P H Y S I C A L R E V I E W V O L U M E 1 4 8 , N U M B E R 1 5 A U G U S T 1 9 6 6

Interaction of Electromagnetic Waves with Quantum and Classical Plasmas. Correlation Effects*

NGUYEN QUANG DONG

Groupe de Recherches de VAssociation EURATOM-CEA sur la Fusion, Fontenay-aux-Roses, Seine, France (Received 1 September 1965; revised manuscript received 31 January 1966)

A quantum statistical approach is used to study the interaction of electromagnetic (EM) waves with a hbt plasma. The quantum calculation takes into account the correlation effects. The Coulomb interaction between the charged particles is renormalized so as to take care of the collective effects. This renormalization also reduces the number of diagrams to be considered to obtain any desired accuracy. The results for a classical plasma are obtained by passing to the limit -h —> 0. Some applications are considered for a classical hot plasma. In the damping of EM waves, the contributions of correlations are more important, at the limits of high and low frequencies, than the Landau contribution, which prevails at plasma resonances. Also, the appearance of the ion Doppler spread, instead of the electron Doppler spread, in the data for the scattering of EM waves observed in the ionosphere backscattering experiments, is explained in terms of correlation effects.

I. INTRODUCTION

TH E simplest microscopic approach to the problem of EM waves and plasma interaction is based on

a kinetic description of the plasma in a self-consistent field approximation. Here, the Boltzmann-Vlasov equa­tion is used for the one-particle distribution function, and the Poisson's equation for the field. Landau1 and several others2-4 have found the solution to the linear­ized problem. This solution describes the behavior of a collisionless plasma which is perturbed out of the ther­mal equilibrium situation. I t is found that EM waves are dampened in such a plasma at a rate given by the Landau damping coefficient.1

The dynamical behavior of the plasma can be de­scribed with a better accuracy by the Liouville equa­tion, or the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations5 derived from it by integration over the coordinates and momenta of all but one particle, two particles, etc. Usually, one breaks off the chain of equations in approximating the higher distribution functions by combinations of the lower ones.6-8 Ichikawa9 and Willis10 studied the problem of the interaction of EM waves with a plasma in basing their calculations on the BBGKY hierarchy. They, however, made some ad hoc approximations to truncate it, so that the dispersion relations obtained in the two calculations are somewhat different and their damping

* This paper is based on a thesis submitted by the author in August 1964, to the University of Michigan, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Plasma Physics.

1 L. D. Landau, J. Phys. (USSR) 10, 25 (1946). 2 N. G. Van Kampen, Physica 21, 949 (1955). 3 K. M. Case, Ann. Phys. (N.Y.) 9, 1 (1960). 4 F . L. Shure, Ph.D. thesis, University of Michigan, 1963

(unpublished). 5 N. Bogoliubov, Problems of Dynamical Theory in Statistical

Physics, transl. by E. K. Gora (State Technical Press, Moscow, 1946).

6 R. L. Guernsey, Phys. Fluids 5, 322 (1961). 7 R. Balescu, Phys. Fluids 4, 94 (1961). 8 N. Rostoker and M. N. Rosenbluth, Phys. Fluids 3, 1 (1960). 9 Y. H. Ichikawa, Progr. Theoret. Phys. (Kyoto) 24, 1083

(1960). 10 R. Willis, Phys. Fluids 5, 219 (1962).

148

coefficients are not comparable in magnitude. Never­theless, from these calculations the fact has emerged that in the limit of long wavelengths the Landau damping being exponentially vanishing, another damp­ing mechanism becomes predominant which arises from the correlation effects between particles. This is also confirmed by the calculation of Oberman el al.n using the Guernsey procedure6 to truncate the hierarchy. They obtained a correct form for the conductivity in a classi­cal plasma, in the limit of high frequencies and infinite wavelengths. Indeed, in this limit, the Landau re­sistivity is rigorously null and the plasma resistivity is due to the correlation effects.

Of course, the correlation effects may be accounted for by other methods of evaluation of plasma transport parameters. Dawson and Oberman12 have considered a simple model for a classical plasma, and have calculated the high-frequency conductivity at infinite wavelengths. Berk13 has extended the calculation to the case of finite wavelengths, and has given the correction to the Landau coefficient. By their nature, these computations are less systematic than those derived from the hierarchy, and the results are applicable only when the specific condi­tions of the problem, i.e., infinite ion mass, etc., are met.

