In deriving expressions for the electrical conductivity

in a gas, Kerrebrock, Cann, and others

have recently recognized the importance of the

elevated electron temperature to the conduction process. The present note attempts to point out

that the role of the local electric field in maintaining

the temperature difference ("nonequilibrium") between

the electrons and the other species, and a

threshold in the field strength below which the

electronic (and ionic) mobilities are independent of

field strength, can be derived from elementary

considerations

Demetriades, Anthony

English

Caltech Authors - Main

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1134 RESEARCH NOTES 
_ n1 (:!!::)!(wv)2(wv _ k·u) 
no8 kv kv kv 
·exp [-! (wv _ k·U) 2]. 
2 kv kv 
For (wv - k·u) < 0 and !'Y1khokl > 1, waves 
will be unstable. From Fig. 1 for k~ = 0 this is 
the region near v. "" u for which v.(ajjav.) > 0, 
where k1. is the component of k which is perpendicular 
to u. Thus 
To determine the range of k~ for which unstable 
waves exist we have 
-'Ylk = 
'Yok 
{ 1 [w~. - (wv - kuu)
2
] 2 o} 
· exp 2 (kuv)2 cos , 
where k 11 = (k·u/u) = k cos 0. We denote the 
0 = 0 growth rate by I'Y1~'Yokle = a. Since we are 
dealing only with very weak instabilities, we shall 
assume that a is only 2 or 3, i.e., the growth rate 
due to the beam is only slightly greater than the 
damping rate without the beam. We then have 
-'Yll~ - {-! [w! - (wp - ki1U) 2 ] • 2 o) 
- a exp 2 (k ) 2 s1n , 'Yok uV 
and the range of k1. for -'Y;k/'Yok > 1 is given by 
The unstable waves are thus confined to a small 
region in phase space of volume 1r(.:lk1.)2 (.:lk 11 ). 
As discussed in reference 3, the equilibrium 
spectrum to which the unstable waves grow is 
given by 
:E IE(k)2 1 = :E &(k) 
k.L 811" k.L 
= (~Y J dkl.&(k) ""s:~ (~)(;Y'Y· 
Only those waves for which (k1./k) ~ (.:lk~/k) ::::::= 
(k"An) are initially unstable, and after e-folding 
many times the spectrum is even more strongly 
peaked such that k1. ~ ok1. = .:lk~/(2R) 1, where R 
is the number of e-folds through which the wave 
amplitude has gone before reaching the equilibrium 
spectrum. The average value of &(k) can thus be 
estimated by 
"- ( 21r ) 2 v2 w (m)2 &(k) = L ok~ 81rL k --;- 'Y 
81r2 R 'Y 3(KT) 
= (k"An)5 Wv n"An 2£3 . 
The energy per mode in thermodynamic equilibrium 
is just KT /2£3 and thus the wave energy, and hence· 
the cross section is enhanced by the factor 
where 
2R ""l 81r 'Y ,a 
= n (k"An)5 Wv n"n· 
For a typical case ('Y/wv) = 10-2 , n"Ai = 106, and 
(k"An) = 0.25 and we obtain R "" 10 and {3 ::::::= 1010• 
With this large enhancement of the cross section 
it appears possible to carry out laboratory experi-
ments on weakly unstable plasmas. It should be 
mentioned that cross sections larger than equilibrium 
by a factor of about this magnitude have already 
been observed in atmospheric scattering experiments4 
from the equatorial electro jet, where instabilities 
are likely to occur; however, the instability involved 
appears to be somewhat more complicated than the 
one discussed above. 
The author would like to thank Dr. N. Rostoker 
and Dr. M. Rosenbluth for stimulating discussions 
on this topic. 
This research sponsored by the Advanced Re-
search Projects Agency, Department of Defense, 
under project Defender, Air Force Contract No. 
AF29(601)-5338. 
1 N. Rostoker, Phys. Fluids (to be published); J. P. 
Dougherty and D. T. Farley, Proc. Roy. Soc. (London) A259, 
79 (1960); E. E. Salpeter, Phys. Rev. 120, 1528 (1960). 
2 K. L. Bowles, Phys. Rev. Letters 1, 454 (1958); National 
Bureau of Standards, Boulder, Colorado, Rept. 6070 (1959). 
8 W. E. Drummond and D. Pines, Nuclear Fusion (to be 
published). 
4 K. L. Bowles and D. T. Farley (private communication). 
Electric-Field Heating Threshold for 
Charged Particles 
ANTHONY DEMETRIADES 
California Institnte of Technology, Pasadena, California 
(Received May 23, 1962; revised manuscript received 
June 29, 1962) 
JN deriving expressions for the electrical conduc-
tivity in a gas, Kerrebrock/ Cann,2 and others 
have recently recognized the importance of the 
elevated electron temperature to the conduction 
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RESEARCH NOTES 1135 
process. The present note attempts to point out 
that the role of the local electric field in maintaining 
the temperature difference ("nonequilibrium") be-
tween the electrons and the other species, and a 
threshold in the field strength below which the 
electronic (and ionic) mobilities are independent of 
field strength, can be derived from elementary 
considerations. 
Many years ago Compton, 3 assuming a billiard-
ball collision model and a Maxwellian velocity 
distribution for each specie, pointed out that a 
charged particle drifts with a velocity proportional 
to the field strength E at lowE, but with velocity 
v ~ E! when the field strength is sufficiently high. 
