REFERENCEICTP/64/1^

MAR 1365'

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

ON RADIATION PROCESSES IN PLASMAS

T. BIRMINGHAM

J. DAWSON

C. OBERMAN

1964PIAZZA OBERDAN

TRIESTE

ICTP/64/15

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON RADIATION PROCESSES IN PLASMAS

T. Birmingham*

J . Dawson*

C. Oberman

(To be published in Phys. Fluids)

^Princeton University, Plasma Physios Laboratory,Princeton, N.J., U.S.A.

Part of this work was supported under contractAT(3O-l)-1238 with the US Atomio Energy Commission

TRIESTE

November 1964

Abstract

The problem of arbitrary test current sources embedded in an

infinite spatially homogeneous Vlasov plasma is considered. By computing

the rate at which the test sources do work on the plasma, the energy emis-

sion into transverse and longitudinal waves is obtained. A variety of

radiation problems can be handled by appropriate choice of the current

sources. Use of the time varying dipole moment due to encounters

between shielded electrons and ions leads to expressions for the brems-

strahlung spectra. If the currents are produced by the interaction of waves

with density fluctuations, the scattering and coupling of waves is obtained.

By equating the rate of emission of longitudinal or transverse waves to

their rate of absorption, we obtain expressions for the steady state energy

densities of such waves.

On Radiation Processes in Plasmas

by

T. Birmingham, J. Dawson, and C. ObermanPlasma Physics Laboratory, Princeton University,

Princeton, New Jersey

I. INTRODUCTION

In recent years considerable attention has been directed

toward the problems of the interaction of.radiation with and the emission

of radiation by plasmas. The earliest work ' on bremsstrahlung

considered only bare binary interactions and neglected the collective

aspects (proper shielding of the colliding particles and the modification

of the emitted waves due to the dielectric properties of the plasma).

3 4 5

Likewise work on the scattering ' ' of radiation by density fluctua-

tions in plasmas has generally neglected the effects of the plasma

dielectric properties on the incident and scattered waves (the longi-

tudinal dielectric properties were used,hpwever7to compute the density

fluctuations).6 7

In previous work two of the authors ' {hereafter denoted by

D. O. ) obtained the bremsstrahlung emission coefficients from a

thermal plasma. This calculation utilized a simple (but valid) model

of the plasma to compute the absorption coefficient and then invoked

- 2 -

Kiirchhdff'.sllawto obtain th.e emission. This work took proper account

of the longitudinal and transverse dielectric properties and included

the coupling of longitudinal and transverse waves and the scattering

of longitudinal waves by density fluctuations.

8.9,10Numerous attempts have been made to treat the plasma

radiation problem from a kinetic description of both the particles and

the radiation field. Only recently Dupree, utilizing a modification

of the Klimontovich formalism, has been successful in this attempt

by proceeding to second order in a systematic expansion in the plasma

parameter. His results on bremsstrahlung are in agreement with

those of D. O.

It is the purpose of this paper to show how a number of radia-

tion problems can be treated utilizing a simple model. The term

radiation as used here includes both transverse and longitudinal

waves. The radiative processes considered here are: (1) the emis-

sion of radiation due to particle encounters (this is a generalization

of the usual bremsstrahlung calculations and includes both transverse

and longitudinal wave emission), (2) scattering of transverse and

longitudinal waves by density fluctuations, (3) the coupling of longi-

tudinal and transverse waves by density fluctuations, and (4) the

generation of transverse radiation by the interaction of longitudinal

waves with each other.

-3-

The calculations are given in the classical limit, i.e. when

f\ id £^$- I • This approximation is valid for a wide range

of frequencies for hot thermonuclear plasmas. The correct

quantum mechanical treatment (even in the presence of a uniform

steady magnetic field) including effects due to the uncertainty

and exclusion principles can he given a parallel treatment

along the lines of Oberman and Son utilizing the reduced

Wigner distribution. We do not pursue these quantum mechanical

modifications further here.

Our procedure is the following. First we com-

pute the radiation, both transverse and longitudinal, from test

sources embedded in a Vlasov plasma. We then assume that the

sources are the currents due to accelerated particles, where the

acceleration is due either to encounters between particles or

has its source in coherent waves propagating in the plasma.

The phase relations between the fieldsproduced by different

particles must be taken into account here. The acceleration

due to encounters is obtained from the fluctuating electric

field which a particle sees. The principle of the superposition

of dressed particle fields due to Hubbai

is used to obtain the fluctuating field.

of dressed particle fields due to Hubbard and Rostoker

Our procedure amounts to carrying Hubbard's

method for obtaining the fluctuating fields in a plasma one

step further in the plasma expansion. Hubbard computes these

fields by adding up the shielded fields due to all the individ-

ual particles moving along their straight line orbits, now

taking the shielded particles to be uncorrelated. We add to

these fields the shielded fields due to the accelerations of

the particles. The radiative processes we oonsider have

their origin in the accelerative corrections to a particle's

orbit. For. bremsstrahlung the accelerations result from

shielded binary interactions and are thus of higher order in

the partiole interaction expansion. The secondary fields

produced by these events are themselves in turn modified1 ;

(shielded)by the dielectric properties of the plasma.

This work was to a large extent stimulated by the approach of

Mercier to the problem of bremsstrahlung. Mercier computed

radiation from the fluctuating dipole moment per unit volume as

14obtained by Hubbard. For plasmas free of static magnetic fields

this is the1 dominant radiative process. We shall see that this is

equivalent to summing up the radiation due to individual shielded

electron ion encounters.

- 5 -

II. RADIATION FROM A TEST SOURCE

We consider an infinite homogeneous Vlasov plasma consisting

of mobile electrons with an isotropic velocity distribution and an

infinitely massive ion background. We consider the perturbation

induced by given charge and current densities p (r,t) and j (r,t)s —* — * s ^

(i. e. , p , j are not regarded at this time as belonging to the

plasma). We take the fields in the plasma to be given by the linearized

Maxwell-Vlasov equations:

£ ^2--.«. (2.1)

SV . E s -47Ten ff d3v + 4ir p , (2.(2)

•""• "™*• O J S

= o .

T7 v TT — -Z. — —

- - " " c at ' (2.4)

8 E 4 7 T e tor'VXB=--— fvd j v+- j . (2.5)— — c 9 t c . J — c — s

Here n • and f are the unperturbed electron density and

distribution function with f normalized to unity. The small damping

term ef has been introduced in the usual way only for mathematical

convenience in deciding the proper paths for contour integration and

is ultimately put to zero. If we now Fourier analyze, in both space

-6-

a n d time',--'w4..obtain"..." •8f

f(k. cu. v) = 2 1 dIm 0) - k . v + i€

k . JE(k,OJ) = 4lTijn J f (k , t t , v ) d3v -J ( k , , v

k . B(k,0>) = 0

kXE(k,0)) a — Ii(k, (2.(9)

(jj 47T i f ( 3 •k X B ( k , O J ) = - - E ( k , u i ) + — b . W f ( k , w , v ) v d v - j (k , w) . ;.

(2.10)

We now decompose E and j into longitudinal and transverse

parts and write, for example ,

E(k,W) = E{k,O>) . k k - k X k X E ( k . W ) , (2.(11)

where; fc ;is. a' unit ihithe direction of k . We thus find

k . E(k,W) = -

k XE(k.W) =s

477

47T

i p

D L

id )

(k,

k

: , W )

W)

x i s

i

CD)

. 2 2k c

(2.(13)

where the longitudinal dielectric function D (k, W) is given by0

DT (k.CD) = 1 +OJ - k . v + i €

d 3 v

- 7 -

and the transverse dielectric function D (k,OJ) by

2 (a 2W f f

2. 2 . 2 2 f tt - k • v + i€c k k c J — —

We have assumed the sources satisfy charge continuity to write

Wps(k,W) = k • iB(k,W) in (2. 12).

