Scattering and Transformation of Waves on Heavy Particles in Magnetized Plasma

doi:10.4236/jmp.2011.23025

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Journal of Modern Physics, 2011, 2, 162-173

doi:10.4236/jmp.2011.23025 Published Online March 2011 (http://www.SciRP.org/journal/jmp)

Scattering and Transformation of Waves on Heavy

Particles in Magnetized Plasma

Hrachya B. Nersisyan, Hrant H. Matevosyan

Institute of Radiophysics and Electronics, Ashtarak, Armenia

E-mail: hrachya@irphe.am

Received November 2, 2010; revised January 3, 2011; accepted January 5, 2011

Abstract

The scattering and transformation of the waves propagating in magnetized plasma on a heavy stationary

charged particle located at a plane plasma-vacuum boundary is considered. The scattering (transformation)

occurs due to the nonlinear coupling of the incident wave with the polarization (shielding) cloud surrounding

the particle. It is shown that the problem is reduced to the determination of the nonlinear (three index) di-

electric tensor of magnetized plasma. The angular distribution and the cross section for scattering (transfor-

mation) of high-frequency ordinary and extraordinary waves and low-frequency upper-hybrid, low-hybrid,

and magnetosonic waves are investigated within a cold plasma (hydrodynamic) model.

Keywords: Scattering, Transformation, Plasma Modes, Magnetized Plasmas

1. Introduction

It is well known that in a medium with a certain fluctua-

tion level, the propagation of electromagnetic waves can

lead to radiation of waves with new frequencies and

wave numbers, i.e. scattered waves and also a new type

of wave: transformed waves. The investigations of elec-

tromagnetic waves scattering and transformation proc-

esses are very important for studying such problems as

plasma diagnostics, wave transformation mechanisms in

plasma, definition of dispersion properties of plasma

wave processes etc. In addition the study of the electro-

magnetic wave scattering spectra (both in laser and mi-

crowave wave ranges) is an efficient method of plasma

diagnostics in laboratory fusion research devices as well

as in the near and outer space.

Electromagnetic wave scattering is caused by thermal

fluctuations of plasma density and other plasma parame-

ters such as current density, electric and magnetic fields,

etc. Spectra of scattered waves provide information on

the density and temperature distributions in the plasma.

A peculiarity of electromagnetic wave scattering in

plasmas is coherent scattering by collective plasma exci-

tations-combination scattering that occurs along with

Thompson incoherent scattering by individual plasma

particles. Wave scattering by collective plasma fluctua-

tions, in particular, makes it possible to find relative

concentrations of charged particles and temperatures of

individual plasma components. The phenomenon of

electromagnetic wave combination scattering by collec-

tive plasma excitations has been considered for the first

time by Akhiezer et al. (see, e.g., [1]). Subsequently a

theory of electromagnetic wave scattering in plasmas has

been developed [1-7]. The detailed theory of scattering

and transformation of waves in magnetized plasma has

been worked out in [8] (see also [9,10] where useful re-

views of the electromagnetic wave scattering problem

have been presented). The theory has been further de-

veloped and extended in the papers [11-15] (see also

references therein), in particular, in the case of strongly

magnetized turbulent plasma [14]. The scattering and

transformation of high-frequency waves in dusty plasmas

due to electron density inhomogeneities have been inves-

tigated in [15]. The scattering and transformation cross

sections for two cases (the electrons in the shielding

clouds around the charged dust particles and induced

electron density fluctuations discreteness) have been

calculated and it has been shown that both can be en-

hanced with respect to scattering from thermal fluctua-

tions.

Another important mechanism for the wave scattering

and transformation could be provided by the nonlinear

interaction of the incident wave with the non-thermal

density fluctuations (wakefield excitations) generated by

the charged test particles moving in plasma. In particular,

if such particle is at rest and heavy and does not oscillate

H. B. NERSISYAN ET AL. 163



in the electromagnetic field of the incident wave the

scattering occurs due to the nonlinear oscillation of the

polarization (shielding) cloud surrounding the particle. It

is clear that the cross section of this process essentially

differs from the standard Thomson cross section when

the wavelength of the incident wave is comparable or

smaller the size of the polarization cloud which is typi-

cally given by the Debye screening length. Moreover, in

a nonlinear regime the polarization cloud surrounding the

particle may be affected by an external strong magnetic

field which introduces a strong anisotropy in the screen-

ing properties of the plasma and as a result in angular

distribution of the scattered (transformed) waves.

In this paper we investigate this process in detail as-

suming that the heavy test particle is located at the plane

boundary of plasma-vacuum interface. The plasma is

assumed to be strongly magnetized so that the cyclotron

frequency of the electrons is comparable or even larger

than the plasma frequency. The case in which the inci-

dent wave propagates across the external magnetic field

is considered. Within cold plasma model general expres-

sions are obtained for the angular distribution and the

cross section for the scattered and transformed waves. The

explicit calculations are done for specific high-frequency

ordinary and extraordinary waves as well as for

low-frequency upper-hybrid, lower-hybrid, and magne-

tosonic waves.

