Perpendicular propagating modes for weakly magnetized relativisticdegenerate plasmaGohar Abbas, M. F. Bashir, and G. Murtaza Citation: Phys. Plasmas 19, 072121 (2012); doi: 10.1063/1.4739223 View online: http://dx.doi.org/10.1063/1.4739223 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v19/i7 Published by the American Institute of Physics. Related ArticlesOn the existence of Weibel instability in a magnetized plasma. II. Perpendicular wave propagation: The ordinarymode Phys. Plasmas 19, 072116 (2012) Three-wave coupling in electron-positron-ion plasmas Phys. Plasmas 19, 072114 (2012) Effect of electron density profile on power absorption of high frequency electromagnetic waves in plasma Phys. Plasmas 19, 073301 (2012) Lower hybrid current drive at high density in the multi-pass regime Phys. Plasmas 19, 062505 (2012) Generation of high power sub-terahertz radiation from a gyrotron with second harmonic oscillation Phys. Plasmas 19, 063106 (2012) Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

Perpendicular propagating modes for weakly magnetized relativisticdegenerate plasma

Gohar Abbas,1,2 M. F. Bashir,1,2 and G. Murtaza1

1Salam Chair in Physics, G. C. University Lahore, Punjab 54000, Pakistan2Department of Physics, G. C. University Lahore, Punjab 54000, Pakistan

(Received 12 February 2012; accepted 3 July 2012; published online 26 July 2012)

Using the Vlasov-Maxwell system of equations, the dispersion relations for the perpendicular

propagating modes (i.e., X-mode, O-mode, and upper hybrid mode) are derived for a weakly

magnetized relativistic degenerate electron plasma. By using the density (n0 ¼ p3F=3p2�h3) and the

magnetic field values for different relativistic degenerate environments, the propagation

characteristics (i.e., cutoff points, resonances, dispersions, and band widths in k-space) of these

modes are examined. It is observed that the relativistic effects suppress the effect of ambient

magnetic field and therefore the cutoff and resonance points shift towards the lower frequency

regime resulting in enhancement of the propagation domain. The dispersion relations of these

modes for the non-relativistic limit (p2F � m2

0c2) and the ultra-relativistic limit (p2F � m2

0c2) are

also presented. VC 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4739223]

I. INTRODUCTION

The Bernstein wave, O-mode and X-mode are the elec-

trostatic, the electromagnetic and mixed mode branches of

the perpendicular propagation, respectively. These waves are

regarded as magnetically split versions of the longitudinal

and transverse waves in isotropic plasmas.1 Such waves in

highly dense electron plasmas having number densities of

the order of 1026–40 cm�3 are expected to exist in compact

astrophysical objects, e.g., in massive white dwarfs, in the

core of pre-supernovae stars, in inertial confinement

fusion2–5 and in pair-plasmas,6 etc. The degeneracy of par-

ticles becomes inevitable when the Fermi energy EF exceeds

the thermal energy, i.e., EF � KBT (where KB is the Boltz-

mann constant) and consequently the thermal de Broglie

wavelength kBe becomes comparable to the inter-particle dis-

tance. In this situation, the electron degeneracy in a plasma

is expected to play a significant role in describing the propa-

gation characteristics of different electrostatic and electro-

magnetic waves.7–10

Analysis of the linear waves in degenerate plasma was

performed by Drummond11 and by Klimontovich and Silin12

who investigated high frequency degenerate electron plasma

oscillations with applications to metals and semiconductors.

Such an analysis has also been done with applications to rel-

ativistic plasmas occurring in laboratories13–15 as well as in

astrophysical environments.16–18 It may be noted that the rel-

ativistic effects occur when the electron number density

increases beyond a certain value and the electron Fermi

energy becomes comparable to the rest mass energy.19,20 For

example, in white dwarfs having electron number densities

exceeding 1029 cm�3 and Fermi temperature of the order of

107 K, the electrons become relativistic. In degenerate plas-

mas, the ambient magnetic field also effects the thermody-

namic and kinetic properties in various ways, depending on

density, temperature, and field strength.21 Several

investigations22–25 have examined the impact of an arbitra-

rily strong magnetic field in degenerate electron plasma.

