be distinct from zero. However, if the value of D j_e is sufficiently small, then d j_eC and d j_iC can be represented in the form

Having determined the derivatives ¶ d j_eC /¶ T_eA, ¶ d j_eC /¶ n_A, ¶ d j_iC /¶ T_eA, and ¶ d j_iC /¶ n_A with the computer, we can solve the system of equations (3.9) for D T_eA and D n_A. By adjusting n_A and T_eA by the values -D n_A and -D T_eA, we obtain new values for n_A and T_eA. Having repeated this procedure several times, we can minimize d j_eC and d j_iC to a zero degree discrepancy.

In concluding this section, we note that approximate theories of the arc mode were developed at different stages in the study of the TIC. There were a number of simplifications in posing the problem which made it possible to obtain approximate expressions for the distribution of the plasma parameters and current-voltage characteristics. The different variations in the approximate theories were usually based on one of the following two opposite assumptions: 1) the rate of ion generation is constant across the gap [10-13] and 2) ion generation is concentrated in an infinitely narrow region near the cathode [4, 6]. The approximations used in these theories greatly simplify the mathematical aspect of the problem and permit a qualitatively correct description of the current-voltage characteristics [14, 15], but they do not make it possible to obtain sufficiently correct distributions of density, potential, and temperature in the plasma volume. Therefore, in the theoretical analysis of the state of a TIC plasma that follows, we will proceed with the complete system of equations with boundary conditions formulated above.

As current j through the converter increases, the temperature T_e and charged particle density n in the plasma increase. Correspondingly, the

rate of ionization increases linearly as n increases and exponentially as T_e increases. However, loss of ions to the electrodes (by ambipolar diffusion) increases linearly as n increases and has a comparatively weak dependence on T_e. Thus, ion generation may no longer be balanced by loss to the walls as T_e increases, so that the density n increases until ion generation begins to be balanced by recombination. In this case, as follows from (1.9), the charged particle density n, at each point, is equal to its own thermodynamic-equilibrium value n(T_e). In short, if the current density is large, the TIC plasma changes to a state of local thermodynamic equilibrium (LTE). The LTE state in a TIC plasma was investigated in [16-21].

Since the ion current j_i is equal to an order of magnitude, to eD_i(1 + T_e/T)x(n/L?), where L is the distance over which the plasma parameters vary significantly, then the left side in (4.2) is also equal, to an order of magnitude, to eD_i(1 + T_e/T)x(n/L?)². It then follows from (4.2) that

The parameters of a TIC plasma usually vary appreciably over a distance the order of the width of the gap, so that L » d. We denote by L_i the following value, having the dimension of length:

Thus, the condition for the presence of LTE in a plasma is the smallness of the length L_i compared to the interelectrode distance d. If the condition

is fulfilled, then a non-equilibrium plasma is retained only with narrow pre-electrode regions of width the order of L_i, through which the ions formed move toward the electrode. As can be seen from (5.3.14) and (4.4), the higher the electron temperature T_e, the narrower are the pre-electrode non-equilibrium regions. The length L_i, which characterizes the width of the pre-electrode non-equilibrium regions, is called the ionization length. Having substituted the value of the ion diffusion coefficient D_i = (kT/e)t _i from (4.7.15) to (4.4) and using (5.3.13) and (5.3.14), we find that the condition for the existence of an LTE plasma (4.5) can be written in the following form:

where p and T are the pressure and temperature of the neutral gas. It is obvious from (4.6) that the conditions for the occurrence of an LTE plasma are large values of the parameter pd and the presence of a high electron temperature T_e. The latter is especially significant, because the left side of (4.6) decreases exponentially as T_e increases.

Transport equations for an LTE plasma. In an LTE region, the condition n = n(T_e) expresses n as a function of T_e, thus reducing the number of unknowns that need to be calculated. The remaining unknowns, T_e(x) and V(x), should be calculated from the transport equations for electron energy flux S_e and for electron current j_e. The latter equation may be used, because the electron current can be assumed constant for the conditions considered here. The ion current j_i(x) should now be calculated using equation (1.5) after solving for the distribution of plasma parameters: n, V, and T_e. To an order of magnitude, in a larger part of the equilibrium plasma, the ratio of ion to electron current is equal to the ratio of their mobilities, i.e., j_i » j_eÖ m/M@. Therefore, the change of the electron kinetic energy flux as a result of ionization is D S_e^(C) » (E_ion/e)D j_i » (E_ion/e)j_eÖ m/M@.* Since the energy transported by the electron flux is S_e^(C) » (j_e/e)kT_e, then D S_e^(C)/S_e^(C) » (E_ion/kT_e)Ö m/M@ « 1. This means we can assume that the electron energy flux is constant in the region of an LTE plasma. By using the Saha formula (5.1.11), we obtain

Equations (4.8) and (4.9) determine T_e(x) and V(x) in the LTE region.

We now consider how the plasma parameters n, V, and T_e vary within the non-equilibrium precathode region. Because of the narrowness of this region, the electron temperature varies little in the region. Electron thermal conduction can equalize the temperature drops that would result from the heating of the electrons in the electric field and the energy losses to ionization of atoms. Therefore, we will subsequently assume that T_e is constant within the precathode ionization region.

Transport equations (1.14) and (1.5) for electron and ion current must be solved jointly with the continuity equation (1.7) for ion current in order to find the distributions n(x) and V(x). In this case, the small terms proportional to dT_e/dx, dT/dx, and the frictional force R_ie can be disregarded in (1.14) and (1.5). Then,

The potential distribution V and the density distribution n in the precathode ionization region vary in such a way as to most effectively remove the generated ions to the electrode. The mobility of the ions is considerably less than that of the electrons. For these distributions, the field and diffusion components of ion current are added to each other (- dV/dx > 0 and dn/dx > 0), whereas the field and diffusion components subtract with the electrons, and compensate to a significant degree. Thus, we have j_e/m _e « j_i/m _i.

since j_e/m _e « j_i/m _i, this becomes

Disregarding the value of j_e(x) on the left side of (4.10), we have

where n(T_eC) is the value of plasma density at the interface of the equilibrium plasma and the precathode ionization region, D V(x) = V(x) - V_dC is the potential drop calculated from the cathode boundary potential,

V_dC = - (V_C - V_d ) is the cathode boundary potential with respect to the anode surface (which was selected above as the zero of potential (see Fig. 9.4), and T_C is the ion temperature in the precathode sheath, equal to cathode temperature.

The solution of this yields the density distribution n(x) in the non-equilibrium sheath. The boundary conditions must be established for the edges of the non-equilibrium region in order to solve equation (4.14). According to (2.2) the following condition should be fulfilled at the interface of a non-equilibrium plasma and the precathode space-charge region (x = 0):*

As the distance from the cathode increases, for x » L_i, the charged particle density asymptotically approaches equilibrium density. Therefore, the second boundary condition is written in the form

Condition (4.16) assumes that dn/dx ® 0 as the distance from the cathode increases (x ® ¥ ). At the same time, according to (4.2), ion current approaches 0.

The density n_C at the interface of the space-charge region is calculated from the boundary condition (4.15). It fo11ows from (4.15) and (4.18), that if l_i « L_i, the relation n_C/n(T_eC) » l_i/L_i holds, where l_i is the ion mean free path. Then n_C² can be disregarded in (4.17) compared to n²(T_eC). In this case, (4.17) and (4.15) give

The change in the electron current in the ionization region is also equal to j_iC.

We now calculate the change in the electron energy flux D S_eC due to ionization in the precathode region. Since the potential change D V_C in the ionization region is usually small compared to V_C^', the expression for D S_eC corresponding to (1.11) can be written in the following form:

Formulas (4.19)-(4.22) make it possible to link the values of the plasma and flux parameters at the plasma-cathode interface to the corresponding values at the boundary of the equilibrium plasma. By substituting these expressions into (2.1) and (2.3), we obtain the following boundary conditions for current j_e and energy flux S_e at the precathode boundary of the equilibrium plasma:

where j_iC is calculated by formula (4.21). In modes with ionization equilibrium, the electron density near the cathode is sufficiently high in most cases so that the effect of the kinetic electron reflection at the precathode boundary of the plasma can be disregarded. Therefore, the reflection coefficients r₁ and r₂ are omitted in (4.23) and (4.24).

This simple consideration may no longer be made for the pre-anode non-equilibrium region, because the electron temperature near the anode is below that near the cathode, and the length of the pre-anode region is considerably greater. Therefore, the variation of the electron temperature must be taken into account within the pre—anode non-equilibrium region. Since the electrons near the anode move in a decelerating field, the electron temperature decreases as the electrons approach the anode. This leads to a decrease of n(T_e) and to a predominance of ion recombination over ion generation.

However, because of the comparatively low charged particle density near the anode, the rate of recombination in the pre-anode region is rather small, and therefore, variation of the values of j_e and S_e can

generally be disregarded in this region. Calculations with formulas (4.8) and (4.9) can then be carried out up to the anode itself. In this case, the boundary conditions for equations (4.8) and (4.9) at the pre-anode potential barrier are expressed by (2.4) and (2.6) as before, if we assume in them that n_A = n(T_eA). Comparison of the distribution of the plasma parameters obtained in the pre-anode region with the data from accurate calculations, which do not use the assumption of ionization equilibrium, shows good agreement [16].

Distribution of plasma parameters and the current-voltage characteristic. Calculation of the phenomena in the equilibrium plasma region reduces to a solution of equations (4.8) and (4.9). Since the variation of the atom density N_A across the gap in a weakly ionized TIC plasma is comparatively small, the term proportional to (1/N_A)dN_A/dx can be disregarded in (4.8). If dV/dx is now calculated from (4.9) and if the result is substituted into (4.8), we obtain one second-order differential equation in T_e(x):

In general (with an arbitrary dependence of D_e on T_e) equation (4.25) cannot be solved analytically. However, if only the largest term, proportional to E_ion/2kT_e, is retained on the right side of (4.25), then the following equation is obtained

Equation (4.27) expresses in explicit form the dependence of T_e on x, and moreover, the integration constant can be calculated from the boundary conditions. Having calculated the electron temperature distribution across the gap by (4.27), we can find the potential distribution using (4.9). If (4.26) is substituted into (4.9), it is obvious that the last term in the right side, i.e., electron thermal conduction, can be disregarded in (4.9) with an accuracy up to terms on the order of 2kT_e/E_ion. We then obtain

Equations (4.26) and (4.28) were investigated in [17-19].