2. Theoretical Model

The main problem in calculating the quantitative charac-

teristics of electromagnetic wave scattering and trans-

formation in plasmas is to find the current produced by

the nonlinear interaction of the incident wave with the

fluctuations of plasma parameters caused by the test par-

ticle. This current determines the scattered and trans-

formed wave fields.

We consider an incident wave

(where 0

(0)

0c.c.

iit









is the complex amplitude) which propagates

in magnetized homogeneous plasma and a heavy particle

with charge

e ( is the charge of an electron) at

rest. The amplitude of the magnetic field of the incident

wave is determined by the Maxwell’s equation and has

the form

e













. In the linear approxima-

tion the incident wave and the electric field

ck









produced by the test particle are independent, and the

Fourier transformed total electric field in plasma is



 



(1)

00 0











 

 



Ek kk

kk Ek

(1)

The amplitude and the frequency of the incident wave,

for given values of the wave vector, are determined from

the equations and



000

, 0

ij j



k





det, 0

Mwk, respectively, where

 

22 2

ijij ij

Mkc k



 kk



(2)

and ij



are the Maxwellian and unit tensors, respec-

tively, and









is the dielectric tensor of the mag-

netized plasma.

The electric field of the stationary heavy particle is

expressed by the equations [16,17]

  

iZe







Ek k



. (3)

Here









ijij

kk k



kk



is the longitudinal

dielectric function of the plasma. We assume that the

particle is heavy and does not oscillate in the field of the

incident wave. Thus the scattering originates from oscil-

lations of the polarization cloud surrounding the particle.

To find the electromagnetic field of the scattered

(transformed) wave, we consider the second order ap-

proximation in Maxwell’s equations. As a result, for the

electric field





(2) ,



k in the second order approxi-

mation we obtain the equation

 

(2)

,, ,

ij ji

ME J







kk k

(4)

where

 



(1) (1)

i ijk

Jdd













  





kkkk







(5)

is the nonlinear current, associated with the nonlinear,

three-index, dielectric tensor



,; ,

ijk







kk of the

magnetized plasma,







kkk, and



 



.

The scattered waves originate from nonlinear coupling

of the incident wave with the electric field of the test

particle. The scattering current corresponding to such

coupling is easily obtained from (1) and (5) if in the ex-

pression obtained for the current we neglect the terms

proportional to 00



and , which

determine the second harmonic generation and the sec-

ond order electric field of the test particle, respectively.

The total scattering current is thus determined by the

equation

 



 



 



()

00 0

00 0000

,,;,

,;,

i ijl

ijlj l







 







 







kkk

kk kk





(6)

where the tensor ijl characterizes the nonlinear prop-

erties of the medium [18]:









,; ,,; ,,;,

ijlijl ilj











kkkkkk





. (7)

The electric field







k of the scattered wave is

H. B. NERSISYAN ET AL.

164



obtained from Maxwell’s Equation (4), in which

is replaced by the scattering current

. Thus



,Jk





sJk

 

()

iijj

ETJ













k. (8)

Here







is the tensor inverse to the Maxwel-

lian tensor, .



,,TM



kk

ij lj

Since the intensity Ws of the scattered radiation is

equal to (with the minus sign) the work performed by the

source of the scattered radiation per unit time, neglecting

damping of the scattered wave, we obtain





22 3

22 22

() ()

(2 )() ,0

Im,, ,

ji ij

iZ ed

















kkkk

W

(9)

where 00 00





,;,e



islss l, AS kkkkk 00





being the complex unit vector along the direction of the

polarization of the incident wave. As one would expect,

it is seen from (9) that the scattering (transformation) on

a stationary charge occurs with no frequency change



).

The total scattering cross section



is the ratio of the

intensity

W of the scattered radiation to the energy

flux





2



S in the incident wave, where





gijij

 











(H) . (10)

Here



(H)





k is the Hermitian part of the dielec-

tric tensor and 00g



 vk is the group velocity of

the wave.

Assuming the group velocities of the incident and

scattered waves to be considerably larger than the elec-

tron thermal velocity, we use cold plasma approximation.

Within this model we write the expression for the linear

dielectric tensor in the form [17, 18]

 

12 3ijijijll

bbie b

 

 





, (11)

where b is the unit vector in the direction of the external

magnetic field, is a fully antisymmetric unit tensor,

and

ijl

 

 

pa pa

pa a











 











(12)



 

(), (),

() .

paca pa

ca ca

ca pa



 



 

















(13)

The summation in (12) and (13) is carried out over all

plasma species ,



and 0/

ca aa

eB mc



 are the

plasma and cyclotron frequencies of particles of the kind

, and



is the effective frequency of electron-ion

collisions.