The present authors previously studied parallel and per-

pendicular propagating waves in a weakly magnetized rela-

tivistic Maxwellian electron plasma.26,27 Robinson28,29

discussed the perpendicular propagating waves in a magne-

tized, weakly relativistic Maxwellian plasma. In particular,

the generalized cyclotron harmonic modes were discussed to

compare the non-relativistic and weakly relativistic reso-

nance broadening, frequency downshift, and damping. These

results were further used to study the effects of weak ambient

magnetic field on plasma waves30 in which the harmonic

structure vanishes with an increase in temperature.

The high frequency parallel propagating modes were

investigated in a weakly magnetized relativistic degenerate

plasma having number densities of the order of

1026�34 cm�3 and an ambient magnetic field of the order of

109�10G.32 This investigation was carried out in the limits,

i.e., x > k:v; X, and X0=x0p < 1; where X; X0 are the rela-

tivistic and non-relativistic gyro-frequencies, respectively,

and x0p is the non-relativistic plasma frequency. In the pres-

ent investigation, we extend this work and derive the disper-

sion relations for perpendicular propagating modes for a

weakly magnetized relativistic degenerate plasma. These

results may prove useful in describing the linear behavior of

high frequency plasma waves that may exist in the super

dense environments of astrophysical and laboratory plasmas.

The plan of the paper is as follows. In Sec. II, the deri-

vation of the perpendicular propagating modes is presented

in relativistic degenerate plasma along with their limiting

cases, and the summary of results and discussion is given in

Sec. III.

II. MATHEMATICAL FORMALISM

In this section, we extend the analysis of Abbas et al.32

Using the Vlasov-Maxwell system of equations, we obtain

the general linear dispersion relations for the X-mode, the

upper hybrid-mode, and the O-mode for the arbitrary distri-

bution as

1070-664X/2012/19(7)/072121/7/$30.00 VC 2012 American Institute of Physics19, 072121-1

PHYSICS OF PLASMAS 19, 072121 (2012)

ðx2 þPxxÞðx2 � c2k2x þPyyÞ þ ðPxyÞ2 ¼ 0; (1)

x2 þPxx ¼ 0; (2)

and

x2 � c2k2x þPzz ¼ 0; (3)

where

Pxx ¼�16p2e2x2

k2x

ð10

p2

v@f0@jpj dp 1� x

2kxvlog

xþ kxv

x� kxv

��������� x2X2k2

xv2

3ðx2 � k2xv

2Þ3

" #

Pyy ¼8p2e2x2

k2x

ð10

p2

v@f0

@jpj dp

"1� x2 � k2

xv2

2xkxvlog

����xþ kxvx� kxv

����� X2

12

9x2 � 23k2xv

2

ðx2 � k2xv

2Þ2� 9

2xkxvlog

����xþ kxvx� kxv

���� !#

Pxy ¼i8p2e2x3

k2x

ð10

p2

vX@f0

@jpj dp

"1

x2 � k2xv

2� 1

2xkxvlog

����xþ kxv

x� kxv

����#

Pzz ¼8p2e2x2

k2x

ð10

p2

v

@f0

@jpj dp

"1� x2 � k2

xv2

2xkxvlog

����xþ kxv

x� kxv

����� X2

12

3x2 � 5k2xv

2

ðx2 � k2xv

2Þ2� 3

2xkxvlog

����xþ kxv

x� kxv

���� !#

(4)

are the polarization tensor components26 obtained by taking

the ambient magnetic field along z-direction and restricted to

the frequency condition x > k:v in which the pole contribu-

tion is not relevant. X ¼ eB0=cm0c ¼ X0=c is the relativistic

gyro-frequency, c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jpj2=m2

0c2

qis the relativistic Lor-

entz factor, m0 is the rest mass of electron, jpj is the magni-

tude of momentum vector, x is the wave frequency, and kx is

the wave vector perpendicular to the ambient magnetic field.

Rewriting the above components in terms of relativistic

energy, i.e., E ¼ cm0c2 ¼ cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijpj2 þ m2

