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167

The study of pre-electrode phenomena occupies a very important position in the physics of gas discharges. Phenomena in the precathode portion of the discharge that lead to ion formation and the development of a positive space charge near the cathode surface are usually of greatest interest. Development of a positive space charge leads to the phenomena, typical for most types of discharges, of a precathode potential drop, which accelerates electrons and imparts to them the energy required for ionization.

Pre-electrode phenomena play a special role in a TIC. This is so because the pre-electrode, non-equilibrium regions in a TIC plasma can occupy a considerable part of the interelectrode gap. The voltage drop in the precathode sheath of the practical arc mode TIC is comparatively small (on the order of 1 v), i.e., the arc discharge in the TIC is a low-voltage discharge.

In the simplest case, when a plasma is in thermodynamic equilibrium with the electrode (both at temperature T) the distribution of charged particle density can be calculated by formulas from equilibrium statistics. If the plasma and electrode are in a state of thermodynamic equilibrium, then their levels of the chemical potential must coincide

(Fig. 6.1). The potential barrier eV_T at the plasma-electrode interface is accelerating (Fig. 6.la) or decelerating (Fig. 6.lb) for emitted electrons depending on the relative magnitude of the electrode work function f and the chemical potential of the plasma m In this case @. The "electron potential" eV_T = ê f - m ê , which is equal to the negative of the ordinary electrostatic potential, is depicted in Fig. 6.1.

where n_i(x) and n_e(x) are the ion and electron densities in the pre-electrode sheath. At a sufficient distance from the electrode (i.e., as x ® ¥ ), a region of neutral plasma is found in which n_i(x) = n_e(x), and the potential V is constant. We can take the potential of the neutral plasma as zero and let the charged particle density there be designated as n¥ . The potential at the interface with the electrode, i.e., at x = 0, is then equal to V_T = (f - m )/e (see Fig. 6.1a). With thermodynamic equilibrium, the electrons and ions are distributed according to the Boltzmann law:

We substitute (1.2) into (1.1) and introduce the dimensionless potential V = eV/kT and the dimensionless length x _D = x/L_D(n¥ , T), where the value of

and the boundary conditions assume the form V(0) = eV_T/kT º V_T and V(¥ ) = 0.

Equation (1.4) can be integrated immediately if we multiply both parts of (1.4) by 2dV/dx _D. As a result we obtain

When integrating (1.5), we note that dV/dx _D ® 0 at dx _D ® 0, i.e., as V ® 0. Then integration of (1.5) reduces to the expression

Consider the limiting cases. If V_T « 1 i.e., eV_T « kT, then, by rearranging the right and left sides in (1.7) by powers of V and V_T, we obtain

In the opposite limiting case, when V_T » 1, i.e., eV_T » kT, unity in expression (1.7) can be disregarded compared to the hyperbolic cosine in almost the entire region where V » 1. In this case, equation (1.7) has the solution x = 0, i.e., the variation of the potential curve essentially proceeds vertically. This is related to the fact that the space charge of ions in thermodynamic equilibrium increase exponentially as V increases, so that large fields may form over a short distance.

When the potential V decreases, the left side of (1.7) begins to deviate from unity, x _D begins to decrease, and at V < 1, the potential drops exponentially in a way similar to that given by (1.8). The course of V(x ) for different values of electrode potential V_T is shown in Fig. 6.2. It is obvious from the curves presented that an increase of V_T alters the

the dimensions of the space charge region very slightly, but in a given case the width is always of order L_D.

As indicated above (see §4, Chapter 2), in many cases, the electrode work function is not uniform over the entire surface. If the dimensions of the heterogeneities, i.e., if the spots with different work functions are large compared to L_D, then the contact of each spot with the plasma may be considered independently (Fig. 6.3a). Then the field on the surface may have a different sign for different spots, as a function of the relationship between the values of f and m (Fig. 6.3b).

If the spot sizes are small compared to L_D, then the spot field has the same form as in a vacuum (see §4, Chapter 2). If the difference of the work functions f ₂ - f ₁ in this case has a sufficiently large value, then the field intensity of the spots may exceed the field intensity created by the space charge in the Debye sheath.

The pre-electrode sheath may be divided schematically into a number of regions. The space charge region I is directly adjacent to the electrode. In this region (for this case), the ion density n_i(x) exceeds the electron density n_e(x). If there is no emission from the electrode, the space charge in region I is created primarily by ions moving from the plasma toward the cathode, pulled in that direction by the electric field. In this case, the ion density decreases as V increases, and the space charge sheath with large potential drop should have a thickness comparable to L_D.

Unlike the equilibrium case considered in the preceding section, the width of region I is now dependent on the voltage applied across the pre-electrode sheath. A case is considered below where the width of this region is considerably less than the ion mean free path l_i; therefore, ions pass through region I essentially without collisions.

A quasi-neutral plasma begins beyond region I, where the electron and ion space charge can be assumed to be mutually compensated to a first approximation, i.e., ç n_i - n_e ç /n_i « 1. However, the electric field and the density gradient at the beginning of the quasi-neutral plasma (region II) are still too large for the motion of the ions to be described in terms of a mobility and a diffusion coefficient. Only with sufficient distance from the electrode (region III) is the ion flow expressed by local plasma parameters n and V and their derivatives.

The pre-electrode sheaths formed in the plasma with negative voltage on the electrode were considered theoretically in [1-11].

where i_i is the ion flow; D_i and m _i = (e/kT) D_i are the diffusion coefficient and mobility, respectively, for the ions; T is the ion temperature, which must be assumed equal to the neutral gas temperature; and n » n_e » n_i

If the potential barrier in the Debye sheath eV₁ is large compared to the thermal energy of the electrons (i.e., eV₁ » kT_e ) then the electron flow is small and the electrons have a Boltzmann distribution:

Equation (2.3) appears to be a diffusion equation in which the role of the diffusion coefficient is played by the quantity D_i(1 + T_e/T). However, the term - D_i(T_e/T)dn/dx, proportional to the ratio T_e/T, actually corresponds to the field component of the current rather than to the diffusion component. Since T_e in a real plasma is usually larger than T, the field component of the ion current always exceeds the diffusion component.

The condition for (2.3) to be valid is the smallness of the directed current ç i_i ç compared to random current (1/4)(nv_i@ ). This condition can be rewritten in the form of the inequality

Two terms may be distinguished in the left side of (2.4): (l_i/n)dn/dx and (T_e/T) l_id(lnn)/dx = -(e/kT) l_idV/dx The smallness of the first term compared to unity gives the ordinary condition for the applicability of the diffusion approximation. The smallness of the second term defines the condition for a weak field. It is obvious from (2.4) that for T_e > T, as the distance to the electrode decreases, the second condition is violated sooner, i.e., the main cause for equation (2.1) to become inappropriate near the electrode is the presence of a strong electric field penetrating the quasi-neutral plasma.

And if the dependence of mobility m _i on the field E = dV/dx is known, formula (2.5) can be used for the case of intermediate and strong electric fields, where condition (2.4) is not fulfilled. By combining equations (2.2) and (2.5), we obtain a non-linear differential equation with respect to V. The potential distribution V(x) in region III is obtained as a solution. We note that if the current i_i is constant with plane or cylindrical symmetry, the concentration n and the potential V increase without limit as the distance from the electrode decreases, so that the boundary of the pre-electrode sheath must be established somewhere.

The mobility of m _i(E), calculated for a uniform electric field E (see §7, Chapter 4) can be used in equation (2.5) if the field changes very little over an ion mean free path, i.e., if

By using (2.2) and (2.5), if i_i = const, we can rewrite the latter inequality in the form

Since (E/m _i)dm _i/dE » 1, condition (2.6) is equivalent to the requirement eEl_i/kT_e « 1. For T_e » T, the latter inequality is less rigid than (2.4). Therefore, as the distance to the electrode decreases, criterion (2.4) is violated sooner than (2.6). This means that for T_e » T, approaching the electrode, there should come a region near the electrode in which, on the one hand, the electric field is strong with respect to ions (eEl_i/kT » 1), and on the other hand, a local analysis of the problem is also applicable, i.e., expression (2.5) may be used, But then, approaching the electrode closer in the region of strong fields, the local approximation and expression (2.5) should become invalid, i.e., when eEl_i/kT_e » 1. Thus, we come to region II.

These circumstances greatly simplify the analysis of this next region-region II-for which equation (2.5) should be replaced by the kinetic equation for the ion distribution function. Now, instead of the kinetic equation, other equations can be generated directly for the desired quantities: the charged particle density n and the potential V.

Accordingly, we consider some point x in region II, and proceed to calculate the ion concentration at that point. To do this, we consider all points x' > x and calculate the number of ion scattering events at those points, as well as the probability that these scattered ions will pass over the path x' - x without collisions. Ion scattering from neutral atoms in an alkali metal plasma occurs primarily as a result of resonance charge-exchange. The scattering cross section Q^ia is rather weakly dependent on the relative velocity g of the ions and atoms, and it may be assumed approximately constant. In a strong electric field, the ions move much faster than the atoms, and the ion velocity with respect to atoms, g, may be replaced by the ion velocity v. The number of collisions per unit volume is then equal to n< v> Q^iaN_a where the brackets denote averaging over the ion distribution function.

