INCCOLLECTIVE BREMSSTRAHLUNG IN A ...AD-Ai36 595 THE TOTAL FIELD INCCOLLECTIVE BREMSSTRAHLUNG IN A...

AD-Ai36 595 THE TOTAL FIELD INCCOLLECTIVE BREMSSTRAHLUNG IN A 1/1MNONEOUILIBRIUM RELATIVISTIC BEAM-PLASMA SYSTEM(U HARRY

DIAMOND LABS ADELPHI MD H E BRANDT qEP 81 HD PL-T-996

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r. REPORT MNMERt2 GOVT ACCESSION No. 3. RECIPIENT'S CATALOG NUMBER

HDL-TR-19M vip6s4. TITLE (sind ubans)S.TPOFRPTItEIDCVRD

The Total Field In Collective Bremsstrahlung in a Nonequilib- Technical Reportrium Relativistic Beam-Plasma System 6- PERFORMING ORG. REPORT NUMBER

7. AUTHN(s) S.CONTRACT OR GRANT NUMBER(m)

Howard E. Brandt

9. PERpORamING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK

Harry Diamond Laboratories AREA & WORK UNIT NUMBERS

2800 Powder Mill Road Program Ele: 61101 AAdeiphi, MD 20783

It. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

U.S. Army Materiel Development and September 1983Readiness ommand IS. NUMBER OF PAGES

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kiDL Project: A10225DRCMS Cods: 611101.91 A9011IDA Project: 1 L161 1OIA91A

IS. KEY WORDS (Conamie an revrse aio It necesarmW 1~11 b~iy h block mmbor)

Radiation Relativistic electron beamsMicrowaves Radiative InstabilityPlasma turbulence BremestrahiungRelativistic plasma Beam-plasma systemsPlasm physics Nonlinear plasmas _____________________

ISTACr (bio mevese alf It ueemmy aind fdowidIr hr block number)

S!vAn expression Is obtained for the total electric field associated with collective bremsstrahlungIn a nonequilibrium relativistic beam-plasma system in the case of a slowly varying, nearlyspatially Independent background distribution with no external fields. The field is assumed 'to con-slst of an unperturbed regular part corresponding to the nonradlative part of the field of the par- Iticls and a stochastic part corresponding to the bremsstrahlung radiation field. The relationsderived here are needed In calculations of collective bremsstrahlung processes and the condi-tionis for the occurrence of bremsstrahiung radiative instability In relativistic beam-plasmasystems.,

DD PUS7S EITION OW f NOV 155 O@GOLETZ NLASFE

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.~~iA--e -11-

CONTENTS

Page

2. DISPERSION RELATION FOR THE BRE!4SSTRAHLUNG FIELD ........... so......... 6

3. THE STOCHASTIC PROPERTIES OF THE BREMSSTRAHLUNG FIELD ..... o.......... 11

4. THE FIELD DUE TO THE PERTURBED TEST PARTICLE TRAJECTORY ... ........... 13

5. THE FIELD DUE TO THE DYNAMIC POLARIZATION CURRENT .... *................ 18

6. THE TOTAL FIELD o ... o.os.......... ......... 22

7DIS CONCU ION . .. . . .. .. o .. .. . . . ... o.* . .. o . . ... .. .......... o ............ ** ** ** 23

Accession For

GTT OA&IPTTC TABuz'annou-ced

Distrib~ution/

Availability Codes

jAvail and/or

Dist Special

3

1. INTRODUCTION

In a nonequilibrium relativistic beam-plasma system with a slowly varying,nearly spatially independent background distribution without external fields,the total field, excited in the beam-plasma and associated with a collectivebremsstrahlung process involving a test particle, has two parts. First, ithas a regular part with a relatively slowly varying phase associated with thenonradiative field of the test particle and its induced dynamic polarization,and it also has a stochastic part with an irregular rapidly varying phaseassociated with the bremsstrahlung radiation field.1 Thus, the Fourier trans-form of the total field involved in the bremsstrahlung process is given by

Ek= + t I (

where and 9t are the regular nonradiative component and the stochasticbremsstrahlung component, respectively. The regular part, , of this fieldis assumed to consist of four parts in sufficient approximation; namely,

+R4(1) +(2) +1) +(2)kEk + F ) + Edpk + 4pk . (2)

Here jI) is the self-field associated with the unperturbed motion of therelativistic bare test particle, ignoring its induced dynamic polarization.The f ield 2) is the field associated with the perturbation in the bareparticle'motion. The fields E(1 and t(2) are fields produced by the dynamicdpk dpkpolarization current induced by the test particle to second and third order inthe total field, respectively.

In sections 2 through 5, the dispersive characteristics and stochasticproperties of the bremsstrahlung field are reviewed, the field due to theperturbed bare test particle trajectory is calculated from the associatedcurrent, and the dynamic polarization current is used to calculate the fieldwhich it induces. In section 6 an expression for the total field is obtained,and in section 7 the results are summarized. Expressions for these fields areneeded in calculations of the total nonlinear force on a relativistic testparticle involved in collective bremsstrahlung in a nonequilibrium beam-plasma

IA. V. Akopyan and V. N. Tsytovich, Bremsstrahlung in a NonequilibriumPlasma, Fiz. Plazmy, 1 (1975), 673 [Soy. J. Plasma Phys., 1 (1975), 371].

5

IPREVIOUS PAGE

IS BLNKI ESS8 LLA:N K

a,~ ~ ~ ~~~~~~~~ % .-. -- ***Ch~* '~*C

system.' *'$ The latter is needed in calculations of collective bremsstrah-lung processes and the confitions for the occurrence of a bremsstrahlungradiative instability. 1-4'ot

2. DISPERSION RELATION FOR THE BREMSSTRAHLUNG FIELD

In zeroth approximation the dispersive properties of the bremsstrahlung..,radiation field 2V are governed by the poles of the linear photon Green'sfunction G (k) for the beam-plasma. The latter determines the mode disper-sion relat ns. Thus,

+st +0(0)(3Ek - Ek(3

where the electromagnetic waves q(0) of mode o satisfy the following disper-sion relation:

IUG (k)j = 0 • (4)

The linear photon Green's function Gmn(k) follows from the linear Maxwell'sequations in the beam-plasma system, namely,

(5)

and

IA. V. Akopyan and V. N. Tsytovich, Bremsstrahlung in a NonequilibriumPlasma, Fiz. Plazmy, 1 (1975), 673 [Sov. J. Plasma Phys., 1 (1975), 371).

2A. V. Akopyan a;d V. N. Tsytovich, Transition Bremsstrahlung of Relati-vistic Particles, Zh. Eksp. Teor. Fiz., 71 (1976), 166 [Soy. Phys. JETP, 44(1976), 87).

3A. V. Akopyan and V. N. Tsytovich, Bremsstralung Instability of Relati-vistic Electrons in a Plasma, Astrofizika, 13 (1977), 717 [Astrophysics, 13(1977), 423).

V. N. Tsytovich, Collective Effects in Bremsstrahlung of Fast Particlesin Plasms, Comments, Plasm Phys. Conf. Fusion, 4 (1978), 73.

*H. E. Brandt, Nonlinear Force on an Unpolarized Relativistic TestParticle to Second Order in the Total Field in a Nonequilibrium Beam-PlasmaSystem, Harry Diamond Laboratories, HDL-PRL-82-7 (May 1982) (to be publishedas HFL-TR-1995, 1983).

