TDFTD-HT-23-671-67

* FOREIGN TECHNOLOGY DIVISION

PROPAGATION OF WAVES IN PLASMA ACROSS THE

MAGNETIC FIELD

by

A. B. Kitsenko andX. N. Stepanov

/UDC

r p flrrv;ript

9 @1967 D1917,2 ' GO LDE[N A NNIV R ARIY

FOREIGN TECHNOLOGY DIVISION

Distribution of this document is unlimited'.It may br released to the Clearinghouse,Department of Commerce, for sale to thegeneral public.

Reproduced by tho,,oCLEARINGHOUSEi.IM E for Federal Scientific & Technical -"--- -

Information Springfield Va. 22151

77 -_ _. . . . . ' . . . . . - - - _2 2 . _ ' _ _ -- 2 -2

UNEDITED ROUGH DRAFT TRANSLATION

PROPAGATION OF WAVES IN PLASMA ACROSS THE MAGNETICFIELD

By: A. B. Kitsenko and K. N. Stepanov

English pages: 22

SOURCE: Yademyy Sintez (Nuclear Synthesis), No. 4, 1964,pp. 272-278.

Translated by: L. Marokus/TDBXT"

TT700i819

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RASIROSTRIINYE VOLN, V PLAZME POPEREK MAGNITNOGO POLYA09 Bi~lish Titli.PROPAGATION~ OF' WAVES IN -PT.ASMA- ACROSS THE -MAGNETIC FIELD

-43- Source-YA2DERIIYY SINTEZ (USPJ ___________________i

,0,:Atlwr 98 D ocunent Lic&tion

aITSENKOJ A. -R'.16-C~to --47- Subet-Codet

STEPA19V K-. "N. 20_____________

16Cb- uthor - .39, TOic Tagsi waI proation, magneticNOWE f ield, xied iistic pl1a si oscilato, !

l6':TCo-Authokr cycatonfrquc, -dispersion equation,plas.a atn

-NONE _ _ _ _ _ _ _ _ _ _ _ _

'AS7A:T-,*prh -pagaton of Waves in a plasma across a magnetic field is 4beJ:ng chatracteriied -by intere',ting featdires -hear gyro-frequencies of electronsand ions,, as well as frequencies5 multiple to them. Frequencies of ordinaryand extraordinar'y waves which propagate across a magnetic field in a non- jrelatiistc plasma are determined, and tha excitation of high frequencybiian~hes of thesep os'C'illations are investigated when an ion cyclotron wavepasses, through the IlasmaL. Various dispersion equations for extraordinary'waves are~giyen. Thei behavioral frequency of various branches of extra-ord'inricy wave, i" -hot -plasma 'are described-.(E,-.gih, tii~lation:- 22' pages.

FTD ES 07 0-90 f

PRO ATION OF WAVES IN PLASM ACROSS, THE MAGNEIC FIED

A. B. Kitsenko and K. N. Stepanov

I

-I

IThe effect of the thermal movement of plasma electronsiiand ions on wavre propagation of a low pressure nonrelativis tic

plas a t_ 47rf n0 TajHi2 ;S i) acrose a magnetic field is iti-

vestigated. The dispersioni equation for ahX -ordinlary wave hassolutions-co%(k)-. iimilar to k sw a ;2,u. at any given

wavelength ratio (A . i/k) to the Larmorlan radius Ioa ype particles having a thermal velocity (see Fig. i).

(c% and mi gyro-frequency of electrons and ions,)eiThe dZ'spersion e4uation for an extra&ordinary wave also

has Val aalogoua so) ution in the field of high frequenciis(CO ftce), (see Fig. 5), as well as a solution,, correspornding

at k I 2 .> , o the longitudinal. plasma oscillations.

FreauencJ.es of longitudinal oscillations, in the case of a

Langmuirs frequency of electrons) as a function of the wavevector, decreaae at a rise in kt e1 decreasing from a value

a) a e( s - 2, 3,,.. at k to w ft (s i)We at*

(%Sl e be the behavior of plasma frequencies is shown inFig, 2 and 3.

In the field of, low frequencies (w < mi) the dispersion

equation for en extraordinary wave (in case of a cold lasma)dote rmines the frequency w - kV~j which cor.kespond to hmagneto-sonic wave (Vi - is the Alfven velocity). This

expression appears to be unacceptable at kQ i o s(s n-opos.when co~ a mi~. 121 this field it is necessary to compute the

FTD-HT-23-67h-67! ____ _ _ __ _

71

thermal movement of the ions. The behavior frequency ofvarious branches of the extraordinary wave in a hot plasma at

w S i is shown in Fig. 6.