We now compute the energy emitted by the source. That is,

we compute

W = lira / I E(r,t) • i (£. t ) d3r dt . (2.16)

By Parseval's theorem Eq. {2.16) can be written as

W = —^-rl /EKk.W) • j_V,W) d3kdC0 , (2.17)Z(Z-n) JJ

while the energy emitted per unit frequency is

r/ E(k.«) " j (k,

W = ^— I E(k.«) • i Jk.W) d^k . (2.18)

'; can: ahd"'&hall "rjestriciv onrsi«lves. to the.- ; i.-. " :. .

case of W > 0 , However the Kour ie r t ransform: : is defined for

_oo < 0) < +oo . \Wejaaakerth£s;cliangeiby nfr.ultLpiy^ng Eqtf L(2. L7)by afa'ctor

of 2 andhefeaftJeat^ntftTE|4ret W as | C4J | .

When the value's of the e lec t r i c field a r e substi tuted f rom E q s .

(2.12) and (2.13)

f J k ' ^ ! 2 « l £ X i <k,W)|2[ 3^ T - ^ ? d k '

D (k.iO) k c D (k.W) . -(2.19)

- 8 -

Of particular interest is the case where j is the current

density of a test dipole of moment p embedded in the plasma,

j (r,t) = p(t) 6 [r - r j .— g — — — — Q

(2.20)

When Eq. (2. 20) is Fourier decomposed and the result inserted in

Eq. (2.18), the energy spectrum reduces to

.2 r i i k . p ( w > r o>F i k . p ( u ) | 2 - k 2 i p

k4 c 2 D T (k.w)

The energy emission at frequency W and wave number k is

given by the integrand of Eq. (2. 21), viz,

2

w - .W[|k- p(OJ)|2-k2 | pi

(2.22)D (k.W) k c DT(k,OJ)

We note from Eq. (2.13) that the transverse wave is always polarized

with its electric field in the plane determined by k and j_

Equation (2. 21) gives the total energy expended by the dipole.

This includes the energy transfer to individual plasma electrons by

encounters with the dipole (witness the logarithmic divergence of the

first integral of (2. 21) for large k, see D. O. ) as weiU^stheemis&toaib:

transverse and longitudinal waves. The wave emission is manifested

by the contributions to the integral (2. 21) from the spikes occurring

because of the near vanishing of D and D for certain values of

k and d) (see Fig. 1). This wave emission is obtained only for

-9-

phase velocities large compared to the electron thermal velocity.

Hence to obtain these contributions we may utilize the asymptotic

forms of D and D for kv /co « 1 (where v is the rms value of v)

y^ f2 T , (2.23)c k c k

a>2 3 k 2 u 2 w 2

D.(k,W) 1 j — +L or id L

(2.24)

Here Im D and Im DT a re small slowly varying functions of k

and CO in the regime considered. Equation (2. 23) admits of solutionfor all W > 0) , while D. O. have found that the simultaneous nea r -

P

vanishing of the rea l and imaginary par t s of Eq. (2. 24) occurs only in

the approximate frequency range U> _< CA) < 1.4 CO. (for an

equilibrium plasma).

Integration of Eq. (2. 21) over the resonances yields for the

wave emission at frequency to

The upper coefficient represents the energy emission in longitudinal

oscillations, while the lower describes the transverse spectrum.

While the longitudinal emission is restricted to a narrow band

near the plasma frequency, in this regime its effect dominates that of3

transverse waves by a factor of order (—)o • . .

-10-

III. FLUCTUATIONS IN THE CURRENT DENSITYOF A SYSTEM OF CHARGES

The radiation emitted by a plasma may be written in terms of

effective current sources within the plasma. (This can in fact be

taken as a definition of the effective current sources. ) We hypothesize

that these current,sources are just the currents arising from the

accelerated motion of the charges due to their interactions with the .

other particles of the plasma. (We also include acceleration due to

waves propagating through the plasma when we consider scattering

and coupling of waves, ) For very low plasma densities, where the

plasma does not influence the radiative processes, this is clearly

correct. It has also been shown to be correct for the Cerenkov

emission of longitudinal waves by fast particles moving along their

straight line trajectories. '

The current produced by a system of n charges is

n.} 6 (3.1)

where q« is the charge on the i'c particle.- and vJt) and £»(t)

are its velocity and position. If (3 .1) is Fourier analyzed in space

we obtain

j(k,t) = S q. v.(t) exp{-i > • r .(t) } . (3.2)

-11-

Now as mentioned above if straight line orbits are used in (3.1), one

obtains Cerenkov radiation of longitudinal waves. (The transverse

waves have phase velocities greater than light for the case we con-

sider, so there is no Cerenkov emission of these waves.) Since we

are here primarily interested in radiation due to particle acceleration,

we shall not consider this but look directly at the j resulting from

the acceleration. Thus we look at d^(k,t)/dt .

d _j(k,t)/dt = Sn -.S q£ F jr^ + v£ — exp{- i k •• _r (3. 3)

The dominant part of the vfl d/dt term in (3. 3) arises from the

straight line motion of the particles.

-12-

IV. THE EMISSION OF LONG WAVELENGTH RADIATION

Equation (3. 3) is the general equation for the rate of change

of j_ with respect to time and is too complicated to employ directly.

We shall begin by looking at radiation at long wavelengths or small

k1 s. This radiation is generated by the small k part of _j . To the

extent that we are looking only at the radiation produced by the

acceleration, the vfl d/dt term in (3. 3) can be neglected (its ratio

to the first term is roughly k • v./w, see Appendix I). Substituting

q« E_(£A/m. for vp we thus write (3. 3) as follows

2d,j_(k,t) n q— ~ = S -JL—E(r.) exp{- i^ k- rAt)} . (4.1)

1=1 £ "

Let us first consider j£ as being produced by particle en-

counters. We write for JE(r^) •

where E-,-, is the elecitr'asfatic fiield atv^o'iinsol

2

dt

r . i % •exp{- i; k • r £(t)} - - ^

(4. 3)

-13-

where use has been made of the fact that

q£—IV ~~q£'— VI . (4.4)

• Now the dominant contribution to E(rfl) will come from particles

within afewDebye lengths. This will be particularly true when we

analyze in CO , since distant encounters do not give high frequency

contributions to E . For those £ and V which contribute to (4. 3)

we may treat k • £ . as equal *o _k • _r-, to lowest order in k.

Thus we write

dj_(k,t) l

dt 2 iv i iV

rq4m £ -

e x p { -• i k - £ ( t ) } . (4.5)

We see immediately that the interactions between like particles make

no contributions to (4. 5). Equation (4. 5) amounts to the dipole

17approximation used by Mercier. The like particle interactions go

out because they produce no net acceleration of the dipole moment*

If we have electrons and one species of ions of charge Ze

and mass M then (4. 5) becomes

2 Z m e r i

(i £) S E exp{ -. i_k • £ . (t)} , (4. 6)dt m " ' M. ' . —ie I e l

where sums are over electrons e and ions i and _E. is the elec-

tric field at the i ion due to the e electron.

