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Density and energy distribution of epithermal secondary electrons in a plasma with fast charged particles
Volume 139, number7
PHYSICS LETTERS A
14 August 1989
DENSITY AND ENERGY DISTRIBUTION OF EPITHERMAL SECONDARY ELECTRONS IN A PLASMA WITH FAST CHARGED PARTICLES R. JAYAKUMAR and H.H. FLEISCHMANN Department ofApplied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA Received 8 September 1988; revised manuscript received 22 March 1989; accepted for publication 1 June 1989 Communicated by R.C. Davidson
The production of intermediate energy secondary electrons in plasmas through collisions with fast charged particles is investigated. The density and the distribution of the secondary electrons are obtained by calculating the generation, slow down and diffusion rates, using basic Rutherford collision cross sections. It is shown that the total density of secondaries is much smaller than the fast particle density and that the energy distribution has roughly a 1 /..,[E dependence. The higher generation secondary populations are also obtained.
In several plasma experiments, a large population of fast electrons or ions, well separated in energy from the thermal distribution of the plasma, exist in the plasma. Examples are: relativistic electron ring or ion ring experiments [1,2], particle beam injection cxperiments in toroidal or mirror plasmas [3—6] and tokamaks with runaway or slideaway electrons [7,8]. In such situations, some plasma electrons will gain energy through Coulomb collisions with the fast partides and would be elevated to energies intermediate between the plasma and fast particles. The electron population thus generated are henceforth dubbed secondary electrons. However, these are not secondaries in the sense of new electrons, but are background plasma electrons which have gained energy through fast particle collisions. Additional “higher generation” secondaries are created when the epithermal (secondary) electrons themselves transfer energy to plasma electrons through collisions. The density and energy distribution of the intermediate energy electrons is of interest in the study of plasma spectra in presence offast charged particles [9], runaway production rates in toroidal plasmas [10], Xray emission studies etc. Somewhat related calculations have been carried out [11,121 in the context Present address: General Electric Magnet Systems, P.O. Box F23, Florence, SC 29501, USA.
of backscattering of electrons from solids and secondary electron yield from solids due to fast ion impact. This Letter presents the results of the calculations on the steady state spectrum of secondaries produced by constant energy, constant density, monoenergetic fast charged particles interacting with a cold plasma. In these calculations, therefore, the collisional equations follow only the evolution of plasma electrons to secondary electrons. The changes in the energy and density of fast particles is ignored and the generation of intermediate energy particles from the slow down of fast particles is neglected. While the calculations can be extended to fast partides with a distribution of energy using numerical techniques, the cold plasma assumption limits the applicability of the results at secondary energies close to plasma temperatures. However, the collisional diffusion due to the thermal spread of the target plasma will not significantly influence the energy distribution of the secondaries at energies significantly higher than the plasma temperature, since only relative velocities are of interest in elastic scattering [131. For most experiments, the assumptions of constancy of fast particles is adequate, since the relaxation times for fast particle energies are quite long. But by the same token, the steady state secondary densities at energies close to fast particle energies are also not achieved. Approximate time-dependent dis-
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
313
Volume 139. number 7
i’HYSICS LETTERS. ~
tribution functions are obtained in a later section. The energy differential cross section for energy transfer in a collision between a fast charged particle with velocity u, mass 1n~and charge and a stationarv electron can be derived from Rutherford cross section to be [14] da 2itZ~e4 d~= in~u~e where in~is the electron mass and
(I)
2~i2v~ sin2(O/2)
with reduced mass =
mf + fl1~
The production rate of secondaries with an energy distribution F(E). through such collisions, is given by ______
dt
)pr~1 =
n 2 1~ZE a \/~
(2) ‘
with a=\/2/m,
n~1te4.
where n~is the plasma electron density, t the time variable and E 1= ~ Similarly the production rate due to a secondary electron (energy E’ =E+e) collision with a plasma electron resulting in energy transfer of e is F(E’) 1 =a r~ dE’ j~ ..~/E ~
/dF(E)\
~ dt
(3)
.