In a strong field, all the ions move in the direction of the field and nç < v> ç = i_ip, where i_ip = - i_i is the absolute value of the ion flow from the plasma to the electrode. If the ion flow is constant, the number of collisions is not dependent on x. The probability that an ion after a collision at point x' will reach point x without scattering is given by the exponent exp[-(x' - x)/l_i], where l_I = (Q^iaN_a)^-1 is the ion mean free path. The ion density at point x is obtained by division of the ion flow from point x' by the ion velocity Ö (2e/M)[V(x) - V(x')]@ and by integration over all scattering points x'. As a result we obtain

By substituting n(x) from formula (2.2) into (2.7), we obtain an integral equation for calculating the potential V(x):

The distribution of parameters in the region of a strong field in a quasi-neutral plasma. We are now in a position to obtain the distribution of plasma parameters in the region of a strong field. The quantity g (0) has been defined

where v_i(0) = i_ip/n(0) is the drift velocity of the ion at the point where V = 0. By converting (2.8) to dimensionless variables, V = eV/kT_e and h = x/l_i, we obtain

Equation (2.10) contains the parameter g (0) which is dependent on the drift speed of the ions vdi(0) at the point of the plasma which is arbitrarily selected for the zero of potential. An increase or decrease of vdi(0) and g (0) means that the origin for the potential shifts accordingly to the region of larger or smaller fields E, i.e., closer to the electrode or farther from it. As can be seen from (2.10), a change in the value of g (0) by a factor of k is equivalent to a change in the origin of potential V by a factor of lnk. Therefore, the solution of equation (2.10) is actually not dependent on the value of g (0) and is universal.

We note that at the transition to region III (see figure 6.5) (h > h ₂ = x₂/l_i), where field intensity varies slightly over a mean free path, the numerator of the integrand can be simplified by setting V(x) - V(x') » ç E ç (x' - x). In this case, the integral (2.10) can be evaluated and we arrive at expression (2.5), in which m _i now represents the ion mobility in a strong field which has become

The zero of potential in (2.12) is selected so that V = 0 at h = 0. Since expression (2.12) is valid only in region III, the point of h = 0 now lies to the right of point h ₂ = x₂/l_i in Fig. 6.5. The values of h are now negative for the points located to the left of the selected origin for x.

We should now return to equation (2.10). The solution of equation (2.10) is initially known only for the points located in region III, where it is expressed by formula (2.12). We can proceed in the following manner to solve equation (2.10) for region II [7]. We replace the integral on the right side of (2.10) by a sum with value weights from the integrand at specific nodal points h = h _K. V(x) If point h , in which the value of

potential V(h ) is sought, is located near the arbitrary boundary of regions II and III, i.e., if h hardly differs from h ₂, then all nodal points are located in region III. In this case, all the values of X'(h _K)@ are known. As a result, the integral equation (2.10) is transformed to a transcendental equation in which the unknown values of V(h ) is contained only in the exponential exp[-V(h )] left side of the equation. By calculating V(h ) from this equation, we can expand the region in which the function V(h ), etc., is known.

The solution of equation (2.10), obtained by this method, is shown in Fig. 6.6a. It is obvious from the figure that the electric field intensity E = dV/dx begins to increase sharply in absolute value as V increases. At some value of V, the field intensity goes to infinity. Greater values of the potential (the dashed part of the curve) no longer have physical meaning. From this we conclude that only a limited voltage (of the order of kT_e/e) can be applied to the region of quasi-neutrality.

The above result is explained by the fact that collisions are required to maintain quasi-neutrality, because the electron and ion densities, in their absence, are dependent in different ways on the potential. Therefore, as ê dV/dx ê increases, when a large potential drop occurs over a distance small in comparison to the ion mean free path l_i, and when collisions cease to be significant, then quasi-neutrality would be violated.

To evaluate the degree of deviation from quasi-neutrality as distance to the electrode decreases, we calculate the ratio D n/n » (n_i - n_e)/n_e , assuming that potential distribution V(x) in region II is unknown. By using Poisson’s equation (1.1) and by converting to the dimensionless variables V and h , we find

where L_D(n(0), T_e) = Ö kT_e/4p n(0)e²@. The smallness of the ratio L_D/l_i provides fulfillment of the condition of quasi-neutrality in the region of strong fields where dV/dh » 1, i.e., when voltage drops on the order of kT_e/e over the length of the ion free path l_i. If the value of L_D/l_i is sufficiently small, significant deviations from quasi-neutrality

(D n/n ³ 1) occur only in direct proximity to the point at which dV/dh approaches infinity. Therefore, the point where ê dV/dh ê = ¥ may be taken as the arbitrary boundary between the quasi-neutral region II and the space-charge region I. The value of expd²V/dh which characterizes, according to (2.13), the degree of deviation of the plasma from quasi-neutrality, is depicted in Fig. 6.6b.

This limitation of the potential drop means that the drift speed of the ions in a quasi-neutral plasma is also limited. The dependence of the parameter g = v_di/Ö 2kT_e/M@ on the value of potential V is shown in Fig. 6.6c. The value of electric field intensity V' = dV/dh is plotted as a function of V (curve 1). The maximum value of g , equal to g ₀ is reached at the boundary of a quasi-linear plasma where V' approaches infinity (g ₀ » 0.76). The latter value determines the maximum value of the drift speed (v_di)₀ = g _0Ö 2kT_e/M@ which ions may have in a quasi-neutral plasma for a given value of T_e.

The value of V', obtained as a result of differentiation of expression (2.12), is also plotted in Fig. 6.6c (curve 2). It is obvious that formula (2.12) correctly maps the course of the potential curve far from the electrode. At the same time curves 1 and 2 diverge near the boundary of a quasi-neutral plasma. It is essential that the value of V calculated by formula (2.12) approaches infinity at h = - [2p g ²(0)]^-1 , at the same time as the precise solution of equation (2.10) leads to a singularity, not in V but, in V'.

Potential distribution in the space-charge region. The potential distribution in region I should be calculated from Poisson’s equation (1.1), in which the electron density is expressed as before by formula (2.2) and the ion density is expressed by the right side of formula (2.7). As a result, Poisson’s equation is written in the dimensionless variables V and h in the form

Equation (2.14) contains the small dimensionless parameter L_D²(n(0), T_e)/l_i² in front of the derivative d²V/dh ². Therefore, the term which contains the derivative d²V/dh ² is significant only in direct proximity to the electrode, where d²V/dh ² » 1. It may be disregarded far from the electrode. In the latter case, we then pass to the region of quasi-neutrality, region II, which is described by equation (2.10).

Thus, equation (2.14) describes both the space-charge region land the strong field region II in the quasi-neutral plasma. However, since equation (2.14) is dependent on the parameter L_D²(n(0), T_e)/l_i², its solution is no longer universal (unlike the solution of equation (2.10)).*

We shall use the values of V(h ) and V'(h ) at some point in region II separated by a sufficiently large distance from the boundary of the plasma as the boundary conditions to equation (2.14) so that decompensation

of the space charge is insignificant there, i.e., D n/n, calculated by formula (2.13), is much less than unity.

Since only the values of the potential V(h ') for h ' ³ h are contained in the integrand of (2.14), the usual methods for integrating second-order differential equations, for example, the Runge—Kutta method [12], can be used for the numerical solution of equation (2.14).

The solutions of equation (2.14) for different values of the parameter L_D²(n(0), T_e)/l_i² are presented in Fig. 6.7.

The width of the space-charge region is small compared to l_i (for L_D « l_i and for values of V not too large). Therefore, the dependence of V' on V rather than the dependence of V(h ) is usually of practical interest (since the potential barrier in region I may be assumed simply right-angular). The corresponding dependencies are shown in Fig. 6.8. These dependencies permit the calculation of the field intensity E at the electrode surface if the potential drop in the pre-electrode sheath is known: E = (kT_e/el_i)dV/dh .

Consider now briefly the different approaches used in calculating the pre-electrode sheath. The smallness of ratio L_D/l_i is usually employed in the analysis of the pre-electrode sheath, and regions I and II are sharply delimited. Collisions are disregarded in region I, whose width is of the order of L_D if the potential drops are not too large. In this case, if the distribution functions for the charged particles on the boundary of the regions are known, the charged particle densities are expressed rather simply as functions of the potential V.* Afterward, Poisson‘s equation is integrated by squares in a way similar to that used in §l. In most cases, the analytical function E(V) can be obtained.

The field intensity in a quasi-neutral plasma E » kT_e/el_i is small compared to the field in region I; therefore, V' = 0 at V = V_B (where V_B is the potential at the boundary of regions I and II) is usually taken as one of the boundary conditions. As Langmuir (see page 140 in [13]) and Bohm [1,14] first pointed out, this approach has certain difficulties, because the solution of Poisson’s equation near the boundary of regions I and II does not always exist. In particular, if the ions at the boundary are assumed to be a single-velocity beam, Poisson’s equation will have a solution at values of V close to V_B only for an ion velocity (v_di)₀ > Ö kT_e/M@ at the boundary (the Bohm criterion) [1, 14].

Thus, the presence or absence of a solution for V, close to V_B may depend on the type of approximation for the ion distribution function at the boundary. As an example, the dependence of ï V_B' ï on V for the case where the ions at the boundary have a Maxwellian distribution, with a temperature equal to the electron temperature, is presented in Fig. 6.9

(curve 1). The region within which Poisson’s equation has no solution is evident. For comparison, curve 2, which corresponds to a mono-velocity ion beam of (v_di)₀ = Ö kT_e/M@, is also presented.

(compare (2.14)), where account has been taken of the fact that all the ions incident to region I arrive as a result of charge exchange and are scattered only from region II, i.e., from h ³ h ₁,

(V(h ') is the variation of the potential in region II, obtained from the solution of equation (2.10)). The boundary conditions for the differential equation (2.15) are the values of V = V_B and V ' = V_B' at h = h ₁. It is natural that, with this approach, the boundary between regions I and II, i.e. point h ₁,

should be selected at the point of the quasi-neutral plasma where v_di < (v_di)₀, since V ' = - ¥ at v_di = (v_di)₀.