-14-

We now take the ions to be infinitely massive so that the ion

positions are no longer time dependent. The source term which we

will insert in Eq. (2.19) thus becomes

j (k,OJ) = + — = —— 2 E. exp l - i k • r..} .— s — (JO m V — i e ••v — —l

e ie(4 .7)

In accord with the superposition principle of dressed particles

of Hubbard and Rostoker, ' the total fluctuating electric field of

all the electrons is the sum of the shielded fields (shielded by elec-

trons only) from all electrons when these electrons are treated as

statistically independent. We consequently approximate Eq. (4. 7) by

i Z e "•* r ij (k.OJ) = - - S EL exp { ; - ik - r.. } , (4.8)

— s — m W — i e r • — —i• J x 'e i e

""• t hwhere E. is the shielded electric field of the e electron at the

—ie

r t , .th .site of the l ion.

We now insert Eq. (4.8) as the source in Eq. (2.19). The

wave emission is then extracted by local integration over the reson-

ances. Taking an ensemble average of this result over the ions and

electrons, we obtain

2 4 •?

87T m P

e

-15-

k- L ,

1 1

ni

ee

ee

(4.9)

As in Sec. II, the upper coefficient represents the emission of

longitudinal waves and the lower the emission of transverse. The

remaining integration in each case is to be carried out over all direc-

tions of emission. The resonant values of ) jc | and | _k | are

determined by the near vanishing of Eqs. (2. 23) and (2. 24) respectively

and are given by

{co2 - u Z)(4.10a)

o a2 2 ^

(or - a; )P (4.10b)

-16-

To obtain the shielded field of an electron, we solve the test

particle problem of an electron moving through an infinite uniform

plasma with fixed ions. The plasma is described by the Vlasov

equations (2.1 - 2. 5), where the source is an electron moving uniformly

along a straight line. We retain only the longitudinal electric field,

the t ransverse interaction being relativistically small and therefore

negligible if the thermal energy is small compared to the electron

res t mass . The longitudinal electric field is obtained from Eq. (2.12)

and is given by

4irik p (k, CO)E (k.U) = - — - . (4.11)

6 k DL(k,W)

The source charge density is given by

p s - e f i ( r - r - v t ) , (4.12)s — —eo —eo

with r and v the init ial position and velocity of the t e s t pa r t i c l e .—eo —eo

F o u r i e r analys is of Eq. (4.12) gives

p (k, CO) = - £7Te expf- i k • r 1-5(0) - k • v ) . (4.13)s,— ^ — ~™ecj *~" —eo

If we now subst i tute Eq. (4.13) into Eq. (4.11) and inver t the k

t ransform, we obtain

6(U>- k- v )— —eo •

(4.14)

-17-

To evaluate 4.9 we must compute (exp{ik- (r. - r.r)} E j r . Ui)

E , (_r.uw)) The superposition principle of dressed particles insures

the statistical independence of the E_' s. In order to perform the

average over i and i1 we must know the ion correlations. D. O.

have pointed out that the presence of non-thermal ion density corre-

lations, as may be induced, for example, by large amplitude ion

waves or instabilities*can significantly enhance the absorption

coefficient and consequently, in the steady state, the wave emission.

This can also be seen from Eq. (4.9),

1 1 ,

8ff m " J -L,T^6

3(3)V3

" \

(4.15)on density and the ensemble average is to be

over ion positions.

where n is the ion density and the ensemble average is to be extended

As an example of the procedure by which ion correlations can

be included in the formalism we have worked cotttiU ito -j. Appendix II

the case in which the ions are thermally correlated. In the present

context, however, we assume that the ions are uncorrelated.

With this assumption of uncorrelated ions, the only contributioni

to Eq. (4. 9) occurs when i = \ . After integrating over all directions,

-18-

the wave contribution is thus given by1

i f 9(3)2U) u

3Wc3

the summations to be carried out over all ions and electrons of the

plasma.

The wave emission from a single encounter can be obtained by

extracting one term from the double summation appearing in Eq. (4.16)

and inserting the expression for the shielded field derived as Eq. (4.14).

We do so, choosing the origin of time to be that time corresponding

to closest approach in a collision; hence (r. - n. ) is the impact

parameter, b. , in the straight line approximation. The result is

w -

6 ( o ) - k - v ) 6 ( w - k | . v )' (4.17)— —eo — —eo' • * '

The total number of e lectron- ion collisions per unit t ime per

unit volume, charac ter ized by impact pa ramete r b . and electron

velocity v > is

dN = n n I v I dA b. d b . f ( v ) d 3 v . (4.18)+ o ' - e ' r le le — e e •

-19-

In Eq. (4,18) <{> r e p r e s e n t s the orientat ion of b. in the plane t r a n s -

v e r s e to v , and n and n are the average number dens i t i e s of—e + o

ions and electrons respectively. :

The total power radiated per unit volume is obtained by

multiplying Eq. (4.17) by Eq. (4.19) and performing the indicated

integrations.

- 3/l/9(3)*« UQ

5'

\2/3O) c3

/ d b . b . J | d k d k 'XG

Z e n n

2TT m

b ]6 ( W - k - v ) 6 ( w - k ' - v ) .

(4.19)

The geometry of the problem is depicted in Fig. 2. The 6-

functions occurring in Eq. (4.19) insure the identity of the components

of k and k1 parallel to v ' , since

6(0J - k • v ) 6 ( 0 ) - k 1 - v ) a

" 6 " ""e

- k- k • v ) .

e (4. 20)

Interchange of the k , k' with the $ , b. integrations generates the

two dimensional 6-function,

d0 / db. b . expfi (k - k«) • b . l - (2ff)2 5(k. • k ' ) . (4.21)i ie ie P — — "- iej ~ x —x

o o

-20-

Eq. (4.19) therefore simplies to

\ 2 /3 0)-v i 4Z2e n+n£

d3k d3v f(v )e —e

7T m

5(W - k • v )

D.(k,W)li-i

{4. 22)

The transverse part of Eq. (4. 22) ip in agreement with the

results of Mercier. For the thermal equilibrium case, it reduces

to the expression obtained by D. O.

-21 -

V. SCATTERING AND COUPLING OF WAVESBY DENSITY FLUCTUATIONS

The electrons of a plasma are accelerated by waves propagating

through it. The accelerated electrons in turn emit radiation (both

transverse and longitudinal) in accordance with Eq. (2.19)- If the

plasma were completely uniform the radiation emitted would simply

reproduce the wave (Huygen's principle). However, the presence of

density fluctuations results in scattering and coupling of the two types

of waves.(Appendix V).

Since only the electrons are appreciably accelerated, only .

electron density fluctuations are involved. The electron density

fluctuations can have their origin in a number of ways. First the elec-

trbns tend to follow the ions so as to maintain charge neutrality and so

there will be electron density fluctuations associated with ion density

fluctuations. Second there are electron density fluctuations due to the

random motion of the electrons. These are important for scattering of

waves whose wavelength is short compared to a Debye length. Third

there are density fluctuations associated with longitudinal electron

oscillations.

Here we shall confine ourselves to the scattering and coupling

of long wavelength waves. We shall assume that we have a spectrum of

waves. If we take the plasma to be contained in a box of volume L ,

then we may write the electric field in the form

- 2 2 -

^(£,t) = 2 E(k) exp{.i£k • £ - 0>(k)$ . (5.1)

The waves can be either longitudinal or transverse depending on whether

J2(k) is parallel or perpendicular to k;;(W(k), of course, • depends-on

the type of wave we have).

