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SOVIET PHYSICS JETP

VOLUME 19, NUMBER 5

NOVEMBER, 1964

RADIATION OF TRANSVERSE ELECTROMAGNETIC WAVES DUE TO SCATTERING OF CHARGED PARTICLES BY PLASMA WAVES
A. GAILITIS and V. N. TSYTOVICH P. N. Lebedev Physics Institute, Academy of Sciences, U.S.S.R. Submitted to JETP editor October 11, 1963 J. Exptl. Theoret. Phys. (U.S.S.R.) 46,1726-1739 (May, 1964)

It is shown that the emission of transverse waves by epithermal electrons takes place in the field of a plasma wave (in the classical limit) as the result of dipole radiation due to the oscillations of the electron in the wave (Compton effect on plasma waves), as well as a result of passage of the electron through density inhomogeneities created by the plasma wave. The emission of transverse waves by electrons is forbidden in the nonrelativistic case v = 0 (n = c = 1) by interference of these two effects. The forbiddenness does not hold for particles whose masses differ from that of the electron. The radiation spectrum of electrons and ions is calculated in the broad energy range from nonrelativistic to relativistic energies. The graph technique is used to calculate quantum effects that become significant for secondary quantum energies close to the energy of the charged particles. Possible astrophysical applications are discussed, as well as the possibility of determining particle energy and mean energy density of the plasma waves on the basis of the radiation intensity. It is also shown that the frequencies of transverse waves produced in the scattering of cosmic ray electrons by plasma waves may considerably exceed the frequency of waves generated by the synchrotron mechanism.

INTRODUCTION
1. The problem of the conversion of longitudinal
waves of a plasma into transverse waves is of interest from the viewpoint of the study of nonlinear effects in plasma, [t] and also for possible astro-

K>-< p, Pe

.. Kl~ / ···1(2
I I
4 p2A4

a

b

c

physical applications (see [ 2•3]).

FIG. 1

In the present work we consider the conversion of a plasma wave into a transverse wave by scattering from an isolated, epithermal charged particle. It is necessary to keep in mind that any plasma

terest as to the presence of transverse waves in a beam of fast particles as the result of scattering of the plasma waves by the beam particles. For a low-density beam, the result can be obtained by

particle can be regarded as a test particle. If the means of the scattering probabilities from single

analysis is generalized by taking into account spa- particles as found below.

tial dispersion, then the resultant probabilities will

2. We shall consider a set of weakly interacting

enter directly into the "kinetic equation" that des- charged particles, i.e., the so-called collision-free

cribes the nonlinear effects of wave conversion in plasma. The effects of interaction of waves in such

the plasma. Here we are interested in elementary a system with charged particles can be considered

processes and carry out a detailed study, including by perturbation theory. If the nonlinear effects in

the ultrarelativistic limit. The study of an isolated a vacuum correspond to closed electron loops, [4]

fast charged particle in a medium is essential also then in a plasma, in the presence of real particles,

for questions of the passage through matter of fast the conversion of waves comes about in the first

° particles whose temperature differs from zero.

approximation of perturbation theory as the result

Under these conditions, there are excitations in the of scattering from real particles. For a fast

medium (longitudinal waves in the plasma), scatter-

ing from which produces additional radiation which is superimposed on the other radiation. In Sec. 4, some possibilities are discussed for the observation of such radiation. The problem here is of in-

l)ln what follows, in the graphical representations of the process, the plasma longitudinal wave is pictured as a wavy line and the transverse wave by a dotted line, while the virtual quantum is represented by the dashed line.

1165

1166

A. GAILITIS and V. N. TSYTOVICH

\ t. .· . . ······~=r{ ~\,,,+~ ••• .< +~_rS+~ 2, ...+ .. \~_+ sS ~ ~··} /

. :

{::f

P,

:
1

P1

P1. :

P1 .q: I

1

:

I

.q .q: 1 t

.q .q

: P, .q
I I

:.q
I •

FIG. 2. Interaction between waves by means of plasma electrons. Summation is carried out over all electrons of the plasma. Pz is the momentum of the test electron which does not change as the result of the interaction.

