Braz. J. Phys. vol.34 número3B

(1)

The Effective Longitudinal Dielectric Constant for

Plasmas in Inhomogeneous Magnetic Fields

R. Gaelzer

1

_{, L. F. Ziebell}

2

_{, and R. S. Schneider}

2

1_{Instituto de F´ısica e Matem´atica, UFPel, Caixa Postal 354, 96010-900 Pelotas, RS, Brazil,} 2_{Instituto de F´ısica, UFRGS, Caixa Postal 15051, 91501-970 Porto Alegre, RS, Brazil}

Received on 26 January, 2004

We present a detailed derivation of the effective dielectric constant to be used in the dispersion relation for electrostatic waves in the case of a plasma immersed in a inhomogeneous magnetic field, with inhomogeneity perpendicular to the direction of the magnetic field.

1 Introduction

We have recently discussed the correct form of the disper-sion relation for electrostatic waves in inhomogeneous plas-mas, considering for simplicity the particular case in which the magnetic field is homogeneous and other plasma param-eters can be inhomogeneous, and deriving a general expres-sion for the dielectric constant, valid for arbitrary direction of propagation [1-3] The situation in which the magnetic field is homogeneous has been chosen for these studies be-cause it features a relatively simple geometry, which has been useful for the discussion of basic features, like the sym-metry properties of the effective dielectric tensor. More gen-eral situations which also feature inhomogeneity in the mag-netic field introduce considerable difficulty to the derivation of the effective dielectric tensor and to the effective dielec-tric constant, since the inhomogeneity in the magnetic field affects the resonance condition [4,5].

From the derivation presented in Refs. [3] and [2], it is important to remember here that the dispersion relation for electrostatic waves in inhomogeneous plasmas can be writ-ten in a general form which is well known from the literature [6],

k2εl−ik·

³

∇·↔ε´= 0, (1) where the↔ε is the dielectric tensor andεlis the longitudinal

dielectric constant, εl=

X

ij

kiεijkj

k2 . (2)

It is also important to remember here that it has been demonstrated that for an inhomogeneous plasma the dielec-tric properties are given by the so-calledeffective dielectric tensor, instead of the conventional dielectric tensor [2,7,8]. When studying electrostatic waves, therefore, Eq. (1) must be utilized along with an effective dielectric tensor and an effective dielectric constant, derived according to Eq. (2), with the components of the effective dielectric tensor ap-pearing in the right-hand side.

In the present paper we derive the effective dielectric constant considering the case of a magnetic field featuring perpendicular gradients, therefore complementing the for-mulation appearing in Refs. [2,3]. The geometry to be used is the same geometry which has been adopted in Ref. [5], which has been concerned with the general dispersion re-lation for electromagnetic waves. The derivation requires a considerable amount of algebraic work whose main steps are illustrated in the following sections. Due to the details which must be provided, we emphasize analytical features of the derivation, leaving detailed applications with numeri-cal results to forthcoming publications.

The structure of the paper is the following: In section 2 the formulation employed in Ref. [5] is adopted and applied to the derivation of a general expression for the effective di-electric constant, and in section III the general expression is particularized to the case in which the plasma particles possess a Maxwellian distribution function.

2 Derivation of the longitudinal

di-electric constant

Let us assume a plasma immersed into an inhomogeneous magnetic field, with weak inhomogeneity perpendicular to the direction ofB. Using Eq. (3) from Ref. [5]:

↔

ε=↔1 −iX

α

4πq2

α

mαω

∞

X

n→−∞

Z ∞

0 dτ

Z

d3u u⊥L(fα0)

×

e

iDnατ

[Fnα(τ)](|n|−1)

Π−

nαΠ+nα

(x−nx+n)|n|

−ezez

X

α

4πq2

α

mαω2

Z

d3_u uk

(2)

⌋

Π±nα=±

·

nJ|n|(x±n)G±nα(τ) +i

J|n|+1(x±n)

x±n

Fnα1/2(τ)bαsinψ

¸

ex

+i

·

|n|J|n|(xn±)G±nα(τ)∓

J|n|+1(x±n)

x±n

Fnα1/2(τ)(bαcosψ± Knτ)

¸

ey

+uk

u⊥

J|n|(x±n)Fnα1/2(τ)ez

⌈

G±

nα(τ) =i

s

Knτ∓bαcosψ+iSnbαsinψ

Knτ±bαcosψ+iSnbαsinψ

G+nαG−nα=−1

Fnα1/2G±nα=−(Knτ∓bαcosψ+iSnbαsinψ).

with

Fnα(τ) =b2α− K2nτ2−i2SnbαsinψKnτ

In these expressions we have

Dnα=γω−ckkuk−nΩα(1 +ǫBx)−ǫB

k⊥u2⊥c2 2Ωα

sinψ

bα=

k⊥u⊥c Ωα

L(fα0) =∂u⊥fα0− Nku⊥

γ L(fα0) L(fα0) =

uk

u⊥

∂u⊥fα0−∂ukfα0

The geometry utilized has been the following. The mag-netic field has been considered pointing in thez direction, and inhomogeneous in the x direction, B0 = B0(x)ez.

The waves were assumed propagating in arbitrary direc-tions, withkkandk⊥as the components of the wave vector

respectively parallel and perpendicular to the magnetic field. The wave angular frequency has been denoted asω. ψ de-notes the angle between the vectork⊥ and the direction of

the inhomogeneity. The inhomogeneity has been assumed to be weak, such that the cyclotron frequency has been written as

Ωα(x′α)∼Ωα(x)[1 +ǫB(x′α−x)], (4)

whereΩα(x) = (qαB0(x)/mαc)is the particle cyclotron

angular frequency,mαis the particle rest mass,qαis the

par-ticle charge,cis the velocity of light,ǫB ≡

h

1

B0

dB0

dx′ α

i

x′ α=x

, andx′

αis the unperturbed position of particleα.