In the long wavelength limit the current produced by the accelera-

tion of the £th electron is

ie2_E(k).(5 .2)

e —

The _k Fourier component of J_ due to all electrons is given by

ie2E(k')

Now ,

S exp^-ik • r .(t)} = n <k,t) ,

S. ~ " £ e " . . . - .

where n (k,t) is the Jk. Fourier component of the electron number

density. We may thus write Eq. (5. 3) in the form

& j ' 1 " ) t • (5,4)(e ^

Fourier analyzing j(k,t) in time gives

)) ' (5*5)w > t t S mo>(k') n e { - " - • w

Substituting (5. 5) into (2.19) and extracting the wave contribution by.

integration over the resonant values of |k | , we obtain

1/2-23-

w(Ji8ir2m 2

e

d Q ,L,T

k « E ( k ' ) n (k - k 1 ,™ " X J *—•• • * • e • •—XJ —

A) - W(k')) k

JE ( - k " ) n (-k +k l l ,W{k")- W(k ))*™ — e ~"j_j — — — 'JLJ

<»3/2V - -W k X E ( k ' ) n (k - k ' , W{k ) - « ( k ' ) ) k X E ( - k » ) a ( - k _ + k " , CU{k")-OJ{k ))

~ x "~" ~— e — x •"• ^ x — ~*X' ~— — e ~ ~ X " ~ ~ — x

c 3 cu(k') a;(k")

The integration is to be carried out over angles of longitudinal emission

dSl, for the upper multiplying factor.and angles of transverse propa-

gation dfy for the lower. The values of |k | and jk j are those

given by Eqs. (4.10a) and (4.10b).

If we ensemble average (5. 6) and assume that waves of different

k are randomly phasedy then the,, only .terras ,whichidontribute: to (5. 6) •

are thos'e'with ^k'. equal to k" .-

1/2

87T2mk'

.Ak. | n (k - k ' , 0>(k

c3cu2{k»)|k XE(k')l |n (k - k k ) (5.7)

If the box size is taken very large then the sum over k' may be con-

verted into an integral in th^usiiahway..

-24-

Equation (5. 7) gives the total energy emitted at frequency (t) . If we

further assume the process goes on from t =-T/2 to t s T/2, then we

obtain the average power emitted per unit volume by dividing (5. 7)

by L3T ,

e V -647T m

2

T ^ — dfik d

U>

3 3 / 2 U o3

W V ) &• = e —J_I — — Li

3 2c W ( k 1 )

- k' , w(kT) (5.8)

2In general |n (k,W)| will be proportional to T , as we shall see

shortly for some special examples, and hence P is independent of T

For frequencies OJ much higher than the plasma frequency

where dielectric shielding can be neglected, that part of Eq. (5. 8)

which represents the scattering of transverse waves reduces to the

results obtained by Rosenbluth and Rostoker. However, the dielectric

corrections are important near the plasma frequency.

We now proceed to consider the particular cases where the

electron density fluctuations appearing in Eq. {5. 8) result from (1)

electron neutralization of ion density fluctuations and (2) longitudinal

electron, oscillations on the plasma.

- 2 5 -

Case I: Electron Neutralization of Ion Density Fluctuations

The electron density fluctuations can here be related to the ion

density fluctuations. We have done so in Appendix III, once again disre-

garding ion motions so that the perturbed electron distribution (perturbed

by interactions with discrete ions) is independent of time. The result is

277 Z CO n (k)

n (k, w) =- n_ / f(k, to) d v = -2- f — r. ~ d v . (5.9)

where n (k) is the Fourier transformed ion number density, F is the

one dimensional electron distribution normalized to unity, D_ (k,o) isXJ

the static longitudinal dielectric function, and the principal value of

the integral over v is to be taken.

If we insert Eq. (5. 9) for the electron density fluctuations, Eq.

(5. 8) then gives the average power per unit volume scattered and coupled

by such density fluctuations.2 4, 2 ,., 2 1/2,., 4

16TT m Te

r

( 3 ) " u

2

oo

c3 ai2(k')

\

'd3k'

n , ( k T - k ' ) 6 (CO(k ) - C O ( k ' ) )^ J _ j —

(k kT —XJ —

. 0)

(5.10)

-26-

Here the limiting procedure which is formally represented by 6 (CJ - U> )

Tis to be interpreted as — 6(0) - W ) where T is the duration of the

emission process (cf. Appendix IV). The power emission is thus seen

f 3to be independent of T . For a single wave the integral d k1 is

replaced by

If the ions are uncorrelated, the ensemble averaged value of

| n (k - k ) | is just equal to N , the number of ions in the volume

L being considered. However, as mentioned in conjunction with the

bremsstrahlung calculation, the presence of strong ion correlations (ion

7waves) can enhance this factor by many orders of magnitude.

Equation (5.10) is valid for any isotropic distribution of electrons.

For the particular case where F is a Maxwellian,

CLBFv 8v

dv = - — 2 , (5.11)u

and

2D_(k,o) = 1 + iL , (5.12)

L k20)

where K - — is the reciprocal Debye length.uo

Thus for a single wave with wave vector k and frequency U>(k )

interacting with density fluctuations associated with the interaction of a

Maxwellian electron distribution with discrete ions, we can write Eq. (5.10)

a s

- 2 7 -

Z 2 e1/2

477 m OJ(k )e x — '

^VI T X - ( k o )

k )—Li

w(k )—O

k )— 1

)—O

(5.13)

For long wavelength radiation | k_ - k | « K and hence can—Lt, T —O

be neglected in evaluating the integrals in Eq. (5.13). Doing this and

integrating over frequencies, we pick up only the resonances at

oJ = w ( k J .

Z 2 e 4 (U>2(k ) - 0) 2 )= 5 2

1/2

nm (e —o

+ l B

wv

3 c"

(5.14)

If E{k ) represents a longitudinal wave, the upper coefficient

gives the wave scattering, while the lower gives the coupling to trans-

verse waves of frequency C0(k ) = 0J . The energy density per frequency

interval of a longitudinal wave of frequency W(jc ) and wave vector k

can be expressed in terms of its electric field amplitude by

- 28 -

2 ^ ' 2dk, , k. 2 dk.

(5.15) .

The first term in Eq. (5.15) represents the energy density associated with

the electric field, while the second is the energy density of particle motion.k 2 dk

The factor •••• ' represents the density of k states per. frequency

interval. If we sum. Eq. (5.14) over all waves in a frequency intervals

ddJ , we determine the scattered power to be

2 2 W 1/2T/z — 5 ( w " w _ ) CT(O))dw , (5.16)0) 172 3 p

L 9(3) rn u p

e o

while the power coupled to transverse waves is

7 2 CD 2 2P w ~da> = " T ~ 3 <to - O SL(W)dw. (5.17)

T m e p

e

In deriving Eq. (5.17) we have noted the fact that the longitudinal

wave couples to only one of the two transverse polarizations, viz. to

that wave polarized in the plane determined by the wave vectors of the

incident longitudinal and outgoing transverse waves. D. O. have derived

these results from absorption calculations and detailed balance argu-

ments. Our results are in agreement with theirs.

In a very similar manner we can compute the scattering and

coupling when E(k ) is a transverse wave. For a given polarization••MB **m^J

18the energy density of such waves is

Z k 2 dk£T<W» * . = , ; I E(fcT) | -w- - 5 5 - d» . (5.18)

-29 -

Equation (5.14) includes the energy associated with the electric field,

magnetic field, and particle motion. The latter two combine to give a

contribution just equal to that due to the electric field alone.

