Resonant Joule phenomena in a magnetoplasma

RESONANT JOULE PHENOMENA IN A MAGNETOPLASMA

A.B. SHVARTSBURG Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation US. S.R. Academy of Science, Izmiran, Troitsk, PlO. 142092, Moscow Region, U.S.S.R.

1

NORTH-HOLLAND-AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 125, No. 5 (1985) 187—252. North-Holland, Amsterdam

RESONANT JOULE PHENOMENA IN A MAGNETOPLASMA A.B. SHVARTSBURG institute of Terrestrial Magnetism, ionosphere and Radio Ware Propagation U.S.S.R. Academy of Sciences, izmiran, Troitsk. P,/O. /42092, Moscow Region, US.S.R.

Received January 985

(‘ontents’; I. Introduction 2. Joule phenomena near Langmuir resonance 2.1. Resonant losses of radiation energy in non-magnetized collisional plasmas

189 191) 192

2.2. Temperature hysteresis and magnetic control of the extraordinary wave field near the Langmuir frequency in a magnetoplasma 2.3. The thermal transformation of the waves in the vicinity of Langmuir resonance 3. Cyclotron heating in a collisional plasma 3.1. Cyclotron absorption line 3.2. Thermal bistability and negative absorption of the electron gyroresonant wave 4. Localization of dissipation regions in the heterogeneous and anisotropic plasmas 4.1. Surface Joule effects in semiconductor magnetoplasma

198 204 206 207 212 22)) 22))

4.2. Resonant collisional dissipation in heterogeneous plasma 4.3. Joule absorption near resonant cones in a magnetoplasma

224 230

5. Geometrical resonances in Joule dissipation 5.1. The resistive cigenmodes in one-dimensionally houn-

234

ded plasma configurations 5.2. Geometrical shift of absorption lines in bounded plasma volumes 6. Relativistic and quantum resonant phenomena in co)lisional absorption 6.1. Joule effects in laser driven plasmas 6.2. Absorption resonances in quantum plasmas Concluding remarks List of symbols used References

235 24)) 242 243 246 248 248 249

Abstract; The present state of research of resonant Joule interactions of collisional plasmas with electromagnetic waves., including both the problems of plasma heating and wave dynamics, is reviewed. The controlled development of non-linear wave processes in gaseous and solid-body plasmas in radiowave and microwave ranges via resonant heating is discussed. The series of thermal histahility effects, produced by electron-temperature hysteresis near Langmuir and cyclotron resonances, is considered. The geometrical resonances ot’ absorption in layered structures and bounded volumes are illustrated. Localization of dissipation phenomena near resonant regions in heterogeneous and anisotropic magnetoplasmas is analyzed. Relativistic and quantum effects in resonant collisional attenuation of waves in a plasma are shown. Some analogous tendencies in Joule wave phenomena are marked in plasmas characterized by very different physical conditions

from laboratory devices up to cosmic objects.

Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 125, No. 5 (1985) 187—252. Copies of this issue may he obtained at the price given below. All orders should he sent directly to the Publisher. Orders must he accompanied by check. Single issue price DII. 42.00, postage included.

0 370-1573/85/$23.10

©

Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

A.B. Shvartsburg, Resonant Joule phenomena in a inagnetoplasnta

189

1. Introduction Numerous aspects of resonant Joule phenomena in collisional plasmas have attracted attention for more than fifty years. The information about these phenomena was obtained in different fields of physics and dispersed in the series of publications. At first, such effects were mainly investigated in gaseous plasmas in the framework of ionospheric radio propagation. Here the Joule effects were considered as pernicious ones due to collisional absorption of the radiowaves in the ionospheric magnetoplasma [1]. Later on the nuclear fusion problem attracted attention to the possibilities of collisional damping of some instabilities in the course of plasma heating [2]. On the other hand the efficiency of resonant plasma—wave interactions for heating was determined in a series of phenomena by Joule processes [3]. Cross-modulation effects for powerful radiowaves in a collisional plasma [4] gave a rise to the development of non-linear electromagnetic wave theory long before the “laser era”. The concept of resonant Joule plasma—wave interaction was applied during the last two decades to the explanation of a series of phenomena in cosmic and laboratory plasmas. Analogous effects were described recently in relativistic [5] and quantum [6] plasmas, including non-linear and surface phenomena. One may note the considerable similarities in the methods of interpretation of these phenomena, in spite of the difference in the wavelengths (from long radiowaves up to optical ones) and the spatial scales of resonant plasma volumes (from semiconductor crystals up to astrophysical objects). One of the possibilities of resonant amplification of Joule phenomena in a collisional plasma is connected with the closeness of pump wave frequency to one of plasma eigenfrequencies (i.e. Langmuir resonance) or to the frequencies, characterizing the motion of plasma particles (i.e. electron and ion cyclotron resonances). Such resonant conditions may be produced in some regions inside the heterogeneous plasma or, in particular, near its surface. Resonant dissipation may be localized near some cones of directions in an anisotropic magnetoplasma, the pump wave frequency being given. Unlike these effects, possible both for unbounded and bounded plasmas, another possibility of resonant Joule phenomena is connected with geometric factors. Such factors may produce the deformation of absorption lines in bounded plasma volumes, such as plasma column or plasma spheroid. Moreover, one may note some other resonant absorption effects, occurring in periodic structures or near the sources of radiation, situated inside the plasma. Modern theoretical and experimental interest in resonant Joule phenomena in a magnetoplasma is three-fold. Firstly, it is connected with resonant plasma heating in radio, microwave and optical frequency ranges, in different laboratory installations from plasmachemical reactors [9] to fusion devices [10]. These processes are also of importance in energy-related technology and in innovative development of sophisticated devices utilizing such phenomena as non-reciprocity, thermal coupling of the waves and geometrical resonances in bounded plasmas. On the other hand, resonance Joule effects are utilized widely in electronics for the governed re-building of microwave and optical beams and pulses through interaction with plasmas. Such resonant effects may be very sensitive to the frequencies of collisions, even though these frequencies are much lower than other characteristic plasma frequencies. The thermal dependences of the frequencies of collisions show the possibilities of governed development of non-linear wave phenomena. Lastly, the employment of resonant absorption lines in the spectroscopy of gaseous, semiconductor and metal plasmas, including such new fields as non-linear and surface spectroscopy is a powerful diagnostic tool. Modern theory of resonant plasma—wave interaction abounds in bistability phenomena. The hysteresis jumps of electron temperature, reflection coefficient or dissipation length, characterized by —

19))

A. B.S/t t’art.churg. Resonant ,Ioule pitenooiena In

0 15 agnewpla.ona

relatively weak strength of the wave field near plasma resonances, show the possibility of temperature monitoring in gaseous and semiconductor plasmas. Side by side with these examples of action of man-made radiation sources, one may note the attempts to include resonant Joule absorption in the theoretical models of some phenomena, which can not presently be reproduced in the laboratory. Such models describe, for instance, wave interaction with magnetoplasma. characterized by high radiation pressure [12] or with self-gravitating plasma layer ]13J. To describe the effects of collisions, we used a very simple, yet somewhat realistic model, such that the mathematical complexity, usually associated with collision operators. is avoided. Nevertheless, it is possible to obtain useful quantitative information even with only an extremely small number of calculations. Such an approach is utilized below in order to consider the current situations and perspectives, connected with the various Joule phenomena. Even the simplest analysis of resonant absorption effects in the model of unbounded homogeneous plasma (chapters 2 and 3) permits one to emphasize the series of bistability and non-linear phenomena near Langmuir and cyclotron frequencies. The generalization of these tendencies (chapter 4), plasma heterogeneity and anisotropy being essential. shows the possibility of governed localization of a resonant absorption region inside the plasma volume, in the vicinity of its surface or near the resonant cone. Unlike this, Joule dissipation in bounded plasma volumes (chapter 5), described by a quantized spectrum of eigenmodes, is characterized by the influence of plasma-volume geometry on the shape of its resonant absorption lines. The model of an unbounded homogeneous plasma is utilized again in chapter 6 in order to emphasize the role of relativistic and quantum phenomena in resonant absorption processes. The considerable analogies in behavior of wave dissipative systems attract attention nowadays to the attempts of generalizing these tendencies in the framework of synergetics [143]. From the theoretical viewpoint one may emphasize the very actual synergetic problems of self-organization or. vice versa, chaos generation in dissipative systems. The formation of periodic structures in conducted media [14] and the occurrence of strange attractors, describing the resonant dissipative systems [100],gave the first examples of such phenomena. The subsequent elaboration of the generalized theory of such phenomena seems to be important in plasma problems and cross-disciplinary physics.. Using the picturesque phrase of Professor L.I. Mandelstam, one may characterize the prediction of the tendencies of dissipative processes in oscillative phenomena as a constituent part of a peculiar ~‘oscillativeintuition” [11.

2. Joule phenomena near Langmuir resonance The energy exchange between electromagnetic radiation and plasma includes processes of energy transfer from electromagnetic field to plasma and vice versa. Here some part of the wave energy, dissipated in a plasma, may become a source of plasma heating. Such Joule losses of the wave energy are produced by collisions between plasma particles, responsible for plasma conductivity. The contribution of such ‘elementary” processes to the dynamics of plasma—wave interaction may be described by means of the wave refractive index and absorption coefficient. Recently the interest in collisional effects was restricted to transport phenomena in a plasma and to Joule absorption of electromagnetic waves. Other wave phenomena in plasma electrodynamics, connected with refraction and polarization of waves in a plasma, were considered usually in a collisionless approximation. In the framework of these ideas the collisions, usually assumed to be sufficiently infrequent, were responsible only for the small corrections to the principal collisionless effects, such as the phase shifts in the wave propagation through the dissipative medium [17],the slow collisional relaxation of solitons in a plasma

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

191

[181or the perturbation of the wave’s group velocity due to its weak attenuation in a plasma [19], however, the dependence of wave refraction, polarization and reflection upon collisions proves to be very essential near plasma resonances. Such a dependence, illustrated long ago in the model of ionospheric propagation of radiowaves [201,was refined and re-verified over and over in laboratory experiments, connected with microwave interaction with gaseous and semiconductor plasmas [211.Here the influence of collisions between similar particles (electron—electron, ion—ion) and different ones (electron—ion, electron—molecule, ion—molecule) on plasma refraction and absorption is quite diverse. The interest of investigators is connected in the first place with the collisions of electrons with heavy particles, because such processes are responsible for Joule losses of electromagnetic waves, produced by powerful sources in radio, microwave and optical frequency ranges. Such high-frequency processes may be analyzed by means of the complex refractive index (n iK); here n is the refractive index and K ~S the absorption coefficient. This complex quantity for the travelling wave with frequency w, the ions motion being neglected, is —

(n—1K)

2

=1—

2v(1—V—is) V—is)— u sin2 a ±Vu2 sin4 a + 4u cos2 a (1— V— IS)~

—

is)(1

—

The frequency of electron collisions with heavy particles, are described here by the dimensionless parameters s

peIW ;

(2.1)

—.

2(1

~e,

and the electron cyclotron frequency,

u=w~lw2.

~0H,

(2.2)

Here a is the angle between the wave vector k and the magnetic field H. The vectors k and H are assumed to lie in the same plane and the signs “+“ and in (2. 1) relate to the ordinary and extraordinary polarizations. The neglect of the ions motion restricts the wave frequency w to the range w > w~VmlM, where m and M are electron and ion masses. Electron—electron collisions are neglected in (2.1), because their contribution to the absorption of high-frequency waves (w Ve) is approximately in w ~ times less, than the contribution of the collisions of electrons with heavy particles [23]. The dimensionless parameter V in (2.1) is connected with the Langmuir electron frequency “—“

~‘

V

122/w2

=

4~e2NIn,

,

~,

(2.3)

where Ne is electron density. The mean-field plasma frequency (2.3) is related to a two-component plasma. Unlike this, the plasma frequency of binary ionic mixture has been shown to be temperature dependent [20]. Here the f2~value may be shifted with respect to (2.3) as was illustrated in the behavior of time-dependent fluctuations in H~—H~ mixture. The value v = 1 relates to Langmuir resonance. This resonance in gaseous and semiconductor’s plasma will be considered mainly in this chapter. The resonant Joule absorption and the thermal phenomena, accompanying this absorption, may be illustrated in general outline by means of the simple model of non-magnetized plasma. These phenomena are assumed as a basis of thermal self-action and interaction effects, occurring in the wave fields near Langmuir resonance in a magnetoplasma. Such non-linear resonant phenomena produce temperature hysteresis, governing coupling of the waves in collisional plasma. The analysis in this chapter is restricted to the case of spatially uniform plasma.

A.B. Sh i’art.shurg, Resonant Joule p/teno,ncna in a inagnetopla.ona

192

2.1. Resonant losses of radiation energy in non-magnetized collisional plasmas

The variety of thermic resonant phenomena of self-action and interaction of lectromagnetic waves in a collisional plasma is stimulated by the different temperature dependence~of plasma parameters. responsible for wave absorption. It is convenient to outline these parameters directly in the energy balance equation. governing the Joule heating of a plasma. Such an equation, describing the growth of electron temperature from T~0to T~.may he written as [24] -

~kN~ ~~e(Te

—

T~11)= ~

(2.4)

Here k is Boltzmann’s constant. ~ is the mean energy fraction, transferred in elastic collisions between electrons and heavy particles, characterized by the wave frequencies, ~em andbyt~ so that 0ei- The energy losses of the monochromatic are determined therespectively, plasma conductivity = “em + tensor u 9, E is the complex wave amplitude. The variations of ion temperature are neglected in (2.4): this neglect is justified if the frequency of ion—molecule collisions i’~, is high enough [24]: i’~,>

6i’~-

(2.5)

The electron gas is supposed to be not degenerate here with the exception of some problems, discussed in chapter 6. The electron density N~is restricted by the condition [25]: 3’2. (2.6) N0 ~ (kTet’fl/h) where h is Planck’s constant. Let one consider the density Ne to be independent of the temperature; then the conductivity o-,.~ depends upon the temperature T 0 via the frequencies of the collisions e~and Oem the influence of the temperature on the parameter may also become important in energy balance. The relaxation of the electron component in collision-dominated plasmas, located in the electric field, is determined by energy and impulse transfer from the electrons to heavy particles due to elastic and inelastic collisions, e.g. excitation and ionization, with additional heating by an electric field. The relation between the portions of the dissipated energy, connected with thermalization of the electrons and with the inelastic effects, may be very different even in the simple case, including only one inelastic process, the excitation of molecules in weakly ionized plasma. The simple model, connected with the energy input, produced by an external source of radiation, shows the possibilities of the occurrence of peculiar thermic phenomena, including bistability and time-dependent inelastic excitations in plasma. The possibilities of plasma—wave interactions may be expended due to the “inner” source of radiation, produced, for instance, by turbulent wave field excitation in a plasma by means of an electron beam and the resulting absorption of this field by electrons. Such turbulent waves are excited often in the vicinity of the Langmuir frequency .12~[411, w=v-v~.

Turbulent heating seems to be useful for plasma-chemical reactors of beam-plasma type, because such processes promote the formation of non-equilibrium plasma with high electron temperature [39]. Some examples of the energy dissipation of the electromagnetic field in collisional plasmas will be discussed below. For simplicity, plasma in this chapter is supposed to be non-magnetized.

A.B. Shvartsburg, Resonant Joulephenomena in a magnetoplasma

193

2.1.1. The temperature dependence of wave absorption in a collisional plasma The motion of an electron in a plasma under the action of the monochromatic electric field E~is characterized by the thermal velocity VT and the velocity v~, produced by the electric field. The frequency of electron—ion collisions ~ei depends upon the correlation of these velocities. If the field E is weak enough (VT>> v~),the frequency t’e~is defined as [24]

~eI =

(2.7)

11c 7rmv~

3

here e, is the ion’s charge and A~is the Coulomb logarithm. The temperature dependence of the frequency ~ei may be written in a form: =

; f= e~f3~

T~/T~1

‘

(2.8)

here f is the dimensionless value of electron temperature, ,4~is the value of ~el, related to an unperturbed temperature (f = 1). In an opposite case (v vT), when the field E is powerful, the temperature dependence of ~“e practically vanishes [26]: ~‘

16Nee~mw3I~ei

=

eE

eE 1

IL 1 + ln 2mwv-fi

(2.9)

The tendencies of variations of electron—molecule collisions, depending on the molecular properties, are much more various. The frequency of these collisions ~em may be evaluated as (2.10)

i’em~VQ.

Here p and Tm are the pressure and the temperature of the neutral gas, v is the relative velocity of the neutrals and the electrons, 0 is the collisional cross section. The dependence of 0 on the velocity v is determined by the sort of molecules and the velocity range. Here the low-energy electron—molecule collision process provides a context for some interesting phenomena, resonant and threshold effects. The dependence of Q for electrons in N 2, the most abundant atmospheric constituent up to 100 km, may be approximated in the temperature range Te 100—300 K as [271 5v). (2.11) 0 = 6.7 x 10_21(1 + 3.3 X 10 Here Q and v are measured in m2 and m-sect respectively. If the electric field strength E, accelerating the electrons, is weak enough (v~ VT), the frequency of electron—molecule collisions ~em (2.10) may be modeled by means of a simple power formula [38] ‘~

~em

= P~mf”.

(2.12)

The power n for the cross section Q, supposed to be independent of the velocity, is n = 0.5; the mixture of N 2 and 02, related to the air and lower ionosphere, is characterized by the value n = 516 [28].

A.B. Sht,’artsburg, Resonant Joule phenomena in a magnetoplasrna

194

The electron—ion collisions in plasma heating problems may usually be considered as elastic ones; therefore the mean energy fraction, transferred in such collisions, is 6 = 2mM~. Unlike this, the analogous parameter 6, characterizing the energy transfer in electron—neutral collisions, depends essentially upon the electron temperature. The analysis of this parameter indicates the descending part of the curve 8(Te), connected with the weak excitement of the rotational levels in nitrogen and oxygen [29]. The experimental measurement of 6(Te)values permits one to approximate the dependence 6(T~) in the range T 3K for the air by means of an empirical formula [30] 0= (0.3—l.5)x 10 6(Te)

=

6~6(f)

6(f)

f’2+f~’4

I .25f°”;

6~= 6.5 X It)

~.

(2.13)

The dependence (2.13), illustrated in fig. 1. describes the considerable decrease of the parameter 6 due to the growth of the electron temperature. The temperature dependence of the frequency of electron—ion collisions (2.8) and the mean part of transferred energy (2.13) may lead to the hysteresis dependence of the electron temperature on the heating wave intensity even far from plasma resonances. One of the first examples of such an effect was connected with the model of low-frequency (w ~ o~)heating of completely ionized plasma [31]. However, a limited quantity of electron—molecule collisions proves to be permissible [35].These effects are similar formally to cyclotron resonant heating in a high-frequency powerful field (w ~ “~); however, such fields are of much more interest for the experimental point of view and so the physical mechanism of such a hysteresis will be analyzed in connection with electron cyclotron resonant effects in chapter 3. A peculiar hysteresis effect in the high-frequency field is connected with the temperature decrease of the magnitude 6 (2.13). This effect leads to the jumps of the energy, transferred from electrons to heavy particles due to their collisions, in a definite range of heating wave intensity [32].The dependence 6(f) in this range has an S-like character. Let us consider the simplest case, connected with the propagation of the linearly polarized high-frequency wave F in the transparent region of an isotropic weakly ionized plasma (w ~ ~ i’~).In this case the plasma conductivity may be written as e2N

ii

(2.14)

-

a? (0~

The substitution of (2.14) in the energy balance equation (2.4) permits one to rewrite this equation in a dimensionless form:

(f~l)6(f)=

(2.lS)

ü2;

5_. i

I

Z

Fig. I. The mean energy fraction d. transferred in electron—molecule collisions in the air, is pkstted versus the electron temperature (see ref. 3))]).

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

195

here a2 = EI2/E~. The characteristic electric strength E~(2.15) is the so-called “plasma field” [24]. The decrease of the temperature dependence in 6(f) leads in some range of the intensity values to a S-like character of the graph 6 = 6(a2). The analysis of this dependence shows a formal analogy with the Van der Waals equation in thermodynamics and with the model of non-linear pendulum in mechanics. The occurrence of the S-like curve f = f(a2) is possible in some domain of intensity a2 values, determined by means of the set of equations (2. 15) and aa2/af=0.

(2.16)

The roots of the system (2.15)—(2.16) determine the bend points of the S-like curve (fig. 2a) fi=1.9;

f2=4.9.

The hysteresis dependence f = a~a2a~

f(a2)

a~0.22;

occurs in the domain a~=0.15.

