THE UNIVERSITY OF MICHIGAN 2756-1-F ARL-TR-60-274, Part II THEORY OF PLASMAS, II LINEAR OSCILLATIONS IN RELATIVISTIC PLASMAS by K. Imre February 1961 2756-1-F Final Report on Contract AF 33 (616)-5585 Prepared for AERONAUTICAL RESEARCH LABORATORY AIR FORCE RESEARCH DIVISION AIR RESEARCH AND DEVELOPMENT COMMAND WRIGHT-PATTERSON AIR FORCE BASE, OHIO ARL-TR-60-274, Part II

THE UNIVERSITY OF MICHIGAN 2756-1-F PATENT NOTICE: When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications or other data is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto. Qualified requestors may obtain copies of this report from the ASTIA Arlington Hall Station, Arlington 12, Virginia. ASTIA Services for the Department of Defense contractors are available through the "Field of Interest Register" on a "need-to-know" certified by the cognizant military agency of their project or contract. ARL-TR-60-274, Part II ii

THE UNIVERSITY OF MICHIGAN 2756-1-F CONTENTS ABSTRACT iv I. INTRODUCTION 1 II. BASIC EQUATIONS 3 1. General Formulation 3 2. Moment Equations 5 III. THE FORMAL SOLUTION OF THE LINEARIZED SYSTEM 11 IV. A SPECIAL CASE 19 V. RELATIVISTIC PLASMA OSCILLATIONS 24 1. Zero External Field Case 25 1. 1. Longitudinal Oscillations 26 1. 2. Transverse Oscillations 31 2. Constant External Magnetic Field Case 34 VI. DISCUSSION AND CONCLUSIONS 45 APPENDIX + The Integral Equation for f 52 BIBLIOGRAPHY 55 iii ARL-TR-60-274, Part II

THE UNIVERSITY OF MICHIGAN 2756-1-F ABSTRACT The linear oscillations in a hot plasma which is representable by the relativistic Vlasov equation with the self-consistent fields are investigated. The method which is used by Bernstein in the nonrelativistic case is generalized to obtain the formal solution of the linearized problem. Particular attention is given to the case when the unperturbed distribution function is of the MaxwellBoltzmann-Juttner type (i. e., the relativistic equilibrium distribution) in which case the integrations involving the velocity space are carried out explicitly. The dispersion equation is derived and studied to some extent, considering the spatial dispersions explicitly in some special cases of interest. The ordinary and extraordinary modes, and the magnetohydrodynamic waves are investigated when the propagation vector is along the unperturbed magnetic field. The asymptotic expansions are developed corresponding to the dispersion relations of the cases considered, and they are shown to be in agreement with the results of previous studies in their respective order of approximations. It is found that circularly polarized transverse waves propagating along the unperturbed magnetic field are evanescent if v2> 1 - o2 / W2, where v is the index of refraction (kc/w) and 2 is the gyrofrequency. In the absence of the external field the cut-off frequency is found to be a monotonically decreasing function of the temperature. iv ARL-TR-60-274, Part II

THE UNIVERSITY OF MICHIGAN 2756-1-F I INTRODUCTION In the recent literature, the problems involving the hot ionized gases have increasingly attracted the plasma investigators. The relativistic Vlasov equation together with Maxwell's field equations have been used in most of these approaches. Since the particle-particle and particle-photon correlations are ignored completely in this model, the validity and the applicability of this representation are somewhat restricted. The extent to which this imposes limitations has not yet been made evident in the literature. However, leaving these questions unanswered, in this work we shall assume that the above-mentioned model can properly represent the system of interest to some extent. Furthermore, to study the oscillatory behavior of the plasma a linearized theory will be employed. In Section I we give the basic formulation of the mathematical problem, mainly for the purpose of introducing the notation. Also some discussion involving the moment equations is presented. The linearization procedure is introduced in Section III, and the formal solution of the system of equations is obtained using the integral transform technique, which is the direct generalization of Bernstein's wellknown procedure to the relativistic case. ARL-TR-60-274, Part II 1

THE UNIVERSITY OF MICHIGAN 2756-1-F Particular attention is given to a special case in which the unperturbed distribution is of the Maxwell-Boltzmann-Juttner (MBJ) type. The study of the dispersive phenomena is considered in Section V. The results of the earlier investigations are examined. Transverse oscillations propagating along a constant magnetic field are considered, and the dispersion relation corresponding to the relativistic magnetohydrodynamic waves is derived. A short discussion of the results deduced in the context of the present formalism is given in Section VI. It is pointed out that in the absence of the external field the cut-off frequency decreases monotonically with increasing temperatures. The longitudinal mode possesses the same cut-off frequency as that of the transverse mode. ARL-TR-60-274, Part II 2

THE UNIVERSITY OF MICHIGAN 2756-1-F II BASIC EQUATIONS 1. General Formulation The relativistic collisionless Boltzmann-Vlasov equation for a species of type N is given as. Of, e af N + N N DfN u + F (1) mp N k The summation convention both for Greek (1 to 4) and Latin (1 to 3) indices are used here. The reduced velocity, u, and the field tensor, F, are defined as uk = vk u4 = i cy v2 -1/2 u2 1/2 T=1 (l- ) = (1 +- (V =VV 2 2 kk C c F.. E.. H F =- F iE.. (2) i ijk k 4j j4 = i We denote by mN and eN (=Ze) the rest mass and the electric charge N N N of the species of type N, respectively. In what follows we shall suppress the suffix N when no ambiguity arises. The fourth component of the space-time is chosen as x4 = ict. * Only Cartesian tensors will be used throughout this work; no distinction will be necessary between the covariant and the contravariant components. ARL-TR-60-274, Part II 3

THE UNIVERSITY OF MICHIGAN 2756-1-F In addition to equation (1) we have Maxwell's field equations ___. 47 F = - J ax z/u c v A 47T Y C d u / e -d- u f c h v e~V -L FF = 0. (3) a/Cv X ax Xv Here the summation is understood to be extended over all species in the plasma. Henceforward, we shall use the symbol/ always in this meaning, unless it is otherwise specified. The symbol c denotes the completely antisymmetric (Levi-Civita) tensor density in the 4-space. The last of equations (3) implies that there exists a vector potential, A, such that -p D3aA DA F (A = i) zv/u ax ax 4 v _p Now, with the Lorentz condition aA A = 0 ax equation (3) can be written as 2 4v __ ______2 r2 A 4r 2 a ) D A c /p ax 3x (4) ARL-TR-60-274, Part II 4