If the state of the plasma is described by equations (4.26) and (4.28), the electron temperature should decrease monotonically, and the potential electron energy should increase monotonically in the direction of the current j_e, i.e., from the cathode to the anode. We note that under these conditions the electron current is primarily a diffusion current, with the electrons moving from the cathode to the anode in a decelerating electric field.

This conclusion is actually valid for a considerable portion of the interelectrode space. However, changing from equation (4.25) to equation (4.26) constitutes a decrease of the order of the differential equation. As a result, the number of integration constants to be determined decreases, so that the system of boundary conditions cannot be satisfied. In order to satisfy all the boundary conditions, it is necessary to retain the term in (4.25) with the highest derivative. Analysis of the solution shows that the term with the second derivative d²T_e/dx² is more significant in the precathode region. If this region is sufficiently thin, and if the variations of plasma parameters in it are insignificant, then an analytical solution of the problem can be obtained [18, 20].

Consider this in an example where the electrons are scattered only by ions. With this restriction, the product D_en is dependent only on electron temperature (D_en ~ T_e^5/2) (not on n), which considerably simplifies the solution. To solve equation (4.25), we turn to dimensionless variables and denote

where n(T₀) and D_e(T₀) are the equilibrium density and the electron diffusion coefficient at temperature T₀. The latter is selected as a unit of measure. By disregarding the value of 3/4 in (4.25) compared to E_ion/2kT, we obtain the equation

Now we introduce a new independent variable z = t ^7/2 and denote d = a _e/I_eW and b = 7I_e/2W. Equation (4.20) is then rewritten in the form

Since the temperature normalization unit T₀ can be selected so that parameter b is equal to unity, equation (4.31) is essentially dependent only on a single parameter d , where d « 1. If the width of the precathode region (in which the term proportional to the small parameter d is significant) is small, the term with the higher derivative in (4.31) may be disregarded in the main part of the gap. As a result, we obtain the equation

which is a special case of equation (4.26) for D_en ~ T^5/2.

The integration constant of equation (4.32) can be calculated by being given the value of T_eA, i.e., z⁽⁰⁾(1). As a result, we obtain

Now consider equation (4.31) near the cathode boundary of the equilibrium plasma, i.e., for small values of x . We assume that the value of z in the precathode region, where the term with the second derivative is significant, varies insignificantly and hardly differs from z⁽⁰⁾( x ). Thus, z( x ) may be replaced by the constant z( 0) in the coefficients of equation (4.31) and in the free term.* As a result, we obtain the equation

Equation (4.34) is linear in z. The common solution of equation (4.34) is the sum

The integration constant C₂ is calculated from the condition that, as the distance from the cathode increases, when x > d z(0), z(x ) changes to z⁽⁰⁾( x ). To do this, it is necessary to assume that C₂ = z⁽⁰⁾(0). Then, we have

The integration constant C₁ is calculated from the boundary conditions at the cathode (at x = 0).

In contrast to z⁽⁰⁾( x ), which is a comparatively smooth function of x ,

varies rapidly at small distances x ~ d . Therefore, the assumption that z is constant, used in the solution of equation (4.4), is fulfilled only if z⁽¹⁾( x ) is a small addition to z⁽⁰⁾( x ), i.e., if the integration constant C₁ « z⁽⁰⁾( x ). However, even if this condition is fulfilled at the precathode boundary of the plasma, consideration of the term z⁽¹⁾( x ) significant, since the derivative dz⁽¹⁾/dx ½ x _{= 0} = C₁/d can be of the same order as the derivative dz⁽⁰⁾/dx ½ x _{= 0}.

Calculation of the term C₁/d when calculating the derivative dz/dx leads to the fact that the distribution of the parameters of the equilibrium plasma in the precathode region is generally non-monotonic. In this case, the field component of current dominates the diffusion component and the electrons move in an accelerating electric field.

*However, the derivative of z with respect to x can differ considerably from dz⁰/dx , because the variation of z occurs in a narrow range.

The integration constants C₁ and t (l) and also the potential barriers V_C¹ and V_C can be obtained from the boundary conditions (4.23) and (4.24), and (2.4) and (2.6). The boundary conditions near the anode can be satisfied approximately by assuming that the electron current j_e and energy flux S_e are expressed by formulas (4.26) and (4.28). The boundary conditions near the cathode can be satisfied accurately by taking into account the current component proportional to dV/dx, and the component in the energy flux proportional to dT_e/dx.

The current-voltage characteristics and the distribution of plasma parameters in the ionization equilibrium region for two points on the current-voltage characteristic are shown in Fig. 9.5. As current increases, the distribution of the plasma parameters in the precathode region becomes non-monotonic. The maximum density and minimum potential energy eV occur near the cathode. With further increase in current, the non-monotonic aspect of the plasma parameter distributions becomes more pronounced. However, the method of calculation ceases to be valid, because the relation z⁽¹⁾(0)/z⁽⁰⁾(0) increases.

With ionization equilibrium, the assumption that the electrons are scattered only by ions is usually well satisfied in the precathode region of the TIC. As the distance from the cathode increase, a mixed scattering mechanism usually comes into effect. When this is the case, instead of (4.33), the general solution of (4.27) can be used, where the diffusion coefficient D_e(T_e) is interpolated between two scattering mechanisms.

it is convenient to specify the total current through the device j = j_e and to calculate the remaining values, including the voltage drop in the gap, V_d. Some value of T_eA is selected, and equations (4.8) and (4.9) are solved for the derivatives dT_e/dx and dV/dx. These are then integrated from x = d to x = 0,* giving the solution for the system of equations. The values of T_eC and

are then calculated. The values of j_iC, V_d and V_C' are eliminated from equation (4.21), (4.24), and (4.41) and are substituted into (4.23). If the value of T_eA is arbitrary, condition (4.23) is not satisfied. The difference between the left and right sides of (4.23) yields a discrepancy of d j_eC, which is a function of T_eA. Since there is now only one discrepancy, it is easily minimized, and the desired solution is quickly obtained. Then, by changing current j or some other parameter by a small value, and having linearized equation (4.23) for small variations D T_eA, we can calculate the value of D T_eA, i.e., we can find the new solution.

We note here that when the electron temperature T_e in the plasma is sufficiently high, calculation using the system of LTE equations is not only convenient, but necessary. In the present case, the value of t _i^-1 is so large that the ordinary calculating scheme, outlined in the preceding section, becomes unstable. This is related to the fact that even a very small error in the boundary values at one end (for large values of t _i^-1) prevents the integration of the system of equations to the other end (if the gaps are large).

The transition of the plasma to a state of ionization equilibrium is illustrated by Fig. 9.6, where the ratio of electron density in the gap to the calculated [16] equilibrium electron density n(T_e) is plotted. It is obvious that the plasma in the main portion of the gap is already in LTE at a current of j » 15 amp/cm² and voltage drop of V_d » 0.6 volt. In this case, the distributions of the plasma parameters, obtained by accurate calculation and by calculation with the LTE equations, essentially agree, i.e., if the finite length of the precathode ionization region is taken into account. As the current increases further, and as T_e increases, the precathode ionization region shrinks so that its width can be disregarded compared to the size of the gap. We note that when calculating the current-voltage characteristics for larger current densities and higher cathode temperatures the presence of ionization equilibrium may be assumed from the very beginning.

a deviation of the distribution function from Maxwellian in the high energy range, as was shown in §3, Chapter 6. The primary mechanism for electron energy relaxation in a high density TIC plasma is electron—electron interaction. Inelastic electron—atom collisions usually play an insignificant role in the electron energy relaxation of the arc mode. In a high density TIC plasma, there is a rapid symmetrization of the distribution function. Therefore, the plasma is stable with respect to plasma oscillations, and this relaxation mechanism also plays no important role.

Deviations of the electron distribution function from Maxwellian due to cathode beam injection have been observed experimentally [9]. Some results from probe measurements for one of the TIC operating modes are shown in Fig. 9.7. For these conditions, the second derivative of the probe current with respect to voltage is related to the distribution function by formula (7.1.10). It is obvious from the figure that a clearly marked "hump" is observed for the curves near the cathode, which indicates that the electron density in the high energy range exceeds the equilibrium

value. As the distance from the cathode increases, the electron beam relaxes and the electron distribution functions change to a Maxwellian distribution. The dependence on electron density of the distance from the cathode at which the Maxwellian distribution is established, is shown in Fig. 9.8. The solid line corresponds to the Maxwellization length L_E for electron-electron collisions at electron energy E = 1.5 eV, calculated by the formula L_E = (Dt _E)^1/2 where D is the fast electron diffusion coefficient and t _E is the energy relaxation time, calculated by formula (3.1.14). Agreement of the theoretical calculations arid the experimental data confirms that the main mechanism of beam relaxation is electron—electron collision.*

The dependence of ion current j_iC (from the plasma to the cathode) on the current j through the device is plotted from experimental data in Fig. 9.9. The addition to the ion current, D j_iC due to the non-equilibrium character of the electron distribution function, calculated by formula (6.5.1), is also plotted here. It is obvious that component D j_iC is relatively small. Data obtained at comparatively low current densities were used to calculate D j_iC. Similar relations have also been obtained at high current densities [21]. The effect of the non-equilibrium character of the electron distribution function on the rate of ionization in a TIC is also illustrated by Fig. 9.10. The solid curves are the calculated values of V(x), n(x), and T_e(x), obtained by taking into account the non—equilibrium mechanism of ionization. @ The dashed curves correspond to the results of calculations in which the rate of ionization was calculated by formula (1.9). It is obvious that the consideration of the non-equilibrium mechanism of ionization in TIC modes does not lead to appreciable change in the distribution. of the plasma parameters. @ Voltage across the gap of V_d » 0.8 volt and V_d » 0.92 volt and a precathode potential barrier V_C » l.31 volt and V_C » 1.36 volt correspond to the data presented in Fig. 9.l0 a and b.§ Accounting for the non-equilibrium mechanism of ionization does not change V_d and V_C significantly. The relative contribution of the correction D j_iC, (for the data presented in Fig. 9.l0 a and b) to the value of ion current j_iC is D j_iC/j_iC » 17% and D j_iC/j_iC » 7.7%, respectively.

*However, with a decrease in pressure, when the electron distribution function of the beam is asymmetrical in velocity directions at a considerable distance from the cathode, two relaxation regions are observed. The distribution function in direct proximity to the cathode relaxes by plasma oscillations. As the distance from the cathode increases, the main relaxation mechanism becomes binary Coulomb collisions (see [88]).

@In this case, the value of D j_iC was calculated by formula (6.5.1) and was added to the right side of boundary condition (2.2) for the ion current j_iC from the plasma to the cathode. At the same time, the energy flux S_e changed abruptly at the precathode plasma boundary by the value [E_ion + e(V_C -V_d)]D j_iC/e.

