IL N U O V O RIVISTA 0RGAN0 SOTTO E
DEL
DELLA
GLI AUSPIGI COMITATO
CIMENTO
IIqTE RNAZIO SOCIETA
DEL
CONSIGLIO
NAZIONALE
I~ALE
ITALIANA NAZIONALE PER
L'ENERGIA
Serie decima
VoL. X L I I I B, N. 2
DI
FISICA
DELL]~ RICERCttE NUCLEARE
11 Giugno 1966
Relativistic Plasma in a Magnetic Field (*). B. Ku~sv~o~Lv
Center for Theoretical Studies, University of Miami - Coral Gables, Fla. (risevuto il 3 Settembre 1965)
Summary. - - Transverse and longitudinal oscillations of a low-density relativistic plasma in bhe presence of a constant magnetic field are considered b y neglecting the collisions. The most general form of the relativistic dispersion relations for linear plasma oscillations is derived and the two extreme eases of zero and infinite equilibrium t e m p e r a t u r e s are discussed.
1.
-
Introduction.
T h i s p a p e r is ~ f u r t h e r g e n e r a h z a t i o n of t h e r e l a t i v i s t i c t h e o r y d i s c u s s e d b y t h e a u t h o r p r e v i o u s l y (1) w h i c h will b e r e f e r r e d t o h e r e as R P . T h e n o n r e l a t i v i s t i e t h e o r y of t h e p l a s m a o s c i l l a t i o n s in t h e p r e s e n c e of a n e x t e r n a l m ~ g n e t i e field h a s b e e n i n v e s t i g a t e d i n s o m e d e t a i l b y B ] ~ S ~ E I ~ (3) a n d b y A KH,~:ZER et aL (B) a n d also b y o t h e r s . I n t h i s p a p e r w e s h a l l cliscuss d i s p e r s i o n r e l a t i o n s f o r h i g h - t e m p e r a t u r e p l a s m a u s i n g r e l a t i v i s t i c t h e o r y . I n t h e p r e s e n c e of a m a g n e t i c field t r a n s v e r s e (') This work is supported b y USAEC Contract No. AT-(40-1)-2761 and b y Air Force Otiloe of Soioatific Research, Washington, D.C., U. S. Air Force Contract No. A F 49(638)-1260. This research was completed in the late summer of 1961 while the author was visiting at Max Planck I n s t i t u t fiir P h y s i k and Astrophysik, Munich. (1) B. KUnSUNO~LU: Natl. Pusion, 1, 213 (1961). (3) I. B. Bv.nNSTEI~: Phys. ttev., 109, 10 (1958). (a) A. I. AKm~znu, Y. B. FAINBERG, A. G. SIT]~NKO, K. STEPANOV, V. KVmLI(O, •. GOI~BATEN]~O and U. KIROCItKIN: Proceedings of the Second United Nations Conference ou the Peaceful Use~ of Ator~ie Energy, vol. 31 (1958), p. 99 (Geneva, 1958). 14 - I I N u o v o Uimento B .
210
B. KURSU~0~-LU
a n d longitudinal oscillations a t zero t e m p e r a t u r e are decoupled ~nd longitudinal oscillation% in this case, are i n d e p e n d e n t of the m a g n e t i c field. One of the b~sie differences b e t w e e n relativistic a n d nonrelativistic t h e o r y m a y b e e x p e c t e d to arise in t h e excitation of higher harmonics in t h e f o r m e r case (for high t e m p e r a t u r e s ) a n d of t h e f u n d a m e n t a l f r e q u e n c y in the l a t t e r case.
2. - D e r i v a t i o n of dispersion relations.
I n t h e presence of a c o n s t a n t external m a g n e t i c field B t h e relativistic Boltzm a n n - V l a s o v e q u a t i o n is, as discussed in R P , given b y
~
(~.~)
ie
p ~,~ + ~ t'~'L,,~e = o,
where the m o m e n t u m space a n g u l a r - m o m e n t u m operators L ~ are defined b y
L~,,-~i p , ~ - - p ~
(2.2)
,
a n d the t o t a l field f ~ ] ~ =/(~)/,~ + -~,t(~)'
,~,f(~= external field.
