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Spatially dependent electron relaxation near a thermionic emitting electrode
Energy Conversion. Vol. 13, pp. 41-47. PergamonPress, 1973. Printed in Great Britain
Sp ially Dependent Electron Relaxation Near a Thermionic Emitting Electrode+ F. T. WU :§ and D. T. SHAW, (Received 12 June 1972)
Introduction
An investigation of the physical processes in the boundary plasma region near electrodes often requires a simultaneous solution of the Boltzmann transport equation for free electrons and the rate equations for bound electronic states. Much effort has been expended to obtain a self-consistent solution from the Boltzmann and rate equations in a homogeneous plasma [1-6]. However, little work has been done to extend these studies to investigate the spatially-dependent relaxation of electrons near an electrode surface. This problem is especially complicated in the boundary region near an emitting electrode because a rapid ionization build-up is taking place. In this case, a strong coupling between free electrons and bound electronic states is expected to play an important role in the relaxation of the electron energy distribution. A number of papers have appeared on the calculation of the spatially-dependent electron energy distribution function in weakly ionized plasmas [7-9]. Most of these papers have been based on at least one, or both, of the following two assumptions: (1) The interaction of the free-electron distribution and the bound electron distribution over the excited states can be completely ignored and (2) electron-electron Coulomb interactions and electron-neutral inelastic collisions have very little effect on the electron relaxation. These assumptions are usually justified more for mathematical convenience than practical realities, and their removal would be very desirable for a better understanding of the ionization relaxation near the emitter. Parker [6] has treated the problem of spatial relaxation of electrons. The solutions were obtained for the cases of energy-independent collision frequency and of energydependent cross-sections in terms of energy and spatial modes. Goldstein [8] and later Salinger and Rowe [9], have investigated the problem in an inert-gas field plasma diode by the use of Monte Carlo simulation. The major difference between reference [8] and [9] is that the latter considered a two-dimensional discharge while the former considered only a one-dimensional case. One of the major disadvantages of Monte Carlo t This research was supported in part by the National Science Foundation under Grant No. GK-1706 and GK-16372. School of Engin~ring and Applied Sciences, State University of New York at Buffalo, Buffalo, New York 14214, U.S.A. § Now at Comarco Engineering, Ridgecrest, California 93555, U.S.A. 5
41
technique is its inability to handle the electron-neutral excitation and ionization collisions. This is because the number of events of such collisions is very much lower than that of electron-neutral elastic and electronelectron Coulomb collisions; the number of events z~ so few that a meaningful estimate of the ionization rate is very difficult during the sampling processes [9]. Several papers have appeared lately concerning the nonequilibrium ionization and simultaneous free-electron relaxation. In the work by Stakhanov [10], the inelastic collision is neglected and the variation of the distribution is determined by Coulomb and electron-atom collisions. The Wiener-Hopf method is employed to solve the simplified Boltzmann equation. Sonin [11], on the other hand, has assumed that the electron-electron Coulomb collision can be dropped in the consideration of non-equilibrium ionization in thermionic converters. The importance of both the electron-electron Coulomb interactions and the electron-atom inelastic collision has been pointed out by Baksht et al. [12]. In their paper, a parameter 7ett is defined as the ratio of the probability of exciting an atom by fast electron to the probability of thermalization of the fast electron. When 7err ~ 1, we have the case of Stakhanov [I0] in which the Coulomb interaction is sufficiently frequent to mainrain the Maxwellian distribution. When 7e~f>~ 1, we have the case of Sonin [II] in which large deviations from the Maxwellian distribution can be expecited In most practical thermionic plasmas, the Coulomb .nteraction and the inelastic collision are two competing factors which should be considered simultaneously. The purpose of the present paper is to investigate the effects of elastic, inelastic, and electron-electron Coulomb collisions on the spatial rate of electron relaxation. The solution of the spatially-dependent electron energy distribution is used to evaluate the interactions between free electrons and the bound electrons, leading to a better understanding of the excitation and ionization relaxation processes near the electrode region in devices such as thermionic converters and MHD generators. Theory It is well known that one of the most difficult problems in the theoretical study of discharge phenomena is usually connected with the calculation of the electron energy distribution. Considering the steady-state relaxation in a spatially nonuniform plasma, the Boltzmann
42
F.T. WU and D. T. SHAW
equation, from which the electron energy distribution function f(x, v) should be determined, can be expressed
as
41r A2ffo) = - - | ne J
v2fo(vl) dvl.
0
vz ~. + - - - -m N
+
(.)
+
(1)
is"
The distribution function fo is defined in such a way that o0
f v2fo(x, v) dv
Here we have assumed tbr simplicity that the plasma 0 relaxation phenomena can be described as a one- gives the electron density ne(x) at position x. Substituting dimensional problem, x is the distance normal to the f l from Equation (3) into Equation (2), we obtain the electrode and v is the electron velocity in this direction. following equation forf0 The two terms on the left-hand side of the equation represent the change of the distribution function by V2 ~ [)t~1 (1)afo eE 0f0]] spatial diffusion and field acceleration respectively. 3 Ox - ~ + m ~-]J These changes must be balanced by the effect of collisions 1 0 eE on the right-hand side. The collision terms include elastic, electron-electron, and inelastic collisions. In this paper, we formulate the problem on the assumptions m l 0 ] that the distribution function is almost spherically = mg v20v fo symmetric in velocity space and therefore can be adequately represented by the first two terms of an expansion v20v ~eeVZA1 ~ + A2vfo + ~ f o . (4) in spherical harmonics which can be written in the following form. where the first term on the left-hand side represents the effect of the spatial variation of f l on the net flow of f(x, v) -----fo(x, v) q-fl(x, v) cos ¢. electrons into the volume element dxdv. The second When this is substituted into the Boltzmann equation term represents the net gain of energy from the field with elastic, inelastic and electron-electron Coulomb through the asymmetrical part of the distribution fl. interactions, it can be shown that the following equations These are balanced on the right-hand side of the equation by the energy loss to atoms in elastic collisions, energy can be obtained for f0 and f l [7, 13] exchange in electron--electron collisions, and energy loss in inelastic collisions. +Y: N A small, but important fraction of the electrons described in Equation (4) will cause transitions between m l a ( v 4 ) v bound electronic states. Such electron-impact transitions mg:av provide a strong coupling between the electron distribuover the excited states and the continuum. To +12~[Teevz (Al ~ + A~fo) ] (2) tions determine the effect of the free electron relaxation on the population of the excited states, we use the rate 8fo eE Ofo v equations which according to Bates, Kingston, and (3) +m £A. McWhirter [14], can be written in the form (6)