THE UNIVERSITY OF MICHIGAN 2756-1-F 2. Moment Equations Let us multiply (1) by an arbitrary function - g(x,u ) and integrate y tJ k in the u space. a d3u d3u --- u fg = d fDg (5) ax T / Here we have used the relation F a (,)= o k au u k In particular, for g = m, equation (5) gives the continuity equation' j ~ 0 (6) ax where the mass current j is defined as Frgmu oe d j - m 3 u f. (7) For g =mu, one obtains after some manipulations 8 e p e F j = o ax v, me v A AL where the energy-momentum tensor, d3 P = m - u u f (8) v pw j 7a v e Now, with the aid of equation (3), we have'We assume the rest mass, m, to be a constant. ARL- TR-60-274, Part II 5

THE UNIVERSITY OF MICHIGAN 2756-1-F 4 F J F - F c V p,p v p ax, p a 2 1 a 2( Xa- F -i (trF) ax vp 4 ax - P v Thus,introducing the traceless, symmetric tensor, (known as the electromagnetic stress tensor), v1 4 F2 1 (t2 / S F - L (trF2) S (9) Av 47T l v v 4 - ) v we obtain the well-known energy and momentum conservation equation a- ]P +S =0. (10) ax V / v The five equations given by equations (6) and (10) are far from being sufficient to provide a closed system of equations. This difficulty customarily is relaxed by introducing a set of systematic approximations which leads to the relativistic magnetohydrodynamic equations. To do this let us introduce the quantities n _S fd3u imc j -1/2?1+X = j/ (-j j ) - nm / A aa pnmc -1 V A — +P k so that we have the relations ARL-TR-60-274, Part II 6

THE UNIVERSITY OF MICHIGAN 2756-1-F x =0 X X = -1 [Xz.= X =0 p^a cav p v Now the energy-momentum tensor may be decomposed (cf. [1] ) EP =wXX +1 q + 1-q X + p v p v c P v p V pv where w = XP X p pv v is the invariant energy density, q -cZ P k is the heat flow vector, and p' - -a P 1a Tp is the stress tensor. We observe that X q =, =0. The energy-momentum tensor corresponding to a "perfect" gas may then be obtained by assuming: (i) q = 0, and (ii) I = p Z (cf. [2, 3, 4]) so that p pxv p. v p P =(w+p)X X +p where p is the invariant pressure ARL-TR-60-274, Part II 7

THE UNIVERSITY OF MICHIGAN 2756-1-F p =-t P 3 p v p' v The terminology introduced above may be justified as follows. First, let us consider the invariant energy density d3 w = X P X= m \ -- X u X u f /P Iv v J / pI v v Since w is an invariant we can choose X. = 0(X =i) without losing any'4 generality. Thus 3 w = du ~ f'~ 2 where 6 = mc T is the total energy. Similarly the heat flow vector introduced above may be written 3 3 q. -i cP4 = d u v. f ( v. = u.) J j4 JJ J where we set X (0, i). This means that q. is to be measured with respect to a frame moving with the instantaneous average velocity of the system. We note that if f(O) is isotropic in u then q = 0. Finally the tensor 45 i introduced above reduces to j P.k = d3(mu.)v f jk -jk j ( j k when measured with respect to a frame which moves with the instantaneous average velocity of the plasma. Obviously Pjk is diagonal when f is isotropic. The invariant pressure reads then (in this frame) ARL-TR-60-274, Part II 8

THE UNIVERSITY OF MICHIGAN 2756-1-F 1 V m dI 2, d3u u2f p 3 3 u In Section IV we shall show that these conditions are satisfied for the equilibrium distribution (cf. equations (9), (10), (il)). It is customary to introduce the proper (invariant) mass density o o nm p = n m = / and the internal energy per unit rest mass o 2 w -p c - ~o P so that we get ax, 3x /V - (P + S ) = 0 Ox v/a v/ where P P ~( E +c +p/pO)X V +P v The latter set is known as the relativistic magnetohydrodynamic equations. Clearly, these equations still are not complete. However, they 0 may be closed by an equation of state, relating F, p and p, and with some additional considerations concerning the electrical conductivity properties of the medium [4, 5]. Harris [4] has shown that for the case of infinite conductivity the well-known adiabatic law is satisfied. ARL-TR-60-274, Part II 9

THE UNIVERSITY OF MICHIGAN 2756-1-F The linear plasma oscillations may be studied by these macroscopic equations. However, here we shall employ the Vlasov equation directly to obtain the necessary relationships. Although additional assumptions concerning the unperturbed state (and the initial perturbations) will be assumed, no phenomenological specific requirement will be adopted explicitly. ARL-TR-60-274, Part II 10

THE UNIVERSITY OF MICHIGAN 2756-1-F III THE FORMAL SOLUTION OF THE LINEARIZED SYSTEM Small amplitude oscillations may be studied by the usual perturbation technique employed on equation (1). Let us introduce (0) (1) f = f() + f(1) F = F + F(1). v pv p/u (0) (0) F = constant, F =. (11) kj 4j The zeroth order equation, now can be written as af(0) e (0) af( + F(0) afe 0 (12) u + — e F u. = 0. (12) /u ax me kj j au 1A k In this case equation (12) may be solved by Lagrange's method. The corresponding characteristic equations are given as dx du = -u, -k = e F() u (13) ds /P ds me kj j The latter may be solved easily. Introducing the quantities (0) eH( me b - F (14) and using the properties ARL-TR-60-274, Part II 11

THE UNIVERSITY OF MICHIGAN 2756-1-F 2 1 H(0) b.. =h.h.-, (h. (0 H.) iJ 1 J ij 1 () 1 H(0) 3 b3 = -b., (15) ij i1 one can write the solution as u = exp [0 s b1] r Here CD [ ( n s exp 1 s b= n b n=0 2 I + sin 2 s b + (1-cos 2 s) b where b = I = ( g ). (16) = I = ( ~jk Then, the unperturbed orbit equations are found to be u [= + sinn sb +(l-cosns)b ]* r,,uv / v,v ~ v x r. 1 - cos 2s Os - sinus 2 x =s + b + b +,uv S2,uv Q,uv /v, (17) Inversely, we have = u [ +sinns b +(l-cosSns) b],u v vu v u via = -u Is + l-~s- Sb +-s-sins b2 (18) In the above formulation, it is understood that ARL-TR-60-274, Part II 12

THE UNIVERSITY OF MICHIGAN 2756-1-F 2 2 b b b b =0 b4 4l= b 4 4= Therefore, the general solution of the unperturbed equation is f(<)(x,uk) = g( k(s), rk(s) where g is an arbitrary function of its arguments, and s is the proper time: t-t x -x o 4- 4 S = = ~Y U4 ~(~~o~~~) a/~~f(0) In particular if f) is space-time independent, i. e. = 0, then we simply ax have f() = g (u * h, u) (19) A special distribution of considerable interest, which satisfies the latter condition is the relativistic Maxwell-Boltzmann (Jiittner [6]) distribution[2] f(0) = C e3 (20) rMBJ which will be given more attention in our future discussions. The first-order equation is found to be _f(1) af(1) e (1) af(0) u + Q bk u - = F uk(21) ax kj j iuk mc kpg au which now can be written in the integral form as usual: f()(x u) - f(l) ( ( ) - ( s, r/) ds4, (22) ~ (x, U.) - ) (- (SI), rl (sI) ds' (22) p u j u4' where ARL-TR-60-274, Part II 13