From Eq. (5.18) we find the power coupled to longitudinal waves

to be

_ 2 CO l / 2p dtu = Z e j - £ <u>2 - CO 2) S (co) dco , (5.19)

L 9(3)1/2 m e u o3 ?

and that scattered as transverse waves (again remarking that the prevail-

ing transverse wave can couple to only one of the two possible polarizations

of the scattered transverse wave) to be

2 U 2 (CO2 - CO 2 )Pco daJ = J ^ - - Z — 3 cT S T ( ">d"« (5.20)

L m ee

19The latter two results are in agreement with those obtained by Berk,

who extended the D. O. model to include finite wavelength as well as

electromagnetic corrections to the absorption coefficient.

We remark in passing that by applying the principle of detailed

balance to a thermal equilibrium plasma, we may equate the coupled

powers, Eqs. (5.17) and (5.19), to find that the steady state wave energy

densities, £ (CO) and £_(&;), are in the ratioLi X

L Uo 6{3)

and that we have equipartition, viz.

- 3 0 -

|E(k )| = lE(k )| . (5.22)

In Eq. (5. 21) £ (CiJ) includes both polarization of the transverse wave.

We shall verify Eq. (5. 21) directly in Sec. VI by balancing emission

against absorption.

-31-

Case II: Longitudinal Electron Oscillations on the Plasma

As a second example illustrating the application of Eq. {5. 8)

we look at the generation of transverse waves by the interaction of two

or more longitudinal waves. This problem has been investigated by

20 21Sturrock and quite recently by Aamodt and Drummond as well as

T, * 22Boyd.

These latter authors solved the problem of the coupling of

longitudinal to transverse waves in an infinite homogeneous collision-

less plasma. They considered only the case where the longitudinal

wave energy is much greater than the transverse wave energy so that

they could neglect the interaction of the transverse waves back on the

longitudinal waves or with each other. Aamodt and Drummond computed

the rate of increase of electric and magnetic field energy in the trans-

verse waves and equated this to the rate of emission of radiation. This

is actually in error, for there is also particle kinetic energy associated

with these waves. This correction is included in our calculations.

The density fluctuations appearing in Eq. (5. 8) are now those

electron density fluctuations associated with the passage of longitudinal

waves through the plasma. They can therefore be determined from a

solution of Poisson's equation

ik •• E (k,«)

If we assume that we have the spectrum of longitudinal waves

given by Eq. (5.1), then E (k, Ci)) is given by . ..

- 3 2 -

CD) = (27T)4 S _E(k") 6{k - k") 6{cu - CO(k")) . (5.24)k"

By combining Eqs, (5. 23) and (5. 24), we find that the quantity

n (k_ _ - k' , UJ(kT ) - W(k')) which appears in Eq. (5.8) can bee — X J , l — — 1 J > l "-

written as

k L J - k- , L / r

T - l \ T - k 1 ) ' Z E(k-) 6(k - k ' - k " ) 6 ( w ( ke — ± j , i — * j - ^ t — LJ, I — — — -Lit

e L k"|E(k")j 5 ( k L T - k' - k")fi(W(kLtT)-«(k')-w(k"))#

(5. 25)

We have already assumed that k" is the wave vector of a

longitudinal wave (and therefore k". is parallel to E(k") in Eq. (5. 25)),

We now further confine ourselves to the situation where k1 also repre-

sents a longitudinal wave. The 5-functions appearing in Eq. (5. 25)

physically represent the conservation of momentum and energy for the

"collision" of these two longitudinal waves.

k _ = k< + k" . , (5.26)— j _ , , i — —

Since k1 and k" a r e both longitudinal waves , £J(k') and

a)(k") each l ie in the range 0) < U ^ 1 . 5 W . The quantity W(k )— p p —i-r, 1

is thus at l ea s t as la rge as 2 CO . Since longitudinal waves can not

- 33 -

propagate atthis frequency, we see that it is energetically impossible

for two longitudinal waves to couple to a third. We thus drop the sub-

script L in Eqs. (5. 26) and (5. 27).

We further note that since k' and k" are much greater in

magnitude than k (except for relativistically energetic plasmas),MOT ^

transverse waves can only be produced by the nearly "head-on collision"

of the two longitudinal waves.

If we introduce Eq. (5. 25) for the density fluctuations appearing

1% «tt ' > I

in E q

J/

. (5.

e XT

4me

81. we obtain

<Ai:kTJy.

c

for the1/2

transverse

« ( ^ T >remission

d \ J d"

, » T - k ' - k " ) 6 ( k T - k - - k " ' )

fi(O)(k )-0)(k1)-W(k"))6(W{k_)-

Assuming that the k"'.s and the k1"'. s are dense, we can con-

vert the summations appearing in Eq. (5. 28) into integrals and use the

6-functions on k" and k"' to carry out these integrals. The result is

1 " 2 1/2

2 (w ( l k _ | ) - OJ ) T 6

4 m T c <2TT) J ~T

22 2.

| k - k- | |E(k - k ' ) |- T T

- 3 4 -

Since Jc1 is a variable of integration, we can rewrite the inte-

grand of Eq. (5. 29.) as

" 2|kTXE(k'}|k - k ' | | j E { k - k - ) j fi(w(k ) - w ( k « ) - c u ( k T - k ' ) )

T|k XE<k')||k - k ' | J E ( k - k ' ) |

(5.30)

In so splitting the k1 integrand we are considering in pairs the

effects produced by wave k1 scattering from density fluctuations

produced by wave k - k1 and wave k •- k1 scattering from density

fluctuations produced by wave k1 . Since E(k') and ( k™ - k1)

represent longitudinal waves, Eq. (5. 3G) can be reexpressed in the

form

k T X E ( k ' ) ||k - k ' | | E ( k -k

JkT-k'

. w ( k ' ) « ( k - k ' ) | k - k ' | | k '

T ) - W(k') -W(k T - k'.))

E(k')||^(kT-_k«

2

- k')) (5.31)

From Eq. (5. 3£) we note that since k « k1 , the two collisional

processes interfere destructively. To lowest order in the ratio

I k T | / | k' I . Eq. (5. 3£) can be written as

- 3 5 -

|k XE(k-)|

2 2

(5.32)

If we now insert Eq. (5. 32) into Eq. (5. 29) and carry out the

integration over directions of emission of transverse waves, we obtain-> . o 3/2

?=• e

2m c

e- T > T < 1 S > ' d k '

2 2J E ( k ' ) | | E ( - k ' ) |

k ) - 2w{k')),

(5.33)

2 \ T

where again 6 (W(k ) - 2OJ(k')) is to be interpreted as -|-6(W(k )-2w(k'

(cf Appendix IV). We have substituted for | k | the value given by

Eq. (4.10b).

When we now integrate over frequencies of emission Cti , we

a r e l e f t w i t h ' ..• ' . , . . ' : , . . . .• • : • ; . - • •':• • •• r , •••••: •• .• • " • • . • ..• '

, 3/2

(5.34)

-36-

If we estimate P by assuming that the average value of W(kJ )

for all waves is given by to(k' ) = OL W , where 1 < a < 1, 5 , then

_ 3 .. Z .,3/2 e20) 2 _ if , 2 2

^ • ^ 2 ( " a 2 ^ < i i > A . | E ( k . ) | J E ( - ^ | . (5.35)

6O7T m c

We see that for a ~ 1. 5 , the power emission is a factor — ( — ) ~ Z. 9

greater than for the choice of a = 1 . To achieve exact agreement

with Aamodt and Drummond, when their result has been corrected

to include particle energy density, we must choose Of S 1. 08.