(2.17)

The graph 6 = 6(a2) in this range is also of hysteresis type (fig. 2a). So, the dependences f = f(a2), 6 = 6(a2) are single-valued outside the range (2.17) and are three-valued inside this range; here, the middle root in the latter case is unstable. The occurrence of the three-valued region is connected with the two states of the plasma a low temperature and a high temperature with a possibility of jump-like transitions between them. The origin of such discontinuities is determined by the simultaneous processes of increasing electron temperature and decreasing parameter 6, due to plasma heating. The electric field being sufficiently intensive, the electrons can not transfer to the molecules a considerable fraction of electric field energy, which is dissipated in a plasma. Here the electron temperature will grow until the decrease of the mean portion of the electron gas energy, transferred to neutrals, will be compensated by the increase of this gas energy itself, the quantity of the transferred energy being constant. This is the reason for the occurrence of the second state. i.e. the high-temperature one. —

—

~

I I

g

~

05

2

$

2 _____

c.~f5

c~2O

0,25

~1c

o,~o

2 6~~

q~c

Fig. 2. The complex of thermic hysteresis phenomena in a weakly ionized plasma: the hysteresis dependences of the electron temperature f (a) and

the mean energy fraction, transferred due to electron—molecule collisions (6). upon the heating wave intensity a2. The jumps of the parameters ~ ~I are shown by the arrows; b~=

f and

A.B. Shvartshurg. Resonant Joulephenomena in a magnetoplasma

196

The hysteresis jumps of the electron temperature f will lead to the jumps of the absorption coefficient of the intense high frequency wave (V< I, eq. (2.3)) V. 4wV

~(f) I

-

(2.18)

V

In other words, this effect gives use to the hysteresis of high frequency wave absorption in a plasma, the ionization degree being low. This example illustrates the influence of weakly inelastic collisions on non-resonant plasma—wave interaction. However, the variety of bistability phenomena is extended considerably in the vicinity of plasma resonances, when the ionization degree is high and the elastic electron—ion collisions are dominant. On the other hand, the contribution to the wave’s absorption. produced by inelastic collisions. may also be characterized by the peculiar temporal behavior and resonant phenomena. 2. 1.2. Inelastic collisional effects in the course of resonant plasma heating The growth of the electric field in a collision-dominated plasma may lead to the reinforcement of the role of inelastic collisions in the energy dissipation processes. The energy losses of the powerful field are redistributed in this case between the thermal and non-thermal phenomena. Thus, the aforesaid c—N2 collisions are characterized by resonances in scattering, in which the resonant excitation of vibrational levels of the temporary negative ion is responsible for oscillations in the cross sections. Here, the cross section increases by a factor of three to four in the range of energies 1.8—2.5eV for e—N2 collision, the resonant energy being 2.3 eV. A similar phenomena of electron resonance trapping occurs in the rare gases due to essential dependence of cross sections (total cross section and momentum-transfer cross section) upon the electron energy e. Such processes may be observed owing to light production, connected with the drift of an electron swarm in a gas (number density Nm) in an electric field E, the F/Nm values being so low, that ionization processes are absent [34].To account for such electroluminescence processes, one must suppose, that a significant fraction of the electrons have energies, close to that, i’R, at which the collision, accompanied by electron resonant trapping, occurs. Such a collision proceeds via the formation of a compound state A+e—sA’--~A+e,

(2.19)

A’ being a temporary ion with mean life time i-c. The aforesaid electron trapping occurs, if the sum of the mean energy change per path plus the mean energy loss K60e) per collision is positive. The energy loss in the vicinity of resonance (2.19) depends upon the initial electron energy r1: 2iit =

Q1(~11)

M r0 Q,~(Fo)

.7

‘

Q~and Qm being the total and momentum-transfer elastic cross sections respectively. Both Q~and Q~, have narrow and deep minima, producing the trapping of electrons, accelerated by the field E. The quantity (6~~) characterizes the Joule heating by this field. The resonant character of this plasma—wave interaction is connected with the kinetic effect (fig. 3) the field E being constant.

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

197

\ 12,25’

19,36

~5

eV

Fig. 3. Elastic cross section of He molecules as a function of electron energy ~.

io’~

~

Fig. 4. The relaxation of the normalized energy input R (eq. (2.21.)), produced by aperiodically altering of the electricfield in a weakly ionized plasma with exciting collisions versus the time parameter 1= tpo (ton see), p~being the neutral gas pressure. The curves 1, 2 and 3 correspond to the values of parameter i~= Too = 10~ l0~ l0~ (torr see) respectively. T is the adjustment time for the aperiodical electric field (see ref. 136]).

Unlike this resonant effect, produced by the constant field, the temporal alteration of an electric field may produce very different situations, depending on the relation of time constants, described by the relaxation phenomena and the field alteration. It is worthwhile illustrating the effects of aperiodic and periodic alterations of the electric field separately. Limiting cases, characterizing the aperiodic alteration, are on the one hand the nearly jump-like change of the electric field and on the other hand the quasi-stationary change simultaneously with the field. The change of the time behavior between these two limits is demonstrated by the variation of the quantity R (fig. 4), connected, for simplicity, with only one type of inelastic phenomena, the excitation collisions [36] R

=

U+/U.

(2.21)

Here U.. is the energy input from the electric field, U_ is the total energy loss from the collisions; this ratio characterizes the degree to which the system is non-stationary. The stationary value of this ratio is R = 1. The dynamics of the establishment of a new stationary state of the electron component depends upon the pressure of the heavy plasma components. However, in the range of small adjustment time T, related to the jump-like change of the electric field, one may notice a strongly non-stationary behavior: the values of R are far away from the stationary value R = 1. This tendency is weakening due to increasing T, related to the quasi-stationary change of the field. Thus we observe that, via the rapidity of an electric field alteration, the time-dependent field exerts a noticeable control on the relaxation processes. The difference between the energy input and the energy loss in the periodic electric field is shown on fig. 5. Such a difference at the greatest chosen period time t0 is small; thus one may find disturbances of only minor importance in the stationary energy balance. However, the deviations continue to grow with decreasing t~. These phenomena illustrate the contribution of time-dependent electric fields to electron component heating, the inelastic exciting collisions being taken into account [36]. The

195

A.B. Sh vartshurg. Resonant Joule pheitoinena en a niagets’toplasota

~1+i~&

ó

d,r

i

ó

~

4

Fig. 5. The tirne.dependeitt normalized energy input ( “, curve I I and etterg~ tess U (curve 2). produced h~ periodical electric lick) in weakly-ionized neon plasma with exciting collisions, are plotted against the dimensionless tOne tT (T is the period of field oscillations). U.,ps, (V torr see ~(are the volt equi\ ale tt t s of the mean elect ron energ~per ic Ut raI gas pressure ~,i. The sit sb inc a) the dilTerc nec hetweeti f’ and f’ due to the growth of period T is illustrated ,en (a) attd (6), related to the s alues of the parameter r~ i~t~ = 2 ref. ]36J(.

II)

and 5

‘~

lIt’ respceiiselv (see

generalized calculations of cross sections for impulse transfer in elastic, exciting and ionizing collisions, performed for a neon plasma [44],can give a deeper insight into the rivalry of these phenomena in the course of plasma heating. The qualitatively similar phenomena, connected with the redistribution of the absorbed field’s energy, between the different degrees of freedom, attract attention in the problems of non-linear spectroscopy. Thus, the shape of the absorption line for a weak wave in the presence of a powerful one may be deformed pronouncedly due to collisions, altering the atomic velocity and the magnetic quantum number [401.Such non-linear collisional phenomena, depending upon the polarization of both waves, may be employed in spectroscopy of the gas molecules [7]. The analysis of such inelastic plasma—wave interactions is based usually on the exact kinetic theory. However, the series of important features, habitual to thermic plasma—wave interactions near the Langmuir and cyclotron resonances in gaseous and semiconductor plasmas in the presence of a magnetic field, may be described in the framework of simplified theory, neglecting kinetic effects. Such an “elementary” approach proves to be valid, in particular, for investigation of thermal effects, based on Coulomb collisions. 2.2. Temperature hysteresis and magnetic control oJ the extraordinary wave field near the Langmuir frequency in a magnetoplasma

The anisotropy of a magnetoplasma leads to a considerable difference between the Joule absorption of ordinary and extraordinary waves. The magneto-ionic theory, describing such an absorption in gaseous plasmas, the angle a between the magnetic field H11 and the wave vector k being arbitrary, leads to very complicated analytical formulae [22]. Both the refractive index and the absorption coefficient of high frequency waves in such plasmas may depend pronouncedly near the Langmuir resonance on the frequency of the electron collisions, even though this frequency is less than other characteristic magnetoplasma frequencies. The magnetoplasma of n-type semiconductors, such as e.g. the well-known InSb, may be characterized by the analogous resonant effects in absorption of microwaves and infrared radiation [45].However, the refractive index of such semiconductors, determined in this range mainly by the lattice’s properties, varies insignificantly due to collisions of the carriers with the lattice. The angular and spectral peculiarities of the pumping wave interaction with the magnetoplasma near the Langmuir resonance depend on the electron’s temperature via the frequency of electron collisions

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

199

with heavy particles. The detuning of this resonance V = 1 (2.3) may be characterized in the general formula (2.1) by means of parameter A: V=1+A ;

A~1.

(2.22)

It seems to be expedient to emphasize, firstly, the angular peculiarities of Langmuir resonant interactions, connected with the wave oblique propagation (0 < a <~r/2),the detuning being neglected (A = 0), and, secondly, the spectral ones (A 0), the direction of wave motion being given, e.g. a = rrI2. Moreover, the analysis of some tendencies, habitual to such Joule phenomena, is simplified in the aforesaid limit V= 1, A = 0. Thus this case will be considered at the outset. 2.2.1. The magnetic modulation of resonant absorption of the heating wave The losses of the heating wave energy in a magnetoplasma depend pronouncedly upon the direction of wave propagation. The combined action of plasma anisotropy and electron’s collisions in the course of an oblique propagation may be characterized by means of normalized value of the frequency of electron collisions p~ [22] t’e

q=—;

~0Hsin

v~=

2cosa

a (2.23)

.

The value q = 1 is in the resonant case V = 1 related to phase synchronism between the ordinary and the extraordinary waves in a magnetoplasma; here the frequency of collisions p~is equal to its critical value v. (2.23). Let us consider the Joule absorption of the extraordinary wave. Unlike the non-resonant case, characterized by the proportionality between the absorption coefficient and the frequency of collisions ~e, the resonant absorption depends non-monotonically on the parameter ~e’ The changes of the absorption coefficient K in the range q ~ 1 are shown on fig. 6. The maximum value of this coefficient in a limited range of angles, when ~p~~jiji tg2a 2, is reached at the point L421 ‘~

‘~

2V3uIcos a~ q

(2.24)

0= 1+3ucos 2 a

__

N

= 0.1 andextraordinary 0.01 respectively). Fig. 6. The effect of electron heating on the absorption coefficient (a) and the refractive index2 a(b) of the wave near the Langmuir resonance in a magnetoplasma (curves I and 2 correspond to the values of the parameter u cos

201)

A.B. Shc’arzshurg. Resonant Joule phenomena in a magnetop/a.sma

The absorption coefficient increases in the region 0 ~ q ~ q15 from K region q0 ~ q ~ I it decreases from K = Km to K = K0, where K0 = ~

=

(p0

~ Po ~i/2 ) :

PO

=

1

+ (U C05

a)

=

0 to

K = Km =

..~

0.35, whereas

ill

the

—

(2.22)

-

The critical value of the temperature T,, is determined from the equation q(T~)=qo.

(2.26)

The essential peculiarity of the curves on fig. 6 is connected with the occurrence of the descending parts on the graphs K(Te): aK/df<0.

(2.27)

The condition (2.27) may be fulfilled both in slightly and totally ionized plasmas. When the electron neutral collisions are dominating (a~emIaf>0). the decrease (2.27) may occur in the range T> T~.On the contrary, the prevalence of electron—ion collisions (9t.’e,/af<(.)) leads to the fulfillment of the condition (2.27) in the range T< T~.Using the dependence ae(Te), fig. 6 may be applied to both aforesaid cases. The heating in the vicinity of the Langmuir resonance will lead to a considerable change of the refractive index of the extraordinary wave and its polarization also [45]. The above-mentioned related the tolimited values of the angle a in the range 2a ~ 1. In an analysis opposite iscase, whenwith a tends IT/2, the resonant dependence K = K(Pe) may PeWHi ~0.5tg be more complicated. Thus, the resonant propagation (V= 1) of the extraordinary wave across the magnetic field may be accompanied by two absorption maxima Ki and K 2 (fig. 7). The critical frequency v~(2.23) tends to infinity in this case, and it is convenient to utilize the dimensionless parameter S = Pef2~t. The absorption maxima are located at the points Si.2 =

~(1 2\/3

±Vl 12u).

(2.28)

—

S’2

~

533

—~S

Fig. 7. The 3Q~) coefficient versus theofdimensionless Joule resonant frequency absorption of electron of the collisions extraordinary a. The wave, curves propagated I. 2. 3 correspond across theto magnetic the resonance field detuning in a gaseous i = 0;plasma 10’

(wl~ l)Y~respectively. = l0 —

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

201

There is an absorption minimum between these two maxima at S

3

=

Vu.

(2.29)

These effects may occur in sufficiently ionized plasma (u < l2~,12f2~>w~);in the case u> 12 these three points are merged into one point s~= S2 = s3 and the graph K = K(s) has one maximum at s = Vu. The difference between the maximum values of the absorption coefficient (Ki)max = (K2)max = 0.35 and its minimum value 1(3 increases due to the decreasing magnetic field on the condition that u <12_I. Here the first maximum K~= Ic(s~) is narrowing. The depth of the minimum K3 may be very considerable: e.g. K1 : K3 = 3.5 in the case u = i0~.When the magnetic field tends to zero, the extrema K~ and 1(3 vanish, and this analysis predicts the sole maximum at the point S~= 3_t/2; this result may be obtained independently from the consideration that the resonant absorption is a non-magnetized plasma. In the vicinity of the resonance (A 0) the dependence K = K(A) has also a non-monotonic character (fig. 7). These phenomena, accompanying the transverse propagation of the resonant wave, are typical for a gaseous plasma. The analogous plasma—wave interaction in semiconductor’s magnetoplasma is characterized, as will be shown below, by only one maximum of the graph K = K(s). The temperature dependence K = K(q) in the above-mentioned situations is single-valued. However, the inverse dependence q = q(K) in some ranges of the value q is not single-valued (fig. 7). This circumstance may become a source of difficulties in the determination of the electron collision frequency by means of measurement of resonant wave attenuation. Occurrence of the negative derivative (2.27) in a high-frequency range attracts attention to the thermal hysteresis and instability phenomena, accompanying the resonant Joule heating. The sensitivity of resonant Joule phenomena to the magnetic field, illustrated, in particular, on fig. 6, shows the possibilities of magnetic control of such phenomena. On the other hand, these phenomena depend upon the thermic effects. Here the magnetic and thermic effects may be utilized for the controlled evolution of the pumping waves in both linear and non-linear regimes. anisotropy 2). The This magnetoplasma will be considered below. attracts attention to the transverse propagation of the waves (a = in 2.2.2. The complex of microwave bistability effects in a plasma Joule dissipation of the electromagnetic wave in the vicinity of Langmuir resonance depends essentially upon the wave’s polarization. Let us consider, for simplicity, the wave, incident normally on the plasma surface (fig. 8); the electric field component, orthogonal to an electric field, is parallel to this surface. This wave excites an extraordinary polarized wave inside the plasma layer. The refractive index and the absorption coefficient of such a wave near the resonance (V= 1) depend upon the electron temperature via the frequency of electron collisions. The heating of electrons may lead to the

Fig. (1. The geometry of resonant interaction of the pumping wave with the magnetoplasma layer.

202

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

occurrence of coupled bistability phenomena in the absorption, refraction and reflection of the pump wave. These phenomena occur simultaneously in some range of the pump wave intensities. Such a range may be revealed by means of the energy balance equation (2.4). When the wave’s frequency is high enough ~

w2>w~>t~:

(2.30)

the anisotropy of the Joule dissipation is insignificant a 2eeNe(mw2)t however, the polariza1 = o~= e tion structure of the refractive extraordinary wave in a geometry, shown on fig. 8, depends essentially upon plasma anisotropy via the polarization factor /( [24]: E~=XE 5

E~=0; 2ut~t1:

J{=pexp(—i~):

Arctg(u/s). (2.31) p = (u + s Here the longitudinal component E~of the resonant wave field (V= 1) is much more than the transversal one E~(JpI2~I). The dissipation of this wave is connected mainly with the longitudinal component. In a totally ionized plasma (s = s 312) the polarization factor ~ (2.31) may be written as =

15f =

(u

+

s~u’~f3)’1/2 exp[—i Arctg(usJf~2)].

The energy balance equation (2.4) may be written in a dimensionless form, similar to (2.15): (f._l)(u+s~uif 3)a2

The energetic parameter a

15

(2.32) connected with the normalized v-component of the electric strength in a

plasma E~: Er

=

RE)

a2 = IEr~2IE~

(2.33)

where E~is the electric strength of the incident wave (fig. 8), the reflection coefficient R is connected with the refractive index n and the absorption coefficient K of the resonant wave

(n +

(2.34) 1)2 +

The values n and n,

K

may be written as

[~(± 1

K =

1(2

f42

~

+ ~2+ ~

/

u2

u2 +

1/2 ,ç2)j

—

-

(2.3~)

The occurrence of the hysteretic dependence of the electron temperature f on the heating wave intensity a2 will lead to related hysteresis jumps in the values n. K (2.35) and R (2.34). Let one consider a totally ionized plasma. The energy dependence of the temperature f (2.32) has a S-like character in some domain of the values a~~ a2 ~ a~:eq. (2.32) has three roots in this domain. At

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

203

a2= a~and a2= a~the curvef=f(a2) has vertical tangents, given by (2.16): f=2f-3.

(2.36)

Equation (2.16) has two different roots f12 and f2i, determining the inflection points of the S-like curve, provided that (2.37)

u2
This temperature hysteresis is analogous to the one, shown in fig. 2a; here the temperature jumps ft2—~f22 and f 2s/~2.So, in the case u2s~/2= 4 x 10~these values are 2i —~f1~ depend on the parameter u ~ = 1.3; f12= 1.65; f21=2.9; f22=4.1 (2.38) .

The resonant absorption is considerable, both for heating and unperturbed plasmas, and it is worth while looking for the hysteresis jumps of the intensity of the refracted wave, characterized by the coefficient R (2.34). Such a graph, related to some dimensionless values of plasma parameters, is shown in fig. 9. Let us consider, e.g., the wavelength of the pumping wave A = 1 cm; fig. 9 relates in this case to plasma parameters Ne 10t3cm3 Ho=0.7kG; Teo= 104K; Pe 5l))< lOnsec_i; the threshold 0(0. amplitudes are at = 17.6 V cm~a 2 = 18.6 V cm’; R12 = R(f~2)and so on; the values off are given in eq. (2.38). The characteristic time of electron temperature variation, t (2.39) ,

TT

(&‘eY

is (2—5) x 10~sec. So, a relatively small energy flow P (1—2)W cm2 may become responsible for the aforesaid bistability of the refracted microwave. It is instructive to emphasize, that this resonant Joule effect occurs, when the frequency of electron collisions is much less, than the pumping wave frequency (2.30). Here, the intensity of the refracted wave, proportional to T = 1 RI2, changes via the electron heating from f = 1 up to f = f22 = 4 from T = 0.75 to T = 0.98. The Joule dissipation length L~ —

R

~ Fig. 9. The bistability of the reflection coefficient

s~ = 5 x 10~).

~2 IR~of the extraordinary wave near by Langmuir resonance in a magnetoplasma (u = 1.4 x iO~,

2))1

.4 .B. Sheartshurg. Resonant Joule phenomena in a magnetopla.ona

L,~=

(2.40)

C/WK,

is small, L,. = 0.5cm
The coherent interaction of electroniagnetic waves in plasmas attracts attention in view of frequency mixing and wave amplification processes in gaseous and semiconductor’s plasma. The series of such processes, being governed by collisional phenomena, depends on heating of the carriers. One of the first results in this field predicted the possibility of amplification of surface waves in a system, consisting of a thin semiconducting layer and a semi-infinite piezoelectric (or vice versa) f48]. Amplification occurs when the directed velocity of the carriers in the semiconductor v0 is greater than the phase velocity of the surface wave v~.The amplification increment yt, proves to he proportional to the frequency of collisions u

(~

ew v~ Ya~

1).