Thus for equilibrium states the last t e r m in (2.1) vanishes (*) I n the a b o v e we used s u m m a t i o n convention for r e p e a t e d indices (/~, v = 1, 2, 3, 4). I n t e r m s of Fourier t r a n s f o r m s of f ~ a n d e eq. (2.11) can be linearized in t h e f o r m
(2.3)
( p~k~'+cB.L e ) e~--c~-~K~p~.dp~,j. ~o~ ~,~ [(k~p,~ k~p,~,)~dk' p,)d4p,,
where we h a v e p u t e(k, p) = (2~r)~n0p~(p) ~ ( k ) / - ~(k, p ) . All the symbols are to be u n d e r s t o o d in 4-dimensional sense, a n d @~ is small deviation f r o m equilibrium s t a t e ~m" I n t r o d u c i n g the current v e c t o r
(2.4)
Jr :/p~ ~l(k, p) d4p
(*) It is interesting to note that if p , and JS~ are treated as operators in I~ilbert space then they satisfy the commutation relations for the inhomogeneons Loren£z group. In this case, one can develop a quantum version of the eq. (2.1) in the form ps(~p/~x~)+ (ie/2~e)fs~Js,,e= O, where, now, Js~ shall operate on the momentum coordinates of the q u a n t u m distribution function ~.
I:~ELATIVlSTIC PLAS2~[A I:N h
MAGNETIC F I E L D
211
eq. (2.3) can be written as
(2.5)
uk" +
B..L
O* --c=k~k;~
gpt,"
We introduce the polar co-ordinates
(2.6)
[ p~ = p sin~f c o s ? ,
k~ = k sin T cos ~ ,
I p~ = p s i n ~ s i n ~ ,
k2 = k sin T sin ~ , ka =/~ cos T .
Pa = P c o s ~ ,
In polar co-ordinates the operator (e/c)B..L can be written as
eB. ~
ie B ~
8
where we assumed B to be along the z-direction. Equation (2.5) can now be written as (2.7)
8 +Z_~Xcos~_~Ysin~
]
~
=
imm~c ~k~£~
k,j.]p
De
~p~'
where X
-= iIcp sin F sin T cos q~, ~'b(D c
y =
u--ikPsin ~f sin ~ cos ~5, m(D c
Z =
mi mco~
(Po leo-- kp cos y~ cos T ) .
Our basic b o u n d a r y condition consists of assuming t h a t co = eko contains a small positive irnagina.ry part. Thus the single-valued solution of eq. (2.7) is
(2.8)
~1 = .
1
exp [~-A(~)]fexp [A(y)] tg(y) d y ,
where
eg~ (k, j , "Q(~) -- c~k~,k ~
kv j z ) P,
8p~J ~y"
212
:s. K ~ I ~ S U N O ~ L U
Using the t r a n s f o r m a t i o n
we obtain
~ol = .
(2.9)
1 ~m69c
exp [-- A(m)]fexp [A(~o-- ~)] g2(.q~--n) d,/, o
where dA -- Z - ~ X cos ~ d- Y sift ~ -d~0
i p~/~, mo)~
so t h a t
A(~o) = Zq) + il~p sin ~o sin T sin (~o -- q~),
(2.10)
m(D c
where the constant of integration is ignored since it will n o t appear in (2.9). The expression (2.10) can also be written as i
A(~) = z~ + - -
(2.H)
mo)~
(pike-- p~k~),
where supercripts and subscripts are related according to P~ = gz, P ~ ,
g ~ = - - ~i~,
g~ = g~ = 0,
g~ = 1 .