H. E. Brandt, Nonlinear Dynamic Polarization Force on a Relativistic TestParticle in a Nonequilibrium Beam-Plasma System, Harry Diamond Laboratories,HDL-PRL-82-6 (May 1982) (to be published as HDL-TR-1994, 1983).

f. B. randt, Theoretical Methods in the Calculation of theBremestrahlung Recoil Force in a Nonequilibrium Relativistic Beam-PlasmaSystem, Harry Diamond Laboratories, HDL-PRL-83-6 (April 1983) (to be publishedas HDL-TR-2009, 1983).

6

., Bi ,i l I - - ' ' ' w '' ''.%'% %' % %% ' % % %

" ..'." .'.." ,'-.-."-.'.." .'.,"

, -

+ 3B(6

The Fourier decomposition of the fields is given by

E J dkei(tr+ it) (7)

and

B dk B ei("4 Wt) ,()

where

dk =d V d w .(9)

The assumed forms of the constitutive relations for the beam-plasma system are

+ B (10)Hk-)J

where U0is the permeability of free space, and

D = £-ij(k)Ekj ,(1

where C* (k) is the dielectric tensor for the beam-plasma system, and sum-mation over repeated indices is understood. From equations (5), (10), and(11) it follows that

I.

.~(ixBkj 1 + iwcij(k)Ekj = )ki ,(12) .

where Tkis the Fourier transform of the nonlinear and external current densi-ty, ard from equation (6)

+ (13)

Here 85 is a small imaginary part of the frequency. Next, substituting equa-tion (13) in equation (12), then

i 4B + k ),i + iwcijEkiki ( (1)

But one has the following vector identity,

1txxk) = k2+ (15)

Substituting equation (15) in equation (14), then

%'-

whre i~k i te ieecri tnsr orth bamplsm sstman sm-7

mat~~~~~~~~lon~~~~ ove reeae inie sunesod romeutos() n(~~ % .6 % tflosta

-(Z'Eki k 2 Eki) + iwcijEkj = ki 16)

Equation (16) may be rewritten as

G!jJ - j (17)

where 'VP

I) -P(w + ia) 2 6ij) + Cij(18)

where 6.. is the Kronecker delta. For a spatially isotropic system, thedielectric permittivity tensor is given by

5'67

ck,+1) = (6ij - k2 )et(Z,w) + kikj (kw) , (19)"

where t and cL are the transverse and longitudinal permittivity, respec-tively. Substituting equation (19) in equation (18) and combining terms, then

-1 2)tk2 k2 kikj

0"~w + idL2 -1 ( + i~i k 2 *(0

Treating equation (17) as a matrix equation, then

iEki - GijJkj . (21)

w+ i6The matrix G inverse to Gj' is easily obtained from equation (20); namely,i i

Gij - - + --k. (22)

G Ct Ct(w + i6) 2 k I2 j) (22'J m k.. 6j -[Po

Checking that equation (22) is in fact the inverse, one notes, using equations(20) and (22), that

SA. I. Akhiezer, I. A. Akhlezer, R. V. Polovin, A. G. Sitenko, and K. N.Stepanov, Plasma Electrodynamics, 1, Linear Theory, Pergamon Press (1975),206.

6V, N.. Tsytovich, Nonlinear Effects in Plasma, Plenum Press, New York(1970), 31.

7V. N. Taytovich, Theory of Turbulent Plasma (Consultants Bureau), PlenumPress, Now York (1977), 63-65.

8

,.: , ) o,:- 2 .- '..N : -,..-.:-.--.,-.'..:-.-- .. .& ':..,-..': ,:-..-..- ... .. ...-..... :,-: -= , _.. . ,_ , " ' ;ie ; ' .: :.A. .' .. :..k .. :. ;... :.::.2.:.. . : .:. .::.-...;:...: .:-2.:'-it..'

Gm Gi j- C.t. -- (Ct C -. .

K~ G +i IJ(w + I5 kc2 J (23)

-1 - 2\ -i - I2 k

x k-€,+ ij _'-] = 6mjLk2 C Ct(W +9 ia) 2 k 2 ( 6 i - -;*)-,

110'

as it must. Equation (21) can be equivalently written in the following form:

n-- iF+ i GnuJkm , (24)

since the linear photon Green's function equation (22), for the isotropicplasma is a symmetric tensor. According to equation (4), the poles of thisfunction determine the zeroth order dispersion relation for the bremsstrahlungin the isotropic case.

More generally the dielectric tensor is related to the linear conductivitytensor by

i (k)

Ci= j o6 i_. + w i (25)

Equation (25) follows through a comparison of equation (12) with the vacuum

form of Maxwell's equations; namely,

if +

Pi (Z x B)i + iwEFr6ijEkj = (ij(k)Ekj + Jki ' (26)

where the explicit linear current in terms of the linear conductivity has beenintroduced on the right,* Jki designates the nonlinear and external contribu-tions to the current, and the vacuum dielectric tensor has been introduced.Thus, moving the first term on the right of equation (26) to the left andcomparing with equation (12), then equation (25) follows.

By equation (47) of HDL-TR-1 994,* the linear conductivity tensor oij inthe beam plasma system is given by

+ [ k.vs kmvsj 3fR(O)dOs vsi 6jm1 w+ i6/ + + i0i Ps (27)oi j -Ze .2 7 ,i,, " -

if- (2) i(w k-vs + i6) aPsm

*H. E. Brandt, Nonlinear Dynamic Polarization Force on a Relativistic TestParticle in a Nonequilibrium Beam-Plasma System, Harry Diamond Laboratories,HDL-PRL-82-6 (May 1982) (to be published as HDL-TR-1994, 1983).

9

R ] ~ m m r ,r, ,,., ; ...,,..,.., .., .. .. .. ..... ,,,....... ..,....-.,:.,..,..,.,. ',,,.,,.,.,.,..,- ..v. .-r] € .& "&. %-%.' % '. . .... _.' '..% .- % . -. ' .. - ... % .. - . . .. .,-N1 .Ask ,- -,.-. , ."."v , .,:',',' ""-C- N".:' '' ..74 '"'-..,. , , ',. ,r: ,", ',". , '," % .," ,""N "'

Here, a is a species label. Substituting equation (27) in equation (25), thenthe linear dielectric tensor for the beam-plasma system is given by

+ Fl kvjL_ + kmvsj R(0'Ci %ij+ -a1 e fd s V 16j,~ T d + i] P

CWj + jAdI f (2s)3 (a- v +i6) 3Psm '(28)5 I..

in which case equation (18) becomes

1] +-i; ) (kk- - k 2 .j ) +

Fl 1t ~ kmVs*1 aR(0)I ~svsi LSjmYl + i6 W + afj P (29)+W+- i6rles ( - "s + i w "

T~)' W+ i6 sm

It should be noted that equations (27), (28), and (29) are in fact symmetricin i and j which follows after a simple integration by parts. The zeros ofthe determinant of the matrix equation (29) must give the dispersive prop-erties of the bremsstrahlung field.

From equation (19) one has that

kik.