epithermal plasma particle, the processes of conversion of longitudinal waves into transverse are represented by Fig. 1.
Processes a and b are similar to the Compton effect for a fast particle, while process c describes the nonlinearity of the plasma, the circle corresponding to the nonlinear interaction between the incident plasma wave, which is scattered by the transverse wave, and the field of the electron. In the final analysis, the indicated nonlinearity is connected with scattering processes on the plasma electrons and is determined by the set of graphs shown in Fig. 2. 2l Account of the graph c is very important, since in the nonrelativistic limit it completely compensates the effects arising from a and b.
3. The conservation laws for graphs a, b, and c of Fig. 1 are:
(1.1)
These connect the frequency w 2 = wt (k2) and the direction of the scattered photon with the initial momenta of the electron p1 and the momentum of
the plasma quantum k 1 of frequency w~ = wl (k1)
(the angles are defined in Fig. 3). For frequencies w 2 that are much larger than the plasma frequency w 0, we have

(1.2)

( E"p 1 , Vt = Pt/E"p1 and, mare the energy, velocity, and mass, respectively, of the electron before scattering). For not very energetic electrons,

Bp,/m ~min {mlffi 1, m!f k 1 [},

(1.3)

(1.2) becomes simplified: I Wt- k,v, cos l'h I
(1)2 = 1 - v, cos tt.

(1.4)

The maximum frequency for v - 1 corresponds to .J 2 = 0 and .J 1 - 1r (Fig. 3) and is equal to

2)The electron line pz represents any of the electrons of the plasma (l = 1, 2, ... , n). To obtain the vertices c, the graphs on the right side of Fig. 2 are summed over all the electrons of the plasma (denoted by a summation sign). As a result of summation of a large number of diagrams of very high order, the graph c becomes of the same order as a and b.

+ ffi:,ax;::::; 2e~,m-2 (ffi 1 (k1)

k1).

(1.5)

Upon satisfaction of the inequality (1.3), the quantum effects become negligibly small. The intensity of scattering, with (1.3) satisfied, is calculated in the next section. The opposite limiting case is considered in Sec. 4.

FIG. 3. Scattering kinematics. l't 1(1't 2) - angle between the longitudinal (transverse) quantum and the total momentum.

2. CLASSICAL LIMIT

It follows from what has been said above that the quantum effects do not play an important role when (1.3) is satisfied. Therefore, in the present section we carry out a purely classical calculation of the radiation of transverse waves by an electron scattered from a plasma wave.
Let the electric field of the plasma wave have the form

(2.1)

The method of successive approximations is used

to find the interaction between the wave (2.1) and the electron. In the zeroth approximation, we shall consider the electron to be moving uniformly and rectilinearly with velocity v. A force acts on it

from the wave (2.1) of the form eE 0 cos (k1 • v- w 1)t. In the first approximation to uniform motion, the small oscillations shown below are added 3l

+ r = vt R cos Qt,

(2.2)

R

=-

~
Bp1

_Q_§2_k_t_

(k1

-

v (k1v)),

(2.3)

If the electron moved in a vacuum according to the law (2.2), then for calculation of the resultant radiation it would be sufficient to find the oscilla-
3)Equations (2.2)- (2.6) were obtained as the result of
solution of the equation dd ~ = eE 1 under the assump-
t yfl-u2
tion that E0 is small so that [u -vi « v.

RADIATION OF TRANSVERSE ELECTROMAGNETIC WAVES

1167

tory part of the dipole moment cR with the aid of (2.3) and to calculate of the intensity of the dipole radiation; in the language of graphs, this would mean a restriction to diagrams a and b (Fig. 1). However, the electron under consideration moves in an essentially inhomogeneous plasma. Its inhomogeneity is brought about by oscillations of the electron density n in the wave (2.1), connected with the field of the wave by the equation

div E = 4:rre (n - n)

(2.5)

The last condition is satisfied if the velocity of the charge is appreciably larger than the phase velocity of the plasma wave and also the mean thermal velocity of the electrons of the plasma, and if the angle J.1 is not close to rr/2. The condition (2.9) is necessary so that Eq. (2.6) can be used; this equation does not take spatial dispersion into account. 4l
In accord with (2.5), (2.6), and (2.9), E changes in the plasma wave according to the law

