September 20, 2011

Anomalous skin effects in relativistic parallelpropagating weakly magnetized electron plasma waves

Gohar Abbas1,2, M. F. Bashir1,2 and G. Murtaza1

(1) Salam Chair in Physics,G. C. University Lahore 54000, Pakistan.

(2) Department of Physics,G. C. University Lahore 54000, Pakistan.

Abstract

Fully relativistic analysis of anomalous skin effects for parallel propagating waves ina weakly magnetized electron plasma is presented and general expressions for longitudinaland transverse permittivites are derived. It is found that the penetration depth for R-& L -waves increases as we move from non-relativistic to highly relativistic regime. The ambientmagnetic field reduces/enhances the skin effects for R- wave/L-wave as the strength ofthe field is increased. In general, the weak magnetic field effects are pronounced for theweakly relativistic regime as compared with other relativistic cases. The results are alsographically illustrated. On switching off the magnetic field, previous results for field freecase are retrieved.2

1

I. Introduction

The skin effect is a process in which the electromagnetic fields are screened out ina conducting medium. In collisional plasma when the thermal motion of the electrons isneglected (cold plasma), then in the frequency range ω << ν, the skin depth is determinedby δn = (c/ωpe)

√2ν/ω, where c is the speed of light, ωpe =

√4πnoe2/m is the electron

plasma frequency and ν is the collisional frequency. This is known as normal skin effect. Inthe collisionless plasma having high frequencies of range ωpe >> ω >> ν, the collisionlessskin depth δl is determined by electron inertia, δl = c/ωpe. This is known as the localcollisionless skin effect. In this local skin effect regime, the frequency of the electromagneticfield is greater than the thermal frequency i.e., ω > vTe so that the electrons do not movefar during the single wave period and therefore experience a local field.

In the frequency range ω < vTe, when the electrons move distances comparable tothe skin depth during the field period without colliding each other, then the conductivitybecomes a function of ω throughout the entire skin layer. The skin effect under thiscondition is known as anomalous. Also the electron response to the applied field becomesnonlocal due to the thermal motion. In this range of non-local anomalous skin depth, thelongitudinal field is screened within the radius equal to the Debye length and transversefield shields out to give penetration depth.1–4

The problem of anomalous skin effects for transverse electromagnetic waves was firstsolved by Reuter5 using Fermi distribution function for semi infinite metals. Weibel6

extended this work for gaseous plasma using Maxwellian distribution function. Since thena significant amount of research work has been done in describing this theory (see e.g.,7–11).It has also become important to numerous plasma processes e.g., plasma heating duringthe interaction of ultra-intense ( I ≥ 1019 W/cm2) short laser pulses with the overdenseplasma created in solid material targets and plasma sources such as inductively coupledplasma (ICP) discharges.12,13

The application of weak external magnetic fields on a plasma tends to enhance skindepth of the electromagnetic fields.14,15 In the high frequency limit they may be signif-icantly important in a number of situations (see e.g.16–19). The influence of an ambientmagnetic field on skin effects in metal and gas discharge plasmas have been studied pre-viously.20–25 Kim26 represented a one dimensional analysis of electron heating processin a weakly magnetized plasma. Experimental results indicate that a proper applicationof weak external magnetic field to a planar type Inductively Coupled Plasmas (ICP) canimprove heating efficiency, plasma potential and impedance matching etc.27

The main aim of our study is to derive explicit expression for the anomalous skineffects in parallel propagating waves. Our work is restricted to weakly magnetized, col-lisionless, isotropic and relativistic plasma. However, it is applicable to a large varietyof astrophysics, space and laboratory plasmas17,29–31,1 etc. On neglecting the ambientmagnetic field our results are in agreement with the results reported in.2

The paper is organized as follows:

Polarization tensor for parallel propagating waves and investigation of anomalous skineffect is described in Section II. Section III gives summary and conclusion.