[V)ie,[ ~fo
In Equation (3), we have made the assumption that the effects of inelastic and electron--electron collisions on the symmetric part of the distribution function ft is small in comparison with that of elastic collisions. In Equations (2) and (3), e, n, v and x are the charge, mass, velocity, and position of an electron, respectively. ms is the mass of neutral atoms, E is the electric field. Aei and Axe, are respectively the elastic and inelastic mean-free-path. ~'ee is the electron--electron frequency defined by 4~rnee4 In [ ~ j . ")lee
m219 s
A1 and As are coefficients for electron-electron collisions defined by
O:o ]
-- N(p)nek(p, c) -- N(p)ne Y~ k(p, q)
-dtdN( p ) =
q*p
-- N(,p) ~ A(p, q) + n, "E N(q)k(p, q) q<~
q*P
+ ~ N(q)A(q,p) + n2,[nek(c,p) + #(p)]
(5) where N(p) is the density of atoms in level p, A(p, q) is the radiative decay rate from p to q, K(p, c) is the electron impact ionization rate from level p, K(c, p) is the threebody recombination rate to level p, fl(p) is the radiative recombination rate. K(p, q) is the impact excitation (de-excitation) rate from p to q and can be calculated by oO
Al(fo) = ~
v~fo(vx)dvl + vs 0
: t,
]
vlfo(vl) dvx
ne
g(p, q) = f Q(p, q)v~o(v) dv epq
(6)
where Q(p, q) is the cross-section corresponding to a
Spatially Dependent Electron Relmmtion Nest a Thermionie Emittlng Electrode
collisional transition from p to q. vv~ = [ 2 ( ~ q -
I0 o
e~)/m,]l/s,
~~
43
'
S~e-
i0-1 ~ is the energy of level p. When q is in the continuum, Equation (6) can also be used to calculate the impact ionization rate K(p, c). The semiciassical Gryzinski 10-2 cross-sections were used for the calculation of all electron impact ionization and excitation collisions [15]. The radiative recombination rates 3(P) were taken from reference [16] where an adjusted quantum defect method ._~ ]0-3 was used. The radiative decay rates A(p, q) were determined from the calculation of reference [17]. All reIG 4 sonance radiations are assumed to be completely trapped. g The cesium atomic model in the present paper is a 5-level model (6S, 6P, 5D, 7S, lumped) similar to the :5 to-5 one used by Dugan [3] and later Shaw [5, 6]. In this model, all excited states higher than 7S are combined into a single level. Various values of the binding energy go ,,-n id 6 eL and the statistical factor gL have been tried and the results are compared with the 26-level calculation of Norcross and Stone [17]. For the plasma conditions studied in this paper, we assume eL = 0"6 eV and g r . = 100. One of the difficulties in the kinetic theory description i0-8 of thermionic plasmas is the boundary condition near the electrode. Gaussian-type functions have been used by a number of authors [9, 18]. Such boundary condi! I I 2 tions are useful in the study of the effect of various Energy, eV collision processes on the relaxation rate but are inadequate in analyzing the interaction between free electrons Fig. I. Boundary distrilmtioa fuactions. S: Caimlated bo,md~ry and the bound states during the relaxation toward condition./) ---- 1 tort, n, = 10 ll era-s, E =ffi50 v/cm. D: equilibrium. Maxwellian functions have also been used Dmyvesteyn distrilmtion function with same mean mergy and same electron density. as the boundary distribution function [8]. The use of this boundary condition involves a number of difficulties. First of all, without any knowledge of the plasma vesteyn function with a very large depletion of the highstate near the boundary, it is difficult to assign a Maxenergy electrons due to the inelastic collisions--including wellian temperature. Furthermore, it is well known both excitation and ionization collisions. For cesium, that in the vicinity of the emitter, large gradients of the distribution begins to depart from the Maxwellian temperature, and potential exist. The thermionic electrons function (curve 2) at an energy of 1.4 eV which is the streaming into the plasma under combined effects of threshold energy for inelastic collisions. all these gradients cannot be truly Maxwellian [19]. The numerical procedure used in the present paper In this paper, the boundary condition is determined by has been described elsewhere and will not be repeated actually solving the Boltzmann equation based on a here [5, 18]. After the distribution is obtained from homogeneous plasma model [5]. Figure 1 shows the Equation (4), it is used in Equation (6) from which new results of such calculations. Here, the calculated bounvalues of the transition rates K(p, q) are evaluated. dary distribution (curve 1) is plotted together with a Then, from Equation (5), we can write five rate equations Maxwellian distribution (curve 2) and a Druyvesteyn for the states 6P, 5D, 7S, lumped, and continuum. distribution (curve 3). It is seen that the boundary They have the form given by distribution can be approximated by two-groups o f A N + FN(1) = B (8) electrons. The low-energy electrons look almost like a Maxwellian function with an effective temperature for 5-levels used in the present atomic model, A is a T* = 2(e)/3k where (~) is the average kinetic energy 4 X 4 matrix describing transitions between the energy defined by levels. B is a column matrix describing 3-body and co radiative recombinations. Fis a column matrix describing ( e) = m ~ f v4fo(v) dr. (7) impact excitation from the ground state, and N is a matrix of the population of all levels except the first 0
The high-energy electrons can be approximated by a Maxweltian distribution at a m u c h lower temperature. Over all, the distribution looks very much like a Druy-
ground
state (2 ~ p
~ 5).
From
obtain N = A-iB
-
-
A - i FN(1)
Equation
(8),
we
44
F.T. WU and D. T. SHAW
or
N ( p ) = R0(p) -- R~(p)N(1)
(2 ~
(9)
where R0 and Rx are independent of N(1). Using Equation (9), the rate of increase of electron density can be written as 5
drl~
N 0 ) n , ~ RI(p)K(p, c)
- -
dt
p=2 5 +
5
n~ Y, Ro(p)K(p, c)
--
n 26 ~ fl(p)
~:2
pf2
5
(10)
--n~ ~ K(c,p).
