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Electron distribution function of a weakly ionized gas in magnetic and time-dependent electric fields
Electron
Distribution Magnetic and Jox.4~ Physics
Section,
Function of a Weakly Ionized Time-Dependent Electric Fields*
S. ZMUIMSAS
.I\ND CHING-SHEKG
Jet Propulsion Lnborutoq, Paliforniu ‘I’echnolr~g:~~, Pasadena, Pnliforxia
Gas
in
Wrr Institute
of
An investigation is made of the electron behavior in a weakly ionized gas in n magnet.ic field and arbitrarily time-dependent. electric field. This problem is of interest eit,her in astrophgsics or the study of t,he basic phenomena in gas discharge. The electron distribution function is studied based on the Boltzmann equation u-hich takes only the electron-atom collisions into awount. A “moment method” is devised to solve the case nhen the electron collision frequency is uniform. In t.he subseyuent, analysis, the electron distribution function is first expanded in terms of hoth the “moments” and the associate I*aguerre polynomials and then a generating equation of the moment,s is derived. This generating equat,ion which is linear and of first order in time ran tie integrated readily. The persistent solution is obtained in closed form for the quasi-steady case. In caw of high-frequency oscillating electric field t,he drift velocit.?-, t,he mean p-rtrrgy, t,hr conduc:iivit,.\r and the dielect,ric constant. :xe ttll diswsxed. I. INTROL)UCTIOiL’
The present inwstigation rowerus a slightly ionized gas in thr prcstncc of a uniform magnetic field and a time-dependent1 electrical field. The plasma is postulated to he homogeneous and before t,he external fields are imposed, to possessMaxwellian distribution correspondiug to the gas tcmpcraturc 7’. The problem under st)udy is the determinat,ion of the clc&on vclocit,y distribution funrtion of such a plasma, as well as the corresponding average elertron energy and drift velocity. The Boltzmann equation for the present problem may he simplified by dropping t,hc collision int.egrals desrribing th(x rlr~t,roil-clcc~.ror1 and electron-ion encount,crs. In the following discussion, only the rlectron-atom collisions, for which the “con~cnt~ional” close binary collision model may be assumedvalid, are considered. Again, we shall assunw that, most, of t,he c~ollisions arc’ c>lastica. This itnplics that the :rwragc clwtroil cnrrgy is low and the rsttwrul field is not wry strong. * This papclr presents the results sion I,aboratory, California Institute nored 1,~ the n’atiottal Act ona1ttic.s
of one phase of research of Technology, under and &Space Administration. 387
rarried Contrwt
ollt
at t,he Jet PropuNo. NASn-6, spon-
In the following analysis the fields generat.ed by the int.wnal dist,rib\ltions and motions of t.hr charged part iclcs art’ post.ulatctl to kc ncgligiblc compared to the applied fields. Hence ill the Boltzmann ccluation both E and B are wnsidered to hc knon-11 cluantities and hcrraftrr wc will write the clcctric field E in the form
E = Eodf), where cp(t) is any given function of time which is hounded and continuous at all time. With the initial Maxwellian distribut,ion n-c may obtain a rolution for the subsryr~ent, time (t > 0) based on the assumpt.ion that. the anisotropic part of the dkkrihution fun&m is a small perturbation of the isotropic dist,ribut,ion. In the following discussion a “moment met,hod,” which will be dcwonstrated lat,er, is used. Our main interest, is to invest,iguk the “persistent, sohkion” (t >> 7, where 7 is the mean collision time). A closed form solution for the quasi-steady case (or some other special cases, for instance, the ac field) can bt found by considering the asympt,otic behavior of the distribution function. The analysis of t,his part is displayed in Section IT:. The calculat,ions for the electron drift velocit,y, mean energy, conductivity, and dielectric constant, based on t,he “moments” are also given in the final part of t#his paper. II.
DEI>CC;TIOK
OF
THE
GOVEILSI?r’G
IN’~EC;ltO-I~IFFElt~~~~I~4L
EQUz4TIO?:
Under the assumptions stated previously, t.hc eltctron distrihut,ion function ,f(v, t) is considered t,o satisfy the Bolt,zmann equation of the following form ( 1) :
E,cp(t) + v X B).;
1
.fCv,O
In ( 1)) the primes denote quant~it,iesafter collisions, and u,( g, J/) is the differential cross sect,ion for elastic scatt’ering of electron and neukal parbicles through an angle #. The solid angle dQ may he expressed in the form dQ = sin 1c,d# d+,
wliere 4 is the polar angle measured in a plane normal to the vect,or g which is t,hr velocity of clect,ron relative to t,he colliding atom, i.e., g = v - v, . Furthermore the subscript m denotes quantities which belong to the neut’ral at,oms. In order to solve (1) for .f(v, i), we shall postulate t,hat, the elect,ron distribu-
ELECTROS
tion funct~ion may hc written j’iv, t) = .f”‘%,
I~ISTRIBUTION
in t.he following
t) + E&“‘(P,
form:
t)
+ (B X EoJ .vf@)(~j, f) + (B.v)(B.Eoj.f”“(2’, The init,ial conditions given as follows:
889
FCSCTION
of t,hc unknown
functions
t).
