P/2300 Ukrainian SSR

High-frequency Plasma Oscillations

By A. I. Akhiezer, Y. B. Fa in berg, A. G. Sitenko, K. Stepanov, V. Kurilko,M. Gorbatenko and U. Kirochkin *

It is well known that the electrical conductivity ofa plasma, the ion-electron equilibration time, and thetime required to heat the electron component of aplasma all increase greatly with increasing tempera-ture. Consequently, the usual method of Jouleheating a plasma may be difficult to apply in theregion of high temperatures (> 106 °K), especially ifthe plasma current alone, without any additionalmeasures, is used to generate magnetic fields for theconfinement of the plasma. Therefore, it is ofinterest to study methods of plasma heating that donot directly use Joule heat, especially methods bywhich energy is directly supplied to the ion componentduring the time between collisions. Some of thesemethods make use of ionic resonance as well as otherresonance phenomena which can occur in a plasma inan external magnetic field. This paper deals withcertain aspects of the theory of high-frequency plasmaoscillations.

KINETIC THEORY OF THE OSCILLATIONSOF AN UNBOUNDED PLASMA

IN A MAGNETIC FIELD

The high-frequency properties of a plasma may bestudied most completely by means of the kineticequation, in which it is possible to omit the collisionoperator. This equation may be expressed as followsfor particles of the oc species: ly 2

+ v¿4 . - ^ - = 0 (1)

where /a(r, V, t) is the perturbation of the equilibriumdistribution function, which we will denote as /0a(^2);ел and w a are the charge and mass of the oc species;<x>Ca = \ea\H0Jmac, and H o is the external constantmagnetic field intensity; E is the electric field inten-sity; the top and bottom sign in =р are for ions andelectrons, respectively; and the angles are shown inFig. 1.

It is not difficult to see that the electric field inten-sity E satisfies the following equation :

Original language: Russian.* Academy of Sciences of the Ukrainian SSR.

We will look for the quantities fa and E in theform of plane waves

/a, E ~ exp ¿(fc>r — co't), Im со' > 0. (3)

By substituting these forms into (1) and (2) weobtain

S [n'2 {щкъ - oik) + «*]£* - 0, i = 1, 2, 3, (4)

where

v ч A Trie С, к ) = ôik + \ 4 ^ V i/ ' ¿¿a«8in* + ift«*

/ | СО COca J

X

= -^ kv± sin

f dvjce - iaasin f - ib<x& dipdv,

'a = ± (kv^cosd — a)')/a)a,

n' = kclco', x = k f t / ' = -^«, в = шг;2/2.

The quantities 8ц form the dielectric tensor. Wesee that eik depends on the wave vector as well ason the frequency. In other words, there is spatialdispersion in the plasma as well as time dispersion.

The following relation may be derived fromEq. (4):»

det[n'2(x&k - da) + 8iJc] = An'* + Bn'2 + С = 0, (5)

where

A = s „ sin2 0 + e33 cos2 0 + 2e13 cos 0 sin 0,

В == 2(e12e23 — £22e13) cos 0 sin 0 + e132 —

С = det(fi»).

(5) is used to determine the index of refraction ofwaves propagating in a plasma.

Electron OscillationsIf the ion motion is not taken into account in the

above formulas, we obtain high-frequency electronoscillations.

99

100 SESSION A-5 P/2300 A. i. AKHIEZER

Oscillations at Zero Temperature

Let us consider, first of all, such oscillations whenthe plasma temperature Te is equal to zero. Thecoefficients in the dispersion equation (5) do not dependon к when Te — 0. Therefore the solution to Eq. (5) is

n ± * = [ - Bo ± {В2 - АА0С0Щ/2В0 (6)

where

Ao = 1 — и — v + uv cos2 0,

Bo = u(2 — v)— 2(1 - v)2 — uv cos2 0,

C o = (1 - v)* - u(l - v),

cope2 = Ajte2n/me.

This equation determines the refraction indices forthe ordinary and extraordinary waves in the " hydro-dynamic" approximation.

