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).