= k2 ,j •(30)£ k2 1)i

and

'C = (Cii - si)(31)

Substituting equation (28) in equations (30) and (31), then in the spatiallyisotropic case, the longitudinal and transverse dielectric permittivity aregiven bV

+ + k)3s~ (32)(0+ i9 k 2) W- Z's + i6 P

and

1 ye fd? + kvs kav -km fR(O)

2( 1) aT (2w)3 L -t4 5 + 6 m apsm

If the background distribution functions fR(O) are also isotropic in momentumPsspace, then they depend only on particle energy; namely,

fR(O) = fR(o)(6 ) (34)p P s

-

10

. A

where

=(p2c 2 + a r1c4)1/2 (35)

Then

afR0 afR(O) s R(0) pfR(0P"a sC2 V (smapsM = s s3 - Vsm 6)cs

Substituting equation (36) in equation (28) and simplifying, then for the caseof isotropy in momentum space, one has

R(0)1 d~p's v afcij -=oi + Ee 2 - VsiVs Ps

W + i6- s- (2w) 3 v • (37)s (w) (W - kOvs + is) acs

Substituting equation (37) in equations (30) and (31), then in the case ofisotropy in momentum space, the longitudinal and transverse dielectric permit-tivity are given by

1 d 3pksvs)2

El = £0 + - (38)+ '2/ J2 (2w) 3 w- kvs + is

and

1 1d- 3 p v 2 - P se t_ _ _ _ _ ( jv i -) 2 / k 2 _ _ _

i.- (2W) 3 W - kov s + is ae (39)

and the zeros of the determinant of equation (20) along with equations (38)and (39) then give the dispersion relation for the bremsstrahlung in thiscase*

3. THE STOCHASTIC PROPERTIES OF THE BREMSSTRAHLUNG FIELD

As stated earlier, it is to be assumed that the bremsstrahlung field re-sulting from the particle scattering in the system is stochastic, having anirregular rapidly varying phase. 1 To the needed order, the phase of thestochastic bremsstrahlung field is assumed to change randomly during a typicalinteraction period, and the time average is assumed to be the same as theaverage over the stochastic ensemble. In particular it is assumed that forthis stochastic field the following averages over the statistical phasedistribution apply approximately; namely,

1A. V. Akopyan and V. N. Tsytovich, Bremsstrahlung in a NonequilibriumPlasma, Fiz. Plazmy, 1 (1975), 673 [Soy. J. Plasma Phys., 1 (1975), 371).

%%N' %- %,* N. .

****V*. ~ ie e..'..%'%

oei'

st -"<i> 0 (40)0st stj t 2 6(k + k ()ekie~j (41)<Ekijkl > = j (41 "e"

where e is the unit polarization vector for the stochastic field.

Equations (40) and (41) are deduced as follows. Representing the stochas-tic electric field in terms of its modulus, Ak, and phase, , one has

stisn •4,-E = ekiAke = ekiAk(cos k + i sin (42)

stTherefore, the average over the statistical phase distribution <Eji> is givenby

<(E> = ekiAk(<cos k + i sin k>) (43)

But assuming random phase to the needed order, then clearly

<cos $k> = <sin > 0 (44)

and equation (43) becomes equation (40). Similarly,

stst -i ,<Eji 0JZ = eke.,. AA,.,e e > . (45)

But

<e e > =<e > = 6'k (46)

since it is vanishing unless k = k' and unity when k = k'. Substitutingequation (46) in equation (45), using the property of the Kronecker delta,+* + ."-denoting k' by -kl, and usiag the reality property of the field (E k =Ek

then

st st (47)Eki El j > = e,. (47)< ek cietjAqc .kk.

Next, one may define the quantity IEtI in terms of the modulus Ak byIt126(k + k1) = 6 klk " (48)

Then substituting equation (48) in equation (47), equation (41) follows.

Furthermore, using equation (3) in equations (40) and (41) one has for thestochastic properties of the bremsstrahlung radiation field the following:

<( 0)> 0 (49)a.(o) 0(o) y , * 0 (0)1 6(k + kj)(0)-:

eii elj > = Iekiekj I+ ) (50)

where + is the unit polarization vector for mode a.

12

...-.. ...... •... •............%

4. THE FIELD DUE TO THE PERTURBED TEST PARTICLE TRAJECTORY

The trajectory of a relativistic test particle of species a moving throughthe beam-plasma system is given by

ra =,at + ArQ , (51)

where va is the unperturbed velocity and At and &A describe the perturbationin the trajectory and the velocity, respectively. Provided the conditions forthe plasma Born approximation are satisfied--namely, that the relativisticparticle momentum be much greater than the electromagnetic impulse received bythe particle in a time interval given by the inverse electron plasma frequen-cy--then the perturbed motion is given by equation (36) of HDL-TR-1995:*

ea f d3Jt din- 4.. +i ~~S} (n44)tJom (Wit -V+a + i6)2 W+i 6

(52) IN

,+ +"-'v

Va*, e

yd 2 i t.+. + i6) 2 .,a'k

Here ea, ma, and v. are the charge, mass, and unperturbed velocity of the testparticle, respectively. (For notational convenience, the naught in HDL-TR-1995 is dropped here.) The unperturbed relativistic factor ya is given by

- fi - (2 (53)

The perturbation in the velocity is given by equation (35) of HDL-TR-1995:*+ -i (W44.v)t...

Av~a =e d d +t e

f V di- na + i (

-~ '~ ca 2 - Z.+ + is

The ith component of A&,, equation (54), can be equivalently rewritten asfollows:

*H. E. Brandt, Nonlinear Force on an Unpolarized Relativistic Test Parti-cle to Second Order in the Total Field in a Nonequilibrium Beam-Plasma System,Harry Diamond Laboratories, HDL-PRL-82-7 (May 1982) (to be published as HDL-TR-1995, 1983).

13

% A :

. . .. . . . . . . . . . . . . . . . .

-ZA ~ ~ ~ n.%x-v2V 7 - T - d T -' 06 . -M Y 771F 4 ": -9 M - - - 77 -7- -

ee

-ie 0 + Vvi r_

w+6Ja w - k.v a + id - .

0a

The current density due to the actual charge of the test particle is then J'

given byAV e( + V )va a ( -vt- (56)

Expanding the delta function in equation (56) for small a' then to lowest

v a

order in a'= e + a -k+t) -v. (57)a3) - +ata (59)

= e 63 (- -+at) - va.63( +t)] (5)

The Fourier components of these currents at the position of the test particle

+ +0 +(2

,() = [_3 dt j()( , e Ci1 .4-w) .(2

Jc~~i ] (2) j ~ - i (1

Substituting equation (59) in equation (61) and integrating over the deltafunction, then

(1) eav a .dt -) t

J(2) I -i 0ti e (63)

~ai=(2w)3

Next, recognizing the integral form of the delta function appearing in equa-tion (63), then

ak i f (2) 3 (

_- .i $ ,,. * , ',' ;..' ..'. .. . * ... ... -. - .- ".-. ..... *..,.** . ,-*., ,, ,-. ," ,.. -" , * ,, . . v ,. ' ,..- --. ". ,-.'- ,.- '

Susiuigeuto 5) in eqato (6 )adita tigovrte et

function, thens

.- 1) e~ ava t-~j-+-

Next, substituting equation (60) in equation (62) one has

(2) fd3r dt 6 3 + _+ (k.-t)=] f(2) e, AV i (r V t)e- i

- 3 Jt v63 ( + t)]e-i(k 'r-t) (65)

The second integral in equation (65) can be simplified by integration overspace by parts and dropping surface terms. Equation (65) becomes

,(2) d 3r+.<dt e+_ [a. -3,+,+(ie fAv (6 (r -Vt)e

- i

fd 3r dt eevj6 3 (r _ t)it-A e-i@(Zo Wt) (66)(2w,)4 aa

Integrating over the delta functions, then

• (2) [ dt " 4

3ki = + (26))

- i f dt evie-i(kov.-w)t oN (67)

( 2 1r) 4

'

Next, substituting equations (52) and (55) in equation (67) and integratingover the time, then

Jki = vair C - f - [E)d kl j+

ci IJ(2- )I (w)3 + kiso)

"k 1 -W, t i )J-+a)+ k1 i +i6 +

+ - kvI + is (68)

+ i ., d _, dw, [lt °%k (1 + + 1__ei6) d +

d] ___ ___ _ 1___ _°_1+_i_ __

+ 0 f (2w) 3 ( t - +i K . i6] 2

x 6(- "1- (- } ]° )¢it--- ~~(i - 1*v + i6)2I ' (1%

6(- W, it +

yVmap2 ( 2w) 3 (w, - + tlc+ i6) 2

Simplifying equation (68), then

15 .