@A similar result was obtained in [22] for the case where electron energy relaxation was accomplished by inelastic electron-atom collisions § The selected values of V_C are very close to the excitation potential of the cesium resonance level. Under these conditions, the non—equilibrium of the fast electron distribution function is reflected primarily in the resonance transition, which makes it possible to use the formulas obtained in §5, Chapter 6.

We also note that the effect of the non-equilibrium ionization mechanism on the calculated results are actually much less than would be expected from the value of the relation D j_iC/j_iC. This is related to the fact that inclusion of the non-equilibrium ionization mechanism and the related electron energy losses into the calculated equations is compensated for by a very insignificant decrease in the electron temperature T_e, which leads to a corresponding decrease in ionization, because of the effects of the equilibrium electrons.

Concluding the kinetic reflection coefficients r₁ and r₂ in the boundary condition (2.1) has a much greater effect. The values of r₁ and r₂ for different points on the current-voltage characteristic, calculated according to Figs. 6.12 and 6.13, are presented in Table 9.5 as an example.

Calculation with the kinetic reflection coefficients leads to a shift of the current-voltage characteristic to the right, i.e., to an increase of V_d and to some decrease in the slope of the characteristic. This effect is manifested most strongly in the lower segment of the current-voltage characteristic at comparatively small values of n_C. This is illustrated by Fig. 9.11, where the lower segments of two calculated current-voltage characteristics of a TIC are compared. One segment is calculated accounting for the reflection coefficients r₁ and r₂ in the boundary condition (2.1). The other is calculated not accounting for them. We also note that consideration of the reflection coefficients at comparatively small emission currents is usually significant in a developed discharge as well at J » J_S⁽⁰⁾.

We now review the main results from experimental investigations of the state of a TIC plasma in the arc mode [9, 23-36] and compare these results with calculations.

The current-voltage characteristics are shown in Fig. 9.13. To compare the calculated and experimental data, the value of emission current j_S⁽⁰⁾ on the calculated current-voltage characteristic was selected so that one of the points of this characteristic (j = 1.5 amp/cm²) coincided with the corresponding point of the experimental characteristic.

density n(x), potential V(x), and electron temperature T_e(x) can be seen from Fig. 9.12. The density distribution has a maximum in the precathode region of the plasma. As current increases, the maximum density approaches the cathode, but the motion of the maximum toward the cathode is slowed as the discharge develops. The value of maximum density increases rapidly as current increases and then takes on a linear dependence.

plasma to the anode, is small and has a value of 0.1- 0.3 volt. As the current increases, a greater part of the voltage applied to the converter appears in the precathode barrier, whereas the pre-anode potential barrier and the distribution of the plasma potential vary insignificantly.

Variations of electron temperature across the gap are small: 300 - 400° K from the results of probe measurements and 400 - 800° K from the results of spectral measurements and calculations [3, 9, 25, 37]. In this case, the sharpest variations of T_e are observed in the pre-anode region of the plasma. The absolute values of T_e in the main part of the plasma are usually within the range of 2300 - 2800° K. As the current increases, the electron temperature in the plasma increases, but this increase is comparatively small.

Ion current in the plasma. The distribution of ion current obtained under comparable conditions, from calculations (curve 2) and from experiments. (curve 1), are presented in Fig. 9.14. It is obvious from the figure that the ion current is positive, i.e., it is directed toward the cathode in the half of the gap adjacent to the cathode. The ion current changes sign at some distance from the cathode, and it is directed toward the anode in the anode part of the gap. The distributions obtained for the density and potential are such that the field and diffusion components of the ion current in the main part of the gap add to each other. This creates conditions for the most efficient removal of ions to the electrodes.

The rate of ion generation (and recombination) in the interelectrode space of a TIC can be determined from the experimental distribution of the ion current. Ionization (G > 0) usually prevails near the cathode and recombination (G < 0) usually prevails near the anode. The ionization is a maximum in the precathode region, whereas the electron temperature is rather high. On the other hand, the electron temperature in the pre-anode region is low, and there ion recombination prevails. However, the rate of recombination near the anode is usually low, and ion current in this region varies weakly.

We note that the comparatively low gas temperature near the anode may lead to an increase of the rate of recombination in the pre-anode region because of dissociative recombination, with the participation of molecular Cs₂⁺ ions. Therefore, comparison calculations were carried out in which the recombination coefficient in the region where recombination dominates ionization was increased by an order of magnitude compared to that given by formula (1.9). The calculation showed that the distribution of plasma parameters is very weakly dependent on the rate of recombination near the anode. The distribution of j_i(x) near the anode remained approximately constant, as in Fig. 9.14 (curve 2).

If the voltage on the converter increases as current increases, the cathode potential drop and electron temperature also increase. An increase of T_e leads to an increase of the rate of ion generation and, accordingly, to an increase in the plasma density and ion current to the cathode. However, an increase of charged particle density also leads to an increase in collisional recombination. As a result, the thickness of the region, from which the ions can reach the cathode without recombining, is reduced. As the discharge develops (current increases), the region of effective ionization contracts, and the maximum density and minimum potential energy move toward the cathode. Since the rate of ionization is exponentially dependent on electron temperature T_e, a small increase in T_e leads to a strong increase in the rate of ionization and to a considerable increase in the density and the ion current. Therefore, the electron temperature in the discharge varies only slightly as current increases, whereas the density, the ion current, and the precathode potential barrier increase rapidly.

The mechanisms of electron scattering in the arc plasma of a TIC. Strong variations in density n across the gap mean that different mechanisms of electron scattering occur in the different regions of the plasma. Electron scattering from ions prevails in the precathode part of the plasma, where the ion density is greater (at sufficiently large currents). The diffusion coefficient has the dependence D_e ~ 1/n with this scattering mechanism. Therefore, density should increase more rapidly than linearly as current j_e increases, i.e., to ensure the passage of electron current beyond the density maximum, where the current is primarily by diffusion. This is confirmed by Fig. 9.12.

Scattering from neutral atoms (a _e = 2 and b _e = 2) prevails in the pre-anode part of the plasma if the currents are not too large. This is shown by the fact that the pre-anode potential barrier is weakly

dependent on the converter current. Actually, according to (1.6), (2.4), and (2.6), at j_sA = 0, the equation of energy balance near the pre-anode potential barrier has the form

In the absence of emission from the anode (j_sA = 0), from (6.1) and (2.4), it follows that

Since the distribution and absolute value of electron temperature varies slightly as current increases, and the diffusion coefficient D_e near the anode is not dependent on current, then, according to (6.2), the value of the pre-anode potential barrier V_A remains approximately constant. Under these conditions, the electron density near the anode should increase linearly as current increases. This has been confirmed both experimentally and by calculation.

where the pressure p is measured in torr, the interelectrode distance d is measured in mm, and kT_e is measured in eV. Calculations show that, for pd > 1 torr× mm, this condition is fulfilled for j > 10 amp/cm².

Transition to the ionization equilibrium mode for large currents and rather large values of pd has been observed in experiment [19, 23, 24].

The presence of a Boltzmann distribution for the excited level population, observed with large currents through the converter and for sufficiently large values of pd, also affects the transition to a state of ionization equilibrium [19,24].

Calculated distributions of plasma parameters [21] in the interelectrode gap of a TIC in the presence of ionization equilibrium in the plasma is shown in Fig. 9.16. It is obvious that the overall character of the

distributions remains the same in the absence of ionization equilibrium. The ion current distribution j_i(x) in the equilibrium plasma, calculated by formula (1.5), is also shown in the figure for these conditions. It is obvious that an LTE plasma may also be subdivided into two regions: that where ion generation (d j_i/dx < 0) prevails, and the recombination region (d j_i/dx > 0). In this case, the main ion generation is concentrated in the non-equilibrium precathode region, because the value of ion current in an LTE plasma is comparatively small.

The nature of the transition from the diffusion to the arc mode is determined by the external parameters of the TIC, T_C, P_Cs, and d (with great sensitivity to the cathode temperature). At low cathode temperatures (T_C < @ 1600 - 1700° K), the transition to the arc mode occurs abruptly (curves 1 and 2 in Fig. 9.17), so that the diffusion and arc branches of the characteristic are disconnected, and the ignition voltage V_ig considerably exceeds the minimum arc, or extinction voltage V_ex. An intensive glow in the pre-anode region is observed prior to ignition in this case. As the temperature increases, the difference V_ig - V_ex first decreases smoothly and then more sharply (Fig. 9.18) until the separation between the branches of the characteristic essentially disappears, and transition from one mode to another occurs on the vertical section of the characteristic (curves 3 and 4 in Fig. 9.17). At even higher temperatures, the transition to the arc mode is accompanied only by a break (or bend) in the current-voltage characteristic (curve 5 in Fig. 9.17). At T_C > 2000° K, transition from the diffusion to the arc mode is continuous.

Ionization in the anode barrier, in actual TICs, begins at voltages on the converter the order of several volts (see Fig. 9.18). The comparatively low values V_ig £ 2 volt, the range of TIC operating temperatures being what they are, indicates, that ionization is multi-stepped for these conditions as well. Direct ionization can occur only for V_ig > E_ion, i.e., for T_C < @ 1000° K. The beginning of ionization is accompanied by a glow in the pre-anode region. The ions formed cannot travel to the anode because of the anode barrier and, therefore, are passed through the plasma to the cathode. This leads to an increase of positive space charge and, consequently, to an increase in the plasma voltage drop and to an increase of the conducted current. An increase of electron density and energy leads to the beginning of ion generation in the volume, as a result of which breakdown begins, i.e., there is a rapid transition into the arc mode.

stepped ionization in the pre-anode region. Variation of the ignition mechanism as cathode temperature increases is determined primarily by changes in the precathode barrier, and accordingly, of the compensation parameter a = exp[2(f _C - m )/kT_C]. It is also determined by the increase of electron density in the diffusion mode, which contributes to Maxwellization and a decrease of the relative radiation losses. A direct increase in the energy of emitted electrons as T_C increases plays a lesser role.

