P/2215 USSR

Absorption of High-frequency Electromagnetic Energyin a High-temperature Plasma

By R. S. Sagdeyev and V. D. Shafranov

In a plasma of high temperature, the collisionfrequency is very low and there is a decrease in theeffectiveness of known heating mechanisms whichare based on collisions. The problem then arises ofnoncollisional plasma heating with high-frequencyelectromagnetic fields. By the term high-frequencyfield, we mean electromagnetic oscillations whoseperiod is much smaller than the characteristic confine-ment times of the hot plasma.

The interaction of a high-frequency field with aplasma can also be used as a probe for studying itsproperties.

The solution of these two theoretical problems isassociated with an investigation of the penetrationof electromagnetic waves into a plasma, i.e., with ananalysis of the electromagnetic absorption spectrumcaused by various mechanisms and with the investi-gation of the rôle which these mechanisms play inheating of plasma.

In this paper an analysis of the cyclotron andCherenkov mechanisms is given. These are twofundamental mechanisms for noncollisional absorptionof electromagnetic radiation by a plasma in a magneticfield. The expressions for the dielectric permeabilitytensor, sao, for a plasma with a nonisotropic tempera-ture distribution in a magnetic field, are obtained byintegrating the kinetic equation with Lagrangianparticle co-ordinates in a form suitable to allow acomprehensive physical interpretation of the absorp-tion mechanisms.

Using the tensor ea/? and taking into account thethermal motion, the oscillations of a plasma columnstabilized by a longitudinal field have been analyzed.For a uniform plasma, the frequency spectrum hasbeen obtained together with the direction of electro-magnetic wave propagation when both the cyclotronand Cherenkov absorption mechanisms take place. Theinfluence of nonlinear effects on the electromagneticwave absorption and the part which cyclotron andCherenkov absorption play in plasma heating have alsobeen investigated.

Any mechanism responsible for the absorption ofelectromagnetic energy involves the accumulation

Original language: Russian.

of energy by a charge in an electric field and thesubsequent redistribution of this energy among thevarious degrees of freedom in the random motion.Absorption mechanisms can be physically classifiedas noncollisional or resonant processes and collisionalor nonresonant processes.

A continuous accumulation of energy by a chargeis possible in a static or quasi-stationary field if thereis no dissipation. But if the charge is in a high-frequency field, it is continually accumulating andlosing energy alternately. If plasma heating is tobe effective, it is necessary that the characteristiccollision time required for energy dissipation becomparable with the oscillation period. The corre-sponding absorption mechanism, which is associatedwith frequent collisions, will be called here the non-resonant or collisional mechanism. It includes ordi-nary Joule heating as well as betatron or gyrorelaxa-tional heating proposed by Budker in 1951 andindependently by Schlüter.1

If the collision time is much longer than the oscilla-tion period, the continuous effective energy accumu-lation by a charged particle will be possible onlyunder resonance conditions when the direction of theelectric field coincides with the direction of increasingvelocity. This mechanism will be called resonant ornoncollisional absorption.

In principle, resonances are possible in the followingcases.

(A) If a charge performs a periodic motion therewill be a resonance when the field frequency coincideswith the motional frequency.

(B) For a charge in a magnetic field there is anion or electron cyclotron resonance. If the particlevelocity along a line of force is vz, the frequencyspectrum of the absorbed radiation is determined bythe Doppler effect according to the relation

CO — kzVz = COce, (1)

where kz is the z component of the wave vector,COce = eH/mc and the other symbols have their usualmeaning.

(C) Resonance can also take place in the absenceof a periodic motion provided that the charged particle

118

ABSORPTION OF ELECTROMAGNETIC ENERGY 119

velocity is equal to the phase velocity of the electro-magnetic wave. This is, of course, possible only forslow waves.

(D) If there is no magnetic field, the resonantaccumulation of energy by charged particles takesplace only for longitudinal electromagnetic plasmaoscillations. The corresponding absorption causedby electron oscillations is known as Landau damping.Damping of a similar type takes place when thereare ion (sonic) oscillations.

