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chengitter-Struktur ebenfalls nicht festzustellen (be-züglich Hg vgl. die Pfeile | in den Abb. 1 a und b ) . Auch hier ist die Übereinstimmung der beobachteten Atomabstände mit den berechneten Abstandswerten überraschend gut.

Die umfangreichen FouRiER-Analysen wurden mittels des elektronischen Rechenautomaten (ER 56) wieder von Herrn Dipl.-Phys. R. LEONHARDT durchgeführt. Wir danken ihm hierfür herzlich, ebenso danken wir der Deutschen Forschungsgemeinschaft für die vielseitige Unterstützung der vorliegenden Untersuchungen.

Nonlinear Incoherent Light Scattering A . S A L A T

Institut für Plasmaphysik GmbH. Garching bei München

(Z. Naturforschg. 22 a. 662—670 [1967] ; received 3 February 1967)

Classical equations are used to calculate nonlinear incoherent light scattered from two coherent beams of electromagnetic radiation of different frequencies propagating in a homogeneous plasma without magnetic field. With a suitable choice of the difference frequency, enhanced light scattering occurs near the plasma frequency. In this case the frequency spectrum is mainly given by the thermal ion density fluctuations.

Nonlinear processes, occurring when one or more electromagnetic waves propagate in a collisionless plasma, have been examined both theoretically 1 - 8

and experimentally9. Theoretical work is of two kinds. In the first type, the plasma is treated using a collective model, be it the V L A S O V equation or macroscopic equations. This produces nonlinear waves with frequencies, wave numbers and direc-tions uniquely determined by the incoming (pri-mary) waves 4 ~ 9 because of energy and momentum conservation. (Though, as a result of refraction, a finite plasma volume makes a finite width of the angle of propagation.) In the second type 1 - 3 the discrete plasma structure is considered explicitly, which leads to new effects. Each particle separately can exchange energy and momentum with the waves. Since, moreover, all particles interact via C O U L O M B

forces, this presents an intricate problem which to the author's knowledge has been treated hitherto mainly in terms of quantum field theory with results of various authors that are not in agreement 2. This

1 H . CHENG and Y . C . LEE, Phys. Rev. 1 4 2 , 1 0 4 [ 1 9 6 6 ] , 2 G . BAYM and R . W . HELLWARTH, IEEE J . Quant. Electronics

Q E - 1 , 3 0 9 [ 1 9 6 5 ] . 3 P. M . PLATZMANN and N . TZOAR, Phys. Rev. 1 3 6 . 11 [ 1 9 6 4 ] .

- D. F. DuBois and V . GILINSKY, Phys. Rev. 1 3 5 . 9 9 5 [ 1 9 6 4 ] . - H . L . BERK, Phys. Fluids 7 , 9 1 7 [ 1 9 6 4 ] .

4 H . S . C . W A N G and M . S . L A J K O , Phys. Fluids 6 . 1 4 5 8 [ 1 9 6 3 ] , 5 W . H . KEGEL, Z . Naturforschg. 2 0 a, 7 9 3 [ 1 9 6 5 ] . - D .

MONTGOMERY, Physica 3 1 . 7 8 9 [ 1 9 6 5 ] . c N. M . KROLL. A . RON, and N. ROSTOKER, Phys. Rev. Letters

1 3 . 8 3 [ 1 9 6 4 1 . - A . SALAT, Z . Naturforschg. 2 0 a. 6 9 0 [ 1 9 6 5 1 .

work contains a purely classical description of non-linear wave phenomena in which the particle aspects of the plasma are included, and the difficulties in-volved in quantum field theoretical representations are avoided.

In section 1 the problem is specified and the equa-tions for handling it are given. Sections 2 and 3 give a brief summary of results relating to thermal plas-mas and linear wave effects. In section 4 nonlinear effects are derived, which are discussed in sections 5 and 6.

1. Basic Equations

The nonlinear interaction of electromagnetic waves will be specified here in the following way: Two electromagnetic waves having the frequencies co0, a = 1, 2 the directions Tl0 and arbitrary polarisa-tion propagate in a homogeneous isotropic plasma: What are the nonlinear (quadratic) effects that can be seen outside the plasma?