The polar co-ordinates of p ' s occuring in ~(~o--~]) in (2.9) are of the form p? i = P sin~ e°s ( ~ - - ~ ) ,
¢ P2 ----P sin ~f sin(~--~7),
/
p~ = p e o s ~ ,
and (A(~0 - - V) can be w r i t t e n as A ( ~ o - - ~) = Z ( q ) - - ~) d- i z sin(~o--~b-- ~/),
where (2.12)
z =
kp
sin yJ sin }P.
mrD c
f
Hence we see t h a t the m o m e n t u m variables p , in ~ ( ~ - - ~ ) are related to the m o m e n t u m variables pz in ~(~) b y a Lorentz t r a n s f o r m a t i o n of the form (2.13)
p~,'
~L
,~p v ,
RELATIVISTIC
PLASMA
IN
A
MAGNETIC
FIELD
~]3
where
(2.~)
( L ~) =
I eosu sin~ -- sinU cos~ 0 0 0
0
0 0 1
0] 0 0
0
1
Now, multiplying (2.9) b y pr and integrating over all p~s we get (2n5)
J'r -~ s~ f p~" [exp [--iz sin (q~-- ~))]. co
•fd~/exp [ - - ~ Z ~- izsin ( ~ - 0
where
-, ~p,~ = ~ - ~p,~ (2n6) l~ow putting oo
(2.17)
M~=exp[--izsin(q~--qS)
xp[--~TZ-~izsin(~--qb--~?)]L~Z~d~ 7 0
and carrying out an integration by parts we find
where (2.18) ttence we obtain (2.19)
Jr .=fp~rq~d4 p ~ [" ~" 8~q
as the four equations from which dispersion relations axe ~o be obtained b y eliminating the current densities j r . We may now calculate F ~ b y mo
(2.20)
lVoo = exp [--
iz sin ( ~ - - ~b)]/ exp [-- uZ + ~z sin (~-- ~ - - V)] S '~qd~7 , J 0
214
]~. KU~SU~OGLU
where
(2.2~) Thus S,23 ~ S2a c o s ~ - - ~qa~sin~ 7
S '~ :
S x~ cos~ ~ S ~ sin V ,
S ' ~ = S ~ sin V d- S ~ cos ~] ,
E q u a t i o n (2.10) can b e w r i t t e n , in t e r m s of Bessel f u n c t i o n s , as
(2.22)
/~
= exp [ - - iz sin ( T - - ~b)]. ¢o
--go
0
Hence
lw~ = exp [-- iz sin (9-- ¢)] ~ J.(z) exp [in(9-- #)]
S~3(Z + in) - - S ~ ( Z + in)" + l '
S~z(Z ÷ in) + S ~ ( Z ÷ in)~ + l ' co $1~ F ~ = exp [ ~ iz sin (~ - - ~ ) ] - ~ J,,(z) exp [in(q~-- q~)] Z -4- in?
F 3~ = exp [ - - i z sin ( ~ - - ~5)] i J~(z) exp [ i n ( ~ - - ~)]
(2.23) F ~4 = exp [ ~ iz sin ( ~ - - ¢ ) ] _~i J,~(z) exp [in(qJ - - ~b)] czJ
/~4 = exp [ - - iz sin ( T - - ~b)] ~ J~,(z) exp [ i n ( ~ - - 0)] --¢o
SI~(Z d- i n ) - - S ~
(Z + in) 2 d- 1
S2~(Z d- in) + 814 (Z -~ in)~ + 1
S3a H 84 = exp [ - - iz sin ( ~ - - ¢ ) J _ ~ J , ( z ) exp [ i n ( ~ - - q})] Z + i n "
3. - W a v e s propagating along the m a g n e t i c field.
~ o r w a v e s whose directions of p r o p a g a t i o n are parallel to t h e m a g n e t i c field we shall set ~ = 0 so t h a t kl ~ k2 ~ 0 ,
z ~-- 0 ,
J~' k~, ~ J4ko + Is3 J~ -~ 0 ,
k = k3 •
I~ELATIVISTIG ~LASI~& IN
215
~k ]YI&GNETIC F I E L D
:Hence, with J~(O)
=
a.0
the a b o v e relations yield ZS23__ $31 Z~÷ 1 F~--
(3.1)
ZL~14__ )~24 '
Z ~÷ I
ZS ~ ÷ S ~ F~ ~ Z~ ÷ 1 '
Z S 2~ ÷ 5 ~ Z ~÷ 1
Z'
Z'
and
(3.2)
8H"~
i
P~ 8p o -- meo~Z~ (k3p~-- kopa) S ~ ÷
÷
mo)o(Z~ + 1) 2iZ m o ~ ( Z ~ ÷ 1) ~
+ p~(4s ~-
koS ~ ÷ koz,~, + k ~ z s ~ ) ] .