(n is the mean value of the electron density). The

(2.10)

dielectric constant, which depends on n, changes

simultaneously with the change in density:

Let us find the power which an electron moving

according to (2.2) radiates in a medium with E of (2.6) the form (2 .1 0). This power Q is equal to the mean

In addition to the electron in the plasma, the polar- work per unit time performed by the electron in

ization produced by it also moves. Because of the motion in the electric field E created by it:

inhomogeneity of E, a dipole moment is generated

T/2

which partially (for a nonrelativistic electron, com- Q =-lim~ ~ dt~d3rE(r, t) j(r, t)

pletely) cancels the dipole moment of the oscilla-

T~oo -T/2

tions of the electron. A charge moving in a medium with E that is
variable in time and space radiates transverse waves. The graph c of Fig. 1 corresponds to such

= -lim <2~· \ d3k dw j (- k, - u') E (k, w), (2.11)

T-+co

It"

j (r, t) = e (v - QR sin Qt) 6 (r - vt - R cos Qt),

a mechanism of radiation. We note that the mech-

(2.12)

anism of radiation corresponding to graph c in

and E ( k, w) and j ( k, w) are the Fourier components

Fig. 1 has a well-known analog in the radiation of of the electric field and current density, respec-

a charge in a layered medium (see [5- 8J). We limit tively.

ourselves to the case in which the frequencies of

To find the field E ( k, w), we use the Maxwell

the radiated transverse waves appreciably exceed equation

the frequencies of the longitudinal waves that create

the density inhomogeneities. This case is the sim- ~E +grad div (e- 1) E- fJ2eE/fJt2 plest. Of course, the graph c also describes the

case of comparable frequencies.

= 4:rt (fJj/fJt- grad p).

Of considerable significance is the fact that the phases of the oscillations of the electron and of the change in E are not independent. The total scattered radiation is not the sum of the radiation produced by the oscillations of the electron (graphs a, b) and the radiation due to the inhomogeneities of the medium (graph c). Interference of these radiations is appreciable. A charge moving according to (2.2) radiates (at an angle J.2 to the velocity v) a wave of frequency

By substitution of (2 .10), we go over to the Fourier representation

+ .

kiki - w2&ii { Eokt

zEi (k, w) = (w i·0)2 _ k2 2mw2 [E; (k- kr, w - w1 )

+ + - E; (k k 1• w w1)l - 4: jj(k, w)}

(2.13)

and then solve by the method of successive approximations.
To find j ( k, w), we use (2.2) and (2.12). After substituting the solution (2.13) in (2.11) and keeping

<02 = Qj(i -Vet (w2) v cos 'l't2).

(2. 7) only terms proportional to E5, we find

For simplicity, we restrict ourselves to plasma waves whose phase velocity is much less than the velocity of light (k1 » w 1), and also to the condition (see above)

(2.14)

(2.8)

4)This is not fundamental, and·the results are easily gen-

or

eralized to the case where one must take into account spatial

(2.9) dispersion.

1168

A. GAILITIS and V. N. TSYTOVICH

where
(2.15)
The integrand function (2.15) is the power radiated in the frequency range dw 2 and with wave vectors d 3k2. Account of the change in the plasma density would give second terms in the square brackets of Eq. (2.15). They are appreciable in the scattering of a plasma wave by a nonrelativistic electron (v « 1). As is seen from Eq. (2.15), the first term in it approaches zero simultaneously with v. Therefore, for v « 1, we have {3 ~ v/w 2 and Q ~ v 2. The two mechanisms of radiation cancel one another. It must be noted that this refers only to electrons. For ions, because of their large mass M, the conversion of the longitudinal wave to the transverse takes place only as the result of the inhomogeneous density in the plasma wave and is determined by Eq. (2.14) with the replacement of {3 by
(2 .16)

We assume that the directions of motion of the

plasma waves are distributed isotropic ally. In

astrophysical applications, the latter is not ob-

vious, inasmuch as the magnetic field and the

directional character of the discontinuities can

have an effect on the distribution of the plasma

waves. We note that the inhomogeneous distribu-

tion of plasma waves can lead to polarization of

the scattered radiation. We now limit ourselves to

consideration of the isotropic distribution.