THE UNIVERSITY OF MICHIGAN 2756-1-F,e (1) af_ 0(x,u ) _ e F(1) u - pX v me ku p auk It is convenient to introduce the matrices * 2 R (s) = -sinOs b + (l-cosns) b /v piv,uv 1v G (s) = s -cosOs b + 2s - sin2s b2 (23) JlAv JAyv /,v Q2 / v so that the orbit equations become u = r R /^ 1V GV/ A, v v/u x = rl G +~ Tr =R u I ~ /cuv v = x -G u /g p v v Hence e (1) HOf 0 (, ) = eF (x-G u ) R u f A V ^ mc kg X Xv p Q kY z kv v AzP P arlk where (0) (0) (0) af af0 G + u R an ax Gk au. kj In what follows we shall restrict ourselves to the case in which (0) (0) af o) af o) ax 1 kj au 0~ 0 * Several properties of R.. and G.. (and also Mjk(s) = \- Gjk(s') ds' = \ RJk(s") ds"ds') are given in [7]. ARL-TR-60-274, Part II 14

THE UNIVERSITY OF MICHIGAN 2756-1-F Thus'/ X4/U4 (1) e Hf(0) (1) f e \ F' (x - u )R Rk ds (24) me v au. k/u x Xpp pVkj 3 v0 In this special case it is convenient to introduce the Fourier transformation in the space-time variables: p -ik x X+(k ) = e' X)(x ) dx. (25) (+) The (+) sign indicates that the x integration is to be restricted to the positive x4 range. Clearly this corresponds to the Laplace transform with respect to time, with a transform parameter p = - k4c. Equation (24), then, will transform into 00 + -e af0 + -() f^ = - uu Fk (k) 5 e R (s)R (s) ds, (26) me v ku. k p -V k where y(s) = ik u G (s) p ca pa isk- u+ps -ui -CS (u hxk)+i 2) ( (0) Let us note that if f(0) is isotropic, i. e. af cC u., then all terms in the au. J J summation containing Fj vanish due to the antisymmetric property of the field kj tensor. Thus in this case we simply have'Hereafter we shall suppress the terms involving the initial perturbations explicitly, and then take them into account subsequently as a whole whenever needed. ARL-TR-60-274, Part II 15

THE UNIVERSITY OF MICHIGAN 2756-1-F o0, af^0 p+ -y(s) Id me 4 au. k4 e(s ds j 0 Now, using (26), we shall compute the electric charge current J P which will enable us to eliminate the particle variables, i.e. f+, through Maxwell's equations, + \ \,+ d u J =/ e \ f u - P J = i pk Fkp, where the "generalized conductivity tensor" is defined as' 2 3 (0) Pi iZI e - \ afo) u \ e Y R R ds (27) PJiP me J auk p j ]k v 0 The usual conductivity tensor is related to the one introduced above as k op =. pa. PZ k4 pJ PJi so that J = a E. (E i F 0) p p Z( 4 44 (0) We note that in the case when f is isotropic in u, this reduces to The Kronecker delta n may be defined as Te dt V1... Vn v... Det ( V = S n q Also it is understood that. = 0 if a = 4 and &. = = when a = k(=l, 2, 3). J J J kj ARL-TR-60-274, Part II 16

THE UNIVERSITY OF MICHIGAN 2756-1-F -3 afA o U e-y d ij ij4 - m d u u ui Rjkds auk 0 (cf. for instance, [9] ). Substituting the current computed above in equation (3) we get + 47r + ik F - i a. F. = (28) p pp c pjp jp p Here { is an explicit function of the initial perturbations and the unperturbed P parameters only, and contains all the terms of that nature which have been suppressed in our equations. Furthermore, we observe that it satisfies the relation k r = 0 P P for kkF =0 and k J =0 A p P Pp p P Hence only three of the four equations given in (28) are independent. We shall use the curl equation, i. e. the second of (3), in order to eliminate the spatial components of the field tensor k + Xj X + F = F (29), v Pu v k 4j Substituting in (28) we obtain k Xk 4 7rXk X + k k k c ^ijp ^J \ \ E =E vi k4 c i k k i which may be written in the form ARL-TR-60-274, Part II 17

THE UNIVERSITY OF MICHIGAN 2756-1-F + T E = -. = ik u 2 i 1 k c where 2 T =. +-' (30) ij ij 2 2 ij 2 Here we substituted k = k.k. and kc =- p. The projection tensor Si., and J J 4 " the "dielectric" tensor E.. are given as A A A S.. =. - k.k. (k. = kk.) 47T lJ 1 1 1 Ej ij P+ ij (31) 1J pJ p ij where a.. is the conductivity tensor introduced earlier. iJ Therefore the formal solution of the field equation is obtained as F+ Xj X -1 T- -1 -f t s tV ku jk k ikTkj i j If this solution is used to eliminate the field variables in (26), one obtains the formal solution for the transformed distribution function, which enables one to compute the average values of the physical properties of interest in terms of the initial values and the unperturbed parameters explicitly. Alternatively, one can deduce an integral equation for the particle variables by eliminating the field variables. The latter form is found heIe more convenient in dealing with some particular aspects of the problem. For this reason we shall give it as an appendix. ARL-TR-60-274, Part II 18

THE UNIVERSITY OF MICHIGAN 2756-1-F IV A SPECIAL CASE In this section we shall give special attention to the case in which the unperturbed distribution function is of the Maxwell-Boltzmann-Juittner (MBJ) type [2, 6], namely* (0) n 3 e-J3 (32) MBJ 4 3 K2() where 3 = me /0, ~ is temperature in ergs, K2(13) is the modified Bessel function of the second kind, and n = f(0) d3 Then the first order average of the property g, i. e. 3u n(g)() = f(1) g d may be written in the transformed space as follows: g3 + j f+g d u 00 e 3 a -y(s) a m Ek N d u au g S ej R ds Changing the order of the integrations and integrating by parts, one obtains 00 +n -- + E3ds d3u ef0 f [ik.Gkjg-Rkj u (33) m Ek J0 /3+ps 3 Jkkj u.J In the form of fiMBJ given above we have selected the spatial components of X/a equal to zero [2]. The general case can be obtained by a transformation without much difficulty. ARL-TR-60-274, Part n 19