depression of the population of the high-energy electrons is somewhat overestimated. Results and Discussions
With the boundary distribution as shown in Fig. 1, the spatial relaxation of electrons in an ignited thermionic converter has been determined. As a numerical example, we consider here the relaxation of electrons streaming under the influence of the electric field (50 V/cm) in the sheath into the plasma where the electric field is assumed to be constant (2 V/cm). It should be mentioned once more that the distribution plotted in Figs. 1 and 2 is such that oo
f f(v) v 2 dv
An effective ionization coefficients and recombination coefficients can be defined by [6, 14, 17]
0
gives the local electron density ne(x). In Fig. 2 the
dne -- SneAr(1) -- o~t,s. dt
(11)
When Equation (10) is compared with Equation (11), we can write the effective recombination (,) and ionization (S) coefficients as [2, 4]
io-2
5
s =
y~ Rx(p)Kf:, c)
(12)
Maxwelllan
p=g
po-4
5
T~ ffi4810 *K
5
= ~ fl(p) + n, ~ K(c,p) ._o
10-6
5
_ l
~ Ro(p)K(p, c). (13)
T, : 1780 =K
ne p=2
It can be shown that in the steady-state, the rate of S and ~ must satisfy the following relation [20]
S
nz =
[2=mkT~ 3/2 -6 'kr =
:,
e
,,
i(~to
04) L~
When @e 1, the plasma is in local thermodynamic equilibrium; becomes the true Maxwellian temperature; and Equation (14) is identical to the Saha equation. At low electron temperatures and densities, the depletion of the population of excited state causes the value of n6 below the Saha's value even though the free electron distribution may remain to be Maxwellian. For such cases, ~e ~ 1. Thus, for given values of N(1) and Te, Equation (14) can be rewritten as ~be= ne/n,s where nes is the Saha electron density determined from Equation (14) when ~, = 1. It is appropriate here to discuss the physical constants used in the present paper. The value of Ael is calculated from an 'average' electron-neutral collision crosssection value of 400 A 2 [21]. Here, we have assumed that electron-neutral atom collisions provide the dominant resistance to electron transport in the plasma. Although electron-ion collisions have been shown to be important for cesium plasmas at high degrees of ionization [22], this assumption is justified in the present study because of the electron density is very low near the boundary. The inelastic mean free path Ax~r is determined by Ax~ = [27~ Q(p, q)N(p)]-l. For simplicity, the superelastic collisions are neglected. Thus the =
icy ~2
IC~14
i
i
i 2
¢ 3
Energy,
t 4
,,i
5
~'
",
.'
eV
Fig. 2. Calculateddistribution function at each position. Position 1, the emitter botmdary and position 10, the main plasma boundary.
spatially dependent electron energy distribution functions at the boundary (curve 1) and at nine other positions are shown. The distance between each two consecutive positions is about 1.3 times the electron mean-free-path. The effective temperature T* as defined by Equation (7) is determined primarily by the shape of the distribution of low-energy electrons. Thus, we see the effective temperature decreases rapidly from the value of 4810°K at the boundary to roughly 2000°K at 14 mean-frec-paths away (Fig. 3). At this position, the distribution is a Maxwellian straight line. It is interesting to note that at the boundary, there exists a large depression of the
Spatially Dependent Electron Relaxation Near a Thermionic Emitting Electrode 5000
4000
56
3000
o
cx
E I-
2000
~oooI I I l 2
I
l
3
4
I
I
6
5
I
I
8
7
9
perature (4810°K in Fig. 1). The values of K(p, q) determined from Equation (6) are plotted in Fig. 4. The initial decrease is due to the decrease in effective temperatures. The values of K(3, 6), K(2, 6), and K(1, 6) increase after some relaxation because of the filling-up of the Maxwellian tail discussed above. The positions of the minimum values of the K(p, q) curves moves away from the emitter as we go from the ground state to higher excited states. This implies that in the region where the distribution depression is serious (>1.4 eV), the higher energy part is relaxed at a much faster rate than the lower energy part. The reason for this is not completely clear at this stage. With the information presented above on the free electron distribution, the spatial relaxation of the population of excited states can now be studied. Using the five-level atomic model (Fig. 5) the population of State
Fig. 3. Electron temperature profile, / = 1"4 ~el where ~el is the electron-neutral a t o m mean-free-path.
Maxwellian tail. As the electrons move away from the boundary, the Maxwellian tail is filled up rapidly. Two processes are responsible for this increase in population of high-energy electrons. First, electron-electron Coulomb interactions are v e ~ effective in restoring Maxwellian function, especially at low electron energies. Second, electron-impact de-excitation collisions also have the effect of filling up the high-energy distribution tail. This effect is especially important near the boundary where the de-excitation events are much more frequent than the excitation events because the depressed highenergy part of the distribution has a very low effective temperature (1200°K in Fig. 1); while the low-energy part of the distribution has a very high effective tern-
10 -7 - -
~
- - io-IO
K(5,6) ~D 10-B
I
olo
10-9
- -
oo[
I0-2o
o 10-22
10-11[~X 10-12
~
~
K3-23
¢D
"6 13 ~
"~{'(" ( 2,6,
u
t-
10-14 ~ io-m
10-25 I
2
I
3
1
4
1
t
5
6
T
T
I
8
I
9
I
10
t
Fig. 4. Profiles o f transition rate.
45
g/.
Continuum
2
Lumped
100
7S
2
5D
2
6P
6
6.5"
2
•1'0 --
q~
2.0--
h5 3.0
4'0--
Fig. 5. Simplified energy levels of cesium atom.
each state has been calculated in three different ways (Fig. 6). Based on the calculated effective temperature shown in Fig. 3, the population of each state can be obtained from the Boltzmann relation
N (p ) _ g~ e_(%_~q)/kT~, N(q) gq where g~ is the statistical factor p-state. Since the population is expected to depart from the Boltzmann equilibrium at low free-electron densities, the non-Boltzmann population is determined from Equation (9) with the values of K(p, q) calculated from Equation (6). Fig. 7 shows the spatial variation of F which is defined as the ratio of the calculated non-Boltzmann population to the Boltzmann-equilibrium population. The strong coupling between the free electron and the bound electrons is clearly shown. At the position roughly 4 or 5 mean-free-paths away, the free-electron distribution approaches that of Maxwellian and the boundelectron distribution approaches that of the Boltzmannequilibrium value. For comparison, we also plotted the Boltzmann population of each state using the temperature after the electron distribution is completely Maxwellized (Te = 1800°K). It is obvious that the population is
46
F.T. WU and D. T. SHAW i0 lc6S
I0)6 I 10)5[--
1018
1014 I017
0
iOf3 .o~
c 1012 o o
1016
c .9 10)5
~- loll
o io)0
1014
b
io 9 1013 IOB 1012 107 t ,,
I 2
I
I 3
I 4
I 5
I 6
I ---I----1-7 8 9
2
I 10
L
3
4
.5
6
7
8
9
I0
F i g . 8. R a t e s o f i o n i z a t i o n s , p = 1 t o n ' . ( a ) C a l c u l a t e d , (b)
Fig. 6. Populations of each individual state ;p = 1 t o r r . [a] C a l c u l a t e d values, (b) Boltzmann factor with effective temperature, (e) Boitzmann factor with frozen temperature ( T e = 1800°K).