To find the frequency spectrum of the transverse

radiation Q ( w 2), we average Q over all angles J- 1:

+1

oo

=-} = Q

~ d cos 'l't1 Q ~ Q (w 2) dw2,

(3.1)

-1

0

W - Q (w2)

=

e41E62~;k:2rrco

2 2

(~'

d

cos

'1't1

d

cos

'1't2

dcp

(k 2 ~) 2/k~}

+ x [o (w 2 (1 - v cos '1't2) k1v cos '1't1) + 0 (w 2 (1 - v cos '1't2) - k1v cos 'l't1)]

(3.2)

( cp is the angle between the planes in which the vectors k 1 , u and k2 , u lie).
Calculation of the integral (3.2) leads to the expression

Therefore, a nonrelativistic ion in a plasma wave produces transverse waves that are much larger

(3.3)

than those produced by the nonrelativistic electron.
Only for the limit of ultrarelativistic ions, for
which -J 1 - v2 ~ m/M, is there a dependence of
the oscillations of the ion itself, and (2.16) is

r e = _R;_ m'

ffi2 (1- v)

q =

k1v

'

> cD (r, q) = 0 for q 1,

(3.4)

violated. In the limit v « 1, we obtain the following ex-
pressions for the intensity of radiation of electrons and ions, respectively: 5'

where q is the ratio of the frequency of the radiated waves to the maximum possible frequency for given k 1 and v.
The function <P(y, q) is given by a cumbersome

(2 .1 7) expression given in the appendix. Here we limit ourselves to graphs of <l>(y, q) for a number of

values of y, obtained with the help of (3.3). The (2.18) presence of two maxima in the curves of Fig. 4 is

both ions and electrons radiate at the frequency
w 2 = k 1v Ieos J.l when v « 1.

brought about by the presence of the two mechanisms of scattering considered. At ultrarelativistic energies ( y » 1), the principal role is played by

3. SCATTERING OF CHARGED PARTICLES BY

radiation from the oscillations of the electron.

ISOTROPICALLY DISTRIBUTED PLASMA

This radiation results in a broad smooth maximum.

WAVES

The narrow maximum at small q is brought about

by the mechanism that is similar to the mechanism

If a continuous spectrum of plasma waves is

of transition radiation (radiation from inhomogen-

incident .on the electr.on, rat~er than the ~lane mono- eities of the density produced by the plasma wave).

chromatic wave considered m the precedmg sec- Inasmuch as in the limit y - oo the latter radia-

tion,. the scattered radiation can be found by inte- tion has a constant intensity which does not increase

gratmg (2.14) over all the scattered waves.

with y, and a mean frequency, while in radiation

S)For small v ;S m/M, the effect of ions which disturb the compensation is important. Compensation is also destroyed for failure of the relation (2.9).

from the electronic oscillations, these quantities increase with increase in Y, in proportion to y 2,
the left maximum decreases as y - 00 and shifts

RADIATION OF TRANSVERSE ELECTROMAGNETIC WAVES
"'(J',Ij)

1169

17.5 FIG. 4

17.15

I;

to the left, and <l>(y, q) approaches the limiting expression, given analytically by the formula
+ <I> (oo, q) = fq [(1 - q)3 - 3q2 (1- q ln q)]. (3.5)
In the nonrelativistic limit ( y - 1), the two maxima merge and both radiation mechanisms suppress each other. For comparison, the curves for <l>(y, q) are plotted in Fig. 5 without account of the density inhomogeneities in the plasma wave. These curves have the same limiting curve (3.5} as y - 00 , but behave quite differently as y - 1.
In scattering of plasma waves by ions, Eq. (3.3) holds for the resulting transverse radiation; therein, <1> 2 ( y, q) has the form shown in Fig. 6. These curves were computed with the help of Eq. (2.16).
We find the total power radiated by the electron by integrating (3.3) numerically:

-Q =

2e•s~ 12
---g;noii (v).

(3.6)

A plot of II (v) is shown in Fig. 7. The coefficient for IT(v) in Eq. (3.6) is determined by the condition II(v)- 1 as v- 1.