The steady state secondary electron energy distribution F(E) is then given by the equation (with dE’ =de for a specific E). +
flhZ2
~
E2
r
J
F(E+e)
F(E+e)l
+
-~
j’ F(E) L
de —
=0,
de
—
1 —
tween fast charged particle and a cold electron. The upper energy cut-off E) corresponds to a 180: scattering of the incident fast particle and therefore is given by E=21 u~/rn~. E,~,,is the minimum energy transfer between the particles in the Coulomb collision. The first term represents the production rate of secondaries with energy E through kick-up of a plasma electron by the fast particles. The second term represents the production rate from a secondary electron of energy E+e losing an energy e~through slow down collisions with plasma electrons. The third term represents the production rate when a secondary electron with an energy E+E collides with a plasma electron transferring an energy E to the plasma electron to elevate it to a “higher generation” epithermal secondary electron. The final term represents the loss of secondaries from the population at energy E through Coulomb collisions with plasma electrons. Only the first term depends upon the fast particle density representing the primary source of secondaries and all other terms describe the evolution of the secondary electrons through collisions with plasma electrons and therefore do not explicitly depend on fast particle parameters. The equation. while quite appropriate for fast ion produced secondaries, neglects the contribution due to the slow down of a primary fast electron. An additional term such as 7~ (E 2 would be needed to include this. ~72// 1—E) to the equation including this However, solutions term could not be obtained. The spreading of the fast particle energy due to the Coulomb collisions is also neglected. Therefore, the solution is less applicable at energies close to the fast particle energy. While the steady state equation shows that the secondary population does notdepend on the plasma density. the approach to the steady state distribution will depend on the plasma density. Also, the secondary electron density is assumed to be much smaller compared to the plasma density, so that the depletion of plasma electrons from generation of secondaries is neglected. The value of e,,,~,for incident electrons with energy greater than 50 eV is given quantum mechanically by that corresponding to the minimum
de
-‘~
(4)
scattering angle (1 5), Om,~
where F=0, E=E 0=maximum energy transfer be314
14 August 1989
2/~~2.63x l0~~
Volume 139, number 7
PHYSICS LETTERS A
14 August 1989
where A[) is the de Broglie wavelength for the mcident electron and T is the plasma temperature. 2 ( Or~,r,/2) = E sin
Table 1 Percent difference between numerical solution of eq. (7) and expression (8).
(5)
2n~/T.
E/E() 0.40
A=60 2.0
A~=40 2.9
A 5.10=20
A0= 9.4 10
0.10 lxl0~ lx lo—~ lx lO~ 4x l0~
1.9 1.8 2.2 2.5 2.0
2.7 3.1 3.0
4.7 5.1 5.4 5.3 5.4
8.9 9.7 10.5 10.5 10.6
~3.5xl0
If, for the present, the higher generation secondary production rate is neglected, eq. (1) can be written as
nZ2
C
F(E)
J
+
/~E2
___________________________________________
7~=-—~de
(
+
1
~
a —=)-de+ ~E ~/FEl C1
c
1
3.2 2.8
j’
2
21
a
~
—~--~de=0. e
/ F
)
terms that were neglected will now be calculated and small the method used to calculate the added.and It therefore will be seen that these contributions are total secondary distribution is appropriate. It may be noted that expression (8) is the solution to the
de
(6)
equation
a
fl~Z2
/ F ~
F 0
Again, neglecting for the present the second order diffusion and higher order terms, the equation can be written as 2 F 7 1 l~ fl~Z ~f~E2
a
(
(9) ~ + ~E \~/EJ ~ The diffusion and higher order terms in which the solution (8) is substituted is therefore written as
J
~~\EOEE) F~
E
2
0—E0 C~ 1~
-r ,E
(7)
~ ln Numerical solutions of eq. (7) were obtained for ln(E~/e~,~) = 10—60. It is seen that the solutions
closely approximate a heuristic solution (10) given by 1
~ ~)
F(E) =
—
‘~
a2
ea3
F
F
(_~+~~-~
a~ F
~ 4
24 =
\E
)
~E
A(EO—E) E3
—
A(E 2 A(EQ—E)3 0—E) + 2E4 3E5
=~(ln~ 1 For this additional production rate, the contribution
A 0
(!
=A\/i~
—
1)
A0 = ln (E0/e~,~).
(8)
The % difference in secondary population between the numerical solutions and the above distribution is given in table 1 for various values of E/E0 and A0. The difference even at low energies and an unlikely low value of A0= 10, is about 10% and the distribution is a good representation of the numerical solution for the first generation secondary distribution. Using this distribution, the contributions from the
to the distribution function from the diffusion and higher terms is calculated to be (by substituting ‘D in place of the fast particle term in eq. (9)) A.,/~r 1 E0 (1 1) (10) FD= —~-_[~ln~~—2 —
~J .
