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Modified Bohm criterion for a collisional plasma
Volume 89A, number 2
PHYSICS LETTERS
26 April 1982
MODIFIED BOHM CRITERION FOR A COLLISIONAL PLASMA V.A. GODYAK Moscow State University, Physical Department, Moscow, USSR Received 14 October 1981
A simple boundary condition is suggested to ensure regularity in a hydrodynamical model of a bounded plasma. As a consequence, the generalized Bohm criterion for a collisional sheath is obtained. It is shown that the estimation of the finite Debye length results in a decrease of the ambipolar velocity at the plasma boundary.
the field growth is unlimited. This situation is commonly regarded as connected with the appearance of a sheath at the plasma boundary (x = + d ) and the relation u 2 = 1 + z or v 1 = v s = [k(T e + Tp)/M] 1/2 is employed as a boundary condition between plasma and the sheath (Bohm criterion) [1,2]. This approach yields finite values of the plasma density n 1 and the potential 71 at the boundary while their gradients are infinitely increasing [ 3 - 5 ] . On the other hand it is evident that the field within the sheath and especially at the plasma boundary cannot be unlimited. The field at the plasma boundary E 1 can be evaluated taking into account the following. The electron density at the plasma boundary sharply falls from n e ~ n 1 in the plasma down to n e ,~ n 1 in the sheath. The dimension of the transitional region is equal to the electron Debye length at the plasma boundary D e l . Then it follows from the condition of equilibrium o f electrons in the electric field (the wall being under relatively high negative potential and electrons being practically blocked)
We consider the set of hydrodynamical equations for ions and electrons in a weakly ionized plasma in the VT = 0 approximation V(npVp) = V(neVe) = z n e , vpVvp + (e/M)V~o + (k T p / M ) V l n np
+ ZVp ne/n p + F p / M = O, YeW e - (e/m)V~o + (k Te/rn) Vln n e
+zv e + Fe/m = 0,
(1)
where z is the frequency of a direct ionization, F the, friction force, and other denotations are commonly used. Introducing new variables
u = v(M/kTe)l/2;
~ = z x ( M / k T e ) l / 2 ; rl = - e t p / k T e ;
[3 = ( F p / M Z ) ( M / k T e ) 1/2 ; 7" = T p / T e , and neglecting frictional and inertial forces of the electrons we get from eq. (1) for a quasineutral plasma (n e = n p = n; ve = Vp = v) in slab geometry with the thickness of 2 d
enlE l = -kTeVn .-~-kT e nl/Del ; [E11 = kTe/eDel ;
du/d~ = u dr//d~ + 1 ; du/d~ = [(1 + 7")/u] d r / / d ~ - fl/u - 1 .
(4)
with O~pl the ion plasma frequency at the boundary. The quite natural condition (4) results in a limited gradient of the plasma density and the equation for the normalized ambipolar velocity at the plasma boundary u 1
(2)
Eq. (2) gives for normalized electrical field dr//d~ = (2u + fl)/(1 + 7" - u 2) .
Idn/d~ll = O~pl/Z
(3)
It follows from the last equation, that for u 2 ~ 1 + 7" 80
0 0 3 1 - 9 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 North-Holland
Volume 89A, number 2
1.0
In -1 dn/dXll = D ~ 1 ; 2u 1 +/3(u) = (1 + 7" - u2)6Opl/Z.
26 April 1982
PHYSICS LETTERS
(5)
W (8)~ ~ ' c . o . o ,
o.s
In the free-path regime (/3 = 0), z = 0.57 Os/d ~ { Os/ d and we get from eq. (5) u 1 = (1 + r) 1/2 [1 - {Del/d + -~(Del/d)2 - ...] ,
°' i"
which for the interesting case Del < d coincides with the Bohm criterion. In the collisional case the solution of eq. (5) depends on the form of/3(u), i.e. is determined by the dependence of the ion friction force on the hydrodynamic velocity. Taking into account that for the ion motion in their own gas the ion friction force is determined by the charge transfer process with the cross section practically independent of the velocity (Xp = const) [4,6], we put down/3(u) as follows
/3(u)
=
(UVpO/Z)[1 + 7r2u2148r] 1/2 ,
(6)
where Up0 = (2/Xp)(3kTg/M) 1/2 is the frequency of i o n - a t o m collisions in weak fields when o 2 ,~ kTg/M, Ap is the mean free path of ions in the charge-transfer process, Tg gas temperature (Tg = Tp). From (5) and (6) (at Del < d the first term in (5) may be neglected) we get for the ratio of the ion velocity at the boundary to the ion-acoustic velocity W = VllO s
COpl/Vp0 = [W/(1 - I412)1 [(1 + 7")-1 + W2/5r] 1/2. (7) The dependence of 1¢ o n ~Opl/Pp0 for different parameters r is given in fig. 1. As follows from (7) and fig. 1, the ion velocity at the boundary satisfies the Bohm criterion (W = 1) in the limit of dense plasma (6Opl >> VpO), otherwise ( ~ p l '~ PpO) (8)
In the case of a strongly nonisothermal plasma (r 1) in a rather wide range of w values, where w 2 >> 5~, it follows from eq. (7) that W = [1 + ¼n(3r) -1/2 VpO/Wpl ] - 1 / 2 l /~p]_l/2 = [1 + ~rrDel .