Solution of equation (2.15) is presented in Fig. 6.9 (curve 3). The solution obtained at V ' = 0 (curve 4) is also shown, and curve 5, obtained by solving the accurate equation (2.14), is presented for comparison. It is obvious that the use of the correct ion distribution function at the boundary of regions I and II does not eliminate the difficulties associated with sewing the solutions together. The fact is that, although the right sides of equations (2.14) and (2.15) differ from each other only by a small value, this difference is considerable near the arbitrary boundary of regions I and II (h = h ₁), because the terms approximately equal to (L_D/l_I)^1/2 must be disregarded upon transition from equation (2.14) to equation (2.15), whereas the total space charge near the boundary of a quasi-neutral plasma is approximately equal to (L_D/l_I)², i.e., less than the term rejected. Therefore, the approximate formula (2.15) is not applicable near the boundary of a quasi-neutral plasma. This is obvious, in particular, from the fact that the derivative of V of the first term on the right side of (2.15) approaches infinity at h = h ₁. In this case, the ion density decreases sharply near the boundary, and if decompensation of space charge is not taken into account in the region of quasi-neutrality, there develops a prevalence of electron space charge r _e(V) over ion space charge r _i(V) and to a changing of the sign of d²V/dh ². The corresponding functions r _i(V) and r _e(V) are shown in Fig. 6.10.

This result is general to a significant degree, because it is related to the presence of ions with a zero velocity component v_x on the boundary [6]. Actually, the ion space charge in region I is given in the general case by the expression

from which it is obvious that dr _i/dV ® ¥ as V ® V_B, if f_i(0) ¹ 0.

We note that the approximate solution of Poisson’s equation, obtained by separation of the pre-electrode sheath into two parts, may be improved considerably if the initial decompensation of the space charge is taken into account at the boundary of region I and II, by adding the value of D r _i(V_B) = eD n to the ion space charge (where D n is calculated from (2.13)). The solution obtained in this case is also shown in Fig. 6.9 (curve 6).

We also note that the dependence of V' on V in region I, obtained as a result of accurate calculation, is well approximated by the expression

where n₀ is the density of charged particles at the boundary of regions I and II; V = V - V_B is the potential in the space-charge region, measured from the boundary of the quasi-neutral plasma; and L_D(n₀, T_e) = Ö kT_e/4p n₀e²@ is the Debye length corresponding to temperature T_e and density n₀ at the boundary point.* During calculations by formula (2.17), the point where g = g ₀ = 0.76 or a value sufficiently close to this must be taken as the boundary of regions I and II. In particular, the results obtained from formula (2.17) (if the point at which g = 0.7 is taken as the boundary of regions I and II) are shown by the dashed lines in Fig. 6.8. It is obvious that formula (2.17) most correctly describes the variation of the potential in region I, although it cannot be used at small values of V, i.e., near the boundary of regions I and II.

Sewing together the solutions at the boundary of regions II and III. In order to sew together the solutions of equations (2.5) and (2.8) we can use equation (2.5) and the fact that for T_e » T a strong field penetrates the region where a local description of the plasma is still valid. In this case, equations (2.5) and (2.8) have the general solution (2.12), which provides a smooth transition from one solution to the other.

However, in cases of interest for a TIC plasma, the ratio T_e/T is usually not sufficiently large that the solutions of equations (2.5) and (2.8) overlap appreciably. In this case, the solutions were sewed together in the following manner [9]. We assume that point @ is the arbitrary boundary of regions II and III (see Fig. 6.5). Equation (2.8), to the left of point x₂ (in the non-local region), should now change its form since the non-local region is the limiting one. In this case, integration over x' on the right side of (2.8) should be carried to point x₂ rather than to infinity. Moreover, the contribution to the density from those ions which arrive at a point x of region II from region III without having undergone scattering must be taken into account. Then, instead of (2.8), we have

where f_i(v_x, x₂) is the ion distribution function at the boundary between regions II and III.

In [9] the distribution function f_i(v_x, x₂) was to have the form

where coefficients a_m were obtained from conditions of continuity for the functions of V(x), V'(x) and V''(x), and also from the condition

*The value of L_D(n₀, T_e) is related to the Debye length L_D(n(0), T_e) by the relation L_D(n₀, T_e)² = expV_BL_D(n(0), T_e)².

of continuity of ion flow at point x₂. We note that the derivatives of each term, obtained upon substitution of (2.19) and (2.18), approach infinity at point x₂. However, the sum of the derivatives, with appropriate selection of coefficients a_m, is finite. Parameter c was left unspecified in order to check the independence of the solution of equation (2.18) from the specific form of f_i(v_x, x₂).

With a large potential barrier which retards the electrons emitted from the plasma, the electron flow at the plasma-electrode interface is carried exclusively by fast electrons with energy E » eV₁. Therefore, calculation of the electron flow at the plasma boundary involves finding the fast electron distribution function, which under these conditions may differ considerably from the equilibrium function. On the other hand the distribution function of the main mass of electrons with energy E » kT_e is essentially a Maxwellian function. In order to determine the main characteristics of the problem, consider first the case where there is no electron emission from the electrode to the plasma. If the fast electron distribution function is Maxwellian, then the electron emission from the plasma to the electrode is equal to

where E₀ » eV₁ is the potential barrier of the Debye sheath boundary, and n₀ is the electron density at the plasma boundary, i.e., just before the potential barrier. However, if energy exchange between electrons is not sufficiently rapid, then the fast electron emission from the plasma leads to a depletion of fast electron density compared to the equilibrium density. The range of values of E and x, in which the number of fast electrons is less than equilibrium, is shown arbitrarily in Fig. 6.11. A decrease of density, in turn, leads to a corresponding decrease of the emissive flow i_pi from the plasma to the electrode compared to the equilibrium flow i_Te (calculated by expression (3.1)). The coefficient r2 = 1 - i_pi/i_Te, which determines the decrease of flow i_pi compared to i_Te will be called the kinetic reflection coefficient for electron emission from the plasma to the electrode [15] (by analogy with the quantum-mechanical reflection coefficient, which takes into account the reflection of electrons due to the potential jump at the metal-vacuum interface).

of fast electrons due to diffusion in the plasma or energy losses during various types of elastic and inelastic collisions does not occur rapidly enough, the excess of fast electrons near the boundary leads to an increase in the reverse flow from the plasma to the electrode. In other words, part of the fast electrons return to the emitter, undergoing elastic scattering in the plasma before losing energy and overcoming the pre-electrode potential barrier.* This effect can be taken into account by introducing a kinetic reflection coefficient for electrons injected from the cathode into the plasma. The electron flow from the cathode to the plasma may then be written in the form i_s(1 - r₁), where i_e is the cathode emission current and r₁ is the fraction of the electrons returned to the cathode.

If i_s » i_Te, the injected electrons may make an appreciable contribution to the total electron density n₀ in the pre-electrode sheath. However, the main contribution to n₀ is most often made by the thermal electrons with energies on the order of kT_e. This situation is considered below. Under these conditions, the interaction of fast electrons among themselves is not considered and only their interaction with the slow, Maxwellian electrons of the plasma is taken into account. Then, both effects - a decrease of fast electron density due to emission from the plasma and increase of fast electron concentration as a result of electron injection from the cathode - may be assumed independent of each other. The reflection coefficients r₁ and r₂ may also be calculated independently.

Under the conditions being considered, the reflection coefficients r₁ and r₂ are not dependent on current, but are determined by the cathode temperature T_C (in the case of r₁) and by the plasma parameters in the pre-electrode sheath.

As was shown in §2, the variation of the potential in the pre-electrode region of a quasi-neutral plasma is the order T_e/e. Since E » kT_e, the electric field in the plasma only weakly affects the motion of the fast electrons. This allows the assumption that the potential barrier at the plasma-electrode interface is right-angular, as in Fig. 6.11.

*Similar phenomena may be appreciable when current passes through a contact between two semiconductors of the same type [16], when there is thermionic emission from semiconductors [15, 17-19], and upon emission of fast secondary electrons and photoelectrons from dielectrics.

@The kinetic reflection coefficient at the boundary of a weakly ionized plasma with an electrode is calculated in [15, 20-23, 46].

electrode on the order of the mean free path l_e. After that the electron drift may be considered in the diffusion approximation.

Under these conditions, the electron distribution function should differ very little from a spherically symmetric function, and it is sufficient in the discussion that follows to consider only the energy distribution function n(E, x), which may be strongly non-equilibrium even after a symmetrization over directions has occurred. This is often the situation in a weakly ionized plasma, where the change in the direction of electron motion (momentum relaxation) occurs as a result of electron scattering from neutral atoms, and energy relaxation occurs as a result of comparatively rare electron-electron collisions or inelastic collisions between electrons and atoms.

The equation for the fast electron distribution function. The boundary conditions. Since diffusion current considerably exceeds field current for fast electrons with energy E » E₀ » kT_e, the electron current with energy E is described by a diffusion-type equation

Variation in the value of i(E, x) occurs because of collisions. By using the results obtained in Chapters 4 and 5, we write the continuity equation for fast electron flow in the form

The right side of (3.4) gives the number of particles with energy E which occur as a result of the electron-electron and inelastic electron-atom collisions. As for elastic electron-atom collisions, under TIC conditions they hardly affect the fast electron energy distribution.

Having substituted (3.3) into (3.4), we obtain the equation for the fast electron distribution function n(E, x):

Equation (3.5) is valid for t _E » t _p, where t _E and t _p are the energy and momentum relaxation times, respectively, for fast electrons. Since the energy range of E is about the order of kT_e about energy E₀ (see Fig. 6.11), when finding n(E, x), the electron energy E can be replaced by E₀ in the coefficients of equations (3.5), and also in the coefficients of the boundary conditions (see formulas (3.13) and (3.14) below).

We now formulate the boundary conditions for equation (3.5). Since current i(E, x) should vary little over a distance the order of the mean free path l_e (as a result of the inequality t _E » t _p), the value of the diffusion current i(E, x) a distance on the order of l_e from the boundary can be set equal to the current at the boundary itself, i(E, 0).

When calculating i(E, 0), it is necessary to consider the ranges of energy E > E₀ and E < E₀ separately. For E > E₀, the electrons may leave the plasma and reach the cathode. For this group of electrons, the boundary condition near the cathode surface has the following form:

where the electron energy distribution in the flow from the cathode to the plasma and from the plasma to the cathode, respectively, are denoted by i_?(E) and i_pl(E).