We also used

u=p/(mαc) Sn=sign(n), Kn =nǫBcu⊥/2,

x±n =

£

b2α±2bαcosψKnτ+K2nτ2

¤1/2

The longitudinal dielectric constant can be obtained from the effective dielectric tensor as follows,

εl= 1−i

X

α

4πq2

α

mαωk2

∞

X

n→−∞

Z ∞

0 dτ

Z

d3u u⊥L(fα0)

×

e

iDnατ

[Fnα(τ)](|n|−1)

X

ij

kikjΠ−i Π+j

(x−nx+n)|n|

−k

2

k

k2

X

α

4πq2

α

mαω2

Z

d3u uk

γ L(fα0), (5) where we did not use the indexesnαin the componentsΠi,

for simplicity. We see that it is necessary to evaluate the following product,

⌋

X

i

kiΠ±i =kk

uk

u⊥

J|n|(x±n)Fnα1/2(τ)

±k⊥cosψ

·

nJ|n|(x±n)G±nα(τ) +i

J|n|+1(x±n)

x±n

Fnα1/2(τ)bαsinψ

¸

+ik⊥sinψ

h

|n|J|n|(x±n)G±nα(τ)

∓J|n|+1(x ±

n)

x±n

Fnα1/2(τ)(bαcosψ± Knτ)

¸

=±(Snk⊥cosψ±ik⊥sinψ)£|n|J|n|(x±n)G±nα(τ)

(3)

−ik⊥sinψ

·_J

|n|+1(x±n)

x±n

Fnα1/2(τ)Knτ

¸

+kk

uk

u⊥

J|n|(x±n)Fnα1/2(τ).

Therefore, the quantity of interest is given by

X

j

kjΠ+j

X

i

kiΠ−i

=n(Snk⊥cosψ+ik⊥sinψ)£|n|J|n|(x+n)G+nα(τ)

¤

−ik⊥sinψ

·_J

|n|+1(x+n)

x+n

Fnα1/2(τ)Knτ

¸

+kk

uk

u⊥

J|n|(x+n)Fnα1/2(τ)

¾

×©

−(Snk⊥cosψ−ik⊥sinψ)£|n|J|n|(x−n)G−nα(τ)

¤

−ik⊥sinψ

·_J

|n|+1(x−n)

x−n

Fnα1/2(τ)Knτ

¸

+kk

uk

u⊥

J|n|(x−n)Fnα1/2(τ)

¾

⌈

After some straightforward algebra, usingG+_G−₌₋₁

andF1/2_F1/2₌_F_{, we arrive to the following}

X

j

kjΠ+j

X

i

kiΠ−i =k2⊥n2J|n|(x+n)J|n|(x−n)

−ik⊥2 sinψ(Sncosψ+isinψ)

×

·_|_n_|_J

|n|(x+n)J|n|+1(x−n)

x−n

F1/2

nα (τ)G+nα(τ)Knτ

¸

+kkk⊥(Sncosψ+isinψ)

×uk

u⊥|n|J|n|(x

+

n)J|n|(x−n)Fnα1/2(τ)G+nα(τ)

+ik⊥2 sinψ(Sncosψ−isinψ)

×

·_|_n_|_J

|n|+1(x+n)J|n|(x−n)

x+n

Fnα1/2(τ)G−nα(τ)Knτ

¸

−k2⊥sin2ψ

·_J

|n|+1(x+n)J|n|+1(x−n)

x+nx−n

Fnα(τ)(Knτ)2

¸

−ikkk⊥sinψ

uk

u⊥

·_J

|n|+1(x+n)J|n|(x−n)

x+n

Fnα(τ)Knτ

¸

−kkk⊥(Sncosψ−isinψ)

×uk

u⊥

h

|n|J|n|(x+n)J|n|(x−n)Fnα1/2(τ)G−nα(τ)

i

−ikkk⊥sinψ

uk

u⊥

·_J

|n|(x+n)J|n|+1(x−n)

x−n

Fnα(τ)Knτ

¸

+k_k2u 2

k

u2

⊥

J|n|(x+n)J|n|(x−n)Fnα(τ)

Using now the expression for F1/2_G±_{, and separating}

the terms containing Sncosψfrom those with isinψ, we

obtain

X

j

kjΠ+j

X

i

kiΠ−i =k2⊥n2J|n|(x+n)J|n|(x−n)

+k2

k

u2

k

u2

⊥

+ik⊥2Snsinψcosψ

·_|_n_|_J

|n|(x+n)J|n|+1(x−n)

x−n

×(Knτ−bαcosψ+iSnbαsinψ)Knτ

i

+ik2

⊥Snsinψcosψ

·_|_n_|_J

|n|+1(x+n)J|n|(x−n)

x+n

×(−Knτ−bαcosψ−iSnbαsinψ)Knτ

i

+k2⊥sin2ψ

·_|_n_|_J

|n|(x+n)J|n|+1(x−n)

x−n

×(−Knτ+bαcosψ−iSnbαsinψ)Knτ

i

+k2⊥sin2ψ

·_|_n_|_J

|n|+1(x+n)J|n|(x−n)

x+n

i

−k2⊥sin2ψ

·_J

|n|+1(x+n)J|n|+1(x−n)

x+nx−n

Fnα(τ)(Knτ)2

¸

+2kkk⊥Sncosψ

uk

u⊥

|n|J|n|(x+n)J|n|(x−n)(bαcosψ)

−i2kkk⊥sinψ

uk

u⊥

|n|J|n|(x+n)J|n|(x−n)(Knτ+iSnbαsinψ)

−ikkk⊥sinψ

uk

u⊥

·_J

|n|+1(x+n)J|n|(x−n)

x+n

Fnα(τ)Knτ

(4)

−ikkk⊥sinψ

uk

u⊥

·_J

|n|(x+n)J|n|+1(x−n)

x−n

Fnα(τ)Knτ

¸

. (6) This expression must be used along with Eq. (5), which constitutes a general form of the longitudinal dielectric con-stant. An alternative form can be obtained by use of the

following property of Bessel functions,

Jn+1(x) =

µ_nJ

n(x)

x −J

′

n(x)

¶

.