(2)
J(“), j(l), ,f(‘)‘, and f’“’ may be
.4t t = 0, .f”“(l’, 0) = N(
.f%,
m/2TkT)R~"
CSP (-Vla’/i’liT),
0) = 0,
.f”‘(P, 0) = 0, .f(Yl’, 0) = 0, whrre N is the electron number den&y and /C is expression (2) is merely the first t,wo terms of the the spherical harmonic in t’he component of v. A of this expression is given by Wu (2). Following obtain
the Roltzmann constant,. The expansion of f(v, t) in terms of brief remark on the deduction the similar discussion (2), w
)I af”’ L== at
-- ; f’” + ;
; !!?f;;:cp(t) - gy C df’*’ __ _ f f21 + ; f(l), at af L ‘“’ == - ; f”’ + f f”‘, al
i-l) ’
1 )
where nz and iI/ are the elcctjronk and atomic mass, rcapec~tivrly, 1 the mea11 free pnt,h of electrons, and y the angle hctwccn t,hc field \wtors B and E. Now we shall introdlwe the Laplace transform
where s is t,he variable in t,he transformed space and has th(l property Re[s] > 0. Using the initial conditions (3), we may transform (5)-(i) to the following
390 forms :
(9, (10) (11) where L[(df”“/d~)cpJ denotes t,he Laplace transform of (@““/&~)~(t) and, for simplicity, we hare denoted 1,‘~ = 7. T is t,he collision t,imc and in gcncrsl is a function of 2~. From (9)-( 11 ) we oht,aiil (ST + 117 = (ST+ lj2+W272m2~
-
f”’ ,f”’ pi
CJ[
’
(
af’“’ do ‘p ’ >
( 12)
= ~ 2 (s7 + I;2 + w2+2m’L lfJ ;l L ( ac , p > ’ T3 + 1)’ +
= (~7
~27~
‘1, m3 1’
?Pp (
)
cyclotron
f”’
[cos’y + sin’y cos w(t - I’)]
B2
cos2yf3)
=
v
vu:
t e-“‘-“’ so
(14)
>
where w = eB/m is the electron one may easily show +
(13)
frequency.
From
( 12) and (14),
(15) x
af0’(2!,
t’)
a2) where v = l/7. Combining ( 15) and (4), f’“’ for t 1 0.
For simplicit.y,
hereafter
we obtain
an integro-differential
we denote 2~ = 21/a, a = d2kT/m, E = y(u,
t)
m/A/,
= f’“‘(P,
t).
dt') dt', equation
for
ELECTI~ON
IIISTRIB'CTIOZI
FUNCTION
:391
Then 1:16 1 becomes
Before we close this se&ion, t,wo points should he remarked. (1) It, is possible to derive a general expression of t,hc dist,rihution function .f(v, t) in t,erms of f’“‘(z1, t) or g( u, tj. From (12)-(1-k), we may obtain the inverse i~rsnsform as follows:
Hence f(v, t) = g(u, t) + &
1’ e-““-“’
+ {[e(B X Eo)~u],lmw} + (e”/m”w’)
(BvE,)(B.~l[i
I (E,.u)
cos w(t - 2’)
sin w(t -
~0s
(21)
t’) ~(t
-
t’)])
(ag/att)&t’)
dt’.
Therefore, once g(u, t) is known, t,he distrihut,ion function .f(v, 1) may he immrdiately calculated from (21) . (2) The asymptotic expression of .f( I’, t) ns t + = may he obtained 1,~ showing, from Eq. ( 18)) t,hat,:
Is (“--iw)l’ af’“’ st
e 0 = 2mau
e
A-ao cp(t’1 (if’ +O (A!- id)t
p+Lw)t
a”‘“’ x cp(l’) co’ e”+‘“”
(22)
392
ZMuIDZIS.\S
where g=(t)
represent,s
.4x1>
the behavior
\vc
of y as / +
x . $‘imilarly,
from (19) alld
(23)
(24)
(25)
III. A.
EXPaKSION
OF
THE
METHOD
IcUSCTION
OF g(U,
SOLUTION
t)
In an attempt to solve (17), we shall devise a method which contains two essential steps: (1) expanding t.he fun&on g(q t) in terms of its moments as defined in the following
m nr,,= 2s g(IL,t)lL’L+2 d1L 0
(26)
and (2) t,hen deriving the “generating equation” for t,hese moments. First, we shall expand the fun&ion g( U, t) in terms of the Laguerre polynomial g(u, t) = P?
g &(A, t)L: 2GAL2~,
where A is a paramct’er which may be a function of t. The determination X(t) will be discussed latter. The Laguerre polynomial Lk” is defined as (3) :
(27) of
(28)
ELECTROS
DISTKIBUTIOX
FUSCTIOS
393
The reason for choosing the Laguerrc polynomial of order J$ in particular is mainly based on its orthogonality property. The discussioll of this choice may be found in the Appendix in which it has also been shown that the coefficients ak ma:y he expressed in terms of t,he momcuts
The choice of X will be based on the criterion that the fast,est, convergence (27 ) may be oMained. To illustrate this, we consider the special case at t = 0. Since
of
it may he sho\vn that
Therefore
Lct,ting X = 1, we have the first’ t,erm only. This implies that the fastjest, convergence may be achieved if X = 1. Howevrr, when t # 1 this is no longer true, as we shall see later (Section IV).
In order to drtcrmine t,hc moments, we shall ret,uru to ( 117). In the following discussion, WC shall restrict, ourselves t,o the c’nsc t,hat v ii; uniform; in some cusw t,his furnishes a good approximation for weakly ionized gas ( 4). In this cast, if each term of (17) is multiplied by u”‘+’ and integrated over all values of IL, we obtain a system of integro-differential-difference equations which may be wri tteii as follows:
Since by dctnmion m ill,, = 2 = 2
s0
g(7l,
t)rl’ du = 3
I0
{](7L, 0)u’ du, (,xi )
m
we may therefore proceed to calculate the higher moments d12 , Ail4 , and so forth by integrating (32). At, t.his point, the problem is solved in principle. In closing this se&ion, we may remark that, it, is possible t,o express the distribut,ion function .f(~, t) in terms of the moments,
where Y(u, t) = ((E0.u)
cos w(t - 1’) + (e/mw)(B + (e”/m’w’)(BE0
IV.