For E £ E₀, the electron flow at the boundary is i(E, 0) = 0; therefore, the boundary condition has the form

The electron angular distribution at the plasma boundary must be found in order to calculate the electron flow from the plasma to the electrode i_pl(E). For E £ E₀, where the electrons are reflected from the potential barrier, their distribution function is spherically symmetric. For E > E₀, however, deviations from spherical symmetry occur which depend on the ratio W ₀/4p , where W ₀ is the solid angle within which electrons of a given velocity can leave the plasma. When the energy E of the electrons leaving the plasma exceeds the height of the potential barrier E₀ by a small value of the order of kT_e, the ratio W ₀/4p » kT_e/E₀ « 1, and therefore, the asymmetry of the distribution function at the boundary (for the electrons leaving the plasma) is small.*

In order to calculate i_pl(E), consider the group of electrons with energy E. For an electron with energy E to leave the plasma, it is necessary that the electron velocity component along the x-axis (we shall temporarily direct the x-axis from the plasma toward the electrode) satisfy the condition vm_x²/2 > E₀. Since v_x = vcosq (where q is the angle between the electron velocity v@ and the x-axis), and vm²/2 = E, then at a given energy E, only the electrons leaving the plasma at angles q to the x-axis such that q < q ₀ (where cos^2q ₀ = E₀/E) reach the electrode. If the electron distribution over angles is spherically symmetric, the number of electrons at the boundary of the plasma with velocities in the element of the solid angle dW = sinq dq dy , is equal to (n(E,0)/4p )dW . Therefore, the electron flux i_pl(E) to the electrode is equal to

If the fast electrons in the pre-electrode sheath have a Maxwellian distribution, i.e., n(E,0) = n_M(E), then, by using (5.2.14) and (3.1), we find from (3.8) that i_pl(E) = i_Te(E), where

is the electron energy distribution for equilibrium emission from the plasma. The total equilibrium emission from the plasma, i_Te, is obtained easily by integrating the right side of (3.9) over E from E₀ to ¥ .

*If the excess of electron energy E above the potential barrier E₀ is significant, then the asymmetry of the distribution function at the boundary may be appreciable. For this case, see [24, 47], where the angular distribution of the electrons emerging from the plasma is calculated.

Consider now the energy distribution i_s(E) for the thermionic emission from the cathode. Here we can use the principle of detailed balance and assume that the electron gas in the plasma is in thermodynamic equilibrium with the cathode and that the electron temperature T_e in the plasma is equal to the cathode temperature T_C. At thermodynamic equilibrium all the forward and reverse processes are mutually balanced; therefore, the resulting flux at the plasma-cathode interface should be equal to zero, so that i_s = i_Te and i_s(E) = i_Te(E). Therefore, using (3.9), we have

Since the electron energy distribution i_s(E) in the emission from the cathode is not dependent on the state of the electron gas in the plasma, expression (3.10) is also valid in general, even when thermodynamic equilibrium between the electron gas in the plasma and the cathode does not exist.

Consider now the boundary condition for the distribution function n(E, x) as x ® ¥ . The fast electron distribution function relaxes to a Maxwellian distribution function at large distances from the electrode. Therefore, the boundary condition at infinity has the form*

Thus, finding the distribution function n(E, x) reduces to solution of equation (3.5) with boundary conditions (3.6), (3.7), and (3.11). Since the equation and boundary conditions are linear with respect to n(E, x), we can find the solution in the form of the sum of n_M(E, x) and two other components: n₁(E, x) and n₂(E, x), each of which satisfies equation (3.5):

The term n₁(E, x) describes the increase of density due to the injection of fast electrons into the plasma from the cathode, and - n₂(E, x) describes the decrease of fast electron density as a result of their emission from the plasma, and accordingly, at x = 0, n₁(E, x) and n₂(E, x) satisfy the boundary conditions

*The distribution function n_M(E) is spatially homogeneous and, therefore, does not give diffusion flow. Condition (3.11) at infinity means that the flow far from the electrode is transported exclusively by slow electrons; i.e., the contribution of fast electrons to the flow is negligible.

In (3.13) and (3.14), v₀ = Ö 2E₀/m@ and D(E₀) = 1/2(l_ev₀) are the velocity and diffusion coefficient, respectively, of fast electrons.*

As x ® ¥

It is obvious from (3.13) and (3.14) that the boundary conditions for n₁(E, x) and n₂(E, x) are described identically. In particular, (3.14) is obtained from (3.13) if i_s is replaced by the equilibrium flux from the plasma i_Te, and the cathode temperature T_C is replaced by the plasma electron temperature T_e.

This result is an expression of detailed balance, according to which the net flow at the plasma-cathode interface is equal to zero for T_e = T_C and i_s = i_Te, and the electron distribution function of the plasma is equilibrium, i.e., n(E, x) = n_M(E, x). In this case, the excess fast electrons n₁(E, x) introduced into the plasma by the electrode emission is equal precisely to a decrease of their concentration n₂(E, x), induced by electron emission from the plasma, and the reduction in flux from the cathode due to the return of the electrons from the plasma to the cathode is compensated for by a reduction in flux from the plasma to the electrode as a result of depletion of fast electrons in the distribution function of the plasma, i.e., r₁ = r₂. We may note that in the present case, the possibility of using the principle of detailed balance is the result of the linearity of equations in n and is related to the absence of an interaction between the fast electrons. Under these conditions, the electron emission from the plasma to the wall is not dependent on whether the wall is an emitter or not.

Expressions for the electron energy distribution function for the limiting cases of large and small reflection coefficients. If the main mechanism of electron energy relaxation in the plasma is electron-electron interaction, then, by substituting expression (5.5.6) for Iee{n(E, x)} into equation (3.5), and by using (5.5.8) and (5.5.9) (disregarding the derivatives of E/t _E and D_E with respect to E) we obtain a two-dimensional diffusion equation in space x and energy E:

where L_E(E) = Ö D_E(E)t _E(E)@ is the length of total energy relaxation due to electron-electron collisions. L_E gives, to an order of magnitude, the distance over which the electron totally loses its excess energy. However, the length L_E* = Ö Dt _E*@ = L_EÖ kT_e/E@ is also significant (see (5.5.18) and Fig. 6.ll) - it is the distance over which an energy the order of @ is lost. Since the distribution function varies appreciably in energy space with a change in energy of kT_e, the length L_E* becomes a typical distance over which significant variation of the distribution function occurs.

First, we calculate the addition to the distribution function n₁(E, x) that results from beam injection. Consider for this the limiting cases of the problem. If T_e » T_C the second term on the left side of (3.13) is on the order of (kT_e/E₀)v₀ n₁(E, 0). The right side of (3.13) is on the order of l_ev₀ n₁(E, 0)/L_E*. The ratio of the second term to the right side is given to an order of magnitude by the parameter p, equal to

For p » l, the right side in (3.13) can be disregarded. This means that energy relaxation within the plasma occurs so slowly that we can assume to a first approximation that all the injected electrons return to the emitter. The case of p » 1 corresponds to large values for the kinetic reflection coefficient, 1 - r₁ « 1. In this case, in correspondence with (3.10) and (3.13), we have

On the other hand, if p « 1, the second term on the left side of (3.13) may be disregarded. Then, instead of (3.13), we obtain

The case of p « 1 corresponds to small values of the kinetic reflection coefficient, where r₁ « 1.

The solution of equation (3.17) for n₁(E, x) (see the last paragraph of this section) with boundary conditions (3.15), (3.15), (3.20), and (3.21) leads to the expression

where the function f(y, k , k ₀) for large and for small values of parameter p, respectively, is expressed as with (3.10),

The integration variable a in (3.23) and (3.24) assumes real values and the root Ö a - ik @ is calculated so that at a = 0, Ö a - ik @ = Ö k exp(-p i/4).

It is necessary, as indicated above, to replace i_s by i_Te, T_C by T_e, and k ₀ by k in (3.22)—(3.24) in order to obtain the corresponding expressions for n₂(E, x)

The electron flow and the kinetic reflection coefficients at the plasma-electrode interface. Having calculated n(E, x), we can find the electron flux at the plasma boundary by formula (3.3). Taking into account (3.3) and (3.12), the electron flux at the boundary can be represented in the form of the difference

where i(E) = i(E, 0),

The components of current i₁(E) and i₂(E) contribute to electron injection into the plasma and to electron emission from the plasma, respectively, where the energy integrals over i₁(E) and i₂(E) yield the total flux of electrons injected into the plasma and the total flux of electron emission from the plasma. In correspondence with (3.2), these currents are written in the following form:

Consider initially the case of large reflection coefficients (p » 1). To calculate E, we substitute (3.22) and (3.23) into (3.27). Having integrated (3.27) with respect to E from E₀ to ¥ (accordingly, integration by y is carried out from 0 to ¥ ) and having divided the result by i_s, we find, according to (3.28), that

where the root Ö a - ik @ is calculated so that at a = 0 we have Ö a - ik @ = Ö k exp(-p i/4). Calculation of the integral in (3.29) by the residue theorem (the singularity is a first-order pole at a = ik ) leads to the following result:

It is obvious from (3.30) and (3.31) that the reflection coefficient, in the limiting case we have been considering, is not dependent on the plasma electron temperature T_e and is calculated only for cathode temperature T_C.

For small values of the reflection coefficient, it is more convenient to calculate the value of r₁ by dividing the flow of electrons returned from the plasma to the cathode (the second term in the left side of (3.13)) by the cathode emissive flow i_s:

Having substituted (3.22) and (3.24) into (3.32) and having performed integration over E, we obtain for p « 1

It is obvious that the reflection coefficient r₁ is dependent on T_C and t _e = T_e/T_C. A graph of the function y (t _e) is shown in Fig. 6.12. As t _e increases, the reflection coefficient and the back electron flow from the plasma increase, since the fraction of fast electrons passing through the plasma increases as the plasma electron temperature increases. The distribution function n₁(E, 0) for high energy E and the number of particles returned to the electrode increases accordingly.