0c2

q, we obtain

Pxx ¼�16p2e2x2

c3k2x

ð1m0c2

EffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4

q@f0

@EdE

"1� x

2kxc

EffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p log

����xEþ kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p

xE� kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p ����

� x2E2c2k2xðX0m0c2Þ2ðE2 � m2

0c4Þ3ðx2E2 � k2

x c2ðE2 � m20c4ÞÞ3

#

Pyy ¼8p2e2x2

c3k2x

ð1m0c2

EffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4

q@f0@E

dE

"1� x2E2 � k2

x c2ðE2 � m20c4Þ

2xEkxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p log

����xEþ kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p

xE� kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p ����

� X20m2

0c4

12E2

E2ð9x2E2 � 23k2x c2ðE2 � m2

0c4ÞÞðx2E2 � k2

x c2ðE2 � m20c4ÞÞ2

� 9 E

2xkxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p log

����xEþ kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p

xE� kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p ����

!#

Pxy ¼i8p2e2x3

c3k2x

ðX0m0c2Þð1

m0c2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4

q@f0

@EdE

"E2

x2E2 � c2k2xðE2 � m2

0c4Þ �E

2xkxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p

� log

����xEþ kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p

xE� kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p ����

#

Pzz ¼8p2e2x2

c3k2x

ð1m0c2

EffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4

q@f0@E

dE

"1� x2E2 � k2

x c2ðE2 � m20c4Þ

2xEkxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p log

����xEþ kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p

xE� kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p ����

� X20m2

0c4

12E2E2 3x2E2 � 5k2

x c2ðE2 � m20c4Þ

ðx2E2 � k2x c2ðE2 � m2

0c4ÞÞ2

!� 3E

2xkxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p log

����xEþ kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p

xE� kxcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � m2

0c4p ����

( ) #: (5)

For a fully degenerate plasma, the derivative of Fermi

distribution function takes the form33

@f0@E¼ � 2

ð2p�hÞ3dðEF � EÞ; (6)

where d is the Dirac delta function and �h is the Planck’s

constant. Employing the above distribution function and

performing the energy integration, we may re-write the

components in Eq. (5) in terms of Fermi momentum pF

as

072121-2 Abbas, Bashir, and Murtaza Phys. Plasmas 19, 072121 (2012)

Pxx ¼ 3x2

0p

cF

x2

c2k2x

c2Fm2

0c2

p2F

1� x2kxcðpF=cFm0cÞ log

����xþ ckxðpF=cFm0cÞx� ckxðpF=cFm0cÞ

����� x2ðc2k2xÞðp2

F=c2Fm2

0c2ÞðX20=c

2FÞ

3ðx2 � c2k2xðp2

F=c2Fm2

0c2ÞÞ3

" #

Pyy ¼ �3

2

x20p

cF

x2

c2k2x

c2Fm2

0c2

p2F

"1� x2 � c2k2

xðp2F=c

2Fm2

0c2Þ2xckxðpF=cFm0cÞ log

����xþ ckxðpF=cFm0cÞx� ckxðpF=cFm0cÞ

����� ðX

20=c

2FÞ

12

9x2 � 23k2x c2ðp2

F=c2Fm2

0c2Þðx2 � c2k2

xðp2F=c

2Fm2

0c2ÞÞ2� 9

2xckxðpF=cFm0cÞ log

����xþ ckxðpF=cFm0cÞx� ckxðpF=cFm0cÞ

���� !#

Pxy ¼ �i3

2ðx2

0p=cFÞðX0=cFÞx3

c2k2x

c2Fm2

0c2

p2F

� �1

x2 � c2k2xðp2

F=c2Fm2

0c2Þ �1

2xckxðpF=cFm0cÞ log

����xþ ckxðpF=cFm0cÞx� ckxðpF=cFm0cÞ

����� �

Pzz ¼ �3

2

x20p

cF

x2

c2k2x

c2Fm2

0c2

p2F

"1� x2 � k2

x c2ðp2F=c

2Fm2

0c2Þ2xckxðpF=cFm0cÞ log

����xþ ckxðpF=cFm0cÞx� ckxðpF=cFm0cÞ

����� X2

0=c2F

12

(3x2 � 5k2

x c2ðp2F=c

2Fm2

0c2Þðx2 � k2

x c2ðp2F=c

2Fm2

0c2ÞÞ2

!� 3

2xkxcðpF=cFm0cÞ log

����xþ ckxðpF=cFm0cÞx� ckxðpF=cFm0cÞ

����)#

; (7)

where x0p ¼ 4pn0e2=m0 is the non-relativistic plasma fre-

quency, n0 ¼ p3F=3p2�h3 is the equilibrium number density,

pF ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2

F=c2 � m20c2

pis the relativistic Fermi momentum,

and cF ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ p2

F=m20c2

pis the relativistic factor in degener-

ate case.