(2.41)

The collisional wave excitation may lead both to geneiation of quasi-monochromatic waves due to frequency mixing and to wide-band generation, connected with the parametric instability development. Let us consider these situations near the Langmuir resonance in the model scheme of spatially uniform plasma; the localization of such phenomena in non-uniform plasma will be discussed below. 2.3.1. The collisional coupling of the waves The non-linear effects due to velocity-dependent collision frequencies may lead to significant coupling of the electromagnetic waves in a plasma. The interaction of intense transversal waves E1 and E2 may stimulate the excitation of one longitudinal plasma wave; if their frequencies w1 and w2 satisfy to resonant condition Wt~W2

(2.42)

lie,

such an excitation may become considerable in the presence of an external constant electric field E1 [49]. The evolution of the electric strength of the longitudinal wave EL is governed in this case by the equation: (9EL/at +

EL/TT

=

(2.43)

AXEiE~,

where the heating time of electrons

TT

is determined in (2.39) and the coupling coefficient A~may be

written in a form: A

=

~ E0 WIW2kTE()Pe

~tfl

(2.44)

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

205

It is interesting to compare the coupling coefficient A,,. (2.44) with the coefficient AL, which accounts for the non-linearity of the conventional Lorentz force. Thus, in the interaction (2.42) this coefficient may be written as AL=——~—’-—kL,

(2.45)

2m w1w2

where kL is the wave vector of the longitudinal wave. The wavelengths of the interacting waves E~and E2 in (2.43) are assumed to be much larger than the mean free path of the electrons; unlike this, the quantity AL in (2.45) depends on the Langmuir wavelength. Comparison of the quantities (2.44) and (2.45) shows, that the collisionless generation may be rather weak if the wavelength of the Langmuir plasma wave is large enough: 2v.~/vT;

v...

=

eE/m[2e.

kLTD~

Therefore, resonant collisional excitation may be very effective in the long wave range of plasma oscillations spectrum. Another example of resonant collisional excitation is connected with the third-order frequency mixing under the condition [161 2Wl~W 2fle.

(2.46)

The use of this effect for diagnostic purposes 2w was treated in [49, 56]. The maximum value of the resulting electric field E3 with the frequency 1 w2 may be evaluated as [16] —

2E~E~ 7 Q~a~—~ / fl2s~_i —e 3= 2 2 2 (1~) ~1—-—-~ m c PeWIW2 “ w1! \

E

,

(2.47)

the ratio IE3Ei~Idepends directly upon the frequency of electron collisions t’e. Therefore, the enhanced generation, while weak in general, may be observed, the 1J1 resonant is of the conditions (2.49) or (2.47) being fulfilled. Thus, for the ionospheric plasma the ratio jE3E1 order of magnitude iO~—io~, which might be within the limits of what one can observe by means of a sensitive satellite antenna [57]. The effectiveness of generation, produced by non-resonant pumping waves (w 1, W2 f?~),is small; however, such an effectiveness may increase due to the approaching of the pumping wave frequency to the Langmuir one. The narrow band generation, as discussed above, is not restricted by any energetic thresholds. However, the growth of the pumping wave intensity near the Langmuir resonance may stimulate the wide band generation, if some threshold intensity is reached. Such an effect is based on the development of resonant plasma instability. ~‘

2.3.2. The thermic excitation of instabilities near the Langmuir resonance The occurrence of threshold phenomena, characterizing the development of instabilities, is connected with the competition between the generation of new harmonics and their Joule dissipation. The thermic influence on such processes is based on the different tendencies of temperature perturbations of

21)6

.4. B. SIit’art.rhurg, Resomiant Joule pltettoittena in a sit agtteiopla rena

electron—ion (2.8) and electron—neutral (2.12) collisions. The competition of the aforesaid tendencies in a partially-ionized plasma reveals the particular ranges of heating wave intensity, responsible for the excitation of the parametric instability. Unlike the resonant character of plasma—wave parametric interaction, the frequency of the heating wave niav he chosen over a wide range of’ values, It ts convenient to analyze these effects by means of a simple model, describing the parametric instability in an isotropic collisional plasma. The threshold amplitude E~may he written as [46] E2

C—

(~ 48)

l6i~Nek(Te+ T~)(t)ec+ t~em).

Let us assume that condition (2.5), related to the constancy of the ion temperature T,, is satisfied. In this case formula (2.48) may be rewritten as =

E~

55Z(f);

-,

(I ±

2

f~”(l+ r)(l

cj) 1-

1~

(2.49)

i’~,

q)

E~0is the instability threshold amplitude in an unperturbed plasma; the role of thermic perturbations is described by the dimensionless function Z(f), shown in fig. 1(1. One may see, that the thermal change of an instability threshold may be considerable: in the case 7~= 0.5, q = 100, the function Z(f) reaches its minimum Z(J~)= 0.4. when /~= 6.3. On the other hand. the range Z(J’) > I is connected with the increase of the threshold due to plasma heating; so the scanning of the heating wave intensity may lead to a controlled development of the parametric instability, the resonant wave intensity being constant.

~ Fig. I)). The temperature excitation and damping of parametric instability due to the dependence of the instability threshold on the electreiit temperature f The curves I, 2.3 correspond to the different values of plasma ionization degree p = 1; )(.13; (((IS respectively, and the ratio of electron and ion temperatures is r = )).5. The scales along the F~axisin the ranges F< I and F I arc dilferetit.

3. Cyclotron heating in a collisional plasma Plasma—wave interactions near the cyclotron resonances attracted formerly attention as a diagnostic tool for investigations in gaseous and semiconductor’s plasmas. The peak and the halfwidth of the resonance absorption line contain information about the different mechanisms of energy relaxation.

A.B. Shi.artsburg, Resonant Joule phettov’ie,ia in a tnagnetoplasnta

207

Afterwards such interactions were considered as a method of Joule heating of collisional magnetoplasma; in particular, some ionospheric modification experiments were connected with the heating wave frequency w = 8.5 X 106 rad sec_i, close to ionospheric electron gyrofrequency at the middle latitudes [80]. Recently there has been renewed interest in electron cyclotron resonance heating of plasmas in laboratory installations, stimulated by the creation of powerful sources in the microwave spectral range. Here, both collisional and collisionless processes of wave dissipation are important for the heating of electrons. The series of versions, utilizing fundamental or second harmonics of cyclotron frequency, are considered now as the different geometrical schemes of heating in magnetically confined devices. Thus, perpendicular electron cyclotron heating by ordinary modes, the effect of oscillating magnetic field being included, may lead to the growth of “perpendicular” electron temperature T1, 2, for second harmonic heating, and algebraically for the fundamental one [68]. exponentially in t such as high power ELMO BUMPY TORUS [811,utilizing a 28 GHz gyrotron at a Another geometry, power level up to 200kW, is characterized by T~=1.1 keV, Ne= (1—2)x l0~cm~3. In several versions of thermal-barrier tandem mirror machines, electron cyclotron resonance heating is applied for increasing the electrostatic potential, which confines solenoid ions [82]. Some possibilities of ion cyclotron heating were examined recently in [70]. The dynamics of collisionless heating depends, however, on the Coulomb collisions via the perturbations of resonant particles density, produced by such collisions. By means of resonance cyclotron absorption one may only transform the energy of radiation into the energy of resonance particles, which are confined in a plasma volume during a long period of time and finally transmit their energy to the main component of the plasma by collisions. Therefore the collisional phenomena may influence resonant plasma heating both directly and indirectly. Moreover, the cyclotron plasma—wave interaction, the plasma ionization degree being small, attracts attention in view of the search for non-equilibrium plasma states, which may be considered as the media with negative temperature and negative absorption. Collision frequency of charged particles in a weakly ionized gas in a strong magnetic field depends on the density of the particles and the relative velocity between the pair of colliding particles. However, the motion of a charged particle is affected by the magnetic field, whereas the motion of a neutral particle is not affected by it. So the relative velocity between these particles changes with the magnetic field intensity. The variation of the parameter ~ = T 2m/T1M, T2, M and T~,m being the temperature and the mass of neutral and electron respectively, from ~ = 1 to ~ = 10~leads to the decrease of the collision frequency ~em by approximately a factor two [67]. 3.1. Cyclotron absorption line The peak and the half-width of the spectral lines of electron and ion absorption contain important information about the characteristics of plasma—wave interaction. The electron cyclotron absorption may be analyzed by means of the general formula (2.1) in the limit u—~1. The coefficient K, the kinetic phenomena being neglected, is characterized by a complicated dependence on the mode and the direction of the wave propagation. The spatial damping rate of such a wave, y~,is connected with the imaginary part k2 of the propagation vector k and the angle /3 between k and the Poynting vector y~=2k2cos/3 I ak1 k1 3a

(3.1) (3.2)

A.B. Shvartsburg. Resonant Joule phenomena in a magnetoplasma

208

0

20

~jo

60

~0

~,°

Fig. 11. The direction of the Poynting vector of the extraordinary wave near the electron cyclotron resonance; ~ and a are the angles between the

Poynling vector and the wave vector k and between k and the magnetic field H respectively; Q~= 0.5w~H = 10kG; Te

=

1O~K.

k1 and a being the real part of k and the angle between k and magnetic field H respectively. The longitudinal (a = 0) and transverse (a = ir/2) propagation of the gyroresonant wave are related to the parallel directions of Poynting2)and wave vectorsby(/3the = 0) in non-dissipative media. Unlike this, an is characterized finite angle /3 0 (fig. 11). oblique propagation a klvTcosa.

(3.3)

Here, the Joule absorption of the higher harmonics of electron cyclotron frequency proves to be not a resonant process. The anisotropy of magnetoplasma conductivity is manifested, in particular, in the occurrence of a simple resonant term, determining the transverse high frequency conductivity I 2 l+x

e2Ne

Ui =

2’

1j

(3.4)

0’~~ = ~,

me~

IA

:~ 2\\

5

10

If

a Fig. 12. The coefficient of electron cyclotron absorptton correspond to the values e’ = 1.1: (.5: 1.9 (a) and u = t .1

20

0

20

4o

60

80 ~

8 versus the angle a br ordinary (a) aitd extraordinary (h( modes. The curves I. 2 and ~ .3: .7(6) respectively.

A.B. Shvartsburg, Resonant Joule phenomena in a mnagnetoplasma

where

O’I(

=

209

is the conductivity of the isotropic plasma, and x0 is the normalized detuning of the resonance, (1

(35)

Vu)2,1s~.

—

Unlike this formula, related to electron collisions with heavy particles, the scattering of electrons on photons may determine the high frequency conductivity in a rarefied plasma with high radiative pressure, in which the electron and photon ranges are restricted by Compton scattering. The mutual dragging between the electrons and photons has influence on the dispersive properties of such a plasma. Thus, the electron energy relaxation depends on the effective frequency of electron—photon collisions The longitudinal component of the conductivity tensor 0’)( of such a rarefied non-relativistic plasma, on the condition that v~ w ~ v~,may be written as [121 ,‘~,.

‘~

2Ne 5e

w2i- 2 1w

-

(3.6)

.

mv~(w~—w2)2+25w~iw2p~2

A cyclotron-type resonance is possible, however, at a frequency W = O.SWH. The analysis of some tendencies of cyclotron absorption in both electron and ion gyroresonances is simplified due to “transverse” geometry of plasma—wave interaction (fig. 8). Therefore, some resonant Joule phenomena, accompanying the extraordinary wave propagation across the magnetic field, will be considered below.

3.1.1. Electron gyroresonant Joule phenomena produced by modulated fields

The temporal variation of magnetic fields, well-known in electron and nuclear spin resonance experiments, may also lead to considerable changes of gyroresonant absorption of high frequency waves in magnetoplasma. Let us consider an electron gas in a magnetic field with a variable harmonic component h, parallel to the constant component H 0, directed along the z-axis: H~=H0—hcos.110t.

(3.7)

The transversal component of the conductivity tensor may be written in the form [601: 1

Vt1e2

2+

+

(w

—

w

1 ~f~)2 +

2]’ (3.8) a’ = ~ J~(a)[~ + W~ ~ui°) where J~(a)are the Bessel functions; the parameter a is connected with the depth and the period of the magnetic field modulation: ~2

—

—

(OH

a~.

h

(3.9)

The dependence of a’, on the variable components of the field may become essential, but Re a~.> 0 for arbitrary values of the parameter a (3.9). In the limit a = 0 the conductivity is determined by only one term (p = 0) in the sum (3.8), related to the well-known formula:

A.B. Shvartsburg, Resonant Joule phenomena in a magneroplasnta

21))

I

t1eli~[ a’

1

8in

—

[(to

2+P~ + (to

—

I + (OH)2

(3.10)

+ (OH)

on the other hand this term vanishes if parameter a (3.9) is equal to the first root of the Bessel function J 0 a 2.4. Thus, the modulation of the magnetic field (3.7) may lead to a considerable variation of cyclotron absorption. Side by side with magnetic field modulation, the variations of plasma conductivity may be stimulated by the modulation of electron temperature, produced by the heating wave. The cross-modulation of a weak probing wave, connected with this variation, is amplified if the heating wave frequency is close to toH [77]. On the other hand the modulation of high frequency (to ~ aH) pumping wave amplitude E= E 0[l

+~

cos(lit+ ~)],

(3.11)

stimulates non-linear resonant an amplification of proves the electron if the frequency of modulation 12 is toH [611.Such amplification to be motion very effective if the extraordinary pumping close to oH or 2 wave is travelling across the magnetic field. The stochastic variation of the modulation phase ~ (3.11) leads to the growth of the electrons transverse temperature T,. Such a process may become more effective than the usual collisional heating, if the frequency of the modulation phase variation t.’~ 5 is large enough //teE11 \2

Ppti>t’e

3(~

4nuoc

)

4

2

(3.12)

-

Qe(OHt’rT

This inequality may be fulfilled for high frequency (to > ~°H; ll~ t’e) powerful waves. Such modulated waves penetrate much deeper into the plasma volume than the electron cyclotron waves; therefore the plasma heating, produced by this non-linear effect, is not only located near the plasma surface. One may mention the qualitatively similar resonance effect, connected with interstellar hydrogen atoms heating due to the kick, accompanying the absorption of L~,photons from the Galaxy background and the resulting radiation by these atoms of the photons. The growth of the kinetic energy of the atoms due to these accidental shocks may be considered as a stochastic heating [84]. 3.1.2. Ion cyclotron attenuation The shape of ion spectral lines in a magnetoplasma differs essentially from the electron ones. Thus, the absorption peak, connected with an ion gyroresonant wave propagation along the magnetic field in a rarefied totally ionized plasma (l2~~ “eliH) =

12e/V2t’eQH.

(3.13)

is much greater than the peak of the electron cyclotron absorption line

i~

[71]. In the simplest case of

two-component plasma, consisting of only one sort of ions, the ratio of these maxima is Ki!Ke VM7~2~I -

Such a high effectiveness of collisional damping near the ion gyroresonance attracts attention to the

A.B. Shvartsburg, Resonant Joulephenomena in a magnetoplasma

211

possibilities of Joule plasma heating by means of ion cyclotron waves [72].The attenuation factor has a sharply pronounced maximum in the narrow frequency range in the vicinity of ion cyclotron frequency (10t—102)fl~.The propagation of extraordinary wave across the magnetic field is characterized in this range by the rising of the attenuation factor by several orders of magnitude (fig. 13). The maximum of the refractive index of this wave is less pronounced [86]. In a multicomponent plasma the greatest contribution to the attenuation factor is made by collisions between ions of various sorts and not by collisions between electrons and ions [73]. The considerable absorption impedes the penetration of the ion cyclotron wave deep into the plasma volume during radio frequency plasma heating. However, the collisionless ion cyclotron resonance is considered nowadays as a powerful method of plasma heating in Tokamaks, stellarators and other installations for controlled nuclear fusion [70].The heating time for the large installation, evaluated via the Lawson criterion NTE ~ 3 sec, TE being the energetic life-time, seems to be long enough; so, 10i4 cm one may discuss a preliminary cyclotron heating of a small admixture of light ions (H1) in a deuterium plasma, accompanied by the resulting heating of the background plasma due to thermalization of these heated ions. Such collisionless heating may be based on the utilization of ion cyclotron frequency harmonics, the absorption of the frequency = 0.SulJ4 being very effective [50]. Another method of ion gyroresonance heating of a large plasma volume is connected with the parametrical decay of the high frequency pumping wave, penetrated deeply into this volume. Such a decay may lead to the occurrence of a backscattered daughter wave and an electrostatic ion wave, responsible for Joule heating of ions. The threshold electric field for this to happen is found to be fairly low [79].Thus, at the EISCAT establishment very high pumping wave frequencies (e.g. 334 MHz) are planned to be pulsed in the ionosphere [83]. Here the aforesaid parametrical instability, characterized by relatively small e-folding time (T 0.1 see) and threshold field (E~= 0.1 V mt), may arise at an altitude of Z 200 km. A similar effect, connected with the resonant transformation of the waves in the vicinity of the point of intersection of their dispersion curves, may be utilized in electronics for selective dissipation of the pumping wave; thus, the magnetostatic waves in some magnetic semiconductors may excite the elastic waves, responsible for the connection between the magnetostatic waves and conductivity electrons. The sound dissipation, produced by these electrons, proves to be decisive and leads to selective dissipation —

~pe

-~

-10~

~

II

JZW

Fig. 13. The logarithm of attenuation factor of ordinary (I) and extraordinary (2) ion cyclotron waves, propagated across the magnetic field in hydrogen plasma versus the resonance detuning (Q~= 4 x 10’ rad see’’, v~,= 3005cc’t, N, = 4.5 X 10’ cm3).

212

.4. B. .S’hvartshurg. Resonant Joule jthettomttena in a ntagmtetoplasma

near the resonant frequency f~105 MHz [114]. Such a system may be considered as a band filter for microwaves. Joule dissipation, produced by ions, may become an effective mechanism of stabilization of low-frequency oscillations in plasma. In particular, such a dissipation may stop the development of drift-cyclotron instability in the spectral range, related to the coincidence of the dispersive curves of ion cyclotron and drift oscillations in a magnetoplasma with a non-uniform temperature distribution. Such drift-cyclotron oscillations may be stabilized, even if the frequency of the ion collisions is small enough, ~ (m/M)51~li~ [871.In spite of large values of the ion cyclotron absorption coefficient (3.13) the depth of such a wave penetration into the plasma layer may become commensurable with this layer thickness if the wave frequency is low enough. Namely. such a situation occurs in the course of propagation of low frequency man-made signals (to (30—120) rad see t) from the altitude Z 300 km through the ionosphere to the Earth [88]. 3.2. Thermal bistability and negative absorption of the electron gyroresonant wave In the above consideration of Joule dissipation the kinetic phenomena were neglected. However, the velocity dependence of the frequency of these collisions and electron—electron interactions may also have an influence on collisional dissipation of gyroresonant waves 123]. Here, the term t.’ can not adequately account for such e—e interactions, and the simple formulae, similar to (3.4), are not valid here. The transverse conductivity a 1 near by electron cyclotron resonance, the aforesaid kinetic effects being taken into consideration, may be written, by analogy with (3.4). as 2Ne K e 5(x~~) (3.14) 2mpe I+x~ Here x0 is the normalized detuning of resonance (3.5). The dimensionless coefficient K5(x~)describes the kinetic corrections [24,69] to the “elementary” formula 2N~ e 2’ 8~=—. (3.15) l+x 1~ 2mp~ An example of the influence of these phenomena on the shape of the cyclotron absorption line is shown in fig. 14. Unlike these quantitative corrections to well-known Joule dissipation phenomena, the velocity dependence of the frequencies of electron collisions with heavy particles may lead to the occurrence of series of very surprising effects. Thus, such a dependence gives rise to the formation of electron temperature hysteresis and cyclotron absorption bistability, electron—ion collisions being dominated. This effect seems to be possible both in semiconductor and gaseous plasmas, even if the temperature dependence of the frequency of collisions is taken into consideration in the simplest manner (2.8). More detailed analysis of kinetic phenomena near by electron gyroresonance, electron—molecule collisions being dominated, predicts the possibility of the occurrence of an “inverse” distribution function in some range of electron velocity. Such a situation indicates the resonance wave amplification. It is worthwhile to analyze these phenomena by means of an extraordinary wave, propagated in the z-direction across the magnetic field (fig. 8).

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

“0,5

0

-~

~3

213

~5 X~

Fig. 14. The kinetic corrections to the normalized value of the transverse electron gyroresonant conductivity are plotted versus the resonance detuning parameter xo. Curve a, electron—electron interaction is not considered; curve b, this interaction is considered. The dotted line corresponds to the neglect of kinetic corrections.