The auto-excitation of the electron (high frequency)branches of the eyAmined oscillations, when an ion cyclotronwave passes through 'bhb plasma, due to bunched instabilitywas investigated.

i. Introduction

I*The propagation of waves in a ylasma across a magnetic field

is characterized by interesting features near the gyro-frequencies

of the electrons and ions, and freniencies multiple to them. In

spite of the investigation -of this question by a whole series of

works [i--i3], it cannot be consldred fully explained. In the

present work, the frequt.ncies t of ordinaiy and extraordinary waves

which propagate across a magnetic field in a nonrelativistic plasma,

are determined and also the excl' ;ion of high-frequency (electronic)

branches of these oscillations when an ion cyclotron wave passes

through the plasma is ini;estigated. As is known, in the case of a

transverse propagation of waven in a plasma situated in an outer

magr.etic field HO, the dispersion equation, which determines the

frequencies t of these wwvez as a function of the jave vector k,

breaks into two equk1 .ions

(2)

where)

, . .. .....,,.(3 )

7r=-T- 3-67167.

*e4.4Z: 7i ~2

k c/-- is the index of refraction. -a = (i4ve2n0/ma)/2, to

- eH0/mac -- is the Langmuirian and cyclotronic frequencies of parti-

cles of type -a, e = -- 1, s1 i, )n - Int) -- is the Bessel

function from 'he imaginary arguement, v, (Tama) I /2 -- is the

th~1e~c~y,= 2 2 ,2 2thermal/ velocity, pa = k V 2/a k Q a a -- is the Larmorian

radius of particles with thermal velocity.

Equation (i) determines the frequencies of an ordinary wave

which appears to be purely transverse (the intensity of the electric

field of this wave is perpendicular to the direction of propagation

and paiallel to the outer magnetic field H0 ). Equation (2) deter-

mines the frequencies of an extraordinary longitudinal-transverse

wave (the intensity of the electric field of that wave is perpen-

dicular to K and has a different (f.om zero) projection in the

direction of propagation).

Let us note, that the expressions of (3), for the tensor of

dielectric permeability Eij, can be used only for the fulfillment

of the inequality

where P - va/c. This condition denotes, that the dispersion of

frequencies of the cyclotron radiaticni particles with thermal

velocity, due to the relativ.stic (transverse) Doppler effect, is

small in comiparison with w When fulfilling the conditions of

(4), the cyclotron oscillation damping is exponentially small.

Below, we will investigate plasma oscillations close to

gyroreoonances. If the length of the wave is on the order of the

Larmorian radius of electrons or ions (P e i orp a"i), then the

square of the refractive index, when w f Wa is on the order c

TTD-HT- 23-671-6t' 3

2 2

' Values aij $I ADc2/-2 if (D is not close to sto. It is obvious

i1 the resonances co so~a, are separated only when c,- A> m

in the case when the gas kinetic pressure of the plasma is consider-

Sably lower than the magnetic pressure,

LI (5)

'We w5-ll confine ourselves to the investigation of this case.

When i, the qualitative pattern of the frequency depend-

ence wc the wave vector is also the same when it, i; however, for

the determination of frequencies m(k), a numerical solution of

eq. i and 2 is needed. Upon the fulfillment of condition (5) it

is possible to obtain simple expressions for frequencies w(k), which

are close to s(Da -

2. Ordinary Wave Frequencies

Let us first of all examine Equation i in the case of high

frequency (electronic) oacillations when the movement of ions can

be disregarded. In the case of low density plasma (a. < ,a) the

value F.3 is comparable to w ,e* Pe ' i with (kc/) 2 c2 2/2

only when 1 -- sw / a PC But in this case the nonrelativistic

expressions of equation 3 for sEj cannot be used. Therefore, we

will examine the case of dense plasma (Se A-me). In this casemaintaining only a resonance component in the expression for e33'

we will obtain

(,.-4 )I,=-.,,Ij) (m,,, ...4 (6)

where

h C) f(7)

Expression 6 can be used only for Le W xe' if Pe < Xe then in E.,

it is also necessary to maintain that a conrPonent with n = 0 in

addition to the resonance member. Then we will obtain

where (8)