In the following, we shall use the above components

to derive the dispersion relations of the X-mode, the upper

hybrid mode, and the O-mode and present them graphi-

cally by using the parameters of different relativistic

degenerate environments. We shall also derive the disper-

sion relations of these modes in the non-relativistic limit

(i.e., p2F � m2

0c2) and in the ultra-relativistic limit (i.e.,

p2F � m2

0c2Þ.

A. X-mode

Generally, the X-mode dispersion relation has two dif-

ferent branches, the upper branch ðx > xRÞ and the lower

branch ðxL < x < xRÞ; where xR and xL are the upper and

the lower cutoff frequencies in the x vs ckx plots of the

X-mode, respectively. In order to obtain an expression for

the linear dispersion relation of the X-mode, we use the rele-

vant tensor components from Eq. (7) in Eq. (1) and get

x2 þ 3x2

0p

cF

x2

c2k2x

c2Fm2

0c2

p2F

1� x2kxcðpF=cFm0cÞ log

����xþ ckxðpF=cFm0cÞx� ckxðpF=cFm0cÞ

����� x2ðc2k2xÞðp2

F=c2Fm2

0c2ÞðX20=c

2FÞ

3�x2 � c2k2

xðp2F=c

2Fm2

0c2Þ3

0B@

1CA

264

375

�"x2 � c2k2

x �3

2

x20p

cF

x2

c2k2x

c2Fm2

0c2

p2F

!� 1� x2 � c2k2

xðp2F=c

2Fm2

0c2Þ2xckxðpF=cFm0cÞ log

����xþ ckxðpF=cFm0cÞx� ckxðpF=cFm0cÞ

����� �

� ðX20=c

2FÞ

12

9x2 � 23k2x c2ðp2

F=c2Fm2

0c2Þðx2 � c2k2

xðp2F=c

2Fm2

0c2ÞÞ2� 9

2xckxðpF=cFm0cÞ log

����xþ ckxðpF=cFm0cÞx� ckxðpF=cFm0cÞ

���� !#

¼"

3

2

x20p

cF

x3

c2k2x

X0

cF

c2Fm2

0c2

p2F

1

x2 � c2k2xðp2

F=c2Fm2

0c2Þ�1

2xckxðpF=cFm0cÞ log

����xþ ckxðpF=cFm0cÞx� ckxðpF=cFm0cÞ

����!#2

: (8)

Now, we proceed to analyze it graphically. Figs. 1–3

show the plots of x=x0p vs ckx=x0p for the X-mode by

choosing parameters of different relativistic degenerate

weakly magnetized plasma environments. Fig. 1 depicts the

relativistic effects through variation of electron number den-

sity for a constant ambient magnetic field strength. We note

that as we move from weakly relativistic (n0 � 1028cm�3) to

strongly relativistic (� 1031cm�3) regime, the separation

between lower and upper cutoff frequencies becomes nar-

rower and eventually vanishes for the highly relativistic

072121-3 Abbas, Bashir, and Murtaza Phys. Plasmas 19, 072121 (2012)

case. This is for the reason that the relativistic effects sup-

press the contribution of the magnetic field and thus in the

highly relativistic or the ultra-relativistic case, we obtain the

standard results of decoupled electrostatic and electromag-

netic modes having the same cutoff points as expected. Fur-

ther, the relativistic effects not only shift the cutoff points

toward the lower frequency domain but also change the de-

pendence on the wave number significantly in the region

between the lower cutoff and the upper hybrid frequencies.

Thus, in the said region, the X-mode becomes more disper-

sive with the increase in the relativistic values. Figs. 2 and 3

show the effect of variation of the ambient magnetic field for

the non-relativistic case (n0 � 1026cm�3) and the weakly rel-

ativistic case (n0 � 1028cm�3), respectively. In the former

case, the separation between the upper and lower cutoff fre-

quencies increases with the increase in the magnetic field

making the resonance more prominent. In the latter case, the

increase in the difference of the cutoff frequencies is less

than in the non-relativistic case, i.e., relativistic effects sup-

press the effect of ambient magnetic field and also the reso-

nance starts to propagate.

Now, we derive the dispersion relations for the X-mode

in the non-relativistic and the ultra-relativistic limiting cases.