3.2.1. Thermal bistability of electron cyclotron absorption in semiconductor plasma

The thermal variations of the frequency of collisions due to Joule heating may lead to a considerable temperature dependence of the absorption coefficient near the gyroresonance. This temperature effect is responsible for absorption bistability of the travelling wave, even when the frequency ~e is much less than the electron gyrofrequency H’ Let us consider the normal incidence of the pumping wave on the semiconductor surface, as shown in fig. 8. The refractive index n and the absorption coefficient K of this wave are connected with the complex dielectric permeability of the semiconductor (3.16)

EEL+Ep,

where

EL

n=

and r13 are the contributions of the lattice and the plasma

/

V(1—Vu)

\/EL—

—

2[(1_Vu)2+s21

;

1 K=—

—

;

(3.17)

4n(1_Vu)2+s2

the parameter V is, as usual, the normalized Langmuir frequency u2~,however, this frequency is connected with the effective mass of the conductivity electron 2N~/m”. (3.18) L2~=4rre

It is worthwhile to consider here n-type indium antimonide in view of isotropy and small effective mass of free conductivity electrons: mx = 0.013m [511. The standard scheme of investigation of thermic hysteresis effects leads to the analysis of an energy balance equation in dimensionless form, similar to (2.15) and (2.32). Such an equation, related to electron cyclotron absorption, may be written as

(f— 1) [(1—\‘/u)2+

S2] =

a2s~;

a2

=

Er~2IE~.

(3.19)

Here E~is the electric strength of the pumping wave inside the dissipative medium (2.33) connected with the incident wave and the reflection coefficient R (2.34). The characteristic “plasma field” E~is

4 .B. Shc’art.rhurg. Resonattt Joule phenomena in a tnagnetopla asia

214

3kT~1m’8ti~55

2

(3.20)

e

=

where m’~is the effective mass of the conductivity electron; parameter 6, characterizing the mean energy fraction, transmitted during the collisions between electrons and ionized impurities in the semiconductor, may be written as [SI] 16 /2m”c~ 3

(3.21)

in

The plasma field E~(3.2(1), connected with electron cyclotron heating, does not depend on the pumping wave frequency. Therefore, such E 0 values may become much less, than those of eq. (2.16), related to non-resonant heating, if the collisions are rare enough (e~55< w2); here c~is the velocity of sound. Unlike the master equation (2.15) and (2.38), let us consider first the energy relaxation processes, connected both with electron scattering on ionized impurities and scattering due to electron—phonon interaction. It is convenient to characterize these processes by related frequencies of the collisions t~, and t’~, depending on the electron temperature [51] ~ei

=

Pei)f

= Veof

~ep = 32

1

‘52

(1 + pJ~’) ;

P

=

The equation, determining the turning points of the hysteresis curve (3.19):

xU4

=

(1+ pf 5/2) [2f-3

-

(3.22)

~p0/~e0

pf’2(6f- 5)1,

f

2), may be obtained from =

f(a

(3.23)

where the parameter x 0 (3.5) isonconnected with ofresonance detuning. location ofofthe turning points 2) plane depends the detuning resonance and theThe contribution electron—phonon on the (f, a interaction to energy balance of electron gas. Electron—phonon collisions (i9s!i9f>0) do not lead to a hysteresis effect, therefore the contribution of such collisions to the energy relaxation process. determined by the parameter p (3.22), must be restricted. Thus in the case of exact resonance x 5 = 0, the 2) occurs in the semiconductor plasma in the range p <2.5 x 10~i.The analogous S-like graph f = f(a calculation for gaseous plasma in the model (2. 12) with the power n = 0.5 predicts some broadening of this range: p <8 x 102. In the case of the finite detuning x 11 0, eq. (3.23) must be solved numerically. In view of these limitations let us analyze first the main features of hysteresis effect by means of a model of totally ionized magnetoplasma (p = 0). Here eq. (3.23) may be written as: 4=2f—3. (3.24) x~f This equation is identical with (2.37), governing the Langmuir resonance. Therefore one may obtain the condition of existence of the turning points ft 2 andf2t, determined by the different roots of eq. (3.24), in the form: (1_Vu)2/s~<1/16.

(3.25)

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasmna

215

The decrease of the resonance detuning x0 leads to the increase of hysteresis temperature jumps up to f22 : f12 6 and to the expansion of the range of intensities, responsible for thermic bistability occurrence (fig. 15). The same graph is valid also for the gaseous plasma, the electron—molecule collisions being neglected. It is worth noting that in the particular case of isotropic plasma u = 0 the inequality (3.25) is transformed to the well-known condition 25peo,

(3.26)

w
determining the existence of thermic hysteresis effect in isotropic plasma, located in the intense quasi-static electric field (~2~ ~~ 0)[31]. Unlike this, the aforesaid high frequency phenomena are realized in a field of a travelling wave in a magnetoplasma. Here, restriction 2 5c’ on thethe condition that(3.25) may be fulfilled even in the case of low frequency of the collisions to ~

—

16(1

—

(3.27)

,

Vu)2

so, the spectral range of the waves in a plasma, characterized by hysteresis jumps of conductivity, is expanded essentially due to the magnetic field in a plasma. The thermic variations of the semiconductor refractive index (3.17) prove to be insignificant just as the variations of reflection coefficient R (2.34). However, the perturbations of the absorption coefficient K (3.17) seem to be considerable. The transitions from the low-temperature state to the hightemperature one and vice versa lead to the related jumps in the wave dissipation length L~.Therefore, the transparency of the semiconductor film near the cyclotron resonance may vary in a hysteresis manner (fig. 16). The jump in the Joule absorption, connected with the pumping wave switch-off L 21—~L11is larger than that, which is produced by the increase of this wave (L12—*L22). The pumping wave power, corresponding to the threshold L12 (fig. 16), may be calculated as (3.28)

=

2 0

Xo~10

intensity, responsible for the occurrence of thermic bistability of electron cyclotron with the decrease of resonance detuning xl, eq. (3.5). The curves I and 2 correspond to the threshold intensities ai and a~

Fig. 15. The expansion of the range of extraordinary waves

absorption, connected respectively.

~

216

A.B. Shvartshurg, Resonant Joule phenomena in a tnagnetoplasma

1 a

a2

,a~

2

0,13

Q,~

015

0!6

Q11

Fig. lb. The thermic resonant bistability of the absorption length L of the extraordinary wave in the semiconductor magnetoplasma (v = 6.5 x 162: = 0, 1: = 3 x l0~).(a). the electron temperature hysteresis: the solid and dotted lines correspond to the values p = 2 x l0’ and p = 102 respectively. (6), the hysteretic jumps from low.temperature absorption to the high-temperature one and vice versa.

where the reflection coefficient R may be evaluated as RI =~0.6;thus, the threshold bistability power for the parameters, shown on fig. 16, is P ~ 10’~W cm2 the characteristic time of the local electron heating ‘r~,eq. (2.39), is TT 10~sec. The characteristic time of the heating due to heat transfer Th may be estimated as Th L2D~ here L is the spatial scale, D is the conductivity coefficient for Th iO~—i0~ sec for the 0scale L ~=heat 0.1 cm. A slow process of heat electron plasma; one may estimate conductivity (Th TT) is neglected here. The time TT may be considered as a characteristic time scale for a series of thermal bistability phenomena: (1) The jump-like variations of attenuation length L~may be useful for the intensity selection of microwave signals, their frequency and polarization being given. (2) Microwave modulation of the transverse conductivity a’, in the current-carrying n-InSb film via the temperature jumps ~‘

a’,(f 2~)

1 + (toHV~)f~ (3 29) 3’ a’1(f12) 1+ (toH p~i)f may stimulate a considerable modulation of the current; so, the ratio (3.29), under the conditions shown in fig. 15, is equal to 0.15. (3) The bistability of cross-modulation of microwaves, propagated through the n-InSb film. The absorption coefficient of the wave, polarized, unlike fig. 8, along the magnetic field H, is —

-

K =

(f~\3I2

V12)

Vs

312 0

2nf

(3.30)

The temperature hysteresis, produced by the powerful extraordinary wave, may lead to absorption bistability of the weak ordinary wave (3.30). The small, but finite, values of the electron—phonon collisions frequency ~ep are responsible for the narrowing of the range of wave intensities, stimulated the bistability and the rapprochement of the high-temperature and low-temperature states (fig. 16a). The plasma heating, preceded to bistability occurrence, leads to the attenuation of hysteresis jumps between these states. It is worth while comparing these effects with well-known bistability phenomena, based on the

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

217

growth of free carriers density, stimulated, e.g., by radiation with the fixed frequency to, responsible for transition of electrons to the conductivity zone (to = 3.35 x 10~~ rad sec_i for InSb) [55]. Unlike this rigid condition, the bistability effects, connected with the hysteresis of Joule heating near the electron cyclotron resonance, are not based on electron density variation. Moreover, the occurrence of the thermic hysteresis is not restricted by any fixed frequency of the semiconductor excitation. The possibility of the smooth re-building of the cyclotron absorption bistability regimes due to the magnetic field variation and the relatively small power of the pumping waves attracts attention to the perspectives of the utilization of these effects for the control of short wave radiation in the range 102—1O3GHz. This millimeter-wave range is interesting nowadays in view of far communications, radio astronomy and cosmic retranslation systems. These cyclotron absorption effects, produced by electron—ion collisions were examined above, for simplicity, without the consideration of kinetic phenomena. The general tendencies, common to these effects, remain valid, the kinetic phenomena being taken into account, although the threshold intensities, responsible for bistability occurrence, may change a little, due to the increase of the peak of the cyclotron absorption line (fig. 14). Along with these corrections the kinetic consideration predicts a qualitatively new effect, based on the appearance of negative conductivity in some weakly ionized gases.

3.2.2. Negative cyclotron absorption in a weakly ionized plasma

The variety of kinetic phenomena in the gyroresonant plasma—wave interaction is connected with the different velocity dependences of the frequency of electron—molecule collisions. This frequency is determined by the range of electron energies, the sort of molecules and the neutral gas pressure (2.10). The model of collisions being known, the components of conductivity tensor a’ 9 in a plasma may be calculated [241 as (3.31)

CJf(v)~[v3~j(v)]dv,

a’~=

where C is a normalization constant, f(v) is the electron distribution function; the Hermitean tensor a’,1 describes the electron motion in the wave field in collisional magnetoplasma. Thus, the 2 ‘2t~ (O~transversal (92 The component of this tensor a’, may be written in the form (3.15) on the condition, that v possibility of negative absorption may be obtained from (3.31) on the basis of the simple model V =

v

(3.32) 0T are the velocity of the directed motion and the thermic velocity of the electrons, where v andThe negative conductivity is originated by the 6-function distribution of the electrons respectively. f(v) 8(v vo) in the case h 3. These values of the quantity h (3.32) may be realized in slightly ionized gases Xe, Ar and Kr due to the Ramsauer effect [641. A negative conductivity indicates the appearance of the wave amplification (negative absorption). The absorption coefficient of the gyroresonant wave, propagated in a tenuous plasma across the magnetic field, may be written as [651 0(v/v~

—

—

Ko[X~(h+3)+3h]

3(1+x~)2

v K 0~

(3.33)

218

A. B. Sitvartsbto’g, Resomtant Joule phenomena in a ntagnetoplasm a

where the parameter x0 is determined in eq. (3.5). The region of negative absorption is shown on fig. 17. This kinetic phenomenon is interesting in view of the possibility of amplification of radiation in such plasmas. It is important to note, that the amplification is wide-banded, and the thermal and magnetic regulation of the wave’s resonant evolution is possible. The negative absorption leads to a considerable difference between the radiation temperature Tr and the electron temperature Te, defined by the average electron energy. Here, the radiation temperature may become negative. Near the cyclotron resonance (x~—*0),the radiation temperature is governed by the equation: TriTe

=

(3.34)

3i(3h),

where the value I’. is negative under the condition h > 3. The radiation intensity also displays an important peculiarity in the region h > 3. It is convenient to write the formula for the gyroresonant radiation intensity I(toH), which escapes per unit solid angle from the unit area of surface of the homogeneous plasma in the absence of reflection at the plasma boundary. in a form [65]: I(toH)

B((OH,Te)

3 í 3—hI

=—fl—exp

Ii —

3 (OHKtsd 11 IL 3 3 JJ —

(3.35)

Here B(toH, T~)is the blackbody radiation, emitted by a Maxwellian plasma with average electron energy ~kTe, ‘0 is given in (3.33), and d is the thickness of plasma layer. The radiation intensity may exceed that of the blackbody (3.35) due to the effect of negative absorption. The peak of this resonance is very sharp, its width being by one order smaller, than that of the “normal” cyclotron absorption resonance. Another possible origin of negative conductivity is the contribution of the “inverse” distribution function in some range of electron velocities in a weakly ionized plasma. Such a distribution may occur due to violation of the thermic equilibrium between electron gas and neutral molecules, produced by electron cyclotron heating. The non-linear effect, based on the appreciable irregularities in energy dependence of the cross section for electron scattering on heavy particles, may be employed, for certain

-~‘

.iü2

S

3

I

0

0,5

I

1,2

~

-4

Fig. 17. The transition from the positive to the negative resonant absorption coefficient v due to plasma heating and magnetic field modulatton. determined by parameter x~~ /t = 3,25.

A.B. Shvartsburg. Resonant Joule phenomena in a magnetoplasma

219

conditions, for producing a negative electron gas temperature [63]. The behavior of the scattering cross section for energies below 1 eV may be determined by studying the shape of the cyclotron absorption line. It is convenient to approximate the velocity dependence of the frequency of electron—molecule collisions in a form:

V

=

~(v)

Vo ço(v);

=

p;

0
1;

u~< v
0;

quc
(3.36)

Here u~is some characteristic velocity, determining the velocity range of the effective electron—molecule interaction, the function ~(v) (3.36) is connected with the cross section Q of such an interaction. The condition of occurrence of the negative electron gas temperature (9 [ I (9(Qe2)i 1>0

I9ELVE

(9E

(3.37)

~

may be satisfied in the model (3.36). This effect may stimulate the considerable deformation of the cyclotron absorption line. The experimentally observed splitting of the cyclotron absorption line [631 may lead to the more effective energy dissipation at the resonance level wings, than at the center (fig. 18). The fluctuations of the electron density in a plasma with negative conductivity have a tendency to grow. This tendency is restricted due to electron diffusion, the electron mobility being independent of the electric field. Such electron clumps, occurring, in particular, in a weakly ionized plasma (t’~ u2~) with a negative conductivity in a constant electric field will be neutralized gradually due to ion flow from the background plasma [74].The electrostatic energy of the electron clump gives rise to plasma heating in the course of such a neutralization. Therefore, the formation of a negative conductivity state in a plasma requires an energy input. Such plasma states attract attention in view of their possible utilization for the generation and the amplification of electromagnetic waves. ~‘

2

Xe Fig. IS. The kinettc splitting of the cyclotron absorption line in a weakly ionized plasma. The absorption coefficient is plotted against the detuning parameter. The curves 1, 2, 3 correspond to the values p = 5: 9: 15 in (3.36) respectively, q = V’S (see ref. [63]).

22))

A. B. Sh c’artshurg, Re.so,ta,tt Joule p/tenomena in a titagitetaplasttta

4. Localization of dissipation regions in the heterogeneous and anisotropic plasmas The resonant heating of large plasma volumes is usually complicated due to heterogeneities of such plasma parameters. as electron density, ion composition or magnetic field. The resonant conditions prove often to be satisfied only in the restricted part of plasma volume or near the surface of some cone of directions, and the specific class of such effects is connected with the localization of dissipation processes in the vicinity of the surfaces, limiting the volumes, filled in by gaseous or solid body plasmas. Some resonant frequencies of gaseous plasma are changed near these surfaces. Thus, the “surface” Langmuir electron (f1~),and ion (f2~).frequencies are (91] (liei)s

lie /V2.

(4.1)

Here the surface wave spectra may be obtained from the related bulk OflCS due to the substitution (4.1). So, the spectra of ion-acoustic waves, propagated in an unbounded plasma o~in the z-direction, and in a semi-infinite plasma ~ along the surface z = 0 are to~=li1(1—~-—~L 1

(O2T1(l~~, V2~ 4k~v~2

(4.2)

2k~v~

where v~is the isothermal sound speed in the plasma. These effects are manifested, e.g. in the changes of threshold amplitudes, characterizing the three-wave interactions near the plasma surface [92].Unlike these effects, habitual to a free plasma boundary, the special surface waves may propagate along the boundary of the plasma with some other dielectric. The collisional dissipation of such waves occurs in a thin skin-layer, although the thickness of the heat region may be much more due to thermoconductivity [1231. The dissipation of surface waves determine the intensity and the spectrum of transition radiation, produced by the moving current in the waveguides, filled by a non-stationary plasma [89]. The regions of resonant Joule absorption may he situated near the surface, bounded by the homogeneous plasma layer or, vice versa, inside the heterogeneous plasma volume. Moreover, the angular localization of absorption is possible in the vicinity of the surfaces of resonant cones, occurring in a magnetoplasma due to anisotropy of its dispersive properties. The knowledge of spatial and angular distribution of absorption permits one to choose the effective regimes of plasma—wave interactions, connected with both external and internal sources. 4.1. Surface Joule effects in a semiconductor magnetoplasma

The interaction of microwaves with the semiconductor magnetoplasma attracts attention due to possibilities of controlled evolution of microwave radiation in electronic devices. The utilization of n-type InSb for these aims is often discussed in view of considerable mobility of free carriers in this semiconductor. The important role of Joule losses in such a spectral range (from centimeter wavelength up to infrared waves) is connected with electron relaxation times i- 109_10~2sec [51], the related frequencies of effective electron collisions V are overlapped with this spectral range. One may consider the useful analogy between the propagation of high-frequency electromagnetic waves in gaseous and semiconductor plasmas, illustrating the possibilities of resonant interactions, and magnetic and temperature control of such processes; and what is more, some concepts, well-known in plasma physics, were formulated at the beginning conformably to the semiconductor plasma, such as, e.g.. the

A.B. Shvartsburg, ResonantJoule phenomena in a magnetoplasnta

221

characteristic plasma field E0 (2.16) [94]. Unlike the experiments with gaseous plasmas, some of the analogous semiconductor ones seem to be more “clear” due to the absence of errors, connected with the plasma confinement. Due to such a reliability the wave processes in the semiconductor plasmas are utilized widely in radio technique and electronics schemes for controlled reflection, transformation, delay and absorption of microwaves, and, especially, in non-reciprocal magnetoplasma waveguides. The tuning of plasma dispersive properties in such devices is stimulated by means of magnetic field variation; here the Joule absorption proves to be appreciable usually even in the cryogenics range of temperatures. The dissipation length L~,as was illustrated in fig. 16, may be short enough in the infrared range, the absorption coefficient K being much smaller than the refractive index n. due to a short wavelength. The localization of absorption near the plasma boundary occurs, due to surface waves propagation, also. Resonant Joule phenomena, localized near the magnetoplasma surfaces, may be described by means of standard Lorentzian absorption lines. Similar lines also occur in the absorption of surface polaritons [8]. 4.1.1. The reflection of waves near the plasma absorption edge The resonant interaction of electromagnetic waves with non-magnetized semiconductor plasma occurs near the Langmuir resonance. The resonant frequency, determined from the condition,E = 0, e being the sum of dielectric permeabilities of the lattice EL and the plasma E0, eq. (3.16), is 2Ne/mx;

(4.3)

fl~/V~i; Q~= 4rre where lie is the Langmuir frequency of the gaseous plasma, with the effective mass of light carriers mx, the EL value for the n-type InSb is EL 15.7. The reflection coefficient of the wave, incident normal to the plasma surface, changes from 0 to I due to the increase of the frequency in the range [93] =

=

EL EL’

1

-

i).