The effect of the ions formed near the anode is intensified by the fact that the pre-anode barrier retards them, forcing the ions to move toward the cathode. Ionization in the pre-anode region thus leads to redistribution of the potential, and as a result, to an increase in the electron density and electron temperature. Moreover, because of this, the region of maximum ion generation is shifted toward the cathode. There is a rapid increase of current with simultaneous re-distribution of potential in the interelectrode space, which terminates in the formation of a potential well. This redistribution of potential leads to a current jump. The potential difference for extinction and ignition is considerably less than for ignition in the pre-anode barrier. At even higher cathode temperatures, where the precathode barrier is sufficiently large, i.e., where the compensation parameter is large a » 1 and the rate of Maxwellization is sufficiently high, electron heating with the conduction of current leads to the beginning of ion generation even when the field in the volume retards the electrons. Thus, ionization begins in the precathode region, transition from the diffusion to the arc branch occurs smoothly and only a small break can be observed on the current-voltage characteristic. At even higher cathode temperatures the temperature of the emitted electrons is so high that there is no need for additional heating of the electrons for volume ionization, and the diffusion mode, without generation in the volume, generally does not exist. In this case, the electron density in the volume with open-circuit is already sufficiently large and increase smoothly as current increases.

As can be seen from Fig. 9.19, the qualitative character of the variation in the theoretically calculated characteristics as cathode temperature varies, accompanied by a variation in @, is the same as that obtained by experiment. The fact that the sign and the value of the precathode barrier (i.e., the compensation parameter) and also the carrier density in the plasma have an appreciable effect on the type of characteristic in this temperature range is confirmed by Fig. 9.20 [37, 38], where the calculated current-voltage characteristics are presented. In these calculations, only values of @ change. The distribution function was assumed to be Maxwellian. It is obvious from the curves that ignition begins in the precathode region and the current-voltage characteristic is slightly dependent on energy losses at @ = 10⁴. At @ = 1, ignition occurs in the volume in the pre-anode region and there is a segment of negative resistance on the characteristic which neglects radiation loss. A further decrease of @ in this approximation hardly affects the type of characteristic. With radiation loss, the diffusion and arc branches at these values of @ are separated. This means that here ignition occurs in the pre-anode barrier. However, ionization in the pre-anode barrier was not taken into account in these calculations.

lifetime, typical in a TIC for the 6P level, of 1/t _eff = 0.84× x 10⁶ sec^-1 [37, 38]. Transition from ignition on the pre-anode barrier to ignition in the volume occurs at T_C = 1600 - 1700° K, where the carrier density in the diffusion mode is on the order of 10¹² cm^-3 for the experimental characteristics depicted in Fig. 9.17. Also, Maxwellization of the electrons is slowed considerably at these concentrations. Therefore, the transition to ignition in the volume occurs at a » 1 for these characteristics.

Variations of P_Cs and d affect the ignition voltage less than T_C. Although V_ig has a minimum both for variations in d and in P_Cs, the dependence of V_ig on pd is not universal, i.e., Paschen's Law is not fulfilled [39, 40, 41].

This is due mainly to the effect of P_Cs on the cathode work function and on the value of a . It is due also to the variation of the carrier density in the pre-anode region as P_Cs and d vary.

When the energy applied to the plasma becomes insufficient to maintain the high electron temperature required for ion generation, the discharge is extinguished [9, 37, 38. 42].

In actual experiments, if the electrodes have large enough areas, the discharge is pulled into a column during extinction [40, 43]. This begins at the points of the current-voltage characteristic where segment cb with a comparatively small slope changes to the almost vertical* constriction segment ba (see Fig. 9.17). The discharge glow varies on this segment, is pulled into a column in passing toward point a (and is expanded over the area of the electrodes by reverse changes). Thus, the discharge on the constriction segment breaks up into two regions, into arc and diffusion modes. The plasma parameters of the arc region, according to probe [31] and spectral measurements, remain essentially constant, so that the variation of the current on the constriction segment, as indicated by investigations with a device having a sectional anode [4], occurs primarily by the redistribution of the areas for the arc and diffusion regions.

At the point where constriction is observed, the theoretical current-voltage characteristics calculated on the assumption of uniform current distribution over the entire area of the electrodes have segments with negative resistance (see Figs. 9.19 and 9.20). However, the arc, homogeneous in the transverse direction, becomes unstable on this segment of negative resistance. Because of this instability, as indicated by calculations [44] there can be a separation of the plasma into arc and diffusion regions. And, the vertical segment of the characteristic observed in experiment can occur. @

*The voltage difference between points b and a is less if there is less non-uniformity in T_C and f _C at the cathode and if the electrodes are more parallel.

@The transverse non-uniformity in the TIC plasma was also considered in [45]. Calculations of systems with this type of instability were previously carried out for the plasma of a semiconductor [46]. These also indicate that constriction of the current should occur.

As indicated by measurements, if cathode temperatures at the extinction point are not very high, the precathode barrier is sufficiently large, so that the back current from the plasma is relatively small, and the current at the extinction point j₀ is close to the emission current. Therefore, the relation j₀(T_C) follows the temperature dependence of cathode emission currents in Cs vapor, i.e., it is similar to the usual S-curve (Fig. 9.21) (compare Fig. 2.15). The variation of these curves with changes in cesium pressure is similar to the shifts that occur with the usual S-curves.

The potential drop V⁰_d across the gap at the extinction point, over a wide range of external parameters, varies only weakly. The dependence of extinction voltage V⁰_d on the parameter pd (Fig. 9.22) has a shallow minimum at pd » 0.1 - 0.4 torr× mm. The increase of V⁰_d as pd increases is due to an increase in the plasma resistance. The increase of voltage for small values of pd is explained by a decrease in the effectiveness of volume ionization and by the increase in the ion loss when the inter-electrode distance becomes comparable to or less than the ionization length.

The calculated dependence of V⁰_d on current at the extinction point for two cathode temperatures [37, 38] and also the experimental values of V⁰_d for different cathode temperatures are shown in Fig. 9.23.

An increase of V⁰_d for small values of j₀ is related to an increase in the loss of excited atoms and to other non-linear effects, which cause a decrease in the rate of ion generation with a decrease in density. As can be seen from Figs. 9.19 and 9.20, accounting for radiation losses increases the calculated values of V⁰_d. It is obvious from the comparison of curves 1 and 2 in Fig. 9.23 that V⁰_d decreases somewhat as cathode temperature increases (for j₀ = const). This is also observed in experiments where modes with identical values for j₀ are compared by using different values of T_C, but values of T_C that correspond to the high- and low-temperature branches of the emission S-curve [4].

The emission current to the plasma in the diffusion mode with compensation parameter a < 1 is limited by the negative space charge formed by the emitted electrons.

As indicated in §2, if there are large emission currents for a < 1, a space-charge barrier may exist in the region of low voltage drops in the arc mode if ion flow from the plasma is inadequate to compensate for the space charge of the electrons [4, 6, 48, 49]. The emission current to the plasma in a virtual cathode mode will increase rapidly as the voltage drop increases, because of a decrease in the height of this barrier as ion flow increases. This trend continues until the barrier disappears and the field near the cathode changes sign. The total converter current also increases rapidly with voltage, but the slope of the characteristic decreases after the virtual cathode disappears.

The extinction current j₀ varies only slightly with temperature, and in plotting this variable, an apparent "cut" appears on the peak of the S-curve (Fig. 9.24a). The current j₁(T_C) of the second break point is also plotted on this figure. The variation of current j₁ with temperature is similar to that of the emission current. The ratio j₁/j₀ may reach values of 10 or more. This significant decrease of the converter current on the segment with great steepness cannot be caused by reverse current, since, according to the values calculated from measured values of electron density near the cathode at point j₀, this reverse current does not exceed half the forward current. However, this large j₁/j₀ ratio can be explained as a decrease in emission current as a result of the appearance of a retarding barrier [4]. On patchy cathodes, the retarding barrier can occur on spots with the largest emission current rather than on the entire surface, i.e., if the patch sizes are greater than the thickness of the space charge sheath. Patches with a high work function are exposed in this case.

Probe and spectral measurements show that the retarding barrier for the segment bc on the characteristics of Fig. 9.24 is present over all or much of the cathode surface.

Formation of a virtual cathode for electrons means the occurrence of a potential well for ions. Ion capture in this well reduces the height of the barrier which retards the electrons [48]. The ion density in the well on patchy cathodes should be low, because the captured ions can move toward the patches with large work function, where the well is not formed.

If the retarding barrier D V_C is less than the precathode voltage drop in the plasma V_C (Fig. 9.2a), then the boundary conditions for the virtual cathode (2.9)-(2.11) differ from the boundary conditions (2.1)-(2.3) only by the replacement of emission current j_sC by the emission current of the virtual cathode j_sv with a "work function" of f _Cv = f _C + D V_C.

*In the foreign literature, the mode with the virtual cathode has been called the double-sheath mode [47] or the obstructed mode [6].

In this case the work function of the virtual cathode f _Cv is found from the condition that field E(V) approaches zero at the point of the maximum potential. Then, if the filling of the well with plasma ions can be disregarded, the segment of the current-voltage characteristic with a virtual cathode is not dependent on the cathode work function f _C. The current-voltage characteristics, calculated in this approximation with ion current j_iC from the cathode equal to zero, are shown schematically in Fig. 9.25 [37, 38]. The solid lines show the characteristics calculated (disregarding the possibility of the formation of a virtual cathode) for different values of f _C, i.e., for different values of j_s⁰. The dashed line shows the characteristic where there is a virtual cathode, calculated with the same parameters, which, as indicated above, is not dependent on f _C. This curve intersects the current-voltage characteristics at the point where E_C = 0; therefore, the formation of the virtual cathode begins at this point. For small currents j_s⁰ (curve 1), point a (where E_C = 0) is located in the unstable branch (the segment of negative resistance) of the current-voltage characteristics. At these points, extinction begins at point b. At large emission currents (curves 2 and 3), point a (where E_C = 0) is located on the stable branch of the current-voltage characteristic.*

*According to the numerical calculations of [37, 38], at P_Cs = 1 torr, d = 0.5 mm and T_C = 1500° K, the value of current j₀, at which the segment with the virtual cathode occurs on the characteristics, does not exceed 2 amp/cm².

With a further decrease of voltage, the characteristic proceeds along the dashed curve after reaching the point where E_C = 0. As can be seen from Fig. 9.25, this curve contains a segment with a negative slope segment de. Upon reaching point d, the mode with the virtual cathode becomes unstable and if current decreases further, constriction of the discharge occurs, i.e., point d is the point of transition from the vertical segment to the first sloping segment, and point a, where E_C = 0, is the point of change in the slope. The curves shown are for j_iC = 0. If the ion currents are finite, with an increase of cathode temperature, the current in the diffusion mode may exceed that of point d. The mode with the virtual cathode is then stable down to the intersection with the diffusion branch and constriction of the discharge is not observed.