Using this property, Eq. (6) can be written as follows,

⌋

X

ij

kikjΠ−i Π

+

j

k2 = k2

⊥

k2 n 2_J

|n|(x+n)J|n|(x−n)

+k

2

k

k2 u2

k

u2

⊥

J|n|(x+n)J|n|(x−n)Fnα(τ) +i

k2

⊥

k2Snsinψcosψ

×

"Ã

|n|2_J

|n|(x+n)J|n|(x−n)

x−nx−n

−|n|J|n|(x

+

n)J|′n|(x −

n)

x−n

!

×(Knτ−bαcosψ+iSnbαsinψ)(Knτ)

i

+ik 2

⊥

k2Snsinψcosψ

×

"Ã

|n|2_J

|n|(x+n)J|n|(x−n)

x+nx+n

−|n|J|n|(x −

n)J|′n|(x+n)

x+n

!

×(−Knτ−bαcosψ−iSnbαsinψ)(Knτ)

i

+k

2

⊥

k2 sin 2_ψ

"Ã

|n|2_J

|n|(x+n)J|n|(x−n)

x−nx−n

−|n|J|n|(x

+

n)J|′n|(x−n)

x−n

!

×(−Knτ+bαcosψ−iSnbαsinψ)(Knτ)

i

+k

2

⊥

k2 sin 2_ψ

"Ã

|n|2_J

|n|(x+n)J|n|(x−n)

x+nx+n

−|n|J|n|(x −

n)J|′n|(x+n)

x+n

!

×(−Knτ−bαcosψ−iSnbαsinψ)(Knτ)

i

−k

2

⊥

k2 sin 2_ψ

·

1

x+nx−n

µ_|_n_|_J

|n|(x+n)

x+n

−J|′n|(x+n)

¶

µ_|_n_|_J

|n|(x−n)

x−n

−J′ |n|(x−n)

¶

Fnα(τ)(Knτ)2

i

+2kkk⊥

k2 Sncosψ uk

u⊥|n|J|n|(x

+

n)J|n|(x−n)(bαcosψ)

−i2kkk⊥ k2 sinψ

uk

u⊥

|n|J|n|(x+n)J|n|(x−n)(Knτ+iSnbαsinψ)

−ikkk⊥ k2 sinψ

uk

u⊥

"Ã

|n|J|n|(xn+)J|n|(x−n)

x+nx+n

−J|n|(x −

n)J|′n|(x+n)

x+n

!

×Fnα(τ)(Knτ)

i

−ikkk⊥ k2 sinψ

uk

u⊥

"Ã

x−nx−n

−J|n|(x

+

n)J|′n|(x−n)

x−n

!

×Fnα(τ)(Knτ)

i

(5)

This is an alternative expression to be utilized instead of Eq. (6), for evaluation of the dielectric constant, given by Eq. (5).

In the homogeneous limit, Kn → 0, x±n → bα, and

Fnα → b2α. In that case it is easy to show that Eq. (7) is

reduced to the following expression,

X

ij

kikjΠ−i Π

+

j

k2 → b 2

αJ|2n|(bα)

·_k

⊥

k n bα

+kk

k uk

u⊥

¸2

and therefore, εl= 1 +

X

α

4πq2

α

mαω

∞

X

n→−∞

Z

d3_{u u}_⊥L(_f

α0)

× 1

DnαJ

2

|n|(bα)

·_k

⊥

k n bα+

kk

k uk

u⊥

¸2

−k

2

k

k2

X

α

4πq2

α

mαω2

Z

d3u uk

γ L(fα0), (8)

where we have performed the integration over the time vari-able,

Z ∞

0

dτ

e

iDnατ

= i

Dnα.

Equation (8) corresponds to the homogeneous contribu-tion appearing in Ref. [2]. It is easy to see that the same limit is obtained using the alternative form, based on Eq. (7).

Another interesting limit to be considered corresponds to the case of propagation parallel to the magnetic field, bα→0, where

x±n → |Kn|τ,

Fnα→ −(Knτ)2=−x+nx−n .

The Bessel functions become all independent of k⊥.

Therefore, for vanishingk⊥, all terms which have

combi-nations of Bessel functions multiplied byk⊥should vanish.

After a small amount of algebra, it is easy to show that Eq. (5) becomes

⌋

εl= 1−i

X

α

4πq2

α

mαω

∞

X

n→−∞ (−1)|n|

Z ∞

0 dτ

Z

d3u u⊥L(fα0)

×

e

iDnατu 2

k

u2

⊥

J|n|(x+n)J|n|(x−n)

−X

α

4πq2

α

mαω2

Z

d3uuk

γL(fα0). (9)

⌈

For propagation parallel to the direction of the inhomo-geneity,

x±n =

£

b2α±2bαKnτ+Kn2τ2

¤1/2

=£

(bα± Knτ)2¤

1/2

=|bα± Knτ|,

Fnα(τ) =b2α−(Knτ)2,

and

εl= 1−i

X

α

4πq2

α

mαω

∞

X

n→−∞

Z ∞

0 dτ

Z

d3u u⊥L(fα0)

×

e

iDnατ[Fnα(τ)] (|n|−1)

(x−nx+n)|n|

n2J|n|(x+n)J|n|(x−n). (10)

The last limiting case to be considered is the case of propagation perpendicular to the direction of the magnetic field and of the inhomogeneity.

x±n =

£

b2α+K2nτ2

¤1/2

,

Fnα(τ) =b2α−(Knτ)2−i2Snbα(Knτ),

The dielectric constant in these cases is given by Eq. (5), with

X

ij

kikjΠ−i Π+j

k2 = n 2_J

|n|(x+n)J|n|(x−n)

−

"Ã

|n|2_J

|n|(x+n)J|n|(x−n)

x−nx−n

−|n|J|n|(x

+

n)J|′n|(x−n)

x−n

!

×(Knτ+iSnbα)(Knτ)

i

−

"Ã

|n|2_J

|n|(x+n)J|n|(x−n)

x+nx+n

−|n|J|n|(x −

n)J|′n|(x

+

n)

x+n

!