-37-

VI. WAVE SPECTRA IN A STEADY STATE PLASMA

If we have a plasma maintained in the steady state, we can com-

bine our emission calculations with the D. O. absorption calculations

to find the steady state wave .spectra,., in the plasma. Some steady

state plasmas which might be considered are thermal equilibrium

plasmas and non-equilibrium but steady discharges. Here we shall

confine ourselves to isotropic electron distributions.

The absorption coefficient can be obtained from the plasma

conductivity. With slight modification, the D. O. high frequency con-

ductivity calculation is valid for any isotropic electron distribution.

They find for the conductivity

CT = or + CM1

where O =o

iO,

47TCi>

(6.1)

(6. 2)

If the ions are uncorrelated, or. is given by

22Ze dk k

1(6-3)

m uThe cut-off in Eq. (6. 3) at k

max 2 o - electron velocity)

is the usual cut-off introduced in order to prevent the divergence for

small impact parameters ( — ) .

-38-

In the paper by D. O. it is shown that the dispersion relation

for transverse waves propagating through the plasma is

2 2 2k c - w + 477 i 0) cr *= 0 . (6.4)

Substituting 0" from (6.1) and (6. 2) into {6. 4) gives

2 2 2 2 2 «-k c - W + W + 0 0 C ; = 0 .

P P 1

(6.5)

If Eq. (6. 5) is solved for W for real k , then the imaginary part of 0>

gives the time rate of decay of the amplitude of this pure k mode

(decay going as exp -(Im Ct>t)). Twice the imaginary part of W gives

the rate of energy decay in the wave because the energy is quadratic

-1 , 1in the amplitude. The e time for the energy is thus ? Trriii) ' ^ e

may solve Eq. (6. 5) for the imaginary part of CU by making use of the

fact that | Of | is small. We thus find

ImOJ = —+ (W " + k c- P

S j - (a; 2 + k 2 c 2 p 2

Hence the decay time for transverse waves is

3

TT =2 ~

0) Imor.P 1

377 m COe

2 Z e 2 U >P

maxdkk Im

(6.6)

(6.7)

where we have used the fact that Im D (k,o) = 0 .

-39-

In the steady state, the rate of absorption of transverse waves

is balanced by their rate of emission provided we can neglect the escape

of radiation from the plasma. The average rate of absorption of trans-

verse wave energy per unit volume per unit frequency interval is just

the steady state energy density £ divided by the decay time forT

such transverse waves as given by Eq. (6. 7). If we balance the absorp-

tion by the emission of transverse waves as represented by Eq. (4. 22)

we may write down the expression for £ ,

4Ze4n n CO2 l/2

^ 2 23

T me COP

d v f(ve x e ' 2 I | 2

k-lH , (.6.8)m a x , IraD,

Equation (6. 8) can be somewhat simplified by integrating the numerator

over velocity space and introducing the value of Im D obtained from

Eq. (2. 14), viz.

7T CD OJ

tax D_ \k, co) = —-E-FM-r) • (6.9)L kZ k

In Eq. (6. 9) F is once again the one-dimensional dis t r ibut ion function

normal ized to unity. Equation (6. 8) consequently becomes

*-40*

4 2, .2 • 2 . 1 / 2 ' d k

l 6 Z e 4 n n W2(O)2 - <o 2) J k | D _ ( k ,TO P "*""'

WT 3 , , 4 rk

• • - m e CO f maxf max -r.', w

dk

(6.10)

We proceed in a quite analogous fashion to determine the steady

state energy density of longitudinal waves. From Maxwell1 s equations,

we determine the dispersion relation for such waves propagating

through the plasma to be

Oj2 = a? 2(i + of) , (6.11)

where cr has been given by Eq. (6. 3). Since | (J \ « 1 , we see that

such waves propagate only at frequencies near the plasma frequency,

a fact that has been used repeatedly in this work.

The decay time for such longitudinal waves as determined by

solving Eq. (6.11) for the imaginary part of Ci) turns out to be

. . 3irm COT _ _ i _ l _ -•• e P 1

t(L 2ImW ,, T ' ~ - , - 2 _ „01 Imff. 2 Z e / max

P *

Landau damping has not been included in this calculation. For

k > 5 X , X = Debye lengthy Landau damping is unimportant.

• 4 U

One could include it in these calculations. However1 if one"does this,

Cerenkov emission of longitudinal1 waves {the inverse process to Landau ;

damping) must also;be included,•:-F!rom Eqs. (4. 22) and:(6.12) we: thiusiobtair

the"st.eady state energy level of longitudinal oscillations.

k h)m a X F ( - ^ )dk *

- 8 Z e 4 n n . _ , l/2 J k | D T ( k ( a J ) j 2

3(3)

|D (k.w )|L, p

In thermal equilibrium

JI\

Jo

k/- max

max

dk

F

k

F

DL(k,W)t2

L (k .co) | 2

2

and the transverse and longitudinal wave energy levels reduce to

) 1 / 2 (6.15)9

2

p«W(«2 « 2 ) 1 / 2

i^.co ( W2 - O J 2 ) 1 / 2 (6.16)

From Eqs. (6.15) and (6.16) , Eq. (5.21) can be immediately verified.

-42-

The density of modes p(W) is given by

k"2(o>) dk

««) = -z- si • «27T

By dividing £ J?< CU and C by P(C0) and using the dispersion rela-

Ttions (4. 10a) for longitudinal waves and (4. 10b) for transverse and

taking into account the two possible transverse polarizations, we find

the average energy per mode is

L | T = m u 2 - KT , (6.18)

where K is Boltzmann's constant.

-43-

VII. DISCUSSION

¥e have shown how many problems involving radiation oan be

handled in terms of effective current sources embedded in a

Vlasov plasma. We first compute the emission of radiation by

arbitrary test sources. By appropriately choosing these sources,

a variety of problems can be handled. If one uses as sources

the time varying dipole moment due to encounters between shielded

electrons and ions, one obtains the bremsstrahlung emission

(including the emission of longitudinal waves). If one uses as

sources the currents produced by the interaction of waves with

density fluctuations or with other waves, one obtains the scatter-

ing of waves, the emission of transverse waves by longitudinal

plasma oscillations, and the generation of longitudinal waves by

transverse waves.

The method used in this paper can be extended to other radia-

tion problems. At present we are calculating the quadrupole

radiation due to electron-electron and to electron-ion encounters.

This work will appear in a later paper. It is also possible to

extend the method to anisotropic plasmas and to plasmas in mag-

netio fields. These problems are also under investigation.

Acknowledgement

One of us (CO.) would like to thank Professor Abdus Salam

and the IAEA for the hospitality extended to him at the Inter-

national Centre for Theoretical Physics, Trieste.

One of us (T.B) would like to thank Dr. H. Berjc fo£ stim-

ulating discussions and advice.

-44-

REFERENCES

1. Scheuer, Monthly notices of the fioyal Astronomioal

Society JL2O, 231 (i960).

2. Oster, Reviews of Modern Physics Ji3_, 525 (l96l).

3. Rosenbluth and Eostoker, Physics of Fluids 116 (l96l).

4. Dougherty and Farley, Proc. Roy. Soo. (London) A259* 79 (i960).

5. Salpeter, Phys. Rev. JL20, 1528 (i960).

6. Dawson and Oberman, Phys. Fluids .5_, 517 (1962).

7. Dawson and Oberman, Phys. Fluids 6,, 394 (1963).

8. Simon and Harris, Physios of Fluids 2.t 245 (i960).

9. Fante, Prinoeton University Plasma Physios Laboratory,

Matt-195 (1963).