7 R. F . WHITMER and E. B . BARRET, Phys. Rev. 121, 661 [1961] ; 125, 1478 [1962]. - L . M. GORBUNOV, V . V . PUSTO-VALOV, and V . P. SILIN, Soviet Phys.-JETP 20. 967 [1965]. — A. SALAT and A. SCHLÜTER, Z . Naturforschg. 20 a, 458 [1965].

8 R. F . WHITMER, E . B . BARRET, and S. J . TETENBAUM, Phys. Rev. 135,369 [1964].

9 B . BLACHIER, J . - L . DELCROIX, and E. LEIBA, C . 11. Acad. Sei. Paris 262. 472 [1966]. - R. F . WHITMER, E. B . BARRET. and S. J. TETENBAUM, Phys. Rev. 135. 374 [1964],

Let the primary waves be

Ep (r,t) = YEaexp{-i(ka-r-a)ai)}; ka Ea = 0 ; Ö

B p ( r , 0 = y Ba exp { - i(ka- r - 0)a t)}; B „ = [kaxEa]; ( 1 )

a = ± l , ± 2 ; E _ ö = Ea*; k_a=-k_a; (o_a=-toa; n 0 = Tl0; k„ njc0.

The plasma consists of 7Ve electrons and Aj ions with charges <7e, a n d masses m e , mi in the volume V and is described by the functions Fa(r,V,t), a = e, i :

Fa(r,v,t) = t ö ( r - r i ) d(v-ri) (2 ) i=i

where rt, r, are the position and velocity of the i-th particle. For Fa we have 10' 11

3a

f ; +v-VFa+ J ; ( E + I [ D X B ] ) - V J „ = O (3)

where E ( r ) , B ( r ) is the effective self-consistent field at the point r . Fa{r,V,t) depends on the values of the particle cordinates at some initial time, say t = 0. It is useful to consider quantities averaged over the initial values of

r f (« = 0 ) = R i ; ri(t = 0)=vi. (4)

With such an averaging procedure we put

(Fa(r,v,t))=fa(r,v,t), ( E ) = E , ( B ) = B . (5)

The particle structure of the plasma is then contained in the fluctuations

dfa(r,v,t)=Fa-(Fa), SE = E - ( E ) , (5B = B - ( B ) . (6)

Averaging (3) there results

+v-v/a+ c!:aA'Vvia= ~ 5 { ( ) A ' V d f u ) ' ( 7 )

+V\7dfa+ q a A-\7vdfa+ q°ÖA-X7vfa=- qa {ÖA-VvÖfa-(ÖA-Vvdfa)} (8) at ma nia ma

with A = E + 1 [ r x B ] , 6A = dE+ 1 [vy.SB]. (9 ) c c

( 7 ) is the V L A S O V equation with a collisional term. (8) is an equation for the fluctuations. For further simplification the following assumptions are made.

a) Correlations involving more than two particles are neglected, which can be done when there are many particles in a D E B Y E sphere. The rigth hand side of (8) is thus neglected.

b) C O U L O M B collisions are assumed to be unimportant, and so the right hand side of (7) is set equal to zero.

e) For the plasma without electromagnetic waves we have

A0> ( » ) = > " ( J i r f e x p { - (v/va)*} =nafa0(V), va2 = 2*-T* . (10)

d) The quantities of interest fa, dfa, E, SE etc. may be expanded in powers of the amplitude of the primary waves.

f a ( r , v , t ) = C + f P + f ? + . . . . (11)

1U YN. L . KLIMONTOVITCH and V . P. SILIN, Soviet Phys.-JETP 1 5 , 1 9 9 [ 1 9 6 2 | . - J. M . DAWSON and T . NAKAYAMA, Phys. Fluids 9 . 2 5 2 [ 1 9 6 6 ] .