The behavior of frequencies depending on the wave vector is

shown' schematically in Fig. io The frequency close to We at low

'e < xe±/2 decreases (Aml'%e - ee[ue + PJ)p it reachese~iunt e ~ t e

minimum at e %e/2 (LM/ e W -. eLI), and then it increases,

approaching asymptotically to we at pe * i. Frequencies close to

S e(s m 2,3, ...), decrease upon an increase in pe. reaching a

minimum at k . kmn nI/Q e, and then increase, approaching se

With an increase in the number s of branch oscillations, value k nin'

increases, and value Ia - soI) e/coe decreases in a minimum, whereby

functions (to -Se)/Ae change slowly at k eft.

I

L --

Fig. i.~ (a) uVo b & e

5-

-- -- "'Ole

I N

Clear expressions for the refraction index as a frequency

functions a w *a se can be obtained in case of long wave oscilla-

t.ions (Pe- i) at s 1 1, 2 and in case of short wave oscillations

( e : i) for any sq j e

If co ce* Pe- , then

At xli i we will thus obtain

-- 0 -' ,.+ U- 'IIkell as olitl - (10)

If w 2 We 'and Pe i, then

2 ( 2 (1 - 2c*i/:WI

At w ace and ILLej' i the refraction index equals

The conclusion about the poa& lf-ty crf wave propagation in

plasma with a greater density at w - we was made by Drummond (i],

who obtained expression 10. (Formula iO was already obtained in

Gershman's work [2). Let us note that the numerical results for

k c/o. pbtained in report I from formula 10, are erroneous, i.e.

they lie outside the applicability area o' expression ±0 either

condition pe i i awrWy from frequency t wo is disrupted, or

6

condition 1 Ix I -- wIIe3/2 W I close to )W

Expredsions 9 and ii were obtained in Stepanov's work [3], and

expression i2 were obtained in Ramashvili and Rukhadze's report [4],

and with the consideration of collisions, they were obtained in the

report by Demidov and Frank-Kamenetskiy [5] also.

An analysis of equation i without decomposition by degreen

Pe 0z- iie for plasma with an arbitrary ratio of gas kinetic pressure

to the magnetic pressure was mentioned in the report by Dnestrov-

skiy and Kostomarov [6], in which the appearance of new brnches

for the index of refraction at c - s'oe was indicated, and the

entire picture of the behavicr of refraction indices is explained

depending on the frequency. Let us note that the results of numeri-

cal calculations of the refraction index in the area com c se , given

in report 6 are incorrect, i.e. they belong to the case 1e < we and

lie outside of the area of applicability of the initial dispersion

equation.

In the low frequency area at a)o sci and.p e4 i we will obtain

the following expresr'on for the frequency from equation i.

The schematic behavior of the frequencies depending on k i is

shown in Fig. I for which it' is necessary to replace we by c i and

e by 'e

3. High-Frequency Longitudinal Oscillations

We will now go to the investigation of equation 2 in the caoe

of high-fr _ency (electronic,) oscillations. If the refraction

index is great (kc/c) 2 rw LJxe > 1, then in equation 2 a branch

of longitudinal oscillations, whose dispersion equation e,, 0O,

7

will be presented in form of the following can be separated

'0'l ey - Pi i o.W .

As is evident from Fig. 2, on which the form of the function

y = f (w) is shown schematically, this equation has solutions

Fig. 2 5

N - t

2 Irk-

e: eIn_ jhe are pe; rm(4)w ilot

(b) L t 0 , /-,.

Fig. 2.

oKEY: (a) o+where

2 CO- . s(k) (s -1i, 2, ... ) corresponding "to intersection points

1, 2, ... of curves y = f '(,O) with a stra ht line y = e e

In the area I e 1. from (hf) we will •obtain

• -o . , ,,.., (o=,, ,..) (15)

or om ab (i + ), where

1 t. 8.3' w' '1 ' -, ')' (16)

v 8

:}-

I.

If Le 1 1. then for frequencies close to s me, we will obtain

the expression

(w- ,,)l ,, aA'sO,.(s2x#,"*()') ..

The behavior of frequencies ,wo = cs(k) is shown schematically in

Fig. 3. at Ifl/we < 3 and in Fig. 4 f2 < 112 2 + f+j2.