Setting the coefficient Ao equal to zero, we find thenatural frequency for longitudinal plasma oscillationsin a magnetic field with TG = 0: 6

Cú±2 = 2 (<*>pe2 + COce)2

± t [(«"pe* + ©ce8)2 - 4cOpe2COee2 COS2 (7)

Thermal Corrections

Now let us take into account the thermal correc-tions, assuming that coce ^> kce, where ce = (Te/me)*is the thermal velocity of the electrons. In this case,the dispersion equation becomes4j 5

Affi + (Ao +

where+ Сг)п* + Co = 0 (8)

20 I

(1 - u)2

'2(1 +U-V

- и) cos* 6

cos2(9 sm 2 (9 +

X — n — v)[3 cos2 0 +

1 — AuV

cos2 О sin2 0 + (1 + cos2 0)

sin 2 0 x

3sin 2

1 — u)2

2 sin2

1 - 1 - и

1 — Аи

- cos2 0

1 — Аи •])•c2

cos2 0

2(1 —u)(l+ 2u1 — Аи

sin2 0

[(1 - v)* - u] ((3 cos2 в + f ^

Figure 1

These equations have three roots n±

2, n2

2, n3

2,which determine the refraction indices of the ordinary,extraordinary and plasma waves, respectively. Thefirst two roots are determined by

= П±(1 (9)

where

while the index of refraction nz for the plasma waveis determined by

% 2 = - AQIAlt ce/c < \AQ\ < 1. (10)

If the frequency is near co+ or со-, then 3

n2 = - Со/Бо,

^2,з2 = [- Л ± (\/^о2 - 4ЛАД/2Л!. (11)

Let us consider the case of resonance, whereсо & coce. Here the indices of refraction for theordinary and extraordinary waves are

2 2 /1 I A \

where

^ ^ i r i , x s i n 2 0 , c o s 2 0 b l ^

- [ ( 1 -v){l - lv)(l + cos2 0)

+ (1 - \v) sin2 0 ] ^ ±

2

— 2v - 2 - sin2 0)- 1 . (12)

We see that the resonance waves damp rapidly.The damping coefficient is of the same order as thequantity ce/c, it is much larger than the thermalcorrections to the refraction indices for the ordinaryand extraordinary waves.

If co ^ wcoce, where m = 2, 3, ... (harmonic re-sonances), then

HIGH-FREQUENCY PLASMA OSCILLATIONS 101

where

v /^» /̂7 r //Л2га —3 Qi-n2w —2 Л Р У Г 1 ( 7 2\

m̂ ^^ ^ x\^±^e/^) (1 ~

(13)

It should be noted that the damping is exponen-tially small far away from the resonance frequenciesand is of the same order as the Landau damping: 2

уъ ~ со exp ( — co2/2k2ce

2).

Let us now consider the longitudinal naturalfrequencies of the plasma. The electromagneticwaves in the plasma cannot be divided into strictlylongitudinal and transverse waves in the presence ofa magnetic field. However, for the limiting case ofn -> oo the longitudinal plasma waves, which satisfythe condition A = 0, can be singled out. Taking thethermal correction into account, the roots of theequation A = 0 are given by 3> 6

where

1 + v±u± [1 — u±)~2 sin2 в

X I 3 cos4 в +(l-^)3

0 sin2 в

3 sin4 в

The damping constant for the plasma oscillations isof the order of уъ-

Let us consider the case where the frequency of theplasma oscillations co1 is close to тсосе, m = 2, 3,... .If the angle between the direction of wave propagationand direction of the magnetic field is not near я/2,the resonant plasma oscillations will be stronglydamped. The damping coefficient is 3

У m —я;*m4 s i n 2 w в (kce/a)ce)

2m-

— l)- 2 tan 2 0]kcG. (15)

Plasma waves with frequencies in the intervalтсосе — sm < со < mcoCQ + £m cannot propagate per-pendicular to the magnetic field. Here, the gap inthe frequency spectrum is 7> 3

Sm = (W2 __ !) (2m+i m ! ) - | (kce/coce)m-2 kce. (16)

Ion-electron Oscillations

In the above analysis, we have considered plasmaoscillations without taking the ion motion intoaccount. Now we shall investigate low frequencyplasma oscillations in which the ions as well as theelectrons are in motion. These oscillations areusually described by means of the magnetohydro-dynamic equations. This is valid only when theoscillation frequency is much smaller than the collisionfrequency v. In reality, however, magnetohydro-

dynamic waves may exist for any relationship be-tween со and v. I t must only be assumed t h a t theoscillation frequency is small compared with t h e ioncyclotron frequency coCi

8> 9

I t may be shown 1 0 from Eqs. (1) and (2) t h a t both anordinary and an extraordinary magnetohydrodynamicwave exist for v <C со <С сои. The ordinary wavehas the frequency

m i = kVA cos в; VA

2 = Н0

2/4лп0ш^ < с, (17)

and the extraordinary wave has the frequency

co2 = kVA- (18)

When в ~ 1, the damping constant for the ordinarywave is given by:

/ \ — 2

( y / M ^ [ ~ ) 4 - ~ - z e X P ( - ^A2/2Ce2),

(y/co)1 ~ co2/coci2,

(УHI ~ CÍBCO2IVA2, CI > FA. (19)

The damping constant for the extraordinary waveis given by

Xexp ( - VA2/2ce

2 cos2 в). (20)

These formulas, as well as Eqs. (12) and (15), donot include the damping caused by close collisions.