II. The dispersion relation for parallel propagating modes

The electric and magnetic fields are governed by Maxwell’s equations given by

∇×B =1

c

∂E

∂t+

4π

cJ (1)

2

∇×E = −1

c

∂B

∂t(2)

Using Eq. (1) and Eq. (2), the dielectric tensor can be written as under

[(ω2 − c2k2)δij+c2kikj+Πij

]Ej=RijEj = 0 (3)

where c is the speed of light, Rij is the dielectric tensor, Πij = 4πiωσij is the polariza-tion tensor and σij is the conductivity tensor defined as Ji = σijEj . We use cgs unitsthroughout.

The linearized Vlasov equation for electron plasma embedded in an external magneticfield is given by

∂f1∂t

+ v · ∂f1∂x

+e

cv ×Bo.

∂f1∂p

= −e

[E1 +

1

cv ×B1

]· ∂fo∂p

(4)

where Bo = (0, 0, Bo) is the ambient magnetic field, v =p

γ mis the relativistic velocity

and γ =1√

1− v2

c2

=

√1 +

p2

m2c2is the relativistic factor.

From Eq. (4), the solution f1 can be written up to second order in Ω assuming weakmagnetic field28 as

f1 =−ie

ω

[1 +

i Ω

ω − k.v

∂

∂φ+

i Ω

ω − k.v

∂

∂φ

i Ω

ω − k.v

∂

∂φ

](E1.

∂fo∂p

+(v.E1)

ω − k.vk.

∂fo∂p

) (5)

Therefore the induced current density is given by

J indi ≡ e

∫dpvif1

=−ie2

ω

∫dp vi

(1 +

i Ω

ω − k.v

∂

∂φ+

i Ω

ω − k.v

∂

∂φ

i Ω

ω − k.v

∂

∂φ

)×

×(E1.∂fo∂p

+(v.E1)

ω − k.vk.

∂fo∂p

)

= σijEj (6)

so that the polarization tensor Πij can be written as

Πij = 4πe2∫

dp vi ×

×(1 +

i Ω

ω − k.v

∂

∂φ+

i Ω

ω − k.v

∂

∂φ

i Ω

ω − k.v

∂

∂φ

)[klvj

ω − k.v+δlj

]∂fo∂pl

(7)

For an arbitrary isotropic distribution function fo, Πij in Eq. (7) simplifies to

3

Πij = 4πe2ω

∫ ∞

0

|v| ∂fo∂ |p|

p2dp

∫ π

0

∫ 2π

0

sinθ dθ dφ

×( sin θ cosφ, sin θ sinφ, cos θ)i ×

×(1 +

i Ω

ω − k.v

∂

∂φ+

i Ω

ω − k.v

∂

∂φ

i Ω

ω − k.v

∂

∂φ

)×

×( sin θ cosφ, sin θ sinφ, cos θ)j

ω − k.v(8)

where we have used spherical polar coordinates. From Eq. (8) it is easy to show that thefollowing components vanish for parallel or perpendicular propagation.19

Πxz = 0 = Πyz = Πzx = Πzy (9)

Thus the dyadic Eq. (3) takes the form

ω2 − c2k2z+Πxx Πxy 0Πyx ω2 − c2k2z+Πyy 00 0 ω2 +Πzz

Ex

Ey

Ez

= 0 (10)

From the above Eq. (10), we obtain R- & L– waves and Electrostatic mode respectivelygiven by

ω2 − c2k2z +Πxx ± iΠxy = 0 (11)

(± signs are for R- and L- wave respectively)

andω2 +Πzz = 0 (12)

where

Πxx =8π2e2ω2

k2z

∫ ∞

0

dpp2

v

∂fo∂ |p|[

1− ω2 − k2zv2

2ωkzvlog

∣∣∣∣ω + kzv

ω − kzv

∣∣∣∣− iπkzv

2ω

(1− ω2

k2zv2

)H

(1− ω

kzv

)]= Πyy

(13)

Πxy = −i16π2e2ω

k2z

∫ ∞

0

dpp2Ω

v

∂fo∂ |p|

[1− ω

2kzvlog

∣∣∣∣ω + kzv

ω − kzv

∣∣∣∣−− iπkzv

4

∂

∂ω

((1− ω2

k2zv2

)H

(1− ω

kzv

))]= −Πyx (14)