10. Rostoker and Simon, Nuclear Fusion: 1962 Supplement, Part 2t 761,

11. Dupree, Physics of Fluids 6,, 1714 (1963).

12. Klimontovich, Zh. Eksperim. i Teor. Fiz. _33, 982.(1957).

£~Engli&h transl. Soviet Phys. - JETP 6., 753 (l958)._J

13. ,C. Oberman and A. Ron, Phys. Rev. 130, 1291 (1963).

14. Hubbard, Proceedings of Royal Society A26P, 114 (l96l).

15. Rostoker, Nuclear Fusion JL.» 101 (i960).

- 45 -

16. Rostoker, Physics of Fluids 1_, 491(1964).

17. Mercier, Proc Phys. Soc. 83, 819 (1964).

18. Daw son, Princeton University Plasma Physics Laboratory

Summer Institute Notes (1964).

19. Berk, Princeton University PhD Thesis (1964).

20. Sturrock, Plasma Physics'(Journal Nuclear Energy, Part C) 2,

158 (1961).

21. Aamodt and Drummond, Plasma Physics (Journal Nuclear Energy,

Part C)_6. 147 (1964).

22. Boyd, Phys. of Fluids 7, 59 (1964).

-46-

FIGURE CAPTIONS

k2 1Figure 1. Graph of Im-=-—. • . against — for various values of the

L

ratio w/wP

Figure Z. Vector diagram of quantities appearing in Eq. (4. 19).

1k 642050

k 1F igu re 1. Graph of I m - — • • ••• • against r- for var ious values of the

Lra t io OJ/OJ .

642051

Figure 2. Vector diagrain of quanti t ies appearing in Eq. (4.19)-

APPENDIX I

In this appendix we are interested in determining the relative

magnitudes of the terras appearing in the equation

<U(k,t) ^

2 M } (3l3)£=1

nIn Sec. IV we considered only the t e r m ) q<* vfl expl- i k • r .( t)L

£—, JL X, ^ JL J --• • — _ .

(term, a)* We there also considered only the acceleration of electrons

and expressed j/ as the sum \ jf lt#» where v*«, represents the

acceleration of the £ ' - electron due to interaction with the £ ' ion.

{Interactions between electrons produce no net change in j{k,t) in the

dipole approximation) ) In the dipole approximation also, we assumed

that for two interacting particles whose positions are £«(t) and jr« ,(t),

k • [r«(t) - £- , { t ) ] « 1 . To this approximation we may therefore

replace the time dependent position £a(t) in the ^exponent - of Eq. (3. 3)

by the stationary position vector jr-, ( of the ion with which it interacts.

Doing this, we write the first te rm to lowest order (order 0) in the dipole

approximation as

When Fourier transformed in time Eq. (1-1) may be written as

E, \ « e x p { " i i ' *

1-2

"We emphasize once more that in adopting the dipole approximation we

have neglected terms of ©{ k • (r - -£« , ) ] • The next such term in

the expansion of the first term of Eq. (3. 3) would be

k • [(r£( - £j8J v£jfj exP[- ik . r^_Jal(k,U>) = i £ q£ k • [(r£( - £j8J v£jfj exP[- ik . r ^ j . (1-3)

Turning our attention now to the second term of Eq. (3. 3) (term

b), we reiterate the remark made in Sec. Ill that if only the unacceler-

ated electron velocities and positions are used, longitudinal Cerenkov

emission is obtained. Since we are interested in computing brems-

strahlung, , accelerative corrections to the electron orbits must be

th'considered. If we express the velocity of the £.• electron as

^ 6 ^££ . , (1-4)

where vfl is the unperturbed velocity and 6v-fl| the perturbation

produced by interaction with the I't particle (and here SL ' can

represent either an ion or another electron), the accelerative part of

the second term in Eq. (3. 3) can be written as

<yb(k,t)^ Uf ex

d-5)

pCik- r

We have, of course, assumed that ) ^Zee i <<: Zo to linearize

Eq. (1-5). * ' . •

) ^Zee i

1-3

For the purpose of comparing Eq. (I<-5) with (1-1), we take the

£ ' ' 5 to represent only ions and once again replace r.(t) by the time

invariant j : . , . If we Fourier analyze the resultant expression andiva(u;)

note that 5v..(C0) = — , we obtain

k • v

We thus see that the terms in Eq. (1-6) are in the ratio-

to those in Eq. (1-2). The phase velocities (Cd/k) of transverse waves

and very long wavelength longitudinal waves are faster than the speed of

light, so that the discarded terms are relativistically small. For very

short wavelength longitudinal waves, however, these terms may be

significant and even dominant.

It can be further shown that the currents represented by Eqs.

(1-3) and (1-6) are of comparable magnitude and when introduced into

Eq. (2.19) combine to give the longitudinal and transverse quadrupole

emission spectra. We shall show this and consider such effects in a

later paper. Electron-electron interaction effects can and should, of

course, be included at this level.

APPENDIX II

We have derived as Eq. (4.15) an expression for the total energy

emission* longitudinal and transverse, at frequency U> from collisions

between shielded electrons and ions,

2 2

m

_, . , ,1 /2 1 , W L^3(3) UQ p e

L3

W

- k .

• - T -

(4.15)

In Eq. (4.15) n is the ion density and E the shielded electric field

due to the e electron. The ensemble average is to be extended over

ion positions. Use of the shielded electric fields due to the electrons

has enabled us to neglect electron correlations.

In Sec. IV we have evaluated Eq. (4.15) for the case where the

ions are also assumed to be uncorrelated. In this Appendix we work

out the case where equilibrium correlations exist among the ions.

If we use Eq. (4.14) for E , then we can write down an expres-—e

sion for n E ,+-e

exp J 6(0)-k-v ) ,eo(II-l)

where the summation is to be extended over the positions of all ions.

II-2

If we Fourier analyze Eq. (II-l) in space, the quantity

\ k • (n E ) . I ) appearing in tfeeUoiB;git<iiditta.U part of Eq. (4.15)

A f^ ' "5

(an exactly analogous procedure holds for ^k^, X(n E ) j )

the transverse emission )can be expressed as

2

,2 .A - r. k

^ r (-k.k"DL(k,OJ)

exp lti(k_ + k) • r. - k • rLi L ~i —* 6(O3-k • v

- -

.3 k

^ • V |d3k' - — j exiJlfcc +k-)- p V - r jDT (k ' .co)

(II- 2)

kv— —eo '

We a r e thus i n t e r e s t e d in evaluat ing

exp i (kT + k ) - r . - - ( k _ + k t ) * r f l \ = \ / exp I n k - k ' J^r. - (k_ + k l ) . r . U S .

l l j 1 < J (II-3)

In wr i t ing down Eq . (II-3) we have equated the e n s e m b l e a v e r a g e of

the sum to the sum of the ensemble averages and have expressed the

position r. .of the j ion in terms of the position r. of the i ion—j — l

and the distance r, . between the two.

If we now arbitrarily select the i ion and ask for the probability

of finding another ion at distance r. . away, then

< e x p F i | ( k - k ' ) i r . - ( k L + k - ) , r . i l > = exp fj (k - k ' ) . r . j

/ P(r . . )expi i (kT + k ' ) , r . l d r . . . (II-4)

II-3

In equilibrium the probability of finding a particular ion in a

volume element d r,. at a distance r.. from a given ion is

mu L muo o

where V is the volume of the plasma and <f> the shielded potential of

an ion given by

• K-, r . . . l l l - o )

u T IJ

In (II-6). :K is the inverse Debye length for both ions and electrons,

2 _ (Z 4- 1) 47Tne2

*T = 2 'mu

o

When Eqs. (II-5) and (II-6) are inserted in Eq. (II-4), we obtain

<exp [ i {{k-k ' ) ' £ . - (kT + k l )«r . l j ) - exp j i ( k - k ' ) . £ . l

—• \ d r . . <1 - — — exp - K— r. \ e x p - i(k_ +k ' )» r . . .V \ ij J 2 r . . T ii ^ —L. — —ij

J J L m u ii JJ ^(H-7)

The first term in the integrand contributes only when j = i .