PERSISTENT
SOLUTIOXS COLLISION
X Eo). usin w(t - t’) cosr)(B
Ip\’ THE FREQUEKCY
. u)[l
CASE
OF
- cos w(t - t’)]). UNIFORM
It is seen from the previous discussion t,hat considerable mathematical plificat,ion may be obtained if the collision frequency may be considered independent of 71. From (32), we may obtain by integration
simto be
(35)
M2,,+2( t) = e--e”(2n+2)t 11f2,z+2(O) where a+,(t)
= (an + a)(211 + 3)
{2&h&
cp(t) I’ e-“l-t,’
. [cos’ y + sir12 y cos w(t - t’)] +O(t’)Mfn(t’) In (35)
t#he integration
constant
dt’ + f wM2n(t)
)
i112,,+2(0) may easily be evaluated:
dl2,,+2(0) = 2 om g(u, o)u2”+4 du I = Yv(~;a)-3r(n
+ 35).
(36)
.
ELECTRON
DISTRIBUTJON
If we denot’e X?,L((t) as the asymptot’ic N.?,, (t ) , t.hen we have:
I’o&l)le further special cases :
The quasi-steady
simplications
form
of (37) and (38) may he found for the following
case is dcfincd p/v”
when
<< cp,
tho following
Jig/v”
conditions
<< II?,, )
hold f,rue :
k = I) 2, 3, . . . )
where 5~(” and Jr;:,’ denote the derivatives of cp and N?,, , respectively. The physiral meaning of these conditions is that the changes of the functions cp and J/, for the time interval T( = llvj are small compared to the values of the functions. This situation enables us to rewrite (37) and (38) as follows: @,?,,~= (2,n + 2)(2n + :3)e%o?p~t) cos?y y + sin’yRe 6mlcT
one may deduce from (:-D-(41)
( )ICpM;,, I-v + iw
(39)
that
where x;*
= M,*(o)
= NC 2/Taj-3.
Hence, the asympt,otic form of g(?s,t) for large time may be given in a rather
simple form :
where
It is clear t,hat the optimum
choice of A, is h,(f)
= [I + n(tj]-I”.
Then grn(u, t)
N( ~MX’)-‘~[~
because
+ Q(t)]-“‘”
exp ( - (u’/[l
Ego (i)
(- l)“L:‘“(x,“u”)
+ Q(t)] )I
(44)
+ Q(l)])).
(45)
= 1.
As a consequence, dg”/du
= --2N(&~)-~u[l
With the suhstit,ut,ion f(v, t) is found. B. HIGH-FREQUENCY
+ Q(t)]-“”
exp { -(2?/[1
of (44) and (45) into (25), the persistent
OSCILLATIXG
solut,ion of
FIELD
The second special casefor which (37) and (38 j may be simplified is t#hat the electric field E varies according to a time function p(l) = eifit. In this case, the approximation used previously is no longer valid if t)hefrequency 6 is of the same order of v. However simplification is still possible, since sometimeswe only need the information of MO and d12. To ill&rate this point, discussionswill be given in the following section. V. APPLICATIONS
The dist,ribut’ion function j(v, t) discussedpreviously t’akes a rather complicated form. From a practical point, of view this is not desirable. However, it should be notred that, in the calculation of the ensembleaverage of certain physical quant#ities (such as energy or velocit8y) it, is really not necessary to first obtain the distribution function. Most of the calculat’ions may be accomplished by direct considerations of hhe “moment.” To demonstrate this, we shall consider t’he following examples1 1 The discussion is still restricted
to the assumption
of uniform Y.
(B X E,) sin ait -. t’)
ws wit - t’)Eo + bu
cos wit -
t’)Eo + &
(B X E,) sin wit -
i-K)
t’) (47)
+ A2
(B.E,)B[I
- cOs
w(t -
t/j]
This implies that, for any give11function C,O( t), t,he drift velocit’y may he rendily computed. It should he remarked t,hst’ if v is not, independent of 21,(47) il;: of court not valid and should be modified to the form
- cos w(t -
t’)Eo
+ >a
(B X E,) sin w(t - t’)
+ ,Lzi5 (B.Eo)B[l
-
ms ~(t -
t’)]
(18)
cp(t’) (It’. !
The cnloulatioll of elcct,ron current density J = -w(v) for the casecp = cos fit has heel1 performed bawd on (47). For simplicity, WConly list the results of the “persistent” part, (large tjime, f >> 1 IV) .~ ,%z ))l
+-
\
[v’
+
~0s’ y[v cos pt + /3 sin v‘?+ 8”
(0
-
wy][v’
+
(P
+
cd)‘] (19)
St]
17
J, = **E. Jr;
sin y cos y m
v- -__.. cos /3t__+ fl sin ~- _@f v'? f P" 7:
_ [v(v* + cd*+ /3”) cos Pt + p(/3’ - cd*+ v’) [v” + (B - w)“][v” + (6 + w)‘] ---r
(51)
sin /Xl\
J Irn, J‘zoc) and ./aa arc t,he t.hree componcnt.s of t,he current density J, in the coordinat’e system specified in Fig. 1. The total current J is therefore
J = JA + J& + J& , where J may he furthermore considered as the sum of conduction and polarization currents. In other words (5)
J= = Je + (Wat), where P is the polarization vector of the plasma P = D - toE, 60being t.he dielectric constant, in free space. In t,he most general case the actual dielect,ric constant Eis expected to he a tensor; WCmay t,herefore write Pi = (Q. - djk)E:I. ) where &+ is the Kronecker deha. Postulating that Al; is independent of time, we have dPi/dt = (tik - ~&)(dEk/dt) for (D= cos Pt Hence J-i = [a,k cos /3t - (Q - E&)@ sin @]Eok.