~
I
t
I
t
t
0.1
0.2
05
t
2
5
Fig.
(9)
Approximations (8) and (9) are given in fig. 1 correspondingly for r = 1.0 and r = 0.01. Let us now consider how the boundary field limitation affects the density at plasma boundary n 1 . In
I )/P* tO
20
1.
the approximation of constant ion mobility (/3 = u VpO/ Z) for quite high pressures (Vp0 >> z) it follows from the solution of eq. (2) [3],
Yl = nl /nO = {rr°s(vpodW)-I • From this relation, in the dense plasma limit the result of the authors [3] follows: Yl = { 7r°s(vpod) -1" Otherwise, in the diffusion approximation (60pl '~ Vp0) we have 1
--
Y l = ~Ir Del /d - [TrDeo [2d]2/3
,
(10)
where De0 is the electron Debye length at the plasma center. In the case of a plasma column of radius R at 6Opl
Vp0 Yl = l'25 DellR = [l'25 Deo IR1213 "
= (1 + r ) - l / 2 u 1 ,
W = (l + 7")1/2 60pl/Vp0 •
f 0,05
(11)
It may be shown that for a cylindrical discharge Yl is determined by the ratio (11) if the axial discharge current I satisfies the inequality
I ¢ (M/4e) (KIKv)rl2(VpO R)3 , where K is a part of energy loss by electrons, Kv = 2m/ M.
References [1] D. Bohm, The characteristics of electrical discharges in magnetic fields, eds. A. Guthrie and R.K. Wakerling (McGraw-Hill, New York, 1949). [2] K.B. Persson, Phys. Fluids 5 (1962) 1625. [3] S.A. Self and H.N. Ewald, Phys. Fluids 9 (1966) 2486. [4] V. Ma~ti~ovitg, J. Phys. B3 (1970) 850. [5] L. Cervanan and V. Marti~ovit~, Czech. J. Phys. B26 (1976) 507. [6] V.M. Zaharowa and Yu.M. Kagan, in: Spektroskopiya gazorazryadnoi plasmy (Nauka, Moscow, 1970) p.291.
81
PHYSICS LETTERS
26 April 1982
MODIFIED BOHM CRITERION FOR A COLLISIONAL PLASMA V.A. GODYAK Moscow State University, Physical Department, Moscow, USSR Received 14 October 1981
A simple boundary condition is suggested to ensure regularity in a hydrodynamical model of a bounded plasma. As a consequence, the generalized Bohm criterion for a collisional sheath is obtained. It is shown that the estimation of the finite Debye length results in a decrease of the ambipolar velocity at the plasma boundary.
the field growth is unlimited. This situation is commonly regarded as connected with the appearance of a sheath at the plasma boundary (x = + d ) and the relation u 2 = 1 + z or v 1 = v s = [k(T e + Tp)/M] 1/2 is employed as a boundary condition between plasma and the sheath (Bohm criterion) [1,2]. This approach yields finite values of the plasma density n 1 and the potential 71 at the boundary while their gradients are infinitely increasing [ 3 - 5 ] . On the other hand it is evident that the field within the sheath and especially at the plasma boundary cannot be unlimited. The field at the plasma boundary E 1 can be evaluated taking into account the following. The electron density at the plasma boundary sharply falls from n e ~ n 1 in the plasma down to n e ,~ n 1 in the sheath. The dimension of the transitional region is equal to the electron Debye length at the plasma boundary D e l . Then it follows from the condition of equilibrium o f electrons in the electric field (the wall being under relatively high negative potential and electrons being practically blocked)
We consider the set of hydrodynamical equations for ions and electrons in a weakly ionized plasma in the VT = 0 approximation V(npVp) = V(neVe) = z n e , vpVvp + (e/M)V~o + (k T p / M ) V l n np
+ ZVp ne/n p + F p / M = O, YeW e - (e/m)V~o + (k Te/rn) Vln n e
+zv e + Fe/m = 0,
(1)
where z is the frequency of a direct ionization, F the, friction force, and other denotations are commonly used. Introducing new variables
u = v(M/kTe)l/2;
~ = z x ( M / k T e ) l / 2 ; rl = - e t p / k T e ;
[3 = ( F p / M Z ) ( M / k T e ) 1/2 ; 7" = T p / T e , and neglecting frictional and inertial forces of the electrons we get from eq. (1) for a quasineutral plasma (n e = n p = n; ve = Vp = v) in slab geometry with the thickness of 2 d
enlE l = -kTeVn .-~-kT e nl/Del ; [E11 = kTe/eDel ;
du/d~ = u dr//d~ + 1 ; du/d~ = [(1 + 7")/u] d r / / d ~ - fl/u - 1 .