F o r T = 0 the equilibrium distribution is given b y (3.3)
e,~ = a(p) a(po - - m e ) .
Using this in (2.19) ~nd noting t h a t the only surviving ~erm in (3.2) is ik3p~ ~q3~ m a4 Z ~
we get, for t r a n s v e r s e waves, t h e equations
j1
~coc ~o~ --
(3.4)
~ , ~ (o.~2 - - ~oo)(o~
- -
(o)~ - ~ ) ( ~ - I f we assume t h a t co ~ w equation
a n d ~o ¢ k c
(__~ k ~
J'--~j~)
c ~)
k ~ c~) these equations
yield the
(o)~-- ~ ) (~o - - ~e~) ~
dispersion
216
B. EUESUNO~LU
or t h e r e f r a c t i v e i n d e x N = kv/co 2
(3.5) l~or l o n g i t u d i n a l o s c i l l a t i o n s w e h a v e 2
j 3 __
ep~, (J3 co -F e k J 4) a ) ( w ~ k s e~)
(3.6) j 4 __
ck~
~o~(~-- l~e~)
(j3e ~+ ckj4).
H e n c e , for w ~ =/=k~c2, w e o b t a i n t h e d i s p e r s i o n e q u a t i o n
o)~-- co~(co: + k~o~) + c ~ k ' ~ = o,
(3.7)
w h i c h for co2va k~e ~ h a s t h e o n l y a c c e p t a b l e s o l u t i o n (3.8)
a)2
=
2~ (-Oto
w h i c h is i n d e p e n d e n t of m a g n e t i c field.
4. - Thermal corrections to oscillations along the magnetic field. For ~ = 0 we have Z
i -(poko--pk cosT) =-mcoc
i mope
p~k ~ .
Vlencc 2~(I _ ~ " =-~1 ~moJcS "
+ I+)
+-~1 mo~o531( I _ - -
I+),
~ 1 = - : 51~ m. ~ o o S 31( I _ - t - I + ) - - ~1- m c o o 8 23( I _ - - I + ) , ~.~12
(4.1)
1 1 ~1~ = 2 imco~ S l q I _ + I+) + -~ m ~ o , S ~ 4 ( I _ - - I + ) , 1 1 F ~4 = ~ imco~ S~4(I_ + I+) - - - ~ m w , 814(1_ ~ I+) , $34
2¢34 = imw~
p~,k~, '
RELATIVISTIC PLASI~I~kIN A MAGNETIC I~IELD
217
where 1
(~.2)
I ± - - P~, k~` ± m o ~
and S~ ~ __
i~o~k
m~oo(~oa-- ka e~) L~l a ~
0
(~.3) m w ~ ( e f - - k a c ~)
mog~(o~)~ - k'c ~)
'
o~o~me
Hence
(4.4)
&~,~ pq ~p~
~ ( ) (~opa-- poek) c% j 3 ~ 1 epoe9 309 p~k ~' - -
- - 2 cp~ p sin T
jx ~
(I+ e - ~ + I _ e~¢) + i J a
(I+ e - ~ -
I _ e ~¢)
.
Hence using the dispersion equation (2.19) we obtain
(4.5)
ji
I
o)2__~a02
:
Z
l
iJ~ ~ _ k~ e~ (It+ + It-) +
(R - - i t + ) - -
--~ --~c~o(Q+ + Q-) + "~' ca eco (Q+-- Q-) '
where the integrals R~ and Q± are given b y It-+ = 2I f p. k ~ ~1 m~oc ~ d4P
(4.6)
(4.7)
- =2Jpo
p~k ~ ~m~o~ e~d4P "
In a similar way
(4.8)
ja
1
÷ iJ 1
e92__ k2 e~
~ 2 _ _ ]~aca
mco~co~ ~ ~ -p ~ (R e + R_) ÷ j a ~% ~ ~ _ _ ka c a "~ ~ (Q+ + Q_) ~ i j 1 % ,
(Q+--q_) .