11 A. SALAT, IPP6/49 [ 1 9 6 6 ] .

Nonlinear effects can thus be found in the second order. It is useful to go over to complex F O U R I E R repre-sentation in the following way:

fa(r,v, t ) = ( 2 J t ) 4 j dco Jd3hexP{-i{k-r~ cot)) fa(oj,k,v), oo

fa(co,k,V)= j dt j* d3r exp{i(k-r-cot)} fa{r,V,t). (12)

o The asterisk indicates that the path of integration in the complex co-plane goes below7 all singularities of the integrand. With a) and b) there results

i(oj-k-v) dfa(co,k,v) -fa(t = 0,k,v) + -^Ao\yvfa = o, (13) ma

i(ü)-k-v) /a(co, k, v) — dfa(t = 0, k, v) + Qa AoXJvdfa+ ( i o M o V t / t = 0 (14) ma ma

where A°B stands for the convolution integral (2 n) ~4 f* do/ J d3A;' A (co', k') -B(OJ — co', k — k'). In F O U R I E R representation M A X W E L L ' S equations can be put in the following form:

E(a>,k) = + {kg(0J,k) + k2f_M2 [kx [ f e x j ( c o , f e ) ] ] | (15)

E A = ~ {oj[kx[kxE(t = 0,k)]] + k2c[kxB(t = 0 , k ) ] } , k = \k\ (16)

and 0 = I o a = Yqa J d*v fu(v), j= I ja = 2 qa Jdhvfjv) (17) a 2

and analogously for the fluctuations. By specifying initial conditions the system (13) — (17) is fully determined.

2. Thermal Plasma

This well-known regime is presented briefly here because the following sections depend closely on the way solutions are found in this section.

After separating into longitudinal and transverse parts the Eqs. (13) — (15) read:

E ( 0 ) (OJ, k)= 0 , i(co-k-v) dfL0) (OJ, k,v)+ Qa (3E'°) (oj, k) • \/v fi0) = df^(t = 0,k,v), (18, 19) ma

k • dE® (OJ, k)= 4JI q W (OJ, k), [kx SE^ (OJ, k) ] = ~ J f ^ [ k x <3 j<°) (co, k) ] . (20)

Taking moments from ( 1 9 ) , (20) there results

= f {öfP(t = 0,k,v)~ £ ÖE^-VvC} (21)

and from this

k-dE(0)(oj,k) = Zk-ÖE<°> = l - ^ w q a J JXv d = 0,k,v) (22)

with

+ oo

— oo

For the approximations to the longitudinal dielectric constant E (OJ, k) see Appendix A. Analogously, from taking the velocity moment it follows that

[k x SE^ (co, k) ] = + I [k x SE(a0) (co, k) ] a (co , A ) a ( 2 4 )

= J * x , S E ? > ] V - 4 . T T ( 0 ) = ^ a ( co , k ) a a ( co , k) J co—k-v > *

-ith I " OO

a(co, k) =k*c*-co9-+ IcoJ [ * V - " f a M . (25) a J co — K v

Again, taking moments of ( 1 9 ) , the following density and velocity fluctuations occur:

(co,k) = f d*vdf<?>= y k-ÖEj?> (co,k) aaß(oj,k), (26) J 41 71 1 lla P

k• <5Va0) (CO, FC) = I k• öE{ß]auß + i Jd3*, <3/<°>« = 0, k, V), (27) 4 1 7 1 1 <Ja P

with

[kxöV(a0Hu,k)] = - A 1 Z.[kxdEf>]bafi(a>k)- [ (28)

L a V 5 / j 4> ti i co qa ß L 7 maa(co,k) J co — kv y '

aaß(co, k) = e (co, k) Öaß + co* J , (co, * ) = a (a>, k) 6aß - J . ( 2 9 ) ) The well-known formula for thermal density fluctuations follows from ( 2 6 ) , (22) by averaging over the initial values (4) of I Y - t . Using

lim I =aid(x), V = Im OJ (30) the result for fc + 0 is

y-jTo r ü W t o W ) . I S ^ I ^ ^ D - f f l - d l t r 2 <31> In the following, the transverse part of thermal fluctuations is neglected.

3. Linear Processes

From the linearized V L A S O V Eq. ( 1 3 ) the frequencies coc and wave numbers kG of waves ( 1 ) are coupled by the dispersion relation

a(coa,k0)=k*c*-co*+ I w / = 0 . (32)

The plasma together with the waves oscillates according to E(V(0J,k) = ZE„ ~i(2 ji)3 d(k — k0) ^

a a> — coa

W(oj,k, v)= I f a a K) , L= ^ E0-Vvf^(v), a co — coa ma coa

NP (OJ, k) = f d3v fV (ft), k,v)= 0

results which follow trivially from ( 1 3 ) , ( 1 5 ) , (16) and the assumption [ co/k'V ^ 1 , which for OJ = OJ0, k = ka and (32) is automatically fulfilled.