For the vector 0

we obtain from (10) and (11)

 







0010

Re| |

[].



 



















ee b

e

(14)

An expression for the tensor ipl in the cold plasma

approximation and in the absence of particle collisions

was obtained in [18]. With allowance for the collisions,

the expression for takes the form

ipl



  

  

 

() ()()()

() ()() ()

() ()()()

,;,

ipl aa

aaa a

ilpilsps

aa aa

iplijpl j

aaa a

ijlpj ipsls



 



  

 





 

 





 

  



 

 



(15)

where













()a

ijaijaijaijl l

hbbile



 b.

It should be noted that at 0



  (





) the tensor

ipl has a singularity, due to the adopted cold plasma

model. Taking into account the thermal motion of the

plasma particles the frequency change, in the scattering

process, is on the order of



v c~Te

kv Te



, where

Te is the electron thermal velocity. We introduce the

truncation parameter T, which is related to the fre-

quency change





 by the relation 1T



 . Obvi-

ously at







we have 0~Tc



Let us consider the case when the test particle is at rest

at a plane plasma-vacuum boundary. We consider the

radiation escaping from the plasma into the vacuum due

to the scattering (or transformation) of the magnetized

plasma waves on this stationary particle. A more rigor-

ous statement of the problem (boundary-value problem)

requires that the generated surface waves are also taken

into account. However, it should be emphasized that

their intensity decays exponentially with distance from

the boundary. Here we are interested only in the scat-

tered bulk waves and the influence of the plasma bound-

ary is neglected.

Consider Equation (9) for the intensity of the scattered

radiation in the vacuum, where . For

the tensor in this limit we obtain



ijij i



 k







222

ji ij

iji j

innkc







 

(16)

H. B. NERSISYAN ET AL. 165

where knk is the unit vector in the direction of the

wave vector of the scattered waves. It thus follows

from (16) that the scattering process occurs with no fre-

quency change, while the wavelength differs from that of

the incident waves (

000

kk kc









) owing to the

difference between the phase velocities of the plasma

waves and the speed of light in a vacuum.

Using Equations (9), (15), and (16) as well as the rela-

tion for the static dielectric func-

tion, where

 

,0 1D





k



is the Debye screening length of the

plasma, for the total intensity of the scattered waves after

lengthy but straightforward calculations we finally obtain



WI d

nn , (17)

where sinddd





 , 0

cos



nn,



is the scat-

tering angle, 000

is the unit vector in the direc-

tion of the incident wave vector 0, and

knk k





,Inn is

the angular distribution of the scattered radiation,

  



00 0

222

1/ 2

IZ T









 



nn nn ,



(18)









() ()

;

,, ,





 





nnnn nn







(19)

 



()

00 0

aaa









nnn nnbn b,

(20)









 



 





 

 









** *

,,1| |

aba b

ig l







 

























 



 











nn ne

eb neb

eb nb

nb ne ebeb

nb neebeb

ebeneneb

ebene neb

nb e





 





()











 

 

bneb

eb neb





(21)

In Equations (18)-(21) we have introduced the fol-

lowing notations: 0



 is the wavelength of the

incident wave,



00 0

2π

cr



, 2

remc2

is the

electron classical radius, and





Gig ,



Hih, aaa

mem e



.

Below, in Sections 3 and 4, in the case of scattering

and transformations of the high-frequency plasma waves

we consider the interaction of the incident wave only

with the electron component of the plasma omitting the

index a in expressions (19)-(21), assuming that the quan-

tities



















, , and G

are related to

the electrons. However, the ion component of the plasma

must be taken into account in the case of low-frequency

incident waves when 0~,

ci pi





(Sections 5 and 6).

Furthermore, we consider the general expressions

(18)-(21) in some special cases, assuming that the inci-

dent wave propagates perpendicular to the magnetic field

direction. We also assume that the incident wave

propagates perpendicular to the plasma-vacuum interface

(i.e., we choose the magnetic field to be parallel to the

plasma boundary).

3. Scattering of Ordinary Waves

We first consider the scattering of ordinary waves from a

stationary charged particle. It is well known [17] that an

ordinary wave is a linearly polarized transverse (00



where 00





E) electromagnetic wave propagating

across a magnetic field. The polarization vector of this

wave is parallel to the external magnetic field, 0||

while its frequency is related to the wave vector by the

usual dispersion equation for transverse electromagnetic

waves propagating in a plasma, . The

amplitude of the magnetic field of the incident wave is

222

pkc



2

determined by the relation







000



Bk

We introduce a spherical coordinate system with the

polar z axis in the direction of the vector and the y

axis in the direction of the vectors 0

and (Figure

1). The angle



is determined from the direction of the

k

0









)2(



)1(



z

y

x

Figure 1. Diagram illustrating the scattering of an ordinary

wave from a stationary charged particle located at the

plane of a plasma-vacuum interface. The wave is traveling

perpendicular to the plasma surface toward its boundary.