t)
218
B. KC~SUXO~LU
H e n c e the dispersion relations for t r a n s v e r s e waves in a m a g n e t i c field are given b y (4.9)
1 - - ~oz - k~e ~
0 9 2 _ k ~ e:
c~ ~
(Q+ + Q - ) =
(q+--q-)--~--~e~(R+ + ~_) or
(4.10)
l--co 2-k:c 2
w~--k=e
~R+ + e2 ~o~
and (4.11)
1
o~ 2m~o~ R +----. o~~ - k 2c ' ~ + ~o2 - k ~e~ e 2 ~o
F o r longitudinal oscillations we obtain (4.]2)
~$ + M c~
~ ~I0
2~$
which is i n d e p e n d e n t of m a g n e t i c field a n d was already Obtained in I~P for the e~se B = 0. I n the eq. (4.12) t h e integral Io is given b y
(4.13)
Io
_1( o
e J p~ k ~ ~ d4p "
W e conclude t h a t for oscillations along the m a g n e t i c field t r a n s v e r s e a n d longitudinal modes are deeoupled a n d longitudinal oscillations are i n d e p e n d e n t of m a g n e t i c field. A t infinite t e m p e r a t u r e (4.24)
lim~(Q++Q_)=~ ,x-~-O C(D
1
[
c 2 cO 1 + ~ (2--#2)~
[
log k ~ - ~
- - 2~z
(4.25)
S u b s t i t u t i n g in eq. (A.20) of the A p p e n d i x we obtain the dispersion relation
(4.]6)
--
=
~ = k~c~ o4JJ
which is i n d e p e n d e n t of the m a g n e t i c field a n d agrees with the one derived for B = 0. This is n o t surprising since m a g n e t i c field does not act on massless particles (in a linear theory).
RELATIVISTIC PLASI~
5.
IN A NIAGN]gTIG F I E L D
Waves propagating perpendicular to the magnetic field.
-
I n this case g * = z 1 2 , k a = O , Z
ipo ko ~o) c
z =
kp . 81II 9) moJ c
i i z sin (~o-- ~b) = ~ (p~k~--p2k~), 'VW~c
8z
k cos ~v
~pl
mco~
3q~
sin q~
'
~p~
p sin~ '
~z ~p2
k sin ~v m~% '
~ ~p~
~Z -~p~ -=0
1
~-~-~= 0 , ~pa
cos q~ p sin ~ '
where p sin ~v = V p ~ ÷ p~, p cos q~ = P3,
W e first discuss t h e T----0 case.
(5.1)
j~ _
(5.2) (5.3)
F r o m (2.19) a n d (2.26) we h a v e
me[ o~ ] I-- e~/co~ S~ ÷ i ~--S ~ 1 -- o~/o;
$ 2 4 - - S14
J ~ = i m c --we S~ a ~0
(5.4)
J4=m~c2
~-
where 2
im~)o( m 2 -
k s e~)
2
N31 __
c% imc%( m 2 -
j3 kl k2 c 2)
2
$1~ __
~% ( j 1 k . ~ _ j ~ k 1) , imc%(eo2 - k~ c~)
219
220
B. KV~SU~OGLV
~o, (J~ k o - - J ~ k ~) , im~o~( ~o~-- k~ c ~) g
imwo((o ~ - k ~c ~) s
imo~(~o~-- ks c ~) ~nd
j~ =---1 (jlkl ~_j~ks) " ko E q . (5.3) y i e l d s t h e d i s p e r s i o n r e l a t i o n (5.5) F r o m (5.1) u n d (5.2) w e g e t
(5.6)
j1 l__A__.4._A
CSk'~
(2) o
iA
=J~A
~0 c
i-----io~
CO
CO
~nd
(5.7) where A
~
- -
°)v°92--
(gO~ --~0~)(~2--k ~c2) " Hence we obtain (5.8)
N s=
--
~l--
w h i c h ~ t e)~--o~ b e c o m e s 2
(5.9)
~V~ = 2 - - - -
W e c o u l d also s o l v e t h e e q u a t i o n f o r ~o
(5.1o)
~o~- o~s(~ + ks c2 +
2
%)~ + %4 + ~s es(o~o + ~o~) = o
i n p l a c e of (5.8) a n d o b t a i n
(5.11)
2~o2 = (~o~ + ~ c s + 2 ~ ) :~ V(~.~-: ksc~)~ + 4 ~ o ~ .