(52)
If the z,U,a-system shown in E’ig. 1 is used, then t,he tensorial conductivit,y and dielect,ric co&ant may he comput!ed by comparing (52) with (49)-( 51) , with the following results: (1) Conductivit,y
uv* 1 2 [ >+(w-p)*
1 u ui/r=-uq, =~2v[v*+w-P (w +p@ (w- 0)'+v*+?+w 1' u,, = url/ = -
uz.-
=
u[v2/(v2
+
@‘I],
+
1 v* + (w + py
un
=
uzg
=
0,
’
u
=
We*/mv.
EIACTRON
DISTRIIWTION
FIG. 1. Coordinates
for
399
FUNCTION
the electron
current
denAl>
(2) Dielectric constant P-w v2 + (0 - w)?
%x = E!/.V=
V
V er,y
=
-eyz
=
TTY
II
v? +
(0 -
w)2 -
v2 +
EZZ = co(l+ &)
(P
+
WY
1 ’
6, = $I = 0, wp2 = Ne2/mco.
‘These results agrrc with those given by Margenau (6), Kelly (7), and Al’pert et al. (8). B. ELECTROS
ME.IX
ENERGY
The mean eleckon energy may be expressed in terms of the second moment’, (53) IIomever, from (37’) and (38 ) : ~
= 0
e2f30'co(t) mkl
MO
2 *="
( -
1)"
[ TY$?
+
sin'
y
Re
(7+k)"i']
!I%$!
and
In the following WC shall consider the example cp = cos @r. The mean electron energy in t,his case may be determined by first, finding M,“(t) Since it, can he readily shown that
22 PfV cos2pt cobs2 /3t p cz E2y?v 2w ( zqq? [
sin
2/3t >I
+~[cos2r(~~~)+sil~~rRe((v+i~)2+82)]
I
sin 2/.3t _ pw cos 2pt A2 + p2 A2 + B”
t2v2
I)
+ ;kT.
(“’
401
Icor the limiting caseas p * 0 J ‘Lm(l’2)= ( J/p”Z&/$$v’)
[( yy + cd2ros2r,/(
Y?+ CA] + ~~lie7’.
This rcsnlt agrees with that obt,ained previously (2) VI. I~)ISCUSHION
III t,he discussion of the “persistent” solut,ion, we have implicitly assumed that the time function cp(t) and its derivat)ivrs are continuous and hounded. This causes certain limitatjiolv to the application of the results. However, if one uses ($5) instead of (37) and (S), the derivatives of the funrtion q(t) do not. have to br continuous rind bounded. The discussion of the paramrter x which appears in the expansion (27) is given for the quasi-steady raw. l:or large time (t >> 71, it has been show1 that, if we take (58) the best, convergence may be obt,ained. Although the expression of X,(t) which may give a similar result still remains unknown, for the general case it is yet conceivable that, (58) may be used to obtain good convergence of ( 27). In closing it may be remarked that for P + 0, the drift velocity calculated by the present met)hod agreeswith the solution obtained from a generalized Spitzer’s equation (9)) as expected. APPENDIX.
ESPASSIOS
OF THE
FUNCTION
Because t,he function y( II, t) is expected to hare “Gaussian-type” we propose the following expansion
behavior.
X is an arbikary parameter to be discussedin the paper, Lka( h”n’ J (I; = 1,3, . . . ) are th’c associat,edLaguerre polynomials in which Q is to be determined, and ak( a, t) the expansion coefficients. In an attempt to det~erminr t,he value of CYfor the proper cspnnsion. we cow sider the following integral
=
al;.(X,
k’
f)cr! (
+ li'
a
(3) >
'
Howww, from the definition also be writ,trn as
It may be visualized
of the Lagurrrc
1;
=
,g
+
k’
l:L
(
-1)“X2”f”
or,,,
1L
a,
RECIGIVED:
to thank
Dr.
C.5)
!
)
This enables us t,o express the coefficients
wish
I mw.y
-:,
‘(,
authors
the integral
t)hat, if we set a = f$
I
The sions.
polynnmi:d,
=
1; 2 n=O 0 n
Frank
( -
ak iu terms of the moments 1 )A”“‘”
(n + l:i) !
B. Estabrook
for
M?,,(t).
his interest
and helpful
discus-
April 25, 1Scil REFEREKCES
1. 8. CHAPIIAN AND T. C;. COWLING, “The Mathematical Theory of Nonuniform Gases.” Cambridge Univ. Press, London and New York, 1952. 8. C. 8. Wu, Proc. Roy. Sot. A269, 513 (1961). 3. W. MAGXUS AND F. OBERHETTINGER, “Functions of Mathematical Physics.” Chelsea, New York, 1954. 4. J. L. OELCROIS, “Introduction to t,he Theory of Ionized Gases.” Interscience, Sew York, 1960. 5. J. A. STRATTON, “Electromagnetic theory.” McGraw-Hill, New York, 1911. 6. H. MARGEIVATT, Ph!/s. Reo. 69, 508 (1946). 7. 1). KELLY, Phys. Rev. 119, 27 (1960). Y. L. AL'PERT,. V. L. GINZBURG, AND S. FEINBERG, “Propagation of Radio Waves” 8. (in Russian). Gost,ekhizdat, Moscow 1953. 9. J. ZSII'IDZINAS AND C. S. Wu, Report TR-34120, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, 1960.