Formulas (19) and (20) may be used if the dampingis small, i.e., у <С со. The rate of damping increasesas the phase velocity falls. An ordinary magneto-hydrodynamic wave is strongly damped (уг >—' сог1—'ЬУА)> if £Í3&>I2 ^ VA3CO-2. An extraordinary wavebegins to decay quickly if the Alfvén velocity VA iscomparable with the ion thermal velocity c\.

Finally, let us consider low frequency longitudinalwaves.13 In the absence of a magnetic field, thefrequency of these waves is determined by the Tonks-Langmuir formula n

со = co0 = coVikae(l + k2aG

2)~2 (21)

where

nii; aG

2 = Tc/4ne2n0)

and the damping constant is 1 2

У = Го = со0{жте/8пц)% - covikaG{l + k2a2)~2. (22)

Here it is assumed that TG^> T{.

Equations (21) and (22) may also be used whenthere is a weak magnetic field that satisfies theCondition COce <^i kce.

If there is a strong magnetic field, where coCi ^> kcutwo longitudinal waves may propagate in the plasmaat frequencies сог and co2'-

COl,22 = Í{C002 + COci2 ± [(C00

2 + O)ci2)2

- (2o)ocoCiCos0)2]i}. (23)

These waves have the following damping coefficients

102 SESSION A-5 P/2300 A. I. AKHIEZER

У1.2 = (rc/8)*W l j24/Âftïe8| COS 0|3

X[l + tan2 0w1>24(co1)2

2 - a)i2)-2]cope2wPi2. (24)

In deriving (23) and (24) it was assumed that

\co — > KC\ COS (7,

CO ^ > kC{ COS 6,

CO ^> kce COS 6. (25)

At intermediate magnetic field intensities, wherecoce ^> kce, CDQÍ <C kc{ and в is not close to л/2, thefrequency is obtained from (21) and the dampingconstant is

У = yo/|c°s 0\. (26)

WAVE GUIDE AND RESONANCE PROPERTIESOF A PLASMA CYLINDER IN A LONGITUDINAL

MAGNETIC FIELD

In order to ascertain the possibility of using highfrequencies to heat a plasma, let us look into thewave-guide and resonance properties of a plasmacylinder located in a magnetic field directed along thecylinder axis. It is necessary to consider wavepropagation in a bounded plasma since the dispersiveproperties and the electromagnetic field distributionof an unbounded and bounded plasma may dinermarkedly.

Although only kinetic theory gives a completepicture of wave propagation, the basic features of theprocesses that are of interest to us can be found froma simpler set of equations, the two-fluid hydro-dynamic equations

v e X H o

where

P = [(^i2 -

q = ezd/e1;

Ô =[(8, - e

Q dq dq

- k*]t

1 CO2COce2(l — [¿Y — [CO2 — t(Ov

62 CO2COce2(l - A*)2 - [CO2 — COceCOci + i(Jùv{\ + ,

COPe2(l + /*)

co[co /л)] '

СОсе —eH0

4ле2п— , ¡i = •

1Щ

The quantities e2, e2, e3 form the dielectric tensor

/ e± is2 0 \

em = [ - ie2 ег0 •

\ 0 0 e3/

The solution of Eq. (28) that is regular at thepoint g = 0 is

£2 = ^4/o(̂ i£?) + BJ0(k2Q), (29)

where

The remaining components of the fields inside theplasma cylinder are determined by Maxwell's equa-tions:

= = + « E + ^ v , x H Q -

V x H = H en(vi — Ve) H рт-с с от

V X E — - L * .с Ы

— v e ) ,

(27)

where v e and Vi are the velocities of the electronsand ions, we and m\ are their masses, n is the equi-librium density of the plasma, v is the effectivecollision frequency and Ho is the intensity of an ex-ternal magnetic field parallel to the z axis.

Assuming that all the quantities are proportionalto exp i(k%z — cot), the following equations may bederived for the longitudinal components of theelectric and magnetic fields of axially symmetricwaves

?£г = 0,

- qHz = 0, (28)

where

Я =К =

Ç =S =

^ 2 - /e32) + £^2Af,

HIGH-FREQUENCY PLASMA OSCILLATIONS 103

Using the boundary conditions at the surface of the plasma cylinder we obtain the dispersion equation

""*" *2 T (h R ) \J 0\^2 0/ J2£ l

й К IZR \ kl T (k R~) + ki 1 (k R Í

1 1 ^2 \ ^ 1

_ 1 _

where x2 = ^32 — &2 and i?0 is the radius of the plasma cylinder.