(E) For a sufficiently dense plasma in a magneticfield, resonance for the transverse branch of the electro-magnetic oscillations is also possible. The index ofrefraction is of the order N ~ a)pe/coce, where cope

2

= 4тге2п/т, so that the electromagnetic phasevelocity is ^Ph = (H2l4uim)z. The ratio of the phasevelocity to the thermal velocity of the charged particleis v-pb/vT = (H2/87zP)* = p-i. Therefore, when theplasma pressure P is approximately equal to themagnetic pressure, the wave velocity and the averagethermal velocity are of the same order. Under theseconditions, the inverse Cherenkov absorption mecha-nism becomes substantial.2 A charge moving along amagnetic field is in resonance with the wave fieldwhere the following condition is satisfied

coordinates. The correction to the distribution func-tion which one obtains in this way is given by

CO kzvz = 0. (2)

If co/kz < c, this condition, as well as condition (1),determines a velocity of the charges which will absorbenergy.

Because of the spread of thermal velocities, thefrequency range for which absorption takes placemay become very broad for both cyclotron andCherenkov absorption.

There is one fundamental difference betweencollisional and non-collisional heating. In the firstcase, when collisions are important, the velocitydistribution during the heating process is nearlyMaxwellian while the temperature is rising. In thesecond case, when only the charges moving in reso-nance accumulate energy, the velocity distributionduring heating will differ greatly from the MaxweHiandistribution. In this case, the term heating will meanan increase in the average random energy of theelectrons and ions.

The interaction of an electromagnetic field witha plasma is determined, in principle, by the Maxwellself-consistent equations together with the kineticequations for the electron and ion plasma. Under theassumption that the velocity associated with theregular motion of the charges in a wave field is smallerthan the average thermal velocity, these equationsare linear in the field quantities. In this case, manyproblems concerning plasma oscillations can besuitably solved by using the dielectric permeabilitytensor eap = s'ap + ie"ap, whose antihermitian part(ie"ap = {eap — £*у?а)/2) determines the absorption.When there are no collisions, the general form of thedielectric tensor can be expressed explicitly providedthat the kinetic equations are integrated in Lagrangian

Xdfo/8p(t')dt'. (3)

If the fields vary harmonically in space and time, i.e.,as exp ¿(k-r — cot), the corresponding tensor ea^will be

/ (4)

with real and imaginary parts

The summation is carried out with respect to thedifferent species of charged particles. The functionKap{t) for each species, is

4ne2n

X [exp i (cot - k*rv(t')dt% (5)

where v(t) is the particle velocity in the unperturbedplasma expressed in terms of the initial velocity v andthe time t. The integration is carried out with respectto the initial momenta. For the particular case ofthe Maxwellian distribution, the expression in bracketsreduces to the simple expression 2 Vpfo/T.

For a plasma with no collisions in a uniform mag-netic field directed along the z axis, we have

Vx(t) = Vx COS COcet — Щ SÙ1 COcet

Vy(t) = Vx Sin COcet + Vy COS COcet (6)

The quantity in the exponent of (5) can then beexpressed in terms of (6)

O)t — k- | tf = (CO — kzVz)t — (kxVx/u)ce) Sin

+ {kxVy/cúce){l — COS COcet), (7)

where к = (kx, 0, kz).

If the unperturbed distribution function, /0, is aproduct of Maxwellian distributions with differenttemperatures corresponding to particle motions paral-lel and perpendicular to HO(TII and T±), the x, уcomponents of the expression in the brackets of (5)will be

and the z component is

• +1 kxvx(

Г,,

120 SESSION A-5 P/2215 R. S. SAGDEYEV and V. D. SHAFRANOV

By substituting these expressions, together with(6) and (7), into (5) and (4), closed formulae arereadily obtained for the tensor eap.

In a strong magnetic field and for relatively lowfrequencies, со < coCe, the expression for ea^ is greatlysimplified by expanding the exponent in powers ofkxv±/(oCe' In the zero-order approximation, we have

cot — k«Jv^¿ — (со — kzvz)t and the tensor will be

cop 2 [ h i) ( T ХЛ

COp" / Ш V/

со \ Г „ô+ (со ~ kzvz) \

where the possibility of anisotropic unperturbedtemperatures is taken into account. The brackets< У denote an average over a Maxwellian distributionalong the magnetic field and the quantity ô+ {%)= i/x + 7iô(x) is defined in terms of the Dirac deltafunction. For given со, k, the argument of the deltafunctions determine the particle velocities for whichthese particles are resonant with the wave field.