More recently, people are interested in the evaluation of the transport parameters for hot quantum plasmas. Dubois et al.u have set up a diagrammatic method for the calculation of the absorption of EM waves. PereP and Eliashberg,15 and Ron and Tzoar16 also have con­sidered a diagrammatic technique to evaluate the plasma conductivity in the limit of infinite wavelengths. In the classical limit, their results correspond to those ob­tained for a classical plasma by direct calculations.16

11 C. Oberman, J. Dawson, and A. Ron, Phys. Fluids 5, 1514 (1962).

12 J. Dawson and C. Oberman, Phys. Fluids 5, 517 (1962). 13 H. L. Berk, Phys. Fluids 7, 257 (1964); 7, 917 (1964). 14 D. F. Dubois, V. Gilinsky, and M. G. Kivelson, Phys. Rev.

129, 2376 (1963). 16 V. I. PereF and G. M. Eliashberg, Zh. Eksperim. i Teor. Fiz.

41, 886 (1961) [English transl: Soviet Phys.—TETP 14, 633 (1962)].

16 A. Ron and N. Tzoar, Phys. Rev. 131, 12 (1963).

151

152 N G U Y E N Q U A N G D O N G 148

The correspondence is thus made between the physics of quantum and classical plasmas, and the possibility of interconnecting the two seemingly different domains of studies of plasma physics, partially accounts for the interest in these calculations.

In the same manner, we consider here a quantum-mechanical approach. We use a diagrammatic technique which is similar to the one used by Ron and Tzoar.16

However the calculation is extended to the case of a finite wave number k. This will allow us to deduce a few new results for a classical plasma concerning the relative importance of the Landau and the correlation dampings, and also, the presence of the ion Doppler spread in the cross section of the incoherent scattering of EM waves. As compared to the calculation of Dubois et al.u with their chosen technique of "open" diagrams, it is seen that with the "closed" diagram technique we do not have any difficulty in the determination of the true order (in the plasma parameter) of any given diagram.16 Also, here, the concept of renormalization of the interaction is introduced in a natural manner into the formalism.

II. GENERAL FORMULATION

We consider the many-body problem approach in the second quantization representation.17-19 To simplify the writing we take h=l by using an appropriate system of units. The several species of charged particles in the plasma are treated equivalently. We denote a species by a subscript s with mass ms and charge es. The field operators in the Heisenberg representation, \pa(r,t), obey the well-known anticommutation rules for fermions for equal time arguments. The Hamiltonian of the system in the absence of an external field 5C consists of two parts, the independent particle energy,

5C0=E IdxW(t,t)[-V*/2m,y,.(x,t), (2.1)

and the charges Coulomb interaction

v=h E E [dtdt'+;(rMH*',t)v(t-T') a s' J

X+Af,t)+a(r,l), (2.2) with

i>(r—r') = e.e.'/|r—r'l.

The charge-density operator is

17 P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959). 18 S. S. Schweber, An Introduction to Relativistic Quantum Field

Theory (Harper and Row, New York, 1961). 19 See Nguyen Quang Dong, dissertation, University of Michi­

gan, 1964 (unpublished) for further details.

and the current-density operator is

- W ( r , * ) > . ( r , / ) > . (2.4)

Before the external field is applied, the plasma is assumed to be in thermodynamic equilibrium, i.e., the system density matrix 3D is given by

a)=exp{-/3(3C-M-9l-Q)}, (2-5)

where 0= (kT)"1, k being the Boltzmann constant and T the temperature; H'^l=^28fxs?fl8, JJLS being the chemi­cal potential of species s, and 3l8 its number operator; and SI is the thermodynamic potential of the system. The equilibrium average of any dynamical operator 0 is then given by

<0) = Tr{£>e}=Tr{exp[-|S(3C-M-9I-O)]e}. (2.6)

The external field is described in a gauge in which the scalar potential vanishes, thus the excitation Hamil­tonian is, in the nonrelativistic case,

3C<A )=---/drj-A(r,0

+ i E — [drp8(r,t)A?(ryt), (2.7) a m8c

2J

where j(r,/) is the current-density operator given by Eq. (2.4) and A(r,/) is the vector potential of the ex­ternal field.

For finding the response of the system to the inter­action of EM waves to first order of the field strength, we apply a formalism19 analogous to Kubo's.20 The re­sponse is measured by the induced current

J(r,t) = JdT'[ dtf K ( r , / ; r ' / ) - A ( r y ) , (2.8)

where K is the retarded current-density polarization tensor

-8im5(t-i')8(t-t') Z(es2Ns/ms), (2.9)

s

I, m=l, 2, 3. In the above formulas, the operators j in the commutator brackets are defined by Eq. (2.4). The 8 functions are the well known Dirac and Kronecker 5's. The function rj is defined as

The second term on the right hand of formula (2.9) is already given in terms of physical constants and ob-servables. Then the calculation of K is reduced to the