The magnetic field is parallel to the interface and is di-

rected along the polarization vector of the incident wave.

H. B. NERSISYAN ET AL.

166

axis. Then, taking into account the dispersion law for

the incident ordinary wave and neglecting the ion com-

ponent of the plasma, from (18) we obtain



222

,,,

1/ 2cos



 

 



  (22)

where 222





 ,





,



,12cossinsGH

 









 , (23)



,,1sin sin













 

, (24)

and the angle



varies in the range 02





 . The

collision frequency



can be omitted from (24), since





 for any.

The wave vector of the scattered wave is easily deter-

mined by equating to zero the argument of the delta

function in (16),

kck



. This relation indicates

that a long wavelength ordinary wave is transformed into

short wavelength electromagnetic radiation in a vacuum.

Let us briefly consider the results which follow from

(22) in the absence of a magnetic field

( ,

0Hpe





). In this case and for





the

radiation is concentrated mainly in the direction perpen-

dicular to the zy plane, i.e., the scattered wave escapes

into the vacuum almost parallel to the vacuum-plasma

interface. For scattering of a long waves (



) the

radiation is uniformly distributed (i.e., it does not depend

on the angle



) in the xz plane.

In the limit of wavelengths that are large compared to



the intensity of the scattered radiation does not de-

pend on wavelength, in accordance with (22) and (23),

and has the form



4222 2

,1si

1sinsin ,

IIGZT

nsin



 

















(25)

where 1









. From (25) it is seen that in the

absence of a magnetic field () the scattering oc-

curs just as from a point charge Ze (Thomson scattering)

having an effective mass

0H



eff

mZpe [16].

Thus, the term proportional to

m TG



in (25) determines

the scattering of long waves due to plasma anisotropy.

It will be shown below that a sufficiently strong mag-

netic field (ce





 or

G ) can significantly affect

the scattering pattern observed in the absence of an ex-

ternal magnetic field. The angular distribution of the in-

tensity of scattered radiation in this case has a maximum,

the position of which is determined by the relation



222

sin sin3









 nb . (26)

From (26) it is seen that the maximum of the intensity

exists only for sufficiently strong magnetic fields,



. In the opposite case with 2



 the

intensity decreases monotonically, while for sufficiently

small angle



(



222

sin2 /3



 ) but for



 it increases monotonically with



. From

(25) and (26) it follows that the maximum of the inten-

sity decreases slowly (by a factor of about 2.2) as the

magnetic field increases from zero to the values of

ce pe



 . The function (, )I



is shown in Figure 2

for the scattering of long waves (4



) as a function



and



. It is seen that the scattered radiation is

concentrated mainly near a contour on





plane de-

fined by (26). We also note that these equations define

two cones



with apices at the point

2constnb

0xyz



 (see Figure 1).

With a decrease of the incident wave wavelength the

intensity of the scattered radiation increases rapidly, ap-

proximately as 4





(see the denominator of (22)), up to



. Here the intensity has a maximum, the position

of which is determined by Equation (26), in the wave-

length range







. It should be noted that the

features of the angular distribution of the scattered waves

discussed above for



 are retained in the case of

small wavelengths.

In the limit of very short waves (



) the angular

distribution









is changed significantly. Under the

condition ce





 , sin/ ce





, for example, (22)

takes the form



022

sinsin

,4sin2

1sin sin.

IIZT



 

























(27)

The intensity maximum is shifted toward smaller



in this case, while the position of that maximum is de-

termined by the equation

Figure 2. Angular distribution (normalized to 10-7J0, where

J0 = I0Z2(peT)2) of the scattered ordinary wave in a long

wavelength range (



= 4



p). The calculations were done for



= 102,





pe = 0.1, and



ce/



pe = 3.

H. B. NERSISYAN ET AL. 167





1/2

max 2/12sin

 











. (28)

As follows from (27) and (28), the maximum intensity

increases rapidly with increasing



. In Figure 3 we

demonstrate the angular distribution (, )I



for the

scattering of the short waves (0.1







). Thus,

the scattering in this case occurs mainly in the direction

of propagation of the ordinary wave.

The total cross section for scattering from a stationary

particle is obtained from (22) after integration over the

angles



and



, where for ordinary waves from (12)-(14)

we obtain



1/2



 and 0. Since

the general expression for the cross section is cumber-

some, below we consider only some particular cases. In

the case of scattering of very short waves (

Svc



 ) the

cross section is almost constant and is given by





222

11 1

Tpe

ZTaGbGHcH

 





 (29)

where





 is the Thomson cross section,

11a13964b, 145 256c.