Πzz =−16π2e2ω2

k2z

∫ ∞

0

dpp2

v

∂fo∂ |p|

[1− ω

2kzvlog

∣∣∣∣ω + kzv

ω − kzv

∣∣∣∣+ iπω

2kzvH

(1− ω

kzv

)](15)

Where H (a− b) is the Heaviside Unit Step function. The above components forparallel propagation (i.e., Eqs.(13-15)) are generalized for any distribution function todescribe the waves in weakly magnetized plasma. In the high frequency limit i.e., ω > kzv,the dispersion relations of the ultra-relativistic18 and the relativistic19 weakly magnetizedplasma for parallel and perpendicular propagations have been discussed in detail. In thepresent investigation, we shall derive the parallel propagating dispersion relations in theopposite limit i.e., ω < kzv.

4

A. Fully relativistic parallel propagating modes for ω < k.v

Expanding the logarithmic expressions in Eqs.(13-15) in the limit ω < kzv, retaining

the terms up toω2

c2k2zand then employing relativistic Maxwell-Boltzman-Juttner distri-

bution function

fo =no

4πm3c3a

K2(a)Exp

(-a

√1 +

p2

m2c2

)(16)

we obtain the following components

Πxx = −ω2op

[a

ω2

c2k2z− iπω

4ckz

e−a

K2(a)

(2(1 + a)

a

)](17)

Πxy = iΩoω2op

[ω

c2k2z

(aK1(a)

K2(a)

)+ iπ

ω2

2c3k3z

(1 + a) e−a

K2(a)

](18)

Πzz = ω2op

ω2

c2k2z

[a + iπ

ω

2ckz

(2 + 2a+ a2

)e−a

aK2(a)

](19)

where we have used the integrals

∫ ∞

1

dγe- a γ(γ, γ2, γ2 − 1,

√γ2 − 1, γ

√γ2 − 1

)=

((1 + a2) e−a

a2,(2 + 2a+ a2)e−a

a3,2(1 + a)e−a

a3,K1(a)

a,K2(a)

a

)(20)

Here a = mc2/KT, K1(a) and K2(a) are the modified Bessel functions of first and second

kind respectively and ω2op =

4πnoe2

mand ωoc =

eBo

mcare the non-relativistic plasma and

cyclotron frequencies.

B. Longitudinal mode for ω < k.v

Longitudinal dispersion relation in the range of small phase velocities can be obtainedby using Eq. (19) in Eq. (12) given by

∈L= 1 +a ω2

op

c2k2z

[1 + iπ

ω

2ckz

(2 + 2a+ a2

)e−a

a2K2(a)

]= 0 (21)

We note that the relativistic longitudinal permittivity for ω < k.v gives screening effects

given by λ2D =

c2

a ω2op

and these effects increase as we move from weakly relativistic to

highly relativistic case.

C. Transverse waves for ω < k.v

The transverse electromagnetic waves in a homogeneous plasma can be obtained usingthe dispersion equation

k2 − ω2

c2∈T= 0 (22)

5

Using Eqs. (17) & ( 18) in Eq. (11), we obtain dispersion relations for R-& L-wavesgiven by

ω2 = c2k2z + ω2op

[a

ω2

c2k2z− iπω

4ckz

e−a

K2(a)

2(1 + a)

a

]±Ωoω

2op

[ω

c2k2z

(aK1(a)

K2(a)

)+ iπ

ω2

2c3k3z

(1 + a) e−a

K2(a)

](23)

or

(∈T )R,L ≡(c2k2zω2

)R,L

= 1− aω2op

c2k2z

[1± Ωo

ω

K1(a)

K2(a)

]+iπ

2

ω2op

c2k2z

e−a

K2(a)

(1 + a)

a

[ckzω

∓ aΩo

ckz

](24)

where the upper sign is for R-wave and the lower sign is for L-wave. In order to proceedfurther first we will discuss the relativistic transverse permittivity for field free case.

1. Field free case

The relativistic transverse permittivity is given by

∈T≡c2k2zω2

= 1− aω2op

c2k2z+

iπ

2

ω2op

ωckz

e−a

K2(a)

(1 + a)

a(25)

Using Eq. (25) in Eq. (22) we obtain

ω = −i2

π

c3k3zω2op

aK2(a)

e−a(1 + a)(26)

where the real part is neglected under the condition c2k2z > ω2, a ω2op.