cos 6P 5D --7S
I0 0 .,.,,,
f
Bolt~mmm factor for excited states with effective temt~rature, (c) Boltzmnn factor for excited states w i t h fl'ozen tempe~tore. ioo
1o-I
v
i0-1
f
~r2 Lumped
10-2
f
io~
l
2
I
3
I
I
4
5
)
6
I
7
I
I
8
9
I0
10-3 F i g . 9. C a l e a i n t e d
s p a t i a l profile o f ~e(Te).
io ~3
1o-4
I
2
I
3
I
4
I
I
5
6
1
7
I
8
I
9
1
I0
(
Fig. 7. Spatial variation of profiles of r values. seriously underestimated. In Fig. 8, the spatial variations of the ionization rate are determined based on the
values of the three types of the bound-electron populations shown in Fig. 6 and the values of K(p, q) in Fig. 4. An examination of this figure shows the calculated ionization rate lies between those determined from the local effective temperature and from the temperature measured at some distance away from the emitter. In Fig. 9, the value of ~be as defined in Equation (13) is determined from the calculations of the values of S and ~ defined in Equations (12) and (13), respectively. Here, we see that the electron density departs significantly from the Saha-equilibrium value near the emitter.
"~
1012
rn
i0 )t
I
2
I
3
I
I
4
5
I
6
1
7
I
8
I
9
l
Fig. 10. Electron density profile, p = 1 tort.
IO
Spatially Dependent Electron Relaxation Near a Thermionic Rmitting Electrode The spatial build-up of the electron density obtained by the integration of the distribution function is shown in Fig. 10. Reichelt [23] has determined spectroscopically the electron temperature and density profiles. In his experiment the electron density was determined by the line broadening measurements; while the temperature was determined from measuring the relative intensity of the radiative recombination spectrum. Intensity measurements were in the range from 5000°A to 4000°A which roughly corresponds to the electron energy range from 0 to 0"7 eV. His measurements showed a steady decrease of the electron temperature determined in this lowenergy region. This is in agreement with the present theory which predicts a decrease of the effective temperature based on the distribution of low-energy electrons. It should be noted that the conclusion about the decrease of the effective temperature as defined in Equation (7) does not necessarily depend on the condition of electron density build-up as shown in Fig. 10. In fact, even if the electron density decreases due to the conditions at the collector--as is the case shown in Reichelt's paper--the temperature of the low-energy part of the distribution would still relax toward a Maxwellian function at a lower temperature just as shown in Fig. (3). The existence of a group of high-energy electrons with a much lower Maxwellian temperature near the emitter was shown by Baksht et al. [24] using electric probe techniques. Figure 11 shows their typical results 5000
'E 6
t~ N
_o
4000
o,.
E. I--
3000
i
_g m
I
0-2
1
0"4
I
0"6
I
0"8
I'0
Fig. 11. Spatial profiles of electron temperature and electron density of Ref. [19].
47
reproduced here for comparison. As seen in this figure, near the emitter there are two electron temperatures which are determined from the change of the slope of the probe current characteristics. The upper curve corresponds to low-energy electrons and the lower curve corresponds to high-energy electrons. The two temperatures merge at a distance roughly 0.35 mm away from the emitter. Figure 11 agrees qualitatively with the temperature and density profiles shown in Figs. 3 and 10. In Fig. 3, the dashed-line presents the spatial variation of the temperature determined from the slope of the high-energy ( > 1.4 eV) part of the distribution functions shown in Fig. 2. Although the above comparison at best is only semi-quantitative because of the lack of experimental data on the plasma conditions at the surface boundary, the probe measurements shown in Fig. 11 strongly support the present theory on the two-group electron distribution and its effects on the spatial relaxation of the bound-electronic states. References
[ll J. F. Shaw, C. H. Kurger, M. Mitcber and J. R. Viegas, Electricity from MHD II, 77. International Atomic Energy Agency, Vienna (1966). [2] J. F. Shaw, M. Mitcher and C. H. Druger, Physics Fluids 13, 325 (1970). [3] J. Dugan, Jr., F. A. Lyman and L. V. Albers, Electricity from MHD, II, 85. International Atomic Energy Agency, Vienna (1966). [4] E. V. Sonin, Soviet Phys. tech. Phys. 13, 319 (1968). [5] D. T. Shaw and S. G. Margolis, J. appl. Phys. 40, 4377 (1969). [6] D. T. Shaw and F. T. Wu, Energy Conversion, 11, 114 (1971). [7] J. H. Parker, Jr., Phys. Rev. 132, 2096 (1963); 139, A1792 (1965). [8] C. M. Goldstein, J. appl. Phys. 38, 2977 (1967). [9] S. N. Salinger and J. E. Rowe, J. appl. Phys. 39, 3933 (1968). [10] I. P. Stakhanov, Soviet Phys. tech. Phys. 12, 7, 929 (1968); 12, 7, 935 (1968); 12, 11, 1522 (1968). [11] E. V. Sonin, Soviet Phys. tech. Phys. 13, 3, 319 (1968). [12] F. G. Baksht, B. Ya. Moizhes and V. A. Nemchinskii, Soviet Phys. tech. Phys. 13, 10, 1401 (1969). [13] The detailed derivation of this equation can be found in the article by W. P. Allis in Handbook of Physics, edited by S. Flugge. Springer (1956). [14] D. R. Bates, A. E. Kingston and R. W. P. McWhirter, Proc. R. Soc. A267, 297 (1962). [15] M. Gryzinski, Phys. Rev. 115, 374 (1959). [16] D. W, Norcross and P. M. Stone, J. Quant. Spectrosc. had. Trans. 6, 277 (1966). [17] D. W. Norcross and P. M. Stone, J. Quant. Spectrosc. Rad. Trans. 8, 665 (1968). [18] F. T. Wu, Ph.D. Thesis, State University of New York at Buffalo (1970). [19] W.B. Nottingham, Advanced Energy Conversion2, 467 (1962). [20] D. Wflkins and E. P. Gyitopoulas, J. appk Phys. 37, 2892 (1966). [21] J. M. Houston, Conference Record Tbermionic Conversion Specialist Conference, p. 300 (1964). [22] W. Nighan, Physics Fluids 10, 1085 (1967). [23] W. R. Reichelt, Appl. Phys. Lett. 14, 382 (1969). [24] F. G. Baksht, G. A. Dyuzhev et al., Soviet Phys. tech. Phys. 13, 893 (1969).