Using (3.19), expression (3.34) for r₁ can be rewritten in the following form:

If the plasma electron temperature T_e is large compared to cathode temperature T_C, i.e., t _e = T_e/T_C » 1, then, according to (3.35) y (t _e) = t _e^3/2/8. In this case, according to (3.36) and (3.18), we have

(3.37)

It is obvious that, in this case, r₁ is not dependent on the cathode temperature T_C, but is determined only by the plasma electron temperature T_e, because electron energy dispersion in the cathode beam is insignificant

for T_e » T_C.

Let us now consider the reflection coefficient r₂. The value of r₂ is dependent only on the plasma electron temperature T_e, where, according to the foregoing, the reflection coefficient r₂ can be calculated from (3.31) or (3.36), if it is assumed in these expressions that T_C = T_e (t _e = 1), i.e.,

All terms are retained in boundary conditions (3.13) and (3.14) at intermediate values of parameter p, when the reflection coefficient is of the order of unity, which complicates the problem. However, if the factor E - E₀ is replaced by the constant value D E₀ in the second term of the left side of (3.13) and (3.14), the problem can be solved by using the Wiener-Hopf method (see [22,25]). The solution obtained in this case will be approximate, but parameter D E₀ can be interpolated so that it matches the accurate solutions obtained above in the limiting cases of p « 1 and p » 1.*

The dependence of the reflection coefficient r₂(T_e) (for electron flux from the plasma) on the parameter p(T_e), calculated in this manner, is presented in Fig. 6.13; and the relation r₁(T_C, t _e)/ r₁(T_C, 1) is presented for several values of t _e in Fig. 6.14 as a function of parameter

p(T_C). In this case, according to (3.38), r₁(T_C, 1) = r₂(T_C). It is

*If the relation t _e = T_e/T_C is small, the limits between which D E₀ must be interpolated are also small. In particular, at t e = 1, D E₀ is interpolated from (8/3)× kT_C for p « 1 to kT_C for p » l.

obvious that for p(T_C) » 1, the relation r₁(T_C, t _e)/ r₁(T_C, 1) is not dependent on the value of t _e and is equal to unity, whereas large values of r₁(T_C, t _e) correspond to larger values of t _e as p(T_C) decreases.

Derivation of expressions (3.22)-(3.24). If we write n₁(E, x) in the form

We can expand F(x@ ,y) as a Fourier integral in y:

the component F(x@ ,a )@ is given by the inversion theorem:*

From (3.40) we now obtain the differential equation for the Fourier component F(x@ ,a )@:

The solution of (3.43), which approaches zero as x@ ® ¥ , has the form

The integration constant C(a ) should be calculated from the boundary conditions at x = 0, i.e., from (3.13) and (3.15). It is simplest to calculate C(a ) for p « 1. In this case (3.13) is replaced by (3.21), and both boundary conditions (3.21) and (3.15) are superimposed on the derivative ¶ F(x@ ,y)/¶ x@. Having applied the Fourier transform (3.42) to the boundary conditions (3.21) and (3.15), we obtain

By differentiating (3.44) with respect to x@ and by equating the derivative to the right side of (3.45), we obtain

By substituting (3.46) into (3.44) and by performing the reciprocal Fourier transformation by formula (3.41), we obtain in the case of p « 1

*The convergence of integral (3.42) is determined by the asymptotic behavior of the function n₁(y, x@ ) as y ® ± ¥ . At large values of y, function n₁(y, x@ ) decreases no more slowly than exp[-(E-E₀)/kT_e] = exp(-2k y). At small values of y, as indicated by the result, n₁(y, x@ ) does not increase

In the opposite limiting case, when p » 1, the boundary conditions at x = 0 (see formulas (3.15) and (3.20)) are mixed: for E > E₀ (y > 0), the boundary condition is superimposed on the function itself, and for E < E₀ (y < 0), it is superimposed on its derivative.

The Wiener—Hopf method [26] is usually employed to solve differential equations with mixed boundary conditions. However, in the case which we considered, the integration constant C(a ) can be calculated by the following rather simple method [15].*

It follows from the boundary conditions (3.15) and (3.20) at x = 0 with regard to (3.39) that the integration constant C(a ) should be calculated from the integral equations

where C(a ) should be an analytic function in the region ï Im a ï < k (see footnote 10). To fulfill (3.49) and (3.50), it is sufficient that C(a ) have a simple pole at a = - ik ₀ in the lower half-plane of the complex variable a and that C(a )Ö a ² + k ²@ be regular, and that it be a function that approaches zero as ï a ï ® ¥ in the upper half-plane of a . These requirements are satisfied by the function

where C₀ is some constant. Having substituted (3.51) into (3.49), we calculate the integral in the right side by the Jordan lemma. As a result, we find that

Therefore, by calculating C₀ and by substituting it into (3.51), we obtain

exponentially. Then, as follows from (3.39), F(x@ ,k y) ~ exp( -k y) as y ® ¥ and F(x@ ,y) ~ exp( k y) as y ® -¥ . Therefore, integral (3.42) converges and determines the analytical function of a in the band ï Im a ï < k , including the real axis.

*A similar problem was solved by the Wiener-Hopf method in [27].

Cathode beam relaxation by electron-electron collisions. As indicated in the preceding paragraph, the distribution function n(E, x) can be represented as a superposition of the distribution function n₁(E, x) of the injected beam and the plasma electron distribution function n_pl(E, x) = n_M(E) - n₂(E, x) which is Maxwellian except for a distortion due to electron emission to the electrode. Consider first the limiting cases which correspond to large and to small electron reflection coefficients at the boundary.

In the first case (p » 1), n₁(E, x) can be calculated from formulas (3.22) and (3.23) (see the last paragraph of this section), which yields

where w(z) is a tabulated function of complex argument *:

In order to obtain the correction n₂(x@, y) to the electron distribution function in front of the non-emitting electrode from expression (4.1), i_b = i_Te = (1/4)(n₀v_e@ )exp(-E₀/kT_e) and t _e = 1 must be substituted into (4.1). Then,

where n_M(E) is the equilibrium, Maxwellian, electron energy distribution function and erf(z) = (2/Ö p )ò ₀^xexp(-t²)dt is the probability integral

*The main properties of function w(z) and numerical tabulations are presented in [28].

It is obvious from (4.5) that for E ³ E₀, the fast electron density n_pl(E, x) ® 0 as the distance from the plasma boundary decreases, x@ ® 0 and q ® 0, since fast electron production because of energy exchange between the emitted electrons in the main mass of plasma electrons occurs very slowly for p » 1.

Distortion of the distribution function n_pl(E, x) near the boundary is extended into the energy range E < E₀ only to a value on the order of kT_e, and the distribution function remains Maxwellian for smaller energies. For example, at the plasma boundary (x = 0, q = p , cosq = -1, and r = ï yï ), we have, according to (4.4),

Even for E₀ - E = kT_e, the factor in the parentheses in (4.6) is equal to 0.157, and the correction of n₂(E, 0) is a small fraction of the equilibrium distribution function n_M(E). The region in the (E, x)-space within which the electron emission from the plasma distorts the electron distribution function is shown schematically in Fig. 6.11. The range of values of E and x where the distribution function is close to equilibrium is covered by the cross-hatching. We note that as the parameter p(T_e) increases, i.e., as the rate of energy exchange increases, distortion of the distribution function as a result of electron emission from the plasma becomes less significant.

In order to obtain the expression for the total distribution function, we substitute (4.1) and (4.4) into (3.12). We can then write the result obtained in the form of the ratio of the distribution function n(E, x) to its equilibrium value n_M(E):

where the parameter f₀ = t ²i_s/i_Te is proportional to the ratio of the cathode emission current to the equilibrium electron emission current from the plasma. Parameter f₀ shows the relative importance of the cathode beam electrons. If the plasma is in contact with the cathode, then during the passage of current, f₀ > 1, i.e., injection of electrons into the plasma dominates emission of them from the plasma. However, for energy E which exceeds the height of the potential barrier by several kT_C, the electron emission from the plasma may dominate the injection into the plasma, because the contribution of n₁(E, x) decreases much more rapidly than n₂(E, x) as E increases (for T_e > T). For example, at x = 0 (at the plasma boundary), n₂(E, 0) ~ exp[-(E - E₀)/kT_e], whereas n₁(E, 0) ~ exp[-(E - E₀)/kT_C].

The ratio n(E, x@ )/n_M(E), calculated by formula (4.7) as a function of x@ for E = const, is shown in Fig. 6.15. The dashed lines 1 and 2 depict the first and second terms, respectively, in (4.7), which indicate the relative contribution to the total distribution function @ of the electrons emitted from the plasma and of the cathode beam injected electrons. It is obvious that with the chosen parameters we obtain

n(E, 0) < n_M(E) close to the electrode; but as distance from the electrode increases, this inequality reverses, because of the increase in the beam electron density n₁(E, x) with distance from the electrode.

It is obvious from Fig. 6.15 that the dependence of the distribution function n₁(E, x) (and consequently of n(E, x)) on coordinate @ is non—monotonic. The latter is explained by the fact that, as the injected electron cloud moves away from the plasma boundary, two processes occur simultaneously: the deceleration of the electron cloud as a whole and the electron diffusion to higher energies. The result is that the total number of fast electrons decreases, but the number of electrons in the "tail" of the distribution function increases because of their diffusion in energy space. To illustrate this, the distribution function n₁*(E, x@ ) for an injected beam of electrons is presented in Fig. 6.16 as a function of E at different distances from the cathode.