We note that upon formation of the virtual cathode the voltage drop across the gap is generally not dependent on current. In a linear approximation, all the additional voltage applied to the TIC appears at the precathode barrier, reducing the height of the retarding barrier D V_C. The current through the converter should increase in proportion to the current emitted from the cathode into the plasma j_sv. Therefore, the current on the segment of the current-voltage characteristic with a virtual cathode, in this approximation, should increase exponentially as follows

where j₁ and V₁ are the value of the current and voltage at the point of formation of the virtual cathode (V < V₁).

The non-linear phenomena (recombination and scattering from ions) at large currents lead to a decrease in the slope of the current-voltage characteristic for the virtual cathode segment. Therefore, the experimental voltage drop across the gap, V_d', at the point of the second break on the current-voltage characteristic, may considerably exceed the voltage drop at point j₀ (see Fig. 9.23). On the other hand, energy losses increase the slope at small currents, and this can also lead to a segment de (Fig. 9.25) with a negative slope.

We now consider the reasons for an increase of current in the developed arc when there is a precathode potential drop that accelerates electrons away from the cathode [4, 21, 40, 50, 51]. As can be seen from the experimental characteristics presented above, saturation is not observed on this segment; current increases as voltage increases. The increase of current on this segment is caused by a number of factors: an increase of emission current because of the Schottky effect, an increase of ion current to the cathode, a decrease of reverse flow of Maxwellized electrons from the plasma to the cathode, and a decrease in the return of hot electrons emitted by the cathode to the plasma. Moreover, when moving along the characteristic, the cathode work function may vary because of changes in the total flow of atoms and ions to the cathode.

Here r₁ and r₂ are the kinetic reflection coefficients for emission currents to the plasma, D f _Sch is a decrease of the work function because of the Schottky effect, which, according to (2.15), is equal to D f _Sch = b ^3/2E^1/2 and D f q is an increase of the work function because of a decrease in the cathode coverage (see §12).

The results of calculations that show the effect of these factors (for P_Cs = 2 torr, d = 0.6 mm, T_C = 1830° K, and j_s⁽⁰⁾ = 4.7 amp/cm²) are presented in Fig. 9.26. It is obvious from Fig. 9.26 that if the voltage drop in the plasma is small reflection of the hot emitted electrons is important, and reflected current is 50% of net emission current. That is, the back current from the plasma is up to 30% of the forward current to the plasma. As the voltage on the plasma increases from 0.4 to 1.4 volt, half the increase of the total electron current is caused by a decrease of electron reflection and decrease of back current from the plasma.

As indicated by calculation and by direct calorimetric measurement [52, 53] in a developed discharge, the main portion of the energy expended in the plasma is expended in the formation of ions, the majority of which return to the cathode. Therefore, as already indicated in §9.1, the ion current to the cathode can be calculated by the formula

It is obvious from this formula that ion current may be up to 30% of electron current in a short—circuited converter, i.e., at eV_d » 1.5 volt. The ion contribution to the current is considerably less in a practical operating converter.

A decrease in the flow of atoms and ions to the cathode as the electron density (and partial pressure) near the cathode increases leads to a comparatively small decrease in the work function. The contribution of this effect at low cesium pressures may be more substantial as is shown in §11.

The change of emission current because of the Schottky effect is approximately 30% under these conditions (b was assumed equal to 1 in the calculations, which correspond to the normal Schottky effect). The Schottky effect increases as the ion current and voltage increase (see (2.8)). It is obvious from Fig. 9.26 that the current-voltage characteristic, calculated taking into account the enumerated factors, is close to the experimental characteristic taken under identical conditions [54].

At higher cathode temperatures, with greater emission current densities and greater electron densities near the cathode, the reflection coefficients r₁ and r₂ decrease and become insignificant. But the role of the Schottky effect increases as a result of an increase of the field near the cathode, which in turn is due to an increase of ion current (see Fig. 9.26b).

The role of the Schottky effect on the segment of the characteristic where not all the patches are exposed is greater for patchy cathodes than for homogeneous cathodes. This can be taken into account formally by increasing the coefficient of anomaly b [4].

or V_d. Under these conditions, according to (2.8) (disregarding ion current from the cathode), we have E_C^1/2 ~ j_sC. Thus, the relative increase of current through the converter z = j_sC/j_s⁽⁰⁾ as a result of the Schottky effect, according to (2.15), is given by the transcendental equation [4, 50]

where a ₀ = b _e^3/2E_C0^1/2/kT_C. Here j_s⁽⁰⁾ is the emission current, determined from the characteristic neglecting the Schottky effect (i.e., for b = 0) and E_C0 is the field near the cathode at b = 0. By using this formula, one may, with the characteristic calculated at b = 0, construct the characteristic for an arbitrary value of b . It is necessary to include a shift of the characteristic to the right due to a decrease in the cathode work function by the value (ln z)kT_C/e because the voltage on the converter increases by that value. (Plotting of points a₁ and a₂ for b @ = 0.4 from point a₀(b @ = 0), where equation (9.3) has two roots z₁ and z₂, is shown in Fig. 9.27.)

As indicated in §8, if the emission currents near the discharge extinction point are large, there is a decelerating field over much of the cathode surface, i.e., a virtual cathode is formed. As the height of the retarding barrier decreases, moving along the current-voltage characteristic, the patches having smaller work functions are exposed sequentially. Under these conditions, it is not possible to distinguish strictly between a mode with a virtual cathode and one with the anomalous Schottky effect, with a gradual exposure of the patches. There is actually a gradual decrease along the curve of b from b » 1, where the main emission is from closed patches above the barrier (the virtual cathode), to b = 1, where all the patches are exposed. Large values of b correspond to the steep section on curves which haye no segment with a large slope corresponding to the virtual cathode. Comparison of the experimental and theoretical curves, calculated at different values of b , shows that the degree of non-uniformity, which must be introduced

As can be seen from Fig. 9.27, including the dependence of cathode emission current on the field can result in a segment nr on the curve with negative resistance [4]; upon reaching this segment an abrupt transition to a second, stable branch can occur. This transition would lead to current constriction to a portion of the cathode surface if a constant external voltage is maintained, i.e., to the formation of a cathode spot.

The effect of cathode temperature. The primary effect of changes in the cathode temperature is change in the emission current. With a fixed emission current in a developed arc, temperature variation of the emitted electrons has little effect, because the electrons are heated in passing through a precathode barrier which considerably exceeds kT_C. This is illustrated by Fig. 9.29a, where the distributions of the plasma parameters are shown for two cathode temperatures which correspond to the ascending and descending branches of the S-curve, so that they have approximately identical values of emission current. These data agree well with the calculations whose results are presented in Fig. 9.29b.

Fig. 9.28 Calculated current-voltage characteristics of a TIC with different cathode emission and different interelectrode spacings. P_Cs = 2 torr,

The effect of interelectrode spacing. An increase in the spacing leads to an increase of plasma resistance; therefore, at large values of d and with a constant current, the total voltage drop and precathode drop increase as the gap increases, and the maximum density in the volume increases.

An increase of maximum density as the interelectrode distance increases is explained by the fact that the current beyond the density maximum is primarily by diffusion, and a large drop in density is required for the passage of the same current through a large interelectrode distance [6].

between the peak of T_e and the anode decreases, which leads to a decrease in the thermal conduction to the anode and to a decrease of the value of V_A. This is easily seen on the experimental curves of Fig. 9.30, where the distributions of plasma parameters at different spacings are shown.

The indicated variation of the parameters as d varies occurs only if the interelectrode distance exceeds the dimensions of the precathode ionization region. At small values of d, when the width of the gap becomes less than the ionization length, an inverse dependence can occur: V_d and the mean electron temperature in the plasma increasing as d decreases. This is explained by the fact that the total number of ionizing collisions in small gaps begins to decrease and higher values of T_e and, accordingly, of V_d are required to provide the necessary rate of ion generation [10, 11]. Therefore, at a given current, the variations of V_d(d) is nonmonotonic, and there is an optimum gap width d_opt for which V_d is minimum, and consequently, the output voltage is maximum. The value of d_opt is dependent on the value of current, i.e., it varies along the curve (Fig. 9.31). When analyzing the experimental data, one should keep in mind that the flow of atoms to the cathode and the degree of coverage (at a fixed value of P_Cs) should vary both as current and as d vary (see below).

The effect of Cesium pressure. The primary effect of changes in cesium pressure is the change in the degree of coverage at the cathode, which leads to a change in the work function and cathode emission current. With a fixed work function, an increase of pressure increases the scattering in the plasma and, in this respect, acts similar to an increase in the spacing. However, changes in the plasma parameters as P_Cs and d increase are not quite identical because of non-linear effects: ion scattering, changes in the kinetic reflection coefficients, and recombination.

The value of the load power as a function of P_Cs (at a fixed value of T_C and d) has a maximum, because on the one hand, emission increases as cesium pressure increases, and on the other hand, the losses due to an increase of scattering also increase. This can be seen from Fig. 9.32, where a series of characteristics at different pressures is shown.

The effect of the anode work function and temperature. A decrease in the anode work function D f _A leads to an increase in the output voltage of the load by a value equal to D f _A. An increase of voltage occurs until thermionic emission from the anode becomes comparable in value to the current through the device. A further increase of anode emission leads to a decrease of the pre-anode barrier, and then to a reversal of it. Therefore, with large emission currents from the anode, a decrease of its work function is completely compensated for by an increase of the pre-anode barrier, which reflects the electrons emitted from the anode and in no way affects either the plasma parameters or the current-voltage characteristics.

Variation of the anode temperature affects primarily its work function: as anode temperature increases, its work function usually decreases because of a decrease in the degree of coverage. The optimum coverage, which corresponds to the minimum work function, occurs at T_C/T_Cs » 1.6 - 2.0 [54], which in turn corresponds to the range T_A » 900 - 1100° K. Therefore, an increase of T_A initially leads to an increase of the output voltage. A further increase of T_A leads to a decrease of the output

However, in a number of experimental cases, the shift of the current-voltage characteristic is less at small anode emission currents than the independently measured variation of the anode work function [54]. This fact, together with direct measurements of the electron heat of condensation to the collector [52, 53, 89], and some other anomalies in the current-voltage characteristics related to the anode, are now being explained by the hypothesis of the existence of a so-called virtual anode. It is assumed that a shortage of ions (j_iA < j_e(m/M)^1/2) is typical for the pre-anode space charge sheath in a developed arc, because the pre-electrode voltage drop has the sign that it accelerates the electrons travelling from the plasma to the anode (Fig. 9.34, solid curve). Therefore, the effective anode work function seems to increase by D f _A. However, a non-monotonic potential distribution is not excluded (Fig. 9.34, dashed curve). In view of the importance of this problem, we dwell on it in more detail.