The electron and ion velocities for the general case are given by

(30)

+ (oce[o)2 — coci2

Уеф = д— {— œCG[œ2 — COCÍ2 + ia)vju(l + /л)]Е + гсо[со2 — œCi2 + *cor(l +

i(ov{\ + /г)]£ р + а)се[[л{п)2 ~ ojce

2) + '̂cov(l

2) + t(Ov(l + /i)]E p — ÍO)[CO2 — COce2 + icov{\ (31)

where Дх = — //)]2 — [(JO2 — o)cecoci + icor*]2; r* = v(l + ^ ) .

Let us consider Eq. (30) for several limiting cases.I f со ^ C COCÍ, cov <C cjOceCúch t h e n

Here, the components of the electric and magneticfields inside the plasma are

JoikxQ)

£ 2 = COCOpe2/cOceCOci2,

k 2 = £ з (e k 2 — k 2 ) .

. ¿^ / 2acocecoci^

(32)

Assuming (V?o)2 < 1, (^2^o)2 5 Ь ( ^ o ) 2 > 1,and cope2 ^> coce^ci, we obtain the dispersion equationfor magnetohydrodynamic waves in a boundedmedium :

e3kI0{kzRQ) Г

JAhR,) = 0, (¿2#o)2 = V ; ¿P = 3.8..., ..., (32)

from which we obtain a phase velocity

1 +

/с J -i \*^зР ) I

7i(*i<!)

+ ^ | . +.^pe"

у = •»>/«. (33)

If the phase velocity is small (FPh = co/k3 <C c),the solution to the dispersion equation close to theion cyclotron frequency is

1 +

X

со = сон I 1 —С 2 Яр 2

(35)

• (34) w h e r e a = (со — coCi)/a)Cij#Ph2.

— co)/coCi <C 1,

where ^ p h = F p h /c

For ^ -> 0, v = 0 (fi8 -> 00), Eqs. (30) and (34) goover into the results obtained by Stix.14

In the case of very fast waves (j8Ph->oo), thedispersion equation (30) breaks down into two

104 SESSION A-5 P/2300 A. l. AKHIEZER

equations corresponding to two different types ofwaves

£ г — const = 0,Hz = const = Ho,

Ai

1 i( T

A Kj( Т ikR0)

= 0,

: 0. (36)

wherek* = eJi2, k2

2 = etk2;

and for Ro > g > 0,

с le a

Waves with phase velocity c cannot propagate inthe plasma cylinder since the radiation condition atinfinity is not satisfied in this case. However, if theplasma cylinder is surrounded by a metal casing, ofradius Rc, these waves may propagate and the dis-persion equation is

_ s2

»

,2 H'2 1

* 2 Я „ i

£^£я

я„o) _ 0X ~~ u *4- h 7

The corresponding components of the electric andmagnetic fields are, for Rc > Q > i?0,

- V

- V

M-

* - V \

(37a)

/ofeg)

N ¿oy^uj_ iT / г. rt \ { >

(37b)

If the frequencies of the waves propagated in the plasma cylinder are large (со >̂ (coecoi)̂ , the ion motionmay be neglected and the dispersion equation (30) reduces to 1 5

| \ Г »Ç)

i)-e22^] i iffigp) Г Л ( ^ р )

(йй) L 1 / ( ^ )(38)

where

«i = 1 + •

СО ( СО2)

g о — 1 —СОре

Thus, slow and fast waves, as well as waves withphase velocity c, may propagate in a plasma cylinderlocated in a magnetic field.

Equations (28), (29), (29a) describe the penetrationand distribution of the field in the plasma cylinder.It may be seen from these relationships and fromnumerical calculations, that electromagnetic wavespenetrate quite deeply into the plasma if the plasmadensity is such that со2- < соСе

2 <С соре

2. This is con-nected with the gyrotropic properties of a plasmacylinder. From Eq. (29), it follows that the radialdistribution of the field depends on ег and s2 as well

as on £3. Therefore the field penetrates the plasmaeven when sz ы 1 — (cope/co)2 < 0.

Let us now discuss the energy obtained by theparticles in a high frequency electromagnetic fieldnear resonance.

The resonance conditions for a dense plasmadepend on the plasma density and geometry, incontrast to the resonance conditions for free electronsand ions, whose cyclotron resonant frequency doesnot depend on these two factors. This is due to adisplacement of the resonant frequencies and achange in the way the field penetrates the plasma.