In the intermediate regime with









the

cross section decreases as











, where





2222

122

20 pe

ZTaGbGHcH









 (30)

with a2 = 1, b2 = 17/44, c2 = 6/77.

In the case of long wavelengths (



) the cross

section increases linearly with the wavelength of the in-

cident wave,



2Tp





, where





2222

233

2pe

ZTaGbGHcH











(31)

with a3 = 1, b3 = 2/5, c3 = 3/35. Such a behavior of the

cross section is explained by the fact that the incident

Figure 3. Angular distribution (normalized to J0) of the

scattered ordinary wave in a short wavelength range ( =

10-3 p). The values of the other parameters are the same as

in Figure 2.

and scattered waves have different group velocities, and

for



 the energy flux in the incident wave is

~1S



. Therefore, in the long wavelength range the

total cross section does not coincide with the Thomson

cross section for scattering from a point particle with a

mass meff, as it occurs in the absence of a plasma bound-

ary and an external magnetic field [18].

Using (30) and (31), the scattering cross section for

ordinary waves at



 can be represented in the

approximate form



12Tp p



 













 . (32)

From (32) it follows that at



1/5

min12







the

cross section has a minimum, the value of which is given





1/5

min1 2

1.25 4



.

The dependence of the scattering cross section on the

magnetic field can be traced from (29)-(32). The cross

section decreases monotonically with increasing mag-

netic field. This behavior is especially pronounced for



 and is one order of magnitude over the range of

variation of the magnetic field from zero to

ce pe







. The decrease of the cross section is due

to the reduction of the transverse cyclotron motion of

plasma electrons with the magnetic field. In the limit of

very strong magnetic fields (1



 ) the plasma behaves

like a one-dimensional fluid, the motion of which is con-

fined to oscillations along magnetic field lines.

4. Transformation of Extraordinary Waves

In this section we consider an extraordinary incident

wave with a complex amplitude



(1) (2)

000

(where

12 i



(1)

and (2)

are the real amplitudes), propa-

gating across the magnetic field. It is well known [17]

that in general an extraordinary wave is elliptically po-

larized in the

y plane (Figure 1), i.e. , 0





. With

no loss of generality, we choose the vectors and

(2)

such that and 0x

(1)

0x 0E(1)(1)

0y 0z

EEE

( 2)

E)0



 (Figure 1). For this choice, the amplitude of

the magnetic field of the incident wave is determined by

the relation







(1)







and is directed

along the external magnetic field.







The relation between the components and

is given by the equation [17]

(1)

E(2)





 

(2)

(1)







, (33)

while the relation between the frequency and the wave

vector is given by the dispersion equation for the ex-

traordinary waves [17],





 

2RL











 (34)

H. B. NERSISYAN ET AL.

168

with



13R





 and

 





13L





.

The energy flux of the incident wave and the intensity

of the scattered extraetermined by

the expressions

ordinary waves are d



2c





, (14), and (18)-(21),

respectively, where

 



0010

00 0



 

 















S





, (35)

 

22 2

,,1sin cossinqQ















 





, (36)



22 222

22 24222

pe pece

Hpece pe

 



  





.

(37)

We investigate the expressions obtained for









for high-frequency (electron) extraordinary waves. The

ion component of the plasma can be again neglected in

this case. The two solutions of the dispersion equation

(34) then have the form [17]

 

1/2

() 1/22

00101020

2kfkfkfk







4







(38)

where



222

1012 0,



c



22222

20 120,

 

c

and 22

ce pe







is the upper hybrid frequency. The

quantities



and 2



are the cutoff frequencies which

are the solutions of the equations







 

 

and , respectively, and under the

condition







ce ci







0







(which is fully justified for both

laboratory and astrophysical conditions) have the form

[17]

22 2

24,

cecepepe 22





 



. (39)

In this section we consider only the scattering of the

high-frequency mode ()



. We briefly recall (see also

(38)) that for this mode





()







k increases monotoni-

cally from 2

()





 at to

at 0. Since (or

) in this frequency range the high-frequency

wave has, in general, right-hand elliptic polarization in

the

00k









()

00 0





(2)

0z 0Eck P

 P

plane (in the positive direction). In the case

of long waves (0), , the wave is almost

circularly polarized, whereas in the case of short waves

k), 0, this mode consists of a linearly

polarized, transverse electromagnetic wave. In the latter

case the only difference between an extraordinary and an

ordinary wave is that the polarization vector of an ex-

traordinary wave is perpendicular to the external mag-

netic field.





0P









The wave vector of the scattered wave is determined

by the expression 0





. From (38) we conclude that



 in the entire wavelength range of the incident

wave. Thus, as in the case of an ordinary wave, the

transformation of extraordinary waves into electromag-

netic radiation in a vacuum is accompanied by a decrease

of the wavelength.