At x = 0 (the boundary of the plasma), and for the limiting case (p » 1) which we are considering, the electrons have a Maxwellian distribution at the cathode temperature T_C. Moving into the plasma, the nature of the distribution changes, and the effective beam electron temperature increases. It should be noted that the occurrence of a non-monotonic dependence of n(E, x) on x for E = const, due to electron diffusion to higher energies, is related to the fact that the electron temperature T_e of the plasma with which the beam interacts is higher than the beam temperature T_C. This is illustrated by Fig. 6.17, where the value of n₁(E, x@ )/n₁(E, 0) is plotted as a function of x(for E = const) for different values of t _e = T_e/T_C. It is obvious that the density of electrons with a given energy (t _e = 1) decreases monotonically with distance from the emitter. This monotonic character of the distribution disappears as t _e increases, and a peak appears in the pre-cathode region of the curves.

A similar pattern occurs in the opposite limiting case of p « 1. In that case, the electron distribution function of the injected beam n₁(E, x) is expressed in the form (3.22) and (3.24). The value of n₂(E, x) is obtained as above if t _e = 1 and i_s = i_Te are substituted into (3.22)

and (3.24). In the limiting case of p « 1, as a result of intensive energy exchange, the "tail" of the Maxwellian distribution is restored rapidly as the electrons move in energy space, and the electron emission from the plasma hardly distorts the electron distribution at an energy of E » E₀ + kT_e, i.e., for those energies where n₂(E, x) « n_M(E). However, for i_s » i_Te, the contribution of fast beam electrons n₁(E, x) to the total fast electron density in the pre-electrode sheath may be significant.

The values of n₁*(E, x@ ) at different distances from the cathode, obtained for p « 1 as a result of calculating the integral in (3.24), are presented in Fig. 6.18. The corresponding character of the distribution has the same form as in the case p » 1. The distribution function has a maximum at x@ = 0, where the greatest number of electrons has an energy of E close to E₀. As the distance from the electrode increases, the peak smooths out and disappears, and the "center of gravity" of the injected electron cloud moves into the energy range E < E₀. In the high energy range, the density n₁*(E, x) varies non-monotonically as x@ increases. This is indicated, for example, by the value of the curves corresponding to x@ = 0 and x@ = 0.5 in the higher energy range.

The limiting case (p « 1) being considered is the one of greatest interest for the low-voltage arc in a TIC. It is also of great interest for arc discharges.* Therefore, this case is considered in more detail below. Along with continuous energy relaxation because of electron-electron collisions, relaxation with discrete energy losses as a result of inelastic collisions between electrons and atoms is taken into account. These losses may be significant with a comparatively low density of the plasma thermal electrons.

*If the discharge parameters are such that p » 1, the results of [25] can be used.

The effect of inelastic collisions on the electron distribution function in the pre-electrode sheath. In order to take into account inelastic collisions, it is necessary to retain the term for electron-atomic collisions I_kl{n(E, x)} (see formula (5.5.1)) on the right side of equation (3.5). As in §5 Chapter 5, we assume that the electrons interact effectively with only two atomic levels, k and l, the distance between which is E_kl « E₀, and that E_kl is comparable in value to kT_e. In this case, the relative populations of levels k and l are determined by the thermal electrons with energies on the order of kT_e and are not dependent on the fast electron density. We also assume that the relative level population is Boltzmann. By using expression (5.5.35) for I_kl{n(E, x)}, we reduce equation (3.5) to the form

Equation (4.8) with boundary conditions (3.15) and (3.21) is solved, as above, by a Fourier transform in E (see formulas (3.41) and (3.42)). In this case, the expression for the addition n₁(E, x) to the concentration for beam injection, is written in the form 29

where R(a ) is used to denote the expression

Here y_kl = E_kl/Ö E₀kT_e@ and b ² = I_E²/L_ea². Parameter b determines the relative contribution of inelastic collisions to beam relaxation in the pre-electrode sheath. For b = 0 (b « k ), expression (4.10) becomes expressions (3.22) and (3.24), which correspond to beam relaxation due to single electron-electron collisions. For b » k , relaxation occurs only by inelastic collisions. In that case

To compare the two beam relaxation mechanisms, we integrate both expressions, (3.22) plus (3.24) and (4.12), over x from 0 to ¥ , i.e., we calculate the total number of non-equilibrium fast electrons in the pre-electrode layer in both cases. The quantity we calculate is

for b » k

The function y _ee(E) and y _ea(E), calculated by formulas of [29], are presented in Fig. 6.19a and b. It is obvious that relaxation by inelastic collisions leads to a distribution function which oscillates in energy space. However, the envelope of these oscillations is similar in shape to the distribution function obtained by beam relaxation from electron-electron collisions. When the value of E_kl decreases, the difference between the two relaxation mechanisms is less apparent, and for E_kl « kT_C, both relaxation mechanisms lead to the same result (in this case, formula (4.16) turns into (4.15)). If the electrons interact with a large system of levels, then the overlapped oscillations become smoothed out.

The kinetic equation was solved, together with the balance equations for a large system of discrete levels, by numerical methods in [30]. The method of moments was used to calculate the electron energy distribution function and the atomic distribution function for the discrete levels. The calculations showed, in this case, that the non-Maxwellian electron distribution function, which occurs near the electrodes, is manifested primarily in the relative population of the ground (6S) and first excited level (6P). The reason, of course, is that the 6S ® 6P transition occurs because of fast electrons. The relative populations of the excited levels (6P and above), among themselves, were the same as for the Maxwellian electron distribution function. The transitions between these excited levels are induced by slow, Maxwellized electrons.*

We should also point out a series of investigations [40-45] in which the kinetic equation was solved numerically for the electron distribution function appropriate to a TIC, including the effects of elastic and inelastic electron-atom collisions and also electron-electron collisions.

*Under some conditions, significant deviation from the Boltzmann distribution may occur for the relative population of higher levels as well (see, for example, [31].

Relaxation by inelastic collisions under TIC conditions becomes significant only with a low level of ionization (n_e/N < 10^-3 – 10^-4). The reason for this is that the inelastic interaction of electrons and cesium atoms occurs primarily in the transitions 6S « 6P. The relative population of levels 6S and 6P under TIC conditions is not an equilibrium population; it is determined by the fast electron density. If this density exceeds the equilibrium Maxwellian value, then accordingly, the population of the excited 6P level increases compared to the Boltzmann value until processes of atomic de-excitation compensate for excitation processes. With balanced rates, the number of fast electrons does not vary during inelastic interactions with the atoms.* As far

*These collisions alter only the energy distribution of the fast electrons, because the fast electrons formed as a result of collisions of the second kind have a Maxwellian distribution at the plasma temperature T_e rather than at the cathode temperature T_C as in a primary electron beam.

Kinetic reflection coefficient for energy relaxation by an inelastic process. We now consider the kinetic reflection coefficient in the case where the inelastic mechanism accomplishes the fast electron energy relaxation. Calculation of the kinetic reflection coefficients for these conditions was carried out in [46] for the limiting cases of large and small reflection coefficients. We note that if the distribution function of free electrons far from the plasma boundary is Maxwellian and if the Maxwellian distribution temperature T_e coincides with the temperature of the population of the levels, then relation (3.38) occurs, as before, between the reflection coefficients r₁ and r₂. Therefore it is possible to limit ourselves to the calculation of only one reflection coefficient r₁. In the limiting case, when 1 - r₁ « 1, the value of r₁ is obtained by the following expression:

where h @ = E_kl/kT_C. When we have h _kl@ « 1 and h _kl « 1, the discreteness of the levels ceases to be manifested and the expression derived for r₁ turns into (3.31) (i.e., if the relaxation time t _E is replaced by the energy relaxation time (5.5.37) for inelastic collisions, t _E^(kl), when calculating p(T_C) in (3.31) by formula (3.18)). If we assume T_C = T_e, i.e., h _kl@ = h _kl, in the expression derived for r₁(h _kl@, h _kl), then the expression for r₂ is obtained.

The expression for the reflection coefficient r₂ in the opposite limiting case, when r₂ « 1, is also reported in [46]:

The graph of function c (h _kl) is presented in Fig. 6.19c. For h _kl = 0, c (0) = 1, and

t _E must again be replaced by the energy relaxation time for inelastic collisions, t _E^(kl), when calculating p(T_e) by formula (3.18). As should be, t _E^(kl) coincides with the expression obtained by formula (3.36) for T_C = T_e. In the opposite limit, when h _kl » 1, c (h _kl) » 3p h _kl/16. Then, from (4.18) we obtain

Derivation of expression (4.1) [20]. To calculate f(y, k , k ₀) in (3.23), we introduce in the x@ - y plane the polar coordinates:

Having substituted the parameters x@ and y from formulas (4.21) into (3.23), we now deform the integration contour in (3.23) to a hyperbola a = -ik cos(q + it). The values of 0 < q < p /2 correspond to the energy range of E > E₀. The values of p /2 > q > p correspond to the energy range of E < E₀.

We now change (3.23) from integration by a to integration by t, and obtain

where we define coshy = k ₀/k = 2t _e - 1. Using the expression

We represent the right side (4.24) in the form of the sum of integrals for positive and negative values of t. Having replaced the integration variable t by - t in the last integral, we reduce (4.24) to the form

where l = q + iy . Let us introduce the change of integration variable t = sinh(t /2) into (4.25). Then, using the relation cosht - cosl = 2[t ² + sin²(l /2)], from (4.25) we obtain

To calculate (4.27), we differentiate F by b :

Taking into account that F(0) = p /4 sin(l /2) and integrating (4.28), we obtain

By substituting (14.30) into (3.22) and expressing y by formula (4.21), using the relations

and the property of function w(z) [28]

If an electron beam is injected into a plasma, then an excess of electrons with an energy on the order of the height of the potential barrier, E₀, at the plasma-electrode interface forms near the cathode. If the atomic excitation or ionization energy is close to E₀, the excitation and ionization processes are accelerated as a result of an increase of the number of inelastic collisions of first kind or as a result of an increase of the number of ionizing collisions. On the other hand, electron emission from the plasma to the electrode reduces the number of fast electrons and at the same time reduces the rate of atomic excitation and ionization in the pre-electrode layer. Concerning the rate of the reverse processes (atomic de-excitation and ion recombination), the rate is not dependent on the number of fast electrons, but is determined only by the concentration of thermal electrons of the plasma with energies on the order of kT_e. Under these conditions, electron injection into the plasma and emission from the plasma should lead to an increase or decrease, respectively, of the excited level populations and of the ionization rate [20, 21, 23, 27, 30-35].