Here we shall limit ourselves to the two mostinteresting cases, i.e., when &3—>-oo and whenkB -> 0. In the first case, the phase velocity of thewave approaches zero and the corresponding oscilla-tion frequency coincides with the ion or electroncyclotron resonance frequency. When со -> сось as-suming

«C — C0)/c0ci|

HIGH-FREQUENCY PLASMA OSCILLATIONS 105

the electron and ion velocities are equal to

_ еЕф œQi_ еЕф

ci - СО

= — we/0,

= — ivip.

(39)

We see that the velocities of the ions are muchgreater than those of the electrons. Let us note thatin the case considered, the resonance frequencydepends only slightly on the plasma density and thegeometry of the cylinder (see Eq. (34)) and is thesame as the cyclotron frequency for a free ion.

W h e n со -> coCe> a s s u m i n g

I К ~> 1,

the velocities of the particles are equal to

еЕфеЕф,

— Щ р >

v\p = =

v p = = (40)

I n t h i s c a s e t n e electron velocities are much greaterthan the ion velocities.

Let us now consider the second limiting case kB -> 0, which corresponds to oscillation of the plasma cylinderas a whole. Here the electromagnetic fields do not depend on the z coordinate. The dispersion equation (30),with &3 = 0, reduces to two equations (see Eq. (36)). The dielectric constant e± corresponding to a purelytransverse oscillation i s 1 6

1 (

2(1

(cOce — O)2)(CO2 — COci2) + (1 + ¡bl){cú2 — COceCOci) (cope2 — 2tCOv) + tCOv(l +

With v = 0, the zeros and poles of e± will be located at the following frequencies:

Zeros :

Poles:

C0+ = (Oc ( 1 H\

CO± =

CO+ =

^

dz (i(coce2 + coci

2 + 2cope2)2 - (cocecoci + (Ope2)2)*

2 CO- = COci ( 1 ~\ ^ — I , CO p e2 < C COCGCOcl,

\ C O C O /

CO p e2 COpe ~ J COce,

COci2

CO c e2 , CO_ = COci

~ (cope2

COve2

, COpe2 > COce2, = (cOceCOci)^ ( 1 — £ ' ^ ) , COpe

2 > COce2

\ CO /

(42a)

(42b)

The ion and electron velocities and the corresponding penetration depths are expressed by the followingrelations :

CO = CoJ 1 + ^ — ) ,COc eCOc i /

— cope2

CO =

(

COpe \COc,i

Cují +

\

геЕф сос

_6p

2v

Ope

COpe2

coc

' l p

, V <C

гсосе

V < COce

lp

со р е

2 — icocev

i 2icocev

CO2 = COceCOci(l — C 0 c e 2 / C 0 p e 2 ) , COce

2

f V < C COC

(43a)

coVQ

2 eEp coc

' ^

еЕф сосе

2v ' ^

2v

(43b)

jLlCO^ + icOceV

2coceV

(43c)

106 SESSION A-5 P/2300 A. I. AKHIEZER

геЕфl

_ геЕф (cúce(Oci)%еф

ш с е _ еЕф coCi1ф

v

\COCeCOci

СО2 = COpe2 + COce

2, V < C СОсе'-

VCOve

(43d)

coCi

C O p e

where Еф is given on the boundary.When the external electric field Ep is given on the boundary of the cylinder, the electron and ion velocities

and the corresponding penetration depths are given by the following expressions:

CO = ОМ 11 H ^

1JeE

• \_

CO p e 2

eE (cope4 + 3ivcocicove* - 2v2coci

2) eE (cove

2 + 2icoGiv)eEr

__

^

1 + — ^ I , COpe2 <CCO

eEpeEp(coVe

2

__

2mecoe

2v

eEp(c0ve

2 + 2icOciV)__ eEp

\ COc e, CO p e,

2mevcove

2

^ _ eEp(cove

2 + 2icociv)

V <C COc e :

l

COCe'

2imev(ove

2'

eEp(œ1?e4: — ^eEp

_ _

4t.

p

+

ecoCGE

(44a)

_ с / шр е

2 \*

со à \2vcoCi)

(44b)

2cOCeCOciV2) __ С / COpe2

' COce 2

C O p e

(44c)

2v

CO = COpe — C O p e , V <^ COCe'-

COpe

ieœGeE coGe

COpe

(44d)

со с е\

2v I

EXCITATION OF WAVES IN A PLASMA

We shall now consider the problem of wave excitation in a plasma by means of external currents. Let usbegin with the simplest problem of exciting hydromagnetic waves in a fluid of infinite conductivity. Thestate of the fluid is described by the usual equations

V^ + 'xH f + * ( ) °

HIGH-FREQUENCY PLASMA OSCILLATIONS 107

VxH = (i+iob (46)

where j 0 is the external current density. Assumingthe current to be sufficiently small, we may linearizethe set of equations (45). As a result, an equationwhich determines the velocity is obtained:

82v - V A X { V X [ V X ( V A X V ) ] }

(47)

where Q0 is the equilibrium density of the fluid, cs is thespeed of sound and VA = Ho/ (ÂnQ0)*.