Besides these collective effects, the primary waves produce fluctuations in the plasma. The linear approximation to Eq. (14) reads

i(co-k-v) ÖfP (oj,k,v) + ^a~ÖEU-Vvf(e?)=df£)(t = 0 ) - A ^ o V ^ / f - Qa SE(°> oVvA1}-ma ma ma

(34)

In comparison to the thermal case, only the inhomogeneous right hand side has changed. Since also MAX-W E L L ' S equations are the same in any order, the solution to Ecj. (34) can immediately be written down by comparing with the last section. Choosing df^ (t = 0) =c3E(1)(/ = 0) = 0 , omitting the B-field of the waves and unfolding the convolution integral, there results with the help of (22) and (24)

k-SE'1) (OJ, k) = [ . ^,,k-{E0dfM(AJ-co0,k-k0,v) +faadEM(oj-oja,k-ka)}, S {CO, K) a o TTlfx J \CO —K V)-

[ f c X <SE " (<0, k ) ] = - Z I f fc-{E.W»(«-».,k-fc.)

+/„«<«. (»-» . , fc-fc.)} - i i <36>

• [ k x { E „ d/<°> (OJ - a>0, k - fe,) + /„., <5E<°> (o> - « „ , fe - k „ ) } ] .

( 3 5 ) , (36) considerably simplify under the restriction \oj/k-V | 1 . Then the contribution of the pole k v — OJ is exponentially small, and from the expansion of the denominator there results

fc-<5Efl>(«,fc) = I f c - < 5 E £ l > = 4 - t Z Z ^ 1 k E a e{co,k) n a ma CO2 "

• | (OJ — coa ,k — kn) + « ,la l-2dN(0){(0 _ a ) o i k _ k o ) ) .

[ f c x d E W ( a > , f c ) ] = - Z Z ^ { [ f c x E a ] (oj-oj0,k-k0) + o (38)

+ 1 qnß l-H[kxE0]k-l0+k-E0[kxl0])dN^(oj-oj0,k-ka) ß 0J la

with l0 = k-kr,, /0 = |I„|. (39)

(38) is the well-known formula for the scattering of electromagnetic waves 12 on density fluctuations in a plasma, written in fc-space. The second term describes scattering of thermal fields on induced plasma po-larisation. Since the poles of dN^ (OJ — OJ0) are close to o) = oja, this term is smaller than the first by a factor of O)(2/OJ2 and is generally omitted for OJ2 OJ2.

The connection between induced density etc. fluctuations and fluctuations of the electrical field because of the analogous form of the equations is the same as in the thermal case:

< W > (<u, fc) = . \ Zk-dEW(°>,k) aaß(o),k) (40) 4 .t i cja ß

etc. as in ( 2 7 ) , ( 2 8 ) .

4. Nonlinear Processes

Nonlinear collective processes, e. g. periodical density changes induced by two electro-magnetic waves were discussed in 5- 6 ' 8. There was proposed 6 to detect these density variations by aiming a third beam at them and looking for the light which is „scattered'' into very special angles only. It is therefore suffi-cient here to summarize very briefly the collective processes. Eq. (13) reads for f*2)(t = 0 ) = 0

i(0J-k-V) fi2) (0J,k,v) + Q n E ^ - V . / f = - Q a A ^ o V r f (41)

ma ma

Again by analogy with section 2, the solution is k-EW(0J,k) = - Y*.. Z [ (,3t. |e ( 1 ) + 1 [ r x B W j j - o V r / ^ . (42)

S{OJ, k) I ma J CO—k-v ( c J J

For ' oj/k • V I 1 i . e . j OJs \/ks va ^ 1 (43) the expansion gives E(2)(co,k) = ZZE,a <5 ( f c - f c , ) , Es a= - iQa — \ , * fc,E„-Er (44)

v ' a o.r • O-COs ' ma 2concor (Os2 £{cos,ks) 3 0

with OJS = 0JA + 0JX, ks = k0 + kT, ks = I ks I . (45)

The transverse part of E ^ vanishes identically. The amplitude of E1— may become large, when the dif-ference frequency of OJS is close to the plasma frequency (op = ( S