(4.4)

This range (flu V) relates to plasma absorption edge (fig. 19). The switching-on of the magnetic field, directed normally to the plasma surface, leads to the ~‘

IRf

0,5 1 1,1 )~(‘mm) Fig. 19. The magnetic splitting of the edge of the plasma absorption spectrum (curve I) in the n-type lnSb. The modulus of the absorption coefficient R is plotted against the wavelength .4, N, = 4 x 3, p = l0~sect. T = 77K. The curves 2 and 3 correspond to ordinary and extraordinary waves respectively, H = 0.9 kG. 10t3 cm

222

A. B. Sitt’arcshurg. Rs’.sonatti lassie jt/ts’ttotst~’stain

a

tttagtts’to/t/a.s’nta

occurrence of ordinary and extraordinary polarized waves. Here the reflection curve (I) (fig. 19) is split into two, corresponding to the two polarizations. This splitting is described by the transversal components of the dielectric permeability tensor [ E~FEL

1

________

E.E~ to(o + it’). 2

________ _____

I—— wf(w

2— (O~J

+ it’)

(4.5)

l~LQ 0(OH

2-- (9~]

= to[(o + it’)

where the frequency range between these curves is of the order of H’ It is worthwhile emphasizing the important effect. which appeared in the vicinity of the magnetoplasma reflection minimum, typical for the ordinary wave. The minimum of the reflection coefficient, related to the conditions shown in fig. 19. is RI 5%; therefore the ordinary wave may penetrate into the InSh layer. Unlike this. the reflection coefficient for the extraordinary wave, the frequency being the same, is close to 1; and the extraordinary wave penetration into the InSb layer is impeded. Thus, such a layer may he considered as a non-reciprocal element. The minimum of the reflection coefficient R,,, 1,, in the conditions under question (ii V ~.) is 2 t’. (4.6) IR ~,, K! The variations of frequency of collisions i’ due to electrons heating show the possibility of thermic tuning of such a non-reciprocity [95]. Appreciable dissipation in the vicinity of plasma frequency (to ~= 12v’)’ influenced on the incident wave reflection, occurs also in some metals. However, the collisional absorption of intense wave can not vary noticeably the energy of the electron gas in metals due to large values of the Fermi energy. The other mechanism of non-linear dissipation, connected with the pumping waves conversion into the Langmuir ones, proves to be effective in this case. The absorption coefficient, describing such a conversion near the step-like plasma surface, is [78]

K

=

((~~2

4in,’2E~

(47)

\mv~’ li~(9li~+4i’)

where v~is the sound velocity. In particular. such a Langmuir generation may become effective, if the plasma density is close to the metallic one and the cut-off frequency fl~,is related to the boundary between the Röntgen and the ultraviolet ranges. This skin-deep absorption, accompanied by the decrease of reflection, is assumed to be responsible for the aforesaid conversion of pumping wave, the wave power being evaluated as p t W cm 2 Unlike these surface effects, determining the penetration of bulk waves into the plasma, OliC may consider the wide class of resonant Joule phenomena, habitual to surface modes, propagated along the plasma boundary. —

4.1.2. Resonant absorption of surface plasm ons and polaritons The dispersive properties of surface magnetoplasma modes in the n-type semiconductor are obtained from the dielectric permeability tensor (4.5). However, these properties vary pronouncedly, if such

A.B. Shtsartsburg, Resonant Joule phenomena in a magnetoplasma

223

modes are propagating along the boundary between this semiconductor and some dielectric. Such systems attract attention in connection with the perspectives of smooth magnetic tuning the velocity and absorption of these modes. The main tendencies of such processes may be illustrated by means of a simple model (fig. 20). Let us consider an extraordinary wave propagation in the z-direction (fig. 20a) along a plane y = 0 and across the magnetic field H (x-axis). The semiconductor with the dielectric permeability tensor E~J (4.5) is filling the half-space y > 0; the half-space y <0 is occupied by the dielectric, characterized by the scalar Ed [96].The electric (l~and l~)and magnetic (hr) components of such an extraordinary wave field are shown in fig. 20b. The influence of dissipation and external magnetic field on such a wave propagation is illustrated in fig. 20c. The condition Im k5~> Re ~ emphasizes the surface character of the propagation (for simplicity, the analogous curves Im k2~and Re k2~,satisfying the same condition Im k25 > Re k2~,are not shown on fig. 20c). The value Re k~is much larger than Im k~in some range of magnetic field values (0 < H0 < 0.2T). This condition indicates the pronouncedly non-reciprocal character of the surface wave propagation and the possibilities of their spectral transformation due to Joule absorption. Such an arrangement of semiconductor and dielectric layers and magnetic field is connected with an occurrence of a peculiar resonance, related to virtual magnetoplasma oscillations [97]. The resonant value of the magnetic field in the low-frequency range (~2 ~ [l~), responsible for the phase synchronism between the waves in both media, is cm” Hresj =

e

EL

Q2

(4.8)

—p. EL+Ed

to

The aforesaid non-reciprocity is manifested especially obviously in the vicinity of the resonant fields ±Hres.Figure 20c is related with the case Ed ~ EL; in the opposite case Ed> EL the new branches of keK.?o-3 -q (in)

_

__

a _~mg

__—-.~~

X

-O,Iu

Il

.,__//l1

~

~1

Q~ ~o(T)

Fig. 20. The propagation of the surface magnetoplasma wave along the boundary between the semiconductor and the dielectric. (a), The geometry of wave propagation in z-direction; (b), the structure of the extraordinary surface wave field; l~,I, and h~,the components of electric and magnetic fields; k, and k5, the components of the wave vector; (c), the values of Re k~,Re k,5, Im k~,Im k,5, shown by the curves 1, 2. 3, 4 respectively, are plotted against the external magnetic field H0. fields Hres are marked by the vertical dotted lines; ‘lie = 1.2 x 1012 rad sec~ 3, 8;The e,=resonant 17.4; T=77K. v ~2xlOHsec_i; w=4,4x 10iOradsec~Ed=

224

A.B. Shr:artsburg. Resonant Joule phenomena in a magnetoplasma

8, ~

36 Fig. 21. Effective values of real

(curse I)

:sttd

imaginary c

2 (curse

2) parts of the dielectric futtction

of

the itatural aluminium oxide in the AR

range, plotted against the waveitunsher 6.

surface high frequency (ü.t ll~)magnetoplasma waves occur in the range H <0 owing to dissipation effects. Somewhat analogous absorption phenomena happen also with the surface plasmons in non-magnetized media. Such plasmons exist in a very broad frequency range, starting at zero and going to sufficiently high frequencies, occupying the radio. millimeter, submillimeter, infrared and visible range. The use of a simple single-oscillator model permits one to obtain the dielectric function for one dipole-active oscillator in the form [8] -=

F = F~+2

~

~ —

+

y~to

(üio

—

(49)

F4to~oWYo, to) +

here t~. and F0 are the high-frequency and the static dielectric constants respectively, o1~5 is the frequency of transversal optical mode, y,. is the phonon damping. The resonant absorption occurs near the frequency to toTo (fig. 21). The eigenfrequencies of surface plasmon—polariton oscillations are determined by the roots of equation F(to~)=

—

I

(4.1(1)

;

2) is to, = l2~!V’2.The thus, the to., value for the boundary of isotropic collisionless plasma (F = 1 l1~o~ dissipation length L~of the surface waves is restricted usually due to appreciable values of the ratio y~ ~ l0’—lO 2 [8]. The measurement of the frequency dependence of L~is used nowadays as a basis of surface electromagnetic waves spectroscopy. The resonance of the surface plasmon polariton oscillations, produced, e.g.. by laser radiation, with excitonic states of the film allows the study of the spectra of some monomolecular films and metal surfaces. Along with the resonant dissipation of the above-mentioned magnetoplasma waves, localized hear the semiconductor’s boundary, the absorption spectra of the surface plasmon modes form a powerful diagnostic tool for the investigation of plasma—wave interactions in solids. —

—

4.2. Resonant collisional dissipation in heterogeneous plasma

The localization of dissipation regions inside the plasma volume may be produced by the heterogeneities in density or magnetic field distributions. It is essential, that the location of these regions may

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

225

be changed due to utilization of the suitable frequency, polarization or direction of propagation of the pumping wave. The amplified Joule absorption of the travelling wave energy in these regions gives rise to many types of plasma instabilities [76]. Such plasma—wave interactions are strengthened near the reflection points of the wave in a plasma. The spatial scales of these resonant regions are determined by spatial dispersion and gradients of density and magnetic field in a plasma. The contribution of each of these phenomena to resonant region formation depends on the correlation between the wavelength of the incident wave A, the characteristic scale of the density profile L, the Debye radius rD and the angle between the wave normal and the magnetic field H. The normal incidence of the pumping wave on the plasma layer, the electron density profile being linear in the half-space z ~ 0, (4.11)

NeNeo(l+zIL),

leads to formation of a standing wave, described by the Airy function. Here the distance / between the reflection plane z = z0 and the position of the first node of the standing wave depends on the wave polarization. Thus /

1o2.3(,,

(92 2

\1/3 2

‘

c sin a

)

(4.12)

for the ordinary polarized wave [58],while the same parameter / for the extraordinary mode is (92

1e2.8(2(2))

i/3 .

(4.13)

Here the limit a Velto in (4.12) needs special consideration [24]. The scale 1 may be considered as a characteristic thickness of a resonant layer in a plasma with smoothly varying density. These results are justified, the spatial dispersion being neglected, if the wavelength of the incident radiation A is large enough: A > r~. Unlike the above examples of hysteresis, arising in a homogeneous plasma volume, the effect under consideration is localized near some surface z = z0 inside the plasma volume. In the simple case, when the magnetic field is absent, the position of this surface is determined by means of the equation 2 (4.14) ‘~

.

~

to

The electric field E leads to the perturbation of electron density; thus the dielectric permeability F may be written near the surface z = z 0 in the form [89] 2/E~, (4.15) E=E,—is; E1X/L+E where

x=

2/c2. E~= 8lTmkTestw The spatial dispersion phenomena are neglected here, because the characteristic scale of plasma density z

—

z

0

226

.4.B. Shvartshurg, Resonant Joule phenomena in a magnetop/a.cma

heterogeneity L is supposed to be large enough: 3

eE

/r0\~’ I—)

1fl(OV~

t’. to

The field E in (4.15) is assumed to be weak enough: E2
(4.16) 2 from eq. E0 being the the fieldnon-linear on the boundary plasma layer, us to the quantity E may be (4.15). Thus equation,ofgoverned by thepermits real part of eliminate the dielectric permittivity, written in the form: 5~.

=0.

Ls

s

(4.17)

I + (Ft/S)2

Here the dimensionless parameter

I

(4.18)

‘5’)

s~E~

describes the non-linear effect. The analysis of eq. (4.17) similarly to eq. (2.15) shows, that the dependence Ft = Fi(x/Ls) is single-valued, if the wave amplitude is small enough: (4.19)

3~~~/3

Unlike this, in the region

~j ~

~ the single-valuedness is violated and the dependence F

1 = E-t(xiLs) has a hysteresis behavior (fIg. 22). It is essential, that such a hysteresis leads to the spatial discontinuities

-2

3’~ respectively Fig. 22. The spatial hysteresis of the non-linear resonant dependence of the dielectric permittivitv Fi upon the dimensionless distance .r(Lsy’ heterogeneous plasma; the curves 1, 2 and 3 correspond to the dimensionless field parameters i~= I): = ij,,: ~s=

itt a

AR. Shvartsburg, Resonant Joule phenomena in a tnagnetoplasma

227

formation in the initially smooth profile of dielectric permittivity. The characteristic spatial scale of this effect is of the order of x Ls. The parameters of these discontinuities depend pronouncedly upon the frequency of collisions lIe (4.18). Thus, the electric field near the hysteresis threshold E~.behaves itself as Ecr V~2. Such an effect attracts attention in view of perspectives of its utilization in non-linearly tunable capacitors in electronics and localized Joule heating in a heterogeneous plasma. The resonant plasma—wave interaction in a heterogeneous plasma is localized mainly in a relatively thin layer, comparable with the wavelength. The processes of waves transformation and reflection in this region may be very sensitive to Joule effects. The series of geometrical resonant phenomena in the thermic wave coupling is connected with the resonant scattering of the probing wave on the periodic structure, formed by the pumping wave inside a plasma. It is worthwhile discussing these processes by means of two models, related to heterogeneous plasma density, the magnetic field being constant, and to heterogeneous magnetic field, the plasma density variations being neglected. =-

4.2.1. The waves coupling in the resonant region inside the collisional magnetoplasma Unlike the waves coupling phenomena, occurring near the Langmuir frequency in the isotropic plasma (2.3.1), the possibilities of wave transformations, both linear and non-linear, are extended essentially due to the magnetic field. The traditional example of linear transformation in a heterogeneous magnetoplasma is connected with the triple splitting of ionospheric waves [37]. Such an effect, associated with the small angles a between the wave normal and the magnetic field, occurs near the Langmuir resonance layer (lie(Zo) = to). Here the ordinary wave, incident upon this layer, is partially reflected and partially transformed to an extraordinary one, the effectiveness of this conversion depending on the frequency of electron collisions. The amplitude coefficient R of this mode conversion may be written as çexp[ R=-~

—

const(l 1

—

q)]

; ;

q~ 1

q>l,

(4.20)

where the collisional parameter q is defined in (2.23). Joule heating of a slightly ionized plasma, stimulating the growth of the parameter q (4.20) may increase substantially the effectiveness of such a conversion. The coefficient R (4.20) reaches its maximum value due to heating up to the temperature fo=q~i>1

(4.21)

related to the closeness of the complex refractive indices of both modes. Another resonant wave process, produced by collisions, is connected with the waves percolation in the vicinity of the maximum of the smooth plasma layer, characterized by the critical value of the Langmuir frequency (Ile)max = 11cr. The temperature decrease of the frequency of collisions leads to the growth of the reflection coefficient amplitude (several times earlier) in the resonant range to > lie: (toQe)l1~~105[24]. The absence of the threshold amplitudes of the pumping wave is typical for such thermic effects. Unlike this, the wave coupling in a collisional magnetoplasma, connected with the parametric phenomena, is characterized by threshold amplitudes of the pumping wave. These amplitudes are

228

.4. B. Sh t’artshurg. Resonant Joule phenomena in a snagnetoplassst a

proportional to the frequency of electron collisions with heavy particles; here these collisions may also have an influence on the non-linear frequency shift of the pumping wave [99].The different tendencies in temperature variations of frequencies of electron—ion and electron—molecule collisions may stimulate the non-monotonic thermic variation of threshold amplitude due to plasma heating, similar qualitatively to fig. 10, related to a non-magnetized plasma. The influence of the magnetic field leads to the difference in the location of the reflection points of ordinary and extraordinary modes in the heterogeneous plasma. The excitation of parametric instability near the reflection point of the extraordinary wave is possible if the magnetic field is strong enough [58]:

Q~<2to~.

(4.22)

The aforesaid difference may become important in the course of ionospheric modification experiments. Thus, the location of reflection points of ordinary and extraordinary modes of the same pumping wave in the E-layer and F-layer respectively may lead to a difference in the threshold amplitudes, responsible for parametric instabilities in these points, in one to two orders of magnitude [58]. The dependence of the group velocity of plasma eigenmodes, generated in the course of parametric instability development, on the magnetic field, may give rise to deceleration of these modes. In the case to 2~Hthe group velocity of longitudinal waves decreases in some limits proportionally to (to 2toH)2, and their attenuation decrement tends to lIe [76]. The appearance of the resonant region inside the heterogeneous plasma volume, located in the heterogeneous magnetic field, proves also to be perspective for plasma heating in laboratory devices. Thus, the collisional absorption of radio frequency fields at the Alfvèn resonance layer is one of the principal features of Alfvèn wave heating in Tokamak plasmas [129]. The damping of such a wave is centered at the spatial location, where the continuum frequency matches the local Alfvèn velocity. Such a resonant absorption may be considered as a consequence of the generation of a continuous frequency spectrum by disturbances in a non-uniform plasma [82j. —

4.2.2. The influence of magnetic field heterogeneity on resonant Joule phenomena in a plasma

The heterogeneities of magnetic field in a plasma attract attention in view of many types of plasma instabilities [2]. The variations of refractive index of low-frequency electromagnetic waves, produced by these heterogeneities, may become responsible for the occurrence of cosmic waveguides near the plasmapause boundary [32].These waveguides spectra depend on the geomagnetic field heterogeneity. The sensitivity of resonant processes near the reflection point of the pumping wave to the magnetic field shows the possibility of spatial restriction of resonant region due to magnetic field heterogeneity. An interesting example of such a restriction is connected with the aforesaid parametric instability near the double gyrofrequency to 2WH. Let one consider the pumping wave incident orthogonally to the magnetic field. The parametric excitation of the upper hybrid resonance (u + v = 1) due to this wave is possible in the region, characterized by the ratio

u2~=3to~.

(4.23)

The essential peculiarity of such plasma eigenmodes is connected with the restriction of the region of their existence by two reflection points Zi and z 2 and the accumulation of these waves inside the resonant region z1
A.B. Shvartsburg, Resonant Joulephenomena in a magnetoplasma /

v

=

Z\

/

vo(51——) ;

u

Li

=

Z\

uo( 1+~.t—); \

Li

229

i2 c9w~\

9zi , ~u= ~,—————)

(4.24)

coHs

where the values v

0 and u0 relate to the point of pumping wave reflection (v0 = 0.75; u0 = 0.25). The resonant region for the longitudinal plasma waves, propagated across the magnetic field, may be obtained from the following wave dispersion equation, the spatial dispersion near the double gyrofrequency being taken into account [101]: 2p~=(1

k

3uv u v)

4u)(1

—

—

—

;

(4.25)

here Pc is the electron gyroradius. One may see, that the plasma waves may exist below the reflection point of the pumping wave (z = 0) on the condition that u > a 0 = 0.25. Therefore such an excitation may occur, if the gradients of plasma density and magnetic field are antiparallel, as is shown in (4.24). Equation (4.25) permits us to determine the reflection points Zi and z2. Thus, the distance between these points z~and z2 in the ionospheric F-layer (L 100 km, p. = 0.05—0.08) is Zi (1.5—2.5) km. The wavelength of the longitudinal plasma wave is A (20—100) m, so such a natural resonator will contain many wavelengths. The accumulation of these waves in this resonator is connected with their repeated interactions with the pumping wave. The attenuation of such long wavelength (A > rD) plasma waves is produced mainly by Joule dissipation, characterized by frequency of electron collisions t’e. This resonant heterogeneous plasma volume is formed by smooth variations of density and field distributions. Anomalous Joule heating in a heterogeneous magnetic field occurs when plasma becomes turbulent, which happens, when the electron drift velocity exceeds the ion thermal velocity. This effect becomes possible, if the magnetic gradient scale-length / is small enough [110]: —

1
12~

\/ kTe’

in order that turbulence is excited. This mechanism, effective in small volumes, may be realized in the outer solar atmosphere. Here scale lengths of less than I km occur probably only under solar flare conditions. The shear of magnetic field is supposed to restrict the scale length of the localized dissipation region also. This restriction proves to be important in the effects of Joule heating and radiation of magnetic tearing. The spontaneous generation in a sheared field of a very narrow layer, characterized by field reconnection, can occur. The rate of this process depends directly the value of conductivity in the 67 [1111.The localizedofJoule phenomena can quench the tearing layer, scaling as o~, where 0.4< V
A .B. Shv’artshtsrg. Resonant Joule phenotns’na in a sssagnetopla ssna

231)

the linear profile of maffhietic field = wo( I

:/L)

—

(4.26)

;

the frequency of electron collisions v~.being constant, for simplicity. The point of cyclotron resonance is characterized by coordinate = L0 < L. If the direction of magnetic field growth coincides with the directicn of the wave propagation. the

./1 = I =

—

2

exp[

L L0

—

(~(I

—

ya)I

—

absorption

coefficient may he written as [102]

C~exp(v~C~): (4.27)

i’~

—- ——

~

ix)

where the constants C. C~and C2 do not depend on the parameter “e, The opposite case, when these directions are antiparahlel. is described by another value of this coefficient:

X= 1

—

exp[

—

C(I

-

y~)].

(4.28)

These effects, specific for the one-dimensional problem (a = 0 and a = ~/2) are cohinected with heterogeneous magnetic fields; the angular anisotropy, essential for an oblique propagation of the wave in the cone (1< a < ~/2, was not considered. However, the variety of Joule phenomena, depending on the magnetic field, is enlarged pronouncedly due to such an anisotropy influence.

-/.3. Joule absorption ticar resonant cones in a inagnetopla.s’ma The anisotropy of plasma, located in a magnetic field, leads to the formation of two systenis of peculiar cones of directions inside the plasma volume. One system, stipulated by resonant growth of the refractive index of the wave in some directions, characterized by the angle ar of their inclination to the magnetic field =

Arctg

\

—

F;/Fj,

(4.29)

forms so-called resonant cones. Another system. the so-called Storey cones, connected with the group velocity maxima, is determined by the angles cc,, calculated from the equation 11031 =

0,

(4.30)

where k~and k1 are the components of the wave vector, parallel and orthogonal to the magnetic field. Joule effects in the vicinity of these cones are accompanied by a series of peculiarities. These effects will be considered below for the frequencies, much more than for the lower hybrid frequency. The resonant cones attract attention in view of the attenuated dependence of the fields, propagated near the direction ccc, on the distance from the source r 113]: 2 H r~2. (4.31) E r_t/ ~-

~=

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

231

instead of the usual dependences E r3 H r2, customary for directions, far from the resonant ones. Large values of the refractive index in the vicinity of the resonance (cc —* ccr) are accompanied by considerable absorption: the complex refractive index in this case may be written as —

—

(n—IK) 2

sin2a(E~.+g2cos2a) 2a 1 sin 2a ~a + 2i1 cos

=

.

.

;

z~a=a—a~

(4.32)

F

~

1v (l+u).2 2

vcos2al

i,,.ar

.