T

\ 0 0 , /

The components e, g and r\ are given by the expressions

Гч kZVZ(^ T A I - , , 7 Ч \

- te) + i i — T— o+ (со + coc — ад >,L M \ rii/J /

A comparison with (1) and (2) shows that thecyclotron absorption resonance is determined by theimaginary part of г and g, the Doppler effect beingtaken into account; the Cherenkov absorption, asso-ciated with particle motion along the lines of magneticforce, is determined by the imaginary part of rj.Averaging over the longitudinal velocities yields thecomponents of the tensor sap

7II /

X 1 — -

COH\\ 17Z»(O( CO — СОН

_ T4- ^ 1 ^ —

kzv „

yzv{l

A-77 = 1 +

where г;,, = (2TJm)* and

W(z) = e-z\l +

is the probability integral with a complexargument.3 The approximation k±v±lcoCQ <C 1 madein (10) is valid provided that the Larmor radius issmall compared with the characteristic transverselength in the problem. In this approximation it iseasy to obtain a solution to several problems associatedwith plasma oscillations in a uniform magnetic fieldeven when the extent of the plasma is limited in thedirection perpendicular to the field.

Let us consider the oscillations in a pinched plasmastabilized by a longitudinal magnetic field, taking intoaccount the thermal motion of the charged particles.The plasma pressure can be counterbalanced by themagnetic pressure of the longitudinal and azimuthalexternal fields. The azimuthal field is produced by

a longitudinal current flowing on the surface of theplasma. The oscillations are taken to be of the formexp i(kz + тф — cot). The basic equations are theMaxwell equations

V(V«D) - ДЕ = (co/c)2D (И)

where Dr = sEr + igE;^, Вф = — ¿g£r + s£^ andD2 = r\Ez. The boundary conditions are :

(1) The tangential components of the electric fieldon the surface of the pinch are continuous and vanishon the surface of an infinite conductivity (metal)cylinder which surrounds the plasma column.

(2) The normal component of the pressure tensoris continuous at the surface of the perturbed plasmacylinder.

Solutions of the equations for the electric field areexpressed in terms of Bessel functions whose argumentis кЬ2 v> where

ABSORPTION OF ELECTROMAGNETIC ENERGY 121

*i, 2

2 = [J?(e«2/c2 - k2) + ig(ig - еЯъ 2)(са/с)2]/е

and the variables Ax and кг are roots of the equation

¿2 + ^ [ g2 + [{kc/œ)2 __ в ] ( ч _ e )]( g e )-l

+ [(^/co)2 + ч - e] = 0.

Neglecting the thermal motion with k = 0, a disper-sion equation first deduced by Kôrper 4 can be ob-tained.

Let us now investigate cyclotron absorption byions. For frequencies satisfying the condition

kc, coc со

(where VA is the Alfvén velocity), an expansionin 7]~1 can be made. If the plasma pressure at thepinch boundary is also neglected in comparison withthe magnetic pressure, the dispersion equation can bewritten as follows for the m = 0 mode

(12)1г{(ха)

= 1 - he

2ka -

whereh = (Нг){/Нф(а), he = (Ще/Нф(а)

- k2],

and a, Ъ are the radii of unperturbed pinch andexternal conductor, respectively. With Нф = 0,b = oo, (Hz)e = {Hz)\y and neglecting the thermalmotion in s and g, this formula agrees with a resultdeduced by Stix5 for the oscillations of a cold plasmacylinder in a longitudinal magnetic field.

In contrast to other work,4' 5 the influence of thethermal motion on noncollisional cyclotron dampingis taken into account. The dispersion equationpermits us to find the imaginary part of the frequencywhich determines the rate of cyclotron energy ab-sorption by the plasma. Then, knowing the expres-sions for the electromagnetic field, one can determinethe amount of energy absorbed by the plasma.