Distribution Magnetic and Jox.4~ Physics
Section,
Function of a Weakly Ionized Time-Dependent Electric Fields*
S. ZMUIMSAS
.I\ND CHING-SHEKG
Jet Propulsion Lnborutoq, Paliforniu ‘I’echnolr~g:~~, Pasadena, Pnliforxia
Gas
in
Wrr Institute
of
An investigation is made of the electron behavior in a weakly ionized gas in n magnet.ic field and arbitrarily time-dependent. electric field. This problem is of interest eit,her in astrophgsics or the study of t,he basic phenomena in gas discharge. The electron distribution function is studied based on the Boltzmann equation u-hich takes only the electron-atom collisions into awount. A “moment method” is devised to solve the case nhen the electron collision frequency is uniform. In t.he subseyuent, analysis, the electron distribution function is first expanded in terms of hoth the “moments” and the associate I*aguerre polynomials and then a generating equation of the moment,s is derived. This generating equat,ion which is linear and of first order in time ran tie integrated readily. The persistent solution is obtained in closed form for the quasi-steady case. In caw of high-frequency oscillating electric field t,he drift velocit.?-, t,he mean p-rtrrgy, t,hr conduc:iivit,.\r and the dielect,ric constant. :xe ttll diswsxed. I. INTROL)UCTIOiL’
The present inwstigation rowerus a slightly ionized gas in thr prcstncc of a uniform magnetic field and a time-dependent1 electrical field. The plasma is postulated to he homogeneous and before t,he external fields are imposed, to possessMaxwellian distribution correspondiug to the gas tcmpcraturc 7’. The problem under st)udy is the determinat,ion of the clc&on vclocit,y distribution funrtion of such a plasma, as well as the corresponding average elertron energy and drift velocity. The Boltzmann equation for the present problem may he simplified by dropping t,hc collision int.egrals desrribing th(x rlr~t,roil-clcc~.ror1 and electron-ion encount,crs. In the following discussion, only the rlectron-atom collisions, for which the “con~cnt~ional” close binary collision model may be assumedvalid, are considered. Again, we shall assunw that, most, of t,he c~ollisions arc’ c>lastica. This itnplics that the :rwragc clwtroil cnrrgy is low and the rsttwrul field is not wry strong. * This papclr presents the results sion I,aboratory, California Institute nored 1,~ the n’atiottal Act ona1ttic.s
of one phase of research of Technology, under and &Space Administration. 387
rarried Contrwt
ollt
at t,he Jet PropuNo. NASn-6, spon-
In the following analysis the fields generat.ed by the int.wnal dist,rib\ltions and motions of t.hr charged part iclcs art’ post.ulatctl to kc ncgligiblc compared to the applied fields. Hence ill the Boltzmann ccluation both E and B are wnsidered to hc knon-11 cluantities and hcrraftrr wc will write the clcctric field E in the form
E = Eodf), where cp(t) is any given function of time which is hounded and continuous at all time. With the initial Maxwellian distribut,ion n-c may obtain a rolution for the subsryr~ent, time (t > 0) based on the assumpt.ion that. the anisotropic part of the dkkrihution fun&m is a small perturbation of the isotropic dist,ribut,ion. In the following discussion a “moment met,hod,” which will be dcwonstrated lat,er, is used. Our main interest, is to invest,iguk the “persistent, sohkion” (t >> 7, where 7 is the mean collision time). A closed form solution for the quasi-steady case (or some other special cases, for instance, the ac field) can bt found by considering the asympt,otic behavior of the distribution function. The analysis of t,his part is displayed in Section IT:. The calculat,ions for the electron drift velocit,y, mean energy, conductivity, and dielectric constant, based on t,he “moments” are also given in the final part of t#his paper. II.
DEI>CC;TIOK
OF
THE
GOVEILSI?r’G
IN’~EC;ltO-I~IFFElt~~~~I~4L
EQUz4TIO?:
Under the assumptions stated previously, t.hc eltctron distrihut,ion function ,f(v, t) is considered t,o satisfy the Bolt,zmann equation of the following form ( 1) :
E,cp(t) + v X B).;
1
.fCv,O
In ( 1)) the primes denote quant~it,iesafter collisions, and u,( g, J/) is the differential cross sect,ion for elastic scatt’ering of electron and neukal parbicles through an angle #. The solid angle dQ may he expressed in the form dQ = sin 1c,d# d+,
wliere 4 is the polar angle measured in a plane normal to the vect,or g which is t,hr velocity of clect,ron relative to t,he colliding atom, i.e., g = v - v, . Furthermore the subscript m denotes quantities which belong to the neut’ral at,oms. In order to solve (1) for .f(v, i), we shall postulate t,hat, the elect,ron distribu-
ELECTROS
tion funct~ion may hc written j’iv, t) = .f”‘%,
I~ISTRIBUTION
in t.he following
t) + E&“‘(P,
form:
t)
+ (B X EoJ .vf@)(~j, f) + (B.v)(B.Eoj.f”“(2’, The init,ial conditions given as follows:
889
FCSCTION
of t,hc unknown
functions
t).