(4.33)

2w L (1— a)

The energy flow along the surface of such a cone is determined by the non-zero frequency of electron collisions lIe; the propagation length of such a surface wave, restricted by collisional losses, may be evaluated as several hundreds of wavelengths [103]. The Storey cones are discussed often in view of the problem of effective energy transfer through the plasma media. The electromagnetic beams, propagated in these directions (cc ar). are diffused more slowly, than in another directions, in particular, in a free space [104]. These resonant phenomena occur mainly in the radio range of frequencies, which is sensitive to the influence of magnetic field on the dispersive properties of plasma. Such an interaction of radio waves with a magnetoplasma are analyzed usually from one of two viewpoints: (a) the peculiarities of resonant propagation of the wave in the far zone of the source; (b) the structure of resonant field near the source. The role of collisional phenomena in both of these situations is illustrated below. 4.3.1. The resonant field in a far zone in a collisional magnetoplasma The spatial structure of the resonant field near the source depends, generally speaking, on the source geometry. Let us consider, first, the model of a small source, whose dimensions are less than the wavelength of the harmonic electromagnetic oscillations, produced by this source. The potential of the source depends upon the angle a between the magnetic field and the wave vector [108]: =

e r

exp(—iwt) 2 a + ~1 sin2 a)

V’F±(F 11cos

The model of the plane waves is suitable for this geometry. It seems to be instructive to emphasize the role of anisotropy by means of comparison of the fields, irradiated by an electric dipole I in a free space and inside a plasma. The distribution of this field modulus in free space may be written as

IE(r)I

Ic sin a =

2wr

-

(4.34)

Here a is the angle between the dipole axis (z-axis) and the wave vector k. Now let us consider the

model of homogeneous magnetoplasma, the z-axis being directed along the magnetic field. The angular pattern of the dipole in a magnetoplasma may contain maxima in the directions, corresponding to the magnetic field, the resonant and the Storey cones. The electric field, irradiated by

232

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

the same dipole (4.34) in the resonant direction (a /3\2

iVi

—

E

11/F1

1

ar),

is restricted by the collisions [86]

to

IE(r)I= I—I

\e’

=

~=—rcosa c

~4y~F1~Vf~_g2/(E11_ff)

:

(4.35)

/~ (Im F1 ~

Im F11 ~ReF1Rer~1

The radiation near the resonant surface is characterized the gigantic electric 11H cone < to <(OH (fig. 23). Here thebyresonant coneamplification is supposed of to the be located field in the frequency range inside the Storey cone. The angular anisotropy of such a radiation is illustrated in fig. 24. One may see, that the electric strength E 7 near the dipole z-axis (a = 0) excels the resonant field E. (a = ar) far from the source. The partial localization of the electric field near the Storey cone is connected with some growth of the field amplitude near the conic surface a = a,. Unlike the resonant electric strength at the cone a = a,., the fields, related to the directions a = 0 and a = a, are not restricted by the Joule phenomena. The collisional dissipation of the resonant field may be utilized for plasma heating. Here the absorption impedes the direct penetration of the pumping wave deeply in a plasma volume. Therefore it seems to be useful to utilize the beat frequency heating of a plasma by two anti-parallel laser beams with the frequencies to1 and to2, their difference being close to the plasma eigenfrequency w~. The electric and magnetic field, electron density and velocity contain the beat frequency terms. The major contribution to the time-averaged power absorption density comes from the z-component (zIJHo) of the non-linear second-order forces [105]. The growth of electron temperature (Te ~ l0~K) in the course of an oblique propagation (a 30°—60°) of the laser beams in weakly ionized plasma may be accompanied by a resulting increase of collisional dissipation [105]. Unlike the aforesaid anisotropy phenomena, which occurred in a model of homogeneous plasma, the heterogeneity of magnetoplasma gives rise to some new resonant effects. Thus, the existence of the

U__

__

0

0,6

0,9

~

Fig. 23. The gigantic amplification of the electric strength E1 of the dipole field in the vicinity of the resonant cone (a = ares) 2WH. inV =a 100 sec~)compared with the(a,., field= E7.5 x I0~rad sect: 12, = collisional magnetoplasma 55. produced by the same dipole in the samethepoints against frequencies. in free space. The quantity t~= lg~EiE6’~is plotted

_lt Fig. 24. The influence of magnetoplasma anisotropy on the angular distribution resonant coneof (E,) the and electric nearstrength the dipole of the z-axis dipole (Er).field The near quantity the q = lg~E,E5’~ 2w,.,; is plotted lie = 2wA; versusv =the100dimensionless seet). distance ~ (4.34): (a = 4.2 x lO

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

233

dielectric susceptibility pole of the order p (p> 1) shows the possibility of the amplification of the travelling resonant wave due to some arrangement of magnetic field, density gradient and wave’ vector in the non-equilibrium collisional magnetoplasma [851. 4.3.2. Joule effects in the resonant field near the source The structure of the resonant field in a collisional magnetoplasma in the quasistatic region near a source of finite dimensions, depends upon its geometry. Here the volume distribution of the dissipation of the wave, irradiated by this source, depends upon its dimensions via the field structure. The maximum of such a distribution is localized near the resonant cone (4.29). The form of the signal envelopes proves to be sensitive to the spatial source distribution and, moreover, these envelopes are different for electric and magnetic fields. The dynamics of the establishment of these envelopes is shown in fig. 25. The characteristic time of this process, t, is determined by the source dimension a, orthogonal to the magnetic field, and the distance between the source and the observation point [106]

rV~

-

awVv

+

(4.36)

,

E

1

The delay time of the signal is evaluated via the group velocity Vg as t = rv~,i.At the beginning of the field establishment (t t) the interference effects, produced by the contribution of the far points of the source, are inessential, and the field does not depend on the source dimensions. Such interference effects, accumulated in the course of the field formation during the time t, govern the appearance of the peculiar shadow zones in the radiation field inside the plasma. Unlike the collisionless dissipation, distributed all over the plasma volume, the Joule absorption is localized mainly in the vicinity of resonant cone a = a,. + L~a ‘~

z~a= -f-sin ar,

(4.37)

2r

where 1 is the scale length of the source. The power absorption density P per unit volume in the region (4.37), irradiated by the linear current ..9, located in a plasma, may be calculated in the far zone as [89] 2 toco: a~ (4.38) =

2 g

ITT V~

e

Fig. 25. The temporal relaxation of the resonant fields near the source in a collisional magnetoplasma. The magnetic (curve I) and electric (curve 2) amplitudes, normalized on their established values, are plotted against the dimensionless time r = t1’ (4.36).

A. B.,51 tart.vhurg. Rexonastt Jsnile phenomena as a Os agnetopla

234

The decrease of the frequency of collisions “e leads to the growth of the field near the resonant co~ie and to an increase of the absorption (4.38). The field of the source depends on the resonance, excited by the standing waves in the plasma layer near the source. If the characteristic scale of such a source is large enough I>

2 7TC to

cos a~

/ (I)

the power absorption density P decreases, the other plasma parameters being equal 1106]. The peculiar effect of the source. located in a plasma, may be connected with the return of some part of the radiated energy to the source with its resultant dissipation on the surface of the source. So. such an effect may he produced by the electron beam, injected from a spacecraft to the upper ionosphere [98]. The excitation of the oscillations of the potential of the spacecraft with a frequency, close to

Langmuir’s one, leads to the radiation of plasma waves. The electrons in the region of spatial charge, adjoined to the spacecraft. are accelerated due to the absorption of the energy of these waves. The electron energy losses, stipulated by their collisions with plasma particles in the upper ionosphere. are negligible in the energy balance: however, the effective mechanism of energy losses of electrons. oscillating with the Langmuir frequency, occurs due to their collisions with the spacecraft. accompanied by the transformation of the energy of these oscillations to the heat. Such effects of anisotropic Joule absorption attract attention in view of problems of antennae

radiation in plasmas and microwave heating of large plasma volumes.

5. Geometrical resonances in Joule dissipation The restriction of plasma volume in some direction leads to the formation of a quantized spectrum of wave modes, propagated across this direction. The differences in polarization structure of each mode govern the dependence of modes dissipation upon their spectra. determined, in its turn, by the geometry of the plasma volume. Such geometrical effects are responsible for the appearance of absorption lines near the eigenfrequencies of bounded plasma, situated. e.g., in the transparency range of an unbounded plasma. Moreover, the absorption near the resonance to l1~. occurring iti the unbounded plasma. may be strongly modified. Such geometrical phenomena attract attention in connection with wave processes in gaseous and semiconductor plasmas in laboratory devices and electronics. The simplest resonant effects occur in the course of alternative electric current establishment in “small” plasma clouds, the scale length I of such clouds being much less than the wave length, related to the characteristic time T of this current variation: I ~ c~.The wave phenomena are neglected here, and one may consider the model of current oscillations in some electric circuit. The self-inductance, the capacity and the resistance of such a circuit depend on the scales of the current-carrying region. An interesting example of this effect in cosmic electrodynamics is connected with the resonating loop in the

outer solar atmosphere [115].The energy generation and dissipation in a coronal loop maybe evaluated by means of a model of the circuit, containing the self-inductance M = 41(irc2)_i, the capacity C= lc2(ITv~~t and the resistance R; here I is the major radius of the torus-like loop, VA is the Alfvèn velocity. The circuit is driven by an external source I = Hl~v 1c where l~is the minor radius of the ~

A.B. Sht.’artsburg, Resonant Joule phenomena in a ,nagnetoplassna

235

loop, v1 is the perpendicular photospheric velocity and H is the magnetic field, orthogonal to this loop. 2R_tc_h/2 is very large (0 l0~), the system will resonate at a Since the quality factor Q = MuJ characteristic frequency (5.1) Hence, only the energy in a narrow frequency band ~wto4~ ~ is picked up from the photospherie source spectrum. The model of the current circuit proves to be useful in the analysis of the Polar Electrojet modification by the powerful ground-based source of the radio waves. The thermic variations of the frequency of the collisions, t’e, in the current-carrying region, produced by the Joule heating, produce the modulation of this conductivity region. The corresponding oscillations of the Electrojet are responsible for the generation of the alternating magnetic field H_. Thus, the modified area may he compared with a gigantic antenna, disposed in the lower ionosphere. The frequency of this antenna radiation to is equal to the repetition rate of the heating pulses. The characteristic scale lengths of this perturbed disk-like area are determined by the half-width of the beam a and the thickness of the current-carrying layer h. One may evaluate the self-inductance of this area M = 2ac~2,the ratio ah~ being of the order of unity. The vortical current, induced in this area, is restricted by its resistance R. The intensity of the radiation of such an antenna, considered as a Hertzian dipole, tends to a maximum. when the frequency to is high enough [116] w2~’Q~=RM~’.

(5.2)

The values of the frequency ll~,calculated for the scales a = h = 20 km, are ho = l0~—l0~ rad sec_i. The effective radiation (H~= 0.6 x iO~G), observed on the Earth surface due to such a generation, was connected with the frequency to = 1.5 X 10~rad sec_i [117]. Unlike these small scale oscillating systems, characterized by one eigenfrequency, the wave phenomena in large scale-bounded volumes, containing many wavelengths, are described by a series of eigenmodes. In order to emphasize the influence of the geometrical factors on the absorption processes in such systems, it is worthwhile considering at first the one-dimensionally bounded plasma configurations. This model can represent the common features of the wave fields of different physical nature, from Alfvèn modes in large scale slabs of cosmic plasmas up to microwaves in thin dielectric films with Joule losses, utilized in electronics. The peculiarities of collisional absorption in more complicated configurations, bounded in two or three dimensions, such as plasma waveguide or semiconductor specimen of n-type with free carriers, are considered later. 5.1. The resistive eigenmodes in one-dimensionally bounded plasma configurations

The simplest models, describing the wave structure in collisional plasma volume, bounded in one dimension, are connected with the plasma layer. Such models include the travelling eigenmodes, propagated along the layer, and the standing waves, formed due to wave propagation across the layer. The role of the finite conductivity in such systems is not restricted by waves dissipation only. Thus, the dissipative properties of the layer’s boundaries may determine the spectrum of resonant oscillations inside the layer, and what is more, the eigenfrequencies of some modes depend directly upon the

.4. B. Sittart.shurg. Rs’,to,sa,st Jostle p/te,tsssnena in a otagns’topla,ssssa

23ts

frequency of collisions in this volume. Similar geometrical resonances occur in microwave slowing systems. based on spatially modulated dielectric slabs with high frequency absorption. It seems to he useful to discuss the dissipative properties, accompanying the wave propagation along and across the layer. separately.

1.1. A/ften c’igeninocic’s in concluded magiictoplasnici slab The spectrum of low frequency eigenmodes. propagated along the collisional magnetoplasma slab, depends upon the boundary conditions and the magnetic field geometry. It seems to he useful to classify these eigenmodes in connection with the orientation of the magnetic field. The simplest spectrum is connected with the plasma slab, bounded by the conducted planes, and the magnetic field, directed ~.

parallel to these planes. Such a configuration attracls attention due to the Tokarnak experiments. In order to achieve the plasma parameters. suitable for nuclear fusion, one may consider the injection of high-energy atoms into Tokamak and the resulting formation of the beam ol’ high-energy ions due to the ionization of these atoms. These beams may become a source of the Alfvèn waves, propagated along the slab. The real and the imaginary parts of the wave vectors of these waves, responsible for the energy losses of the ion beams. may he written for each mode as [119]

Re k,t

=

w

d

—

~

/(k~+ k~+~)2 I 4k~R,~ —

-

—;

Im k,

k~+~

w =

c

-

2kR,

-~———~~:

5,

=

ir(p

+ I

)/2a :

R,,, = 4ir0L’..\/H:

p

=

U. 1. 2...:

(5.4)

2a is the distance between the conducted planes: k; and k : are the projections of the wave vector. parallel and orthogonal to the magnetic field. H being parallel to the boundaries of the slab; o- is the longitudinal plasma conductivity, VA is the Alfvên speed.

Some more complicated model, connected with the magnetic field, orthogonal to the plasma layer. becomes important for cosmical physics in connection with the problem of interaction between neutral

and ionized gases. One may suggest, that a high ionization rate in such an interaction appears froln the collision between plasma and neutral gas clouds and because of the absorption of plasma waves due to ion—neutral collisions as in photospheres and chromospheres and in cool interstellar clouds [13]. The above model, representing the strata of a heavy. inviseid, self-gravitating and finitely conducting cloud. may he utilized for the analysis of stability of such strata. The conditions of instability of the heterogeneous layers of a fluid, obtained by Rayleigh. were generalized to a self-gravitating medium in [13]. The joint effects of neutral gas friction and finite conductivity are shown in fig. 26. The dimensionless parameters N. B and S, determining the dynamics of the departure of equilibrium froni the initial state, shown on this figure. are connected with the frequency of ion—neutral collisions

i’~,,,.

viscosity i~and gravitation constant g vmd N=—;

WI]

tA d B=——--’:

5=

.,

4Gpd ITL’7~

:

to’d

‘j~=—:

(5.5)

ITVA

where d is the thickness of the layer, p is the plasma density. and to is the growth rate of the perturbation with the wave vector k. One may see the increase of the growth rate due to an increase of the viscosity; on the other hand, the ion—neutral collisions have a stabilizing influence. The maxima of

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplassna

237

K Fig. 26. The instability of the waves, propagated along the self-gravitating partially ionized plasma layer: the dimensionless growth rate y of the departure of the equilibrium from its initial state is plotted versus the perturbation wave number k, normalized on the quantity ird (d is the thickness of the layer). The curves 1 and 2 correspond to the values B = 5, N = 20 and B = 20. N = S respectively: S = 5. The dimensionless parameters B. N and S are determined in (5.5). (See ref. [131.)

the curves in fig. 26 illustrate the strengthening of the instability near the geometrical resonance kd I. The influence of collisional dissipation and finite gyroradius on the Alfvèn wave spectrum of a plane slab in a sheared magnetic field leads to the occurrence of the discrete eigenfrequencies of lowfrequency resistive modes. The thickness of this slab may be evaluated by means of the shear length L,. The spectra of the Alfvèn waves with wave vector k, propagated along the slab, depends directly upon the frequency of ion collisions e and ion gyroradius p~via the dimensionless parameter 0. [120] ‘—

u2~,

=

~- ~

(V3~ i) (2p +

~VAL,Jn ek 2p3kT, ‘

5)2/3.

p

=

0, 1,2 -

(5.6)

(5,7)

‘

The discrete eigenfrequencies of the resistive eigenmodes (5.6) tend to zero, in the limit of weak collisions, as e t/3 The periodic corrugation of the layer may stimulate the growth of the energy of some propagated modes. This effect may be utilized in electronics for making the planar dielectric resonators for the surface waves [125].The thickness of these dielectric films must be commensurable with the wavelength A, and the resonant effect occurs, when the spatial period of corrugation d is connected with A by means of condition d = 0.5Ap (p = 1,2,3. .). Unlike the microwave resonators, formed by metallic cavities. the Q-factor of such planar resonators is determined by absorption, distributing all over the film’s volume. The small values of high-frequency conductivity of the film may lead to the large values of the 0-factor of such a resonator. Thus, the segment of the film with thickness h = 0.6 x l0~~ cm and length = 0.3 cm, corrugated at the ends, the period of corrugation being equal to 0.3 x iO~ cm and its depth being equal to 0.05 x l0~~ cm, forms a planar resonator for the wavelength A = 0.6 X 10~~ cm; the attenuation factor in this case is 0.2cm_i. The 0-factor of this resonator may reach the value O = 5 x l0~[120]. These resonant effects are due to eigenmodes, travelling along the conducted slab. Unlike this, the . .

A.B. Shvartsburg. Resonant Joule phenomena in a magnetoplasma

238

incidence of the wave on the boundary of the conducted layer is characterized by a peculiar system of geometrical resonances in dissipation. 5.1.2. Geometrical resonances for the waves, incident on the boundary of conducted slab The interference of the fields, reflected from the boundaries of a conducted layer, leads to the formation of the standing wave, the amplitude of which depends on the spatial distribution of attenuation in the conducted layer. The simplest models of spatially corrugated absorption and its localization near the boundaries attract attention to the role of geometrical factors in phase effects, determining the structure of the standing waves in dissipative systems. The geometrical resonances of absorption appear due to the scattering of the microwaves on the metallic lattices, located inside the layered structures. Such a geometry of scattering is utilized in radio technique for polarizers, antenna fairings and slowing systems. The resonant absorption of the slowing system, consisting of metallic strips, may be amplified due to the filling of the slots of this system by a dielectric with a relatively high value of permittivity F > ~:.. where e~is the dielectric permittivity of the surrounding medium (fig. 27). Such filled grooves may be considered as a matching transformer between the medium with permittivity ~O and the metallic surface [121]. The losses per unit length P of this system depend on the period of the system d, the thickness of the dielectric layers i and h and the angle of incidence 0. The influence of the aforesaid matching layer on the growth of the absorption may be characterized by the ratio P(0; d;i;h’. e~~)

P5=limP(O;d;i:h;F5F)~~_,5.

(5.8)

The resonant maxima of the absorption parameter y5, are shown on fig. 27. The localization of the absorption near the boundaries of the plasma layer may be considered as a convenient model, describing the Alfvèn wave propagation along the geomagnetic tube through the magnetosphere. The successive reflection of these waves upon the South and North ionosphere determine the structure of this low-frequency field; here the values of the reflection coefficients RN and R~depend upon height-integrated Pedersen conductivities ~ in both these hemispheres [109]: 4ITUA

RNS

=

l—o’~ l+0NS

UNS =

2

(5.9)

(-~P)N.s.

C

L~

Fig. 27. The resonant absorption of the incident waves by the slowing structure, immersed in the layered dielectric nsediuns (e absorption rate y (5.8) versus the ratio of the structure parameters. The curses I and 2 correspond to the values ade

ref. [t2lj).

=

5/es1 = lit): the (1.4 and (0.6 respectively (see

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

239

The resonant frequencies of the magnetic tube, fixed at its bases and characterized by the length 1, are 2

liITVA(

(5.10)

p=O,l,

2l)

The length I is much more, than the thickness of the collisional ionospheric layers, which form the bases of the tube; so, one may consider the Joule dissipation of such Alfvèn waves to be localized at these bases. The excitation of such waves by means of heating perturbation of the bases is described by the spectrum, depending on the Pedersen conductivity (5.9). Thus, the contour of the spectral line, observed near the North hemisphere, is [109] A(to)=

A(too) (1

—RN) [

1

N-

2

R~exp(2ITiwtojt) RN R~ exp(2iritotoJt)

+

.

5.11

The wave absorption between the ends of the tube is neglected here. The resonant growth of the wave’s amplitude is shown on fig. 28.