THE UNIVERSITY OF MICHIGAN 2756-1-F where as before p = - kc and y=i k G _ u+ pys. In deriving equation (33) we have used the relation bf 3 -Y 0) ag (+ ps) s3 y af(u e g = d3u e () ( k G1j g - au The integral*,3 La a K1( / 1 \ d u c a KA 1 (l a), 3 J T e = - 4rc u /a a - v p JIH and its derivatives are useful in carrying out the velocity integrations corresponding to the first-order moments. For instance the first-order mass current (which is obtained by setting g = mu ) is found to be +- K2 K2 ( K3 (E) en E ds R. -c G k kG k j K 2 ) q qk 23 D00 + 2 K(z) P + 3'- t- encE. k ds(g +sp)G ) 2 where s 2 (3+p)2 +2c2 k' M-k. M= G(s')ds (34) *Strictly speaking, a singular term, proportional to 8 (aM aM), should be added to the right-hand side. However, we shall ignore this term for the time being. In the next section, while discussing the spatial dispersions, we shall give more attention to the contributions due to this improper part. ARL-TR-60-274, Part IT 20

THE UNIVERSITY OF MICHIGAN 2756-1-F It can be shown without much difficulty that the continuity equation k j = 0 is satisfied by the form derived above. The energy-momentum tensor may be computed similarly; however it will not be given here. Let us note that the nonrelativistic limit may be carried out easily by using the relations /3- -L^oo 1 Jlim (3-,) - sp - k. M k lim) 1 ) 00 m lim = l exp (3sp - k-M-k (35) — Coo K(P) - - = - Thus it can be seen readily that equation (34) approaches the results obtained in the nonrelativistic analysis (see for instance [7] ). We also note that it is possible to introduce the diffusion and the mobility.tensors, as discussed in [7, in such a way that k -iDkkj knm j (36) where oO + e 2 j m K2(P) jk I0 o3 O D( + -.05G.K Kds + 0 G o jk 3 p ds D + kj -m + ARL-TR-60-274, Part II 21

THE UNIVERSITY OF MICHIGAN 2756-1-F 00 oo 2 K2 K ^ + f 3 P K( 3 (3+ sp) 3 p ds (37) 2 - K2 a) The electric current density is computed as usual,>C e P - m J so that one obtains + + J = u E. (38) k kj where 7e2n 12 CK K K ^-^r = ^(3 ds 2 kj m K2() 50 [k ) K() k (k)2 - - G kk G K;3 (). m jA q qk 3 This result is in agreement with Trubnikov [8], and in the nonrelativistic limit reduces to Mower's conductivity tensor (cf. for instance, [7] ). For the electric charge density we have + x e + Q = -- p m -iE kZ ~2 12 S(3+KG = -i m K(3) ( + sp) G ds (39) m K2(/3 j 3 We shall return to these relations later in connection with the study of the dispersion relations. ARL-TR-60-274, Part II 22

THE UNIVERSITY OF MICHIGAN 2756-1-F In closing this section we give some of the zero order properties corresponding to the MBJ distribution. The stress tensor (in a reference system which moves with the local velocity of the plasma) (0) d 3u (0) Inmu. u, f p' jk JY jk U jk 2 (0) = n = (40) p The energy density (0) 2 c du,2 f(0) w = = mcS e T f = 44 2 3 n 2 K3 1 2 (0) = nmc - = nmc K -p (41) K22 Thus in an arbitrary (Galilean) frame the energy-momentum tensor can be written immediately 2 K3( P = nmc K ( X X + p a. (42) P.v K2(3) / 2 v (cf. [2]). (We see that these properties satisfy the conditions imposed in Section II for the perfect fluid). ARL-TR-60-274, Part II 23

THE UNIVERSITY OF MICHIGAN 2756-1-F V RELATIVISTIC PLASMA OSCILLATIONS It follows from (30) that, assuming Yi is analytic, the dispersion relation for the linear plasma oscillations is Det iS..+.- 0 (43) This may be written in the form 2. + 22 L tr (E) + (E2 ]Det (E) 0, (44) 22 44 kc L J kc where cE =kE k ((E) kE E k and tr (c) E=.. We note that in the nonrelativistic limit (i. e. c -- oo) this equation reduces to the well-known longitudinal (Landau-Vlasov) dispersion relation 0 E 47T A A 0 =~ 1+ k.cy- k We shall return to this point later. The biquadratic equation (44) may be solved as kc (E - 24

THE UNIVERSITY OF MICHIGAN 2756-1-T As is well known, the transverse and the longitudinal waves are usually coupled. Due to this fact, in the general case the study of the dispersion relation given above is somewhat complicated. However, there exist some special cases of interest for which a decoupling can be established, so that equation (45) reduces to the purely transverse dispersion relationship. These cases will be studied in the following subsections. 1. Zero External Field Case Here we shall consider the case in which the external field is zero, i. e. 2 =0. Then we have R -- and G -- s. Furthermore, if we /V / v.uv,uV' A a f(0) assume that the unperturbed distribution function is isotropic, i. e. dcC uk, auk the conductivity tensor becomes:f(o). =. -- d3U u u (46) ij - m ik u +p 7 s $ It may be readily seen that in this case the longitudinal and the transverse parts are completely decoupled, so that the dielectric tensor Q A A t E.. = k.k. + S.. 1J 1 J 1J ij i 3 ij A A t 1 where e =k * e k, and e = - S... 2 i] L3 Thus the dispersion relation becomes [ij 2 k22 2 k 2 2 kc kc kc so that we get the familiar relations ARL-TR-60-274, Part II 25

THE UNIVERSITY OF MICHIGAN 2756-1-F e = 1+ 4 oR = 0 (47) P for the longitudinal oscillations, and 2 2 ~ t p2 1 22 e = 1+ -+' =O (48) k c kc ( for the transverse oscillations. These relationships have already been examined to some extent in the literature [ll, 13, 14]. We shall study them in some detail. We have 20) 2 2E. (u )2 1 df ~ A ^.) e 3 V u du ao- k- cr k = i ---, - =- d-"' mu J 3 um ku -ipy Z and t 1 1 t _ o - S:o =- (tr a - ) 2 = 2 (u 1 df(0) i e2 i d3 (ux) u du i - d u (49) m J 3 k u - ip y z In the above integrations, we have selected the u -axis along k. 1.1. Longitudinal Oscillations We shall restrict ourselves merely to the study of the solutions which behave as outgoing waves at + oo. In this case one substitutes p = Z-iu and considers the limit as — e 0 +, so that' In the general case, the factor iw in the last term is to be replaced by an arbitrary function of w and k, cf. U15]. ARL-TR-60-274, Part II 26