Let us consider the angular distribution of the scat-

tered waves in the limits of small and large



. In the

limit of very short waves (



) the angular distribu-

tion has a maximum at the values of the small angle



determined by (28), in which sin



is replaced by

cos



. All the properties obtained for ordinary waves in

the range of



under consideration are retained in this

case. In this limit the cross section is almost constant and

is determined by (29) in which the numerical coefficients

are 11a



, 1

b61 64



, 1381 1280c.

In the intermediate wavelength range with







, where



is a number on the order

of unity, the intensity of the scattered radiation decreases

rapidly with increasing



as 4



. The angular distribu-

tion of the scattered waves is also changed. The intensity

maximum is shifted toward larger angles



, and for





, sin ce





, and 2

cos2 3



 the position

of that maximum is determined by the expression

sincos23







. However,







increases mo-

notonically with further increasing



cos2 3



)

and reaches the maximum value at 2





 (or







) (Figure 4). In the same wavelength range the

scattering cross section has the form







1Tp









, where



1 is determined from (30)

with coefficients 21a



, 239 44b, and 218 77c



In accordance with (22), (23), (36), and (37) in the

limit of the wavelengths larger than c/



ce, the intensity of

the scattered radiation is



,1s

sinsin,

IIZT



insin















(40)



02 21 4



. (41)

It is seen from Equation (40) that the intensity of the

scattered radiation increases monotonically with





sin sin





nb and scattering occurs mainly in the

direction of the external magnetic field.

In the limit of ce





, from (35) we obtain













, where

 



145 2

143













. (42)

Then the cross section reads





2Tp





,

where 2



is determined from (31) with coefficients









301

afF



, 33

45ba



, and 33

935ca.

H. B. NERSISYAN ET AL. 169

Figure 4. Angular distribution (normalized to J0) of a scat-

tered extraordinary wave with a frequency ()





in the

intermediate wavelength range ( = 5D). The values of the

other parameters are the same as in Figure 2.

From these expressions and (40)-(42) it follows that at





 and 1





the angular distribution and the

scattering cross section of the extraordinary waves are

proportional to



and 11



, respectively, and

increase considerably with the magnetic field. At





 an extraordinary wave has right-hand circu-

lar polarization in the xz plane and at 1



 its fre-

quency is 2ce



. Thus in this case a specific cyclo-

tron resonance may occur which, however, differs from

the usual one so that the incident wave is polarized in the

plane of incidence and propagates across the external

magnetic field [17].

5. Transformation of Waves of Intermedia te

Frequency

In this section we consider the transformation of the ex-

traordinary waves with a frequency spectrum





()







determined by (38). The frequency increases

monotonically (see (38)) from



()





()





 at 0 to 0k

()







at 0. Since the frequency of this mode

is high compared to the characteristic ionic frequencies

(see, e.g., [17]), we neglect here the contribution of plas-

ma ions both in the scattering current and in the wave

dispersion. From (33) it follows that in this frequency

range (or ), i.e., this mode, in gen-

eral, has left-hand elliptical polarization in the xz plane

(Figure 1) and cannot resonate with plasma electrons. In

the case of long waves (0), , and the

wave is almost circularly polarized, whereas in the case of

short waves (0), , this mode consists

of a longitudinal wave (upper-hybrid oscillations). In the

latter case the upper-hybrid waves are transformed into

electromagnetic radiation in a vacuum.

k

0





(2)

0z 0E

k









11P





From (38) for ()



 one concludes that 00







 and 00



 at p



. Thus, at p





and p



 the scattering of the mode ()



 is accom-

panied by an increase or a decrease of the wavelength,

respectively.

General expressions for the angular distribution of the

scattered extraordinary waves have been obtained in Sec-

tions 3 and 4 (Equations (22), (36), and (37)). In the

range of very short wavelengths (



), from these

expressions we obtain the angular distribution of the

transformation of upper-hybrid oscillations,









2222

sinsin1 .

IIZTG















 





(43)

The transformation cross section in this wavelength

range it is obtained from (35) and (43). After integration

of Equation (43) with respect to the angles, one obtains







 

, where



2222

ZTG

 















. (44)

From this expression for the cross section it is seen that,

in contrast to the scattering (transformation) of

high-frequency waves, in which the cross section for





is constant, in the case of intermediate up-

per-hybrid waves the cross section increases essentially

(as





) with decreasing the wavelength of the incident

wave. This feature is due to the strong reduction of the

energy flux (3

0~S



) in the incident wave.

Consider now the opposite limiting case of the long

wavelengths,



. We first note that for sufficiently

strong magnetic fields, 2

ce pe



 the frequency of

these waves at 221/

(2 )

ccepe

 

 2

coincides

with the electron cyclotron frequency, ()

0ce



. On

the other hand,









 at ce



 and the incident

wave is circularly polarized in the xz plane (see Figure

1). Near the cyclotron frequency, 0ce



, the energy

flux of the intermediate wave has the form



422

22 2

~cepecepe

Hce







 (45)

and increases strongly due to the cyclotron resonance.

