IL :NUOVO CIMENTO VoL 26 B, N. 1 11 YIarzo 1975

Effect of Collisional Models on the Dispersion Properties of Electromagnetic Waves (*).

G. P. GUPTA and R. ~ . SI~GH

Applied Physics Section, Institute o] Technology, Banaras Hindu University - Varanasi

(ricevuto il 22 Luglio 1974)

Summary. - - Following Van Kampen technique the dispcrsion equa- tions for high-frequency elcctromagnetic waves have been derived for two collisional models. The indices of refraction and absorption have been computed. It is shown that the maximum change in the dispersion fcature results in the vicinity of ~o~/0)2~ 1.

1 . - I n t r o d u c t i o n .

The k ine t ic - theory formula t ion with appropr ia t e collisional integral depicts the det~filed dispersion fea ture of e lect romagnet ic waves p ropaga t ing through ionized media . I n the case of isothermal , homogeneous and weakly ionized p l a sma the B G K collision.fl integral (1) is of ten used. The B G K collision

re laxat ion model does not conserve the charge ins tantaneously . Therefore, the simple B G K collision model is not able to depict the e lec t romagnet ic-wave p ropaga t ion th rough tu rbu len t and fas t changing ionized media . The disper-

sion equat ion for e lec t romagnet ic wave with B G K collision model was derived

b y 1)RADHAN and MIS~A (2) and by VARMA (a). The choice of an appropr ia te col-

lision model affects the e lec t romagnet ic -wave propagat ion. I n this paper , wi th a view to s tudy rad iof requency wave p ropaga t ion th rough tu rbu len t and f~st

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (1) ]I). L. ]~IIATNAGER, E. ]). GROSS and M. K~OOK: Phys. Rev., 94, 511 (1954). (~) T. PRADHAN and I ). Mlsnh: Phys. Rev., 119, 1878 (1960). (a) N. L. VAlZMA: Acta Phys., 21, 384 (1966).

57

5 8 G . P . OUPTA and m N. SINGH

changing plasma, we have chosen a collisional model which accounts for instantaneous and local conservation of charge in the weakly ionized plasma (4). Following Van Kampen's (5) technique we have accounted for the local charge conservation and obtained the dispersion equation for a weakly ionized plasma. The integrals in the general dispersion equation have been analytically eval- uated with the assumption of plasma initially obeying the Maxwellian velocity distribution function. The indices of refraction and absorption for high-fre- quency waves have been separated. The refraction and absorption index for two collisional models have been computed and their variations with ~o~/co * and for different Z values are depicted. I t is shown that the choice of appro- priate collisiona] model results into a small change in the index of refraction, whereas the index of absorption changes appreciably. As depicted, the changes are seen to have a maximum in the vicinity of co*~/~o2-- - 1.

2. - F o r m u l a t i o n o f dispers ion equat ion .

In the neutral plasma we assume the ions to be immobile providing a uniform positive background. The electron component of the plasma conforms to an isotropic and stationary velocity distribution function. The effect of external field and other constraints produces temporal and spatial variations in the velocity distribution function and is expressed by the Boltzmann equa-

tion

~t 4- V. + E(y, t) Jr- -c ( V • " ~ - V : coil "

I n order to solve this equation we write the perturbation scheme

(2) ](y, V, t) = A~To/o(V) -~ ll(Y, V t),

where ]1 is the small perturbation due to the electromagnetic field and No the is number density of electrons in the unperturbed plasma. The normaliz- ing condition used is ffo d 3 V z 1. The collisional model which conserves the local charge is written as (4)

lxa! (3)

In this model v is the collision frequency of charged particles with neutrals of the weakly ionized plasma which is independent of velocity. This is justi-

(4) J . P . DOUGHERTY: Journ. l~luid Mech., 16, Part 1, 126 (1963). (s) N. G. VAN KAPPa, N: Physica, 21, 949 (1955).

