Majumdar, Saroj K

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A NOTE ON THE DISPERSION RELATION FOR VERY HIGH TEMPERATURE PLASMA SAROJ K. MAJUMDARSaha Institute of Nuclear Physics, Calcutta {Received Novertiber 1(], 1960)ABSTRACT. The aBympiolic expaiiRiou of the fniirtion Sj,{z) wliich foiupletoly dos- cribf>8 the plasmM (lia])or8ion law and intrudueod by Majuinar (lOGo) has been earricjd out for very high temporaturo and low magnetic field. From this ex]>ression th(' wt'll-known results of zero magnetic field case are recovered.I N T R O D U C T I O Nla  an earlier work on tli(‘ dispersion relation for ('leedron jilasma in an external magrudic field, (Majiinidar, 1965), it^  was sliowii that the whole ])rol)lem of plasma dispersion can be throughly soIyihI in terms of a now function given by61<9„(Z) =  J f M + Z e o a t ( l i _  0 • • (1)wliero V — to/coc "I (2)and Z  -  JHere, (Oc — vH^^ime is tlui (cyclotron frecpiency of the plasma (‘lectroiis, T  is their kineti(‘ temperature, <.> and are resj)eeti\ ely the fr(‘(|U('!U‘y and wave number—>(perpcmdieular to the applii'd magmdic fic‘hl //„) of tht'. disturliaiKii's.It has b(Mui shown (Mazumdar 1965), that the ^vho h  problem of dispid’sion of waves in a magnetii; plasma can Ik* solved if om* i*an find the various pro­perties of Si,(Z) as a fumdion of v and Z.In tJiis report, we wish to study the asymptotic beliavioiir of the funetion S i , { Z )  for very high t(‘mpin’auro and low magnetic field. In that case both v and Z  assume large values, but v/Z remains small. From the result so obtained, we sliall recover the well-known ('ase of z(KH) magnetic field.A S Y M P T O T I C  E X P A N S I O N  O F  .Sv(Z)It has boon discussed (Mazumdar, 1965) thatr the function can bethought as to belong to the family of Bessel's function. Calling v as the order and Z  as the argument of Sy{Z)y we necnl to find the asymptotic expansion for607508 Saroj K. Majumdarlargo order and argument. For this purpose, we follow the method of steepest desccmi. To do this, wo first transform the function Sp(Z) in a suitable form. Setting t =  iw in oqn. (1 ), we get—< «0Sy(Z) =  i J 0..  (3)The contour of the integral (3) runs from 0 to — oo along the imaginary axis as shown in fig. 1 . We divide this contour at all points which correspond to oddFig. 1. Contour of integration in eqns, (3) and (4),multijdc of 7T. At each of these points we add and subtract straight line contours going to +00 in a dinHdiou parallel to the real axis, as sliown in fig. 1. The result is that oiu‘ (-ail write the function S^(Z) in the following form :[ oo-<7r t o - S i n  oo~5*7r •»/  4- J +  /  +  • • I dwe^0 ao~<7r oo-3<»r Jo~5<7r0 oo + tV^  f cosh w-i;w d ^ c + i  — ____  f dUK . . (4)oo-i»r 2 sin V7T m-inThe form of the integrals in (4) is very similar to that of the integral r(^prcsenta- tion of the modified B(^ssel function of the second kind. Therefore, following Watson (1952), wo calculate tln  ^asymptotic expansion of Sy(Z) :8y(Z) =  ie^^a<f>(ll2, 3/2; aM )+  I  '“ L  '’'“ +*>+ 5 ' " > ‘" ’1 .  2 - n  '’<"•+*>=  ™ same order. (6)A Note on the Dispersion Relation  ^ etc. 609In (^ )), we have used the following :a — sinli^N/Z A  — ( Z I 2 )  cosh a B ^  cosh a —a siiih a.(fi)the quantities 's are certain coeflSci(^nts of expansion whoso values arc tabu­lated by Watson, (1052). H(‘,rc we shall use only ijie and terms, wliieli have the following values :ao =  (— i  oosh ^2 tanli a \ (7)a, = 3 cosh aTJie function  (f> appearing  in eejn. (5) is the  eoiifluc^nt hyi)erg(H)metric function (Erdclyi et al 1053.)Eqn. (5) is th(  ^ complete expression for tlu' asymptotics expansion of 6^(2). This is a quib^ (roniplicated expression, but in most cas(*s of interest the summations ov(‘r m can be replaced by their respective^ first t(Tms. Eor exam])l( ,^ iix the case of high temperature plasma in a low magnc^tic field, th(‘ (juantity Z  is very large. In that ease wc' are justified in using only the 7n — 0 termL O W  M A G N E T I C  F I E L D -  H I G H  T E M F K H A T U K E  P L A S M A  In tliis case, v/Z < <  1 . Hence from eqns. (0) and (7) we can write, a0<=ssv/Z; .4 e=sf Z/2+v^/4Z;i 4Z*) ’2 v/Z3Using th(\se approximati? values, the quantities ^3 and whose sumis tlie asymptotic expression of Si,.(Z) given in eqn. (5), can be calculated. Theresult is81 ea i^ ^ ( l /2 ,  3/2; v*/2Z)e^-""/2^^ V 13 eZ-v*/2ifV'2 (8)and S,(Z) =  s ^ + s ,+ s ^ .510 Saroj K. MajumdarTo cluM-k t h v  result of th(* as^ i^nptotic^  expansion, wo shall now show that this valn<‘ of S„ {Z )  givi'ii hy ecpi. (S) h a^ds to tlie oorroot expression of the dispersion relation for z(^ ro magnetie fi(‘Id. It lias been shown (Mazumdar, 1965) that th(^  dispersion law for plasma wavt^  for z.ero magnetie field is given hy=  0,wlion^Cl)* [l+w -^S,(Z )J,(9)( 1 0 )is th(‘ electron-plasma frec]uoncy. Using eqns. (8) and (10) we ealeulato the limit of for co^ .—> 0, and tlien use the resulting expression in (9), This gives us tlu; following dispersion relation :(0^  , V*+ [■ -^(1, 3/2; -v * /2 Z )+  ^(2, 6/2; */2Z)]V -r(i/2)co3 [ ( 11)Writing o> — where cof donotijs the real part of the frequon(*y and o,*gives the imaginary part, and using the asymptotic form of the fumdion wc separate* the real and imaginary parts of oqn. (11). The final result isC0^ 2 ~  CO 2-fand COi exp ( ■ ( 12)" \ 2fc*A*z)/wlioro <V'^'>^av average electron spewed and A/) is the well-kmmm Debyelenglh. The. first (apmtion of (12) is tin*, dispersion relation in the familiar form, wlien^as the second relation is tie* wt l^l-known expression for Landau dampliiig.For a non-vanishing magnetic field one c.au, in a similar manner, calculate the dispersion law for liigh temperature plasma placed in a low magmdic field. It is also possibles to extend the metlioil for other values of the ratio v/Z and thus may cover a whole range of magnetic field.A C K N O W L E D G M E N T SThe author expresses liis sincere gratitude to tlie Director Prof. B. D. Nag for providing him with the facilities to carry out the work.R E F E K E N C E SErdolyi, A,, 1953, Higher Transcendental Functions V c ^ l ,  McGraw-Hill Book Com­pany, Inc,, [Chapter 6].Majumdar, S. K., 1965, Indian J . Phys.^ 89, 511.Watson, G. N., 1952. Theory of Bassel Functions^*, ^nd Ed., Cambridge.

A Note on the Dispersion Relation for Very High Temperature Plasma

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A Note on the Dispersion Relation for Very High Temperature Plasma

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