20 R. Kubo, J. Phys. Soc. Japan 12, 570 (1957).

p(M) = E P.(r,/) = E ^ / ( r , 0 ^ ( r , 0 , (2.3) H(/-/') = 1, ^ = 0, if t<t\

148 I N T E R A C T I O N OF W

calculation of the matrix elements of the average of the retarded current-density commutator:

Hlm+(i,f, t',t') = rl{t-t')i([Ji{tAUt',t')l). (2.10)

One may easily verify that this tensor depends on time only through the difference t—t'. We further assume that the system is isotropic and homogeneous. The field correlation functions then depend only on the differences of the arguments and it is then possible to transform to momentum space. Equation (2.8) is transformed into

J(k,co) = a(k,co)-E(k,co), (2.11)

where E(k,co) is the external electric field, and the con­ductivity tensor is given by

alm(k,o>) = Illn+(k,a>)-dlmi: «.V4ir, (2.12) 8

where IIjw+(k,tt) is the Fourier transform of IIjm

X ( r - r ' ; t-t'), and o)2 = ^we2N8/m8. For the scalar conductivity defined by

o-(k,o>) = k-<7-k/&2, (2.13) we have

a(k,a>)=(a>/ik2)Q+(k,a>), (2.14)

where <2+(k,co) is the transform of the retarded charge-density commutator

Q+(r-r';t-t') = v(l-Oi([p(t,t)At'/)l)- (2.15)

In the problem of scattering of EM waves, the ex­ternal field will be purely transverse. The phenomenon is governed by the second part of the Hamiltonian 3C(A) [Eq. (2.7)], which indeed is involved in the direct transitions in which incoming waves defined by their wave number ka, and polarization vector ea, are scat­tered into waves of wave number k&, and polarization vector e&, while the system of particles change from state n to state m. The transition matrix is found to be

Tha = \mc2 (2coa2co6)-1/22 (ia • eb) E (es/2m8c

2) 8

X(tn\p(kfl)\n)2T&(Em-En+a>), (2.16)

where k=k&— ka, to=co6—coa, the Em,n are eigenvalues of 3C and where the p(k,2) are the Fourier transforms of the density operators p(r,/) . Then the differential scat­tering cross section is given by

d E(k,cu) = i(Wwa)(l+cos26>)5(k,co)J0^6 , (2.17)

6 being the scattering angle, and

5(k,co) = E exp{P(a+n-Nn-En)} m,n

xjz — E—<»IP.(M)I«>

I s msc2 «' mS'C2

X(tn\ps>(-k,0)\n)\2Td(En-Em+u). (2.18)

V E S W I T H P L A S M A S 153

Now let us introduce the charge-density correlation function

G(T,f,t',l') = {p(x,t)p(r'/)), (2.19)

the Fourier transform of which is

G(k,co) = i : exptfiQ+n-N.-En))

m,n

X { < » | p ( k , 0 ) | » ) < « | p ( - k > 0 ) | » »

X2w8(E„-Em+o>). (2.20) The cyclic properties of the traces, and other properties of the field operators, yield

1 rx do>' Q+(k,o>) = - r ( l - e - ^ ' ) G ( k , o > ' ) ,

2irJ^_oo co — co — it €->0+ . (2.21)

In the momentum representation, where the opera­tors of creation ap,st and annihilation aPf8 of a particle of species s, having momentum p are defined, we have

p(k,0)=p(k) = ]T esap+b,MVtS. (2.22)

Then we see, in replacing Eq. (2.22), in Eqs. (2.18) and (2.20), that these expressions are known when we evaluate the "partial G functions"

Gss';pp'(k,C0)

= Z exp{0(Q+/i-iV»—En)}e8e8r(n\a9+kJap,s\m)

m,n

X ( w | a p ^ k , 8 ^ , s ^ ) 2 7 r 5 ( E n - E m + c o ) , (2.23)

or, the corresponding "partial Q functions"

1 r00 Jco' Q+ss,;pp,(k,co) = - / r ( l -6r^ ' )G. . , : p p (k ,co / ) >

2TJ-WO) — co—ie e->0+. (2.24)

We note also that 2

G„,;PP*(k,«)= ImQ„*;pp,+(k,«). (2.25) 1 — e-P"

Thus when the functions GS8>-tVV> are known, so are the functions Q«S';PP'+, and vice versa. We choose to evalu­ate the Q functions. This is a fundamental step, since we have

P p' n+(k,co)= Z G « ' ; p p ' + ( M , (2.26)

SS';PP' ms tns>

cr(k,co) = L -^e s 8 ' ; P P '+(k,co), (2.27) ««';PP' ik2

and

S«';PP' msc2 m8*c2 l — e~P°}

XIm0S8, ;pp,+(k,co). (2.28)

154 N G U Y E N Q U A N G D O N G 148

For the computation of the Q functions, we use a diagrammatic method19 which is derived from the work of Abrikosov et al,n Bloch and De Dominicis,22 and Luttinger and Ward.23 This allows us first to obtain the Q functions which are defined by

&«';pp'(k,«n)

= — / - ( l - ^ ' ) C ; ^ ( k y ) , (2.29)

with <an=2Trni/P, « = 0, ± 1 , •••. On comparison this relation and relation (2.24) it is seen that the function Qss' pp'+(k><^) is the analytic continuation of the func­tion QSs>;w(k,o0n), in the variable o>, from discrete points on the imaginary axis (»>0) to the real axis.