The alternating magnetic field h = H — Ho and thechange in density caused by the wave are determinedby

8Tx/dt = V x ( v x H 0 ) ,

дд/dt = £0V-v. (48)

The change in the total energy of the medium perunit time is

/=(l/c)Jv.(HoXjo)¿r. (49)

The Fourier component v(k, со) of the velocity isobtained from

[со2 - (k.VA)2]v - [(cs

2 + F A

2 ) k - VA(k.VA)]k.v

+ kk.VAVA-v = — Ho x j 0 (50)CQ0

where j o(k, со) is the Fourier component of the externalcurrent density.

Putting the determinant of Eq. (50) equal to zero,we obtain the dispersion equation for the free os-cillations of an infinitely conducting fluid located ina magnetic field. After obtaining v from Eq. (50),we can derive the following general equation for theradiation intensity of three types of waves, onehydrodynamic wave and two magnetosonic :

dj = 8тгБ-^,CO

* )

u2

2 — cs

2 cos2 0

u2

2 — u2 - , в , фsin2 ф

c s

2 cos2 в, 2

« S *

в, ф do (51)

where ulf u2 and щ are the phase velocities of thehydromagnetic and magnetosonic waves:

2 = (VA COS в)2,

в is the angle between the direction of wave propaga-tion and the external magnetic field and ф is the anglebetween j o x H o and k x H 0 . The external currentis considered to be a harmonic function of time.

For a surface current j ,

j = jsà{z) exp — icot,

the total radiation intensity of the hydromagneticwaves is

J. = nVAjs2lc2. (53)

This quantity does not depend on the frequency ofthe current.

For a line current the radiation intensity is

Ji = ncoji2¡2c\ (54)

Equations (44) and (45) may only be used todescribe oscillations at a frequency much less than theion cyclotron frequency.

To determine the intensity of excited waves nearthe ion cyclotron frequency, we may use (27) or thesimpler set17

E + r ^ x H o — (milQec)j x H 0 = mime/Qe2)dj/dt,

VX (V XE) = - c~2d2E/dt2 - Аж-Щ] + jo)/8t. (55)

This set of equations is already linearized. Thesecond equation takes the place here of the relationE + (vxH 0 )/C = 0, which is Ohm's law for aninfinitely conducting medium. Collisions are nottaken into account in Eq. (55) for the sake ofsimplicity.

It may be shown that the Fourier components ofthe electric field are determined by

- и2 sin2 в)]} (56)

X w.2 sin2 в sin ф cos ф

- /3А

2Й1е(1 - «2)(1 - И2 Sin2 в Sin2 ф)]

+ рАЦ1 - f,fe)[l - /SA2fiíe(l - П* Sin2 в)]},

where

Д = An* + Впг + С, v = kc/co,

/Зд = У А/С, fi = a>/cOá, fe = Cú/cOce

А = £Д

4{(1 - /SA2fife)[(l - fife)2 - fi2]

- (1 - f i f e - f i 2 ) Sin2 в},

B=- /SA2{2(1 - /?A2fife)[l - fife

+ £А

2(1 - fife - fi2) Sin2 в]

- [1 + ДА2(1 - fife - fi2)] Sin2 в,

С = (1 - j8A8£,fe){j8A*[(l - íife)2 - fi"]

+ 2^A2(1 - fife) + 1}, (57)

± VA2) — (2CSFACOS б)2]*}, (52) where the directions of the axes are given in Fig. 2.

108 SESSION A-5 P/2300 A. I. AKHIEZER

Figure 2

Taking A to be zero, we obtain the indices of re-fraction for waves that can propagate in the medium(pressure effects are neglected here) :

%,22 = 1 + Jl - l i f e

2(1 -

sirr- /3A2|ile

2|i(l - iÎA2|ile

X ,3A-2 \ (1 - lile)2 - li2 - --f—

'sin 4 6»V

(58)

The total radiation intensity J is determined byEq. (49). Substituting Ey from Eq. (56) into (49),we obtain the following general equation for / :

- ^ A 2 l i l e Sin2e SÍnV][(l - lile)2 - li2]

+ «2/3A2lile(l - lile Sin2 в)}

1 1X . (59)

where n = kc/co.Let us consider in more detail the excitation of

waves by the surface current jo(r) = Jod(z) exp—¿со/.In this case, the intensity of radiation per unit surfacearea is