×(Knτ+iSnbα)(Knτ)

i

−

· ₁

x+nx−n

µ_|_n_|_J

|n|(x+n)

x+n

−J|′n|(x+n)

¶

×

µ_|_n_|_J

|n|(x−n)

x−n

−J|′n|(x−n)

¶

Fnα(τ)(Knτ)2

¸

(6)

In the same limit, an alternative expression can be ob-tained using Eq. (7), resulting the following,

X

ij

kikjΠ−i Π+j

k2 → n 2_J

|n|(x+n)J|n|(x−n)

−

·_|_n_|_J

|n|(x+n)J|n|+1(x−n)

x−n

(Knτ+iSnbα)(Knτ)

¸

−

·_|_n_|_J

|n|+1(x+n)J|n|(x−n)

x+n

(Knτ+iSnbα)(Knτ)

¸

−

·_J

|n|+1(x+n)J|n|+1(x−n)

x+nx−n

Fnα(τ)(Knτ)2

¸

3 The dielectric constant for the case

of a Maxwellian distribution

func-tion

Let us assume a Maxwellian distribution function for parti-cles of typeα,

fα0=

nαµ3α/2

(2π)3/2e

−µαu 2

/2_, _µ

α=

mαc2

Tα

(11) ∂fα0

∂u⊥

=−µαu⊥fα0

Choosing the coordinate system in order to havex= 0, and considering, for simplicity, the non-relativistic limit, we obtain

εl= 1 +i

X

α

ω2

α

ω µ5α/2

(2π)1/2

∞

X

n→−∞

Z ∞

0

dτ

e

i(ω−nΩα)τ

×

Z ∞

−∞

duke−µαu

2

k/2_e−ickkukτ

×

Z ∞

0

du⊥u3⊥e−µ

αu 2

⊥/2_e−iǫBk⊥u 2 ⊥c

2

sinψτ /(2Ωα)

×[Fnα(τ)]

(|n|−1)

(x−nx+n)|n|

X

ij

kikjΠ−i Π

+

j

k2 . (12) The quantitiesx±

n andFnαdo not depend on the

paral-lel momentum, which only appears in the productΠ−_i Π+_j. Therefore we are left with the following integrals over par-allel momentum,

Ij =

Z ∞

−∞

duke−µαu

2

k/2_e−ickkukτ₍_u_k)j_,

withj = 0,1,2. These can be easily performed, with the following results,

j= 0 : I0=

(2π)1/2 µ1α/2

e−c2k2kτ 2

/(2µα)

j= 1 : I1=−i

(2π)1/2 µ1α/2

ckkτ

µα

e−c2k2kτ 2

/(2µα)

j= 2 : I2=

(2π)1/2 µ1α/2

1

µα

Ã

1−c

2_k2

kτ2

µα

!

e−c2k2kτ 2

/(2µα)

(13) Using these results,

εl= 1 +i

X

α

ω2

α

ω µ5α/2

(2π)1/2

(2π)1/2 µ1α/2

× ∞

X

n→−∞

Z ∞

0

dτ

e

i(ω−nΩα)τ_e−c 2

k2 kτ

2

/(2µα)

×

Z ∞

0

du⊥u3⊥e−µ

αu 2

⊥/2_e−iǫBk⊥u 2 ⊥c

2

sinψτ /(2Ωα)

×[Fnα(τ)]

(|n|−1)

(x−nx+n)|n|

Λ, (14)

where

⌋

Λ = k

2

⊥

k2 n 2_J

|n|(x+n)J|n|(x−n)

+k

2

k

k2

1

u2

⊥ 1

µα

Ã

1−c

2_k2

kτ2

µα

!

+ik 2

⊥

k2Snsinψcosψ

·_|_n_|_J

|n|(x+n)J|n|+1(x−n)

x−n

×(Knτ−bαcosψ+iSnbαsinψ)Knτ

i

+ik 2

⊥

k2Snsinψcosψ

·_|_n_|_J

|n|+1(x+n)J|n|(x−n)

x+n

(7)

+k

2

⊥

k2 sin 2_ψ

·_|_n_|_J

|n|(x+n)J|n|+1(x−n)

x−n

×(−Knτ+bαcosψ−iSnbαsinψ)Knτ

i

+k

2

⊥

k2 sin 2_ψ

·_|_n_|_J

|n|+1(x+n)J|n|(x−n)

x+n

i

−k

2

⊥

k2 sin 2_ψ

·_J

|n|+1(x+n)J|n|+1(x−n)

x+nx−n

Fnα(τ)(Knτ)2

¸

+2kkk⊥

k2 Sncosψ

1

u⊥

µ

−ickkτ µα

¶

×|n|J|n|(x+n)J|n|(x−n)(bαcosψ)

1

u⊥

µ

−ickkτ µα

¶

×(Knτ+iSnbαsinψ)

−ikkk⊥ k2 sinψ

1

u⊥

µ

−ickkτ µα

¶ ·_J

|n|+1(x+n)J|n|(x−n)

x+n

Fnα(τ)Knτ

¸

−ikkk⊥ k2 sinψ

1

u⊥

µ

−ickkτ µα

¶ ·_J

|n|(x+n)J|n|+1(x−n)

x−n

Fnα(τ)Knτ

¸

⌈

In order to simplify the ensuing calculations, we intro-duce the following notation,

x±n ≡y±nu⊥,

Fnα(τ)≡fnα(τ)u2⊥, (15)

where y±n =

£

c2k2⊥/Ω2α±2(ck⊥/Ωα) cosψ(ncǫB/2)τ

+(n2ǫ2Bc2/4)τ2

¤1/2

, fnα(τ) =

£

c2k2⊥/Ω2α−(n2ǫ2Bc2/4)τ2

−i2(ck⊥/Ωα) sin(ψ)(|n|cǫB/2)τ)] .