This resonance is stabilized taking into account the elec-

tron-ion collisions. Here the energy flux can be very

large but finite quantity. Thus, at c



 the transfor-

mation cross section is vanishingly small, 0



.

In the limit of the long wavelengths, for the angular

distribution from the general expressions (22) and (36)

we obtain









022

,si

1sinsin.

IZT

IGH



nsin















(46)

H. B. NERSISYAN ET AL.

170

In this limit











, where

 



20 2

Ff f



 



































(47)

The cross section in the limit



 is determined

from the expression



2Tp





, where 2



given by (31) with

 

30 2

1afF



, 33

45ba



and 33

935ca. A comparison of these expressions for

the angular distribution and the cross section with the

similar expressions obtained in the case of the transfor-

mation of a high-frequency extraordinary wave shows

that in the former case a strong external magnetic field

can strongly suppress the transformation of an intermedi-

ate wave, the intensity of which decreases as 8





with

increasing of the external magnetic field (see (46)).

From (1) and (4) it is seen that the intensity of the

transformation of an intermediate wave increases mono-

tonically with



and takes a maximum value at





 and 2





 (or 32





). Therefore, the

radiation mainly escapes from the plasma parallel to its

boundary in the direction of the external magnetic field

(Figure 1).

The intensity (, )I





of the transformation of the in-

termediate wave as a function of the wavelength and the

angle



is shown in Figure 5. From this figure it is seen

that the intensity has a maximum in the short-wavelength

range, whereas the intensity of the scattering (transforma-

tion) of the high-frequency waves decreases monotoni-

cally with wavelength of the incident wave.

At the end of this section we note that the restriction

vv on the group velocity (see Section 2) leads to a

limitation



1/2

Te p

cv f











of the

wavelength of the incident wave.

Figure 5. Dependence of the intensity I (





) (normalized to

J0) for a transformed intermediate wave on the wavelength

and the angle



for



= /2. The values of the other pa-

rameters are the same as in Figure 2.

6. Scattering of Low-Frequency Waves

In this section we consider the scattering (transformation)

of low-frequency magnetosonic and lower-hybrid plasma

waves, the frequencies of which are much lower than the

characteristic electron frequencies (ce



and pe



) and

are comparable in order of magnitude with the

ion-cyclotron and Langmuir frequencies ci



and pi



respectively. In this low-frequency limit one must take

into account the dynamics of the plasma ions and their

partial contributions to the dispersion equation and the

scattering current.

From the general relation (34) we obtain an expression

for the frequency of the low-frequency waves (see also

[17]),



00 22 2

ALH

kku





, (48)

where 222

HcecipeH



 

 is the lower hybrid fre-

quency, 22

AA A

uV Vc, and is the Alfvén

velocity.

From (48) it follows that (or ),

i.e., this mode in general has right-hand elliptical polari-

zation in the xz plane (Figure 1) and can resonate with

plasma ions. In the case of the long magnetosonic waves

(0) we obtain



00P



(2)

0z 0E

k





0P0



and the wave has trans-

verse polarization, while in the case of short low-

er-hybrid waves (0), and this mode

consists of a longitudinal wave. In the latter case we have

the transformation of the lower-hybrid waves into elec-

tromagnetic radiation in a vacuum.

k P







From the expression for 0



it follows that 00





for any



. The transformation of the low-frequency

mode is therefore accompanied by an increase in wave-

length. Let us make some estimates. In astrophysical con-

ditions for a density g/cm3 and a magnetic field

kG we obtain that the transformation of the

magnetosonic wave in a vacuum generates radiation with

a wavelength exceeding that of the incident wave by two

orders of magnitude,





10B

00 106kc cu

.



Consider now Equations (18)-(21) for the intensity of

the transformation of the low-frequency waves. Taking

into account the dynamics of the plasma ions (19)-(21)

become











 

()2

() ()

2()2

,,, ,,

2,, ,

,,,,

eee

iii





 









 

 

(49)









()2222

,12cossins

aaa

GHin,

 









 

(50)

H. B. NERSISYAN ET AL. 171

 

22 2

, ,1sincossin,

ab abab



 



 





(51)

   







aa bb

glPglP



 









,

(52)

 





  

aa bb

lgPlgP

QglPglP



 



 

















(53)

In (50)-(53) the indices a and b take the values e or i to

denote the contributions of plasma electrons and ions,

ii i

Zmm



 , where i

, i, and are the

charge number and mass of an ion and the electron mass,

respectively, and the quantities a

G, a











, and





are determined by (13) and (33),

respectively.