There is no Cherenkov absorption in Eq. (12). Totake this into account, the higher-order terms in therf1 and kjvJcoH expansions should be evaluated.The Cherenkov absorption of low-frequency waves in apinched plasma is discussed elsewhere.6

The analysis of the complete dispersion equationfor a plasma cylinder is a rather complicated proce-dure. It is of interest to investigate first the absorp-tion spectrum for the simpler case of a uniform plasma.In this case, the wave damping is determined by theimaginary part of the index of refraction, N —p{\ + ix). With the above assumption, namelyk±v±/coc <C 1, the indices of refraction of both theextraordinary and ordinary waves, Nx and N2, areexpressed in terms of eap by the same formulae whichwere employed when thermal motion was not takeninto account. Therefore, for the propagation acrossthe magnetic field of a linearly polarized wave withits electric vector perpendicular to the constant

magnetic field, one finds N-f = (s2 — g2)/e; whilejV2

2 = rj for waves with the electric vector along thedirection of the magnetic field.

In the case of transverse wave propagation, kz = 0,there is no absorption and the formulae for sao takethe same form as when the thermal motion is not takeninto account. However, even for purely transversewave propagation, there exist absorption regions in theplasma that are beyond the range of validity of theassumptions made in order to obtain the expression(10) for the dielectric tensor. For example, thecalculation of the relativistic Doppler effect shows thatthere is cyclotron absorption with higher harmonics.

For frequencies substantially greater than coce andin a frequency range where Re N ^> 1, Cherenkovabsorption will also occur when the thermal motionof charged particles across the magnetic field is takeninto account. Formally, this result follows from thefact that with со ^> œce it can be assumed in Eq. (7)that sin cocei — CJOCQÍ and, as a consequence, one obtainsсо = kxv± for the resonant frequency. This is justthe condition for Cherenkov radiation and the cor-responding Cherenkov absorption. For waves pro-pagating in the direction of the magnetic field therefraction index squared is iVlj2

2 = s ± g and, there-fore, has no component r¡ which can give rise toCherenkov absorption. The absence of Cherenkovabsorption for both longitudinal and transverse waveswhen со < coce is connected with the fact that accord-ing to the theory of Cherenkov radiation, there can beno (Cherenkov) radiation in these directions from themotion of a charged particle along a line of force.For frequencies со ^> coGe> the conception that thecharge is " attached " to a line of force is not validand, therefore, transverse Cherenkov radiation is alsopossible.

Let us now determine the absorption edges whereIm N < R e i V and Im s <^i Re s ; then one can make anexpansion in terms of the imaginary parts of e, gand rj and obtain the index of refraction in the formN = ft(l + ix). For longitudinal wave propagation,when » < 1 , the cyclotron absorption by electrons isdescribed by the following expressions 2

(Jo

ft)pe\2 —(a>c/vtp)3,

where

CO = ((JO — C0Ce)/c0. (13)

For oblique wave propagation, the absorption edgeis caused by Cherenkov absorption by electrons withсо <C coce, where the cyclotron resonance absorptionis no longer significant. With the assumption that

ande2 > 1

Sin 4 0/cOS2 в < 4сОре4/сОСе2СО2,

122 SESSION A-5 P/2215 R. S. SAGDEYEV and V. D. SHAFRANOV

we have

N2 =C0[i¿>ce COS в — O)]

X /i(0 *1* S i n 2 б СоГбОсе COS 6 — 0)] I С

v/ e x D L J j[C0c eC0S в — О)]? COS в C0 p e

2 COS2 0 \V {]

(14)

¿» =

For ion cyclotron resonance, when |0 — я:/2|̂ > (m/M)i, we have the relations

copi coci(l + cos20) ± [coCi2 sin4fl + 4co2 cos26]i

CO c i 2 — CO2

4 \ œ J v у ръ cos iexp -

2coCi cos20

— coCi]\2'

Xcos2(9)[coci

2 ± [coci sin40 cos20]

2 cos2(9[(coci2 si 4w2 (15)

For oblique wave propagation in the regionсо <C (сон)ъ there are also noncollisional absorptionswhich have been studied by Braginsky andKasantzev.* Unity can be dropped from theexpression for 2V2 in sufficiently dense plasmas when(соо/соя)2 ^> 1 ; the absorption then depends only onthe ratio of the electron or ion pressure to the magneticpressure

j8e = 8лпТе/Н2

Pi =

for ne — m == n.