<3+(k,co) = lim(2(k,co„) con-*to+ie, e-> 0+. (2.30)

As we already know, we cannot directly evaluate the statistical averages, using the density matrix given by Eq. (2.5). However, it is possible to formulate a per­turbation theory which considers the interaction V between the charged particles of the plasma as a per-

(e) ( f )

(g) (h) ( i ) ( j ) FIG. 1. The diagrams up to second order in Uqt to be

considered (a-f), and discarded (g-j). 21 A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, Zh.

Eksperim. i Teor. Fiz. 36, 900 (1959) [English transl.: Soviet Phys.—JETP 9, 636 (1959)].

22 C. Bloch and C. De Dominicis, Nucl. Phys. 1, 459 (1958). 83 J. M. Luttinger and J. C. Ward, Phys. Rev. 118. 1417 (1960).

turbative potential. The existence of a theorem19'22

analogous to the Wick theorem for ordinary Green's functions will allow us to evaluate any Q function as a sum of the contributions of all the diagrams that we may draw between two outer vertices. On Fig. 1, these are marked by a cross.

We also renormalize the interaction. The difficulty of divergence at small q for the "bare" Coulomb inter­action Vq is avoided, although it should be noted that we do not introduce any further approximation in the perturbation theory. Thus, we perform a partial sum­mation in the series by replacing all "bare" interaction by an "effective" interaction which is represented by a knotty line,

Uq(am)= (47r /^)[ l - (47r /^)Q ( 0 ) (q ,« w ) ] - 1 , (2.31)

where am=27rmi/j3y m = 0, ± 1 , • • •, and <2(o) is the con­tribution of the zeroth-order diagram. In contrast to the bare Coulomb interaction, the effective interaction carries an "energy" am. In a sense, this renormalization is equivalent to the introduction of the collective effects.

We finally arrive at the following rules for the com­putation of the Q functions:

(1) Consider all possible ^th-order diagrams drawn in one piece between the outer vertices, n being the number of renormalized interaction factors U\;

(2) with each solid line labeled s, p, f i, associate a free-particle propagator

GP..G-0 = [fi..-ep (.J-1, (2.32) where

f i . .= (2/+l)7r;//H-Ms, / = 0 , ± 1 , ± 2 , • • -,

a n d ev,s = p2/2ms;

(3) with each knotty line, associate an effective interaction Uq(am);

(4) restrict the number of independent f's by the "conservation theorem";

(5) For each interaction vertex, put a factor es. For each charged density measuring vertex (outer vertex), marked by a cross sign on the graphs, put a factor ies, and for each closed loop, a factor (—1);

(6) sum over all indices and independent variables, except k, o>n, P, p', s, and s'.

II. CORRELATION EFFECTS IN QUANTUM PLASMA

When (1) the number of particles N in a Debye sphere is large, i.e., £20iV1/3«l, and (2) the Born ap­proximation is assumed, i.e., e2/#(z>)<3Cl, where (v) is the average particle velocity, we may limit ourselves to a few simple diagrams. They are sufficient to give us an estimation of the correlation effects,15-16 correct to the first order of the plasma parameter. In this paper, we perform the calculation for the scalar conductivity.

The zeroth-order diagram, Fig. 1(a), has no inter-

148 I N T E R A C T I O N OF W A V E S W I T H P L A S M A S 155

action line, and corresponds to the Vlasov treatment of energy of the species s, respectively, the plasma. In applying the rules given at the end of _ the last section, we have the well-known result wp > s-Lexp{^€p , s-Ms)}-r-lJ , €p.»-? I lm&* \P'L)

o) f dp np+ktS—nPt8 Next, the first-order diagrams, Fig. 1(b), 1(c), 1(d), o^o)(k,w) = — £ es

2 I ~, allow us to consider the "one-species interaction." In ik2 s J (27r)3 ep+k,s— ep>s—o>—ih factj t n e 0 ther species of the medium intervene through