As is evident from Fig. 4, frequencies as(k) at s < f decrease

monotonrically upon an increase of k, tending pe k i and s < f

toward s we" Frequency wi(k) (f we <'(fe2 + Me2)i/2 < ( + ) We)

at low pe rises with an increase of k, if J > i, it reaches

maximum at k Qe - i, then it monotonrically decreases, tending

toward f ae" At f = i, this frequency monotonrically decreases,

approaching we with a rise in k Qe" Frequencies greater than )s(k).

at s > f increase at low k e, reaching maximum at k e f i, and

then again approach s we. Expressions 15 s - f or s - f + i and

equation 16 are inapplicable, when the hybrid frequency 'o0 is close

to (f + i) we. In this case, maintaining the components in equation

14 with n -+ i and the resonance member n -f or n f + i, we will

obtain0m- .(1+e) (±7)

where

' *T\ _ .. WL .. 1)'(±8)

,,?; equations 17 and ±8 it follows, that frequency Wo, as well

.4 9 _

iii

,i ____

Fig. 3.KEY: (a) frequency; (b) wave

"vec o.I

o|

(b) HiSH .. ,,d~ . ,

~Fig. .E:. (a) requcy (b)_wave

~vector.

wer obaied n Gos reot[]ada I ,3 . -i h

Siteko nd tep nrert[]

; iO

The expressions for the refraction index of longitudinal

oscillations can be obtained in a clear form at p i and e )b i.

If CD sw, e then,.e e

kc - 1'- 2.-- ,- o -ts I I

At ae kiand w s w we have[l]e e

--- ge (20)

4. An Extraordinary Wave (High Frequencies)

Let us now examine equation 2 in the general case of plasma with f

greater density in the field of high frequencies. At p N e in

expressions 3 for 6iW. E22 and s12$ it is possible to leave only

the resonance components -/(,-- s coe)" Then we will obtain

"- = : ,() 1, ,,.. (21)

where

97, (;k) = 8 (MP - 1) VOW1 + 1 ) 1, - zI, 1] ( 22) :

At small pe the function ys( e) a P IeS/2E (s + l)1 increase

in ±.eI reaching maximum at ILe i i, and then decreases, approachingTs (Pe) - s/(27)i/2,3/2. (At pe > i function % (pe) and f

coincide asymptotically, therefore the refraction indeu of the

extraordinary and ordinary wave is determined by one and the very

same formula 12 [1]).

Li

II

in the area p e hen e tensor Eij it is necesry to

consider also the components with n =+ i in addition to the resonance

wo.on nts N / _s to In this case we will obtain

• , o,, .,,)(23),+)

At 4 Le Xe this formula coincides with formula 21

In addition to the oscillations branch in the field of low

there is still another branch, whose frequency is determined by

expression

.,-,,,.. , . +.) + i,. )(.] ,3(24)me (Xe

The frequency of that branch settles away much farther from s eethan the frequency determined by formula 21. In the area mt

e efexpression 24 transforms into expression 15 for a frequency of

longitudinal waves (in the denominator of equation i5 in the case

question of dense plasma e 2e in comparison with i/(s2 - i)in 2/(

can be disregarded. In this way waves with the frequencies of

equation 24 are the long wave part of the branch of plasma

oscillations.

The behavior of frequencies w (k) depending on kQe for both

branches is shown schematically in Fig. 5.

Equation 2 has been numerically solved at low ILe and W f e

in Drummond's report (i], which mentioned the possibility of wave

passage with a frequency of w P we through dense plasma. (The

numerical results of the report [i] are erroneous, because at pe ± 1

for calculations a formula suitable only for p 4 i is used). The

expressions for the refraction index of an extrordinary wave at

12........

I

Fig. 5.

wave vetor.

e e n e I were obtained

in report [3.

Dnestrovski-Kctmarov report [9] a-- analys-is of equation 2 in

the

general case (g" < is em. i, le > i) was made and

shown for the

existence- of n . oscil -lation branches at a) s we Let us note,

that the numrical calculations of the refraction index, dvlpd

in report 9 at - s we are invalid, i.e. for them ondition 4 of

the applicability of the expressions 3 for sij used in this case

is disrupted, Schematic graphs for" k c/, which correspond to

waves with the frequencies equation 21 are given in the Demidov

report CIO].