Fig. 28. The dependence of the amplitude of the geomagnetic tube oscillations on the conductivity’, located at tts ends. The dimensionless values of

amplitude a

=

A(w)A ‘(a 55) (a

=

2w,,) are plotted against the dimensionless absorption parameter o~= o-~= a-s (5.9).

An interesting non-linear effect, produced by conductivity localization in the vicinity of the boundaries of the system, is discussed nowadays in connection with the scattering of the powerful radio waves from metallic objects, covered by oxide films. The metal—oxide—metal contacts, habitual for metallic objects, may become a source of harmonic generation due to non-linear corrections to Ohm’s law 3)/R (5.12) (U + /3U where g, U and R are the current, the tension and the resistence, respectively. It is essential that these =

parameters correspond to the thin film near the object’s boundaries. Such a mechanism, responsible for the non-linear variations of scattering and for the third harmonic generation by a half-wave dipole, may become important for radio location [124]. 5.1.3. Geometrical resonances, produced by plasma lamination The coupling of the waves in the conducting media may he realized by means of the artificial

240

A.B.Shr’artshurg. Resonant Joule phenom ena in a ,nagnetop/a.sma

geometrical resonances. Such interactions occur due to resonant scattering of probing waves on the periodic structures, formed by the powerful standing waves. Thus, the thermic perturbations, produced by a standing wave in a collisional plasma, may form a spatially periodical profile of electron temperature. Plasma diffusion, stimulated by temperature perturbations and striction forces. leads to the formation of plasma lattice, whose period is close to the distance L12 between the standing wave nodes. Such lattices, utilized as antennae for ionospheric radio propagation, may he characterized by the ratio of the scattered wave amplitude E’ to the incident one E, proportional to the maximum value of plasma density perturbation ~NN5, ]122]. ‘This ratio is evaluated as 10 ~—l0~~ [32]. So, the amplification of such a scattering scenic to he Important. One of the possibilities of such an amplification is connected with the resonant amplitude modulaticn of the pumping wave at the ion sound frequency to = 12. = kr’, cos a [101]. Here k = 2ITL0’ and u, are the wave vector and the velocity of the wave, a is the angle between the magnetic field and the vertical, the normal incidence of the pumping wave on the plasma layer being considered. The comparison of the amplitude values of the density perturbations ~N.related to modulated (~N11)and unmoclulated (~N,) pumping waves, indicates the growth of ~N12, restricted by the attenuation decrement of ionic sound y,, 2,/y., =

-

h

Thus, the amplification of plasma perturbation even by three times (hi. = 3y,) may lead to approximately a 10-fold increase of the scattering wave. The thermic variations of the electron density inhomogeneities spectrum give rise to a peculiar mechanism of waves interaction in weakly ionized plasma and, in particular, in the low ionosphere. The heating of electrons leads to the attenuation of small scale turbulence and to the growth of the large scale one. These variations modify the characteristics of the signal scattered in the heated region. The smallest scale length of the turbulence 1(f) increases due to heating as [113]

/(f) I

-

~I ~fy.s ‘\ 2)

A more complicated periodic structure may he formed in conducting media by two coherent electromagnetic waves of the same frequency, propagated in different directions. The properties of this structure are determined by dragging of the carriers [15]. Such geometrical resonances seeni to he important for the tunable control of radiation in the conducting media in optical, microwave and radio frequency ranges.

5.2.

c;eontdtridal s/tiff 0/ absorpluni lutes

in boundc’d pkisnw to/tunes

The dependence of plasma eigenmodec in hounded volumes on the geometrical parameters of these systems displays itself in two groups of phenomena: (I) The shift of resonant frequencies in comparison with those in an unbounded plasma: (2) The absorption lines deformation due to geometrical factors. The importance of these effects for Jule phetiomena in hounded volumes of gaseous and semiconcluctors plasmas is illustrated below for plasnia volumes. hounded in two and three dilnensions. Here the first model corresponds to the plasma waveguide, the second one to the plasma spheroid.

A.B. Shvartsburg. Resonant Joule phenomena in a magnetoplasma

241

5.2.1. Joule phenomena in the two-dimensionally bounded plasma column The absorption effects in the radially restricted plasma volume attract attention in connection with the waves propagation in waveguides, filled with collisional plasmas. Some tendencies of such a plasma—wave interaction may be considered on the basis of a simple model, connected with the cylindrical metallic waveguide, filled with homogeneous plasma, the radii of waveguide and plasma column being equal to a. The threshold wavenumber k~,restricting the transparency of the waveguide k > k~,increases due to plasma influence from k~= vto/a up to k~= v10/(aVF-) where e~is the first root of the Bessel function I~.On the other hand, the attenuation coefficient varies from Tm k (the unbounded plasma) to Tm k~ Im k

=

1e 2 2cV~~~ ~e ‘~

~e~1e

~1p

c~2 v~) Tm k~= 2w(w2+ a2 Re2 k

1~’

p

=

0,1,2,...

(5.13)

Thus, the coefficient Tm k~has different values in each mode. The shift of the value Tm k~in comparison with Im k depends directly upon the mode structure. The influence of collisions on the steepness of the dispersion characteristic of the waveguide proves to be especially considerable near the resonant frequency h1~and the threshold wavenumber k~[126]. An additional increase of wave attenuation in a narrow waveguide is produced by the growth of the frequency of electron collisions with the waveguide walls [127]. A somewhat analogous effect occurs in the ionospheric plasma: the plasma waves excited by the electron beam, injected from the spacecraft, are damped due to background electrons colliding with the spacecraft’s surface [98].Similar phenomena for ion-acoustic waves in finite geometry with absorbing boundaries are analyzed in [128]. The presence of boundaries leads in all these problems to the shift of plasma resonant frequencies and additional dissipation of the waves, different in each mode.

5.2.2. Resonant dissipation in plasma spheroid The influence of the boundaries of the plasma volume may lead to plasma depolarization inside this volume. Such effects are important for three-dimensionally bounded particles of a crystalline material, containing the conduction electrons, e.g. n-type semiconductors with free carriers. The simple case, related to the spheroidal geometry of such particles. may be characterized by the depolarization factor D, depending on the axial ratio of the spheroid. Here the effective value of the electric field E~0is connected with the field E, determined by refraction laws: E—_eN,.r E~0= I +~D

.

(5.14)

Here Ne is the density of conductivity electrons, the vector r describes the electron’s displacement, and x is the electric susceptibility 4irX. The electron per unit Langmuir volume,frequency connectedinwith the the specimen dielectric mayconstant he written F byasmeans [107] of the equation F = 1 + j~2

~ eO

u1~= 1 +~D

,

where the value

hle()

(5.15)

corresponds to an unbounded plasma (3.18). Thus, Langmuir frequency values for

242

A.B. Shvartshurg. Resonant Joule phenomena in a snagnetoplasma

two samples of the same material, only distinguished by their shapes, are shifted considerably (fig. 29a). Neither do the absorption lines contours of these samples coincide. The influence of depolarization may be taken into account by the substitution to-sto-u2~/w.

(5.16)

where the hl~ value (5.15) is connected with the above geometrical effect. Thus, the transverse conductivity u, of the collisional magnetoplasma may he written as (1

=

(T~~

2+ to~

______

[12

t’~[t’

+ wj~+ (to —

—11~to~j hl~w t)2]2 + 41,2( — u2~w~ I)2

(5.17)

1+(to

where a’ is the longitudinal conductivity. One may see, that the conductivity a’, related to the frequency to = ul~in a sample, coincides with the conductivity a’1, related to the constant field in an unbounded plasma (curve I, fig. 29b). The curve 3 exhibits a distinct magnetoplasma resonance. connected with the geometrical shift of the frequency lie up to the value ll~= 2to [107]. The picture of such shifts becomes more complicated due to rotation of the spheroid’s major axis relative to the wave vector of an incident field via the variations of the depolarization factor D. These phenomena attract attention to the possibility of smooth evolution of electron absorption lines of the given sample.

2i,i-.lQ~~

~5O

~2~2

0,2

0,5

0

1

3

5

a Fig. 29. The geometrical effect in plasma resonances 3, m’ = 10 tn2mn.type indium atitimonide specirnett. (a): A plot of the Langmutr frequency values 5(5.17) versus on the the axial ratio of frequency a. the Thespheroid curse I jz(N~ corresponds = i0~ cm to the frequency’ 55.a e =Q~. 13.6, the,~‘curse = 1). (b): 2 exhibits The dependence the cyclotron of the resonance normalized (Q~—~(((. conductivity and thea-1o~s~ curve 3 shosxv the magnetoplasma reson since, related to the value 11. = 2w (is — si (sec ref. 111(71 I.

6. Relativistic and quantum resonant phenomena in collisional absorption Relativistic effects in Joule dissipation of electromagnetic waves in gaseous plasmas seem to he important for the series of problems, arisen in astrophysics and nuclear fusion. Thus, these effects influence the waves absorption in hot plasma with finite conductivity. when the thermic energy of electrons, kT~.is not small in comparison with their rest energy aid2 1130]. The dissipation in very hot plasma (kT~> m,c2) is determined by the modified values of the Langmuir resonant frequency and electron—ion collision frequency 1131]

A.B. Shvartsburg, Resonant Joulephenomena in a magnetoplasma

hi 2 = 4~e2Nec2 _______

‘

p~” 15

3kTe

243

ce4Ne liiA~

6.1

2 (kTe)2

where A~is the Coulomb factor. Along with cosmical plasmas, relativistic phenomena occur often in laser-driven plasmas [5]. Moreover, the wave propagation in moving plasmas may be accompanied by some relativistic effects; here the eigenvalues of the dielectric permittivity tensor for a moving cold electron plasma are given by [132]: p=O,±l,

y(w—kv)[(to—kv)y—pwH—It’]

(6.2)

where the dashed quantities to~ and e’ are measured in the comoving system, and y = (1 — v2c2)”2 is the relativistic factor. The relativistic-mass effects give rise to a series of peculiar non-linear and bistability phenomena in gaseous plasmas. Quantum resonant mechanisms of Joule dissipation of light and microwaves in solid-body plasmas were not analyzed above. However, such mechanisms are important for a series of semiconductors effects, for IR waves propagation in the current plasma of semiconductor with a superlattice [133],or for cyclotron resonances in metals [134],for the determination of the Fermi surface. For simplicity, the peculiarities of resonant Joule absorption in relativistic and quantum plasmas are considered separately below, although the simultaneous display of both these groups of effects is essential in some natural and man-made hot dense plasmas [135].

6.1. Joule effects in laser driven plasmas The acceleration of electrons in a plasma up to relativistic quiver velocities, produced by a strong electromagnetic wave, may become a source of two mechanisms of rapid energy losses, effective in this range of velocities. One mechanism is connected with the non-thermal perturbations of electron—ion collision frequency t’e by the pumping wave E [5]: t’e

In AR

r

lnAc

L

~eti~(l+b2Y°4l

m 2 3kTe

i_3/2

1+~~~~__(\/1+b2_ 1)] 0c

;

b=

eE .

(6.3)

m 05toc

Here u~0is the “classical” value of the frequency of collisions, related to a weak field (b Coulomb factor, AR is its relativistic generalization: AR

=

A~{1+ m0c (Vi

2 — 1)];

+

b

Ac

=

0.37

~

—~

0), A~is the

(6.4)

m 0 is the electron rest mass, and Te is the temperature. Another mechanism of losses is produced by the radiation reaction, characterized by the frequency 2to2 2e 3. (6.5) 3m

0c’

.4.B. S/scart.slsurg. Resonant Jostle p/u’s,os,sena us a ssu agtss’top/a ssna

244

The losses, produced by this effect, prove to he comparable with the collisional dissipation, only in a very rarefied plasma [54]. However, the frequency CR (6.5) manifests itself in the series of dissipation phenomena similarly to the frequency of electron—ion collisions~~, thus some hysteresis phenomena near the electron cyclotron resonance. produced by a small, hut finite value of the frequency 0R’ will he

analyzed below.

6. 1. 1. The sell—ac/ion o/ a pumptng wade’ in relaitt’i.stic’ collisional pla.s’inas It seems to he useful to mark here the relativistic peculiarities of non-linear wave processes by means of some effects, also occurrilig in non-relativistic plasmas. One may consider for this aim such phellomena as the harmonics generation and the self-focusing of the pumping wave. An interesting example of the first effect is connected with the generation of a constant longitudinal electric field by means of a circularly polarized plane wave, propagated through a plasma near the Langmuir resonance w — h2~.The longitudinal electric field E~arises from the displacement of the electrons from the immobile ions and constitutes a return force. which exactly balances the forward Lorentz force due to the radiation reaction (6.5), the relativistic mass-effect being taken into account. This balance indicates that the field E~.may he calculated as f75] “R

c’E~ co~4~ kc2

‘

--~——-).

(6.6)

kc’w

/nwd

where “R is given by (6.5). To get a feeling of the magnitude of the longitudinal field, one may take 0) 102 rad see “R = 10” sec ‘, h2~,= l0~rad see and N~= 1025 cm 3; the density of the incident radiation power is P = 10’” W cm 2 In this case the field E~.is evaluated as E~.= lO7E 0 = 300 V cm l’his field exactly cancels the effect of the forward Lorentz force. so that the electrons have only

-.

~.

—.

‘.

transverse motion. The self-focusing of powerful laser beams in plasmas may happen due to the relativistic change of the 2. The non-linear electron lnass, the quiver energies of electrons being close to or above inc Inodification of the refractive index of the collisional plasma is given by

~=

(i

h1’~

2

~

+

~-.

(0+

IT

b1’~

2

(—~) to (O~

Ii)

where c~is the energy-dependent frequency of the collisions (6.3). The self-focusing may lead to the

formation of a channel behind the focus inside the plasma. The collisional absorption in this channel, the relativistic effects being taken into account, exceeds pronouncedly the absorption in the nonrelativistic limit [136]. Self-focusing of the laser beam attracts attention in connection with irradiation of targets and plasma acceleration, where the effect had to be avoided for the application of laser nuclear fusion. The relativistic-mass effect is responsible for periodic variations of electron mass due to electron velocity oscillations. produced by an intense laser field. This effect may give rise to a peculiar resonant mechanism of relativistic parametric instability [137]. A similar velocity dependence of electron gyrofrequency varies the conditions of electron cyclotron resonant absorption. The collisional phenomena restrict the efficiency of the transition radiation. produced by relativistic motion of electron beams in plasma waveguides. Thus, the spectral density of this radiation W(k),

A.B. Shvartsburg, Resonant Joule p/teno,nena in a magnetopla.sma

related to the case y W(k)=

(k

~‘

245

1, is described by the typical resonant distribution [89]:

— k,)2 + (va/to )k0)2

where k 00= 12e(cy)~tis the resonant wave number and the function F does not depend on i’,~.Such a resonant dissipation is important for the generation of powerful microwave pulses by means of plasma waveguides. which contain the relativistic beams. 6. 1.2. Relativistic bistability in a inagnetoplasma Relativistic generalization electron cyclotron bistability conditions shows, that even a very weak 2c2 of = f32 ~ 1) of the electron can result in a large hysteresis jump of its relativistic-mass effect (v if the wave intensity or frequency is varied. One may consider the feasibility steady-state kinetic energy, of this effect in the simplest case of a single electron immersed in a very strong constant magnetic field H 0 interacting with a strong electromagnetic field E. If the relativistic shift of cyclotron frequency becomes sufficiently larger than the frequency width of resonance, it changes dramatically the resonant conditions of excitation of equation an electron. 2 is governed by the [54] The normalized steady-state kinetic energy of the electron F = ~p

~(EIH 2 F[(Cr/to~)2+ (i + F)2], (6.7) 0) where i is the dimensionless resonant detuning ~ = (to — toH)to~ the frequency e. characterizes the radiation losses, caused mainly by interaction of the electron with other particles and a resonator. The hysteresis jumps occur if the frequency v~is in the range /3V3E2\t13 toH(~~) ‘~

16 H~1

to~to

>~~>

—

V3

Assuming the losses to be, for instance, ~r = iO4toH, one may calculate the incident power to be about 7 W cm2. The kinetic energy of an electron at the onset of hysteresis corresponds to 60 eV; this energy changes by a factor of 3 to 5 due to hysteresis jumps [54]. Scattering of a strong electromagnetic wave by an electron in a heterogeneous magnetic field H(z) gives rise to another relativistic bistability effect. This bistability may become apparent in the course of propagation of circularly polarized extraordinary wave along the static magnetic field H(z), the radiation reaction being taken into account. The decrease of the magnetic field leads to the hysteresis growth of the scattering cross section 0 near the point, determined from an equation toH(Z,) = eE(mc)t, and vice versa, the increase of the magnetic field is accompanied by the jump-like diminution of the cross section [138]. The former effect occurs at the point z = z 2. governed by the equation 37T eE \t/4 to~(z / 2)=toi,—-——-~ \ r,0 mto where r51 is the classical “electron radius” and to is the pumping wave frequency. Such a relativistic non-reciprocity determines the intensity of the radiation, emitted by the plasma volume, immersed in a strong heterogeneous magnetic field.

246

.4.8.55 t’art.s/uurg. Resonant Jsssth’ Js/tenossst’na in a sssag,ueto~sla.ssssa

6.2. Absorption resonances’ in quantum p/a.s’inas Quantum phenomena in resonant Joule absorption occur due to interaction of light and niicrowaves with degenerate solid-body plasmas. Such a plasmna model, based on the Fermi distribution for electrons, was utilized, at first, for the explanation of the anomalous skin effect in metals. accompanying the temperature dissipation of radio frequencies by metallic specimen 1139]. The important dissipation effects, predicted by this theory, were connected with spin resonances in the interaction of EM waves with quantum plasmas in metals and semiconductors. Spin resonance in liletals, connected with the electron energy change ~F = hto~1due to electron spin

flip in the degenerate plasma. immersed in a magnetic field, was considered as a mnechanism of resonant paramagnetic dissipation of EM waves near the frequency WH. The half-width of the related absorption line is determined by the characteristic time of spin flip rs. The magnetic permittivity x(to), stipulated by this paramagnetie effect. is described miear the frequency to w~ by means of the typical resonant curve -=

x(to)

0H =

2[ 1

+

(6.8)

2t Xi T~(W — 0)13) 2 1

where Xv is the static value of the magnetic permittivity. A somewhat similar mechanism may become responsible for resonance variations of the mobility of hot electrons in semiconductors, immersed in a constant lnagnetic field H 0 [6]. These variations are induced by a high-frequency magnetic field H. in a vicinity of the spin resonance frequency 11. The dissipation of EM wave energy leads to the growth of spin temperature, which may exceed the mean kinetic energy of the electrons. A part of the absorbed energy is transformed due to relaxation to the kinetic degrees of freedom of the electrons, where the hot electron mobility ,u is increasing. The relative variation of ~i is restricted by spin-lattice relaxation, characterized by the time scale Ts:

____________________

2]’

geH

to~~=~lc’’

j~ l+r~[to~+(w—hi) where g designates the Landé factor amid [2= to

0H. H1’. For the semiconductors of InSb type the relative variation of the mobility can reach 5—10% ]6]. Along with the spin resonance effects, common both to some metals and semiconductors, it seems to

be useful to note the absorption resonances of another physical nature, which occurred in semiconducter and metal plasmas separately. 6.2. 1. Stark resonance in a semiconductor wit/i super/a/lice The heating of an electron gas in semiconductors is considered often in connection with the prohlems

of self-focusing and self-modulation of the powerful EM waves in these media. The semiconductor with superlattice proves to he an interesting object from the viewpoint of such problems. The non-linear properties of such a system are amplified due to the strong constant electric field E5, directed along the period of superlattice, the value E0 being limited by the condition of the /th Stark resonance ]33] lto=eE,,d/h :

1= 1:2,3,....

(6.9)

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

247

where d is the period of superlattice and to is the EM wave frequency. The high-frequency current, induced in such a plasma by the EM wave, may be written as /

,

1 1 + 1 C1L(l~~hi)~+ ~2 (lto+hi)2+ e2j’

(6.10)

where the frequency t.’ characterizes the electron momentum relaxation. The condition of non-linear waves propagation has a resonant character Ilto—QH~e.