$ _> 0+ ( 3 i ) the effective potential. We have, after performing the / and m summations15 and grouping together the three

where nVtS and ep>s are the Fermi distribution and contributions,

(P /-00 fix <3SS';Pp'(6,c,<i)(Mn) = 5 s s ^ p p ^ s

4 L —; / dx coth— q \-KlJ _ « 2

X Uq+(x) Wp-i-q.8 ^ p , s ^p-fq+k,s ^ p , s |

— '——+b, 11 1 \-u-{x)i5^ _S] q,s— ep,s—x— ib €p+q+k,s— ep>s—x—wnJ

+ Uq+k(x+^n)cs\jip+q,8—nVt8']2Tridlep+(l,8— ep,s—x]\ , S—•O*, (3.3)

where (P means principle part and where the coefficients as, bs, and c8 are functions of p, q, k, and con, and are given, respectively, by

®s—[_\€P+k,8—€p,s—C0n)~ ( e p + q + k , s — e p + q . s — Un) — ( e p . j _ k , s — € p , s — COn)

~~ (ep,s — €p_k)S—w„)"2+ (ePiS —€p_k,s—con)-1(ep+q)S —ep+q_k,s—con)"1] ,

O s = L ( e p 4 - k , s € p > s 0)n) ( e p + q + k , s € p + q , s W „ ) ~ J ,

C s = = L ( € P + q . s € p + q + k , s — W n ) ~ — ( € p _ k , s — £ p , s — 0>„)~ J ,

and

47Tr 47T f dt »r+D.«-Kr..r I

, 5 ->0+ . (3.4) U,Hx) = -4?H~ 47r f dx nr+a,a"-nT,

' i — E ^ 2 / ^ 2 o- J ( 2 7 r ) 3 € r + q , a — € r , a — ^ T i S .

Finally, the second-order diagrams, Fig. 1 (e), 1(f), are the only ones retained. The others, Fig. 1 (g)-l(j), are to be discarded, since their contributions will be negligible because of the assumptions made at the beginning of the section. The contributions of Fig. 1(e) and Fig. 1(f), which consider the " two-species interaction" are grouped together for the total

(P r* fix &s';pp'(e, /)(k,con) = -\e?ej X — : / dx coth—{Uq

+(x)Uq+k(x+a)n) q iiriJ-oo 2

XC^ P , ^ P ^ S ' + i ? P , 8 * i? P ^^* ] - t / q -Wt / q + k(x+co n ) [5 -^ - 5 ] } , (3.5)

where

Wp+q,s ^p ,a ^ p + q + k . s ^ p , »

Rt.. = A. +B. — €p-fq,s € p l S X id € p + q - f k , s €p,s X C0n

A8=[(ev,s— ep_k,s—con)"1— (ep+q+k,s—ep+q>s—o>n)~1~],

5 s = [ ( e p + q + k , s — e p + q , s — C 0 n ) _ 1 — ( € p + k , s — € P f S — W , , ) " 1 ] ,

and

With the formulas (3.3) and (3.5), our calculation of the contributions of the correlation diagrams is completed for the partial Q functions. We may now take the limit a>n —• co+ ie, e —» 0+, to obtain the partial Q functions, and

156 N G U Y E N Q U A N G D O N G 148

then the correlation terms of the conduct iv i ty <r(k,o>). F rom the three diagrams, 1(b) , 1(c) , 1(d) we h a v e

es*oo f dp f dq 6> f™ fix <7(D (k,w) = £ / / / dx co th-

( 2 T ) V (27r)34xO_0O 2

xM*) ^p+q+k.s ^Pt«

- £ V ( * ) [ s - ^ - 5 ] L 6p-fq,s €p,s .£ lb 6p-fq+k,s €p,s # ^ l€-

+ C/q+k+(^+o))c s

+[^p+q, s—Wp, s]27ri5[ep-fq, s—€p, s—^] | , (3.6)

eshs>so) f dp f dp' (P r°° 0x f dq

« r ( 2 ) ( M = - i £ / / / <& coth— / { Uq+(x)Uq+*+(x+a>)

.. ' ik2 J (2w)3J ( 2 7 r ) 3 W - « 2 J (2TT)3

where 5—•> 0 + and € —•> 0+. And, from the two diagrams 1(e), 1(f), we h a v e

eshs>so) f dp f dp' (P r°° fix f dq

/ / dx co th— / (2TT)3./ (2TT)3 4 « 7 - » 2 J (2TT)3

XtR9tS+Rv>,s>++Rv,s*+RP>,s>*+l-Uq- -8-]}, (3.7)

where 5 —> 0+ and t —> 0+ . In the above formulas, for any function /

/ + ( » ) = lim / ( « . ) .

The sum of Eqs. (3.6) and (3.7) is a generalization of the result of Ron and Tzoar16 to the case of finite k.