Sp ially Dependent Electron Relaxation Near a Thermionic Emitting Electrode+ F. T. WU :§ and D. T. SHAW, (Received 12 June 1972)
Introduction
An investigation of the physical processes in the boundary plasma region near electrodes often requires a simultaneous solution of the Boltzmann transport equation for free electrons and the rate equations for bound electronic states. Much effort has been expended to obtain a self-consistent solution from the Boltzmann and rate equations in a homogeneous plasma [1-6]. However, little work has been done to extend these studies to investigate the spatially-dependent relaxation of electrons near an electrode surface. This problem is especially complicated in the boundary region near an emitting electrode because a rapid ionization build-up is taking place. In this case, a strong coupling between free electrons and bound electronic states is expected to play an important role in the relaxation of the electron energy distribution. A number of papers have appeared on the calculation of the spatially-dependent electron energy distribution function in weakly ionized plasmas [7-9]. Most of these papers have been based on at least one, or both, of the following two assumptions: (1) The interaction of the free-electron distribution and the bound electron distribution over the excited states can be completely ignored and (2) electron-electron Coulomb interactions and electron-neutral inelastic collisions have very little effect on the electron relaxation. These assumptions are usually justified more for mathematical convenience than practical realities, and their removal would be very desirable for a better understanding of the ionization relaxation near the emitter. Parker [6] has treated the problem of spatial relaxation of electrons. The solutions were obtained for the cases of energy-independent collision frequency and of energydependent cross-sections in terms of energy and spatial modes. Goldstein [8] and later Salinger and Rowe [9], have investigated the problem in an inert-gas field plasma diode by the use of Monte Carlo simulation. The major difference between reference [8] and [9] is that the latter considered a two-dimensional discharge while the former considered only a one-dimensional case. One of the major disadvantages of Monte Carlo t This research was supported in part by the National Science Foundation under Grant No. GK-1706 and GK-16372. School of Engin~ring and Applied Sciences, State University of New York at Buffalo, Buffalo, New York 14214, U.S.A. § Now at Comarco Engineering, Ridgecrest, California 93555, U.S.A. 5
41
technique is its inability to handle the electron-neutral excitation and ionization collisions. This is because the number of events of such collisions is very much lower than that of electron-neutral elastic and electronelectron Coulomb collisions; the number of events z~ so few that a meaningful estimate of the ionization rate is very difficult during the sampling processes [9]. Several papers have appeared lately concerning the nonequilibrium ionization and simultaneous free-electron relaxation. In the work by Stakhanov [10], the inelastic collision is neglected and the variation of the distribution is determined by Coulomb and electron-atom collisions. The Wiener-Hopf method is employed to solve the simplified Boltzmann equation. Sonin [11], on the other hand, has assumed that the electron-electron Coulomb collision can be dropped in the consideration of non-equilibrium ionization in thermionic converters. The importance of both the electron-electron Coulomb interactions and the electron-atom inelastic collision has been pointed out by Baksht et al. [12]. In their paper, a parameter 7ett is defined as the ratio of the probability of exciting an atom by fast electron to the probability of thermalization of the fast electron. When 7err ~ 1, we have the case of Stakhanov [I0] in which the Coulomb interaction is sufficiently frequent to mainrain the Maxwellian distribution. When 7e~f>~ 1, we have the case of Sonin [II] in which large deviations from the Maxwellian distribution can be expecited In most practical thermionic plasmas, the Coulomb .nteraction and the inelastic collision are two competing factors which should be considered simultaneously. The purpose of the present paper is to investigate the effects of elastic, inelastic, and electron-electron Coulomb collisions on the spatial rate of electron relaxation. The solution of the spatially-dependent electron energy distribution is used to evaluate the interactions between free electrons and the bound electrons, leading to a better understanding of the excitation and ionization relaxation processes near the electrode region in devices such as thermionic converters and MHD generators. Theory It is well known that one of the most difficult problems in the theoretical study of discharge phenomena is usually connected with the calculation of the electron energy distribution. Considering the steady-state relaxation in a spatially nonuniform plasma, the Boltzmann
42
F.T. WU and D. T. SHAW
equation, from which the electron energy distribution function f(x, v) should be determined, can be expressed
as
41r A2ffo) = - - | ne J
v2fo(vl) dvl.
0
vz ~. + - - - -m N
+
(.)
+
(1)
is"
The distribution function fo is defined in such a way that o0
f v2fo(x, v) dv
Here we have assumed tbr simplicity that the plasma 0 relaxation phenomena can be described as a one- gives the electron density ne(x) at position x. Substituting dimensional problem, x is the distance normal to the f l from Equation (3) into Equation (2), we obtain the electrode and v is the electron velocity in this direction. following equation forf0 The two terms on the left-hand side of the equation represent the change of the distribution function by V2 ~ [)t~1 (1)afo eE 0f0]] spatial diffusion and field acceleration respectively. 3 Ox - ~ + m ~-]J These changes must be balanced by the effect of collisions 1 0 eE on the right-hand side. The collision terms include elastic, electron-electron, and inelastic collisions. In this paper, we formulate the problem on the assumptions m l 0 ] that the distribution function is almost spherically = mg v20v fo symmetric in velocity space and therefore can be adequately represented by the first two terms of an expansion v20v ~eeVZA1 ~ + A2vfo + ~ f o . (4) in spherical harmonics which can be written in the following form. where the first term on the left-hand side represents the effect of the spatial variation of f l on the net flow of f(x, v) -----fo(x, v) q-fl(x, v) cos ¢. electrons into the volume element dxdv. The second When this is substituted into the Boltzmann equation term represents the net gain of energy from the field with elastic, inelastic and electron-electron Coulomb through the asymmetrical part of the distribution fl. interactions, it can be shown that the following equations These are balanced on the right-hand side of the equation by the energy loss to atoms in elastic collisions, energy can be obtained for f0 and f l [7, 13] exchange in electron--electron collisions, and energy loss in inelastic collisions. +Y: N A small, but important fraction of the electrons described in Equation (4) will cause transitions between m l a ( v 4 ) v bound electronic states. Such electron-impact transitions mg:av provide a strong coupling between the electron distribuover the excited states and the continuum. To +12~[Teevz (Al ~ + A~fo) ] (2) tions determine the effect of the free electron relaxation on the population of the excited states, we use the rate 8fo eE Ofo v equations which according to Bates, Kingston, and (3) +m £A. McWhirter [14], can be written in the form (6)