Using similar ideas we derive the following formula 
for the drift velocity: 
v = qEA. (~)! 
Vr mM 
(1) 
where q is the particle charge, m its mass, and where 
M and Vr are the mass and thermal velocity of the 
neutral particles in which m drifts; A. is the mean 
free path of the charged particle in the neutral gas 
Oetectron • 1/5 Oion 
--ELECTRONS 
i 
-3 -' __ L_._L-c--'_j__ __ _j__~_J___i___i_~_j_~__j_~-
-4 -3 -2 -I 2 3 4 
FIG. 1. Elementary mobility theory for elastic collisions of 
hard-sphere ions and electrons. 
and is assumed independent of the collision energy. 
Also, the rms velocity c of the charged particle is 
expressed by 
C = Vr(~Y 
such that "equilibrium" is defined by mc2 Mv;,. 
We distinguish two limiting cases. When the 
second term of the radical in the above bracket is 
much smaller than unity ("weak"-field case), the 
drift velocity is 
(3) 
with IL = m/M, and the charged specie equilibrates 
with the neutrals since mc2 = Mvi. When the 
em /sec 
X 10- 4 
10 
E/P (Vo\t/cm·mmHg) 
FIG. 2. Ionic drift velocity as a function of E/p, after 
Hornbeck (reference 5). The critical boundaries calculated by 
the present theory are shown in solid vertical lines. 
second term in the radical dominates, however, 
not only does v become dependent on E!, but 
m(c2 - v2 ) > Mv;, generally, so that the average 
velocity of the charged specie begins to increase also. 
If simple ideas are borrowed from kinetic theory 
to express A., the threshold separating the "weak" 
from the "strong" field (or the "equilibrium" from 
the "nonequilibrium") case is given by 
Eq/pQ = 6.6[~.t!/(~.t + 1)], (4) 
where p is the actual hydrostatic pressure of the gas, 
and Q the charged-neutral collision cross section. It 
is obvious that nonequilibrium is attained at much 
lower values of E/p for electrons (IL ~ 10-4-10-6) 
than for ions (IL ~ 1). Further, very heavy ions in a 
very light gas should also depart from equilibrium 
at very low values of E/p, according to these ideas. 
If the temperature dependence is eliminated by 
normalizing v with vr, the over-all role of E/p can be 
assessed from Fig. 1, where typical values are 
assigned to IL and to the ratio of electron-neutral 
to ion-neutral cross sections. Also, using ion-neutral 
cross sections given by Hornbeck, 4 the ion threshold 
is evaluated from Eq. (4) and shown on Fig. 2 
superimposed on experimental data.4 The agreement 
·{1 + [1 + 3E2 2A.2 (m + M)2]!}! 
q milf"v~ ' (2) is surprisingly good in view of the crudity of the assumptions underlying Eq. (4). 
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1136 RESEARCH NOTES 
The "heating" threshold for electrons occurs at 
extremely small values of E/p; if Lamb and Lin's 
electron-atom cross sections5 for argon are used, 
for example, an electron threshold of 0.8 X w-a 
V /em per mm Hg is found (versus 37 V /em per 
mm Hg for the ion). Therefore, it is not surprising 
that all available data on electron drift, which are 
taken at above-threshold values of E/p, do not 
show a linear dependence of velocity on field 
strength. 6 
The work discussed in this paper was carried out 
under the sponsorship and with the financial support 
of the U. S. Army Research Office and the Advanced 
Research Projects Agency, Contract No. DA-04-495-
0RD-3231. 
1 J. L. Kerrebrock, "Conduction In Gases with Elevated 
Electron Temperature," Proceedings of the Second Symposium 
on the Engineering Aspects of Magnetohydrodynamics (Colum-
bia University Press, New York, 1962). 
2 G. L. Cano, American Rocket Society Paper Number 
2396-62, Berkeley, California (1962). 
a K. T. Compton, Revs. Modern Phys. 2, 210 (1930). 
4 J. A. Hornbeck, Phys. Rev. 84, 615 (1951). 
6 L. Lamb, and S. C. Lin, J. Appl. Phys. 28, 754 (1957). 
G I,, L. Loeb, Basic Processes of Gaseous Electronics (Uni-
versity of California Press, Berkeley, California, 1955). 
Comments 
COMMENTS refer to papers published in The Physics of Fluids and are subject to the same length limitation as 
Research Notes. The Board of Editors wiU not hold itself 
responsible for the opinions expressed in the Comments. 
Erratum: Secondary Flow about a Sphere 
Rotating in a Viscous Liquid inside a 
Coaxially Rotating Spherical Container 
[Phys. Fluids 5, 625 (1962)] 
W. L. HABERMAN 
Office of Naval Research, Washington, D. C. 
THE research note bearing the above title 
contained three typographical errors which had 
occurred in the author's manuscript: 
(1) The expressions for A 1 and B1 should read 
-(w1"Aa - w2)/(1 - X3 ) and (wt - w2)R~/(1 - X3), 
respectively. 
(2) The expression for DR1 should read 
Dt(>-)B~R~3v-1 + D2("A)A1Btv-1 • 
(3) The decimal points in the - D2("A) column 
should be moved one digit to the right [e.g., at "A = 
0.9, -D2 ("A) = 0.379 156 9]. 


Electric-Field Heating Threshold for Charged Particles

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Electric-Field Heating Threshold for Charged Particles

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