IV. CLASSICAL LIMIT

We now take the classical limit16 of the formulas (3.6) and (3.7) to obtain the first and second kind of correlation terms of the scalar conductivity of a plasma. Results of simple transformations are

« esAN,

o-(i)(k,co) = — YL f dq ^ r 0 0 <i£(47r 1

V ( 2 T ) S 2 ^ £ U 2 A(q,;

(k-q)2

[> s+(v, q + k , H - « ) - * . + (v ,q ,0 ]

4TT 1 (k-q)2

q2 A * ( q , ^ ) ( k . v - c o ) 4

4TT 1

€ / g ) ( k . v - « )

[> s+(v, q + k , H - « ) - * r ( v , q , £ ) ]

k - ( q + k )

and

w e.3e,>3N,N,>0 f f r dq <P f <r(2)(k,co) = — £ \dv\dv'\

m„»v J J J (2r)3 2irJ _,

|q+k|M(q+k,{+«/|q+k|)(k.v+«)* C*.+(v,q,|)-$.-(v,q^)] (4.1)

2 ^ . . .

X-(4*)2

d q <P /•" d£

( 2 T ) 3 2 T J _ ?

{ — )L(k-v-coWkv

I

q + k ( 2 A ( q + k , £+a;/1 q + k | )L(k- v-co)2(k- v'-co)2 (k- v+cu)2(k- v / +«) 2 J

1 X [ (k .q)$ .+ (v, q + k , £ + c o ) - k . (q+k)#.+ (v,q,£)]r

1 v—> v'

3»A*(q,$/9) [(k-q)$,+(v, q+k , £ + « ) - k . ( q + k ) * . - (

l?2A(q,?/9)

In these formulas, we h a v e

^ ^ ( v . q . C ) = - / . ° ( v ) C l + * / (q - v - $ = F M ) ] , « - » 0 + ,

where / , ° (v ) is the Maxwell ian dis t r ibut ion of species s, and defining

rs-*s' -i v,q,{)]

Lv—• v J

M=Fie • 0 + ,

(4.2)

(4.3)

(4.4)

148 I NTER ACTION OF WAVES WITH PLASM AS 157

with f8°(u) being the one-dimensional Maxwellian distribution,

A(q,^) = 1 + E - « . W ^ [ l + / . + («)]. (4.5) s q2

Finally A* is the complex conjugate of A. The results obtained here are the most general and systematic ones for a classical plasma which are yet given

in the literature. They are correct to the first order of the plasma parameter. From them we may obtain for ex­ample the results of Berk13 in neglecting electron-electron interactions and in taking the ions to be of infinite mass.

For the case of high frequencies, we may neglect (k- v) before w and on noting that

/ dv *.Hv,*,Q = 1+ Wq)fMS/q), (4-6)

it is now a simple matter of combinatorics to derive from the total of Eqs. (4.1) and (4.2) the high-frequency limit of the plasma conductivity, with k finite:

e8o)82ks>

2/ es es\ f r(M~L 1 J/"

»,»' uW \ms mS'/J

dq 1

(2*-)V|q+k|2

fco2|k-(q+k)|2

X [/.+(«A)/.'+(«/?)/A(q^/9)A(q+k> 0)] \2i q2

V fx .. « k-(q+k)/s-(S/<?) ) /.00

- / dt-2xJ_M (£+co)<7 A*(q,l/?)A(q+k, ( + « / | q + k | )

X[ (k .q ) t t+« / |q+k | ) / . .+ t t / 9 ) -k . (q+k)({/?)/."({/?)]} , (4.7)

where o)s2 = 4:TrNses

2/ms, and ks2 = iwN8es

2/®, 0 being the thermodynamic temperature in energy units. This is a generalization to finite k of the result of Ron and Tzoar for a classical plasma.16

In the case of a two-component plasma composed with AT ions and N electrons, using the fact that the ratio of the ion mass over the electron mass is very large, we obtain the main contribution to the conductivity

e^P2kD

2 f dq 1 (7CllF(k,w) = /

mrfk2 J (27r)Vlq+k l2

coM*.(q+k)|* •[/•+(«/?)/<+(«/ff)/A(q^)A(q+k,0)]

2% q2

co k.(q+k)/r(*/<?) <y r

27ri_ + - dt (H+a>)q A*(q,^)A(q+k, ? +co/ |q+k | )

X [ ( k . q ) a + a ) / | q + k | ) / l + ( ^ ) - k . ( q + k ) ( f A ) / r ( ^ ) ] , (4.8)

which is a generalization to finite k of a result of Oberman et al.11 and Berk.13 Herecop2 = 47riVe2/w, and &D2 = 47riVe2/©.