/ = ./-c(n1 + n Ù \П1П2 V

i 1 !Ii2]j

where

- l i le)2 - !i2]J Г j

(60)

И Шв dz 1) = (1 + /?A2)/̂ A2, the resonance con-dition is satisfied and / tends to infinity. Theresonant frequencies for the case of surface currentexcitation are

where(61)

cop e

2 —

Let us now consider several limiting cases. Iff e < l , then

1 1 + iSA

a(l — ft2) — fife

c(% + «2)\%«2 £ A 2 ( 1 - ! Í )

In this event if | ¡ 2 -C 1, then

, then

(62)

*• (63)

If

If f i 2 > l , then

- fi)], ni = (1 + 2|8А2)/2|8А2,

/ = Я + 2)*. (64)

+ / 1 -

1

х i + -(SA 2 ! Í +

Г"

1/SA 4 ! Í 2

(65)

Let us now consider the limiting case f e -—' 1( f i> 1). Here V = 1 - {2pA4i)-\ п2 = {1+ ¿ЗА 2 )/^ 2

and

л : 7 п

2 1

i +1 +

2,3A2

1(66)

Finally, if fe > 1, then

Пх

2 = П2

2 = 1 — (cOpe/co)2

17 = С [1 - (ft)pe/ft))2]i '

(67)

HIGH-FREQUENCY PLASMA OSCILLATIONS 109

If the plasma is excited by a harmonic line current,we obtain Eq. (54) for the radiation intensity perunit length.

ENERGY LOSSES AND COHERENCE EFFECTSt

The possible existence of slow electromagneticwaves in confined and unconfined plasmas is signi-ficant not only from the standpoint of the absorptionand propagation of external electromagnetic radiation,but also because slow waves in a plasma can increasethe effectiveness of interaction between the chargedparticles and the plasma, for both the ordered anddisordered motion of these particles. The interactionof individual particles, bunches of particles, or beamsof particles with slow waves may turn out to beresponsible for a number of physical phenomenawhich occur in the plasma. The elementary processesof interaction between the charged particles and thefields excited by them result in composite Cherenkovand Doppler effects and also in polarization losses.It is also necessary to point out a peculiar radiationeffect which may take place when the velocity of auniformly moving particle satisfies the inequalityv < FPh but the medium is bounded in the directionof motion of the particle. For this radiation it isessential that the width of the dielectric layer throughwhich the particle moves be less than A/2. This

effect was demonstrated by S. I. Vavilov, but it isvery difficult to observe in the visible region of thespectrum. When there is an interaction between acharged particle and a bounded plasma it may beof considerable importance.$

However, it should be pointed out that the energylosses of an individual charged particle by the excita-tion of various plasma oscillations are very small,owing to the comparatively low plasma density.For example, when the particle moves uniformlythrough a plasma in a magnetic field, these lossesare as follows: 19> 2 0

For weak magnetic fields, the energy loss per unitdistance is

dW __

dx

where q is the charge of the particle, v its speed, and& is a transverse dimension of the plasma.

For strong magnetic fields,

In

dWdx 2v2

_kn№__

For plasma with a density n = 1010 electrons cm"3

and with an energy of 50 kev the loss is

dW—— ^ — 10~7 ev/cm.dx '

The energy lost by a particle moving in a plasma wave guide without a magnetic field has been calculated : 21> 2 2

~lx = Î;2£3(1 — fi3)"

X ixal^a^K^xa) - x2oa*KQ{>ca)I0{oa) + (2/1 - р*е3)1г{аа)К0(ха) J

where

K1(XOL)IQ{GU) = — (e2xl(j)K0(xa)I1((7ci),

x2 = k3

2 — k2, and G = kB

2 — e3k2. This agrees with the measured loss in order of magnitude.

When a particle moves in a spatially periodicplasma (e.g., in plasma in which there are sausage-type coagula) a parametric Cherenkov effect takesplace. In this case the energy loss is: 2 3

Due to Cherenkov loss,

If longitudinal electron oscillations are excited in theplasma, when v > F T the loss is 2 5

dW1 Í ymvz F T 2 \

dW;\ =

/ Cher 2v2L

Lkm2vm

2

ЪАппв2

Due to polarization loss,

/dW\ _ __

\dx)voi ~~ v2LIn

kmb

It should be noted that in the case of anisotropicor gyrotropic plasma, Cherenkov radiation occurseven when v -> 0 since the condition for radiation is 2 4

For plasma with a density n & 1012 — 1014 cm-3,the energy loss of the charged particles is small.However the magnitude of this energy loss (or,alternatively, of the energy gain of the high-frequencyfield) may be substantially increased by utilizingthe coherence effect ; it was Veksler26 who firstpointed out the possibility of using this for particleacceleration. If the wavelength of the wave withwhich a bunch of charged particles interacts is greaterthan the dimensions of the bunch, the energy loss

t Cf. К. D. Sinelnikov et al., P/2211, this Volume of theseProceedings, for related experimental information.