The second generation secondary production rate is .~0—
de—
F(E+e) ~/E+e = A 7
EE0
~ ~ln ~ +
~‘
F(E’)
dE’
—~--
—
E —
i).
(11) 315
Volume 139. number 7
PHYSICS LETTERS A
so that the distribution function for the second generation secondaries is F,(F)= ~_~[ln A~) [
(~‘~(-~-
+
\, F )~,E
.~~_~_2(I —
E
0)
\E
L~1
F0,!] (12)
Similarly, the third generation population is given by .~
F3(E)=
/E
r[~ I F0 ~,2+ln~~~) /
\2
—~-~—
4I~,,r / 1
—
+
2 ~1
.
(13) The above contribution would be nearly doubled when the contribution to the distribution function from the diffusion and higher terms is included. The normalized distributions F(E), F 2(E) and F~(E) are plotted in fig. 1. It can be seen that the contribution from the diffusion and higher generations is smaller than the first
14 August 1989
glected. It is evident that the total secondary density is much smaller than the fast particle density. Time dependent distribution function. The secondary electron population starts out to be a 1/F2 spectrum and relaxes to the steady state distribution given tn eq. (10). This relaxation and the time dependent distribution can be approximately obtained by taking only the leading term in the density balance to ln(E (5) and approximating In [ (E0 —E)/t,,,,,,] equation 0/r,,,,j, so that dE \~~:El r,,,,, ~f~Z 2 + aJ—~=~\ln-~-=0. (15) :p E
~F+a dt
where 4. a=\/ 2/tn.then0ice Defining normalizations
.vL/E 0.
order solution (8) by approximately the ratio A~/ In (E0/E).is In practical situations this means the solution reasonably valid (within 20%) that for seeondary energies E/E 0>~l0~.(Typically ~ l0 ‘ eV.) The total secondary population is -‘.‘
2
I~
43 n1Z A~
E
/
).
l.7+2/A~~
(14)
/10
1
G
~
-
-
~
~
IOU
~
-
II
I
I
I 1
I ~
N0511/LIZ[D [512100 12210
Fig. l.Thc energy distribution function; E~/41
316
(_~._~) J~~/44
x
~:v
ax
(17)
G— ~ (1
(18) 2)2~3 (l+3r/2x~ It can be seen that the relaxation time (corresponding to r= I ) has a very similar expression as the electron—electron collision time. x~
—
References
-
-
-
(16)
eq. (15) reduces to aG 1 1 aG
—
0
z
A solution that satisfies the distributions at t=0 and t=cc is obtained to be (only for x< 1)
Further higher generation populations which are smaller by additional factors of A0/ln (E0/E) are ne-
-
r=aA0t/E~ F 2 F0 ~E’
~
(~—) F~/E,,/A; (-‘
)
[I] HA. Daviset al., Phys. Rev. Lett. 37 (1976) 542. [21 C.D. Kapetanakos et al.. Phys. Rev. Lett. 44 (1980) 1218. [31V. Bailey et al., Physics International Report PIFR-l466-l (1982). l~lK. Narihara et al., Institute of Plasma Physics, Nagoya University Report IPPJ-62l (1983). 15] P.H. de Haanet al.. Proc. 9th Eur. Conf. onControlled fusion and plasma physics, Oxford. 1979, p. 126. [6] R.A. Dandl et al.. Oak Ridge National Report ORNL/TM6457 (1978). [7] H. Knoepfel etal., Phys. Fluids 20 (1977) 511.
Volume 139, number 7
PHYSICS LETTERS A
[81 G. Fussman et al., Proc. Conf. on Plasma physics and controlled fusion, Vol. 3, Baltimore, 1982, p. 295. [9] R. Jayakumar and H.H. Fleischmann, J. Quant. Spectrosc. Radiat. Transfer 33(1985)177. [10] T.E. Everhart, J. Appi. Phys. 31(1960) 1483. [11] E.J. Sternglass, Phys. Rev. 108(1957) 108.
14 August 1989
[12] R. Kulsrud, Private communication. [131 N.A. Krall and A.W. Trivelpiece, Principles of plasma physics (McGraw-Hill. New York. 1973) p. 302. 1141 DV. Shivukin, Review of plasma physics, Vol. 4 (Consultants Bureau, New York, 1966) p. 93.