From Eq.(26), we obtain the relation of relativistic skin depth λsk given by

(λsk)rel =1

Im(kz)= 2

(2

π

) 13(

c3

ωω2op

aK2(a)

e−a(1 + a)

) 13

(27)

where Im(kz) is known as the space scale of damping obtained from the same imaginaryparts of the two complex roots of Eq. (26).

The result for skin depth (λsk)rel Eq. (27) is relativistically generalized and reduces

to the results reported in Alexandrov2 for the non-relativistic case (a =mc2

T>> 1) i.e.,

(λsk)non−rel = 2

(2

π

) 13(c2vthωω2

op

) 13

(28)

and the ultra-relativistic case (a << 1) i.e.,

(λsk)ultra−rel =

(8

3π

c3

ω2pω

) 13

(29)

6

where we have used the expansions of Bessel functions in the two limiting cases as

For a > > 1, Kn(a) ≈√

π

2ae−a

a < < 1, Kn(a) ≈Γ(n)

2

(2

a

)n

(30)

ω2p =

4πe2Nec2

3Tebeing the ultra-relativistic electron plasma frequency and vth =

√Te

meis

the thermal velocity of electrons. We note that results in Eqs.(28 & 29) are in agreementwith the results reported earlier.2

2. R-& L-waves

In the presence of weak ambient magnetic field, the transverse permittivity under thecondition c2k2z > ω2, a ω2

op can be written as

(∈T )R,L ≡(c2k2zω2

)R,L

=iπ

2

ω2op

c2k2z

e−a

K2(a)

(1 + a)

a

[ckzω

∓ aΩo

ckz

](31)

From the above equation Eq.(31), we obtain the quintic equation in ckz as under

c5k5z −iπ

2ω2op

e−a(1 + a)

a K2(a)

(ω c2k2z ∓ a Ωoω

2)= 0 (32)

Treating the equation carefully, we observe that the weak magnetic field effects appear inthe space scale of damping. By switching off the magnetic field, we retrieve the roots of

the transverse cubic equation Eq. (26). We have plottedωopλsk

cvs

ω

ωopin Figs. (2, 3,

& 4) for right and left circularly polarized waves. The relativistic temperature variation

(for a =mc2

T= 1, 5, 10 ) in a weakly magnetized plasma (

ωoc

ωop= 0.1) is shown in Fig.(2 ),

while weak magnetic field effects (ωoc

ωop= 0.01, 0.1, 0.15) for strongly relativistic ( a = 0.1)

and weakly relativistic (a=10) case are shown in Figs. (3 &4) respectively. From Fig.2,it is observed that the skin depth for R-& L- waves shows the same relativistic behavioras observed in Fig. 1 for unmagnetized case. However, the effect of ambient magneticfield becomes different in different relativistic regimes. Fixing the inverse temperature

factor ”mc2

T”, we observe that variation of weakly magnetized parameter

ωoc

ωopshow a

small contribution in highly relativistic regime (a=0.1) as evident in Fig. (3). However inweakly relativistic regime (a=10), the situation become more clear as shown in Fig. (4).

By decreasing the strength of parameterωoc

ωopfrom 0.15 to 0.01, the skin effects increase

in R-wave and decrease in L-wave.

3. Limiting cases (Non-relativistic and Ultra-relativistic)

Applying the non relativistic limit (Eq. (30)), the Eqs. (21 & 31) reduce to

(∈L)Non−rel = 1 +ω2op

v2thk2z

(1 + i

√π

2

ω

vthkz

)= 0 (33)

7

and

(∈T )R,L ≡(c2k2zω2

)R,L

= i

√2

π

ω2op

v2thk2z

[vthkzω

∓ aΩo

vthkz

](34)