THE UNIVERSITY OF MICHIGAN 2756-1-T 1 ^ 1- P + iTT (ku -w ). (50) ku -ipy ku -wy z z z The symbol 9 indicates that the first term is to be interpreted as a principal value. The contribution of the last term, which gives rise to the spatial dispersion, will be studied next. That is, considering the imaginary part of - a on the imaginary axis of the complex p plane (as Z -- 0), we get p ~ 3 3 2 (0 2 2 87 c e d ) 2 Im(jcr) -0(v -1) d u (51) \P 3 2m du U kc Here v is the index of refraction (=-), 0(x) is the Heaviside's step function, and U is the "reduced" phase velocity, that is U fr r- 2 17 1+ kc 2w2/k2 c The step function in equation (51) indicates that (as is pointed out by the former investigators [1] ) no spatial dispersion occurs for the waves propagating faster than the speed of light. However, the nonrelativistic analysis, which gives (cf. for instance, [15]) -Im 47r 8-_ c 3 e f(0) ( im~ Y 38 2 m k cannot predict this point, unless the distribution function is required to vanish outside a sphere of radius c. ARL-TR-60-274, Part II 27

THE UNIVERSITY OF MICHIGAN 2756-1-F In particular when f(0) is of the MBJ type, as discussed in the previous section, one finds P 2 Im, 2 Ee +a 2/l+ A23- 2) 0(v 1), = 3 2i 3K3) 2 v o K2(9) (52) where the plasma frequency 2 47rne ( W = --. (53) p m Silin [14] studied the ultrarelativistic limit, assuming f()= C e13u/c (This corresponds to replacing y by u/c, i. e. c- 0). His result can be obtained 2 by ignoring the terms in equation (52) of the order of 3, which leads to Im 4( a) r P +E( 2), (v2>1) UR (54) P\P / 2V - w In the nonrelativistic limit, i.e. 3 >> 1, one obtains Im 3/2 -(r -1) (P' 2 3 2 This, when v >> 1, reduces to the result corresponding to the Maxwellian distribution 2 (Im = (j = E 1 3/2 e-/2 (56) P, 2 3 which leads to the well-known Landau damping [16. ARL-TIR-60-274, Part II 28

THE UNIVERSITY OF MICHIGAN 2756-1-F Now, we consider the real part of- 4u on the imaginary axis of the complex p plane (i. e. the principal value integral) R4e (4)2 ye2. df(0) 1 Re p -- / - C du yu du [1 - c L tanh-1 v (57) P k2 -m vu CT k 2 0 (0) Again if f is of the MBJ type one finds 02 00 cosh x R47T [ 1 1 2 2-P cx \exdx p 2 2( 3 2 1-22 2 vp v V ).0 1 + -cosh x (58) When v= 1, one can carry out the above integration p 2 o Re 47r p 2 0 + 1 (59) ~2 L'3 + ~2~321 Re -p aQ, - K2(/)- K2 (59)] Alternatively, one may use the relations derived in Section IV. In the absence of the external field (Q = 0), equation (37) reduces to j = a k. k. + a S.. (60) 1ij 1 J j where 4: cr =? f K3( ) ycr' s( + sp) ds P p K2(3p) 2 2 = (+ps) +kcs (k 2- k for2 =0). ~~p~ ~ ~ ~ ~~j 22 2 k 2 2 2 2 iNk 2 jk (61) ARL-TR-60-274, Part II 29

THE UNIVERSITY OF MICHIGAN 2756-1-F The longitudinal dispersion relation then is obtained by setting the first of equation (61) equal to -1 when vl < 1. We have deduced an asymptotic expansion for the integration involved;,t02 1K () n 21( K (1 ) (1-n F2 b 1 E 2 n=0, l,... K (E) n n where p = - i (2n+2)! b (2n)!, (b = ). (62) n a n+l- n n-l 2 (n+) 2 (n-) We see that for v = 1 we obtain,... K1 2 K(() n W2 2 K /2 which is the same as the result obtained from equation (58) for this case. For the cold plasma, i. e. 13 - oo, equation (62) reduces to the wellknown Langmuir and Tonks equation 2 2 J = / Up, and to the first order it gives the fore ollowing result which is first derived by Clemmow and Willson [ll, -2 ( /3 v2- ) +'1). In deducing equation (63) we have used the astotic equation of) w th ee Bessel functions ARL-TR-60-274, Part II 30

THE UNIVERSITY OF MICHIGAN 2756-1-F K G3) 2 2 2 2 2 2 n_ 1 n -m (n-m )(n -m -2) 1 _ Km(P) 1 2 2! (2p)2 3 The ultrarelativistic limit as considered by Silin [14] can be rederived either from equation (58) by replacing 7y — u/c, or from equation (61) by setting 3 K3(~) - 8/3. The result is 3 22 v2 = V 1 (1 tanh v-1 v < 1, UR. (64) For the case I v [ ( 1, approximating successively one obtains 3 2 1 - V = 5 32 2 which indicates that no longitudinal waves can propagate in an ultrarelativistic 2 1 V_ 2 plasma with frequencies w2 < 3 wP It is seen that equation (64) blows up for v= 1. However, an examination of equation (59) shows that the phase velocity corresponding to v = 1 is finite. This observation indicates the limitation of Silin's ultrarelativistic result in the vicinity of v = 1. One may write w 7 I ( — n p, (; UR 1.2. Transverse Oscillations Similarly, one can perform the angular integrations for the transverse conductivity which leads to the results ARL-TR-60-274, Part II 31

THE UNIVERSITY OF MICHIGAN 2756-1-F 00 Im (4 7r 9 = -m) df 2 5 do2 -1 21 47r t) 87r e2 d u df(-) 2 2_1 Re -Im = - m du -d- u In particularif f is MBJ we obtain i+~r- ktvu)2 ( ) Im(pa) 13) 2 (v-).wk (66) Im(pa) 2 - wkc r + (3), (<1). UR (67) The nonrelativistic case is obtained by setting,3 ~ 1, 2U Re - d 4 (0) (0) Imp wk m =k edu (+ 1 _- tanhe- k (65) We found it more convenient to work with the representation developed in n particular if f is MBJ w e obtainhe Re ( ) when f is the MBJ type. As is indicated by equations (48) and (60), the dispersion relation for transverse (66) Jpwr 2 wkc K2 In the ultrarelativistic limit one obtains Silin's result [14] 2 waves when v < 1 becomes m ARL-TR-60274, Part II 32(67) The nonrelativistic case is obtained by setting > >> 1, Im =2 ~ to p -(1 or for v >> 1, Im at ^ 0 V e- NR. (68) Section IV when dealing with the Re (- a )when f is the MBJ type As is ARL-TR-60-274. Part II 32