317
PHYSICS LETTERS A
14 August 1989
DENSITY AND ENERGY DISTRIBUTION OF EPITHERMAL SECONDARY ELECTRONS IN A PLASMA WITH FAST CHARGED PARTICLES R. JAYAKUMAR and H.H. FLEISCHMANN Department ofApplied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA Received 8 September 1988; revised manuscript received 22 March 1989; accepted for publication 1 June 1989 Communicated by R.C. Davidson
The production of intermediate energy secondary electrons in plasmas through collisions with fast charged particles is investigated. The density and the distribution of the secondary electrons are obtained by calculating the generation, slow down and diffusion rates, using basic Rutherford collision cross sections. It is shown that the total density of secondaries is much smaller than the fast particle density and that the energy distribution has roughly a 1 /..,[E dependence. The higher generation secondary populations are also obtained.
In several plasma experiments, a large population of fast electrons or ions, well separated in energy from the thermal distribution of the plasma, exist in the plasma. Examples are: relativistic electron ring or ion ring experiments [1,2], particle beam injection cxperiments in toroidal or mirror plasmas [3—6] and tokamaks with runaway or slideaway electrons [7,8]. In such situations, some plasma electrons will gain energy through Coulomb collisions with the fast partides and would be elevated to energies intermediate between the plasma and fast particles. The electron population thus generated are henceforth dubbed secondary electrons. However, these are not secondaries in the sense of new electrons, but are background plasma electrons which have gained energy through fast particle collisions. Additional “higher generation” secondaries are created when the epithermal (secondary) electrons themselves transfer energy to plasma electrons through collisions. The density and energy distribution of the intermediate energy electrons is of interest in the study of plasma spectra in presence offast charged particles [9], runaway production rates in toroidal plasmas [10], Xray emission studies etc. Somewhat related calculations have been carried out [11,121 in the context Present address: General Electric Magnet Systems, P.O. Box F23, Florence, SC 29501, USA.
of backscattering of electrons from solids and secondary electron yield from solids due to fast ion impact. This Letter presents the results of the calculations on the steady state spectrum of secondaries produced by constant energy, constant density, monoenergetic fast charged particles interacting with a cold plasma. In these calculations, therefore, the collisional equations follow only the evolution of plasma electrons to secondary electrons. The changes in the energy and density of fast particles is ignored and the generation of intermediate energy particles from the slow down of fast particles is neglected. While the calculations can be extended to fast partides with a distribution of energy using numerical techniques, the cold plasma assumption limits the applicability of the results at secondary energies close to plasma temperatures. However, the collisional diffusion due to the thermal spread of the target plasma will not significantly influence the energy distribution of the secondaries at energies significantly higher than the plasma temperature, since only relative velocities are of interest in elastic scattering [131. For most experiments, the assumptions of constancy of fast particles is adequate, since the relaxation times for fast particle energies are quite long. But by the same token, the steady state secondary densities at energies close to fast particle energies are also not achieved. Approximate time-dependent dis-
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
313
Volume 139. number 7
i’HYSICS LETTERS. ~
tribution functions are obtained in a later section. The energy differential cross section for energy transfer in a collision between a fast charged particle with velocity u, mass 1n~and charge and a stationarv electron can be derived from Rutherford cross section to be [14] da 2itZ~e4 d~= in~u~e where in~is the electron mass and
(I)
2~i2v~ sin2(O/2)
with reduced mass =
mf + fl1~
The production rate of secondaries with an energy distribution F(E). through such collisions, is given by ______
dt
)pr~1 =
n 2 1~ZE a \/~
(2) ‘
with a=\/2/m,
n~1te4.
where n~is the plasma electron density, t the time variable and E 1= ~ Similarly the production rate due to a secondary electron (energy E’ =E+e) collision with a plasma electron resulting in energy transfer of e is F(E’) 1 =a r~ dE’ j~ ..~/E ~
/dF(E)\
~ dt
(3)
.