. . . . . . . . . . .. . . . . . .N

(2) ie. f d3kdw --k- 6(w- -. i.,- Ycs% J (2w) 3 w + i 1 I

X vaivaj I * ,a vaivj kliVaj

+ij 2 it + (69)

k8 c (,- t Vv 6c2 W v + i

=1 - i O a + (i - it va + i6) 2 c 2 (w - O +i

Usinj the property of the delta function to replace (w - . a)by(w- f-v.) in the fourth, fifth, and sixth terms of equation (691, then a*

(2)= ie- f-d 3 k~dw E 6(ww- - + w 1 0)d i -y - J- 1 ( 2 ) 3 1 * + i s t a

kj .v(i) + kliV 1°ivivaj v

x aiJ + v+ + _ _ 2 - (70)w- It-V+a + is + (w - ita + i6) 2 c

+ +-~~ ~~~ *1 avaivaj W1i vtia :

(- w t,+. + i6)c 2 C2 (ol - t1 .+ + i6) 2

Changing the integration variable kI = (w1 't1) to -kI, equation (70) becomes

(12) = fe d-x dw -k1j + - + 14iaki Y cImaJ(2w)3 -W1 + is s(w + (it iti)

i + kjvai - kliVaz 4*l1vaivaj viv j

to it (+ + is + i6)2 c 2 (71)

zt 0+ itsI "avvai cj W, "avaivaj }

-2( +t41 -iS + c2 (+a tlc - i 6 )2f

Combining terms and using the property of the delta function, one notes that

_ i vv 1 " avaivo- vit1 'rv )

IcC (w) -2 k~v~ itS)c 2

() .;v (w 1 .+V V.a WW1 (2

+c1 (1 -' i 4" - i+) = v( + - - ") c 2 (w - t.%t + i6)2

Substituting equation (72) in equation (71) and using the reality property ofthe field A C0' then

16

Mp

-,% , ,.\I, . , .. 1. jqv %-. -. ..-...-. .. ,%'.. - -,- . . -r ] . l '.~~~~~ ~~~~~~ V' .''... ...,.,,., ,....'''".2 .'. Z'"'....-'.''"......,,.",''..""

--1*q

J~ ~~ dk ie( 1 °va IVaA *(klfk - (73)f11 _i c . _

of(2w)3 w- i6

where

() + ~ -k 1 1 Vv~j (t-t 1e a V Oik j -kliVaj vcv0 ( £ 2 1 ['[

I ++a i i)W- it.va- is (W - k.Vc- i6) 2 )The curt ( ae(2)i by equations (64) and (73) are the cur-

rents associated with the unperturbed motion and the perturbation in themotion, respectively, of the bare particle. By the same arguments leading toequation (21), the components of the field due to the unperturbed and pertur-bed test particle trajectory are given by

(1) (75) (1Nn+i nJkm (75) v-.

and

(2)i(2Nn+ i nmkm ( (76)

respectively. Therefore, substituting equations (64) and (73) in equations ,:(75) and (76), one obtains

41) ~ie,

En (20)3(w + i6) anm

and

e*

2)K. - e a Gn (k) f (klk)6(w + w,- Z -V+ 1°.)

(2) 3 (w + is) m -i6

(78)

Equations (77) and (78) agree with equations (16) and (17) of Akopyan andTsytovich, 1975,1 except for erroneous overall factors of (2w)- 3 and (2w)- 3i

appearing there in equations (16) and (17), respectively. The indices n and mare interchanged there as they are in the defining relation, equation (14) ofAkopyan,I compared to equation (21) here. However, a spatially isotropicplasma is assumed there, so according to equation (22) the Green's function issymmetric, namely Gmn = Gnm. Also in Akopyan, 1 single-wave particle resonanceis ignored, in which case the small imaginary part i6 in equation (74) is

IA. V. Akopyan and V. N. Tsytovich, Bremsstrahlung in a NonequilibriumPlasma, Fiz. Plazmy, 1 (1975), 673 [Sov. J. Plasma Phys., 1 (1975), 371].

17

ignorable, A(.) becomes effectively real, and the complex conjugate sign inequation (78TJmay be removed. The additional factors of (4w)(2r)3 in equation(16) and (4w) in (17) there are due to the Gaussian form of Maxwell's equationused there compared to the MKS form used here, and to the difference in Fouri-er transform convention. For example, in equation (5) of Akopyan andTsytovich, 1975, 1 the Fourier transform convention employed has a factor of(2w) -3 in the inverse Fourier transform in the integration over the three- '"dimensional wave vector space, and a factor of 1 for the integration overfrequency, giving a total factor of (2w) - 3. In this report, a total factor of1 is used, as in equation (7), for example. Likewise, the Fourier transformitself then has a factor of (2w)- l there and (2w) -4 here, as in equation (61),for example.

5. THE FIELD DUE TO THE DYNAMIC POLARIZATION CURRENT

The Fourier transform of the dynamic polarization current to third-orderinduced in the beam-plasma is given byll*

+0 +(2) +(3) (79)Jdpk = Jdpk + J dpk + J dpk"(79)"•

The linear polarization current densityIk( ) is given byJddpi

( = oft)k)k (80)

Its effects are manifest in the linear Green's function for the beam-plasmasystem as discussed in secton 2. The second-order and third-order polar-ization current densities J(p and j( are given by

( fdki - ki - k2),(idp + i81W2 + i6) ijLtk'kl'k 2 )EljEk(

and

(3) = es dki dk2 dk 3 6(k - k1 - k 2 - k 3 )

- sj (W1 + i6)(w2 + i6)(W3 + i6) (82)

(S Xm(k,k,k 2 ,k 3 )Ekj2 E 3m

respectively. The seconJ-order nonlinear and third-order nonlinear conductiv-ities for species s, S'n, and E! , respectively, are given by

1A. V. Akopyan and V. N. Tsytovich, Bremsstrahlung in a NonequilibriumPlasma, Fiz. Plazmy, 1 (1975), 673 [Sov. J. Plasma Phys., 1 (1975), 371).


18 -

. . -% 1.,

- ~ ~ 1 .7 -777 777 77 - * .

d 36.

+Vjkim a k~ R(O)si] ~sc. w~ 2 vs +1 TPSn P

and

V. 3p.

Efd)= (2k, - - + [ (w, V + k .vsjl

as 1 [,(R(0)

2 - - + is X 2 . (84)

14.(2uvs 2t) q- IIi[S . -( 3 V 5 S

x-su 3 - 't C*S + i6

+ kvsm] fR(o) .o9Psq

The function fR(O) is the assumed slowly varying and spatially independentbackground distribution function for species s, whose Fourier transform is

fR(s) _ fR(0)k Ps 6k) (85)

To determine the fields defined in equation (2) and associated with therespective parts of the dynamic polarization current, one proceeds as inobtaining equations (75) and (76). Thus, one has for the fields produced bythe second- and third-order nonlinear dynamic polarization currents, respec-tively,

3. G (k)' (86)E(1) i G((k6 ( 2

dpkn W + is nm dpkm

and

E ( 2 ) i Gnm (k) jkm (3) (87)dpkn nm + i

Rext, substituting equation (81) in equation (86), one obtains

E sfdk1 dk 2 6(k -k - k 2 )El") _ Gn (k) F e s .-'pk == +i"S (i1 + i6)(w 2 + iS) (88)

x Sm (k,k1,k 2 )1 k kEk

19

........................ !..... .... . .. ..................