(2)
J(“), j(l), ,f(‘)‘, and f’“’ may be
.4t t = 0, .f”“(l’, 0) = N(
.f%,
m/2TkT)R~"
CSP (-Vla’/i’liT),
0) = 0,
.f”‘(P, 0) = 0, .f(Yl’, 0) = 0, whrre N is the electron number den&y and /C is expression (2) is merely the first t,wo terms of the the spherical harmonic in t’he component of v. A of this expression is given by Wu (2). Following obtain
the Roltzmann constant,. The expansion of f(v, t) in terms of brief remark on the deduction the similar discussion (2), w
)I af”’ L== at
-- ; f’” + ;
; !!?f;;:cp(t) - gy C df’*’ __ _ f f21 + ; f(l), at af L ‘“’ == - ; f”’ + f f”‘, al
i-l) ’
1 )
where nz and iI/ are the elcctjronk and atomic mass, rcapec~tivrly, 1 the mea11 free pnt,h of electrons, and y the angle hctwccn t,hc field \wtors B and E. Now we shall introdlwe the Laplace transform
where s is t,he variable in t,he transformed space and has th(l property Re[s] > 0. Using the initial conditions (3), we may transform (5)-(i) to the following
390 forms :
(9, (10) (11) where L[(df”“/d~)cpJ denotes t,he Laplace transform of (@““/&~)~(t) and, for simplicity, we hare denoted 1,‘~ = 7. T is t,he collision t,imc and in gcncrsl is a function of 2~. From (9)-( 11 ) we oht,aiil (ST + 117 = (ST+ lj2+W272m2~
-
f”’ ,f”’ pi
CJ[
’
(
af’“’ do ‘p ’ >
( 12)
= ~ 2 (s7 + I;2 + w2+2m’L lfJ ;l L ( ac , p > ’ T3 + 1)’ +
= (~7
~27~
‘1, m3 1’
?Pp (
)
cyclotron
f”’
[cos’y + sin’y cos w(t - I’)]
B2
cos2yf3)
=
v
vu:
t e-“‘-“’ so
(14)
>
where w = eB/m is the electron one may easily show +
(13)
frequency.
From
( 12) and (14),
(15) x
af0’(2!,
t’)
a2) where v = l/7. Combining ( 15) and (4), f’“’ for t 1 0.
For simplicit.y,
hereafter
we obtain
an integro-differential
we denote 2~ = 21/a, a = d2kT/m, E = y(u,
t)
m/A/,
= f’“‘(P,
t).
dt') dt', equation
for
ELECTI~ON
IIISTRIB'CTIOZI
FUNCTION
:391
Then 1:16 1 becomes
Before we close this se&ion, t,wo points should he remarked. (1) It, is possible to derive a general expression of t,hc dist,rihution function .f(v, t) in t,erms of f’“‘(z1, t) or g( u, tj. From (12)-(1-k), we may obtain the inverse i~rsnsform as follows:
Hence f(v, t) = g(u, t) + &
1’ e-““-“’
+ {[e(B X Eo)~u],lmw} + (e”/m”w’)
(BvE,)(B.~l[i
I (E,.u)
cos w(t - 2’)
sin w(t -
~0s
(21)
t’) ~(t
-
t’)])
(ag/att)&t’)
dt’.
Therefore, once g(u, t) is known, t,he distrihut,ion function .f(v, 1) may he immrdiately calculated from (21) . (2) The asymptotic expression of .f( I’, t) ns t + = may he obtained 1,~ showing, from Eq. ( 18)) t,hat,:
Is (“--iw)l’ af’“’ st
e 0 = 2mau
e
A-ao cp(t’1 (if’ +O (A!- id)t
p+Lw)t
a”‘“’ x cp(l’) co’ e”+‘“”
(22)
392
ZMuIDZIS.\S
where g=(t)
represent,s
.4x1>
the behavior
\vc
of y as / +
x . $‘imilarly,
from (19) alld
(23)
(24)
(25)
III. A.
EXPaKSION
OF
THE
METHOD
IcUSCTION
OF g(U,
SOLUTION
t)
In an attempt to solve (17), we shall devise a method which contains two essential steps: (1) expanding t.he fun&on g(q t) in terms of its moments as defined in the following
m nr,,= 2s g(IL,t)lL’L+2 d1L 0
(26)
and (2) t,hen deriving the “generating equation” for t,hese moments. First, we shall expand the fun&ion g( U, t) in terms of the Laguerre polynomial g(u, t) = P?
g &(A, t)L: 2GAL2~,
where A is a paramct’er which may be a function of t. The determination X(t) will be discussed latter. The Laguerre polynomial Lk” is defined as (3) :
(27) of
(28)
ELECTROS
DISTKIBUTIOX
FUSCTIOS
393
The reason for choosing the Laguerrc polynomial of order J$ in particular is mainly based on its orthogonality property. The discussioll of this choice may be found in the Appendix in which it has also been shown that the coefficients ak ma:y he expressed in terms of t,he momcuts
The choice of X will be based on the criterion that the fast,est, convergence (27 ) may be oMained. To illustrate this, we consider the special case at t = 0. Since
of
it may he sho\vn that
Therefore
Lct,ting X = 1, we have the first’ t,erm only. This implies that the fastjest, convergence may be achieved if X = 1. Howevrr, when t # 1 this is no longer true, as we shall see later (Section IV).
In order to drtcrmine t,hc moments, we shall ret,uru to ( 117). In the following discussion, WC shall restrict, ourselves t,o the c’nsc t,hat v ii; uniform; in some cusw t,his furnishes a good approximation for weakly ionized gas ( 4). In this cast, if each term of (17) is multiplied by u”‘+’ and integrated over all values of IL, we obtain a system of integro-differential-difference equations which may be wri tteii as follows:
Since by dctnmion m ill,, = 2 = 2
s0
g(7l,
t)rl’ du = 3
I0
{](7L, 0)u’ du, (,xi )
m
we may therefore proceed to calculate the higher moments d12 , Ail4 , and so forth by integrating (32). At, t.his point, the problem is solved in principle. In closing this se&ion, we may remark that, it, is possible t,o express the distribut,ion function .f(~, t) in terms of the moments,
where Y(u, t) = ((E0.u)
cos w(t - 1’) + (e/mw)(B + (e”/m’w’)(BE0
IV.