12 E. F.. SALPETER. Phys. Rev. 120. 1528 [I960],

For the nonlinear fluctuations (14) gives

i(co-k-v) dfp (co,k,v)+ qa dE^-Vvf^ ma

= df^(t = (),k,v) - CJa { A W o V . i / i " +dAWoS7vf£> +E'2j ° V » d f + dEW ° V®/a 2 ) } • (46) ma

For df£)(t = 0) = (5Et 2 ) (F = 0 ) = 0 and with M A X W E L L ' S equations the transverse field follows from analogy with the thermal regime (24) :

[ k x AEW(a>f fc)] = , 4 * 2 ^ f ^ ^ . { A d ) L a{co,k) a /na J co — k v y v /ot (47j + <5A(1) o V , + Ei2) o V , <$/£» + <3El0) o V , /<2>} .

As can be seen by the pole a (to, A;) = 0 (47) contains fluctuating electric (electromagnetic) fields that pro-pagate as waves in the plasma, and that are radiated away when the plasma volume is finite. This non-linearly scattered light comes from four different processes: a) Primary waves A ^ ) are scattered by in-duced fluctuations <5/^ . b) Induced fluctuations of the electric field <5A^ are scattered by the plasma polarisation f ^ . c) Nonlinear fields E ^ are scattered by thermal fluctuations <5/a0). d) Thermal fields are scattered by the polarization . The scattered light, ( 4 7 ) , due to the stochastic fluctuation, has a finite time and length of coherence. It may be called partially coherent or incoherent.

Since a (co, k) = 0 for real OJ only permits I co/k j ^ c , the denominator of (47) can be expanded to give

[kxSEW(oj,k)] = - ** I I qnf \[kxE0] dN™ (OJ-OJ„) + 1 K-EFFL[KX W > ( Ü > - O 0 ) ] CL((0,K) OL CT,T ma [ to

+ 1 [kxE0]k-dVil)( OJ-OJ0)+ 1 k-«W(©-ffla ) [kxV.,a] (i) (O

+ I [k x SE(1) (OJ — OJ0) ] k-v0,a+ I [ f c x W ( w - ^ x B J ] + I [ k x [V 0 j a x SBW (a) — OJ0) ] ] + [kx Es] dN^ (co — OJS) < 4 8 )

+ [k x <5E(0) (co - OJS) ] A,, a + I K <5E(0) (co - OJS) [k x V,, J

+ 1 [kxdE^(co-ojs)]k-Vs,a}.

The frequency spectrum of (48) is centered around the frequencies OJS = OJA + cox. Using the results of sec-tions 2 and 3 gives two types of terms:

[ k x dEW (co, k) ] = [ k x dEj] + [ k x dEu] (49)

[kxdE]]= I qß2 q" , 1 ( \ \ l0'Er a((o,k) o,r tx,ß,y mß ma £{co — coa) a)T~ la~ • | / ö 2 [k x E0] + 2 [kxla] k-Ea+ c^k-l0[kxE0] + [kx[l0xBa]] -aaß( to-co,)

+ < ([kxl0]k-Ea+k-la[kxE0])\ (50) co0 COs )

• [dßy + 2 l'y % £ lr/ls} dN<°> (co -OJs,k- k , ) ,

[kxdEu]= **{-2 I q * q* / - v . ^ 9 1 En Er A(CO, A) o,t <x,ß.y ma mß £((os,ks) OJS- 2 co0 cox

•! Qr \ (k,2[k X ls] + k - i j k X k , ] + k - k , [ k x ls]) aaß(cos, ks) (51) I lla 'S

+ Say [ k X k , ] } ÖN™ (CO - ojs, k - k , ) .

The origin of [ k x ^ E j ] are processes of types a) and b ) . [ k x ^ E n ] comes from processes c) and d)

The fluctuations SE^\ d V ^ have been replaced by their longitudinal parts. In 11 the contribution of the transverse parts is shown to be negligible, for co2 > oja2. Depending on the value of the frequency cos, two different approximations to the scattered field are discussed in the next sections.

5. Nonresonant Case

If cos is not near the plasma frequency cop and if, moreover,

m s 2 > 0 ) 2 (52)

only parts of [ k x < 5 E [ ] contribute effectively to the scattering:

[k x SE(2) (OJ, k) ] = ^ [ I l f 1 l[kxEa] la Et + fc-E0[fcxEr] 3 (CO, A.) CT,r a MA- COZ COS I COT (53)

+ [k X Ea] k • Er + [Xe x [E r x [ka x E 0 ] ]] ^ } bA"<0) (co - cos, k - k0).