' ', ' ., . ". '% ,' ' "......... .... .,, ... ..... .. . . ...-.. ,.-....-.... ... ...-.-....... ....-...:< : .. v..4,.;:.C .. ,:v-. .. ,......-..-.... . . . .'..'...... . ..- ,.,. ...- : .... ,'..'--.,......

First symmetrizing equation (88) in (j,kl) and (1,k 2 ), changing variables ofintegration from kI to -kI and k2 to -k 2 , and using the reality property ofthe field, then equation (88) becomes

EmI i d1 kGnmlk) e s 6(k + k I + k2dpkn =2(w. + 1,6) 1 2)-i,),~ X0~f (w - 6)(2 '6 I~ 1 I

(89)

x j+( k - - m ,ik2

Comparing equation (89) with equation (22) of Akopyan, 1 a disparity of afactor of (1/2)(4w)(2w)-6 in the latter .s apparently due to explicit inclu-sion there of a self-field factor of (1/2), the use of Gaussian units, and thedifferent Fourier transform and normalization conventions chosen there. Thefactor of (1/2) is either a typographical error or results from explicitinclusion of a factor of 1/2 associated with self-fields. For the -genericexpression, equation (89), it is preferable that such a factor be maintainedimplicit until the formula is applied explicitly and a self-field factor of(1/2) is needed. The factor of (4w) is due to the use of Gaussian unitsthere. One of the factors of (2w)- 3 is due to the differing Fourier transformconvention. Because of the different Fourier transform convention, the coun-terpart of equation (81) would have an additional factor of (2w) - 3 . The otherfactor of (2w)-3 is apparently due to the different normalizationof fR(O) there. In short, the fR(O) there is evidently (2w)3 times thathere. Alternatively, if the normalization is the same as that here then thereis an erroneous factor of (2R)- 3 appearing there. An isotropic plasma isassumed there, in which case by equation (22) the Green's function is symmet-ric, namely, Gm =Gnm. However as already noted the indices of G. areinterchanged in Akopyan and Tsytovichl in the definition of the Green's unc-tion. It appears, however, that there are typographical errors there. Thecomplex conjugate sign on the fields and a factor of es are left out. Also,there in the second-order nonlinear conductivity, one notes from equation (83)that*

(s) i) ,, s)(k 2 0) _Smj t(k,-kI,-k 2 ) = Smj1 (-k,kl,k 2 ) = -Sij )(kkk 2 ) ,(90)

where S(J(k,k 1 ,k 2 ) designates the conductivity tensor appearing in Akopyanand Tsytovich. l ' In the last. step of equation (90) it is recognized that.(J_,(k,kl,k2) differs from Smdj(k,k,k 2 ) here in that the first, complex de-

nominator w - It s + i6 in equation (83) here is implicitly w - t•s - i6

'A. V. Akopyan and V. N. Tsytovich, Bremsstrahlung in a NonequilibriumPlasma, Fiz. Plazmy, 1 (1975), 673 [Soy. J. Plasma Phys., 1 (1975), 371].


20

S..

~ ~ - -~ -

7 4, = = < = = , . . -..... 7 . - - X:.' - ". . . . . . . . .

there. 8 9 1* Thus, there is also a disagreement in overall sign between equa-tion (89) here and equation (22) of Akopyan and Tsytovich.1 In summarythen, equation (22) of Akopyan and Tsytovichl is in error; it should have anadditional factor of (-es), the fields and the conductivity tensor on theright-hand side should be complex conjugated, and the explicit additionalfactor of (1/2) there should be omitted.

Next, substituting equation (82) in equation (87),

(2) i Gnm(k) dk dk2 dk3 6(k - k- k2 - k 3 )

dpkn f (W1 ~ + '6)(w 2 + '6)(w 3 + i6)(91)

X(s) (k kl k0,k i •""mij £ 1 1' 2 3 )Eki 2

jEk3-

Symmetrizing equation (91) in (i,kl), (j,k 2 ), (l,k 3 ), changing variables ofintegration from k1 to -kJ, k to -k 2 , and k 3 to -k 3 , and using the reality

property of the field (t-k = __, then equation (91) becomes

dp(n) G(00 1 e dk+ dk2 dk 3 6(k + ki + k2 + k 3 )dpkn i6(w + M) s - ')(w 2 - i6)(W 3 - i)

x[ . ( s ) (k,-kl,-k2,-k3) +r Z s) (k,_kl,_k3,_k2) :

mi j mi tj

(92)f (s) (k, k2 _k 1 ,k3) + E (s) (k, k2 ,-k3 ,kl ) (92

mjil mji

+ E i( s ) (k,_-k3,_k1,k2 (s (kk kji *

Comparing equation (92) with equation (23) of Akopyan and Tsytovich, thedisparity of a factor of (1/2)(4w)(2w) - 9 in the latter is again due to anexplicit self-field factor of (1/2), the use of Gaussian units, and the dif-ferent Fourier transform and background normalization conventions there.

1A. V. Akopyan and V. N. Tsytovich, Bremsstrahlung in a NonequilibriumPlasma, Fiz. Plazmy, 1 (1975), 673 [Sov. J. Plasma Phys., 1 (1975), 371).

8H. E. Brandt, On the Nonlinear Conductivity Tensor -for an UnmagnetizedRelativistic Turbulent Plasma, Harry Diamond Laboratories, HDL-TR-1970 (Febru-ary 1982).

9H. E. Brandt, Comment on "Exact Symmetry of the Second-Order NonlinearConductivity for a Relativistic Turbulent Plasma," Phys. Fluids 25 (1982),1922.


21

. . . . . .* ,. . , . . , ,- .. ,,. - " - 4 " ". 4 . . 2 ".4 4 ". ' " . '" - . . "- "." " "

Alternatively if the normalization is instead the same as that here, then

there is an erroneous factor of (2w)- 3 appearing there. There is apparently

an error there. A factor of ies has been left out. Note that by equation -

(84) it follows that

()(k,-k I 0- (S)*-kikk ,k,3 (93)."--k21-3) 'r iv b ut ()

Also in the conductivity tensor Ti4£,(k,kl,k2,k3)given by equation (21) of

Akopyan and Tsytovich, 1 the first go complex denominators are implicitly w

- it.s - i6 and w + w- (1 + I i6, respectively, there.* Also, there

is no overall factor of i as there is in equation (84) here. Therefore,

(s i (_k,kl~k2,k3 = iT(S) "k:'.k(94Zmij] 3)£kklk,3 = (94 21k'

Combining equations (93) and (94) then

it (k,-k-k 2 ,-k3 ) = -iTj0 *( ' k 2 ',k3 ) (95)

In summary then, equation (23) of Akopyan and Tsytovichl is also in error andshould also have an additional factor of (ies), the complex conjugate of thefourth-order conductivity tensor there must be taken, and the explicit addi-tional factor of (1/2) there should be omitted.