4. QUANTUM EFFECTS
In the process under study, the quantum effects begin to play a role when the energy of the secondary quantum becomes of the order of the initial energy of the electron: w 2 ~ Ep 1• For this case, the latter must be sufficiently large [see (1.3)] :
(4.1)
(for example, for a plasma wavelength ~ 1 em, Equ ~ 1015 eV). Because of what was pointed out above (Sec. 2), only the graphs a and b (Fig. 1) are of importance in scattering of such energetic elec-

§L-------L-------~-------L----~

tU5

IJ5

17.75

FIG. 5

1170

A. GAILITIS and V. N. TSYTOVICH

Dfu)

appreciably less than the velocity of light

I

< I k1 I » w 1 l .

Multiplying (4.3) by the energy of the secondary

quantum w2 and by the number of plasma quanta

(4.5)

0.5

/Ot/

FIG. 7

trons. Because of the large energy of the secondary quanta, their dispersion can be. neglected, if we
consider w 2 ~ I k2 1; therefore, these graphs differ
from the graphs of the Compton effect in vacuum only by the replacement of the initial transverse quantum by the longitudinal plasma quantum, which has a momentum k 1 = (k1, iw 1). One can show that replacement of the vacuum photon in the external photon line by the plasma photon leads to the result that in the matrix element (computed in accord with the rules set forth in [4]) it is formally necessary to replace the unit polarization vector of the vacuum photon by the non-unit vector Sl

V2wl(wlkl, iki)
e1 = (k;- w;) I k1l (oe1(w1, k1)/iiwl)'!. ·

(4.2)

We carry out the indicated substitution in the matrix elements represented by the graphs a and b. Calculation by standard methods (see L4J) of the scattering probability dw, averaged over the initial spins of the electron and summed over the finite spins of the electron and the polarizations of the secondary photon, leads to the result

dw -

{2 + + e•w~ dQk,

(~ :x2 ) - 2k~ (~ __!_)

- 2e~,k1 1:x1l ae1;aw1 :x2 :x1

:x1 :x2

r- + : +m2ki ( : 1

2

:x:!. + ((ep, Bp,)2 + ffi~- kiJ

(4.3)

we obtain the intensity of the radiation in an element of solid angle:

w2E~ oel
dQ = 16:n: awl dw.

(4.6)

The integration of (4.6) over the directions of flight of the secondary quantum of frequency w 2 and averaging over the angle of collision between k1 and P1 gives us the spectral distribution of the radiation in the scattering of the electron by the isotropically distributed plasma waves:

Ul2

1

+ : :;) q = '1 (ep,- w,) = (1

wmax!Wo-:::;' (4. 7)

f [a + + ( + <Dk (q, a) = q {(1 - q) (1 q) (1 1 1q:::;)2)

+ + - -1 -t2- q:::; (1

q

4q2) J" - -1 -1-2;-,qq2~:::; ln q} ,.

(4.8)

here Wmax = Ep/(1 + 1/u) is the maximum radiated frequency.
In obtaining (4.8) from (4.3), it was taken into account that the energy of the electrons was ultrarelativistic ( Ep1 » m) while the plasma wavelength was large in comparison with the Compton wavelength ( k1 « m). In the non-quantum region ( u « 1), (4.8) undergoes a transition to (3.5). By integration of (4.7) and (4.8), we find the total radiation in the ultraquantum case ( u » 1):

w,max

= -

\

Q = • Q (ffi) dffi

e4R~ep,

2ep,kl

4m2kl In m2

0

(4.9)

Here d nk2 is the element of solid angle in the direction of flight bf the secondary quantum,
(4.4)
As in the classical calculation, we have limited ourselves in the derivation of (4.3) to the consideration of plasma waves having a phase velocity
6)Equation (4.2) is obtained from the requirement e,ll k,,
e,k, = 0 (if a Lorentz gauge is chosen). The normalizing factor in the quantization assures the equality of the energy of the plasma quantum with its frequency.