Further simplification is possible in considering that the mass of the ion is practically infinite, i.e., that11

fi°(u)-+d(u), then

/<=*=(«)= -(P(l/u)±iw5(u),

and

A(q,«) -> D(q,u)^ l + kD2/q2+ (kD

2/q2)uf+(tt). (4.9)

158 N G U Y E N Q U A N G D O N G 148

Equation (4.8) becomes

p(k,«) : W f ^ q (co 2 |k - (q+k) | 2 q2+kD

2

mo)6

? f dq |

J (2TT)4 2ik2 | q + k | 2 + &

k-(q+k)[g2+&z>23 (P f co

- ( - — — ) n2\q2+kD

2 q2D{q^/q)J (P r

(4.10) | q + k ( Y 2T]^ ( { + « ) { D ( q + k , £+co/|q+k|)Z>*(q,{/(?)l

where &D 2 = 2 £ D2 = 87riVe2/G.

In the low-frequency limit, a><kvth, we may not neglect (k- v) before co. However, in this case the contribution (4.1) is dominant. Some trivial transformations and integration over £ yield

dq (k-q)2/,°(v) eso>2 f r dq (k-q)2/8°(v) ( M ) = E / civ / - — - - — - { [ A - K ^ q - v/g)-A*-i(q ,q- v/g)]

- w8 7 y (27r)3g2(k-v-oJ)4

- J [ A - i ( q , ( Q + k ) - v + « / ( Z ) + A - K q , ( q + k ) - v - « / 9 ) ] } . (4.11)

Here the prescription that co has a small positive imaginary part should be understood. I t is now easy to take the limit of a two-component plasma with large ion mass, and the limit for an infinite ion mass. In the former case only one term of Eq. (5.18) remains which contains the coefficient e2cop

2/w. This will be used in the following. In the latter case A is replaced by D. However, it does not present much interest since it is oversimplified and will not allow us, for example, to explain the scattering experiments.

Finally we consider a few simple applications of the results derived in this section. In particular we examine the effects of correlation in the case of damping of EM waves, and the case of scattering of EM waves by a plasma, which is composed with electrons and ions, the latter having a very large mass.

A. Damping of EM Waves

I t is well known24 that the coefficient of damping of EM waves is proportional to the real (resistive) part of the conductivity. When correlation is neglected, we obtain the usual Landau resistive part, which has the following form, for the case of a very large ion mass

Re(70= (TT/2) 1 / 2 | ( 7 0 | ( C O / C O P ) 3 ( W £ ) 3

Xexp[-J (co /co p ) 2 (W^) 2 ] , (4.12) where

cr°=ia>p2/47ra;.

As has been noted,9-16 this resistive part vanishes for k —> 0. Thus, there is no Landau damping in the limit of long wavelengths. However, in formula (4.10), we see that when k —> 0, the conductivity becomes11

2e2 rqj* q2+kD

2< -r- 1

{•kD2Y 3>Trniu)2J o q2JrkD

2\-q2D{q,o)/q) q2+kL

where QM = (e2/3)-1 is the maximum cutoff which is

24 See, for example, Ref. 14. In this reference, however, the imaginary part of the conductivity is to be considered, due to a difference in definition of the conductivity.

needed in the case of a strictly classical plasma to assure the convergence of the integral. Since D{q,u/q) is a complex function, a real part is obtained for the conductivity. Thus in the linit of long wavelengths, the resistivity is due to correlation effects,

RecrCHFc-(7r/2)1/2(67r2)-11 a°||A InA | (cop/co) , (4.13)

where A=%DZ/N. Similarly, we obtain the approxima­tion to the real part of the low-frequency correlation contribution to the conductivity,

i / M y 2

RecrCLF^ — 1 | a°||A InA | 3 V 2 ( 4 T T ) 2 W

/ " \ 2 / £ I A 3 | \M/u>\2/kD\2)

\a>J \ k J e X P l 2 m \a>J \k) ) '

We then arrive at the following conclusions:

1. The zeroth-order resistivity (4.12), which corre­sponds to the Landau damping coefficient, vanishes when k —> 0, while the H F resistivity due to correla­tion, Eq. (4.13), does not.

2. Also, in the limit of large co, the Landau resistivity is exponentially small, while the HF-correlation re­sistivity varies as the inverse of the frequency. Hence, the latter will exceed the former at a certain frequency.

3. At low frequencies, the ratio between the correla­tion resistivity (4.14) and the Landau resistivity is dependent inversely on the frequency, and thus, may become very large.

4. At plasma resonances, it is seen that the Landau damping is predominant, since RecTc^/Reo-o^ | A InA | <<C1, unless k—>0.