6. THE TOTAL FIELD .

Collecting equations (1), (2), and (3) the total field involved in thebremsstrahlung scattering process and acting on the test particle is given by

(1) + (2) + M) + 1(2) (96)VO +dpk dpk

Here (0) is the bremsstrahlung field whose dispersion relations are given byequations (4) and (29) in the spatially isotropic case and equations (4),(20), (38), and (39) in the case of isotropy in both ordinary space and mo-mentum space. The stochastic properties are given by equations (49) and(50). The fields (1, and (2 given by equations (77) and (78) are theself-fields arisin? from the relativistic test particle's own motion. The

dps and'dpk given by equations (89) and (91) are the increasing orderfields produced by the nonlinear dynamic polarization current induced by the

test particle. The fields It") and ) are excited in nonlinear scatteringof ~ ~ ~ ~ dj tefedb th an pk to u

of the f ield by the induced dynamic polarization current. They involve thethree- and four-plasmon vertex, respectively, and play a key role in deter-mining the nonlinear contribution to the collective bremsstrahlung process.

1A. V. Akopyan and V. N. Tsytovich, Bremsstrahlung in a NonequilibriumPlasma, Fiz. Plazmy, 1 (1975), 673 [Soy. a. Plasma Phys., 1 (1975), 371].


22

:---- • T;,*,:' .... , . .-.. --......-.... .-. ,......... . .. "--

.,,- ..4

7. CONCLUSION

An expression has been obtained, equations (96), (77), (78), (89), and(91), for the total field involved in the collective scattering and brems-strahlung of a relativistic test particle in a nonequilibrium beam-plasmasystem. Dispersion relations, equations (4), (29), (20), (38), and (39), andstochastic properties, equations (49) and (50), for the bremsstrahlung fieldhave also been obtained. These results are in agreement with those of Akopyanand Tsytovichl with the exception of apparent typographical errors, and havebeen used by them in their work on nonlinear bremsstrahlung in nonequilibriumplasmas.

The present work, together with related work by the author, 10-14,*t*§ isimportant for ongoing work in calculating collective radiation processes andconditions for the occurrence of radiative instability in relativistic non-equilibrium beam-plasma systems.

1A. V. Akopyan and V. N. Tsytovich, Bremsstrahlung in a NonequilibriumPlasma, Fiz. Plazmy, 1 (1975), 673 [Soy. J. Plasma Phys., 1 (1975), 371].

10H. E. Brandt, Symmetries of the Nonlinear Conductivity for a RelativisticTurbulent Plasma, Harry Diamond Laboratories, HDL-TR-1927 (March 1981).

11H. E. Brandt, Exact Symmetry of the Second-Order Nonlinear Conductivityfor a Relativistic Turbulent Plasma, Phys. Fluids, 24 (1981), 1760.

12H. E. Brandt, Second-Order Nonlinear Conductivity Tensor for an Unmagne-tized Relativistic Turbulent Plasma, in Plasma Astrophysics, Course andWorkshop Organized by the International School of Plasma Physics, 27 August to7 September 1981, Varenna (Como), Italy (European Space Agency ESA SP-161,November 1981), 361 (also to be published by Pergamon Press).

13H. E. Brandt, Symmetry of the Complete Second-Order Nonlinear Conductiv-ity Tensor for an Unmagnetized Relativistic Turbulent Plasma, Journal of Math.Phys. 24, (1983), 1332, 2250.

1 4H--E. Brandt, The Gluckstern-Hull Formula for Electron-Nucleus Brems-strahlung, Harry Diamond Laboratories, HDL-TR-1884 (May 1980).

*H. E. Brandt, Nonlinear Force on an Unpolarized Relativistic Test Parti-cle to Second Order in the Total Field in a Nonequilibrium Beam-Plasma System,Harry Diamond Laboratories, HDL-PRL-82-7 (May 1982) (to be published as HDL-TR-1995, 1983).

tH. E. Brandt, Nonlinear Dynamic Polarization Force on a Relativistic Test

Particle in a Nonequilibrium Beam-Plasma System, Harry Diamond Laboratories,HDL-PRL-82-6 (May 1982) (to be published as HDL-TR-1994, 1983).

*H. N. Brandt, Theoretical Methods in the Calculation of theBremsstrahlung Recoil Force in a Nonequilibrium Relativistic Beam-PlasmaSystem, Harry Diamond Laboratories, HDL-PRL-83-6 (April 1983) (to be publishedas HVL-TR-2009, 1983).

'other related work prepared in preprint form will be published later andIs available from the author.

23 . -

%.. 1-

~i.]

LITERATURE CITED

(1) A. V. Akopyan and V. N. Tsytovich, Bremsstrahlung in a NonequilibriumPla ma, FiL. Plasmy, 1 (1975), 673 (Soy. J. Plasma Phys., 1 (1975),371).

(2) A. V. Akopyan and V. N. Taytovich, Transition Bremsstrahlung of Relati-

vistic Particles, Zh. Sksp. Teor. Fiz., 71 (1976), 166 [Soy. Phys. JETP,44 (1976), 871.

(3) A. V. Akopyan and V. N. Taytovich, Bremstralung Instability of Relati-vistic Electrons in a Plasma, Astrofizika, 13 (1977), 717 [Astrophysics,13 (1977), 423].

(4) V. N. Tsytovich, Collective Effects in Bremsstrahlung of Fast Particles

in Plasmas, Comments Plasma Phys. Conf., Fusion, 4 (1978), 73.

(5) A. I. Akhiezer, I. A. Akhiezer, R. V. Polovin, A. G. Sitenko, and K. N.Stepanov, Plasma Electrodynamics, 1, Linear Theory, Pergamon Press(1975), 206.

(6) V. N. Tsytovich, Nonlinear Effects in Plasma, Plenum Press, New York(1970), 31.

(7) V. N. Tsytovich, Theory of Turbulent Plasma (Consultants Bureau), PlenumPress, New York (1977), 63-65.

(8) H. Z. Brandt, On the Nonlinear Conductivity Tensor for an UnmagnetizedRelativistic Turbulent Plasma, Harry Diamond Laboratories, HDL-TR-1970(February 1982).

(9) H. E. Brandt, Comment on "Exact Symmetry of the Second-Order NonlinearConductivity for a Relativistic Turbulent Plasma," Phys. Fluids 25(1982), 1922.

(10) H. Z. Brandt, Symmetries of the Nonlinear Conductivity for a Relativis-

tic Turbulent Plasma, Harry Diamond Laboratories, HDL-TR-1927 (March1981).

(11) H. Z. Brandt, Exact Symmetry of the Second-Order Nonlinear Conductivityfor a Relativistic Turbulent Plasma, Phys. Fluids, 24 (1981), 1760.

(12) H. E. Brandt, Second-Order Nonlinear Conductivity Tensor for an Unmag-

netized Relativistic Turbulent Plasma, in Plasma Astrophysics, Courseand Workshop Organized by the International School of Plasma Physics, 27

August to 7 September 1981, Varenna (Como), Italy (European Space AgencyESA SP-161, November 1981) 361 (also to be published by Pergamon Press).

24

1 IN O l iv

. .. .p . . . .-

LITERATURE CITED (Cont'd)

(13) H. E. Brandt, Symmetry of the Complete Second-Order Nonlinear Conductiv-ity Tensor for an Unmagnetized Relativistic Turbulent Plasma, J. math.Phys. 24, (1983), '332, 2250.

(14) H. E. Brandt, The Gluckstern-Hull Formula for Electron-Nucleus Brems-strahlung, Harry Diamond Laboratories, HDL-TR-1884 (May 1980).