Moreover, we define N⊥=

ck⊥

ω , Nk= ckk

ω , NB = cǫB

ω , Yα=

Ωα

ω , (16) and

ζα=NBN⊥sinψ/Yα, t= (ωτ /µα). (17)

Using this notation,

Knτ= (nNBµαt/2)u⊥, bα= (N⊥/Yα)u⊥,

and the dielectric constant becomes the following

εl= 1+i

X

α

ω2

α

ω2µ 3

α

∞

X

n→−∞

Z ∞

0

dt

e

iµα(1−nYα)t e−µαN

2 kt

2

/2

×[fnα(τ)]

(|n|−1)

(yn−yn+)|n|

Z ∞

0

du⊥u⊥e−(µαu

2

⊥/2)[1+iζαt]

Λ, (18) where

Λ = k

2

⊥

k2 n 2_J

|n|(x+n)J|n|(x−n)

+k

2

k

k2

1

µα

³

1−µαNk2t

2´_J

|n|(x+n)J|n|(x−n)fnα(τ)

+ik 2

⊥

k2Snsinψcosψ

·_|_n_|_J

|n|(x+n)J|n|+1(x−n)

y−n

×((nNBµαt/2)−(N⊥/Yα) cosψ

+iSn(N⊥/Yα) sinψ) (nNBµαt/2)u⊥

i

+ik 2

⊥

k2Snsinψcosψ

·_|_n_|_J

|n|+1(x+n)J|n|(x−n)

y+n

×(−(nNBµαt/2)−(N⊥/Yα) cosψ

−iSn(N⊥/Yα) sinψ) (nNBµαt/2)u⊥

i

+k

2

⊥

k2 sin

2_ψ·|n|J|n|(xn+)J|n|+1(x−n)

yn−

×(−(nNBµαt/2) + (N⊥/Yα) cosψ

(8)

+k

2

⊥

k2 sin 2_ψ

·_|_n_|_J

|n|+1(x+n)J|n|(x−n)

yn+

×(−(nNBµαt/2)−(N⊥/Yα) cosψ

i

−k

2

⊥

k2 sin 2_ψ

·_J

|n|+1(x+n)J|n|+1(x−n)

yn+yn−

×fnα(τ) (nNBµαt/2)2u2⊥

i

+2kkk⊥

k2 Sncosψ

¡

−iNkt

¢

×((N⊥/Yα) cosψ)

¡

−iNkt

¢

×((nNBµαt/2) +iSn(N⊥/Yα) sinψ)

−ikkk⊥ k2 sinψ

¡

−iNkt

¢ ·_J

|n|+1(x+n)J|n|(x−n)

y+n

×fnα(τ) (nNBµαt/2)u⊥

i

−ikkk⊥ k2 sinψ

¡

−iNkt

¢ ·_J

|n|(x+n)J|n|+1(x−n)

y−n

×fnα(τ) (nNBµαt/2)u⊥

i

.

In this expression we can find the following integrals over the perpendicular variable,

I⊥1=

Z ∞

0

du⊥u⊥e−(µαu

2

⊥/2)[1+iζαt]

J|n|(x+n)J|n|(x−n)

I_⊥±₂=

Z ∞

0

du⊥u2⊥e−(µ

αu 2

⊥/2)[1+iζαt]_J

|n|(x∓n)J|n|+1(x±n)

I⊥3=

Z ∞

0

du⊥u3⊥e−(µαu

2

⊥/2)[1+iζαt]_J

|n|+1(x+n)J|n|+1(x−n)

(19) These integrals can be easily solved, and are given by simple expressions involving the modified Bessel function In[9]

⌋

I⊥1 =

e

−(ν2 α+χ

2 nαt

2

)/(1+iζαt) µα(1 +iζαt)

I|n|

µ

Snα(t)

1 +iζαt

¶

I_⊥±₂ =

e

−(ν2

α+χ 2 nαt

2

)/(1+iζαt) µ3α/2(1 +iζαt)2

×

·

Tn±I|n|

µ _S

nα(t)

1 +iζαt

¶

−Tn∓I|n|+1

µ _S

nα(t)

1 +iζαt

¶¸

(20)

I⊥3 = 2

e

−(ν2 α+χ

2 nαt

2

)/(1+iζαt) µ2

α(1 +iζαt)2

· _S

nα(t)

1 +iζαt

I|n|

µ _S

nα(t)

1 +iζαt

¶

−

µ

|n|+ν

2

α+χ2nαt2

1 +iζαt

¶

I|n|+1

µ

Snα(t)

1 +iζαt

¶¸

, (21)

⌈

where T±

n =

p

ν2

α±2ναcosψχnαt+χ2nαt2

Snα=

p

ν4

α−2να2cos(2ψ)χ2nαt2+χ4nαt4,

χnα=

nNBµ1α/2

2 ,

να=

N⊥

µ1α/2Yα

= k⊥vα

Ωα , vα=

µ_T

α

mα

¶1/2

.

Using the symbols introduced by Eqs. (19), and the no-tation introduced by Eqs. (21), we can write

fnα=µαHnα(t),

where

Hnα(t) =να2−i2Snναsin(ψ)χnαt−χ2nαt2, (22)

and

yn±=µ1α/2Tn±, (23)

y+

ny−n =µαSnα(t).

The dielectric constant therefore can be written as fol-lows,

εl= 1+i

X

α

ω2

α

ω2µ 2

α

∞

X

n→−∞

Z ∞

0

dt

e

iµα(1−nYα)t_e−µαN 2 kt

2

/2

×[Hnα(t)]

(|n|−1)

(Snα(t))|n|

Λ∗, (24)

where

Λ∗₌ k2⊥

k2 n 2_I

⊥1+ k2

k

k2

³

1−µαNk2t2

´

(9)

+ik 2

⊥

k2Snsinψcosψ

·_|_n_|

Tn−

I_⊥−₂(χnαt−ναcosψ

+iSnναsinψ)

³

µ1α/2χnαt

´i

+ik 2

⊥

k2Snsinψcosψ

·

|n|

Tn+

I_⊥+₂(−χnαt−ναcosψ

−iSnναsinψ)