Eq. (53) describes the scattering of incident waves at induced density and velocity fluctuations. The energy, emitted from the plasma per unit of time into the solid angle d Q and the frequency interval dco

d/ (co , n ) = r2 lirn JL | [<$£(2) (OJ, r) x öB'2) (co, r ) ] | dco d Q (54)

is thus found on averaging over the initial conditions of the plasma (see Appendix B) to be:

<d / (eo ,n )> = 2 [ n x [ n x { e , (n ojs-n0oj0) -Ex

+ E r n - E 0 + E ö n - E r + [ E r x [ n a x E o ] ] l ] 2 (55)

m r i r | ( 2 - ( * „ ) • lim \ (| ^yV(e0) (OJ — cos,k — fes) |2) dco dQ

with k = kn, n = r/r, kc = (co2 - C0p2)y°- sign R e c o . (56)

The spectrum of the scattered light is given by the electron density fluctuations, as in the linear case. The angular distribution differs from a dipole characteristic. The order of magnitude of the intensity is smaller than the linear one by a factor r/:

rj= (e Ejmeoj0c)2. (57 )

6. Resonant Case Let the frequencies cox , co2 be such that the difference frequency is close to the plasma frequency:

| ojs | = | OJx — co2 | « ojp (58)

The amplitudes of the collective oscillations E{2\ A(a2), V^2) will then be high, and so parts of [ fcx<3E I R ] dominate in the scattering process. The low frequency wing of the scattered light will not be observable, however, whenever it drops below the plasma frequency. For me < mi Eq. (51) becomes

[fc x T > E ( 2 ) (OJ , k) ] = ^ Y 2 " ' fr'El x ( 5 9 )

L a(co,k) A,R me~ 1OJ0COX )

X | ( 5 A ( e 0 ) ( C O - C O , ) + ( l - I (qedN^(aj-cos)+qidN\°\oj-ojs))]. Because of k2 c2 k2 c2 < cop2 and k2 c 2 « (nx cox - n2 OJ2) 2

it follows that kfl > 1 + - 1 tJi 4 sin2 a " > 1 (60) k- CO n- 2

except for nearly parallel primary beams. a12 is the angle between beam 1 and beam 2. With (60) the electron density fluctuations disappear from (59) :

[kxdE(2)] = - 2 q i f a ' E r [ k x k „ ] ~ 9 i - . 1 dN[°H<o-(o„k-ka). (61) a(co, k) a.r = me- 2concoT qe e{(os,ks)

±(1-2) The P O Y N T I N G vector gives the energy emitted into the solid angle di2 and the frequency interval dco (see Appendix B)

< d / ( c o , f i ) ) = 2 j E a - E r |2 ( e f lq[)\m2(n,ns) N ' ' \mec2J 4 ji c.t ] T| \me(D0c! coz \ qe J ^^

•2 sin2 a ' 2 . , 1 7 — , i m 7 (\dN^(co-cos,k-k,Y'2)dcodQ. 2 \s(cos, ks)'2 \ co I o Ji 1 v s' s'' ' The order of magnitude of the scattering cross section is higher than in the nonresonant case by a factor of (1 — 0Jp2/c02)I/2-1 e((os, ks) j~2 and "higher" than the linear cross section by (1 — cop2/co2)1/2• j e(cos, ks)\~2~rj. The actual amplitude of the resonance in \e j - 2 depends on the L A N D A U damping or collisional damping, whichever is the larger. The frequency spectrum of (62) is given by the ion density fluctuations. From (31 ) :

/ 1 - 2 f df /eo (v) 2

l - 2 < w a " / , A,~ (63)

" Ak \,ll't{)\Ak ) r dvfM(v) \ a J (Aco-Akv)2

, [ dv ho(v) j~\ , [Aco\ 1 J (Aco — Akv)2 \ „, . .