As in Chapter 5, let us consider a specific physical situation, namely ionization of Cs atoms in the precathode sheath of a low-voltage arc. The precathode potential drop in the arc, E₀, is usually close to E₁-the excitation energy of a Cs atom to the first level. For clarity, we shall assume that E₀ < E₁.* With the free electron density which usually occurs in the pre-cathode sheath of a low-voltage arc in Cs, the fast electron distribution function may be calculated by disregarding inelastic collisions with atoms; and, by knowing the form of the distribution function in the higher energy range, the number of inelastic collisions can then be calculated.

The addition of n₁(E, x) (related to beam injection) to the distribution function has greatest effect in the energy range near E₀. For lower energies, as a result of an exponential increase of the equilibrium part of distribution function n_M(E), the effect of the beam is rapidly decreased, so that if E₀ is close to E₁, it is sufficient in many cases to take into account the non-equilibrium of the distribution function only when calculating the rate of atom excitation to the first level (transition 6S « 6P). Transitions between closer excited levels occur primarily due to interaction of the excited atom with the Maxwellized electrons of the plasma.

With these assumptions, variation of the rate of ionization and of the values of ion flow from the plasma to the cathode because of the non-equilibrium of the electron distribution function were calculated in [34]. The corresponding additional term for the ion flux from the plasma can be written in the form

where g₀ and g₁ are the statistical weights of the ground and first excited states. The function

is also depicted in Fig. 6.20. The first term in (5.1) corresponds to the increase of the rate of ionization as a result of electron injection from the cathode and the second term corresponds to a decrease of the rate of ionization as a result of electron emission from the plasma. Since we have L_E ~ n_e^-1 during electron-electron collisions, the contribution of the ionization mechanism under consideration is inversely proportional to the degree of ionization of the plasma, whereas the rate of ionization because of Maxwellized electrons of the plasma is directly proportional to n_e. We note that an increase of D i_ipl as t _e increases is related to diffusion of injected electrons to the higher energy range, which is intensified when the temperature of the Maxwellized electrons of the plasma increases.

If there is a positive bias on the electrode (Fig. 6.21), a potential barrier is formed in the pre-electrode sheath which decelerates the ions travelling from the plasma to the electrode and accelerates the electrons. Analysis of the phenomena at the plasma-electrode interface may be carried out in this case in a way similar to that which was done in §2 for the case of a negative bias on the electrode. However, there are several significant differences in this case that are related to the fact that the ion temperature T in the gas discharge is usually significantly less than the electron temperature T_e.

It is significant that the electrons in this case are accelerated toward the electrode, while the ions, because of the large retarding potential barrier, have a Boltzmann distribution. As was shown in §2, the potential drop in the quasi-neutral region of the pre-electrode sheath has a magnitude the order of kT_e, where T₀ is the particle temperature whose flow to the wall is limited by the high potential barrier. In this case, these particles are ions, and T₀ = T. However, if T_e » T, this potential drop has hardly any effect on the motion of hot particles-the electrons-so that in the collision region of the pre-electrode sheath the diffusion component of electron flow dominates the field component. The reverse is true for ions.

Fig. 6.21 Potential distribution V(x) and distribution of electron n_e(x) and ion n_i(x) densities at the plasma boundary with a positively biased electrode: I, space charge region; II, region of strong distortion of the electron distribution function; III, region in which the electron distribution function is close to a spherically symmetric distribution.

This inequality is violated as the distance to the electrode decreases. The width of the pre-electrode region in which diffusion approximation (6.1) is invalid (region II) is equal in order of magnitude to l_e and, for T/T_e « 1, it is not dependent on the ratio T/T_e. In the opposite case, considered in §2, deviations from formula (6.1) are related to the occurrence of a large density gradient in the boundary region rather than to the penetration of a strong electric field into a quasi-neutral plasma. This forces us to reject the diffusion approximation at the boundary and to use the kinetic-equation method. In this case, the electric field in a quasi-neutral plasma can be disregarded in the kinetic equation, and we can assume that electron-atom collisions occur without a change in the electron energy.

We now consider the case where the degree of plasma ionization in the pre-electrode sheath is so small that Coulomb collisions can be disregarded. Let us delineate a group of electrons with energy E. Since there is no electric field and since electron—atom collisions, according to the hypothesis, occur without energy losses, the energy E is constant for this group of electrons. Let us also define m = cosq , where q is the angle between the x-axis (directed perpendicularly to the electrode from inside the plasma) and the direction of the electron velocity v. We introduce a particle distribution function f_e(E,m ,x) such that f_e(E,m ,x)dEdm is equal to the electron density at point x, with energy the range from E to E + dE, and with velocity directions in the range from m to m + dm .

According to (4.l.9) and (4.2.4), the kinetic equation for the distribution function f_e(E,m ,x) is written in the form

where Na is the density of atoms and dW = 2p dm ’is the element of the solid scattering angle (dm ’= sinq dq , q is the scattering angle).

If the electrons are scattered by neutral atoms uniformly in all directions, i.e., s (v,q )= Q(v)/4p , where Q(v) is the total scattering cross section, then from (6.4) we obtain

where l_e = 1/Na Q(v) is the mean free path for electrons with a given energy. If Q and l_e are not dependent on electron energy E, then the electron distribution over angles is also not dependent on E.

The conditions at the interface, x = 0, with the non-emitting electrode, have the form

Here i_pl(E) is the electron current(with energy E ) in the direction of the negative axis x.

Equation (6.5) with boundary conditions (6.6) and (6.7) is the Milne problem [36], known from astrophysics. It has been studied intensively for the theory of radiation transport and has been applied later to the study of neutron transport [37]. We shall use some of the results of these studies below.*

on distance x is shown in Fig. 6.22. The density and distance are measured in dimensionless units: z = x/l_e, y ₀(z) = vn(E,x)/i_pl(E). Since i_pl(E) is proportional to v, then y ₀(z) is not dependent on v. Far from the electrode, where z » 1, the solution of the Milne equation leads to a linear distribution for the density, which also follows from the diffusion equation (6.2) (the diffusion coefficient is given by D_e = (1/3)(l_ev), and the derivative dy ₀/dz ® 3). Asymptotically, for z₀ » 1, solution of the Milne equation has the form

where z₀ = 0.7104 [38]. The quantity L₀ = l_ez₀ is called the extrapolation length.

The precise solution of equation (6.5) (curve 1) and the asymptotic expression (6.9) (straight line 2) are shown in Fig. 6.22. It is obvious that the asymptotic expression, i.e., the diffusion approximation, satisfactorily approximates the precise solution up to the electrode itself. Even at z = 0 the difference between the precise solution (1/3) y ₀(0) = 0.5773 and the approximate value (l/3)y ₀(0) = z₀ = 0.7104 is comparatively small.

By analogy with §2, it is interesting to determine the mean drift velocity (v_de)₀ = i_epl/n₀ of plasma particle at the interface with the electrode (here n₀ = ò _0¥ n(E,0)dE and i_epl = ò _0¥ ipl(E)dE are the particle density and current, respectively, at the interface). It is obvious that

where < v> is the average random particle velocity at the interface. Under the conditions considered here, the nature of the energy distribution is not changed during the diffusion of particles from the volume to the surface. Therefore, if the particles far from the interface have a Maxwellian energy distribution, then < v> = v_e@ = Ö 8kT_e/p m@ and (v_de)₀ = 0.578v_e@

*One can become familiar with the calculation methods in [36 - 39].

(compare this with the corresponding formulas of §2).

In some cases, it is necessary to determine the diffusion flow of particles from the plasma to the electrode when their density is known only at a distance x from the electrode large in comparison to the mean free path l_e. In this case, one can use the asymptotic expression (6.9) for the density far from the interface. Since z » 1 in this case, the value of z₀ can be disregarded in (6.9). In other words, if the pre-electrode barrier accelerates the electrons, then for x » l_e the density at the interface compared to its value in the volume can be disregarded, and the diffusion equation (6.2) with the boundary condition n = 0 at x = 0 can be solved, in the usual way of plasma theory. The boundary condition can be refined so that the distribution of density corresponds precisely to formula (6.9), according to which y ₀(0) = 3z₀ In this case, as follows from (6.10),

where g ₁ = 0.47. It is interesting to note that a very similar result is obtained if it is assumed that the particles emerging from the plasma are uniformly distributed over angles. In this case, we have f(E,m ,0) = n(E,0) for m < 0. By integrating (6.7) and by calculating the total flow of particles to the electrode in this manner, we find that i_epl = (1/2)v_en₀, i.e., that (v_de)₀ = (1/2)v_e and g ₁ = l/2,* which does not differ essentially from (6.11). However, this equality is actually accidental, because the angular distribution of the particles is far from uniform in the case considered (in the absence of a potential barrier for the particles emerging from the plasma).

To illustrate this, we turn to Fig. 6.23, where the angular distribution of particles f_e(E,m ,0) at x = 0 is presented. It is obvious that the particle distribution is strongly anisotropy: as the angle of incidence to the surface decreases, the number of particles per unit solid angle increases. The anisotropy of the distribution function at the interface is related to the fact that the boundary of the medium intercepts the electrons which experience their last collision at a distance of the order of the mean free path from the position at which they leave the plasma. Therefore, the electrons, for which the angle of incidence to the surface is nearly equal in value to a right angle, undergo the last collision at the surface itself, while the electrons incident to the surface at small angles, i.e., almost normally, undergo the last collision at a distance on the order of l_e from the electrode, where their concentration is greater. This explains the increase of f_e(E,m ,0)

*It is usually this value of g ₁ that is used in practical calculations.