³

µ1α/2χnαt

´i

+k

2

⊥

k2 sin 2_ψ

·

|n|

Tn−

I_⊥−₂(−χnαt+ναcosψ

−iSnναsinψ)

³

µ1α/2χnαt

´i

+k

2

⊥

k2 sin 2_ψ

·_|_n_|

Tn+

I_⊥+₂(−χnαt−ναcosψ

−iSnναsinψ)

³

µ1α/2χnαt

´i

−k

2

⊥

k2 sin

2_ψ·Hnα(t)

Snα(t)

I⊥3¡µαχ2nαt2

¢ ¸

+2kkk⊥

k2 Sncosψ

¡

−iNkt

¢

|n|I⊥1(µα1/2ναcosψ)

¡

−iNkt

¢

µ1α/2|n|I⊥1(χnαt+iSnναsinψ)

−ikkk⊥ k2 sinψ

¡

−iNkt¢µα

·_H

nα(t)

Tn+

I_⊥+₂χnαt

¸

−ikkk⊥ k2 sinψ

¡

−iNkt¢µα

·_H

nα(t)

Tn−

I_⊥−₂χnαt

¸

Using the integrals given by Eqs. (21),

⌋

εl= 1 +i

X

α

ω2

α

ω2µ 2

α

∞

X

n→−∞

Z ∞

0

dt

e

2 kt

2

/2

×

e

−(ν2

α+χ 2 nαt

2

[Hnα(t)](|n|−1)

(Snα(t))|n|

Λ∗, (25)

where

Λ∗=k

2

⊥

k2 n 2_I

|n|

µ _S

nα(t)

1 +iζαt

¶

+k

2

k

k2

³

1−µαNk2t

2´_I

|n|

µ _S

nα(t)

1 +iζαt

¶

Hnα(t)

+ik 2

⊥

k2Sn

sinψcosψ

(1 +iζαt)

×

·_|_n_|

Tn−

·

Tn−I|n|

µ _S

nα(t)

1 +iζαt

¶

−Tn+I|n|+1

µ _S

nα(t)

1 +iζαt

¶¸

×(χnαt−ναcosψ+iSnναsinψ) (χnαt)]

+ik 2

⊥

k2Sn

sinψcosψ

(1 +iζαt)

×

·_|_n_|

Tn+

·

T+

nI|n|

µ _S

nα(t)

1 +iζαt

¶

−T−

nI|n|+1

µ _S

nα(t)

1 +iζαt

¶¸

×(−χnαt−ναcosψ−iSnναsinψ) (χnαt)]

+k

2

⊥

k2

sin2ψ

(1 +iζαt)

×

·_|_n_|

Tn−

·

Tn−I|n|

µ _S

nα(t)

1 +iζαt

¶

−Tn+I|n|+1

µ _S

nα(t)

1 +iζαt

¶¸

×(−χnαt+ναcosψ−iSnναsinψ) (χnαt)]

+k

2

⊥

k2

sin2ψ

(1 +iζαt)

×

·

|n|

Tn+

·

Tn+I|n|

µ

Snα(t)

1 +iζαt

¶

−Tn−I|n|+1

µ

Snα(t)

1 +iζαt

¶¸

(10)

−k

2

⊥

k2

2 sin2ψ

(1 +iζαt)

×

·_H

nα(t)

Snα(t)

¡

χ2nαt2

¢ ¸ · _S

nα(t)

1 +iζαt

I|n|

µ _S

nα(t)

1 +iζαt

¶

−

µ

|n|+ν

2

α+χ2nαt2

1 +iζαt

¶

I|n|+1

µ _S

nα(t)

1 +iζαt

¶¸

−2kkk⊥

k2

¡

Nkt¢|n|µ1α/2[iSnνα+ sinψχnαt]I|n|

µ _S

nα(t)

1 +iζαt

¶

−ikkk⊥ k2

sinψ¡

−iNkt

¢

(1 +iζαt)

µ1α/2

×

·_H

nα(t)

Tn+

·

T+

nI|n|

µ _S

nα(t)

1 +iζαt

¶

−T−

nI|n|+1

µ _S

nα(t)

1 +iζαt

¶¸

χnαt

¸

−ikkk⊥ k2

sinψ¡

−iNkt

¢

(1 +iζαt)

µ1α/2

×

·_H

nα(t)

Tn−

·

Tn−I|n|

µ _S

nα(t)

1 +iζαt

¶

−Tn+I|n|+1

µ _S

nα(t)

1 +iζαt

¶¸

χnαt

¸

, (26)

where we have taken into account the following

2kkk⊥

k2 Sncosψ

¡

−iNkt

¢

|n|(µ1α/2ναcosψ)I|n|

µ

Snα(t)

1 +iζαt

¶

−2kkk⊥

k2 sinψ

¡

Nkt¢|n|

×(µ1α/2χnαt+iSnµα1/2ναsinψ)I|n|

µ _S

nα(t)

1 +iζαt

¶

= 2kkk⊥

k2

¡

Nkt

¢

|n|µ1/2

α

£

−iSnνα(cos2ψ+ sin2ψ)

−sinψχnαt]I|n|

µ _S

nα(t)

1 +iζαt

¶

=−2kkk⊥

k2

¡

Nkt

¢

|n|µ1α/2[iSnνα+ sinψχnαt]

×I|n|

µ _S

nα(t)

1 +iζαt

¶

.