Similarly, in the ultra-relativistic limit (Eq. (30)), we obtain

K1(a)

aK2(a)=

1

2=

1

a2K2(a)

therefore Eqs. (21 & 31) take the form

(∈L)Ultra−rel = 1 +3ω2

p

c2k2z

(1 + iπ

ω

2ckz

)= 0 (35)

and

(∈T )R,L ≡(c2k2zω2

)R,L

=iπ

4

3ω2p

c2k2z

[ckzω

∓ ωc

ckz

](36)

where ωc =eBoc

Tis the ultra-relativistic cyclotron frequency. On neglecting the mag-

netic field, the expressions for longitudinal and transverse permittivities for both thenon-relativistic and ultra-relativistic cases (i.e., Eqs. (33-36)) are in agreement with theresults reported in.2

8

III. Summary and conclusion

Using the Vlasov-Maxwell’s model, we have derived the generalized polarization ten-sor for weakly magnetized electron plasma and discussed its components for parallel prop-agation. In particular, we have discussed the behavior of parallel propagation under thecondition ω < k.v in plasma. Using the relativistic distribution function, we have derivedthe longitudinal dispersion relation for Langmuir wave and the transverse dispersion rela-tions for R-& L-waves. In the field free case and under the condition c2k2z > ω2, a ω2

op, wehave obtained relativistic screening effects in the longitudinal permittivity and anomalousskin effects in the transverse permittivity.

Looking at the transverse expression in the presence of weak ambient magnetic field,we obtain a complex quintic equation in kz having one pure imaginary and four complexroots. The transverse skin depth obtained from the imaginary parts of the quintic equationhave additional weak magnetic field effects. The graphical representations exhibit thebehavior of relativistic effects and the weak field effects on the transverse skin depth.From the graphs, the skin depth goes on decreasing as one moves from fully relativistic

to weakly relativistic regime. Similarly fixing the inverse temperature factor ”mc2

T”, we

observe that variation of weakly magnetized parameterωoc

ωopshows small effect in the

highly relativistic regime. However in weakly relativistic case, the weak magnetic fieldeffects are pronounced in R-wave causing enhancement of the skin depth. By switchingoff the magnetic field, our results reduce to those reported earlier2 in the non-relativisticand ultra-relativistic limits.

Thus we conclude that the skin depth for R-& L -waves decreases as we move fromfully relativistic to weakly relativistic regime and that the ambient magnetic field enhancesthe skin effect for R- wave but reduces it for L- wave. Moreover the weak magnetic fieldeffects are more pronounced in the weakly relativistic regime than in other relativisticregimes.

Depending upon the physical scenario, our results have wide range of applicability toall temperatures ranging from keV to MeV and to the magnetized plasmas having ratioωoc

ωop< 1. These ranges can be found in a number of environments like gamma ray burst

afterglow plasmas,17 solar flares, solar corona, coronal outflows, and in the plasma sheets29

etc. In laboratory, these results can be applied to highly dense laser induced plasmas,30

radio frequency field penetration effects into the plasma31 and magnetically inductivelycoupled plasmas1 etc where weak magnetic field effects may play a significant role.

Acknowledgments

One author (G. A. ) gratefully acknowledges financial assistance from the ‘Office ofthe External Activities’, AS-ICTP, Trieste, Italy.

9

List Of Figures(i) Fig. 1. A plot of

λskωop

cvs

ω

ωpfor transverse wave at a =

moc2

T= 10 (solid line)

, 5 (large dashed line), 1 (small dashed line), 0.1 (dotted line) showing the relativisticdependence of penetration depth.

(ii) Fig. 2. A plot ofλskωop

cvs

ω

ωpillustrates the relativistic dependence of penetra-

tion depth for (i) R-wave and (ii) L-wave for α =ωoc

ωop= 0.1.

(iii) Fig. 3 A plot ofλskωop

cvs

ω

ωpshowing the weak magnetic field dependence

α =ωc

ωp= (0 (dotted), 0.01 (small-dashed), 0.1 (large-dashed, 0.15 (solid)) of penetration

depth for (i) R-wave and (ii) L-wave in highly relativistic regime a =moc

2

T= 0.1.

(iv) Fig 4: A plot ofλskωop

cvs

ω

ωpshowing the weak magnetic field dependence

α =ωc

ωp= (0 (dotted), 0.01 (small-dashed), 0.1 (large-dashed, 0.15 (solid)) of penetration

depth for (i) R-wave and (ii) L-wave in weakly relativistic case a = 10.

.

10

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