PERSISTENT
SOLUTIOXS COLLISION
X Eo). usin w(t - t’) cosr)(B
Ip\’ THE FREQUEKCY
. u)[l
CASE
OF
- cos w(t - t’)]). UNIFORM
It is seen from the previous discussion t,hat considerable mathematical plificat,ion may be obtained if the collision frequency may be considered independent of 71. From (32), we may obtain by integration
simto be
(35)
M2,,+2( t) = e--e”(2n+2)t 11f2,z+2(O) where a+,(t)
= (an + a)(211 + 3)
{2&h&
cp(t) I’ e-“l-t,’
. [cos’ y + sir12 y cos w(t - t’)] +O(t’)Mfn(t’) In (35)
t#he integration
constant
dt’ + f wM2n(t)
)
i112,,+2(0) may easily be evaluated:
dl2,,+2(0) = 2 om g(u, o)u2”+4 du I = Yv(~;a)-3r(n
+ 35).
(36)
.
ELECTRON
DISTRIBUTJON
If we denot’e X?,L((t) as the asymptot’ic N.?,, (t ) , t.hen we have:
I’o&l)le further special cases :
The quasi-steady
simplications
form
of (37) and (38) may he found for the following
case is dcfincd p/v”
when
<< cp,
tho following
Jig/v”
conditions
<< II?,, )
hold f,rue :
k = I) 2, 3, . . . )
where 5~(” and Jr;:,’ denote the derivatives of cp and N?,, , respectively. The physiral meaning of these conditions is that the changes of the functions cp and J/, for the time interval T( = llvj are small compared to the values of the functions. This situation enables us to rewrite (37) and (38) as follows: @,?,,~= (2,n + 2)(2n + :3)e%o?p~t) cos?y y + sin’yRe 6mlcT
one may deduce from (:-D-(41)
( )ICpM;,, I-v + iw
(39)
that
where x;*
= M,*(o)
= NC 2/Taj-3.
Hence, the asympt,otic form of g(?s,t) for large time may be given in a rather
simple form :
where
It is clear t,hat the optimum
choice of A, is h,(f)
= [I + n(tj]-I”.
Then grn(u, t)
N( ~MX’)-‘~[~
because
+ Q(t)]-“‘”
exp ( - (u’/[l
Ego (i)
(- l)“L:‘“(x,“u”)
+ Q(t)] )I
(44)
+ Q(l)])).
(45)
= 1.
As a consequence, dg”/du
= --2N(&~)-~u[l
With the suhstit,ut,ion f(v, t) is found. B. HIGH-FREQUENCY
+ Q(t)]-“”
exp { -(2?/[1
of (44) and (45) into (25), the persistent
OSCILLATIXG
solut,ion of
FIELD
The second special casefor which (37) and (38 j may be simplified is t#hat the electric field E varies according to a time function p(l) = eifit. In this case, the approximation used previously is no longer valid if t)hefrequency 6 is of the same order of v. However simplification is still possible, since sometimeswe only need the information of MO and d12. To ill&rate this point, discussionswill be given in the following section. V. APPLICATIONS
The dist,ribut’ion function j(v, t) discussedpreviously t’akes a rather complicated form. From a practical point, of view this is not desirable. However, it should be notred that, in the calculation of the ensembleaverage of certain physical quant#ities (such as energy or velocit8y) it, is really not necessary to first obtain the distribution function. Most of the calculat’ions may be accomplished by direct considerations of hhe “moment.” To demonstrate this, we shall consider t’he following examples1 1 The discussion is still restricted
to the assumption
of uniform Y.
(B X E,) sin ait -. t’)
ws wit - t’)Eo + bu
cos wit -
t’)Eo + &
(B X E,) sin wit -
i-K)
t’) (47)
+ A2
(B.E,)B[I
- cOs
w(t -
t/j]
This implies that, for any give11function C,O( t), t,he drift velocit’y may he rendily computed. It should he remarked t,hst’ if v is not, independent of 21,(47) il;: of court not valid and should be modified to the form
- cos w(t -
t’)Eo
+ >a
(B X E,) sin w(t - t’)
+ ,Lzi5 (B.Eo)B[l
-
ms ~(t -
t’)]
(18)
cp(t’) (It’. !
The cnloulatioll of elcct,ron current density J = -w(v) for the casecp = cos fit has heel1 performed bawd on (47). For simplicity, WConly list the results of the “persistent” part, (large tjime, f >> 1 IV) .~ ,%z ))l
+-
\
[v’
+
~0s’ y[v cos pt + /3 sin v‘?+ 8”
(0
-
wy][v’
+
(P
+
cd)‘] (19)
St]
17
J, = **E. Jr;
sin y cos y m
v- -__.. cos /3t__+ fl sin ~- _@f v'? f P" 7:
_ [v(v* + cd*+ /3”) cos Pt + p(/3’ - cd*+ v’) [v” + (B - w)“][v” + (6 + w)‘] ---r
(51)
sin /Xl\
J Irn, J‘zoc) and ./aa arc t,he t.hree componcnt.s of t,he current density J, in the coordinat’e system specified in Fig. 1. The total current J is therefore
J = JA + J& + J& , where J may he furthermore considered as the sum of conduction and polarization currents. In other words (5)
J= = Je + (Wat), where P is the polarization vector of the plasma P = D - toE, 60being t.he dielectric constant, in free space. In t,he most general case the actual dielect,ric constant Eis expected to he a tensor; WCmay t,herefore write Pi = (Q. - djk)E:I. ) where &+ is the Kronecker deha. Postulating that Al; is independent of time, we have dPi/dt = (tik - ~&)(dEk/dt) for (D= cos Pt Hence J-i = [a,k cos /3t - (Q - E&)@ sin @]Eok.