We now introduce the definition of the ”inhomogeneous plasma dispersion function”, for the non-relativistic case [5]

Gr,p,m,l=−i

Z ∞

0

dt(it)

r

_e

2

/2

(1 +iζαt)p

×

e

−(ν2 α+χ

2 nαt

2

)/(1+iζαt)[Hnα(t)]

m

(Snα(t))l

Il

µ

Snα(t)

1 +iζαt

¶

(27) Eq. (26) can be prepared for the use of the inhomogeneous plasma dispersion function, as follows,

Λ∗= k

2

⊥

k2 n 2_I

|n|

µ _S

nα(t)

1 +iζαt

¶

+k

2

k

k2

³

1 +µαNk2(it)2

´

I|n|

µ _S

nα(t)

1 +iζαt

¶

Hnα(t)

+k

2

⊥

k2Sn|n|χnα

sinψcosψ

(1 +iζαt)I|n|

µ _S

nα(t)

1 +iζαt

(11)

×(−iχnαit−ναcosψ+iSnναsinψ) (it)

−k

2

⊥

k2Sn|n|χnα

sinψcosψ

(1 +iζαt)

T+

n

Tn−

I|n|+1

µ _S

nα(t)

1 +iζαt

¶

+k

2

⊥

k2Sn|n|χnα

sinψcosψ

(1 +iζαt)

I|n|

µ _S

nα(t)

1 +iζαt

¶

×(iχnαit−ναcosψ−iSnναsinψ) (it)

−k

2

⊥

k2Sn|n|χnα

sinψcosψ

(1 +iζαt)

T−

n

Tn+

I|n|+1

µ _S

nα(t)

1 +iζαt

¶

−ik 2

⊥

k2|n|χnα

sin2ψ

(1 +iζαt)

I|n|

µ _S

nα(t)

1 +iζαt

¶

×(iχnαit+ναcosψ−iSnναsinψ) (it)

+ik 2

⊥

k2|n|χnα

sin2ψ

(1 +iζαt)

T+

n

Tn−

I|n|+1

µ _S

nα(t)

1 +iζαt

¶

−ik 2

⊥

k2|n|χnα

sin2ψ

(1 +iζαt)

I|n|

µ

Snα(t)

1 +iζαt

¶

+ik 2

⊥

k2|n|χnα

sin2ψ

(1 +iζαt)

T−

n

Tn+

I|n|+1

µ

Snα(t)

1 +iζαt

¶

+k

2

⊥

k2χ 2

nα

2 sin2ψ

(1 +iζαt)

·

Hnα(t)

Snα(t)

(it)2

¸ ·

Snα(t)

1 +iζαt

I|n|

µ

Snα(t)

1 +iζαt

¶

−

µ

|n|+ν

2

α−χ2nα(it)2

1 +iζαt

¶

I|n|+1

µ _S

nα(t)

1 +iζαt

¶¸

−2kkk⊥

k2 Nk(it)|n|µ 1/2

α [Snνα−sinψχnα(it)]I|n|

µ

Snα(t)

1 +iζαt

¶

+kkk⊥

k2 Nkχnαµ 1/2

α

sinψ

(1 +iζαt)

Hnα(t)I|n|

µ _S

nα(t)

1 +iζαt

¶

(it)2

−kkk⊥

k2 Nkχnαµ 1/2

α

sinψ

(1 +iζαt)

T−

n

Tn+

Hnα(t)I|n|+1

µ _S

nα(t)

1 +iζαt

¶

(it)2

+kkk⊥

k2 Nkχnαµ 1/2

α

sinψ

(1 +iζαt)

Hnα(t)I|n|

µ

Snα(t)

1 +iζαt

¶

(it)2

−kkk⊥

k2 Nkχnαµ 1/2

α

sinψ

(1 +iζαt)

T+

n

Tn−

Hnα(t)I|n|+1

µ _S

nα(t)

1 +iζαt

¶

(it)2

Using the definition of the inhomogeneous dispersion function, we obtain the following

εl= 1−

X

α

ω2

α

ω2µα

∞

X

n→−∞

½_k2

⊥

k2 n 2_G

0,1,|n|−1,|n|

+k

2

k

k2

³

G0,1,|n|,|n|+µαNk2G2,1,|n|,|n|

´

+k

2

⊥

k2Sn|n|χnαsinψcosψ

¡

(12)

−ναcosψG1,2,|n|−1,|n|+iSnναsinψG1,2,|n|−1,|n|

¢

+k

2

⊥

k2Sn|n|χnαsinψcosψ

¡

iχnαG2,2,|n|−1,|n|

−ναcosψG1,2,|n|−1,|n|−iSnναsinψG1,2,|n|−1,|n|¢

−ik 2

⊥

k2|n|χnαsin 2_ψ¡

iχnαG2,2,|n|−1,|n|

+ναcosψG1,2,|n|−1,|n|−iSnναsinψG1,2,|n|−1,|n|

¢

−ik 2

⊥

k2|n|χnαsin 2_ψ¡

iχnαG2,2,|n|−1,|n|

−ναcosψG1,2,|n|−1,|n|−iSnναsinψG1,2,|n|−1,|n|¢

+2k

2

⊥

k2χ 2

nαsin2ψ

£

G2,3,|n|,|n|− |n|G2,2,|n|,|n|+1

−ν2αG2,3,|n|,|n|+1+χ2nαG4,3,|n|,|n|+1¤

−2kkk⊥

k2 Nk|n|µ 1/2

α

£

SnναG1,1,|n|−1,|n|−sinψχnαG2,1,|n|−1,|n|

¤

+kkk⊥

k2 Nkχnαµ 1/2

α sinψG2,2,|n|,|n|

+kkk⊥

k2 Nkχnαµ 1/2

¾

+iX

α

ω2

α

ω2µ 2

α

∞

X

n→−∞

Z ∞

0

dt

e

2 kt

2

/2

×

e

−(ν2

α+χ 2 nαt

2

[Hnα(t)](|n|−1)

(Snα(t))|n|

Λ†

where

Λ†₌₋k⊥2

k2Sn|n|χnα

sinψcosψ

(1 +iζαt)

T+

n

Tn−

I|n|+1

µ _S

nα(t)

1 +iζαt

¶

−k

2

⊥

k2Sn|n|χnα

sinψcosψ

(1 +iζαt)

T−

n

Tn+

I|n|+1

µ

Snα(t)

1 +iζαt

¶

+ik 2

⊥

k2|n|χnα

sin2ψ

(1 +iζαt)

T+

n

Tn−

I|n|+1

µ _S

nα(t)

1 +iζαt

¶

+ik 2

⊥

k2|n|χnα

sin2ψ

(1 +iζαt)