In the limit of short wavelengths (



), from

(49)-(53) we obtain the angular distribution of the trans-

formation of lower-hybrid oscillations,



22222

1 sincossin,

IIZT



  





























(54)

where



3/2 2

1/2 2

GG G



 

  (55)

For small external magnetic fields (1



) the con-

tribution of the ions in (54) and (55) is negligible. For



, however, the transformation occurs mainly due

to the ionic current.

From (54) we derive the cross section for the trans-

formation of lower-hybrid waves by integrating the latter

over angles. The result is given by the expression







 

, where



222

ZTG

 

 



















. (56)

From (35) and (48) it follows that for the wavelengths

cAci





 the frequency is close to the ion-cy-

clotron frequency 0ci



. At these frequencies we

obtain . Therefore, near ci









the wave has

right-hand circular polarization and, as noted above, it

can resonate with plasma ions. Ion-cyclotron resonance

occurs in this case and the energy flux of the incident

wave increases sharply as



~ci pi







. (57)

The cross section of the process thus approaches to

zero at c





In the limit of long magnetosonic waves (,







), from (18) and (49)-(53) we obtain









222

12 cossinsin,

pe p

AA A

IIZT







 







 





(58)

where 0,

HHH



 0,

GGG



 AA





It should be noted that in the second term in (58) the

contribution of ions can be neglected for any values of

the magnetic field (ei



), whereas in the first

term the contribution of ions can be neglected only for

weak external magnetic fields.

From (58) we find the cross section for the transfor-

mation of magnetosonic waves. After evaluation of the

integrals over angles, we find







 

, where



3/2

145

eAA

ZT GH































(59)

From (54) and (58) it is seen that the intensity of the

transformation of the magnetosonic waves increases mo-

notonically with angles



and



, approaching the

maximum value at 2







and 2





 (or







). Consequently, as in the case of an intermedi-

ate wave, radiation escapes from the plasma mainly par-

allel to its boundary in the direction of the external mag-

netic field.

In Figure 6 we demonstrate the dependence of the in-

tensity of transformation of a low-frequency wave as a

function of wavelength and the angle



. From (54), (58),

and Figure 6 it is seen that the efficiency of transforma-

tion of lower-hybrid waves far exceeds the efficiency of

transformation of magnetosonic waves. However, the

intensity of the emission produced by the transformation

of magnetosonic waves can be essential if we take into

account that it is proportional to the (large) quantities

and . We note that the estimate of the cutoff

parameter T depends on the specific model of magnet-

ized plasma.

7. Discussion and Conclusions

In this paper, we have presented a detailed investigation

of the scattering and transformation of the plasma waves

on heavy charged particle in magnetized plasma. The

basic idea of this paper is that the scattering (transforma-

tion) occurs due to the nonlinear interaction of the inci-

dent wave with the polarization cloud surrounding the

particle. In the course of this study we have derived some

analytical results for the angular distribution and the

cross section of the scattered (transformed) radiation and

H. B. NERSISYAN ET AL.

172

Figure 6. De pendence of I(





) (normalized to 10-10 J0) for a

transformed low-frequency wave on the wavelength and the

angle



for



= /2 and



= 10-5. The values of the other

parameters are the same as in Figure 2.

we have shown that the problem is reduced to the deter-

mination of the nonlinear (three index) dielectric tensor

of magnetized plasma.

After introduction to the general theory in Section 2,

we have studied some particular cases of the scattering

and transformation processes assuming that the incident

wave propagates in the direction transverse to the exter-

nal magnetic field. The angular distribution and the cross

section for the scattering and transformation of high-fre-

quency ordinary and extraordinary waves and low-fre-

quency upper-hybrid, low-hybrid, and magnetosonic

waves have been investigated within a cold plasma

model which is valid when the group velocities of the

incident and scattered waves exceed the thermal veloci-

ties of the plasma particles. A number of limiting and

asymptotic regimes of short and long wavelengths have

been studied. The theoretical expressions for the angular

distribution of the scattered waves derived in this paper

lead to a detailed presentation of a collection of data

through figures.

We expect our theoretical model to be useful in ex-

perimental investigations of the wave scattering by

plasma as well as in some astrophysical applications.

Going beyond the presented model calculations which

are based on the cold plasma approximation we can en-

visage a number of avenues. One of the improvements of

our model will be to include the thermal effects which

are particularly important in the case of dusty plasmas

[15]. Furthermore, the theoretical model developed here

although is strong but is not adopted for immediate as-

trophysical applications. For this purpose it is required 1)

fully relativistic fluid calculations with appropriate equa-

tion of states and transport coefficients of a strongly

magnetized and dense (degenerated) plasma (see, e.g.,

[19]). 2) Short range quantum effects which appears due

to the tunneling of electrons and positrons through the

Bohm quantum potential barrier [20]. A study of these

and other aspects will be reported elsewhere.

8. Acknowledgements

This work has been supported by the Armenian Ministry

of Higher Education and Science under Grant No. 87.

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