THE UNIVERSITY OF MICHIGAN 2756-1-F p 2 2j f ^, p d2 Ks k2 2p K 2 1= - 2-2 2 L p KZ(1) ~2 ds kc +p 2 2 + 2 222 2 ( 3+ps)2 + kc2s. (69) An asymptotic representation of this relationship may be deduced without much difficulty - 2 K (E) ( E0 2) 2 K (j) w 1=0,1,2.. 2i! where again we set p = -iw. It is seen that for cold plasma, i. e. 13 - oo, one obtains 2 2= 1- -2 z-i 2 which is the well-known Langmuir dispersion relation. If we ignore terms of 2 2 the order of (1- 2) we find* 2 P 2 1+ 2 2) 2. ov 2 ~~~(71) 1+/ 2 3 2 Approximating successively one gets V 0 1 -E 1- _ 2-~ {2' j3 2 2 2 A2 Thus one may write k c w -w where A2 47rne2 2 30 [p = _ _ _ n = 2V = - p < _ m <v2>, m Part II 33 ARL-TR-60-274, Part II 33

THE UNIVERSITY OF MICHIGAN 2756-1-F The latter is in agreement with Buneman's result [13]. The ultrarelativistic limit can either be calculated from equation (65) by setting'y — u/c (cf. Silin [14]) or from equation (60) in the limit of 3 < 1. The result is 2, y7p 3v-2 2 v = 1 - 2 - 2 1 + - tanh v UR. (72) 2 2 v For v << 1 one can obtain by successive approximation' 6 2 2 P 5 3 2 2. Constant External Magnetic Field Case In this subsection first we shall assume that the unperturbed distribution function is of the MBJ type, so that the conductivity tensor given in equation (38) can be used directly. The second term of this tensor contributes the major difficulty in the solving of the determinantal dispersion equation. However, if we restrict ourselves to the case in which the propagation vector is along the unperturbed magnetic field, then again it is possible to decouple the transversal and the longitudinal modes. The latter is found to be exactly the same as studied in the previous subsection, and hence will be omitted here. 1 2 Then, assuming k x h = 0 which implies that R. k =-G k = Mk = k, we get * Our formula differs from the one given by Silin by a factor of 2/3 ARL-TR-60-274, Part II 34

THE UNIVERSITY OF MICHIGAN 2756-1-F 2 oo E.... ds R.. - - k. s ij. = ij +p K2 (3) ji 2 j 3 where 2 + sp2 + c222(73) = (3+sp) +cks. (73) We see that k - E S = S. k = 0 so that A ^A^ t E.. = k.. + e.. 1j 1 J 1j where 2 2 00 K T) i ds 2 2 2k22 K3( ) Ea,, p s( ~~ Kfjds - k s p K (3) 2 3 0 2 00 K3() +K P ds s( + ps) p K- (P) 3 2 0 (cf. equation (61)), and t Ej. = S ik.k Sj ik k.tj 7'2 2 D0 Y 2 K^) =S..+,-P 2 ds (R..-kk.) 2 1J + P K2P) O Ji ij'I~ 2 oo00 =Sij 1+ p ds cos 22s 2 + eR k4 P ds sin 0 s 2 ijXj p2 K 2() 0 C2

THE UNIVERSITY OF MICHIGAN 2756-1-F (Observe that the latter is in agreement with equation (60) for 2 = 0). It is a straightforward task to show that A A A 2 2 Det(A Sij+Bk.k.+ C k) = B (A + ) = B(A + i C) (A-iC) In our problem the first factor corresponds to the longitudinal dispersion relation (B = E ) which has already been studied. The last two factors represent the two (ordinary, extraordinary) modes of the transversal oscillations: kc.2L -d 2 - 0O --- (A+iC) = 1 + (A2 2 ) " K \ dse J 2~ = 0 (74) (cf. equation (69) for 0 = 0). If we consider the nonrelativistic limit [cf. equation (35)], we get 00 T. 0 k2s2 2P 2 2 2 p P1 + V- / s ds e +0s k2c2+p2 P 20 For cold plasma, i. e. 0 = 0, one obtains 2 2 \ ~ u2 1 ~~k 0=1 - v 2 ) - __(75) to 1 +~t ~ t~? which is the well-known transversal dispersion relationship (p = + it). We have developed an asymptotic expansion for the above dispersion relation. The result is' (p = iw) Because of the divergent character of the involved summations, extra care should be exercised in dealing with the above representation. XRL-TR-60-274, Part II 36

THE UNIVERSITY OF MICHIGAN 2756-1-F /2\1 j+2i+l W22 2 Wo \ 7,2 oi 2i K, (K3) 2 41 L <P"1 (l- ci (1+ ___ w i.jO,... 2+i"j or v2 v 1 - P (-1) i' K i +j _ 1 ) / K (13C (76) w i,j=0,l,... where J (j + 2i)! 2 i! j! It is observed that the second of equations(76) reduces to equation (70) for Q = 0. Both equations become equation (75) when j3 - oo (cold plasma). To the first order in 13 we obtain 9 2 [-u2 " i+) 1 5 1 2 2 1 2 2 - 2 NR (77) The.magnetohydrodynamic (Alfven) waves may be studied by the assumption that -- 1, so that the ion dynamic is particularly important. Then for a binary, initially neutral, singly ionized plasma, ignoring the terms of the order of (-) in the first ofequations(76), one obtains ARL-TR-60-274, Part II 37

THE UNIVERSITY OF MICHIGAN 2756-1-F 2 47c2 K3 (3+ K3 (3-) 2 p+ K2 (+) K2 ( ) (77) 0 To the first order in 1/13 we have 2 c2 ( 5 p Iv + —1 2\12 22 a c where p is the total pressure (zero order), and a is the Alfven speed H o a = \/47r(p+p ) Alternatively equation (77) can be written as 2c 47r (0) (0) k = 1+ (w + p (78) 2 2 o w ~ H where w( is the zero order total energy density. In deriving the latter form we have used equation (41) (cf. Harris [4]). Next we shall consider the spatial dispersion for the case discussed above. However, we shall reformulate the problem without specifying the unperturbed distribution function explicitly. To do so let us consider the conductivity tensor (0) A Tij assuming (i) f is isotropic in u, and (ii) k x h = 0. Then e2 3 1 df(0) -iktu s rij- - e R ds (79) ARLTR6074 Part II 3jk ARL-TR-60-274, Part II 38

THE UNIVERSITY OF MICHIGAN 2756-1 -F A A For the sake of simplicity, we select k (and h) along the z-axis. Clearly in this case, due to the azimuthal symmetry, one has a11 =U 22' c12 - a21' c13 =a31 =a23 = 32 =0. Then the transverse dispersion equation can be computed easily from (45), using the relations tr E (e2)~ 47r ___ - =1+-.11 2 2E p Det () 4r 2 4r - =(1+ =(1+)2 + ( 21) -e p P Thus one obtains k2c 4 7 1+ 2 2 (11 + i 21) =. (80) P P (cf. equation (74)). Here, the upper and lower signs correspond to the extraordinary and the ordinary modes, respectively. Let us consider the real It should be kept in mind that Si is defined as an algebraic quantity depending on the sign of the electric charge of the corresponding species, e. g., for electrons f2 < 0. ARL-TR-60-274, Part II 39