If we now introduce the variables x ^ OJ/COH andz = [(со — сон)1сор~\(с1Уц), then Eq. (13) has the form

1(1 - х*У1% = ip^ir* fZ e~i%dt,jo

fiv^/c = (1 — x)/zx,

[§* y\ (13a)

The absorption described by the above formulaecovers a rather wide frequency range, even when theplasma pressure is small compared to the magneticpressure. As an example, for an appropriatelypolarized electromagnetic wave propagating along auniform magnetic field, the absorption edges for ionand electron cyclotron resonance depend on /?i andpe in the manner shown in Fig. 1. It can be seenthat the relative absorption width, Aco/coc, is quitelarge even when the temperature is comparativelylow. Therefore, when fie = 10~2, the absorption atten wavelengths occurs in the range Дсо/сос = 0.3(the " plasma wavelength" is А=2пс/щ). It can alsobe seen from an examination of Eqs. (14) and (15)that there is a broad frequency range where ion andelectron cyclotron absorption can occur for obliquewave propagation.

* Ed. note : See S. I. Braginsky and A. P. Kasantzev,Magnetohydrodynamic Waves in Rarefied Plasmas, PlasmaPhysics and Problems of Controlled Thermonuclear Reactions,AN SSSR, Vol. 4, p. 24 (in Russian), Edited by M. A. Leonto-vich, Moscow (1958).

It seems inviting to use the specific dielectricproperties of a high-temperature plasma in a magneticfield for actively or passively probing the plasma.For example, black-body intensity measurementswould constitute a useful passive probe, since it canbe seen from the formulae and graphs presented herethat there are extensive " black " regions of thespectrum associated with ion and electron cyclotronresonance radiation. It follows from the expressionsfor e^ (see Eq. (10)) and iV2 that " b l a c k " regionsof the spectrum also exist for oblique waves. In thiscase, even when thermal motion is not taken intoaccount, absorption occurs since the index of refrac-tion greatly exceeds unity because of Cherenkovradiation. A black spectral region can also be ex-pected in the vicinity of iV2 = 0, where the longi-tudinal plasma oscillations are transformed into trans-verse electromagnetic waves. The various mechan-isms which give rise to relationships between thetransverse and longitudinal waves at a plasma boun-dary are reviewed elsewhere.9

It is possible, in principle, to develop a probebased on an investigation of the shape of the absorp-tion spectrum. Since the width of the absorptionresonance depends on the parameters P\ and /?e> thiskind of investigation could be of use in measuringthe electron or ion pressure. In practice, however,the nonuniformity of the magnetic field which isalways present will complicate such probing.

It has been demonstrated that electromagneticenergy will be effectively absorbed by a plasma inspite of the absence of collisions between particles.However, our linear approximations fail to give ananswer to the question of how this energy is distri-buted between the plasma ions and electrons. Evi-dently, in noncollisional absorption an increase ofthe energy of the plasma cannot be immediatelyidentified with a temperature rise. To solve thisproblem one must resort to nonlinear considerations.

When one investigates wave propagation along thefield it is possible to slightly improve the linearconsiderations which have been employed previouslyin analyzing cyclotron resonance absorption. Let usconsider frequencies close to the electron Larmor

ABSORPTION OF FI ECTROMAGNETIC ENERGY 123

0.2

0.1

•

1II

f! /1

h l/s /1 X I1 "> I/ 7/ i1 pi 1

1 IIi)\ II*

: / r-0.2 0.4 0.6

U > / ( U > H ) ¡

Figure 1. Dependence of electron and ion cyclotron resonance absorption edges on /?e and /?¡, respectively

0.8

frequency, (сон)е = eH/mGc, when the ions can beregarded as being at rest. In the kinetic equationfor electrons we shall neglect the term containing themagnetic field. This approximation can be partiallyjustified by the fact that in the ordinary linearapproximation, with an unperturbed isotropic Max-wellian distribution, a corresponding term is absent;namely, vxHdfJdv = 0 with /0(v) - /0(va). If allthe variables depend only on one spatial coordinatedirected along the constant magnetic field, then asolution to the problem with the initial conditions/(v,*,0) = /ofe2 + vv\vz*), Ex{z,0) = EJ>{z),Ey(z, 0) = Ey°(z), Ez(z, 0) = 0 can be obtained fortransverse waves by " integrating along trajectories ".The appropriate kinetic equation and wave equationare

df/dt + vdfjdz + {eE/m).8f/dv (17)