1. Non-relativistic and ultra-relativistic X-mode

In the non-relativistic case, the Fermi momentum is less

than the rest momentum of the electron, i.e., p2F � m2

0c2

(cF ¼ 1) and therefore the dispersion relation of the X-mode

in Eq. (8) takes the form

FIG. 1. A contour plot of xx0p

vs ckx

x0pshows the relativistic density variations

(n0 ¼ 2� ð1028 (——–); 1029 (– – – –); 1030 (- - - - -); 1031 (........)Þ cm�3Þin the dispersion curves corresponding to magnetic field strength

(B0 ¼ 1010G) for the X-mode. Resultantly, the numerical values of X0

x0p

become equal to ð0:0220 (——–), 0.0069 (– – – –) and 0.00220 (- - - - -) and

0.0006 (........). The increasing relativistic effects suppress the contribution

of the ambient magnetic field and thus the difference between the higher and

the lower branches of the X-mode becomes narrower. In the highly relativis-

tic regime (n0 � 1030�31 cm�3Þ; the curves almost have the same cutoff

points.

FIG. 2. A graphical representation of xx0p

vs ckx

x0pillustrates the effects of differ-

ent ambient magnetic field strengths (B0 ¼ ð1� 109 (——–);3� 109 (– – – –);5� 109ð����Þ;8� 109ð:::::::::::ÞÞ G) in the dispersion curves in the non-

relativistic density regime (n0 ¼ 2� 1026 cm�3Þ for the X-mode. In this pa-

rameter range, the ratio X0

x0pbecomes¼ (0.0220 (——–), 0.0661 (– – – –),

0.1102 (- - - - -), and 0.176 (........). The cutoffs of the upper and the lower

branches of the X-mode get closer with the reduction of the field strength.

FIG. 3. A contour plot of xx0p

vs ckx

x0prepresents the behavior of dispersion

curves for the ambient magnetic field variations (B0 ¼ ð1� 109 (——–),

3� 109 (– – – –); 5� 109ð����Þ; 8� 109 (...........)) G) in the weakly rela-

tivistic density regime (no ¼ 2� 1028 cm�3Þ for the X-mode. In this parame-

ter range, the ratio X0

x0pbecomes¼ (0.0139 (——–), 0.0418 (– – – –), 0.0696

(- - - -), and 0.1115 (........). Also, the separation between the cutoff points has

become much narrower in comparison with the non-relativistic density regime

in Fig. 2.

072121-4 Abbas, Bashir, and Murtaza Phys. Plasmas 19, 072121 (2012)

x2 þ 3x20p

x2

k2xv

2F

1� x2kxvF

log

����xþ kxvF

x� kxvF

����� x2k2xv

2FX2

0

3ðx2 � k2xv

2FÞ

3

!" #

� x2 � c2k2x �

3

2x2

0p

x2

k2xv

2F

1� x2 � k2xv

2F

2xkxvFlog

����xþ kxvF

x� kxvF

����� �

� X20

12

9x2 � 23k2xv

2F

ðx2 � k2xv

2FÞ

2� 9

2xkxvFlog

����xþ kxvF

x� kxvF

���� !" #

¼ 3

2x2

0pX0

x3

k2xv

2F

1

x2 � k2xv

2F

� 1

2xkxvFlog

����xþ kxvF

x� kxvF

����� �� �2

: (9)

Similarly, in the ultra-relativistic case, the Fermi momentum becomes greater than the rest momentum of the electron,

i.e., p2F � m2

0c2 (cF ¼ pF=m0c) and thus the dispersion relation of the X-mode in Eq. (8) becomes

x2 þ 3x2pF

x2

c2k2x

1� x2kxc

log

����xþ ckx

x� ckx

����� x2c2k2xX

2F

3ðx2 � c2k2xÞ

3

!" #

� x2 � c2k2x �

3

2x2

pF

x2

c2k2x

1� x2 � c2k2x

2xckx

����xþ ckx

x� ckx

����� �

� X2F

12

9x2 � 23k2x c2

ðx2 � c2k2xÞ

2� 9

2xckxlog

����xþ ckx

x� ckx

���� !" #

¼ 3

2x2

pF

x3

c2k2x

XF1

x2 � c2k2x

� 1

2xckxlog

����xþ ckx

x� ckx

����� �� �2

; (10)

where the ultra-relativistic plasma and the gyro-frequencies

are defined as

x2pF ¼

x20p

pF=m0cand XF ¼

X0

pF=m0c: (11)

On switching off the ambient magnetic field, we obtain the

standard non-relativistic and ultra-relativistic field-free

results reported in Refs. 33 and 34.