(~0c
J
221
R E L A T I V I S T I C PLASI~IA II~ A M A G N E T I C : F I E L D
The nonrelativistic form of this equation follows b y assuming that t h e term ke is very large compared to all others. Thus _~ (~)~ + ~ ~ + 2~o~)± (m~--/~ c~) 1 +
2~
2(0. (%
l
_
-~ ((o.~ + ~ c~ + 2 ~ ) ± ( o , ~ - - ~ c~). Hence (5.12)
(o ~ -= m~ -~ o~~
~ o ~ = ~ ~ + k~e ~ .
and
It is, now, necessary to s h o w that eq. (5.4) is computible with the dispersion e q u a t i o n (5.11). To show this consider the m o s t general form of /~*~ as given by (2.20). W e m a y write (5.13)
F "0 = e x p
--m~
]
(P~k2--P2k~)
"
fo[
]
eo
o
H e n c e for zero-temperuture distribution w e need to k n o w co
(5.14) 0
which yields ¢o
/Y~ t
¢o
17 S '~4
"
/(~=°'~°=mc) = . o
exp
~
S ~4 cos ~ - - S 24 sin ~) exp
L
d~ =
O')cJ
D
~
(D ~-
(D
--
(,0~
[~0~~ + i~%~'],
03 c
~nd
8p ~ J0
m(o~-- ~)
+
co
+ =
Ilk2
mo~c .] o
cos ~ ~ - - kl
k~(o~% 14 - - i(o~ S ~4)
+
sin 2 ~ -~ (kl
k2 S 2~) sin ~ cos ~] exp
•
09
~ --
d~
222
B. K U I ~ S U ~ O GLU
÷
8P ~ Jo
-~{~mi[i(k~S~4~k~ S ~ ) --i ° ~ ( f ~ S ~
~-
-~ k ~ S : ~ ) - -k 2w~(k~ s ¢~ S~--k~S~)
"
Equation (5.4) is
Hence (using conservation of current) we get (5.15)
j1 kl -P j2 k~ ---2
60m g(O)2~
~
[i(co2__ ks c2)( j ~ kl _~_ j 2 k2) - - eoeoo(J ~k s - - j 2 k~)].
2
The last term can be calculated from (5.1) and (5.2) as
j~k2_
j2kl
_
_
~;)2 [I
io~(~ ~ -
~I
--
(~--
o,~)(~ ~ -
k~ c~)]
Substituting in (5.15) we regain the dispersion equation (5.11).
APPENDIX
Integrals for oscillations parallel to the magnetic field. The integrals R+ and R_ can be reduced to d [1 f exp [-- ~ cosh y] (# + #o cosh Y)l. . . . . . ~ - - ~ o.y o co
0
where (~
lz = ~ c '
(D ~
#° -
kc '
~2
~z = . ~
.
I~ELATIVISTIC P L A S M A
IN A M A G N E T I C
FIELD
223
Now we e~n write (# cosh y - - tt~)~ - sinh ~ y = ( 1 - - tt~)(~-- eesh y ) ( ~ + cosh y ) , where
~ + ~ = 2 V i i + ~)(1 + ~ ) ,
(A.1)
% =V(1 +p)(1 +~) +~, #
(n.2)
#c ~° - ~/~-~ ~ '
- VT~'
I
(A.3)
[ = 0 at a~=~%],
1 --#~ =
+/~-t t +tie 1 --,u 2
[ = 0 at c o = - - a ~ ] .