5. Low-Frequancy Pdazma "Oscillations

In case of low frequenc pIasma oscillations

( << e equation

-. 0 has the following form at e 'ci

0A

I t i n o b v o u s t h a t oe ff 5. a n w e l l a n e q u a t i o n 1 .4 h a s a

number'of olut o ns W m (k) ( mi, 2, 3, .. ) IyLn g in the interval

.33

I

f C., <CD< (S, i .

in case of long wave oscillations (kQ i-C 1) we finid from

equation 25

J, ,.g. . . (26)

1or

* or 0 aw4f- I) (+(27)

where

" N- 0 + Iw') S-4.S JM

Expression 27 for a plasma oscillation frequency at Ta - 0

was derived by Koerper [ii]. A consideration of the thermal

movement of plasma particles, which leads to the limitation of

refraction indices of an unusual wave at w -.*ai, is given in report

3, £2, and £3. (Let us note, that in report 3 the thermal movement

of electrons was not considired; in report £2 the thermal movement

of ions was n)t considered; in addition the coefficient at N in

formula 12 of report [£2] should be mfltiplied by 2.)

In the area of short wave oscillations (Pi > i) from eq. 25,

we have

The dependence of the frequencies ws(k) on kQ i is schemtically

represented in Fig, 3 and Fig. 4 (it iz only necessary to make a

replacement of we by ci e by i and a0 by wk).

'1 1

If the hybrid frequency wk is close to s ci then expressions

26 and 27 are inapplicable. In this case

-* (29)

where

itsp (30)

6. The Extraordinary Wave (Low Frequencies)

Let us first of all mention, that eq. 2 in a hydrodynamic

approximation (PeC i., P - i) has a solution w wc in the area

corresponding to a magnetic audio wave

W k -A (3:)

where

(3ia)

On the other hand, maintaining the rbsonance components in eij' we

will obtain

M-ee~~~4 1 eq(pf 0- - p..u -+-. (,.,,,0 ...) (32)

where function Ts(Li) is decermined by foryule. 22, or (at low Ii)

..-.. .*(..)a,i CM- ) (* z2, 3, ,..) (

~T~F~L(33)

Funtction 33 has an extirvu at points i "+ where

15

.P4. function 32 -- has in points pi = 2 s i(s, 1,2, ... ),

The branch of oscillations with a frequency of eq,, 33 passes in the

area pi oti into longitudinal oscillations, investigated above(see formula 26).

Expressions 31 and 33 are applicable in the narrow area at2

Pi s 2 i . In this case instead of eq. 31 and 33 we will obtain

WZ2W+, where

--,-- .. = _1r,,i1/3 i](., - j[ 1l"-']

+,-412 8i 1 zi , ..l8 (34)

+ }' ( I-i) ( -. (,

At an increase in differentl I(l!t)"'2 -i formulas

34 and 34, change respectively into formulas 31, 33 and 31, 32.

M .1 + )IIIZ' A V~t W(35)

where91:t WZ=_ W. + [ V18l

8" /' +)! (/4 - 1A,)I//40

Formulas 35 at IxI l 1changes into formulas 32 and 33°

The dependence of frequencies %(k), determined by formulas

31-36, on the wave vector k is shown in Fig. 6. Let us discuss the

individual sections of curves 1, 2, 3; ... in Fig. 5 in greater de-

tail, Curve 1 in area pi < Y. corresponds to magnetic audio

oscillations with the frequency of eq. 31. In the area p. 0 xi the

frequency of this wave is determined by formula 34) for w+ ( in this

- i16 -

6.t

... y; (a) f-requency; b rv -,-tiector.

case w - i ft -- X ii/2 W if < X- l 2) and at Ii > xi (l iXil Y- i ) by formula 32. Curve i reaches maximum at P, s- 2x, (in

2A

this case w - i- -- ii and then drops, At PiI i curve I has

a minimum a) - i -i ) With a further increase of p. curve

i approaches asymptotically to ai.

The frequency of the branch of ohcillations corresponding to

curve 2 is determined by formulas: 32 at p. < xij at Ii " Ili 34'

f or w+, (in this case u>- u m xi i/2wi, if lI .- " Xi < Xi 2 ) ; 31 -

In area x, < Pi < 4 i; 34 for w_ (s - 2) and 33 in the area 4xi <

< i (s 2 ). Curve 2 reaches maximum -;L Pi= P+ - 6.3x,; (in

this case w - 2 11i " - 1 ) -' and then decreases, converting.

into a branch of longiudinal oscillations, whose frequency tends

towar'd w., when pi e 1o

Curves -2 s - i - 3P 5o 7 whose form at i (0 are

22

ftrmined by formula 32. dCrve i reach maximum at 1 Xi (in

' in this case . - U and then drops.eAty cure hase

A i = 0hefrofteecurves2 is determined by formulas 2 t 35<~~ a

i/2fo 1(nthscs if''~i

32. That curve rt-hehes maximum at + (in this - s mi

Xj s+i wi); it then dec -eases, reaching minimum at pi g 1," (co-s

s ii ED and then approaches s Wi"

Point p is displaced to the right with an increase of s, and

value a2 at an increase of s decreases).