By comparison of (4.9) with the non-quantum formula (3.6), we establish the fact that the quantum effects impede the growth of radiation with in-
crease_Jn energy (Q ~ Eb in the non-quantum
case, Q ~ Ep 1 ln Ep 1 in the ultraquantum case). The quantum effects are most important in a
relatively dense plasma, for example, the electron plasma in,metals, where they play a role at low energies. We have to deal here with the additional "bremsstrahlung" generated in the medium, which is superimposed on the long-wavelength part of the ordinary bremsstrahlung. We estimate the relative

RADIATION OF TRANSVERSE ELECTROMAGNETIC WAVES

1171

role of both radiations in the region of application of (4.9).
In the continuous spectrum of plasma waves, (4.9) transforms to
(4.10)
For estimation purposes, we assume that the plasma waves possess an effective temperature
v Teff in the region of weak damping (k1 < 1/Rn
= 47re2n/T); this temperature generally differs from the electron temperature T. In accord with (4.10),
(4.11)
By comparing (4.11) with the bremsstrahlung of the electron
(4 .12)

we find

(4.13)

Z is the nuclear charge of the material, n/ni is
the number of free electrons in the nucleus. We
note that Q/Qb is smaller in the non-quantum region by a factor m 2/k1 Ep1 than in (4.12). For
Teff/T » Z2ni/n in the quantum region, Q » Qb·
In the non-quantum region, Q is smaller than Qb
for energies of the electron 7l

(4.14)

5. SOME APPLICATIONS TO THE DIAGNOSTICS OF PLASMA TURBULENCE
All the methods of plasma diagnostics (see Artsimovich [s]) are in some degree associated with the obtaining of definite information on the plasma parameters, either with the aid of different radiations (in the broad sense of the word) arising in a gas-discharge plasma, or with the aid of changes in radiation from the outside. Here and in most cases, an important loss of information takes place on the boundary of the plasma, which has a complicated structure. The diagnostics of beams of charged particles has an important advantage in the sense that the generated transverse
7)Jn a medium at thermodynamic equilibrium, even under
the most favorable conditions (tiw 0 « T, where Ted= T) the
radiation studied by us does not exceed the bremsstrahlung in intensity, in accord with (4.13). In a recently published
paper,[ 117 a contradictory conclusion is reached, which ap-
pears to be the result of an error made in the derivation of Eq. (2.4) of Ryazanov's paperJ17]

radiation considered in the present work can have a sufficiently high frequency, such that the plasma boundary and the plasma itself are in practice no different than the vacuum for these frequencies. Thus this radiation, which is generated in the plasma, and which carries information on the plasma parameters, in practice loses no information on the boundary of the plasma. The problem touched upon here is also of interest for the socalled "turbulent" plasma (see [ 10- 12]), which is characterized by the high intensity of the plasma waves in it (epithermal noise).
The intensity and the distribution of plasma waves can be determined in principle from the intensity and the spectrum of the transverse radiation which arises in the passage of a beam of charged particles through the plasma. B\ We write down the formula which connects N (w, k) with Q (w, k) for each individual charged particle [compare (2.14), (2.15)1:

= - : - Q (w, k)

4e4k 2 w N (w k)

·

m

& B1

(

1 Ulr,

' 1
kr)f&wr

6

(k2

-

w2 ) 6 (krv cos -frr

+ W- ro (1- v cos-fr2))

(k~) 2jk2 }.

(5.1)

Attention should also be paid to the fact that if Q(w, k) and N(w, k) are known, then measurement of the energy of ultrarelativistic particles, Ep/m » 1, is possible in principle.

6. RADIATION PRODUCED BY THE ELECTRONS OF COSMIC RAYS IN COSMIC PLASMA
At the present time, it is generally accepted that cosmic plasma is in a state of turbulent motion (it is not quiescent). One of the manifestations of turbulence is the presence of plasma waves capable of accelerating the cosmic rays by the mechanism considered by one of the authors in [ 13• 14]. Information on the presence of cosmic rays, for example, in the Galaxy, [tsJ is given us by radio emission [ 16] and by the optical radiation of cosmic objects. The cyclotron-radiation nature of the cosmic radiation, i.e., the radiation of the electrons in the cosmic rays in cosmic magnetic fields, is also generally accepted. Yet the nature of the cosmic radiation at very high frequencies is still not clear. We would like to direct attention here to the fact that a definite contribution to cosmic radiation can be brought about by the transformation of longitudinal plasma waves which exist in a turbulent plasma, into transverse waves as the result of scattering by the cosmic rays.