The steady state secondary electron energy distribution F(E) is then given by the equation (with dE’ =de for a specific E). +
flhZ2
~
E2
r
J
F(E+e)
F(E+e)l
+
-~
j’ F(E) L
de —
=0,
de
—
1 —
tween fast charged particle and a cold electron. The upper energy cut-off E) corresponds to a 180: scattering of the incident fast particle and therefore is given by E=21 u~/rn~. E,~,,is the minimum energy transfer between the particles in the Coulomb collision. The first term represents the production rate of secondaries with energy E through kick-up of a plasma electron by the fast particles. The second term represents the production rate from a secondary electron of energy E+e losing an energy e~through slow down collisions with plasma electrons. The third term represents the production rate when a secondary electron with an energy E+E collides with a plasma electron transferring an energy E to the plasma electron to elevate it to a “higher generation” epithermal secondary electron. The final term represents the loss of secondaries from the population at energy E through Coulomb collisions with plasma electrons. Only the first term depends upon the fast particle density representing the primary source of secondaries and all other terms describe the evolution of the secondary electrons through collisions with plasma electrons and therefore do not explicitly depend on fast particle parameters. The equation. while quite appropriate for fast ion produced secondaries, neglects the contribution due to the slow down of a primary fast electron. An additional term such as 7~ (E 2 would be needed to include this. ~72// 1—E) to the equation including this However, solutions term could not be obtained. The spreading of the fast particle energy due to the Coulomb collisions is also neglected. Therefore, the solution is less applicable at energies close to the fast particle energy. While the steady state equation shows that the secondary population does notdepend on the plasma density. the approach to the steady state distribution will depend on the plasma density. Also, the secondary electron density is assumed to be much smaller compared to the plasma density, so that the depletion of plasma electrons from generation of secondaries is neglected. The value of e,,,~,for incident electrons with energy greater than 50 eV is given quantum mechanically by that corresponding to the minimum
de
-‘~
(4)
scattering angle (1 5), Om,~
where F=0, E=E 0=maximum energy transfer be314
14 August 1989
2/~~2.63x l0~~
Volume 139, number 7
PHYSICS LETTERS A
14 August 1989
where A[) is the de Broglie wavelength for the mcident electron and T is the plasma temperature. 2 ( Or~,r,/2) = E sin
Table 1 Percent difference between numerical solution of eq. (7) and expression (8).
(5)
2n~/T.
E/E() 0.40
A=60 2.0
A~=40 2.9
A 5.10=20
A0= 9.4 10
0.10 lxl0~ lx lo—~ lx lO~ 4x l0~
1.9 1.8 2.2 2.5 2.0
2.7 3.1 3.0
4.7 5.1 5.4 5.3 5.4
8.9 9.7 10.5 10.5 10.6
~3.5xl0
If, for the present, the higher generation secondary production rate is neglected, eq. (1) can be written as
nZ2
C
F(E)
J
+
/~E2
___________________________________________
7~=-—~de
(
+
1
~
a —=)-de+ ~E ~/FEl C1
c
1
3.2 2.8
j’
2
21
a
~
—~--~de=0. e
/ F
)
terms that were neglected will now be calculated and small the method used to calculate the added.and It therefore will be seen that these contributions are total secondary distribution is appropriate. It may be noted that expression (8) is the solution to the
de
(6)
equation
a
fl~Z2
/ F ~
F 0
Again, neglecting for the present the second order diffusion and higher order terms, the equation can be written as 2 F 7 1 l~ fl~Z ~f~E2
a
(
(9) ~ + ~E \~/EJ ~ The diffusion and higher order terms in which the solution (8) is substituted is therefore written as
J
~~\EOEE) F~
E
2
0—E0 C~ 1~
-r ,E
(7)
~ ln Numerical solutions of eq. (7) were obtained for ln(E~/e~,~) = 10—60. It is seen that the solutions
closely approximate a heuristic solution (10) given by 1
~ ~)
F(E) =
—
‘~
a2
ea3
F
F
(_~+~~-~
a~ F
~ 4
24 =
\E
)
~E
A(EO—E) E3
—
A(E 2 A(EQ—E)3 0—E) + 2E4 3E5
=~(ln~ 1 For this additional production rate, the contribution
A 0
(!
=A\/i~
—
1)
A0 = ln (E0/e~,~).
(8)
The % difference in secondary population between the numerical solutions and the above distribution is given in table 1 for various values of E/E0 and A0. The difference even at low energies and an unlikely low value of A0= 10, is about 10% and the distribution is a good representation of the numerical solution for the first generation secondary distribution. Using this distribution, the contributions from the
to the distribution function from the diffusion and higher terms is calculated to be (by substituting ‘D in place of the fast particle term in eq. (9)) A.,/~r 1 E0 (1 1) (10) FD= —~-_[~ln~~—2 —
~J .