At low anode temperatures, the anode work function in cesium vapor is measured in operating devices primarily by two methods: 1) by the thermionic emission current from the anode to a cold (non-emitting) cathode, j₁ = A T_A²exp(-f _A/kT_A); and 2) by the retarding field curve

for electron emission from a hot cathode to a cold (non-emitting) anode; j₂ = A T_C²exp((f _A + eV)/kT_C) [54,90]. The measurements are made with small interelectrode gaps and low current densities, so that the effect of the space charge on the potential distribution can be disregarded. Values of the work function of a molybdenum anode, measured in this manner as a function of T_A/T_Cs, are shown in Fig. 9.35 [90]. The potential distribution for these measurements is also shown. It is obvious that the points corresponding to the two methods are in very good agreement. However, if the current-voltage characteristics of the converter in the arc mode are changed by increasing, for example, the anode temperature, the current-voltage characteristics are shifted by a smaller amount than the variation of the anode work function, measured under similar conditions (Fig. 9.36) [54, 90]. Incidentally, the authors themselves [54, 90] note that some not quite understandable anomalies were observed when measuring the anode work function. For example, the slope of the semi-log retarding current plot did not always yield the correct cathode temperature, and the transition from the exponential to the saturation current segments was somewhat extended. When both electron and ion current from the cathode were being measured, their ratio was incorrect. When measuring emission current from the anode, the characteristic did not have good saturation - the current increased appreciably as voltage increased, even though the Schottky effect should not have been present.

The experimental data from direct (calorimetric) measurement of the electron heat of condensation at the anode in the arc mode as a function of anode temperature are shown in Fig. 9.37. According to theory, if there is a retarding barrier near the anode (Fig. 9.34, dash-dot curve), qA = f _A + 2kT_eA. If one assumed, as one should from other measurements and calculations, that the electron temperature near the cathode is

about 2000° K, then the anode work function has a minimum f _Amin » 1.8 eV. This value is somewhat greater than the value of f _A obtained from other independent measurements. The authors of [90] feel that the large values are easily explained if one assumes that there is an accelerating barrier for electrons in the plasma near the anode, as shown in Fig. 9.34 by the solid curve, rather than a decelerating barrier, as is usually assumed (Fig. 9.34, dash-dot curve).

The increase of the effective heat of condensation for large values of T_A in Fig. 9.37 is partially due to an increase of f _A, but is primarily the result of large electron emission from the anode. The large anode emission should lead to an inversion of the potential barrier near the anode, and thus, should increase qA. With emission from the anode, some increase of qA is also related to the fact that the electron temperature in the plasma near the anode is greater than T_A.

Current-voltage characteristics for the same cathode and different anode materials were studied in [87]. It was expected that a family of current-voltage characteristics would be obtained that would be shifted in voltage by f _A with respect to each other. However, an appreciable shift of the characteristic was actually observed only in the arc extinction (ignition) region. In the developed arc the current-voltage characteristics for different anode materials essentially coincided (Fig. 9.38). It is obvious from Fig. 9.38 that the arc ignites more easily (at smaller plasma voltages) with a monocrystalline anode surface than with a polycrystalline anode surface.

Let us now discuss the main experimental facts outlined above. The authors feel that these facts can be explained if it is assumed that there is appreciable work function non-uniformity at the anode surface in the range of the minimum anode work function (T_A/T_C » 1.6). This non-uniformity should be expected for a polycrystalline surface, because the different faces have a different minimum work function in cesium vapor. For a monocrystal anode, f _A

An accelerating barrier could form near the anode if the anode does not have enough ions j_iA < j_e(m/M)^1/2. This could happen with a very high (hypothetical) rate of ion recombination near the anode. This intensified volumetric recombination could be related to molecular ions and also to the more rapid de-excitation of cesium atoms near the anode. However, specific calculations show that even an intensification of the rate of recombination in the entire recombination region near the anode in a developed arc by ten times has hardly any effect on the current-voltage characteristic of a plasma diode with small gaps.

*Some dependence of f _A on T_A and on the interelectrode spacing may be related to variations in the flow of cesium atoms to the surface in the Knudsen mode (see §12, Chapter 9).

The effect of a structured cathode surface. An increase in the emitting surface of a cathode, when it has an accelerating field, leads to a corresponding increase of the emission current, which accordingly increases the total current through the device. This is obvious from Fig. 9.39, where the calculated characteristics are shown for a smooth cathode and for one with double the emitting surface but the same work function [55]*

*Experimental data on TIC operation with a developed cathode surface is presented in [56, 57].

where v_de, v_di, and v_da are the drift velocities of the plasma components, G is the rate of ionization-recombination, j_e = en_e v_de, and j_i = en_i v_di.

Here p_e, p_i, and p_a are the partial pressures of particles of a given kind (p_e = n_ekT_e, p_i = n_ikT, and p_a = n_akT), R_ea and R_ei are the forces acting on the electrons as a result of electron collisions with atoms and ions, and R_ia is the force acting on the ions as a result of collisions with atoms. Equations (11.2) and (11.3), which express the balance of forces acting on the electrons and ions, were previously obtained in Chapter 4 (see (4.5.2) and (4.7.2)).* Equation (11.4), which expresses the balance of forces acting on the atoms, differs from (11.2) and (11.3) only by the absence of an electric force, which naturally does not effect neutral atoms. In this section, equations (11.2)-(11.4) will be used only for a quasi-neutral plasma, where n_e = n_i = n.

Consider now the equation of heat balance (law of energy conservation). The equation for heat balance for electrons is the continuity equation (1.11) for electron energy flux, the latter calculated by expression (1.6). Energy losses D Q_rad, of minor importance at large current densities and high degrees of ionization, can be disregarded on the right side of (1.11).

The temperatures of the atoms and ions may be assumed to be identical in a partially or in a strongly ionized plasma, as was the case for a weakly ionized plasma. Energy transfer from the electrons to the heavy component of the plasma, because of the large Coulomb scattering cross sections, occurs primarily by collisions between electrons and ions. One must take this into account when calculating the temperature T of

*Equation (4.7.2) was written on the assumption that the neutral atoms are at rest. In order to take into account the motion of neutral atoms, j_i and (4.7.2) must be replaced by - en_i(v_di - v_de). In this case, as can be seen from comparison of (4.7.2) and (11.3), force R_ia = - en_i(v_di - v_de)/m _i - K_i^(T)nkdT/dx.

Here S_T is the energy flux transferred by the atoms and ions and t _Q^ei is calculated by formula (4.6.17). The first term on the right side of (11.5) describes the energy transfer from the electrons to the ions.* The second term takes into account the energy acquired by the ions in the electric field. This term is negligible both for a weakly ionized plasma, where S_T » j_iD V, and for a strongly ionized plasma, where j_i ® 0. Usually, in the intermediate case, as indicated by specific calculations, it is also small compared to the first term, so that we can henceforth disregard the value of j_idV/dx.

where k _eff = k _a@ + k _i@ is the effective thermal conductivity of the heavy component of the plasma. The term k _a'@ = k _a/N_aQ_p^ai/nQ_p^aa + 1)@ is the thermal conductivity of atoms, which approximately takes into account atom-atom and atom-ion collisions (Q_p^ai is the atom-ion scattering cross section and Q_p^aa is the atom-atom scattering cross section @, and the term

is the thermal conductivity of ions, which approximately takes into account ion-ion and ion-atom collisions. (Q_pⁱⁱ is the ion-ion scattering cross section and Q_p^ia = Q_p^ai ). The coefficient k _i = (3.9nkT_{it i}/M)k is the thermal conductivity of the ions [60] (t _i = (2M/m)^1/2t _Q^ei is the collision time for ions). In a cesium plasma, the thermal conduction of ions becomes appreciable for n/N > @ 10.@

*The energy transferred from the electrons to the ions is significant for the calculation of the ion and atom temperature T, but is insignificant for the calculation of electron temperature T_e, since the energy flux S_T is negligible compared to the electron energy flux S_e.

@ The value of k _a = l.65× 10^{-6 12} W/cm× deg [58] was used in the calculations. From this use we have the cross section Q_p^aa = 10^-14 cm². Since Q_p^ai » 10^-13 cm² [59], then Q_p^ai/Q_p^aa » 10.

@ In (11.6) thermal transport by ion and atom flow is not taken into account. This contribution is very small, since the net flow of atoms and ions is equal to zero (see below, p. 350).

The above system of equations should be supplemented by the condition of equilibrium between the working interelectrode gap and the rest of the space filled by the cesium vapor. To derive this condition, we add the equations of motion (11.2)-(11.4) for the individual plasma components. As a result, we obtain dp/dx = 0, i.e., p = const, where p = p_e + p_e + p_e is the total plasma pressure. For the plasma to be in equilibrium with the remaining volume of the gas, it is necessary that

where P_Cs is the cesium vapor pressure in the reservoir outside the working interelectrode gap. In a weakly ionized plasma, condition (11.9) becomes condition (1.1) for a constant neutral gas pressure.

We now point out an additional consequence of the system of equations given above. From equation (11.1), we obtain for the flow of heavy particles n_iv_di = N_av_da = const, where, in the absence of mass transport into the gap, the constant is zero, i.e.,

As a result, the state of a partially ionized plasma in a TIC gap can be described by the equations of motion (11.2) and (11.3) for electrons and ions, by the two continuity equations (11.1) for electron and ion currents j_e and j_i, by the transport equation (1.6) for energy flux S_e, by the continuity equation (1.11) for S_e, and by the equation of thermal conduction (11.7) for the heavy component. These differential equations, together with the algebraic equations (11.9) and (11.10)* are used to calculate the nine unknown distributions: n(x), N_a(x), V(x), T_e(x), S_e(x), v_de(x), v_di(x), and v_da(x).

Local thermodynamic equilibrium in a partially ionized plasma. The above system of equations can be simplified appreciably when used to describe the conditions of a dense TIC plasma. Under these conditions, the transition of the plasma to a state of partial ionization is preceded by the establishment of ionization equilibrium in the main volume. The conditions of ionization equilibrium, n = n(T_e) and N_a = N_a(T_e), make it possible to relate the electron and ion concentration n to that of atoms, N_a, and to electron temperature T_e by using Saha's formula (5.1.11). Therefore, the calculation of the plasma state can be carried out in two stages (as is also the case for a low degree of ionization). In the first stage (in zero approximation), the plasma goes to equilibrium. Then, the values of n, N_a, V, T_e, and T are calculated from equations (l.4),@ (l.6), (11.7), (11.9), and (5.1.11).