THE UNIVERSITY OF MICHIGAN 2756-1-F and imaginary parts of 4 r 4ir t _ 21_ (81) + 11- 21 p - p on the imaginary axis of the complex p plane. After performing the angular integrations in u-space, one obtains 4r 4 3 e2 df() Im - -- - e (k2 2 + 2 w2) du — u p - wk m du U! 7- ( __ ku 4r t 8r2 e2 df(O) Re - du — u p k- mk m du 0 toi+12 C +i2+ku ~W+Q 1 1r -(____ w+ _ + 2ku -j — + -( —--- )2 n.+.k J-(82) L ku 2. + 0 - ku (cf., equation (65)). In deriving the latter we have used the well-known improper integral -ixs i e ds = + iT (x). 0 x ARL-TR-60-274, Part II 40

THE UNIVERSITY OF MICHIGAN 2756-1-F The limits of the u-integration in the first of equations (82) are determined from the condition - +2/w I y+ = ---- < 1. (83) v u/c Again the upper and lower signs correspond to the extraordinary and ordinary modes, respectively. 4ir The imaginary part of - a. vanishes identically on any interval of u for which y+ > 1. In other words, in this case there is no damping * (or instability). Equation (83) is satisfied by the ordinary mode (i. e., y + < 1) provided v > 1 and U1 < u < oo, where Ivn/w l 2 -2 1 i+/W2 U1 = c -. (84) y2-1 On the other hand for the extraordinary mode we have the following cases: y <1 whenever (i)v > 1 and U1 < u< oo (ii)v < 1 and U, < u < U2 Strictly speaking this is the only case when a true dispersion relationship exists [15]. ARL-TR-60-274, Part II 41

THE UNIVERSITY OF MICHI-GAN 2756-1 -F where IvQ/w I + Jv2-1 +Z2/w2 U2 c. (85) 1 -V2 Let us note that c 1 -/w2 2 lim U (86) v - 1 2 I Q/W (cf. Figure 1) In the case when f(0) in the MBJ distribution the u-integration involved in the first of (83) can be carried out explicitly (v 2 > 1 - 2/ 2) 47r v T - 2 e- r Q Im L t = 2 + 3 2 Im —at = ^-P-4 v 2 _ 1 + v -1) 1 +- (^21)+ -- p 2 2v W 1K2 () W U1 r + - j2(v2-1) + 2 -- |2 (87) 2 to w 2 U2 Let us consider the case v > 1 Im - 2 p 2v3 W 2 / K2 (3) *{V 2 -1 + V JV2 -1 +F 2/W2 - 2j/W 0 (+ Q/w)2.2 vJ1/l/J~v2 i+2/t_2-^/2 _ o v2 - 1 J (88) ARL-TR-60-274, Part II 42

THE UNIVERSITY OF MICHIGAN 2756-1 -F 3 I I \zY-(-(1~1:o 1~! \ 8Y+ln =o 2 x \ Y-( I Q 0 = 1. 0 U0 2u 3u 4u 5u c c c c c FIGURE 1 ARL-TR-60-274, Part II 43

THE UNIVERSITY OF MICHIGAN 2756-1 -F Clearly, for 2 = 0 equation (88) reduces to equation (66). In the nonrelativistic limit for v >> 1 and 2/2 < 1 we obtain +47r f cj2 2 Im- / - — a. N R (89) p -w 2v2 We have also ignored the terms of the order of 2 | in deriving the latter. Equation (81) is to be compared with equation (68). Although the other cases of interest can be studied in a similar manner, a detailed discussion will not concern us here. However, it might be of some interest to note that the ultrarelativistic limit also can be considered as was done in the case of the absence of the external field. One finds that the effect of the magnetic field does not appear until to the second order in /3 (rather in J 32/w I 2 -1 ARL-TR-60-274, Part II 44

THE UNIVERSITY OF MICHIGAN 2756-1 -F VI DISCUSSION AND CONCLUSIONS In this report we have attempted the study of the linear oscillations in a hot plasma. It was assumed that the system can be represented by the relativistic Vlasov equation coupled with the Maxwell's field equations. The formal solution of the linearized problem was deduced employing the integral transform technique which was originally used by Bernstein for the nonrelativistic case. Giving particular attention to the spatial dispersion, we examined the dispersion equation for some special cases in which the longitudinal and transversal modes can be decoupled. We developed asymptotic expansions for the cases studied when the unperturbed distribution function is of the MBJ type. The longitudinal oscillations were found to be undamped* when 2 2 w > k c; this is in agreement with the result of the former investigations. It was shown that unattenuated, circularly polarized, transverse waves propagating along the unperturbed magnetic field can exist provided 2 2 w > k2c2 + A' where the suffix A represents the species which has the largest | eA/ mA ratio. Here, it should be pointed out that the collisions, which are ignored in the framework of the present theory, can provide an additional damping mechanism which is different from the one discussed above. ARL-TR-60-274, Part II 45

THE UNIVERSITY OF MICHIGAN 2756-1-F For example, for a simple electron gas, ignoring the terms of the order of 1/ 2, the above condition may be written as follows (cf. (77)) h2 k2c2 p 1 ___ - (,2 1 +Q/( 213 1 + Q/L 0>, 2 1 1 2 ~ l+_W~3 2 1 2 w which reduces to the quadratic form 2 21 1 X2 -(1 — )X + < 0 2 / 2/3 where X= 1 + Q/w. An examination of the latter indicates that in order for the above inequality to hold one must have 2 2 -- < -- (1 - ), w 47r nm c 2 / p or, approximately, (3 >> 1) ARL-TR-60-274, Part II 46

THE UNIVERSITY OF MICHIGAN 2 756-1 -F 105 n 1 H < ( )1/2 (1 -1/3), (inc.g.s.). o 2 An estimate of magnitude of the right-hand side shows that the latter condition is satisfied for almost all plasmas of practical interest. Moreover, one finds that (i) If tW 3 __ P < 1 + n2 23 there is no frequency region for which the ordinary mode is unattenuated, 2 (ii) If - < 1 + -, the extraordinary mode is undamped for frequencies n2 23 1- I > 1+ 1 23 t2 u 3 However, when -- > 1 + -, the undamped frequency region is bounded. p2 2 P We note that the resonance frequency, viz. w = i 2l, lies in the damped region. The magnetohydrodynamic waves may be examined in a similar manner. One finds that there is no undamped frequency range in this case. The results obtained may be further illustrated by the following sketches. (0) Assuming f() is MBJ distribution, the transverse and the longitudinal waves in a simple electron gas are considered in Figures 2 and 3, respectively, when the external field is absent. In Figure 4 the cut-off frequency, w, is plotted ARL-TR-60-274, Part II 47 ARL-TR-60-274, Part II 47