B . Scattering of EM Waves

For a two-component plasma composed with elec­trons and one species simply ionized ions the mass ratio is very large, hence, the principal contribution to

148 I N T E R A C T I O N O F W A V E S W I T H P L A S M A S 159

5(k,o;) will come from the electrons

«S(k,u) = r02 £ Gee ';PP>(M)

ee';vv'

2 = r„2 — £ I m Q « e , ; p p , ( M , (4.15)

1 — e ^ w ee';ppf

where we note the appearance of the "classical electron radius." The simplified formulas which correspond to the approximations made in the latter paragraph are easily obtained. The zeroth-order contribution is

So ( M c ^ A V A (jSw)-1 («/«p)

Xk~l exp[2r1fim((a/k)2'}. (4.16)

At high frequencies, the correlation contribution is

SCHF(k,u)~AVo2! A \nA/N\ (fa)-l{up/o>fk\ (4.17)

And, at low frequencies, we have

SCLF(Ku)^Nro2(M/fn)ll2\A2 lnA| (fa)-1*-1

Xexp[2 -^J f (« /*) 2 ] . (4.18)

In the above formulas, we have omitted the numerical factors to simplify the writing since they are not necessary for the qualitative discussion. In addition, the factor (/fo)-1 comes from [1 —exp(—^co)]-1 in Eq. (4.15), and is nondimensional.

Here, also, we can obtain some qualitative results:

1. In the high frequency, or long-wavelength regions, the ratio between the contributions (4.17) and (4.16) is

r ~ |A lnA\(o>p/a>Y(k/kDy e x p O 1 (co/a>p)2(kDA)2].

Since co/wp>l, or k/kD<\, we note that r will surpass unity for sufficiently large co, or small k, i.e., the corre­lation effects are important.

2. At plasma resonances, o?/cop^l, hence the ratio becomes

r~ | A lnA| {k/kDf e x p p - 1 (kD/k)22,

which is small since A«C1, unless the limit & —> 0 is taken. Therefore, correlation effects are less important in this case.

3. The correlation effects become quite important at low frequencies, because of the factor (/So?)-1 in Eq. (4.18). Then, instead of seeing the electron Doppler spread in the plot of the scattering cross section, as is evidenced by the factor exp£2-l0m(<a/k)2l of Eq. (4.16), we shall see an ion Doppler spread, as is evidenced by the factor expp^/Mf («/*)2] of Eq. (4.18). This con­firms the results of the experiments performed in con­nection with the study of the physics of the ionosphere

[incoherent scattering of radio waves (Refs. 25-30)]. We note that although the ion Doppler spread is seen, the actual scattering is done by the electrons, as is evidenced by the factor Nr0

2.

V. CONCLUDING REMARKS

We have presented here a basic calculation. By this, we mean that in the formulation, only fundamental dynamical variables and parameters, such as the density, the temperature, are given and we refrain ourselves from introducing empirical coefficients, such as the effective collision frequency, etc. The notion of transport parameters is connected with the response of the plasma to an external field, and the statistical averages are obtained by the evaluation of common factors, the functions Q or G.

This calculation also provides a bridge between quantum (e.g., in a conductor) and classical (e.g., in thermonuclear devices) plasmas. We see that from the results obtained for a quantum plasma may be deduced all results for a classical plasma, where a kinetic ap­proach is more usual.

The closed diagram technique is well suited to the necessities of our calculation. In particular, two ad­vantages over the open diagram technique used by Dubois et at. in their calculation14 are to be noted: (i) the renormalization of the Coulomb interaction is more naturally introduced as a partial summation of terms, and (ii) there is no confusion about the order of a diagram and the degree of accuracy of a term of the inner expansion in £/q.

31

The importance of the correlation effects has been stressed by several applications for a hot classical plasma. Although very simplified formulas are used, we have come to the conclusion that Landau damping is only effective at plasma resonances, and that, indeed, the relatively narrow spread in the scattering cross section, in the experiments with radar beams sent into the ionosphere, is due to the correlation effects. Further applications may be envisaged for quantum or classical plasma and are under investigation.

ACKNOWLEDGMENTS

The author wishes to thank Dr. R. K. Osborn, Dr. W. Kerr, Dr. R. J. Lomax, and Dr. F. L. Shure for their interest in this work and many fruitful discus­sions. A fellowship granted by the Agency for Inter­national Development is gratefully acknowledged.

25 K. W. Bowles, Phys. Rev. Letters 1, 454 (1958). 26 V. C. Pines, L. G. Craft, and H. W. Briscoe, J. Geophys. Res.

65, 2629 (1960). 27 T. A. Fejer, Can. J. Phys. 38, 1114 (1960). 28 E. E. Salpeter, Phys. Rev. 120, 1528 (1960). 29 M. N. Rosenbluth and N. Rostoker, Phys. Fluids 5, 776

(1962). 30 D. F. Dubois and V. Gilinski, Phys. Rev. 133, 1308 (1964). 81 Reference 14, see the explanation on the to and k dependence

of higher order diagrams, and Fig. 10.