@The transport equation for electron current (1.4) is equivalent to the equation of motion (11.2).

In this stage the values of j_e and S_e are assumed constant across the gap, and the continuity equations for these values are not used, while the continuity equation for ion current is replaced by Saha’s formula (5.1.11). From the known distributions of the plasma parameters, in the second stage, j_i(x)is calculated by formula (11.3) and atomic flux N_av_da is calculated by formula (11.10). This gives the values for v_di and v_da.

When calculating ion current, the comparatively small thermal force, k I(T)@nkdT/dx, can usually be disregarded. The force R_ia is then equal to

By substituting the velocity of the atoms v_da from (11.10) into (11.11) and by using the equation of ion motion (11.3), we obtain

Equation (11.12) differs from the transport equation for ion current (1.5) (used in the case of the weakly ionized plasma) by the factor (1 + n/N_a)^-1 on the left side. It is obvious from (11.10) that j_i ® 0 in a completely ionized plasma when n/N_a ® ¥ .

We now consider the non-equilibrium pre-electrode regions as was done in §4 for the case of a weakly ionized plasma. Instead of (4.12), we obtain the following expression for ion current in the non-equilibrium precathode region:

where the temperatures T_e and T, as before, can be assumed constant within the sheath. Equation (11.13) should be supplemented by the continuity equation (11.1) for ion current. By using expression (5.3.7) for the rate of ionization in a partially ionized plasma and by substituting (11.13) into the continuity equation (11.1), we obtain

where B = D_iaN_a is a constant, not dependent on the density of atoms. By expressing the density of atoms N_a(x) in (11.14) in terms of charged particle density n(x) and the pressure p by using (11.9), we obtain the equation for calculating n(x):

For practical operation, if the cesium pressure is sufficiently high, we have for the ionization length L_i » L_ia (L_ia is the mean free path for ion-atom scattering), and the plasma remains weakly ionized at the interface with the Debye sheath. Thus, the boundary conditions (4.15) and (4.16), derived for a weakly ionized plasma, can be used. We denote y (n)@ = (dn/dx)² order to reduce the order of and to solve differential equation (11.15):

By solving equation (11.16) with boundary condition y (n)@ ® 0 as n ® n(T_e) (compare (4.16)), we obtain

As with a weakly ionized plasma, the above expressions are valid if the width of the non—equilibrium region of the plasma is large compared to the ion mean free path l_ia. In this case, the charged particle density at the boundary of the plasma is low, and therefore, we can assume that n = 0 in (11.13) and (11.17) at the boundary with the space-charge layer. As a result, we obtain the following expression for ion current j_is from the plasma to the electrode:

If the region considered is the non-equilibrium region near the cathode then we have T_e = T_eC and j_is = j_iC. Having substituted expression (11.18) for j_iC into (4.23) and (4.24), we obtain the boundary conditions for the electron current and energy flux at the precathode boundary of the equilibrium plasma. As is shown below, when the plasma in the TIC changes to a strongly ionized state, the electron temperature near the anode T_eA may increase so much that the ionization in the pre-anode non-equilibrium region may be calculated the same way as the precathode region. The corresponding boundary conditions for the pre-anode region of the plasma has the following form:

where the ion current to the anode j_iA, as above, is assumed to have a negative value.

Distribution of the plasma parameters and the current-voltage characteristics. Consider the nature of low-voltage arcs with partially and strongly ionized plasma [61]. The calculated current-voltage characteristics are presented in Fig. 9.40. The distribution of the plasma parameters with ionization equilibrium, at P_Cs = 0.5 torr and for different values of current j, are shown in Fig. 9.41. The distributions of n, V, and T_e qualitatively repeat the corresponding distributions for a weakly ionized plasma with a comparatively weak current. With a further increase of current, however, the increase of density at the peak is held back because the electrons and ions begin to make a significant contribution to the total pressure p. An increase of p_e and p_i causes atoms to be forced out of the precathode region, where the plasma becomes completely ionized. In a completely ionized plasma P_Cs = nk(T_e + T) = const, so that the minimum n rather than the maximum corresponds to the maximum T_e. Therefore, a break occurs on the distribution n(x) in the hottest, precathode part of the plasma. As the degree of ionization increases, the energy exchange between the electrons and heavy component of the plasma increases. As a result of this, the temperature T of the heavy component near the cathode increases and begins to approach the electron temperature T_e.

Since a decrease of pressure increases the degree of ionization of the plasma, the above phenomena are manifested most strongly at comparatively low cesium pressures. In particular, it is obvious from Fig. 9.41b and c, for a cesium pressure of P_Cs = 0.5 torr, that almost all the plasma in the working region of the TIC becomes completely ionized. A lower ion density in the gap and a smaller ion current accompany a reduced pressure P_Cs. Also, the ionization energy losses decrease, and the electron temperature T_e increases. The latter leads to an increase in the pre-electrode potential barriers that limit the electron currents from the plasma to the electrodes. Therefore, the potential well for electrons is deeper at low pressures.

As indicated above, the electric field in a fully ionized plasma becomes an accelerating field for electrons, and electron current is primarily a field rather than a diffusion current, which is obvious from Fig. 9.41a. We again emphasize that a slight variation of density across the gap (measured in tenths of a percent) is typical for completely ionized TIC plasmas, whereas n varies appreciably across the gap with a weakly ionized plasma (where the density n is also strongly dependent on the electron temperature T_e).

The state of the plasma in the non-equilibrium pre-electrode regions varies also for reduced pressure P_Cs and large currents j. Ionization rather than recombination now occurs in the pre-anode non-equilibrium region (as a result of the high electron temperature). At the same time, if the current is small, the width of the precathode ionization region is shortened so much that it becomes of the order of the ion mean free path. Calculation of the rate of ionization in this case is generally a very complicated problem. However, when the length of the ionization region L_i < @ L_ia, there is no great need to solve the problem, because in this case, a completely ionized plasma approaches the cathode itself. Ion scattering from atoms within the pre-electrode, non-equilibrium sheath can then be disregarded, and the ion current to the cathode may be calculated by the formula j_iC = 0.61eÖ kT_eC/M@n_C, where n_C is the charged particle density near the cathode.*

The calculated current-voltage characteristic corresponding to P_Cs = 0.5 torr is shown in Fig. 9.40 (curve III). The current-voltage characteristic is much flatter in the region of large currents and a segment of current saturation begins to occur. The appearance of saturation on these current-voltage characteristics is related to two causes. First, when the plasma becomes strongly ionized, the charged particle density n_C near the cathode ceases to increase. As a result, the field intensity E_C near the cathode increases only weakly (see formula 2.8) and the Schottky effect is nearly constant. Second, if the pressure P_Cs is reduced, an increase of the work function caused by a decrease of heavy particle density near the cathode begins to have an affect on the current-voltage characteristic (see the following section).@

The distribution of ion current j_i(x) across the gap, calculated by formula (11.12), is presented in Fig. 9.41d. It is obvious that the nature of distribution of j_i(x) varies as the plasma becomes completely ionized. When the plasma is weakly ionized, the entire gap can be subdivided into two regions (see curve 1 in Fig. 9.41d): the ionization region (dj_i/dx < 0) is located near the anode. As the plasma becomes completely ionized and as an electric field, which accelerates electrons, is applied to the volume, the electron temperature in the volume

*This expression is obtained for j_iC provided that l_ii « L_i « l_ia where l_ii is the ion-ion mean free path (compare page ?).

@In the calculations, the cathode emission current was calculated by formula j_sC = j_s⁽⁰⁾exp(- D f _C/kT_C), where D f _C = - D f _Sch + D f q , - D f _Sch = - e@ ^3/2E^1/2 is the decrease of the work function because of the Schottky effect, and D f q = -(¶ f _C/¶ p)p_eC = -(¶ f _C/¶ p)n_CkT_eC is the increase of the cathode work function because of a decrease of the flow of atoms and ions to the cathode. In these calculations, ¶ f _C/¶ p was assumed to be equal to 0.15 eV/torr.

increases and the recombination region disappears. In a strongly ionized plasma (see curve 2 in Fig. 9.41d), ionization prevails over recombination in the entire volume.

The value of d (x) plotted from the above calculations, carried out on the assumption of ionization equilibrium, is shown in Fig. 9.42. At P_Cs = 2 torr (Fig. 9.42a), appreciable deviations from ionization equilibrium occur only near the anode.* If pressure decreases (P_Cs = 0.5 torr, Fig. 9.42b), the deviations from equilibrium are more pronounced. However, it should be noted that calculations by formula (11.21) yield an increased value of ï d ï . Solving the problem by sequential approximations, the energy liberated by recombination and absorbed by ionization should be taken into account in the subsequent approximation. This leads to a decrease of electron temperature T_e in the ionization region and to an increase of T_e in the recombination region. Although variations of T_e in this case are insignificant, they are reflected significantly in the value of G _i, because of the exponential dependence of s ₀ on T_e. As a result, G _i decreases in the ionization region and increases in the recombination region, which brings the plasma very close to equilibrium. Therefore, the true deviation from equilibrium is less than that shown in Fig. 9.42.

*These deviations are related primarily to the fact that at P_Cs = 2 torr the pre-anode, non-equilibrium region was not considered separately in the calculation (because of the weak level of recombination), but was included in the quasi-equilibrium plasma.

A further decrease of pressure leads to a significant departure from ionization equilibrium. However, in this case, the plasma in the entire gap is still essentially fully ionized* when the presence of ionization equilibrium or the absence of it is important only for determining the small number of non-ionized atoms, i.e., to calculate the heat conduction of the atoms k _a@. If the concentration of atoms is very low, when N_a/n « 0.1, k _a@ is negligible. In this limiting case, there is no longer any need to calculate the concentration of neutral atoms N_a, and the calculation of the state of the plasma is simplified considerably.

The cathode work function f _C is determined by the cesium coverage; therefore, any variation of the flow i_C of heavy particles to the cathode can appreciably change its work function. This circumstance was illustrated in the preceding section in the example of a strongly ionized plasma. However, even in a weakly ionized plasma, variation of the value f _C in the discharge due to a variation of i_C can appreciably affect the current—voltage characteristic. The flow i_C is formed at a distance the order of a mean free path from the cathode where the ion distribution function, as well as the atom distribution function (in a strongly ionized plasma), is strongly asymmetric. Therefore, precise calculation of i_C requires a solution of the kinetic equations for the distribution functions of atoms f_a(v,x) and ions f_i(v,x). We shall consider this problem approximately on the basis of a hydrodynamic description of the plasma.