[V)ie,[ ~fo
In Equation (3), we have made the assumption that the effects of inelastic and electron--electron collisions on the symmetric part of the distribution function ft is small in comparison with that of elastic collisions. In Equations (2) and (3), e, n, v and x are the charge, mass, velocity, and position of an electron, respectively. ms is the mass of neutral atoms, E is the electric field. Aei and Axe, are respectively the elastic and inelastic mean-free-path. ~'ee is the electron--electron frequency defined by 4~rnee4 In [ ~ j . ")lee
m219 s
A1 and As are coefficients for electron-electron collisions defined by
O:o ]
-- N(p)nek(p, c) -- N(p)ne Y~ k(p, q)
-dtdN( p ) =
q*p
-- N(,p) ~ A(p, q) + n, "E N(q)k(p, q) q<~
q*P
+ ~ N(q)A(q,p) + n2,[nek(c,p) + #(p)]
(5) where N(p) is the density of atoms in level p, A(p, q) is the radiative decay rate from p to q, K(p, c) is the electron impact ionization rate from level p, K(c, p) is the threebody recombination rate to level p, fl(p) is the radiative recombination rate. K(p, q) is the impact excitation (de-excitation) rate from p to q and can be calculated by oO
Al(fo) = ~
v~fo(vx)dvl + vs 0
: t,
]
vlfo(vl) dvx
ne
g(p, q) = f Q(p, q)v~o(v) dv epq
(6)
where Q(p, q) is the cross-section corresponding to a
Spatially Dependent Electron Relmmtion Nest a Thermionie Emittlng Electrode
collisional transition from p to q. vv~ = [ 2 ( ~ q -
I0 o
e~)/m,]l/s,
~~
43
'
S~e-
i0-1 ~ is the energy of level p. When q is in the continuum, Equation (6) can also be used to calculate the impact ionization rate K(p, c). The semiciassical Gryzinski 10-2 cross-sections were used for the calculation of all electron impact ionization and excitation collisions [15]. The radiative recombination rates 3(P) were taken from reference [16] where an adjusted quantum defect method ._~ ]0-3 was used. The radiative decay rates A(p, q) were determined from the calculation of reference [17]. All reIG 4 sonance radiations are assumed to be completely trapped. g The cesium atomic model in the present paper is a 5-level model (6S, 6P, 5D, 7S, lumped) similar to the :5 to-5 one used by Dugan [3] and later Shaw [5, 6]. In this model, all excited states higher than 7S are combined into a single level. Various values of the binding energy go ,,-n id 6 eL and the statistical factor gL have been tried and the results are compared with the 26-level calculation of Norcross and Stone [17]. For the plasma conditions studied in this paper, we assume eL = 0"6 eV and g r . = 100. One of the difficulties in the kinetic theory description i0-8 of thermionic plasmas is the boundary condition near the electrode. Gaussian-type functions have been used by a number of authors [9, 18]. Such boundary condi! I I 2 tions are useful in the study of the effect of various Energy, eV collision processes on the relaxation rate but are inadequate in analyzing the interaction between free electrons Fig. I. Boundary distrilmtioa fuactions. S: Caimlated bo,md~ry and the bound states during the relaxation toward condition./) ---- 1 tort, n, = 10 ll era-s, E =ffi50 v/cm. D: equilibrium. Maxwellian functions have also been used Dmyvesteyn distrilmtion function with same mean mergy and same electron density. as the boundary distribution function [8]. The use of this boundary condition involves a number of difficulties. First of all, without any knowledge of the plasma vesteyn function with a very large depletion of the highstate near the boundary, it is difficult to assign a Maxenergy electrons due to the inelastic collisions--including wellian temperature. Furthermore, it is well known both excitation and ionization collisions. For cesium, that in the vicinity of the emitter, large gradients of the distribution begins to depart from the Maxwellian temperature, and potential exist. The thermionic electrons function (curve 2) at an energy of 1.4 eV which is the streaming into the plasma under combined effects of threshold energy for inelastic collisions. all these gradients cannot be truly Maxwellian [19]. The numerical procedure used in the present paper In this paper, the boundary condition is determined by has been described elsewhere and will not be repeated actually solving the Boltzmann equation based on a here [5, 18]. After the distribution is obtained from homogeneous plasma model [5]. Figure 1 shows the Equation (4), it is used in Equation (6) from which new results of such calculations. Here, the calculated bounvalues of the transition rates K(p, q) are evaluated. dary distribution (curve 1) is plotted together with a Then, from Equation (5), we can write five rate equations Maxwellian distribution (curve 2) and a Druyvesteyn for the states 6P, 5D, 7S, lumped, and continuum. distribution (curve 3). It is seen that the boundary They have the form given by distribution can be approximated by two-groups o f A N + FN(1) = B (8) electrons. The low-energy electrons look almost like a Maxwellian function with an effective temperature for 5-levels used in the present atomic model, A is a T* = 2(e)/3k where (~) is the average kinetic energy 4 X 4 matrix describing transitions between the energy defined by levels. B is a column matrix describing 3-body and co radiative recombinations. Fis a column matrix describing ( e) = m ~ f v4fo(v) dr. (7) impact excitation from the ground state, and N is a matrix of the population of all levels except the first 0
The high-energy electrons can be approximated by a Maxweltian distribution at a m u c h lower temperature. Over all, the distribution looks very much like a Druy-
ground
state (2 ~ p
~ 5).
From
obtain N = A-iB
-
-
A - i FN(1)
Equation
(8),
we
44
F.T. WU and D. T. SHAW
or
N ( p ) = R0(p) -- R~(p)N(1)
(2 ~
(9)
where R0 and Rx are independent of N(1). Using Equation (9), the rate of increase of electron density can be written as 5
drl~
N 0 ) n , ~ RI(p)K(p, c)
- -
dt
p=2 5 +
5
n~ Y, Ro(p)K(p, c)
--
n 26 ~ fl(p)
~:2
pf2
5
(10)
--n~ ~ K(c,p).