The second generation secondary production rate is .~0—
de—
F(E+e) ~/E+e = A 7
EE0
~ ~ln ~ +
~‘
F(E’)
dE’
—~--
—
E —
i).
(11) 315
Volume 139. number 7
PHYSICS LETTERS A
so that the distribution function for the second generation secondaries is F,(F)= ~_~[ln A~) [
(~‘~(-~-
+
\, F )~,E
.~~_~_2(I —
E
0)
\E
L~1
F0,!] (12)
Similarly, the third generation population is given by .~
F3(E)=
/E
r[~ I F0 ~,2+ln~~~) /
\2
—~-~—
4I~,,r / 1
—
+
2 ~1
.
(13) The above contribution would be nearly doubled when the contribution to the distribution function from the diffusion and higher terms is included. The normalized distributions F(E), F 2(E) and F~(E) are plotted in fig. 1. It can be seen that the contribution from the diffusion and higher generations is smaller than the first
14 August 1989
glected. It is evident that the total secondary density is much smaller than the fast particle density. Time dependent distribution function. The secondary electron population starts out to be a 1/F2 spectrum and relaxes to the steady state distribution given tn eq. (10). This relaxation and the time dependent distribution can be approximately obtained by taking only the leading term in the density balance to ln(E (5) and approximating In [ (E0 —E)/t,,,,,,] equation 0/r,,,,j, so that dE \~~:El r,,,,, ~f~Z 2 + aJ—~=~\ln-~-=0. (15) :p E
~F+a dt
where 4. a=\/ 2/tn.then0ice Defining normalizations
.vL/E 0.
order solution (8) by approximately the ratio A~/ In (E0/E).is In practical situations this means the solution reasonably valid (within 20%) that for seeondary energies E/E 0>~l0~.(Typically ~ l0 ‘ eV.) The total secondary population is -‘.‘
2
I~
43 n1Z A~
E
/
).
l.7+2/A~~
(14)
/10
1
G
~
-
-
~
~
IOU
~
-
II
I
I
I 1
I ~
N0511/LIZ[D [512100 12210
Fig. l.Thc energy distribution function; E~/41
316
(_~._~) J~~/44
x
~:v
ax
(17)
G— ~ (1
(18) 2)2~3 (l+3r/2x~ It can be seen that the relaxation time (corresponding to r= I ) has a very similar expression as the electron—electron collision time. x~
—
References
-
-
-
(16)
eq. (15) reduces to aG 1 1 aG
—
0
z
A solution that satisfies the distributions at t=0 and t=cc is obtained to be (only for x< 1)
Further higher generation populations which are smaller by additional factors of A0/ln (E0/E) are ne-
-
r=aA0t/E~ F 2 F0 ~E’
~
(~—) F~/E,,/A; (-‘
)
[I] HA. Daviset al., Phys. Rev. Lett. 37 (1976) 542. [21 C.D. Kapetanakos et al.. Phys. Rev. Lett. 44 (1980) 1218. [31V. Bailey et al., Physics International Report PIFR-l466-l (1982). l~lK. Narihara et al., Institute of Plasma Physics, Nagoya University Report IPPJ-62l (1983). 15] P.H. de Haanet al.. Proc. 9th Eur. Conf. onControlled fusion and plasma physics, Oxford. 1979, p. 126. [6] R.A. Dandl et al.. Oak Ridge National Report ORNL/TM6457 (1978). [7] H. Knoepfel etal., Phys. Fluids 20 (1977) 511.
Volume 139, number 7
PHYSICS LETTERS A
[81 G. Fussman et al., Proc. Conf. on Plasma physics and controlled fusion, Vol. 3, Baltimore, 1982, p. 295. [9] R. Jayakumar and H.H. Fleischmann, J. Quant. Spectrosc. Radiat. Transfer 33(1985)177. [10] T.E. Everhart, J. Appi. Phys. 31(1960) 1483. [11] E.J. Sternglass, Phys. Rev. 108(1957) 108.
14 August 1989
[12] R. Kulsrud, Private communication. [131 N.A. Krall and A.W. Trivelpiece, Principles of plasma physics (McGraw-Hill. New York. 1973) p. 302. 1141 DV. Shivukin, Review of plasma physics, Vol. 4 (Consultants Bureau, New York, 1966) p. 93.
317