Haskell, R. E.

Holt, E. H.

English

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TROY, mr Y o x  
Qasms Research Laboratory 
TECHNICAL FEPCiiT No. 2,  June, 1561 
This work was par t i a l ly  supported by the 
dktional Aeronautics and Space Administration 
under Research Grant No, NsG 48-60. 
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REPRESENEI€IION OF PROPAGAmON PARAl .am FOR ?IIE PLASMA IN A WlGNE2IC 
FIELD. 
In  a recent Ilaper t h e  -pnse:-tiss ol" a laifom plasma f o r  t he  
p r q w a t i o n  of electroxua@ctic waves izre  i-epresented ~y cwvcs  
p l s t t e d  i n  the  cm.Jc:; p . t?aC%i; i  3ii @ne . 
ma+etic Tieid can -. e sh.,wn by e.cLen6j-a; t;,es,: CJTVBS t o  th;.ee di-  
nensional mx.els ir- the  m y  descr-i. -.e?. -dew. 
.1 
A Tile e f i e c t  of a d.c. 
The c,Jrnplc--;. :.i-;:)sgat ini ,  :\law 1.s ~;.:cae,~teLi by cuordinntes 
A = cx (C/LU) an:', E = > (c/&) wher2 0 arid P ai-e the  reo1 and imaginary 
pa r t s  cjf t h e  coraplcx po-?zt.iJn co1.1stait and (c/w) i s  the  r a t i o  
of the veloci ty  oi" 2.i;ht t o  t.hc I"-el;.ien..y of prc,W.ltion. 
rei-ence 1 tile pwar,e ters p l f - t t n c i  m L * e  t3.e normalized co l l i s ion  fre- 
qcency z = 3/01 wkLzre j -c 'it-. calLisioii frequency and t he  nar- 
ma'.lzcci electroil d c r s i t y  x = ~ o ~ . . t c \ ) g  
c_umcy. 
transfannction of ccrrespoiiding c-a-ves i n  t h e  cm?lex d i e l e c t r l r  
plane. 
a t ion  using the d i e l ec t r i c  coeif ic ient  f o r  CL uniform pl?. L Sma. 
I n  re- 
where w i s  t:-e plasma fm--  
.k P 
-The eqxation,: f o r  the cuL*ves were derived from a conformal 
These eyukions  i n  turn we;'? CeriveC fi-om t h e  wave equ- 
A later. cornmicatioil;! indicated t h a t  by su i t ab ly  choosing 
t h e  noimal ized parrxneters t o  iscl*lbe t h 3  eler.+.ron cyc lo t rm fre- 
quency UA~, t&' ? same curves could Le v.se6 f o r  t h e  case of ;2 plasna 
i n  t h e  p.-esence of a dc mzgnetic tiel$-. 
i s  t o  point out t h a t  t h e  orl,-inal curves plottec? ir. t h e  c:mple;: pro- 
gayi t ion pla.ie can L e  exten4ed t o  tk-ee-aixensional models f o r  the 
c&sc of a dc ri&y,etic Tield by i z t i adcc i ig  a t h i r d  coordtnate 
Thc pa-pQce of thi-s note 
Y = cu/(u. 
0 
A,i>leton's eqLation for tl-e cm2lex rc.?rractive in&x of a mag- 
+? iieto-ionic medium i s  given 0;' 
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-2- 
Yz = u,/a = 
L. 
Y = U\/{O = 
Y "  
oz = (e/n)B 
= (e/rn)B Y 
the norma!.izPd parmeters  dei'iaed above w-3 
the nonml-izeu cyclotron freqaeilcv i n  the  direction 
of prqx:&tiax, z 
the  nol-malized cyclcti*on frcqumcjr iLi tile y-6irection 
ccs 6 
s i n  
@ = angle betveen the n q n e t i c  field B ond tile djrection of 
woVaga% i 01; I z 
For tire C R S ~  of a lon.Tr:uCqd ccqx-etic f i e ld ,  Y = 0, 
J 
Y = Y a m  (1) recuces to 
7, 
2 X 
n = 1 - -  
1 -  j Z ' Y  - 
Since ths sqiare of the x f r s c t i v c  index i s  eJual t o  the di- 
e lzc t r i c  coefficient,  one can r;;tonali?e the righ%--liand side or' (2) 
and equate the r e a l  and i m g i m r y  rrtrts t o  K and K;, the  real and 
imsinary  parts of the dielectr ic  coefficient K. 
fornation frc-n the cmiplz;: dielech-i:: plane t o  the con2lex z;-opa- 
gatioii ::.me SiclJs the following two c p a t i o n s  f o r  t h e  nom-slized 
coll isic, i  freqJency case and f o r  the  r ,onalized electron density 
case. 
r -- 
X coilformal trans- 
! 3 )  
2 
X X 
(A2 + B2; - ( 2  - ) !  # - 2 )  + (1- - ) = G  
1 +  Y - 1 +  Y - 
( 4) 
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Y 
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1.0 
0.5 
0 
/ 
/ 
A I .o 2-0 
FIGURE 1 
COLLLSION FREQUENCY MODEL 
ORDINARY WAVE 
A = (c/u)a = normalized attenuation constant 
B = (c/w)p = normalized phase shift contant 
Y = uz/u = normalized cyclotron frequency 
z = o/u = normalized collision frequency 
Y 
I .S 
I .o 
0 -5 
A 0 I lo 2-0 
FIGURE 2 
COLLISION FREQUENCY MODEL 
EXTRPLORDINARY WAVE 
A = (c/u)a = normalized attenuation constant 
B = (c/u)@ = normalized phase shift constant 
Y = LU /Lo = normalized cyclotron frequency 
Z = o/cu = normalized collision frequency 
2 
Y 
I .5 
1.0 
0.5 
0 A 2 -0 
FIGURE 3 
ELECTRON DENSITY MODEL 
ORDINARY WAVE 
A = (c/cu>a: = normalized attenuation constant 
B = (c/03)B = normalized phase shift constant 
Y = 0 /O3 
X = (03 /a) = normalized electron demity 
= normalized dyclotron frequency Z 
2 
? 
0 1.0 A 2 .o 
FIGURE 4 
ELECTRON DENSITY MODEL 
EXTRAORDTNARY WAVE 
A = (c/cu)a: = normalized attenuation constant 
B = (c/w)f3 = normalized phase s h i f t  constant 
Y = a /w 
X = (a /w) = normalized electron densi ty  
= normalized cyclotron frequency 
2 
2 
P 


Representation of Propagation Parameters for the Plasma in a Magnetic Field

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