depression of the population of the high-energy electrons is somewhat overestimated. Results and Discussions
With the boundary distribution as shown in Fig. 1, the spatial relaxation of electrons in an ignited thermionic converter has been determined. As a numerical example, we consider here the relaxation of electrons streaming under the influence of the electric field (50 V/cm) in the sheath into the plasma where the electric field is assumed to be constant (2 V/cm). It should be mentioned once more that the distribution plotted in Figs. 1 and 2 is such that oo
f f(v) v 2 dv
An effective ionization coefficients and recombination coefficients can be defined by [6, 14, 17]
0
gives the local electron density ne(x). In Fig. 2 the
dne -- SneAr(1) -- o~t,s. dt
(11)
When Equation (10) is compared with Equation (11), we can write the effective recombination (,) and ionization (S) coefficients as [2, 4]
io-2
5
s =
y~ Rx(p)Kf:, c)
(12)
Maxwelllan
p=g
po-4
5
T~ ffi4810 *K
5
= ~ fl(p) + n, ~ K(c,p) ._o
10-6
5
_ l
~ Ro(p)K(p, c). (13)
T, : 1780 =K
ne p=2
It can be shown that in the steady-state, the rate of S and ~ must satisfy the following relation [20]
S
nz =
[2=mkT~ 3/2 -6 'kr =
:,
e
,,
i(~to
04) L~
When @e 1, the plasma is in local thermodynamic equilibrium; becomes the true Maxwellian temperature; and Equation (14) is identical to the Saha equation. At low electron temperatures and densities, the depletion of the population of excited state causes the value of n6 below the Saha's value even though the free electron distribution may remain to be Maxwellian. For such cases, ~e ~ 1. Thus, for given values of N(1) and Te, Equation (14) can be rewritten as ~be= ne/n,s where nes is the Saha electron density determined from Equation (14) when ~, = 1. It is appropriate here to discuss the physical constants used in the present paper. The value of Ael is calculated from an 'average' electron-neutral collision crosssection value of 400 A 2 [21]. Here, we have assumed that electron-neutral atom collisions provide the dominant resistance to electron transport in the plasma. Although electron-ion collisions have been shown to be important for cesium plasmas at high degrees of ionization [22], this assumption is justified in the present study because of the electron density is very low near the boundary. The inelastic mean free path Ax~r is determined by Ax~ = [27~ Q(p, q)N(p)]-l. For simplicity, the superelastic collisions are neglected. Thus the =
icy ~2
IC~14
i
i
i 2
¢ 3
Energy,
t 4
,,i
5
~'
",
.'
eV
Fig. 2. Calculateddistribution function at each position. Position 1, the emitter botmdary and position 10, the main plasma boundary.
spatially dependent electron energy distribution functions at the boundary (curve 1) and at nine other positions are shown. The distance between each two consecutive positions is about 1.3 times the electron mean-free-path. The effective temperature T* as defined by Equation (7) is determined primarily by the shape of the distribution of low-energy electrons. Thus, we see the effective temperature decreases rapidly from the value of 4810°K at the boundary to roughly 2000°K at 14 mean-frec-paths away (Fig. 3). At this position, the distribution is a Maxwellian straight line. It is interesting to note that at the boundary, there exists a large depression of the
Spatially Dependent Electron Relaxation Near a Thermionic Emitting Electrode 5000
4000
56
3000
o
cx
E I-
2000
~oooI I I l 2
I
l
3
4
I
I
6
5
I
I
8
7
9
perature (4810°K in Fig. 1). The values of K(p, q) determined from Equation (6) are plotted in Fig. 4. The initial decrease is due to the decrease in effective temperatures. The values of K(3, 6), K(2, 6), and K(1, 6) increase after some relaxation because of the filling-up of the Maxwellian tail discussed above. The positions of the minimum values of the K(p, q) curves moves away from the emitter as we go from the ground state to higher excited states. This implies that in the region where the distribution depression is serious (>1.4 eV), the higher energy part is relaxed at a much faster rate than the lower energy part. The reason for this is not completely clear at this stage. With the information presented above on the free electron distribution, the spatial relaxation of the population of excited states can now be studied. Using the five-level atomic model (Fig. 5) the population of State
Fig. 3. Electron temperature profile, / = 1"4 ~el where ~el is the electron-neutral a t o m mean-free-path.
Maxwellian tail. As the electrons move away from the boundary, the Maxwellian tail is filled up rapidly. Two processes are responsible for this increase in population of high-energy electrons. First, electron-electron Coulomb interactions are v e ~ effective in restoring Maxwellian function, especially at low electron energies. Second, electron-impact de-excitation collisions also have the effect of filling up the high-energy distribution tail. This effect is especially important near the boundary where the de-excitation events are much more frequent than the excitation events because the depressed highenergy part of the distribution has a very low effective temperature (1200°K in Fig. 1); while the low-energy part of the distribution has a very high effective tern-
10 -7 - -
~
- - io-IO
K(5,6) ~D 10-B
I
olo
10-9
- -
oo[
I0-2o
o 10-22
10-11[~X 10-12
~
~
K3-23
¢D
"6 13 ~
"~{'(" ( 2,6,
u
t-
10-14 ~ io-m
10-25 I
2
I
3
1
4
1
t
5
6
T
T
I
8
I
9
I
10
t
Fig. 4. Profiles o f transition rate.
45
g/.
Continuum
2
Lumped
100
7S
2
5D
2
6P
6
6.5"
2
•1'0 --
q~
2.0--
h5 3.0
4'0--
Fig. 5. Simplified energy levels of cesium atom.
each state has been calculated in three different ways (Fig. 6). Based on the calculated effective temperature shown in Fig. 3, the population of each state can be obtained from the Boltzmann relation
N (p ) _ g~ e_(%_~q)/kT~, N(q) gq where g~ is the statistical factor p-state. Since the population is expected to depart from the Boltzmann equilibrium at low free-electron densities, the non-Boltzmann population is determined from Equation (9) with the values of K(p, q) calculated from Equation (6). Fig. 7 shows the spatial variation of F which is defined as the ratio of the calculated non-Boltzmann population to the Boltzmann-equilibrium population. The strong coupling between the free electron and the bound electrons is clearly shown. At the position roughly 4 or 5 mean-free-paths away, the free-electron distribution approaches that of Maxwellian and the boundelectron distribution approaches that of the Boltzmannequilibrium value. For comparison, we also plotted the Boltzmann population of each state using the temperature after the electron distribution is completely Maxwellized (Te = 1800°K). It is obvious that the population is
46
F.T. WU and D. T. SHAW i0 lc6S
I0)6 I 10)5[--
1018
1014 I017
0
iOf3 .o~
c 1012 o o
1016
c .9 10)5
~- loll
o io)0
1014
b
io 9 1013 IOB 1012 107 t ,,
I 2
I
I 3
I 4
I 5
I 6
I ---I----1-7 8 9
2
I 10
L
3
4
.5
6
7
8
9
I0
F i g . 8. R a t e s o f i o n i z a t i o n s , p = 1 t o n ' . ( a ) C a l c u l a t e d , (b)
Fig. 6. Populations of each individual state ;p = 1 t o r r . [a] C a l c u l a t e d values, (b) Boltzmann factor with effective temperature, (e) Boitzmann factor with frozen temperature ( T e = 1800°K).