+ n M Ä k ) j_ v n ( d . / * M - I = : S(Aco,n) 1 - 2 ^ J (Aco—Akv)2 !

with AM = co - cos, zlA; = | Ak | = | k - fcs |. (64)

It seems appropriate at this stage to discuss briefly the frequency dependence of the ion fluctuations. On the assumption that Te and T[ are not too extremely different the following approximations are found

a) | Aco \/Ak va 1 , a = e, i Using Eq. ( A 4 )

«> - L { ( - / » ( & ) + ( t t I V J - / - ( £ ) } « * >

with a / = l / ( A k y x f , V = " / / 2 V - (66)

Since for small arguments fao(v) goes like va~l, the first term of (65) dominates. Thus for arbitrary a.ß the center of the spectrum has an essentially GAussian profile corresponding to the ion thermal velocity. b) | Aco \/Ak ve<\, j Aco {/Akv, > 1 .

According to Eqs. ( A 3 ) , ( A 4 ) e f A V (\ 1 +a e 2 2 , IAco\ (coi/A co)2/u[ 2 /Zla>\\ ^ (Zl0J' n ) = Äk I l + ae2— (coi/Aco)2 f4,[—i Ay >i0 \ Ak ) + 1 + a e 2 - ( W ^ M V i " ^ | n e / e 0 \ j j ( 6 7 )

In this regime there is a more or less pronounced maximum near (Aco)2 = CO-2 /J-\(\ + a e 2 ) 1 , correspond-ing to ion oscillations. S(Aco, Tl) at this frequency is higher than the corresponding electron spectrum by a factor of (1 + a<12)2 /ae i .

c) | Aco \/Ak va 1 , a = e, i. The ion term can be neglected because of its exponential smallness. Thus

5 (Aco, n) = { k _ _ 2 „ e / e 0 ( j - ) . (69)

The electron satellite line at the frequencies where the denominator is small is smaller by a factor of (mP/m[)2 than in the electron spectrum and will hardly be visible.

7. Conclusions

In the foregoing sections, a microscopic model has been used to calculate scattered light the amplitude of which depends quadratically on the amplitude of the scattering light. It has been shown that by choosing two light frequencies appropriately resonances give higher intensity for the scattering process. A com-parison of the results with those given in 2 is suggested. The effects in 2 seem to be different, however, since there a second order proces is found by combining a linear and a third-order amplitude to give a quadratic effect in the intensity.

Acknoiuledgements

The author wishes to thank Professor Dr. A . SCHLÜTER for suggesting the subject of this paper and for help-ful discussions.

This work has been undertaken as part of the joint research program of the Institut für Plasmaphysik and EURATOM.

Appendix A + oo + 00

The ,unct ion Z . ( » ) ^ - - J ( 1 + ( A D — oo

can be expressed in terms of known functions, see1 3

Za(z) = - * 2 (l-ixVxe-*[l-erf(ix ) ] ) . ( A 2 ) k / a

For lim Im co = — 0 the T A Y L O R series or the asymptotic representation gives

for M M > ] : * . ( { 2 i y S ( - « J e x p { - ( » J } + 1 + \ + . . . ] . ( A 3 )

for | » | / * r . < 1: + . . . ) • ( A 4 )

The corresponding formulas for e(co,k) can be found from

e((o,k)=l-2Za((o/kva). (A 5)

Appendix B The integral Ma (OJ, r ) = * / d3A- exp{ - i k • r} / ,. }, M ( a 0 ) (co - co,, k k,) (B 1)

71) J CL (CO, fx, rC~

is easily calculated with the aid of ( 26 ) , ( 2 ) , and ( 6 ) . For co =1= cos

xVa oo

Mu(co, r) f ch exp{ — i(OJ — OJ,) t) ( 2 ^ ) 3 [ . e x p { i ( f c - ks) • r,(t) } e x p { - i k rj «•=t o

oo 4-1 co

Y j dt exp{ — i(co — OJ,) t )exp{ —iks' r , } * fd cos 0 f dk ( B 2) i 6 - 1 0 k —k

--= V I ch exp{ —i(oj — OJ,) / ) exp{ — i ks• f ; } ^ ^ exp{ - ik i Ö

with k c — (OJ2 — C'JP2)1/1 sign Re co, Rt* = \ r - r((t) \ . (B3)

In the wave zone there results from

| r - r t \ « r - n rt, n = r/r, k = kn, (B4)

Ma(co, r) = ~ - e x P { - ik r) ÖN[0)(co - OJ,, k - ks). (B 5)

13 B. FRIED and S. CONTE, The Plasma Dispersion Function, Academic Press, New York —London 1961.