THE uNIV SITYM IGAN 2756-1-F o' Up0 0 I I ---- L-TR-60- Part1 1FU 2 48 kC dC 0 1 i ARrL-Tl-60- Z74 part IUIE2 48

THE UNIVERSITY OF MICHIGAN 2756-1-F cs I -, —I C C) "-4 1 I I < -0.5.1.'/ctp, — 0.5 1.0 FIGURE 3 ARL-TR-60-274, Part II 49

THE UNIVERSITY OF MICHIGAN 2756-1 -F 1 0woI 5 10 10l 1 10 100 1000 R = mnc2 _ e FIGURE 4 ARL-TR-60-274, Part II 50

THE UNIVERSITY OF MICHIGAN 2756-1-F vs. 3, using (o )2 2K2 (X) ( —o) = --- \ - ^ dX P K2 (3) X2 5 55 3 1 - -- + ---, 8 >> 1, NR 2/ 8/3 R/ 3, << 1, UR. ARL-TR-60-274, Part II 51

THE UNIVERSITY OF MICHIGAN 2756-1-F APPENDIX A The Integral Equation for f It was previously pointed out that instead of eliminating the distribution function through the Maxwell's equation as is done in Section III, one can eliminate the field variables in equation (26) to obtain an integral equation for f. To do this one substitutes equation (29) in the Maxwell's equation + 4 + k 47r d3u ik F ik F - F - e- u f L v Au c v Au vp k 4j c J Only the first three of the above four equations are needed for our purposes 47r + i + 2 2 iJ. F-V (k. k.i k i (k2 -k k) c I k 4 j j i / ij / A/~ / Using the relation -1 (k. k2 S )(k k + k2 )= /i i2l j k4 ~j ij k2 Jkikl u 4 one obtains + _ 4wr 2 + +F 4 (k k + k )J. 4i 2 c i J 4 ij J k4k 4 u and hence substituting back in equation (29) F+ X 47r.i2 k+ vij vi/u k2 2 C i 4 j ) d, 4 ac ARL-TR-60-274, Part II 52

THE UNIVERSITY OF MICHIGAN 2756-1-F The latter is used to eliminate the field variables in equation (26) to obtain the integral equation 47re f k 2 + 47e af x (k k k + ) ff = i 2 u -u 2k2 I p 4 p \e RRk ds m2 v au. k 2 2 kj me J k k 0 4 ct " e d- l u f+. (90) 3 d 3u Now if we multiply the above equation by e u - and integrate over u space, q y then sum over all species, we obtain 47 _) xa 2 J u f0) Spq^ 2 2 P^k p 4 lp 4qk 4 p- q 4 a _ 1 2 1 =Pqq- -4 c' (k + k ) k J [ Pq k4c k2 t kp 4 kp )q q =Initial Conditions =. p Here we added to the right-hand side the initial conditions which were being suppressed. The solubility condition then reads Det i- j -- (k.k +k2 ) a, 0 ij k c 2 it 4 if jQI 4L 4 J J Since Det(A * B) = Det(A) Det(B) we can write ARL-TR-60-274, Part II 53

THE UNIVERSITY OF MICHIGAN 2756-1-F Det [(k ij -k.k.)- k k= = Det k ij k + k4 ij 4c ij La 1J 13 k4c 4 or Det S..j+ 22 e = 0 k2 12 J where again we put AA cij j' S.. =. -.k. = 1J p 1J kc -p, k = k. 4 j [cf. equations (30) and (31)] (0) A(O) af In particular, if f (=-n f ) is isotropic then u a becomes symmetric for v = 1, 2, 3; thus the pair (p, k) is also symmetric for u = 1, 2, 3. Remembering the antisymmetric property of 8 k equation (90) can be written for this case as ^(0) S 3 f+ 2 af iEx 1 d u + f 7 2 i -p \ e R ds e 1 u, 2 2 2 kc where we set v =- -- 2 p ARL-TR-60-274, Part II;54

THE UNIVERSITY OF MICHIGAN 2756-1-F BIBLIOGRAPHY 1. Eckart, C., "The Thermodynamics of Irreversible Processes III", Phys. Rev., 58, 919 (1940). 2. Synge, J. L., The Relativistic Gas, (Amsterdam, North Holland Publishing Co., 1940). 3. Taub, A. H., "Relativistic Rankine-Hugoniot Equations", Phys. Rev., 74, 328 (1948). 4. Harris, E.G., "Relativistic Magnetohydrodynamics", Phys. Rev., 108, 1358 (1957). 5. Zumino, B., "Some Questions in Relativistic Hydromagnetics", Phys. Rev., 108, 1116 (1957). 6. Jiittner, F., "Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der Relativtheorie", Ann. d. Physik, 34, 856 (1911a). 7. Imre, K., "An Investigation of Transport Properties of Plasmas", The University of Michigan Radiation Laboratory Report No. 2756-2-T, July 1960. (Ph.D. thesis at The University of Michigan). 8. Trubnikov, B.A., "Electromagnetic Waves in a Relativistic Plasma in a Magnetic Field", Plasma Physics and the Problem of Controlled Thermonuclear Reactions, III (London, Pergamon Press, 1959). 9. Drummond, W. E. and Rosenbluth, M. N., "Cyclotron Radiation from a Hot Plasma", Phys. Fluids, 3, 45 (1960). 10. Klimontovich, Y. L., "Relativistic Transport Equations for a Plasma I and II", Soviet Physics JETP, 10, 524 (1960) and JETP, 11, 876 (1960). 11. Clemmow, P. C. and Willson, A. J., "The Dispersion Equation in Plasma Oscillations", Proc. Roy. Soc. (London), 237A, 117 (1956). 12. Clemmow, P. C. and Willson, A. J., "A Relativistic Form of Boltzmann's Transport Equation in the Absence of Collisions", Proc. Cambridge Phil. Soc., 53, 222 (1957). ARL-TR-60-274, Part II 55

THE UNIVERSITY OF MICHIGAN Z756-1-F 13. Buneman, O., "Transverse Plasma Waves and Plasma Vortices", Phys. Rev., 112, 1504 (1958). 14. Silin, V. P., "On the Electromagnetic Properties of a Relativistic Plasma", Soviet Physics JETP, 11, 1136 (1960). 15. Van Kampen, N. C., "On the Theory of Stationary Waves in Plasmas", Physica, 21, 949 (1955), and "The Dispersion Equation for Plasma Waves", Physica, 23, 641 (1957). 16. Landau, L., "On the Vibrations of the Electronics Plasma", J. Phys. USSR, 10, 25 (1946). ARL-TR-60-274, Part II 56

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