cos 6P 5D --7S
I0 0 .,.,,,
f
Bolt~mmm factor for excited states with effective temt~rature, (c) Boltzmnn factor for excited states w i t h fl'ozen tempe~tore. ioo
1o-I
v
i0-1
f
~r2 Lumped
10-2
f
io~
l
2
I
3
I
I
4
5
)
6
I
7
I
I
8
9
I0
10-3 F i g . 9. C a l e a i n t e d
s p a t i a l profile o f ~e(Te).
io ~3
1o-4
I
2
I
3
I
4
I
I
5
6
1
7
I
8
I
9
1
I0
(
Fig. 7. Spatial variation of profiles of r values. seriously underestimated. In Fig. 8, the spatial variations of the ionization rate are determined based on the
values of the three types of the bound-electron populations shown in Fig. 6 and the values of K(p, q) in Fig. 4. An examination of this figure shows the calculated ionization rate lies between those determined from the local effective temperature and from the temperature measured at some distance away from the emitter. In Fig. 9, the value of ~be as defined in Equation (13) is determined from the calculations of the values of S and ~ defined in Equations (12) and (13), respectively. Here, we see that the electron density departs significantly from the Saha-equilibrium value near the emitter.
"~
1012
rn
i0 )t
I
2
I
3
I
I
4
5
I
6
1
7
I
8
I
9
l
Fig. 10. Electron density profile, p = 1 tort.
IO
Spatially Dependent Electron Relaxation Near a Thermionic Rmitting Electrode The spatial build-up of the electron density obtained by the integration of the distribution function is shown in Fig. 10. Reichelt [23] has determined spectroscopically the electron temperature and density profiles. In his experiment the electron density was determined by the line broadening measurements; while the temperature was determined from measuring the relative intensity of the radiative recombination spectrum. Intensity measurements were in the range from 5000°A to 4000°A which roughly corresponds to the electron energy range from 0 to 0"7 eV. His measurements showed a steady decrease of the electron temperature determined in this lowenergy region. This is in agreement with the present theory which predicts a decrease of the effective temperature based on the distribution of low-energy electrons. It should be noted that the conclusion about the decrease of the effective temperature as defined in Equation (7) does not necessarily depend on the condition of electron density build-up as shown in Fig. 10. In fact, even if the electron density decreases due to the conditions at the collector--as is the case shown in Reichelt's paper--the temperature of the low-energy part of the distribution would still relax toward a Maxwellian function at a lower temperature just as shown in Fig. (3). The existence of a group of high-energy electrons with a much lower Maxwellian temperature near the emitter was shown by Baksht et al. [24] using electric probe techniques. Figure 11 shows their typical results 5000
'E 6
t~ N
_o
4000
o,.
E. I--
3000
i
_g m
I
0-2
1
0"4
I
0"6
I
0"8
I'0
Fig. 11. Spatial profiles of electron temperature and electron density of Ref. [19].
47
reproduced here for comparison. As seen in this figure, near the emitter there are two electron temperatures which are determined from the change of the slope of the probe current characteristics. The upper curve corresponds to low-energy electrons and the lower curve corresponds to high-energy electrons. The two temperatures merge at a distance roughly 0.35 mm away from the emitter. Figure 11 agrees qualitatively with the temperature and density profiles shown in Figs. 3 and 10. In Fig. 3, the dashed-line presents the spatial variation of the temperature determined from the slope of the high-energy ( > 1.4 eV) part of the distribution functions shown in Fig. 2. Although the above comparison at best is only semi-quantitative because of the lack of experimental data on the plasma conditions at the surface boundary, the probe measurements shown in Fig. 11 strongly support the present theory on the two-group electron distribution and its effects on the spatial relaxation of the bound-electronic states. References
[ll J. F. Shaw, C. H. Kurger, M. Mitcber and J. R. Viegas, Electricity from MHD II, 77. International Atomic Energy Agency, Vienna (1966). [2] J. F. Shaw, M. Mitcher and C. H. Druger, Physics Fluids 13, 325 (1970). [3] J. Dugan, Jr., F. A. Lyman and L. V. Albers, Electricity from MHD, II, 85. International Atomic Energy Agency, Vienna (1966). [4] E. V. Sonin, Soviet Phys. tech. Phys. 13, 319 (1968). [5] D. T. Shaw and S. G. Margolis, J. appl. Phys. 40, 4377 (1969). [6] D. T. Shaw and F. T. Wu, Energy Conversion, 11, 114 (1971). [7] J. H. Parker, Jr., Phys. Rev. 132, 2096 (1963); 139, A1792 (1965). [8] C. M. Goldstein, J. appl. Phys. 38, 2977 (1967). [9] S. N. Salinger and J. E. Rowe, J. appl. Phys. 39, 3933 (1968). [10] I. P. Stakhanov, Soviet Phys. tech. Phys. 12, 7, 929 (1968); 12, 7, 935 (1968); 12, 11, 1522 (1968). [11] E. V. Sonin, Soviet Phys. tech. Phys. 13, 3, 319 (1968). [12] F. G. Baksht, B. Ya. Moizhes and V. A. Nemchinskii, Soviet Phys. tech. Phys. 13, 10, 1401 (1969). [13] The detailed derivation of this equation can be found in the article by W. P. Allis in Handbook of Physics, edited by S. Flugge. Springer (1956). [14] D. R. Bates, A. E. Kingston and R. W. P. McWhirter, Proc. R. Soc. A267, 297 (1962). [15] M. Gryzinski, Phys. Rev. 115, 374 (1959). [16] D. W, Norcross and P. M. Stone, J. Quant. Spectrosc. had. Trans. 6, 277 (1966). [17] D. W. Norcross and P. M. Stone, J. Quant. Spectrosc. Rad. Trans. 8, 665 (1968). [18] F. T. Wu, Ph.D. Thesis, State University of New York at Buffalo (1970). [19] W.B. Nottingham, Advanced Energy Conversion2, 467 (1962). [20] D. Wflkins and E. P. Gyitopoulas, J. appk Phys. 37, 2892 (1966). [21] J. M. Houston, Conference Record Tbermionic Conversion Specialist Conference, p. 300 (1964). [22] W. Nighan, Physics Fluids 10, 1085 (1967). [23] W. R. Reichelt, Appl. Phys. Lett. 14, 382 (1969). [24] F. G. Baksht, G. A. Dyuzhev et al., Soviet Phys. tech. Phys. 13, 893 (1969).