î For the case of interaction between particles and aresonant cavity the question was examined by Sinelnikov,Akhiezer and Fainberg. Cf. P/2211, this Volume of theseProceedings.

110 SESSION A-5 P/2300 A. I. AKHIEZER

of an individual particle increases N times, where Nis the number of particles in the bunch. This followsformally from the fact that the loss ~ q2, whereq = Ne is the charge of the bunch.

Taking into account the conditions of coherence,it can be shown that the maximum value of the fieldwhich can be excited by a bunch is E fía Ne/Lmax

2,where Lmax is the maximum dimension of the bunch.It follows that

dW/dz ъ Ne2/LmSiX.

Thus even in very rare plasma (n ~ 1010 cm~3) theenergy loss of the particles increases 108-1010 timeswhen the number of particles in the bunch isN ~ 109. It should be noted that in many casesthere is no need for initial modulation of the beam inorder to realize the conditions of coherence. Thus,for example, when a beam of charged particles movesthrough a plasma with a velocity v0 > FT, it becomesunstable, the resulting density and velocity fluctua-tions produce increasing fields, and the beam ismodulated at the same time. The rate of growthof these oscillations is 2 7

о2 COQ /COpe, beam\ g"

21 F ïe \ ft>o /

When a beam excites longitudinal oscillations of aplasma in a magnetic field,22

y±co = 3*2-1X[C0pe, beam2 COS2 6co±(cO±

2— COœ2)

X {2CO±2 — COce2 - COpe2)-1]*

where the frequency со is given by

CO2 fía k2V2 COS2 О Ъ i (Шее2 + COpe2)

± i [(cOce2 + COpe2)2 - 4cO c e

2 0) p e

2 COS2 0 ] * .

Similar phenomena occur when there is an interactionbetween several beams. For example, in the caseof two beams with Maxwell distributions there is anoscillation growth rate 2 9

where v is the average velocity (v± + v2)/2 of thebeams, co0 is their Langmuir frequency, coopt

2c52

— |co0

2^2, and ô = vx — v2.

If the densities of the beams are different and if thetemperature of the exciting beam is high, instabilityoccurs at the Doppler-shifted frequency 23

со = coj[l - К Ю ]

where co2Q is the Langmuir frequency of the denserbeam 2 and v1 and v2 are the velocities of the beams.

In a number of cases (for spatially periodic plasma,non-uniform plasmas or plasma in the first stage ofinstability) coherence is realized not because of amodulation of the beam, but because of a modulation(spatial non-uniformity) of the medium.

In any case, when the conditions for Cherenkovradiation, Doppler radiation, or polarization losses arefulfilled for the individual particles, the beam ofparticles and the plasma through which it movesbecome unstable. This follows from the circumstancethat the growth rate of the oscillations is greatest forfrequencies satisfying the conditions of Cherenkov orDoppler radiation. For example, when the beampasses through a plasma, у = ymax for со = cope

(1 - FT 2 /?; 2 )-*.

These mechanisms of collective interaction canexplain instabilities in different types of dischargesand the associated oscillations in the plasma. Theycan also account for the short relaxation time of theMaxwell distribution. Finally, the increase in energyloss of bunches of charged particles can be utilized forinjection into magnetic traps and for measurement ofplasma parameters. These new methods are basedon the fact that the energy loss of the particles of thecoherent bunch can be made very large when theenergy loss of particles of an unmodulated beam arevery small. For example, with n = 1010 cm-3,N = 109, and the electron energy W = 50 kev,dW/dx = 1 0 0 ev/cm when for the unmodulated beamdW/dx = 10~7 ev/cm. The energy loss may befurther increased if coherence between bunches isutilized. In this case the distance between bunchesmust be L = f$X where /? = b̂unch/c and Я is thewavelength of the excited wave. It should be notedthat to facilitate coherent losses it is necessary toturn to longer waves (different types of ion oscilla-tions). By using coherent interaction it is alsopossible to excite a desired frequency in systemswith many frequencies. For this the conditions ofcoherency must be fulfilled: the dimensions of thebunches should satisfy the condition Хг > a > A2 whereAx is the wavelength of the wave to be excited.

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HIGH-FREQUENCY PLASMA OSCILLATIONS 111

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