25

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*ADISTRIBUTION (Cont' d)

UNIVERSITY OF CALIFORNIA CORNELL UNIVERSITYELECTRICAL ENGINEERING DEPARTMENT ATTN R. LOVELACEATTN C. K. BIRDSALL ATTN R. N. SUDANBERKELEY, CA 94720 ATTN J. NATION

ATTN D. HAMMERUNIVERSITY OF CALIFORNIA ATTN H. FLEISHMANNPHYSICS DEPT ITHACA, NY 14853ATTN A. N. KAUFMANBERKELEY, CA 94720 DARTMOUTH COLLEGE

PHYSICS DEPTUNIVERSITY OF CALIFORNIA, DAVIS ATTN J. E. WALSHAPPLIED SCIENCES HANOVER, NH 03755ATTN J. DEGROOTDAVIS, CA 95616 UNIVERSITY OF THE DISTRICT

OF COLUMBIAUNIVERSITY OF CALIFORNIA, IRVINE VAN NESS CAMPUSDEPT OF PHYSICS DEPT OF PHYSICSATTN G. BENFORD ATTN M. J. SMITHATTN N. ROSTOKER 4200 CONNECTICUT AVE, NWATTN M. MAYER WASHINGTON, DC 20008ATTN A. RAYIRVINE, CA 92717 ENGINEERING SOCIETIES LIBRARY

345 EAST 47TH STREETUNIVERSITY OF CALIFORNIA, LOS ANGELES ATTN ACQUISITIONS DEPARTMENTDEPT OF PHYSICS NEW YORK, NY 10017ATTN K. NOZAKILOS ANGELES, CA 90025 GENERAL DYNAMICS

ATTN K. H. BROWNCAMBRIDGE UNIVERSITY PO BOX 2507 MZ 44-21INSTITUTE OF ASTRONOMY POMONA, CA 91769MADINGLEY ROADATTN M. REES UNIVERSITY OF ILLINOIS ATCAMBRIDGE CB 3 0 HA, ENGLAND URBANA--CHAMPAIGN

DEPT OF PHYSICSCHALMERS UNIV OF TECHNOLOGY ATTN N. IWAMOTOINST OF ELECTROMAGNETIC FIELD URBANA, IL 61801THEORY

ATTN H. WILHEL14SSON ISTITUTO DI FISICA DELL' UNIVERSITAS-41296 GOTHENBURG, SWEDEN VIA CELORIA 16

ATTN P. CALDIROLAUNIVERSITY OF CHICAGO ATTN C. PAIZISLAB FOR ASTROPHYSICS & SPACE RESEARCH ATTN E. SINDONIATTN E. PARKER 20133 MILANO, ITALYCHICAGO, IL 60637

ISTITUTO DI FISICA DELL' UNIVERSITAUNIVERSITY OF COLORADO ATTN A. CAVALIEREDEPT OF ASTROGEOPHYSICS ATTN R. RUFFINIATTN M. GOLDMAN ROME, ITALYATTN D. SMITHBOULDER, CO 80309 ISTITUTO DI FISICA GENERALE

DELL' UNIVERSITACOLUMBIA UNIVERSITY CORSO M. D'AZEGLIOATTN S. JOHNSTON ATTN A. FERRARI216 MUDO BLDG 46 TORINO, ITALYNEW YORK, NY 10027

JAYCORATTN E. CONRAD205 S. WHITING STALEXANDRIA, VA 22304

30

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DISTRIBUTION (Cont'd) "

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KAMAN SCIENCES CORP UNIVERSITY OF MARYLANDATTN T. A. TUMOLILLO DEPT OF ELECTRICAL ENGINEERING --1500 GARDEN OF THE GODS ATTN M. REISER .-COLORADO SPRINGS, CO 80907 ATTN W. DESTLER

ATTN C. D. STRIFFLERLAWRENCE BERKELEY LAB ATTN M. T. RHEEATTN A. FALTENS COLLEGE PARK, MD 20742ATTN A. H. SESSLERATTN D. KEEFE UNIVERSITY OF MARYLANDBERKELEY, CA 94720 DEPT OF PHYSICS & ASTRONOMY

ATTN H. R. GRIED4 5*

LAWRENCE LIVERM4ORE LABORATORY ATTN E. OTT .

UNIVERSITY OF CALIFORNIA ATTN C. GREBOGI

ATTN L. MARTIN ATTN K. PAPADOPOULOSATTN P. WHEELER ATTN J. WEBERATTN H. S. CABAYAN COLLEGE PARK, MD 20742

ATTN R. BRIGGSATTN E. K. MILLER MASSACHUSETTS INSTITUTEATTN R. ZIOLKOWSKI OF TECHNOLOGYATTN R. SCARPETTI PLASMA FUSION CENTERATTN R. MINICH ATTN G. BEKEFIATTN S. BURKHART ATTN J. BELCHER .

ATTN G. VOGTLIN ATTN T. CHANGATTN V. W. SLIVINSKY ATTN B. COPPIATTN S. L. YU ATTN R. DAVIDSONATTN G. CRAIG ATTN C. LINATTN J. WYATT ATTN S. OLBERTATTN R. ALVAREZ ATTN B. ROSSI I"ATTN G. LASCUE CAMBRIDGE, M4A 02139ATTN 3. H. YEEATTN H. W. MELDNER MAX PLANCK INST, FUR AERON0OMIE

ATTN 3. S. PETTIBONE 3411 KATLENBGATTN A. J. POGGIO ATTN J. AXFORDATTN Ho LOYD ATTN E. MARSCH :4PO BOX 808 LINDAU 3, WEST GERMANYLIVERMORE, CA 94550

MAX PLANCK INSTITUTE OF PHYSICSCHIEF & ASTROPHYSICSLIVERMORE DIVISION, FIELD COMMAND, DNA ATTN H. NORMANLAWRENCE LIVERMORE LABORATORY ATTN J. TRUMPER.

PO BOX 808 GARCHING BEI MUNICH, WEST GERMANYLIVERMORE, CA 94550 t5.5

MAXWELL LABORATORIES-.LOS ALAMOS SCI LAB ATTN J. S. PEARLMAN VATTN S. COLGATE 8835 BALBOA AVEATTN D. F. DUBOIS SAN DIEGO, CA 92123

ATTN D. FORSLAND -.ATTN C. M. FOWLER MISSION RESEARCH CORPATTN B. WARNER ATTN B. GODFREYATTN J. LAN ATTN D. J. SULLIVAN.ATTN R. BOEBERLING ATTN M. BOLLENATTN T. R. KING ATTN C. LONGMIREATTN A. KADISH ATTN D. VOSSATTN K. LEE 1720 RANDOLPH RD, SEATTN R. W. FREYMAN ALBUQUERQUE, NM 87106ATTN A. W. CHURMATZ - -

ATN L. N. DUNCAN UNIVERSITY OF NAGOYA .-ATN D. B. HENDERSON DEPT OF PHYSICSATTN L. E. THODE ATTN S. HAYAKAWA dq]

ATTN H. A. DAVIS NAGOYA, JAPANATTN T. NAMPO BOX 1663OS ALAMOS, Mt 87545

31

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NATIONAL RESEARCH COUNCIL QUEEN MARY COLLEGEDIVISION OF PHYSICS DEPT OF APPLIED MATHATTN P. JAANINAGI MILE END ROADOTTOWA, ONTARIO ATTN D. BURGESSCANADA LONDON El 4NS, ENGLAND

NATIONAL SCIENCE FOUNDATION RUHR UNIVERSITYDIVISION OF ASTRONOMICAL SCIENCES ATTN W. SEBOLDATTN M. AIZENMAN ATTN W. ZWINGMANN1800 G STREET, NW 4630 BOCHUM 1, WEST GERMANYWASHINGTON, DC 20550