8 )The beam should be so weak that no instabilities develop during the time of observation.

1172

A. GAILITIS and V. N. TSYTOVICH

Although the fluctuating electric fields of the plasma waves are presumed to be weak in comparison with the magnetic field, and consequently only a small fraction of all the cosmic electromag-
netic radiation ( ~ E 2/H 2 ) is associated with the
presence of the plasma waves; the frequency of the radiation arising from the mechanism under consideration is much larger than the frequency of the cyclotron radiation (by ~ 1/v~, where vT is the velocity of the turbulent motion in the plasma). Therefore, the ~ppearance of transverse electromagnetic waves ·in the scattering of electrons by plasma waves can create new bands of non-thermal radiation of cosmic objects. For example, sources having a maximum of radio emission in the region of centimeter waves and v~ ~ 10-5 - 10-6 must also emit in the optical region as the result of the scattering of the electrons by plasma waves.
Actually, let us compare the mechanism considered here, radiation of transverse electromagnetic waves, with cyclotron radiation. Losses of energy by the electron as the result of the cyclotron radiation are equal to
(6.1)
The loss ratio in the ultrarelativistic limit tends to
(6.2)

trons have a power-law spectrum

(6.8)

We now find the frequency spectrum of the radiation of the electrons with the distribution (6.8). In accord with (3.3) and (3.5),
I (w) = ~ dy n (y) Qy (ro)

e''E2K ( 1 (3-a)/2

= b (a)--'.'- -.-)

ro(l-al/2,

m 2 '2k1

(6.9)

The radiation also possesses a power spectrum,
with the same spectral index 1;2 (a - 1) as the
cyclotron radiation. A different dependence on a of the coefficient b (a) and the corresponding coefficient in the formulas of the cyclotron radiation[ 15] leads to the result that when the spectrum of the electrons deviates from the power-law, the spectra of the cyclotron radiation and the mechanism studied by us do not appear entirely similar.
APPENDIX
FORMULAS FOR THE FUNCTIONS <I>(y, q)
By integration of (3.2), we get the following expressions for the functions <I>(y, q) that enter into Eq. (3.3):

Here -f\E2} is the mean fluctuating electric field
of the plasma waves. In accord with (3.3) [see also (1.5)] , the radia-
tion in scattering by plasma waves is concentrated in the region of frequencies of the order

(6.3)
The cyclotron radiation is chiefly coocentrated in the region

Then the frequency ratio has the order

(6.4)

CD (y, q) = <D1 (y, q) + <Dz (y, q) + CD3 (y, q); (A.1)
CD1 (y, q) = '¢1 (y, q, Xmax) - '¢1 (y, q, Xmtn), <Dz (y, q) = '¢z (y, q, Xmax) - '¢z ('y, q, Xmin), <Da (y, q) ='¢a (y, q, Xmax) -'¢a (y, q, Xmin); (A.2)

+ '¢1 = j 6v3 (1 1 v) z4q { - X - Xz2 -

1 2v4j 2

X [ 3X _)_ - 1 ( Z~"V 2 1 X

3- - j2- 2v2 )

(1 - - - v222 x2

1
3xj<l

)

-

2 (3 -

2v2) In x]} ·'

(A.3)

w~lwcr- k1m/eH = k1Rnm!eHRn. (6.5)
Taking it into account that the Debye radius Rn.
k 1, and the spontaneous magnetic field are connected with the plasma temperature T by the relations

+ + '¢z =

v3

(1

+ 1 v)

r•q

{

-

4 x

v2
32 (2x

3z2 -

+ ;~ [(5 - 3v2) z4 - (16- 8v2) z2

+ + 24 - 8v4 ]ln I vcp(x) 2.:x - z2 J

12) cp(x)

H- VsnTn,

Rn = VT!4nne2 , (6.6)

we find that the scattering by the plasma waves produces radiation that is

(6. 7)
greater in frequency than the cyclotron radiation mechanism.
In a number of cases, the ultrarelativistic elec-