EFFECT OF COLLISIONAL ]~[ODELS ON TIIE DISPERSION PROPERTIES ETC. 59

fiable since we are dealing with the propagat ion of low-power electromagnetic waves through ~ weakly ionized cold plasma. Subst i tu t ing eqs. (2) and (3) in eq. (1) a, nd neglcef, ing second-power terms we obtain

(4) (~]1 ~ emNO[ 1 ] ~/o ~[]l--Nl]o] ~ i V. + E+c (V• "8--V=-

In order to fret Na in terms of /o, we assume the first-order per turbat ion harmonic in space and t ime of the form exp [i((~t--k.y)l. The solution of

eq. (4) is expressed as

.[_e5 ,/oNe] (5) L = i(o)--iv--k.V)L m E 'ev + J

While using the B G K collision model, the role of the second te rm depicting the spatial dispersion is ignored. The collision model with local conservat ion of charge results into microscopic t ransfer of charge density. Therefore, the second t e rm tu~s been re ta ined while writ ing eq. (5). This is not an explicit solution of eq. (4). The per turba t ion in the number densi ty of electrons is de- tined in terms of pel. turbalion in veloci ty distr ibution function. We prime V to indicate a d u m m y variable of integrat ion to dist in~lish from tile va- riable V and define

(6) N, =.(I,(V')d~V'.

Using this definition and making use of eq. (5), we write the expression for the per turba t ion number densi ty as

(7) N, ~ .J i(oJ--iv--k.V ) 8V d3V'q wv~J i(~o~ iv--le---V'-) "

The integrat ion of this equat ion is ra ther involved due to the appearance of k .V ' - t e rm in the denominator . Therefore, while expressing N~ explicitly, we ignore the effect of k. V' and take the simpler form

(8) -G ieNo E f ~fo(V') : - m ~ o " ~V; d 3V'.

The linearized Bo l t zmann equat ion now is expressible only in terms of No:

~fl ~ eNo E @/O ievNO foE.f @fo (9) r ? V. -? ,n~- "~V- - * f ~ + m,o ~V ) daV'"

This is the integro-differential equat ion with pr imed and unpr imed vari- ables. The I)ropagation of clectrom.tgnetic waves along the z-axis results in

6 0 G. P. GUPTA and 1~. N. 81NGI~I

anisotropy of the velocity distribution function. Therefore, the transport equation is rewritten in terms of parallel and perpendicular components

(10) 81~(z, V, t) 8/~ eNo E• ~]o( V) 8t -~ V,, -~z + m ~ V• - - ~]1 ~-

In order to account for the changes produced in the distribution function due to velocity components, we multiply eq. (10) by V• and integrate over the perpendicular velocity component. Defining the corresponding distribution function

(11) ]• K,, t)=f f K,, V• t)dV•

we rewrite eq. (10) as

(]2) elk ~/• ~--7 + V,, -6; • # • =

eN~177 ~-ie~~ E'Lff vi/~ f f f ~ dV~dV; "

The dispersion equation for electromagnetic waves propagating through the plasma characterized by a chosen velocity distribution function is obtained by solving the appropriate wave equation

1 82E• 4n ~J• 4ge ~ ~, (13) V2E• c 2 ~t ~ ----'c 2 ~t -- c 2 ~ j ! •

In order to obtain the dispersion equation we need to assume the nature of the modified velocity distribution function due to wave propagation in the collisional and perturbed plasma. For the sake of obtaining a tractable dispersion equation we write

(14) /[~.(z, v,,, t) = g~.'.(v,,) exp [ - ik(z- ust)]

and

(15) ~k,~,,. u• m(ic9 ~• (z, t) = + ~') e

exp [ - - ik(z -- u,t) ] .