+ (e/mc)vxK*df/dv = 0

- V x ( V x E ) = (l/c2)£2E/£¿2

= {4ne/c2)d(ívfdv)/dt. (18)

The following result for the distribution functionis obtained from these equations

f{vx, Щ, vz, z, t)

where

Vl(t} z) = — cos œct I [Ex{f, z') cos

)(19)

J'1Jo — Eyit'.z') sin coHt'W

H Sin Wc¿ I [ ^ ( ^ , Z1) COSw I

Jo

+ Ex(t', zr) sin

^ = z - ^( / - t') (20)

and г>2(̂ , 2") has a similar form.Equation (18) is linear in the field quantities and

can be easily solved by the Fourier-Laplace methodwhen the longitudinal velocities have a Maxwelldistribution initially. From (19) and (20) one canalso see that the average random thermal energy will

not increase during the wave damping, i.e., the fieldenergy will appear in the kinetic energy of the orderedmotion of the plasma electrons. This is because theelectrons which receive energy from the high-frequencyfield are always in phase with the field and, therefore,with one another. As a result, the thermal distribu-tion of electron velocities associated with the distribu-tion of the phases of their Larmor rotation will remainunchanged. To transform the energy of regularelectron rotation into energy of random motion, it isnecessary that the phases be " mixed ". In additionto phase mixing due to the extremely rare collisionsin a high-temperature plasma, randomization of phasescan be caused by the nonuniformity of the magneticfield along a line of force. This can occur becausethe electrons rotating at different places and atdifferent frequencies, шсе, have their phases mixeddue to the thermal motion along the line of force.

In the above case of cyclotron resonance the energyis absorbed by charged particles moving along aconstant field with a velocity vz = (coG — co)/kz.Here, the energy in the transverse motion increasesand, therefore, a change takes place mainly in thedistribution of transverse velocities. Since thenumber of particles which are in resonance is deter-mined by the longitudinal velocity distribution (whichis almost unchanged), the absorption takes placecontinuously.

The situation is different with Cherenkov absorptionsince during the process of absorbing energy there isa change in the longitudinal velocity distributionwhich determines the resonance of the chargedparticles. Here the particles are brought out ofresonance and a balance may occur between theabsorption and emission of radiation for certainparticles. Mathematically, this is expressed by thevanishing of the derivative dfo/dvz in the velocityrange for which the resonance vz = co/kz takes place.

The problem of noncollisional absorption influencedby nonlinear effects is a subject under further investi-gation.

ACKNOWLEDGEMENTS

The authors wish to thank M. A. Leontovich andD. A. Frank-К amené tski}/ for valuable discussions.

124 SESSION A-5 P/2215 R. S. SAGDEYEV and V. D. SHAFRANOV

REFERENCES

1. A. Schlúter, Z. Naturforsch., 72a, 822 (1957). 6.2. V. D. Shafronov, Zhur. Eksptl. i Teoret. Fiz., 34, 1475

(1958). 7.3. V. N. Fadeyeva and N. M. Terentyev, Tables of the Values

of the Integral of Probabilities of Complex Argument, GITTL(State Publishing House for Technical and TechnologicalLiterature), Moscow (1954). 8.

4. K. Kôrper, Z. Naturforsch. 72a, 815 (1957). 9.5. T. Stix, Phys. Rev., 106, 1146 (1957).

R. S. Sagdeyev, B. B. Kadomtsev, L. I. Rudakov andA. A. Vedenov, P/2214, this Volume, these Proceedings.Y. L. Alpert, V. L. Ginsburg and E. L. Feinberg, Pro-pagation of Radio Waves, GITTL (State Publishing Housefor Technical and Technological Literature), Moscow(1953).B. A. Trubnikov, Dokl. Akad. NaukSSSR, 7 75,913(1958).B. N. Gershman, V. L. Ginsburg and N. G. Denisovr

U.F.N., X, 561 (1957).