-

+ (z2--:- 2)2 [(2x - z2) (z~v2 2 - v2)

b(4-z 2)<:p(x)

2

+ - (1 - z•2.)f\v3z2 _r±' _- 2x)J} ·'

{x - x ( + x + '¢a = -

1 z"v" (1

v) qrs

~ In -

2z2

Y1" ) 1

-

v2<p (x)[__!__ -
'2

[_vz22_z/2_2+_-J-4'---;--r-'2-rJ--x"-J

(A.4)

----------

--

RADIATION OF TRANSVERSE ELECTROMAGNETIC WAVES

1173

+ + 2v

[__!___ _ ~

+ + + V z v2z2 4 I 12 12

2

z2 (z2/ 2 ~-2)] v2z2 4 I 12

I+ ( j}. X In

+ - + Y + v2z2 ~~ 2x vcp z(x) v2z2 4jy2)

(A.5)

In the formulas given above, the following notation

7 G. M. Garibyan, JETP 35, 1435 (1958), Soviet Phys. JETP 8, 1003 (1959).
8 V. E. Pafomov, JETP 39, 134 (1960), Soviet Phys. JETP 12, 97 (1961); ZhTF 33, 557 (1963), Soviet Phys. Tech. Phys. 8, 412 (1963).
9 L. A. Artsimovich, Upravlyaemye termoyader-

is used:

nye reaktsii (Controlled Thermonuclear Reactions)

Y V = y2 - 1/y, z = (1 - v)/vq, Xmin = 1 - v,
Xmax = min (vz, 1 +v)

(Moscow, 1961). 10 Vedenov, Velikhov and Sagdeev, Yaderny1
sintez (Nuclear Fusion) 1, 82 (1961).

+ + <p (x) = [z4

4z2 - 4z2

4x2J'/,

v212 v2 x v" .

(A.6)

11 A. A. Vedenov, Atomn. energ. (Atomic Energy) 13, 5 (1962).

The graphs of the functions <l>(y, q) computed by these formulas, are shown in Fig. 4. <1> 1 (y, q) corresponds to a neglect of the oscillations of the density in the plasma wave, and its graphs are given in Fig. 5. The function <1> 2 (y, q) characterizes the radiation by ions and is shown in Fig. 6.

12 Yu. L. Klimontovich, DAN 144, 1022 (1962), Soviet Phys. Doklady 7, 530 (1962).
13 V. N. Tsytovich, DAN 142, 319 (1962), Soviet Phys. Doklady 7, 43 (1962); Izv. VUZov, Radiofizika 6, 918 (1963), Geomagnetizm i aeronomiya 3, 616 (1963).
14 V. N. Tsytovich, Astron. Zh. 40, 612 (1963),

1 B. B. Kadomtsev and V. I. Petviashvili, JETP 43, 2234 (1962), Soviet Phys. JETP 16, 1578 (1963).
2 V. L. Ginzburg and V. V. Zheleznyakov, Astronom. zh. 36, 233 (1959), Soviet Astronomy 3, 235 (1959).
3 V. V. Zheleznyakov, Usp. Fiz. Nauk 64, 113 (1958).
4 A. I. Akhiezer and V. B. Berestetskil, Kvantovaya elektrodinamika (Quantum electrodynamics),

and 41, no. l (1964), Soviet Astron. 7, 471 (1964); 8, in press.
15 V. L. Ginzburg and S. I. Syrovat-skil, Proiskhozhdenie kosmicheskikh luche1 (The Origin of Cosmic Rays) (Moscow, 1963).
16 I. S. Skhlovskil, Kosmicheskoe radioizluchenie (Cosmic Radio Emission) (Gostekhizdat, 1956).
17M. I. Ryazanov, JETP 45, 333 (1963), Soviet Phys. JETP 18, 232 (1964).

2d Ed., 1959.

5 Ya. B. Fa1nberg and N. A. Khizhnyak, JETP

32, 883 (1957), Soviet Phys. JETP 5, 720 (1957).

6 M. L. Ter-Mikaelyan and A. D. Gazazyan,

JETP 39, 1693 (1960), Soviet Phys. JETP 12, 1183 Translated by R. T. Beyer

(1961).

254