2. Upper hybrid mode

Upper hybrid mode (decoupled electrostatic branch in

the X-mode in Eq. (8)) can be obtained in the approximation

ðx2 þPxxÞðx2 � c2k2x þPyyÞ � ðPxyÞ2. Using Pxx from

Eq. (7) in Eq. (2), we obtain the upper hybrid mode given by

x2 þ 3x2

0p

cF

x2

c2k2x

c2Fm2

0c2

p2F

�

1� x2kxcðpF=cFm0cÞ log

����xþ ckxðpF=cFm0cÞx� ckxðpF=cFm0cÞ

����� x2ðc2k2

xÞðp2F=c

2Fm2

0c2ÞðX20=c

2FÞ

3ðx2 � c2k2xðp2

F=c2Fm2

0c2ÞÞ3

!¼ 0: (12)

Fig. 4 describes the behavior of the upper hybrid wave Eq.

(12) for different density regimes (i.e., 1028cm�3 to

1031cm�3Þ. A frequency down-shift is observed as one

moves from the weakly relativistic to the highly relativistic

regime. Moreover, all the curves intersect at a point wherex

x0p’ ckx

x0p, i.e., on the light line x ¼ ckx: The same was also

observed in Ref. 32 for Langmuir waves. On examining the

behavior of the upper hybrid dispersion curve with the varia-

tion of the ambient field strength, we find that the curves sep-

arate out and that the separation is more prominent in the

non-relativistic case than in the highly relativistic case. Simi-

lar results have been obtained for the Bernstein waves when

the harmonic structure vanishes and that the magnetized and

the unmagnetized curves become closer to each other.30

FIG. 4. A plot of xxop

vs ckx

x0pshows the relativistic density variation

(n0 ¼ 2� ð1028 (——–); 1029 (– – – –); 1030 (- - - -); 1031 (........)) cm�3Þ in

the dispersion curves corresponding to the ambient magnetic field strength

(B0 ¼ 1010G) for the upper hybrid mode.

072121-5 Abbas, Bashir, and Murtaza Phys. Plasmas 19, 072121 (2012)

B. O-mode

Now, we write the dispersion relation for the O-mode

(i.e., Eq. (3)) by using Pzz component of the polarization ten-

sor from Eq. (7) as

x2 ¼ c2k2x þ

3

2

x20p

cx2

c2k2x

c2m20c2

p2F

�"

1� x2 � k2x c2ðp2

F=c2m2

0c2Þ2xckxðpF=cm0cÞ log

����xþ ckxðpF=cm0cÞx� ckxðpF=cm0cÞ

����� X2

0=c2

12

(3x2 � 5k2

x c2ðp2F=c

2m20c2Þ

ðx2 � k2x c2ðp2

F=c2m2

0c2ÞÞ2

!

� 3

2xkxcðpF=cm0cÞ log

����xþ ckxðpF=cm0cÞx� ckxðpF=cm0cÞ

����)#

: (13)

Fig. 5 represents the dispersion characteristics of the O-mode

with the variation of the relativistic effects through number

density. From the figure, it is observed that the relativistic

effects shift the cutoff point toward the lower frequency re-

gime and that results in the enlargement of the propagation

domain. We have also checked numerically that the weak am-

bient magnetic field effects do not alter the dispersion curves

of the O-mode. Although their absolute strength is very high

ð� 109�10GÞ but in comparison with the relevant densities

ð1028�31cm�3Þ, the magnetic field effects are negligible.

Taking p2F � m2

0c2 (cF ¼ 1), the dispersion relation of

the O-mode in Eq. (13) takes the form

x2 ¼ c2k2x þ

3

2x2

0p

x2

k2xv

2F

"1� x2 � k2

xv2F

2xkxvFlog

����xþ kxvF

x� kxvF

����� X2

0

12

(3x2 � 5k2

xv2F

ðx2 � k2xv

2FÞ

2� 3

2xkxvFlog

����xþ kxvF

x� kxvF

����)#

:

(14)

When the Fermi momentum is much greater than the rest

momentum of the electrons, i.e., p2F � m2

0c2 (cF ¼ pF=m0c),

the dispersion relation of the O-mode in Eq. (13) becomes

x2 ¼ c2k2x þ

3

2x2

pF

x2

c2k2x

"1� x2 � k2

x c2

2xckxlog

����xþ ckx

x� ckx

����� X2

F

12

3x2 � 5k2x c2

ðx2 � k2x c2Þ2

!� 3

2xkxclog

����xþ ckx

x� ckx

����( )#

:

(15)

On neglecting the ambient magnetic field, the above results

(Eqs. (14) and (15)) in the non-relativistic and the ultra-

relativistic limits are in agreement with the results reported

in Refs. 33 and 34.