As the distance to the electrode becomes less, the quasi—neutral plasma makes a transition to the space-charge region (region I in Fig. 6.21), whose dimensions we disregarded above in view of the condition L_D « l_e. The space-charge region can be calculated by the same scheme as in §2. Only now the electrons and ions must exchange places.

When sewing together the solutions of the space-charge and quasi-neutrality regions (at the interface), generally, the same difficulties arise as in the case considered in §2. That is, if the electric field and decompensation of the space-charge in the quasi—neutral region is disregarded, the potential variation may not continue monotonically into region I, because the electron density at the interface of regions I and II will decrease more rapidly than the ion density as the potential increases, so initially there could be net ionic charge. However, since T < T_e, ion penetration into the region of positive potential is difficult, and the range of potential values within which there is no solution, is small. If the value of T can be disregarded, this difficulty generally disappears.

Since the field at the electrode surface is a decelerating field for electrons emitted from the electrode, and the Schottky effect on the electrode does not occur, it is usually not necessary to calculate the distribution V and dV/dx in region I; and it is sufficient to determine the electron flow from the plasma to the electrode by considering electron diffusion in a quasi—neutral plasma.

When calculating the ion flow from the plasma to the electrode through the retarding potential barrier, the situation is also significantly simpler than in the case of electron emission from the plasma considered in §3. We note that the momentum and energy relaxation times t _p and t _E for ions are the same order of magnitude. Therefore, ion emission from the plasma to the electrode does not lead to appreciable distortion of the distribution function in the high-energy range. As a result, the ion flow from the plasma through the high retarding potential barrier may usually be calculated by the formula

where eV₁ is the potential barrier in the Debye sheath and n₀ is the charged particle density at the interface of regions I and II.

Boundary conditions with a large negative electrode potential. If the electrode has a large potential barrier that retards the electrons from the plasma (see Fig. 6.5), i.e., eV₁ » kT_e, then, according to (3.1) and (3.2), the expression for electron flux i_e at the plasma-electrode interface has the following form:

Here eV₁ is the potential barrier in the space-charge region, n₀ is the charged particle density at the interface of the quasi-neutral plasma and the space-charge region, i_s and i_is are the electron and ion emission fluxes from the cathode, r₁ and r₂ are the kinetic reflection coefficients for electron fluxes at the plasma interface (see Figs. 6.13 and 6.14 and formulas (3.31), (3.35), and (3.36)), T_C is the electrode or cathode temperature, T_e is the electron temperature of the plasma near the electrode, and (v_di)₀ = g _0Ö 2kT_e/M@ is the drift velocity of the ions at the plasma boundary.

With specific calculations, the value of g ₀ should be assumed close to 0.76 (see §2). The value of g ₀ = 0.76 was obtained on the assumption that t _e = T_e/T_C » 1. However, as shown by the calculations, when t _e decreases, g ₀ decreases very slightly, so that this same value of g ₀ can be used at t _e » 1. In this regard, it is interesting to compare the results of calculating the potential distribution and drift velocity of ions at the plasma boundary, obtained on the assumption of t _e > 1, to results obtained in [11] for the opposite limiting case t _e < 1, when the electric field in the quasi-neutral plasma is weak for the ions. In the latter case, the drift velocity of ions at the interface is equal to (v_di)₀ = Ö kT_i/M@. Therefore, if the results obtained for t _e > 1 and for t _e < 1 are extrapolated to the case t _e = 1, i.e., T_e = T_C, the difference in the value of (v_di)₀ will be only about 7%.

The potential distribution for the two limiting cases also does not differ markedly, if the potential is measured in units of kT_e/e.

We now calculate the energy flux transported by the electrons through the plasma electrode interface. Let us consider the collisionless space charge region I. The energy transported by the electrons is different in different sections of this region, because the kinetic energy of the electrons varies by the value ei_eD V (where D V is the potential difference between the sections considered, arising because of the presence of an electric field). Since the total energy flux S_e, which contains both kinetic and potential energy, remains constant, it is more convenient to use in the subsequent analysis.

Let us calculate the energy flux S_e as the difference S_es = S_epl, where S_es is the energy introduced into the plasma by the cathode emission electrons and S_epl is the energy removed from the plasma by the electrons. The simplest expressions for S_es and S_epl are written in the case where the electron reflection at the plasma-cathode interface can be disregarded. In this case, the electrons emitted from the plasma, the same as the electrons emerging from the cathode, may be assumed to have a Maxwellian distribution, and S_es = i_s(eV + 2kT_C), and S_epl = i_Te(eV + 2kT_e), where V is the cathode surface potential and 2kT is the average kinetic energy of the electrons in a half Maxwellian distribution.

where for this case the potential of the plasma has been taken as zero. Formulas (7.1l)—(7.3) for particle and energy fluxes are used below (see §1, Chapter 9) as the boundary conditions for the precathode region of arc discharge in a TIC.

Boundary conditions for small potential barriers. When the retarding pre-electrode voltage drop eV₁ decreases, the solid angle which contains the velocities of the charged particles emerging from the plasma increases. This leads to an increase of the angular asymmetry of the distribution function of the emerging particles. The effects related to an angular asymmetry of the distribution function in a TIC can be significant at the pre-anode plasma boundary in the arc mode (and also in the diffusion mode). Therefore, the phenomena in the pre-anode region of a low—voltage TIC arc, where a potential barrier usually occurs that only slightly retards the plasma electrons, are of great interest.

If the condition t _E » t _p is fulfilled for the electrons which transport the current over the potential barrier, calculation of their distribution function may be carried out in two stages. The angular distribution is independent for each group of electrons with energy E. The electron energy distribution is then determined, where the electron distribution function at this stage may be assumed to be symmetric over velocity angles.

*Since we have neglected quantities the order of kT/eV₁ in the calculation of r₁ and r₂, we should also, for consistency, disregard 2kT_C and 2kT_e. However, we keep them so that, when r₁ = r₂ = 0, (7.3) transforms into the usual, exact expression for S_e.

To find the electron distribution function f(E,m ,x) over angles, it is necessary, as before, to solve equation (6.5), but with the modified boundary conditions: f(E,m ,0) = 0, if m ₀ < m < 1, and

if 0 < m < m ₀. Here, we have m ₀ = Ö eV₁/E@ = cosq ₀, and q ₀ is the maximum angle with respect to the negative x axis with which an electron can emerge from the plasma. This problem was solved in [24, 47].

The main results are illustrated by Figs. 6.24-6.26. The distribution function f(E,m ,0), normalized to unity

at the plasma boundary for m < 0, i.e., for the electrons emerging from the plasma, is presented in Fig. 6.24 [24]. The distribution function for positive values of m is obtained provided that f(E,m ,0) is an even function of m for 0 < m < m ₀. It is obvious that the angular asymmetry of the distribution function decreases as m ₀ increases, i.e., as the pre-electrode potential barrier V₁ increases. At values of m ₀ » 1, the distribution function is almost symmetric in the direction of the velocities.

We now consider the relationship between the flux and density at the plasma boundary, i.e., we calculate the type of boundary condition which must be used with the diffusion equation for the density n(E,x) in the plasma - so that the distribution n(E,x) obtained in this case may be as close as possible to the precise solution of the kinetic equation. Linear extrapolation of n(E,x) to the plasma boundary satisfies this condition.

An example of this extrapolation was presented in §6 for the case where m ₀ = 0. In this case, the extrapolated density (denoted as n(E,x) below) is written in the form

where y ₀(z) is taken from (6.9). An expression for y ₀(z), similar to (6.9), may also be obtained for the case of m ₀ > 0. The extrapolation length z₀, contained in these expressions, is dependent on m ₀. The dependence is shown in Fig. 6.25.

The boundary condition which links the electron flux for a given energy, i_pl(E), to the extrapolated density at the boundary

As was shown in §6, we have z₀ = 0.7104 and g ₁ = 0.47 at m ₀ = 0. When m ₀ ® 1 and the electron distribution function approaches the spherically asymmetric function, then g ₁ ® 1/4. The values of g ₁ for intermediate values of m ₀, according to [47], are presented in Fig. 6.26 (curve 1). The relative difference between the real density at the

plasma boundary, n(E,0), and its extrapolated value n_s(E,0), i.e., the value (n(E,0) - n_s(E,0))/ n_s(E,0), is also shown there.

The use of an extrapolation boundary condition provides rather high calculating accuracy. This is related to the fact that the asymptotic expressions of type (6.9) have exponential accuracy, i.e., corrections for the next order of smallness are proportional to exp(-x/l_e).

However, since the energy relaxation time t _E is considerably less for slow electrons, as V₁ decreases, deviation of the electron distribution function from a Maxwellian distribution (for emission from the plasma) becomes much less important. If the energy distribution function is assumed to be Maxwellian, then, by using the result of [24, 47], we can calculate the electron flux i_epl and energy flux S_epl from the plasma to the electrode. With this assumption, the following expressions are obtained for the limiting cases of eV₁ » kT_e and V₁ = 0:

where i_Te = 1/4n₀v_eexp(-eV₁), and V is the potential anode surface.

where D₀(v) = vc (v) and feM(v) is the Maxwellian distribution function (see formulas (4.2.1) and (4.2.5)). In this case, the asymmetry of the distribution function at the boundary is approximately described by the term D₀(v)cosq in the brackets in (7.9). We calculate the electron flux from the plasma through the retarding potential barrier by using the distribution function (7.9). By treating it in the same way as when deriving (3.8), we obtain

Integrating (7.10) over E from E₀ to ¥ , we obtain the total electron flux from the plasma:

The explicit expression for D₀(v) in the integrand of (7.10) is determined by the dependence of relaxation time t _p on energy (see (4.2.10)). In particular, if the electron mean free path (4.3.13) is not dependent on energy, then, in the absence of a magnetic field and temperature gradient, we have D₀ = const. In this case, by performing the integration in (7.10) and (7.11), we obtain