The modified phase velocity of the wave in the perturbed collisional plasma is denoted by u, = - iu,, where u= a~/k and u~ = v/k. The perpendicular electric-field component of the wave interacting with the velocity distribu-

E F F E C T O F C O L L I S I O N A L M O D E L S ON TILE D I S P E R S I O N P R O P E R T I E S E T C . ~ 1

l ion function is given by eq. (15), where u• is the average velocity of elec- trons along the direction of the electric field. Subst i tu t ing from eqs. (14) and (15) in eqs. (12) and (13), we rearrange for

(~6) '~ ~[ ,q (E,) u ~(i~o u - - V,

+ ~(~" V,I~ -3.J '~ ~JJ~VL dV',, dV2]

anti

(1 7) muL(iO)e + ~') - k(u~--c~')J47deu~-~ ['9.k ,~ ~(V, )dV, .

Now, el iminating g(Vj~) f rom eqs. (16) and (17), we obtain

k ~ ( u ] - - c 2)

~ u s

q-co -}-co

_f dr,, ((. - - o o - - c o

+ c o + ~ + Q o

a, b ff (ff dVii d V a , + ; u ,i v'~loaVl. ' ' - c o - - ~ - c o

where ~ , )= (4:~Noe'~/m) �89 is the plasma frequency. The dispersion equat ion in anMytic form is obta ined by choosing a suitable form for the velocity distri- but ion funct ion ;rod integrat ing eq. (18). The first t e rm on the r ight-hand side of eq. (18) conforms to dispersion equat ion with B G K collision te rm (3). The second te rm on the r ight-hand side arises due to the requi rement of local-charge conservation.

3. - Dispersion equation: Maxwellian distribution.

The dispersion equat ion in analyt ic form is obta ined by e, hoosing an

appropr ia te velocity dis tr ibut ion function. The integrations shown in eq. (18) have been performed with the veloci ty distr ibution function of the form

] . ( V ) = \2~K1'] exp m

The i'csultin~' dispersion equat ion is writ ten as

(19) ro~((o - - iv) o) (o" re z 2 [(o e*) 2 '

62 G.P. GUeTA and R. ~. S~NGH

where C~ = (KT/m) ~ is the the rmal speed of electrons, K being the Bol tzmann constant . The dispersion equat ion in the case of B G K collision model reduces to the familiar equat ion (3)

_ _

(o9--i~)2--c2k ~ 1 ~- C, . (201 ~(~--i~) -- ~ ~'\~]

Equat ions (19) and (20) bo th in the l imit of collisionless, cold and nondrif t ing plasma reduce to the well-known dispersion equation w~= w2, + c ~ k s for elec- t romagnet ic waves. Although in the above formulat ion we have included the k .V- te rm bu t in the case of labora tory and terrestr ia l plasma, this t e rm does not p lay an impor t an t role. I ts role, however, m a y become quite significant in the case of plasmas of astrophysical origin. Since we have ignored the effect of k. V in the evaluat ion of per tu rba t ion in the part icle densi ty IV1, we also ignore its role in the following analysis. The dispersion equations are found to differ due to the choice of collisional term. The result ing changes in the dispersion characteristics of high-frequency electromagnetic waves have

been evaluated for two chosen collision models.

Case i). B G K collision t e rm --vi i .

The general dispersion equat ion (19) for this case reduces to a well-known equat ion with dominant collisional effects

where

o r

where for Z <<1

and

c2 k 2 ~._2 = ( 1 - iZ) ~ --~ ( 1 - iZ)

Z : 7)/(0 7

C2k 2 = (# § iz)~ = el + is~,

2 O9~

e l = i a~2(l + Z 2)

s2---- Z ~ Z ~ ) 2 .

Separat ing the real and imaginary par ts we write the indices of refract ion

and absorpt ion as

1 = ~ [~, + V;~ + d] ~

~FF:ECT OF COI,LISIONAL MODGLS ON THE DISP:ERSION PROPERTIES ETC. 63

and

1

Case ii). Collision t e r m with local conservat ion of charge --~)(/1--~1]0).