Curves f - 2 s 4 1, 6, ... are determined by formulas: 32

i at i-i < P0 ' 35 fol-J at P, " o (in this case c - s '0m S wi ) ,

33 at <s 2 g' <.x s )3 for a+t at p s Xi;II2 (s± 2 ,3atS (si 2

3iat s (i < (B + i) 34 at p - (s + i)2 i (in this case

it is necessary to replace s by s + i in formula 34, 33 at (s + 1)

x i < Pi i (here it is also necessary to replace s by s + i). in

this area gi %i this curve determines the frequency of plasma

oscillations which decreases with an increase in pi approaching

s co Curve f = 2 s has a maximum at pi. p+ (in this case c -

(s + i) y! - X s + i co). which lies below the minimum of curve

= s + i at point pi. p_, where p_ is determined by formula 33,

in which s should be replaced by s + i.

The obtained expressions for frequencies can be used only at

s2 2 2

Let us note that for curve sections w(k), which decrease with

an increase in k, group and phase velocities of the waves are

directed in opposite directions.

7. Excitation of Electronic Oscillations When Ions CyclotronIJ Waves Pass Through Plasma

When an ion cyclotron wave passes through plasma, electrons

and ions acquire a relative velocity u in the field after that

wave has passed, If u W v., then in the plasma beam instability

may originate [i4]. We will examine the excitation of extraordinary

18

and ordinary waves in dense plasma investigated above by an ion

stream caused by the field of a cyclotron wave. We will assume

that the wave length of.the excited waves.is considerably smaller- .

than the length of an ion-cyclotron wave and the time of develop-

ing the instability is considerably less than the time of turning

the ions in the magnetic field H.. Then the value u(t) can be

considered as a constant value and the effect of the magnetic

field on the perturbed movement of ions may be disregarded.

The contribution of ions into the tensor eiJ in the reading

system in which electrons rest, equals

_____Dig Dk it,i'C(w-kux)'+k'Ui'J w{ -k _)'

f2

Disregarding the components C Eij, we will obtain from the

dispersion equation

dot [(h ( -V 6) + (a"@ + 0)J O (36a)

two equations

(kc*~ (37)L

(c,= +'.ll') (,ko/o)'- (ell + #I&,) (#g. +eo.') - ,, 0 (8

where ii Es 2 2 J s33 and E12 are determined by formulas 3, in which

i9

ii[

I

it is only necessary to consider the contribution of the electrons.

Let us first of all examine the excitation of a usual wave.

Assuming

where ,%(k) is a solution of eq. 37 at i= 0, determined by

formula 6, we will obtain w(k) = kxu, at resonance which

-I + i31, -,.k,,,.,(o,- ev,.} 111 (4 0)E = 2 Llm, (1rP - 0Is m) as '"

At ve (kq e" i) by the order of magnitude e (Xe2m/mi)i/3

5.

In case of the excitation of a longitudinal-transverse extra-

ordinary wave, whose frequency is determined by formula 21, we will

find w s(k) - k ux , in resonance conditions that

a ((-1 +i3,)l2] (,,.,,w/,d)1yl~) @.(11_-,7,), (.4 1 )

At uxft Ves X e± we have e 2 me/mi)/we.

Let us now investigate the excitation of longitudinal

oscillations. The dispersion equation of these oscillations, E.l

+ - 0, has the form

o'i'( l-u. ) - ,_,' (42)

a--r ,. .._,-

FTD.-HT- 23-6?l 67 ,

The component M fl, plays a substan-i.! role Ln equation 42 at a )

k . Assuming cD = k ux+, we will fid thatx

___ __ (413)

Hence it is evident that the exciiation of oscillations (IM r > 0)

takes place at s toe <k ux <ms, where -s is the frequency of M

longitudinal oscillations (E,= 0 at w - (k)). The increment of

increase in expression 43 increases at k ux -0 s. However, ex-

pression 43 can be utilized only at Im - k u~I >Is;4" At Ic -

k Uxj j a the increment reaches a maxiium value

.4 +ia2 P(,i y 08itv (44)-s-IX

If i, Ux' Ve, then E (mem i/ic e At vi qux, ve and

11e 3pi formula 44 coincides with the result of report 14.