l%r the integral R+ the corresponding par&meters are obtained b y replacing ~ in the above relations b y --V~ and in this ease
(A.4)
]
• d~ ~ J [ ~ + cosh y + ~ - - eosh y (/* +/t~ eosh y) exI) [-- ~ eosh y] d y , 0
(A.5)
R_-
~K~V1
+ v~"
+v~
da ~zJ [ ~ - - cosh y + ~z~+ cosh y (# -- ~ eosh y) exp [-- ~ cosh y] d y . o
I-Iellce
(A.6)
R+ + R
--
d 2mckK~ d~
(A1 + A~)
&nd
(A.7)
R+ - - I t _
-
-
o: d (11-- I~) 2mckK2 do~
,
224
B. KURSUI~0GLU
where co
(A.S)
A(~,
i V ~
~l) = ~
-~ z
;exP [- ~ coshv] -
=
--go l
=V1
+2, f J [}
-w--VI+~.~--u~
i~ exp [ _ _ ~ v ~ du-- E~
0
(see RP). I n t h e p r e s e n t ease for A1, I1
=~/~-~ w h e r e we used t h e relations
]//1
+r/~ /~--(z~flo _ ~ / / ~ +~'~ /,+~z~/~ o --i.
(A.10)
F u r t h e r m o r e we h a v e t h e relations dA d~
(A.il)
1 A--~/1 +A2I, ~X A
o
i
af --fV 0
Thus
R+ +
~.
R_ --
1 o: (A~ + A~) + - 4mekK2 4mekK 2
(A.12)
(A~ + A~) + 4mckK~
A~ infinite ~emper~ure, i.e. o: ~
(gl/T1 + 0~212--IZ + /'2),
O,
2 K~ ~-~- -
1 K1 --~ -
for A2, I s ,
,
RELATIVISTIC
PLASMA
IN
A
MAGNETIC
FIELD
~925
so that lim R+ = lira R_ = 0.
(A.13)
c¢=0
c~=O
F o r other i n t e g r a l we h a v e ~Q_ ~o.~ =
(A.LI)
_
1
P~2 em 2 J (p~ k~-~ mo)~)~ d~p =
--
[
1 ~ ~fsirth~yexp[__acoshy]dyf tt cosh (y1---- xx~sinh )d x y - - te~
8k ~ K2
0
--1
Using 1
f
(A.15)
( 1 - - x 2 ) dx _ 2(te eosh y--te~) # cosh y - - x sinh y - - #, -sinh 2 Y d-
--1
+ sil~h~ y - - (~ oosh ~ - - teo)~ ~og [te eosh ~ + si.h ~ - - t~o] sinh ~ Y
Lfi ~
Y - - sinh y ~ J
'
we obi~ain (A.16)
&o
4k z co
1J' te - - tee cosh y exp [ - - ~ eosh y] dy . "~ (# cosh y - - tt~) "~- sinh 2 y o
T h e i n t e g r a n 4 can be w r i t t e n as
2 [~-- cosh y + ~ T ~ y ] "
(/~cosh y--te~):--sinh2 y ~enee
(A.~7)
8m
4k ~ ~ - 8 k ~ X ~Ste ( 1 - - / )
~--2##~--(1
v~
+te~)
•
1
[ exp E-- ~ cosh y] d y . d- ~ + cosh y] 0
Similarly,
(A.18)
~Q+_ &o
1 4k 2 + Sk?K ~ ~
(~--~) h~.~ + 2/~o ~ - - (I + ~ )
co
; J [~1 -5 cosh y + o
0
1 5 -- I I N u o v o
Cimento B.
]
~
•
]
c-o~h-yj exp
[--
~
eosh y] d y .
226
B. KU~SUNOGLU
I~ence
(An9)
0
~ (Q+ + Q_) =
21r~ + ~
[(1--#2)(IoI+/o2)]--
4kK,
(A, + A~)
cg~t~ 8
0%~
where I o --
g d2A ekK~ d~ 2"
Also (A.20)
~-~ (Q+-- Q_)
c~
0 [(]_/~) ~z~-- (] + ~) (11-- A)--
2l¢K 2
[/~(A1 + A , ) ]
RIASSUNT0
2kK~ ~o) [/~(~111 -~- g , I , ) ] .
(*)
Trascurando le collisioni si studiano le oscillazioni trasversa,li e longitudinali di un plasma rela¢ivistico di bassa densit~ in presenza di un campo magne¢ieo eostante. Si deduce la forma pifi generale delle reIazioni di dispersione relativistiche per oscillazioni lineari de1 p l a s m a e si discu¢ono i due casi estremi di temperaCure di equilibrio nulla ed infini~a.
(*) T r a d u z ~ o n e a c u r a della R e d a z i o n e .