(52)
If the z,U,a-system shown in E’ig. 1 is used, then t,he tensorial conductivit,y and dielect,ric co&ant may he comput!ed by comparing (52) with (49)-( 51) , with the following results: (1) Conductivit,y
uv* 1 2 [ >+(w-p)*
1 u ui/r=-uq, =~2v[v*+w-P (w +p@ (w- 0)'+v*+?+w 1' u,, = url/ = -
uz.-
=
u[v2/(v2
+
@‘I],
+
1 v* + (w + py
un
=
uzg
=
0,
’
u
=
We*/mv.
EIACTRON
DISTRIIWTION
FIG. 1. Coordinates
for
399
FUNCTION
the electron
current
denAl>
(2) Dielectric constant P-w v2 + (0 - w)?
%x = E!/.V=
V
V er,y
=
-eyz
=
TTY
II
v? +
(0 -
w)2 -
v2 +
EZZ = co(l+ &)
(P
+
WY
1 ’
6, = $I = 0, wp2 = Ne2/mco.
‘These results agrrc with those given by Margenau (6), Kelly (7), and Al’pert et al. (8). B. ELECTROS
ME.IX
ENERGY
The mean eleckon energy may be expressed in terms of the second moment’, (53) IIomever, from (37’) and (38 ) : ~
= 0
e2f30'co(t) mkl
MO
2 *="
( -
1)"
[ TY$?
+
sin'
y
Re
(7+k)"i']
!I%$!
and
In the following WC shall consider the example cp = cos @r. The mean electron energy in t,his case may be determined by first, finding M,“(t) Since it, can he readily shown that
22 PfV cos2pt cobs2 /3t p cz E2y?v 2w ( zqq? [
sin
2/3t >I
+~[cos2r(~~~)+sil~~rRe((v+i~)2+82)]
I
sin 2/.3t _ pw cos 2pt A2 + p2 A2 + B”
t2v2
I)
+ ;kT.
(“’
401
Icor the limiting caseas p * 0 J ‘Lm(l’2)= ( J/p”Z&/$$v’)
[( yy + cd2ros2r,/(
Y?+ CA] + ~~lie7’.
This rcsnlt agrees with that obt,ained previously (2) VI. I~)ISCUSHION
III t,he discussion of the “persistent” solut,ion, we have implicitly assumed that the time function cp(t) and its derivat)ivrs are continuous and hounded. This causes certain limitatjiolv to the application of the results. However, if one uses ($5) instead of (37) and (S), the derivatives of the funrtion q(t) do not. have to br continuous rind bounded. The discussion of the paramrter x which appears in the expansion (27) is given for the quasi-steady raw. l:or large time (t >> 71, it has been show1 that, if we take (58) the best, convergence may be obt,ained. Although the expression of X,(t) which may give a similar result still remains unknown, for the general case it is yet conceivable that, (58) may be used to obtain good convergence of ( 27). In closing it may be remarked that for P + 0, the drift velocity calculated by the present met)hod agreeswith the solution obtained from a generalized Spitzer’s equation (9)) as expected. APPENDIX.
ESPASSIOS
OF THE
FUNCTION
Because t,he function y( II, t) is expected to hare “Gaussian-type” we propose the following expansion
behavior.
X is an arbikary parameter to be discussedin the paper, Lka( h”n’ J (I; = 1,3, . . . ) are th’c associat,edLaguerre polynomials in which Q is to be determined, and ak( a, t) the expansion coefficients. In an attempt to det~erminr t,he value of CYfor the proper cspnnsion. we cow sider the following integral
=
al;.(X,
k’
f)cr! (
+ li'
a
(3) >
'
Howww, from the definition also be writ,trn as
It may be visualized
of the Lagurrrc
1;
=
,g
+
k’
l:L
(
-1)“X2”f”
or,,,
1L
a,
RECIGIVED:
to thank
Dr.
C.5)
!
)
This enables us t,o express the coefficients
wish
I mw.y
-:,
‘(,
authors
the integral
t)hat, if we set a = f$
I
The sions.
polynnmi:d,
=
1; 2 n=O 0 n
Frank
( -
ak iu terms of the moments 1 )A”“‘”
(n + l:i) !
B. Estabrook
for
M?,,(t).
his interest
and helpful
discus-
April 25, 1Scil REFEREKCES
1. 8. CHAPIIAN AND T. C;. COWLING, “The Mathematical Theory of Nonuniform Gases.” Cambridge Univ. Press, London and New York, 1952. 8. C. 8. Wu, Proc. Roy. Sot. A269, 513 (1961). 3. W. MAGXUS AND F. OBERHETTINGER, “Functions of Mathematical Physics.” Chelsea, New York, 1954. 4. J. L. OELCROIS, “Introduction to t,he Theory of Ionized Gases.” Interscience, Sew York, 1960. 5. J. A. STRATTON, “Electromagnetic theory.” McGraw-Hill, New York, 1911. 6. H. MARGEIVATT, Ph!/s. Reo. 69, 508 (1946). 7. 1). KELLY, Phys. Rev. 119, 27 (1960). Y. L. AL'PERT,. V. L. GINZBURG, AND S. FEINBERG, “Propagation of Radio Waves” 8. (in Russian). Gost,ekhizdat, Moscow 1953. 9. J. ZSII'IDZINAS AND C. S. Wu, Report TR-34120, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, 1960.