III. CONCLUSION

The dispersion relations for the fully relativistic perpen-

dicular propagating waves in weakly magnetized degenerate

plasma are derived and graphically presented. For the X-

mode, it is observed that the difference between the lower

and the upper cut-off frequencies (i.e., xL and xR, respec-

tively) becomes narrower as we move from the weakly rela-

tivistic (n0 � 1028cm�3) to the strongly relativistic

(� 1031cm�3) regime and this difference eventually vanishes

for the highly relativistic case. The relativistic effects not

only shift the cutoff points toward the lower frequency re-

gime but also change the dependence on wave number essen-

tially in the region between the lower cutoff and the upper

hybrid frequencies, i.e., in this region, the X-mode becomes

more dispersive, the cutoff points merge and the propagation

domain is enlarged with the increase of the relativistic Lor-

entz factor value. On the other hand, the magnetic field

strength increases the separation between the cutoff points.

Further, this increase is more pronounced in the non-

relativistic limit than in the weakly relativistic regime since

the relativistic effects tend to suppress the role of the ambi-

ent magnetic field. Similarly, the resonance is more distinct

in the non-relativistic case than in the weakly relativistic one

since the resonance starts vanishing in the weakly relativistic

limit and instead the oscillations start propagating.

The upper hybrid wave also becomes more dispersive as

we move from weakly relativistic to the highly relativistic

case. Further, all the curves intersect at the point wherex

x0p’ ckx

x0p, i.e., on the light line x ¼ ckx. It is of interest to

mention here that, in general, the properties of cyclotron har-

monic modes (e.g., Bernstein mode) in the degenerate

plasma differ from those of non-degenerate plasma in the

sense that while the former exhibits in its tail the oscillatory

character in the limit x! nX for large k and the latter

FIG. 5. A contour plot of xx0p

vs ckx

x0pshows the relativistic density variation

(n0 ¼ 2� ð1028 (——–); 1029 (– – – –); 1030 (- - - -); 1031 (.........)) cm�3Þ in

the dispersion curves corresponding to the magnetic field strength

(B0 ¼ 1010G) for the O-mode.

072121-6 Abbas, Bashir, and Murtaza Phys. Plasmas 19, 072121 (2012)

shows no such behavior.33 However, our results in the pres-

ent paper which are derived for high frequency x > X and

weak magnetic field exhibit no harmonic character at all.

This is due to the reason that we have used the technique,31

in which we solve Vlasov-Maxwell’s system and express the

perturbed distribution function as a series of j Xx�k:v j for com-

pletely degenerate electron plasma in the weak ambient mag-

netic field limit, i.e., j Xx�k:v j < 1: Further, we take the

frequency limit x > k:v and therefore as a consequence, the

wave frequency becomes greater than the gyro-frequency.

The series converges only if the Doppler shifted frequency

becomes greater than the gyro frequency. Further, we have

also assumed X0

x0p< 1 for our weak magnetic field limit other-

wise the series would blow up.

For the O-mode, the relativistic effects shift the cut-off

point toward the lower frequency regime and thus results in

the enlargement of the propagation domain. Further, the

magnetic field effects which are rather weak in our case play

no role in the dispersion profile of the O-mode.

We believe that our results may prove useful in studying

the propagation characteristics of the perpendicular propagat-

ing waves in the weakly magnetized relativistic degenerate

electron plasma. The relevant densities (n0 � 1026cm�3(non-

relativistic) to 1031cm�3(highly relativistic)) corresponding

to the magnetic field strengths (B0 ¼ 109�10G) satisfy the

condition x0p > X0 and cover a vide range of astrophysical

environments like white drawfs and neutron stars. Our results

may also be useful to study the coupling of the electrostatic

Langmuir wave and the electromagnetic Z-mode in a weakly

magnetized plasma.

ACKNOWLEDGMENTS

We are thankful to the anonymous Referee for his com-

ments on this paper and Professor H. A. Shah for many use-

ful discussions. We also gratefully acknowledge financial

assistance from the “Office of the External Activities,”

AS-ICTP, Trieste, Italy.

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