As discussed earlier, the conservat ion of local-charge densi ty in a dis turbed

p lasma is es tabl ished due to t r a n s p o r t processes working in a t ime scale of

the order of re laxat ion t ime. I n this case the general dispersion eq. (19)

reduces to

o r

where for Z ~/.. 1

and

~2k2 N 2 - -

0)2

2

v~ ~ 1 - - 092(1 -~ Z) 2

, /w~( 1 § 4/~r 2) } .

The corresponding indices of refract ion and absorp t ion are wri t ten as

1 --7~ �89

and

1

4. - Results and conclusions.

The indices of ref rac t ion and absorp t ion have been computed and their

var ia t ions in the p e r m i t t e d range of high-frequency propaga t ion have been

shown in Fig. 1 and 2. The effect of two collision models has been shown b y

solid and dashed curves. For a given Z ~ ~/w the B G K eollisional model

shows a higher ref rac t ive index (solid lines) as compared to the collisional

model with local conservat ion of charge (dashed lines). The change in the

index of refract ion for lower w~lues of ,)~/(o-" is ve ry small and is not shown

~ 4 G . P . GUPTA a n d R. N. SINGH

in Fig. 1. The noticeable change is seen only in the vicinity of o~/m~= 1. Therefore, we find that the dispersion feature is not significantly affected by the choice of the collisional models. However, the index of absorption of

0.8

,u

0.6

0.4

0.2

Z=0.10

=0.05 ~ _ _ ....

=0.01

\

o l L. I J 1 I I o.4. o.s o.s 1.o 2 2

Fig. 1. - Variation of the index of refraction with ~/o) ~ for different values of z=,,/~. - ~ ' 1 1 : - - , ~ ' [ A - - Y l / o ] : - - -

electromagnetic waves propagating through two chosen collisional models shows a significant change. In the vicinity of w',/o~ ~-- 1, the change in the index of ~bsorption is seen to be m~ximum and is directly proportional to the Z-parameter. The present investigation brings out an important feature and shows that the Z-parameter alone is not able to depict the full dispersion and ~bsorption features of weakly ionized plasma. The changes in the disper- sion and absorption features especially in the vicinity of w~/o)~ = 1 are found

EFFECT OF COLLISIONAL MODELS ONT THE DISPERSION PROPERTIES ETC. 65

to depend on microscopic changes in the collisional models and the chosen

veloci ty d is t r ibut ion function. The present s tudy has been made by assuming the initial veloci ty d is t r ibut ion funct ion to be Maxwettian. The choice of

](

0 .2 - -

i /

/I /

/ , /

/ z /

/J/

I I i I ~ I I I l o o.~ 0.4 0.6 o.~ ~.o

Fig. 2 . - Variation of tile iude~ of ~bsorption wilbh ~o~/a~ 2 for differertt w, lues of , t o ] : - - -

an a rb i t r a ry and ac tua l veloci ty d is t r ibut ion funct ion makes the evaluat ion

of the integrals ra ther difficult. Although, it is obvious t ha t the choice of

the initi~l veloci ty dis t r ibut ion funct ion plays r a the r a dominan t role in con-

troll ing the dispersion and absorpt ion characteris t ics of e lect romagnet ic waves p ropaga t ing through a weakly ionized plasma.

This research was pa r t l y suppor ted by U.S. Air Force Research Projec t

under Gran t :No. E00AR-70-0070. One of us (G. P. G.) is gra teful to CSIR for the aw~,rd of Jun io r Research Fellowship.

5 - I I Nuovo Cimento B.

66 G. P. GUPTA al ]d R. N. SI~GH

�9 R I A S S U N T O (*)

Si sono dedot te le eqtmzioni di dispersione per onde elet t romagnetiche ad a l ta frequenza per due modell i d 'ur to seeondo la tecnica di Van Kampen. Si sono caleolati gli indici di r ifrazione e assorbimento. Si metre in luce come si ott iene il cambiamento massimo nella carat ter is t ica della dispersione in vicinanza eli w~/~2= 1.

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