In conclusion the authors thank A. I. Akhiezer for evaluating

the work and for useful council.

Literature

i,, Drummond, J. E. Phys. Rev. ±10 (1958) 293.

2. Gershman, B. N. "Zh. eksp. teor. Fiz." (Journal ofExperimental and Theoretical Physics.) 24 (1953) 659.

3. Stepanov, K. N. "Doklad na i konterentsii Fizikotekhicheskogo Ingtituta Akademli Nauk Ukrainskoy SSR po fizike plazmy iprobleme ypravlyaemykh termoyadernyx reaktsiy (Khar'kov, iyun,-iyul,±959); v sbornike Fizika Plazmy i ProbleaW Ypravlyaemogo Termoyadernopo Sinteza (K. D. Sinel'nikov, red.). Izdatel'stvo Akademii NaukUSBR, riev i (962) 52" (Report at the ist Conference of the Physics-Technicil Institute of the Academy of Sciences of the UkrainianSSR for Plasma Physics and the Problem of Controlled Thermonuclear

FTD-HT- 23-671-67 2i

Reactions (Khar'kov, June-July 1959); in the Collection of PlasmaPhysics and the Problems of Controlled Thermonuclear Synthesis(K. D. Sinel'nikov, editor). Publiehing House of the Academy ofSciences of the Ukrainian SSR. Kiev i (1962) 52.

4. Ramazashvil!, R. R., A. A. Rukhadze. "Zh. tekhn. fiz."(journal of Technical Physics), 32 (1962) 644.

5. Demidov, B. P., D. A. Frank-Kamenetskiy. "Zh tekhn Fiz."(Journal of Te_-hnical Physics), 33 (1963) 703.

6. Dnestrovskiy, Yu. N., D. P. Dostomarov. "doklad naVsesoyuznoy konferentsii Ministerstva vysshego obrazovaniya poradioelektronike (Xarlkov, noyabrI 1959); Zh. eksp. teor. Fiz" (Re-port at the All-Union Conference of the Ministry of Higher Educationfor Radio Engineering (Khar'kov, November 1959); (Journal ofTheoretical Physics) 40 (i961) 1404.

7.. Gross, S. P. Phys. Rev. 82 (1951) 2342.8. Sitenko, A. G., K. N. Stepanov. "Zh. eksp teor. Fiz."

(Journal of Theoretical Physics), 31 (1956) 642.

9. Dnestrovskiy, Yu. d., D. P. Kostomarov. "Zh. eksp. teor.Fiz." (journal of Experimental and Theoretical Physics), 41 (1961)1527.I0. Demidov, V. P. "Dokl. Akad. Nauk. SSSR" (Report of the

Academy of Sciences of the USSR) 139 (i9jI) 1342.

1!. K*rper, K., Z. Naturforsch. i2a (1957) 815.

i'-. Demidov, V. P., D. A. Frank-Kamenetskiy, V. L. Yakimenko.j "Zh. tekhn. Fiz" (journal of Technical Physics) 32 (1962) 184.

i3. Kitsenko; A. B., K. N. Stepanov. "v sbornike FizikaPlazzy i Problemy Upravlyayemogo Termoyadernogo Sinteza (K. D.

Sinel'nikov, red.). Izdatel'stvo Akademi- Nauk USSR, Kiev"(In the Report of Plasma Physics and Problems of Controlled Ther-monuclear Synthesis (K. D. Sinel'nikov, editor). PublishingHouse of the Academy of Sciences of the Ukrainian SSR, Kiev 3.(±963) 3.

14. Kurilko, V. I., Vi I. Miroshnichenko. "v sbornikeFizika Plazmy i Problemy Upravlyayemogo Termoyadernogo SintezaK. Do Sinel'nikov, red). Izdateltstvo Akademii Nauk USSR. Kiev"(n the Report of Plasma Physics and the Problems of ControlledThermonuclear Synthesis (K. D. Sinel'nikov, editor). PublishingHouse of the Academy of Sciences of the Ukrainian SSR. Kiev 3 (1963)

Manuscript submitted 4/6/64

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