The hydrodynamic equations for the plasma components have the following form [60]:

where E = dV/dx is the electric field intensity and p_exx, p_ixx, and p_axx are the diagonal elements of the total pressure tensors. The total pressure tensor p_{ra b} for the r-component of the plasma is calculated by the expression p_{ra b} = n_rm_r< v_r'a v_r'b > , where v_r'a = v_ra - v_dra are the components of the random velocity of particles of a given kind (r = e, i, a; a ; b = x, y, z). Equations (12.1) and (12.3) are generalizations of equations (11.2)-(11.4) for the case where the directed particle

velocities v_dr are comparable to random velocities v_r'. Equations (12.1)-(12.3) describe the state of plasma components near the boundary, where the drift velocities are high and the distribution functions are very asymmetrical. As the distance from the boundary increases, the drift velocities v_dr decrease relative to the random velocities v_r', and the pressure tensors degenerate into scalars. The system of equations (12.1)-(12.3) then changes to equations (11.2)-(11.4). By adding equations (12.1) and (12.3) and by using Poisson’s equation dE/dx = 4p e@(n_i -n_e) we obtain

Summation in (12.4) is carried out for all three plasma components. At a pressure P_Cs » 1 torr, the term E²/8p is small and we shall subsequently disregard it.

where in a quasi-neutral plasma n_e(x) = n_i(x). In this approximation, the distribution functions have spherical symmetry in a space moving with the drift velocities and the pressures of the atoms and ions are scalars, i.e., p_rxx = p_r = n_rkT(r = a, i). The electron distribution in the precathode layer can be expected to be Maxwell-Boltzmann in the discharge mode. In this case, p_exx = p_e = nkT_e, and v_de = 0. As a result, from (12.4) we obtain

In particular, it is obvious from (12.6) that the total pressure of the plasma components decreases as the distance to the electrode decreases and as the drift velocities of v_di and v_da increase. On the other hand, as the distance from the electrode increases, the terms proportional to v_di and v_da cease to be significant and in this region p_i + p_a + p_e = P_Cs. Taking this into account, and also relation (11.10), we obtain from (12.6)

We can apply relation (12.7) to a weakly ionized plasma. In this case, we can disregard the drift velocity of the atoms v_da in the pre-electrode layer compared to the ion drift velocity v_di. By considering relation (12.7) at the boundary of the quasi-neutral plasma with the precathode Debye layer, and by assuming that n = n_C, T = T_C, T_e = T_eC, and v_di = (v_di)₀ = -g ₀(2kT_eC/M)^1/2 (see page 209), from (12.7)

we can calculate the density N_aC of neutral atoms at the boundary with the cathode.

We use expression (12.5) for the distribution function of atoms at the plasma boundary to find the random flux of atoms i_apl from the plasma to the cathode. Then

where v_aC@ = (8kT_C/p M)^1/2 is the random velocity and (v_da)₀ is the average drift velocity of the atoms at the boundary with the cathode.* Taking into account relation (11.10), from (12.9), we obtain

We note that the x-axis is assumed to be directed from the cathode into the plasma in formulas (12.9) and (12.10), the same as in the initial equations (12.l)-(12.3), so that (v_da)₀ > 0 and (v_di)₀ < 0.

The total random flux of atoms and ions to the cathode i_C is equal to

Here j_is is the ion emission current from the cathode, which is almost completely returned to the cathode in the discharge mode because of the presence of a high precathode potential barrier. The increment D i_a is the increase of the flow of atoms to the cathode because of ion-atom collisions in the precathode space charge sheath. Approximately half of all the atoms in a weakly ionized plasma have random velocities directed from the cathode to the plasma. Therefore, if one assumes that the width of the space-charge sheath is equal to the Debye length L_D and that the ions and the atoms, formed during charge exchange in the space-charge sheath, reach the cathode, then we have D i_a = 1/2 (j_iC/e@ )× L_D/l_ia. Since we have also L_D « l_ia and j_is « j_iC, then D i_a and j_is/e@ are small corrections to the total flow of charged particles i_C.

We now compare the flow i_C to the flow of charged particles from the plasma to the cathode i_C^(T) that would occur with thermodynamic equilibrium between the plasma and the cathode. If the cathode work function f _C is greater than the chemical potential m of the plasma, then

*The current toward the cathode corresponds to a positive value for i_apl.

where n(T_C) and N_a(T_C) are the equilibrium densities of ions and atoms in the plasma at the cathode temperature, and V_TC is the equilibrium barrier between the plasma and the cathode. The equilibrium densities are related by

Comparing expression (12.11) and (12.12) and using (12.8), (12.10), and (12.12), we obtain the variation D i_C of the random flow of heavy particles to the cathode during the transition from the diffusion mode (where i_C = i_C^(T)) to the discharge mode:

The term j_iC/2e@ = n_Cg 0Ö kT_eC/2M@ and the next term on the right side of (12.14) constitute the main contribution to the value of D i_C. The term j_iC/2e@ accounts for the increase in i_C due to the directed ion and atom flows at the plasma-cathode interface. The term - 1/4n_Cv_aC@ [(2g ₀² + 1)T_eC/T_C + 1] accounts for the decrease in the random flow of atoms to the cathode because of a decrease in atom concentration at the boundary with the cathode. It consists of three terms. The term proportional to 2g ₀²T_eC/T_C accounts for the decrease of plasma pressure near the cathode because of an increase in ion drift velocity. The term proportional to T_eC/T_C accounts for the decrease of atom partial pressure due to the increase of electron partial pressure. Finally, the last term in the brackets of (12.14) accounts for the decrease of atom partial pressure because of the increase of ion partial pressure. Although the value of D i_C is small in a weakly ionized plasma compared to 1/4N_a(T_C)v_aC@, the corresponding change in the cathode work function D f q may be appreciable. This is illustrated by Fig. 9.43, where the value of D f q is calculated by using the experimental dependence of the work function on the atom flow i_C for a vapor deposited tungsten cathode [54].

The inert gases (Kr and Xe) from uranium fission are released from the fuel elements in reactor converters and mix with the Cs vapor. Therefore, it is important to know how the output parameters of diode converters vary as the pressure of inert gases increases. On the other hand, it has been suggested [62] that inert gases be specially introduced into converters (Ne and Ar) to improve the parameters of diode converters in the arc mode.

*The cross section for thermal electron scattering from Ar is less than 10^-16 cm², whereas the effective cross section for Cs ions scattering from Ar is on the order of 10^-15 cm².

inert gas pressure, one may achieve a decrease of ion flow to the electrodes (and related energy losses), without introducing appreciably impedance to electron current through the plasma. As will be shown later, several factors affect the diode characteristics simultaneously when inert gas is introduced under experimental conditions. Therefore, to see the effect of changing the ion mobility, special calculations were carried out [63] in which it was assumed that an added inert gas only reduces the ion mobility. Current-voltage characteristics calculated to evaluate this effect are shown in Fig. 9.44. It was assumed that d/l_i = 40 for curve 1, while the ion mobility for curve 2 was decreased by a factor 7.5. It is obvious from the figure that the diffusion segment of curve 2 is shifted to the right and the arc segment is shifted to the left as l_i decreases. The shift of the diffusion characteristic to the right is explained by the fact that the pre-anode barrier, which keeps

Fig. 9.44b shows how the plasma parameter distributions vary in the arc mode as ion mobility decreases. Points with identical current (j = 1.75 amp/cm²) and different applied voltage were selected for comparison. It is obvious that a decrease of ion mobility leads to deepening of the potential well for electrons and to an increase of the ion density in the gap, so that a region of ionization equilibrium and a recombination region develops. Despite the increase of density gradients, the ion current decreases, especially ion current to the anode. This leads to a decrease in the pre-anode barrier and to greater cooling of the electrons in the plasma. It should be noted that radiation losses were not taken into account in these calculations. But consideration of radiation apparently leads more or less to an identical shift in characteristics 1 and 2, because the calculated temperature distribution at identical currents is very similar.

We now examine the available experimental data. Variation of short-circuit current in a TIC at different Kr pressures is shown in Fig. 9.45 [64] as a function of cathode (Mo-foil) temperature (P_Cs = 3.9 torr and d » 1 mm).

The authors themselves [64, 65] explain the results by the fact that the components of the inert gas-cesium mixture is changed because of thermal diffusion, so that the Cs pressure near the heated cathode is much less than that at the interface with the liquid cesium.* This should lead to a decrease in the degree of cathode cesium coverage, and as a result, to a decrease of the thermionic emission from the cathode.

Since thermal diffusion plays an important role in the analysis of the effect of inert gases on TIC operation, we consider in more detail the physical causes which lead to a non-uniform density distribution of the gas components in a temperature gradient [66 - 68]. And we refer to Fig. 9.46 where a heavy particle, for example, a Cs atom or a Cs₂ molecule, is shown among atoms of a lighter gas, for example, Ne.

The flows of Ne atoms from a high-temperature to a low-temperature region and vice versa are equal in steady state, i.e., n_xv_dx = n_yv_dy. However, hot particles, colliding with the heavy atom, impart a greater momentum than the cold particles, since n_y< v_dy> » n_x< v_dx> . Therefore, a force develops which drives the heavy particle to the lower temperature region and which increases the density of Cs in the cold part of the device. Since the total gas pressure is constant, a decrease of Cs density in the hot part would be compensated by an increase of inert gas

It should be noted that this outline description of thermal diffusion is qualitative. In a strict quantitative theory it would be necessary, for example, to account for the dependence of the scattering cross section on energy. If the cross section decreases as energy increases,the resulting force may even change sign, because the hot particles will collide less often with the admixture than the colder particles. One should also take into account the thermal motion of the admixture. In this case, the resultant force occurs only if the foreign particle is somehow distinguished (by mass, cross section, or the law of interaction) from the particles of the main gas. Separation of the gas mixture is characterized by the dimensionless thermal-diffusion coefficient D_T given by the formula

We will not present the rather long theoretical formulas for calculating D_T. We note only that the thermal-diffusion coefficient increases as the ratio of the particle masses in the mixture increases. This agrees well with the fact that the lighter Ne and Ar, which weakly scatter electrons, affect the TIC experimental characteristics

The importance of thermal-diffusion separation in a TIC is indicated by the fact that the optimum Cs pressure always increases with the introduction of an inert gas [69].

Experimentally, the thermal-diffusion coefficient for a cesium-inert gas mixture was not measured. Indirect estimates of the variation of Cs concentration near the cathode as the emission current decreases [64] lead in some cases to excessive values for D_T. This possibly results from inadequate purification of the inert gases.

It was noted in [70] that a relative oxygen concentration of about 10^-6 in an inert gas greatly affects the output parameters of the diode converter.*