(6.11)

Here the typical values of the superlattice period d = 2 x I0_6 cm, the statical dielectric permittivity 10, and v 10~~ sec~tprovide the resonant frequency (~ 10i3 rad sec_i. The /th term in the sum (6.10) is proportional to the EM wave amplitude E, raised to the power 2/— 1. Thus, the Stark resonance of the /th order, connected with the decisive contribution of the /th term to the current (6.10), leads to the formation of non-linearity of order 2/— 1; the value / = 2 relates to the trivial “cubical” non-linearity. Such peculiar non-linearities, which occurred near the Stark resonances (6.11), may become important for self-compression of submillimeter radio waves [133]. 6.2.2. Cyclotron resonance in metals

The knowledge of cyclotron resonance frequency in a metal permits one to determine the effective electron mass in’, which is connected with electron velocity on the Fermi surface. Such a resonant absorption is a sensitive method for determination of the Fermi surface topology [141]. The cyclotron radius is determined by means of the radius p’ of cross section of the Fermi surface by the plane = constant, the magnetic field H being directed along z-axis: R~ = cp~(eH) Let us consider the simplest geometry of such a resonant effect, related to normal incidence of the wave on the surface of 5s far the metal slab, immersed in a magnetic field, parallel to this surface. The skin-layer thickness from the resonance is usually small 8~ 2Re. The peculiar effect near the cyclotron resonance on the conditions under consideration is connected with the anomalously deep penetration of the electromagnetic field into a metal. This anomaly is governed by the conductivity tensor. Thus, the o’~component of this tensor in the vicinity of the fundamental resonance to toH may be written as [134] ‘.

‘~

4ve2N

=

I

., ., 4m to~kRi~+(v/toH)

t [1

I sin(2kR + ii’/4) +—~ 2V~ (kR)312

‘

(6.12)

Here R = cp,,(eH)’ is the Fermi impulse of the carriers, k = ~ and i is the dimensionless detuning of the resonance, i = (toH — to)to~t. One may attract attention to the resonant denominator in (6.12). The oscillating term in (6.12) is responsible for the anomalous penetration of the resonant field, accompanied by the formation of field splashes inside the metal slab at a distance from the surface, approximately equal to an integer multiple of the cyclotron diameter D~.The reason for this effect is the generation of the cyclotron wave at all internal skin layers, the condition Dc ~ 6s being fulfilled. These waves may effectively penetrate through the samples of large thickness ci ~ D~[134]. An important geometrical effect in cyclotron absorption in metals occurs due to a decrease of the thickness of the metallic plate d in the range d
245

A .B. .55vart.shurg. Resonant Joule phes,osnena is, a magnetoplassna

Dc there appear the new cyclotron resonance frequencies 12. The cyclotron radius of the electrons. responsible for such a resonant absorption. is determined by the condition 2Re(p~)=d: here p. is the d<

projection of electron impulse on the magnetic field, which is supposed to be parallel to the plates surface. Thus, the resonant frequencies depend upon the thickness of the metallic plate. The impedance possesses a logarithmic singularity near the frequency 11. The measurement of the shape of the resonant absorption curve, containing the maxima. related to the frequencies toH, 12 and 112 (1 = 1, 2 ). is useful for the determination of electron effective mass [1421.

Concluding remarks

The modern interest in resonant interaction of electromagnetic waves with the magnetoplasma

IS

connected with the problems of governed evolution of wave properties and quick plasma heating. The

achievements in this field emphasize new problems in the powerful waves dynamics in the conducted media. In the capacity of the following investigations in this area of problems one may show: (1) The peculiarities of Joule processes, accompanied by wave propagation in turbulent collisional plasmas. These processes are governed by the simultaneous action of collisional absorption and collisionless losses, produced by the generation of new harmonics. (2) The common action of quantum and relativistic phenomena on the wave absorption in dense moving plasmas. (3) The search for the optimal regimes of self-organizing processes in conducted media. Such an applied aspect of the common synergetical approach may he of interest in some problems of

energy-related technology or laser chemistry.

Acknowledgements The author thanks Pi’ofessom’ H. I bra. Prolessor A. Rukchadie and Professor 1.. Steti lb discussicns of the problems. mentioned in this paper.

List of symbols used to

the frequency:

ii

the refractive index: the absorption coefficient:

K

l2~ Langmuir electroti frequency: the electron cyclotron frequency: the ion cyclotron frequency: i’~ the frequency of electron—ion collisiotis: v~, the frequency of electron—neutral collisions: the frequene~’of ion—rteutral collisions: N~ the electron detisitv: at the electron mass: ~I-l

(If-]

,‘~,,,

11.1

the ion mass:

lot valuable

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma in’

~ Tm

T, Tett

f E k h e•1

the effective mass of the carriers in the semiconductor; the mean energy fraction, transferred in elastic collisions between electrons and heavy particles; the temperature of molecules; the ion temperature; the unperturbed electron temperature; the dimensionless electron temperature; the strength of the electric field; Boltzmann’s constant; Planck’s constant;

the tensor of dielectric permittivity; the conductivity tensor; 0 the collisional cross section; p the pressure of the neutral gas; H,1 the external magnetic field; T1 the transversal temperature; E.L, F11 the diagonal components of the dielectric permittivity tensor in a magnetoplasma; H. the amplitude of alternative magnetic field; a the angle between the wave normal and the magnetic field; ares the angle of resonant cone; a5 the angle, determining the Storey cone; L the characteristic spatial scale of plasma density heterogeneity; r~ the Debye radius; L~ the dissipation length; E~ the characteristic plasma field; L5 the shear length; Pc the electron gyroradius; P the ion gyroradius; y the relativistic factor; in,, the electron rest mass; ~ the thickness of the skin-layer. a’,1

References [1] E.V. [21A.B. 131 V.P. 14] V.A. 151 D.A. [6] A.N.

249

Appleton and G. Builder, Proc. Phys. Soc. 45 (1933) 218. Mikchailovsky, The Theory of Plasma Instabilities (in Russian), (Nauka, Moscow, 1971). Silin, The Parametric Action of Intense Radiation on a Plasma (in Russian), (Nauka, Moscow, 1973). Bailey and D.F. Martyn, Phil. Mag. 18 (1934) 369. Jones, E.L. Kane, P. Lalousis, p. wiles and H. Hora, Phys. of Fluids 25, No. 12 (1982) 2295. Zaitsev and AK. Zvezdin, J.E.T.P. 55 (1968) 966.

[7] A.!. Alekseev,

J.E.T.P.

58

(1970)

2064.

[8] G.N. Zhizhin, E.A. Vinogradov, MA. Moskaleva and V.A. Yakovlev, Appi. Speetr. Rev. 18 (1982) 171. [9] A.B. Mikehailovsky, Beitr. Plasmaphys. 23 (1983) 183. 1101 R.S. Symons and HR. Jory, Adv. Electron, and Electron. Phys. 55 (1981) 2. [11] LI. Mandelstam, Complete works (in Russian), vol. 2 (Nauka, Moscow, 1949). [12] L.E. Gurevieh and VI. Vladimirov, J.E.T.P. 44 (1963) 164.

A.B.Shvartshurg. Resonant Joule phe,ton,ena in a snag,uetoplassna

25))

[13] V.D. Sankhla and J.L. Bansal, Beitr. Plasmaphys. 23(1983) 423. 14] A.B. Shvartshurg. Plasn,a Phys. 23, No. 4 (1981) 283. [IS] A. Ya, Vinnikov and L.E. Gurevich, J.E.T.P. 69 (1975) 23. [161 L. Stenflo and MY. Yu, Phvs. Rev. B 7(1973)1458. [17] L.A. Wainstein. Soviet Phys.-Uspechi IlS. No. 2 (t976) 339. [18] N.R. Pereira and L. Stenflo. Phys. of Fluids 20. Nss. I)) (t977) 1733. [19] MV. Kuzelev and A.A. Rukchadze. Radio Ph~s.22(1979)1223.

[20] M. Bans, Phys. Rev. Lett. 40, No. 12(1978) 793. 121] \‘.V. Vladimirov, A.F. Volkov and E.Z. Meilikchssv. ‘I’he Plasma of 5cmiconductors (in Russian). (Nauka. Moscow. 979). [22] D. ter Haar and V.N. Tsytovich, Phys. Reports 73. No. 3 (1981) 173. [23] Rd. Hwa, Phys. Rev. 110(1958)31)7. [24] V.L. Ginzburg. The Propagation of Electromagnetic Waves in a Plasma (in Russian), (Nauka. Moscow, 1967). 1251 L.D. Landau and E.M. Ltfshitj. The Statistical Phystus (is, Russian). (Nauka, Moscow, 1964). 1261 V.P. Sum, J.E.T.P. 47(1964) 2254. [27] M. Friedrich and KM. ‘I’orkar. J. Atm. Terr. Phys. 45, No, 4 (1983) 267 [28]

A.V.

Phelps and J.L. Pack. Phvs. Rev. Lett. 3(1959)

340.

129]

RD. Hake and A.V. Phelps, Phys. Rev. 158 (1967) 7)). [30] Z.N. Krotova and V.A. Ryzov, Radii, Phys. 24, No. 4 (t982) 363. [31[ [32] [33] [34] [35]

S. Altshuler. J. Geophys. Res. 68 (1963) 4707. A.B . Shvartsburg, Space Sci. Rcs’. 33 (1982) 361. MA. Morrison, Australia,, J. of Phys..38, Nis. 3A (1983) 239. G.M. der Munari, L. Gabba, F. Giuslano and 6. Mambriani, Beitr. Plasmaphys. 23, No. 6 (1983) 625. A.B. Shvartsburg, V.6. Koroheini6sv. M.D. Deminos and A.V. Razmadze, Planet. Space Sc). 27 (1979) 159.

1361

J. Wilhelm and R. Winkler. J. de Phys. 4)) (1979) 251. [37] DR. Ellis. J. Atm. Terr. Phys. 8 (1956) 43. [38] KG. Budden, Plasma Ph~s.25. Nss. 2 (1983) 113.

139]

R. Winkler, J. Wilhelm and V. Stanch, Beitr. Plasniaphys. 19, Ni) .5(1979) 315.

[40] S.D. Rautian. 6.1. Smirnos and AM. Shalagit,. J.E.’F.P. 62. No. 6 (1972) 2)197. [41] 1.6. Krusha and AN. Kondratenko. Beitr. Plasmaphvs. 23. No. 3 (1983) 229. [42] A.B. Shvartsburg, Plasma Phys. 21(1979)663. [43] D.E. Baldwin and B.G. Logan, Phys. Rev. Lett. 43(1979)1318. [44] R. Winkler and J. Wilhelm. Comput. Phys. Commun. 21) (1980) 113. [45] A.B. Shvartsburg, Phys. Lett. 57 A (1976) 435. [46] K. Nishikawa, J. Phys. Soc. Japan 24 (1968) 1152. [47] A.B. Shvartsburg, Phys. Scnipta 2)) (1979) 663. [48) Yu. V. Gulacv and V.!. Pustovoit, J.E.T.P. 47 (1964) 2251. [49] L. Stenflo, J. Geophys. Res. 76 (1971) 5349. [50] A.B. Kitzenko, J.E.T.P. 67 (1974) 1728. [SI] R.A. Smith, Semiconductors (Cambridge. 1959). [521W.G. Spitzer and DY. Fan, Phys. Rev. 11)6 (1957) 882.

[5316. Mourou, W. Knox and

S. Willimsoti. Laser Focus IS. Ni). 4 (1982) 97.

A.E. Kaplan. Phys. Rev. Lett. 48. No. .3 (1982) 138. [55] l.P. Lowenau. S. Schmidt-Rink and H. Hang. Phys. Rev. Lett. 49. No. 2)) (1982) 1511. [56] P.K. Kaw. Phys. Rev. Lett. 21(1968)539. [57] 6. Weyl, Phys. of Fluids 13 (1970) 1802. [54]

[58] A.B. Shvartsburg, Geomagn. and Acre,,. 14(1974)31. [59] CM. Celala. Phys. of Fluids 26. No. 10 (1983) 3029. [60] V.N. Lugovol, J.E.T.P. 41, No. 5(1961)1562.

[61) A.V. Gurevich and A.B. Shvartsburg, Geomagn. and Aeron. 8 (1968) 11)14. 1621 B.N. Gershman. J.E.T.P. 38 (1960) 912. 163] A.V. Rokchlenko, J.E.T.P. 69. No. 1 (1975) 175. 1641 G. Bekefi and J. Hirshfield, Phys. of Fluids 4 (1961) 173. [65] S. Tanaka and K. Mitani, J. Phys. Soc. Japan 19 (1964) 1376. 1661 A.B. Shvartsburg. Beitr. Plasmaphys. 23. No. 1(1983)1. 1671 S. Imazu, Phys. Rev. A23. No. 5 (1981) 2644. 1681 S.P. Kuo and BR. Cheo. Phys. of Fluids 24. No. 4 (1981) 784. [69] I.P. Shkarolsky, Canad. J. of Phys. 39 (1961) 1619. 170] K.N. Stepanov, Plasma Phys. (in Russian) 9 (1983) 45. [71] H.G. Booker, Phil. Trans. R. Soc. A280 (1975) 37.

A.B. Shvartsburg, Resonant Joule phenomena in a magnetoplasma

251

[72] G.D. Aburdjania, N. Braushkovich and D.G. Lominadze, Plasma Phys. (in Russian) 7 (1981) 1288. [73] A.!. Akhiezer, V.!. Lapshin and K.N. Stepanov, J.E.T.P. 70 (1976) 142. ]74[ V.B. Brodskii, Ukr. Phys. J. (in Russian) 23. No. 4 (1978) 597. [75] Hon-Ming Lai and Yau-Wa Chan, Phys. Rev, Lett. 35 (1975) 1226. [76] V.L. Ginzburg and A.A. Rukchadze, The Waves in a Magnetopiasma (in Russian). (Nauka. Moscow, 1973). 1771 F.H. Hibberd, Radio Sci. 69D (1965) 25. [781V.K. Ablekov, Yu. N. Babaev and A.M. Frolov, DokI. Akad. Nauk (in Russian) 254 (1980) 1241. [79] P.K. Shukla and 5G. Tagare. J. Geophys. Res. 84 (1979) 1317. [80] S.S. Shluger. Soviet Phys.-Uspechi 113, No.4(1974)736. [81] NA. Uckan, Plasma Phys. 25. No. 2 (1983) 129. [82] R.S. Steinolfson, Phys. of Fluids 27, No. 4 (1984) 781. [83] T. Hagfors. Europhys. News 7 (1976) 8. [84] V.S. Letokchov and V.G. Mmnogin, Phys. Reports 73(1981)1. [85] N.S. Erokchin, S.S. Moiseev and L.A. Nazarenko, Pisma J. Techn. Phys. (in Russian) 3, No. 12 (1977) 1623. [861B.S. Moiseev, Radio Phys. 20 (1977) 1623. [871L.S. Bogdankevich and A.A. Rukchadze, J.E.T.P. 51, No. 2 (1966) 628. [88] G.V. Givishvili, L.A. Lobachevskii, M.V. Fiskina and A.B. Shvarlsburg, Geomagn. and Aeron. 24 (1984) 9. [89] N.S. Erokchin, MV. Kuzelev, S.5. Moiseev, A.A. Rukehadze and A.B. Shvartsburg. Non-Equilibrium and Resonant Processes in Plasma Radio Physics (in Russian), (Nauka, Moscow, 1982). [90] T.D. Kieu and W.N.-C. Sy, Australian J. of Phys. 36, No. 4 (1983) 491. [91] Yu, M. Aliev and O.M. Gradov, J.E.T.P. 69 (1975) 1203. [92] E. Aby-Asali, BA. Alterkop and A.A. Rukchadze, Plasma Phys. 17 (1975) 189. [93] V.K. Kononenko and E.M. Kuzelev, Radio Techn. and Electron. (in Russian) 25. No. 8 (1980) 1717. [94] W. Shockley, Bell Syst. Techn. J. 30 (1951) 990. [95] D.A. Usanov and V.N. Konovin, Radio Techn. and Electron. (in Russian) 24, No. 8 (1977) 1677. [96] V.V. Popov, M.M. Rezvin and MA. Safonova, Radio Techn. and Electron. (in Russian) 28, No. 10 (1983) 2008. [97] A. l-Iarslein, F. Burstein, I.!. Brion and R.F. Wallis, Solid Stale Commun. 12 (1973) 11)83. [981V.A. Fedorov. J. Techn. Phys. (in Russian) 50 (1980) 1396. [99] L.M. Falk and O.D. Kocherga, Phys. Scripta 9 (1974) 237. [100] M.A. Vishkind, MI. Rabinovich and G.A. Fabricant, Radio Phys. 20 (1977) 218. [101] S.M. Grach, Radio Phys. 22 (1979) 521. [102] A.A. Skovoroda and B.N. Shvilkin, Radio Techn. and Electron. (in Russian) 23. No. 3 (1977) 637. [103] R.L.O. Storey, Phil, Trans. Roy. Soc. A246 (1953) 113. [1041E.M. Belenov, Radio Phys. 21, No, 6 (1978) 877. [105] B.B. Arora, D.R. Phalswal and N.L. Varma. Beitr. Plasmaphys. 23, No. 3 (1983) 261. [106] N.P. Galushko, N.S. Erokchin and S.S. Moiseev, J.E.T.P. 69 (1975) 142. [1071G. Drosselhaus, Phys. Rev, 100, No. 2 (1955) 618. [108] R.K. Fisher and R.W. Gould, Phys. of Fluids 14, No. 3 (1971) 857. [109] AM. Lyatskaya and W.B. Lyatsky, Geomagn. Aeron. 16 (1976) 331. [110] M. Kuperus. Space Sci. Rev, 34. No, 1(1983)47. [111]T. Tachi, R. Steinolfson and G. Van-Hoven, Phys. of Fluids 26, No. 10 (1983) 2976. [112]M. Dobrovolny, P. VelIni and A. Mangeney, J. Plasma Phys. 29 (1983) 393. [113] G.M. Teplin and Yu. M. Stenin, Radio Phys. 25, No, 11(1982)1244. [1141AS. Bugaev and Yu, A. Filimonov, Radio Techn. and Electron. (in Russian) 24. No. 1(1984)141. [115]J.A. Jonson, Astrophys. J. 254 (1982) 318. [116] A.B. Shvartsburg, Space Sci. Rev, 25 (1980) 33!. [1171L.V. Budilin, GD. Getmantsev, PA. Kapustin, D.S. Kotik, NA. Mityakov, A.A. Petrovsky and V.0. Rapoport. Radio Phys. 2)1 (1977) 83. [118] N.S. Bellyustin, Radio Phys. 20, No. 11(1977)1605. [119)V.M. Latygin and AM. Sagalakov, Plasma Phys. (in Russian) 9 (1983) 512. [120]J.W. Connor, W.M. Tang and J.B. Taylor, Phys. of Fluids 26, No, 1(1983)158. [121] A. Ya. Slepyan and G. Ya. Slepyan, Radio Techn. and Electron. (in Russian) 26, No. 4 (1981) 689. [122] A.V. Tolmacheva, Radio Phys. 23, No, 3 (1980) 278. [1231AG. Boev and A.V. Prokopov, Radio Phys. 21, No. 1 (1978) 48. [124]A.B. Steinshleiger, Soviet Phys.-Uspechi 192. No. 1(1984)131. [125] A.!. Gudzenko, Radio Techn, and Electron. (in Russian) 20, No. 2 (1975) 281. [126] A.!. Rogashkova and M.B. Zeitlin, J. Techn. Phys. (in Russian) 48, No, 2 (1978) 420. [127] A.N. Kondratenko, J. Teehn, Phys. (in Russian) 42, No. 4 (1972) 743. [128] R.V. Jensen and lB. Bernstein, Phys. of Fluids 26, No. 4 (1983) 953. [129) K. Appert and J. Vaclavik, Plasma Phys. 25, No. 5 (1983) 551.

252

A.B. Shuartshurg, Resonant Joule phenomena in a magnetoplasma

[130] A. Granik, J. Plasma Phys. 27, Ns. 2 (1982) 343. [131] V.P. SilO,. J.E.T.P. 38 (1960) 577.

35a (198))) 28)). [1321 H. Hehenstreit, Z. Natur6srsch .. [133] A.P. Teter~,v.Radio Phys. 25. No. 11(1982)1231. [134] AS. Rozhavsky and MA. Luryc, J.E.T.P. 670974)1168. )135] H. Hors. Nuovo Cimentis 64 (1981) I. [136] D.G. [,ominadze, S.S. Moiseev and E.G. Tsikarishvili, J.E.T.P. l.ett. 38, Nss. I)) (1983) 473. [137] N.L. Tsintsadze, J.E.T.P. 59, No. I)) (1972) 1251. [138] Ya. B. Zeldovich and A.F. !llarionov, J.E.T.P. 61, Nis. 3 (1971) 88)).

[139] G.E. Reuter and E.H. Sondheimer, Proc. Roy Soc. l95A (1948) 336 [140] F.J. Dyson, Phys. Rev. 98 (1953) 349. [14!] MI. Kaganov and l.M. Lifshitz. Soviet Phys..Uspechi 129 (1979) 48~. [142] MA. Luryc and V.G. Peschanskv. J.E.T.P. 66. No. (1974) 24)1. [143] H. Haken. Synergetics (Springer.Verlag, Berlin, 1978).