(There are N of these terms corresponding to the number of ions in

the volume V. ) The second term can be integrated directly. (There

are N(N-l) = N of these corresponding to N-l choices of j for each

choice of i . ) If we carry out the integration in Eq. (II-7) and evaluate

the double sum yve obtain per unit volume

II-4

(H-9)

We can now insert JDq. (II—9) into Eq. (II-2) and in turn substitute

this expression into Eq. (4.15). The summation over electrons is then

carried out exactly as in Sec. IV. In carrying out the integration over

directions of emission, we neglect k by comparison with k1 in

Eq. (II-9), since the dominant contribution to the k' integration

arises from large k1 (small impact parameters).

The results for both longitudinal and transverse radiation are\

P(OJ) = 9(3)1/2U, up o

it m

\ 3C0

d k1 d v f(ve % e ' , 2

21 -

k1 +(Z+l)K

(11-10)

APPENDIX III

We here evaluate the electron density fluctuations produced by

interaction of the electrons with massive discrete ions. Since density

fluctuations exist among such discrete ions (cf.Appendix II), the elec*

tron distribution in tending to follow the ions itself deviates from

spatial homogeneity.

For immobile ions, the electron perturbation is assumed

stationary in time and the Vlasov equation linearized in the electron-

ion interaction becomes

e ^ f 3fv . i i + __!_ . __2. = . €f . (III-l)— dr m 9v '

— e —

The perturbed potential <p is given by the Poisson equation with the

ions inserted discretely as sources^

,3 4 „ x *,_. _ * (IH-2)n (1 + pf d3v) - Z(

If we solve Eq. (III-l) for the Fourier transformed f , we obtain9fo

f = "Jl ^ - ^ . { m . 3 )k m k • v r- i€

e — —

When this value is inserted into Eq. (Ill-Z), we may solve for £

obtain for k # 0

47TZe n+{k). 2 , , 2 r 9fk - W / o

p / k • ——

, T—- d V .- (HI-4)k • v - i£

Ill-2

where n (k) has been defined as

In terms of the static longitudinal dielectric function (Eq. (2.14)),

Eq. (III-4) can be written as '

47TZe n(k)

and Eq. (HI- 3) asdi

4TZ°2 , n W - " ° r , ,m.7)m Vt DT<k.°> k- v - is

e ij - " —

The electron density fluctuations induced by Coulomb interactions with

the ions thus turnr out to be

n r:lOT1'H4ffZe n r :l-OT1'/ f k d V & " ^ ! n| %^ H T!»

J m k D (k,0) J

e JU

In Eq. (Ill-8) F (v) is the one dimensional electron distribution

normalized to unity. The pole contribution to the integration has dis-

appeared because of the assumed isotropy of the electron distribution f

Since there is no time variation in the problem, the only w-

component present is « » 0 and

k^D^kvO)

APPENDIX IV

In Sec. VI we have introduced the function 6 (Qi) . In this

appendix we explain what we mean by this function.

We define the function 5(W) to be

/

+T/2expiOJt dt . (IV-1)

-T/2

With 6{(J)) so defined, 6 (0)) can be written as

2 i r + T / 2 r + T / 2

5 (W) = l im / dt / d t ' exp i « ( t + t ' ) . (IV-2)T-*« (27T) J J

-T/2 -T/2

Now both S(W) and 6 (W) by their definitions assume non-

zero values only near OJ = 0, so that the integral

I = If(OJ) 62(W) dW (IV-3)

is to good approximation given by

I = f(o) I 52(o;) dW , (IV-4)

where f(CO) is Ct) well-behaved function and the range of integration

includes the point (ti = 0 . '

If we use the definition of 6 (W) as given by (IV-2), I can be

written tT/2 ^ + T / 2

exp iw( t + t ' ) (IV-5)

•T/2 J_T/2

r- ±T/2 + T / 2

/doj/ dt / dt1

J J-T/2 IT/2

IV-2

Since the CO-integrand is spiked at U) = 0 with negligible con-

tribution elsewhere, we can extend the limits of integration to encom-

pass the region - °° < (ti < + °° with negligible error. If we do this

and invert the order of integrations so that the W integration is done

first, we obtain

1= lim —7 I dt/ dt1/ do; expi W(t + t<) , (IV-6)T - * <2ir)2J J J

-T/2 -T/2 -«>

From the definition of the 6-function, Eq, (IV-1), this can be rewritten

as

I = lim ^2l / . d t / dt' 6{t+ t ' )

-T/2 -T/2

I = U m f<°>T / dt 6(t) . (IV-7)

By comparing Eqs. (IV-4) and (IV-7) we see that integrating

Z • T5 (CO) over its resonance value is equivalent to integrating ~- 6(UJ)

c/ti

over the resonance, T being lajcgiejcojnpairi^fi.to l/co .

APPENDIX V

In this appendix we show that by using as source terms in the

linearized Maxwell-Vlasov equations currents and charges produced

by the interactions of first order waves we obtain solutions for E ,

B , and f which are identical with those obtained by solving the non-

linear Vlasov equation to second order.

We first consider the non-linear Maxwell-Vlasov equations.

When Fourier transformed in space and time, these equations are

= 0 f (V-l)

k,w

(V-2)

h' Bk - 0 , (V-3)

kXE. „ = - B . „ , (V-4)— —k, 0) c —k, U) 7

( V " 5 )

If E , B1 , and f are solutions of the linearized version of

these equations, the second order quantities E , B , and f are

given by

8£ ( v X B • 8f.o e } ,^ . — — 1 . 1

(V-6)

V-2

(where we have taken f^ to be isotropic) and the second order Maxwell

equations {i.e. , Eqs. (V-2) through (V-5) with E , , B

P2k,C0 a n d i2k,CO as variables).

If we solve equation (V-6) for f

2k,C0 "* * m

2 k, 0)

v X B d£ 8£_E . * —EL— 2 k , CO 8v

W - k • v

and evaluate

k.Ctf = " n o 6 j f2k,W d 3 v

(V-7)

(V-8)

and

thenCO

- ' - 2 k , CO

and

CO CO

kXE. _ P

-

ifc.

V....

CO

e

u d 3

XB.I.1U.W ^

C

- k •

V

1 '

V

V <

V

X

c

»

d£1

a v J.+ ie

k,

* !8v

CO

k ,

d

CO

3V

* d

(V-9)

(V-10)

3v .

(V-ll)

We now look at the linearized Maxwell-Vlasov equations, incorp-

orating the effects of the first order waves as source terms. The sources

are just those charges and currents produced by solving the non-linear

equation (V-6) without shielding. If we thus take for the source charges

V-3

and currents

and

where f is obtained from the equation

= 0 ,

and insert these into the linearized Maxwell-Vlasov equations, we obtain

Eqs. (V-10) and (V-ll) as solutions. (In Eq. (5. 2) we have used the test

currents given by Eq. (V-13) in the long wavelength limit. ) If we further

take f£ to be the sum of i {from Eq. (V-14) ) and the f produced by .

j and p (the solution of the linearized Vlasov equation) we see that—S 8

the f obtained in this manner is identical with the f_ obtained from

Eq. (V-6).