INSTITUTE OF EXPERIMENTAL PHYSICSOAK RIDGE NATIONAL LABORATORY RUHR-UNIVERSITYPo PoX Y POSTFACH 2148ATTN A. C. ENGLAND ATTN H. KUNZEOAK RIDGE, TN 37830 436 BOCHUM, WEST GERMANY

OCCIDENTAL RESEARCH CORP SACHS/FREEMAN ASSOCATTN D. B. CHANG ATTN R. A. MAHAFFEY2100 SE MAIN 19300 GALLANT FOXIRVINE, CA 92713 SUITE 214

BOWIE, MD 20715OXFORD UNIVERSITYDEPARTMENT THEORETICAL PHYSICS SANDIA LABORATORIES1 KEBLE RD ATTN R. B. MILLERATTN D. TER HAAR ATTN J. POWELLOXFORD OXI 3NP, ENGLAND ATTN W. BALLARD

ATTN C. EKDAHLOXFORD UNIVERSITY ATTN W. D. BROWNASTROPHYSICS DEPARTMENT ATTN S. HUMPHRIES, JR.SOUTH PARK ROAD ATTN G. ROHWEINATTN A. HALL ATTN Ko PRESTWICHOXFORD--OXI 3 RQ, ENGLAND 4244 SANDIA LABS

ALBUQUERQUE, NM 87115OXFORD UNIVERSITYDEPT OF ENGINEERING SCIENCE SCIENCE APPLICATIONS, INCPARKS ROAD ATTN E. P. CORNETATTN L. M. WICKENS ATTN A. DROBOTATTN J. E. ALLEN ATTN E. KANEOXFORD, UNITED KINGDOM 1710 GOODRIDGE DR

PO BOX 1303PP1YSICS INTERNATIONAL MCLEAN, VA 22012ATTN B. A. LIPPANNATTN R. D. GENUARIO SCIENCE APPLICATIONS, INC2700 MERCER ST ATTN F. CHILTONSAN LEANDRO, CA 94577 1200 PROSPECT ST

PO BOX 2351PRINCETON UNIVERSITY LA JOLLA, CA 92038ASTROPHYSICAL SCIENCESPETON HALL SERVICE DE CHEMIE PHYSIQUELLATTN R. KULSRUD ATTN R. BALESCUPRINCETON, NJ 08540 CAMPUS PLAINE U.L.B.

CODE POSTAL n° 231PRINCETON UNIVERSITY BOULEVARD DU TRIOMPHEPLASMAPHYSICS LABORATORY 10 50 BRUXELLES, BELGIUMATTH R. WHITEPRINCETON, NJ 08540 SPECOLA VATICANA

ATTN W. STOEGER, S. J.1-001 20 CITTA DEL VATICANOITALY

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SRI INTERNATIONAL US ARMY ELECTRONICS RESEARCH &ATTN G. AUGUST DEVELOPMENT COMMANDATTN C. L. RINO ATTN COMMANDER, DRDEL-CG333 RAVENSWOOD AVE ATTN TECHNICAL DIRECTOR, DRDEL-CTMENLO PARK, CA 94025 ATTN PUBLIC AFFAIRS OFFICE, DRDEL-IN

STANFORD UNIVERSITY HARRY DIAMOND LABORATORIESINST PLASMA RES ATTN CO/TD/TSO/DIVISION DIRECTORSATTN P. A. STURROCK ATTN RECORD COPY, 81200STANFORD, CA 94305 ATTN HDL LIBRARY, 81100 (3 COPIES) le

ATTN HDL LIBRARY (WOODBRIDGE), 81100STERRERWACHT-LEIDEN ATTN TECHNICAL REPORTS BRANCH, 81300ATTN C. A. NORMAN ATTN LEGAL OFFICE, 97000 " "ATTN P. ALLAN ATTN CHAIRMAN, EDITORIAL COMMITTEE2300 RA LEIDEN, THE NETHERLANDS ATTN MORRISON, R. E., 13500

ATTN CHIEF, 21000TEL-AVIV UNIVERSITY ATTN CHIEF, 21100DEPT OF PHYSICS & ASTRONOMY ATTN CHIEF, 21200ATTN G. TAUBER ATTN CHIEF, 21300TEL AVIV, ISRAEL ATTN CHIEF, 21400

ATTN CHIEF, 21500TELEDYNE BROWN ENGINEERING ATTN CHIEF, 22000 -a

CUMMINGS RESEARCH PARK ATTN CHIEF, 22100ATTN MELVIN L. PRICE, MS-44 ATTN CHIEF, 22300HUNTSVILLE, AL 35807 ATTN CHIEF, 22800

ATTN CHIEF, 22900INSTITUTE FOR THEORETICAL MECHANICS- ATTN CHIEF, 20240

RUG ATTN CHIEF, 11000KRIJGSLAAN 271-S9 ATTN CHIEF, 13000ATTN F. VERHEEST ATTN CHIEF, 13200B-9000 GENT, BELGIUM ATTN CHIEF, 13300

ATTN CHIEF, 13500UNIVERSITY OF TENNESSEE ATTN CHIEF, 15200DEPT OF ELECTRICAL ENGINEERING ATTN BROWN, E., 00211ATTN I. LEXEFF ATTN SINDORIS, A., 00211ATTN J. R. ROTH ATTN GERLACH, H., 11100KNOXVILLE, TN 37916 ATTN LIBELO, L., 11200

ATTN LOKERSON, D., 11400UtEA UNIVERSITY ATTN CROWNE, F., 13200 '

DEPT OF PLASMA PHYSICS ATTN DROPKIN, H., 13200 "ATTN J. LARSSON ATTN LEAVITT, R., 13200 $.%ATTN L. STENFLO ATTN MORRISON, C., 13200 'S-90187 UMEA ATTN SATTLER, J., 13200SWEDEN ATTN WORTMAN, D., 13200

ATTN KULPA, S., 13300UNIVERSITY OF WASHINGTON ATTN SILVERSTEIN, J., 13300DEPT OF PHYSICS ATT FAZI, C., 13500ATTN M. BAKER ATTN LOMONACO, S., 15200SEATTLE, WA 98195 ATTN J. CORRIGAN, 20240 "-

ATTN FARRAR, F., 21100WEIZMANN INSTITULE ATTN GARVER, R., 21100 .*-DEPT OF NUCLEAR PHYSICS ATTN TATUM, J., 21100ATTN AMRI WANDEL ATTN MERKEL, G., 21300REMOVOT, ISRAEL ATTN MCLEAN, B., 22300

ATTN OLDHAM, T., 22300 .WESTERN RESEARCH CORP ATTN BLACKBURN, J., 22800ATTH R. 0. HUNTER ATTN GILBERT, R., 22800225 BROADWAY, SUITE 1600 ATTN KLEBERS, J., 22800SAN DIEGO, CA 92101 ATTN VANDERWALL, J., 22800

ATTN ROMBORSKY, A., 22900

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9.' ' 9 'o ,''."% ° 4" ' -'. -. "•"

DISTRIBUTION (Cont'd)

HARRY DIAM4OND LABORATORIES (Cant' d)ATTN DAVIS, D., 22900

ATTN GRAYBILL, S., 22900ATTN HUTTLIN, G. A., 22900

ATTN KEHS, A., 22900

ATTN KERRIS, K., 22900

ATTN LAMEB, R., 22900ATTN LINDSAY, D., 22900ATTN LITZ, M., 22900ATTN RUTH, B., 22900ATTN STWqART, A., 22900ATTN SOLN, J., 22900ATTN WHITTAKER, D., 22900ATTN ELBAUM, S., 97100ATTN BRANDT, H. E., 22300

(60 COPIES)

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