(4)
with O~pl the ion plasma frequency at the boundary. The quite natural condition (4) results in a limited gradient of the plasma density and the equation for the normalized ambipolar velocity at the plasma boundary u 1
(2)
Eq. (2) gives for normalized electrical field dr//d~ = (2u + fl)/(1 + 7" - u 2) .
Idn/d~ll = O~pl/Z
(3)
It follows from the last equation, that for u 2 ~ 1 + 7" 80
0 0 3 1 - 9 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 North-Holland
Volume 89A, number 2
1.0
In -1 dn/dXll = D ~ 1 ; 2u 1 +/3(u) = (1 + 7" - u2)6Opl/Z.
26 April 1982
PHYSICS LETTERS
(5)
W (8)~ ~ ' c . o . o ,
o.s
In the free-path regime (/3 = 0), z = 0.57 Os/d ~ { Os/ d and we get from eq. (5) u 1 = (1 + r) 1/2 [1 - {Del/d + -~(Del/d)2 - ...] ,
°' i"
which for the interesting case Del < d coincides with the Bohm criterion. In the collisional case the solution of eq. (5) depends on the form of/3(u), i.e. is determined by the dependence of the ion friction force on the hydrodynamic velocity. Taking into account that for the ion motion in their own gas the ion friction force is determined by the charge transfer process with the cross section practically independent of the velocity (Xp = const) [4,6], we put down/3(u) as follows
/3(u)
=
(UVpO/Z)[1 + 7r2u2148r] 1/2 ,
(6)
where Up0 = (2/Xp)(3kTg/M) 1/2 is the frequency of i o n - a t o m collisions in weak fields when o 2 ,~ kTg/M, Ap is the mean free path of ions in the charge-transfer process, Tg gas temperature (Tg = Tp). From (5) and (6) (at Del < d the first term in (5) may be neglected) we get for the ratio of the ion velocity at the boundary to the ion-acoustic velocity W = VllO s
COpl/Vp0 = [W/(1 - I412)1 [(1 + 7")-1 + W2/5r] 1/2. (7) The dependence of 1¢ o n ~Opl/Pp0 for different parameters r is given in fig. 1. As follows from (7) and fig. 1, the ion velocity at the boundary satisfies the Bohm criterion (W = 1) in the limit of dense plasma (6Opl >> VpO), otherwise ( ~ p l '~ PpO) (8)
In the case of a strongly nonisothermal plasma (r 1) in a rather wide range of w values, where w 2 >> 5~, it follows from eq. (7) that W = [1 + ¼n(3r) -1/2 VpO/Wpl ] - 1 / 2 l /~p]_l/2 = [1 + ~rrDel .
~
I
t
I
t
t
0.1
0.2
05
t
2
5
Fig.
(9)
Approximations (8) and (9) are given in fig. 1 correspondingly for r = 1.0 and r = 0.01. Let us now consider how the boundary field limitation affects the density at plasma boundary n 1 . In
I )/P* tO
20
1.
the approximation of constant ion mobility (/3 = u VpO/ Z) for quite high pressures (Vp0 >> z) it follows from the solution of eq. (2) [3],
Yl = nl /nO = {rr°s(vpodW)-I • From this relation, in the dense plasma limit the result of the authors [3] follows: Yl = { 7r°s(vpod) -1" Otherwise, in the diffusion approximation (60pl '~ Vp0) we have 1
--
Y l = ~Ir Del /d - [TrDeo [2d]2/3
,
(10)
where De0 is the electron Debye length at the plasma center. In the case of a plasma column of radius R at 6Opl
Vp0 Yl = l'25 DellR = [l'25 Deo IR1213 "
= (1 + r ) - l / 2 u 1 ,
W = (l + 7")1/2 60pl/Vp0 •
f 0,05
(11)
It may be shown that for a cylindrical discharge Yl is determined by the ratio (11) if the axial discharge current I satisfies the inequality
I ¢ (M/4e) (KIKv)rl2(VpO R)3 , where K is a part of energy loss by electrons, Kv = 2m/ M.
References [1] D. Bohm, The characteristics of electrical discharges in magnetic fields, eds. A. Guthrie and R.K. Wakerling (McGraw-Hill, New York, 1949). [2] K.B. Persson, Phys. Fluids 5 (1962) 1625. [3] S.A. Self and H.N. Ewald, Phys. Fluids 9 (1966) 2486. [4] V. Ma~ti~ovitg, J. Phys. B3 (1970) 850. [5] L. Cervanan and V. Marti~ovit~, Czech. J. Phys. B26 (1976) 507. [6] V.M. Zaharowa and Yu.M. Kagan, in: Spektroskopiya gazorazryadnoi plasmy (Nauka, Moscow, 1970) p.291.
81