T−

n

Tn+

I|n|+1

µ _S

nα(t)

1 +iζαt

¶

−kkk⊥

k2 Nkχnαµ 1/2

α

sinψ

(1 +iζαt)

T−

n

Tn+

Hnα(t)

×I|n|+1

µ

Snα(t)

1 +iζαt

¶

(it)2

−kkk⊥

k2 Nkχnαµ 1/2

α

sinψ

(1 +iζαt)

T+

n

Tn−

Hnα(t)

×I|n|+1

µ _S

nα(t)

1 +iζαt

¶

(13)

This expression can be somewhat simplified, by collecting together some terms and by cancelling others

εl= 1−

X

α

ω2

α

ω2µα

∞

X

n→−∞

½_k2

⊥

k2 n 2_G

0,1,|n|−1,|n|

+k

2

k

k2

³

G0,1,|n|,|n|+µαNk2G2,1,|n|,|n|

´

−2k

2

⊥

k2nναχnαsinψG1,2,|n|−1,|n|

+2k

2

⊥

k2χ 2

nαsin2ψ

£

G2,3,|n|,|n|+|n|G2,2,|n|−1,|n|

−|n|G2,2,|n|,|n|+1−να2G2,3,|n|,|n|+1+χ2nαG4,3,|n|,|n|+1

¤

−2kkk⊥

k2 Nk|n|µ 1/2

α

£

SnναG1,1,|n|−1,|n|−sinψχnαG2,1,|n|−1,|n|¤

+2kkk⊥

k2 Nkχnαµ 1/2

¾

+iX

α

ω2

α

ω2µ 2

α

∞

X

n→−∞

Z ∞

0

dt

e

2

/2

×

e

−(ν2

α+χ 2 nαt

2

[Hnα(t)](|n|−1)

(Snα(t))|n|

Λ†. (28)

Let us examine the terms appearing in the expression forΛ†_{. Initially we consider the following terms,} −k

2

⊥

k2Sn|n|

sinψcosψ

(1 +iζαt)

T+

n

Tn−

I|n|+1

µ _S

nα(t)

1 +iζαt

¶

×(−iχnαit−ναcosψ+iSnναsinψ) (χnαit)

−k

2

⊥

k2Sn|n|

sinψcosψ

(1 +iζαt)

T−

n

Tn+

I|n|+1

µ

Snα(t)

1 +iζαt

¶

×(iχnαit−ναcosψ−iSnναsinψ) (χnαit)

=−k

2

⊥

k2Sn|n|

sinψcosψ

(1 +iζαt)

I|n|+1

µ _S

nα(t)

1 +iζαt

¶

(χnαit)

×

·

−iχnαit

µ_T+

n

Tn−

−T −

n

Tn+

¶

−ναcosψ

µ

T+

n

Tn−

+T −

n

Tn+

¶

+iSnναsinψ

µ

T+

n

Tn−

−T −

n

Tn+

¶¸

. (29)

Now we may consider the following, T−

n

Tn+

= T −

nTn+

Tn+Tn+

=

³_p

ν2

α+ 2ναcosψχnαt+χ2nαt2

´

³_p

ν2

´

×

³_p

ν2

α−2ναcosψχnαt+χ2nαt2

´

³_p

ν2

´

=

p

ν4

α+χ4nαt4−2να2cos(2ψ)χ2nαt2

ν2

= Snα(t)

|ν2

α+ 2ναcosψχnαt+χ2nαt2|

(14)

This expression can be inverted, and we obtain, T+

n

Tn−

=ν

2

Snα(t)

. (30)

Multiplying numerator and denominator byT−

n,

T−

n

Tn+

=T −

nTn−

Tn+Tn−

=

³_p

ν2

´

³_p

ν2

´

×

³_p

ν2

´

³_p

ν2

´

= ν

2

p

ν4

α+χ4nαt4−2να2cos(2ψ)χ2nαt2

= ν

2

Snα(t)

(31) Consider now Eq. (30),

T+

n

Tn−

=ν

2

Snα(t)

= Hnα(t)

Snα(t)

+2ναχnαt

Snα(t)

(cosψ+iSnsinψ) +

2χ2

nαt2

Snα(t)

.

Taking into account the following property, e±iSnψ_{= cos(}_S

nψ)±isin(Snψ) = cos(ψ)±iSnsin(ψ),

we obtain,

T+

n

Tn−

=Hnα(t)

Snα(t) +

2ναχnαt

Snα(t) e iSnψ

+2χ

2

nαt2

Snα(t).

(32) In analogous way,

T−

n

Tn+

=ν

2

Snα(t)

= Hnα(t)

Snα(t)

−2ναχnαt

Snα(t)

(cosψ−iSnsinψ) +

2χ2

nαt2

Snα(t)

=Hnα(t)

Snα(t) −

2ναχnαt

Snα(t) e

−iSnψ

+2χ

2

nαt2

Snα(t).

Collecting together these results, T±

n

Tn∓

= Hnα(t)

Snα(t)

±2ναχnαt

Snα(t)

e±iSnψ

+2χ

2

nαt2

Snα(t)

. (33)

Therefore, we have,

µ

T+

n

Tn−

−T −

n

Tn+

¶

=2ναχnαt

Snα(t)

¡

eiSnψ

+e−iSnψ¢

=−i4ναχnα(it) Snα(t)

cosψ (34)

µ

T+

n

Tn−

+T −

n

Tn+

¶

= 2Hnα(t)

Snα(t)

+4χ

2

nα(t)2

Snα(t)

+2ναχnαt

Snα(t)

¡

eiSnψ

(15)

= 2

Snα(t)

£

Hnα(t) + 2χ2nα(t)2+iSn2ναχnαtsinψ¤

= 2

Snα(t)

£

να2+χ2nαt2

¤

. (35)

Using these results, a small amount of algebra transforms Eq. (29) into the following

2k

2

⊥

k2n

sinψcos2_ψ

(1 +iζαt) I|n|+1

µ _S

nα(t)

1 +iζαt

¶

(ναχnαit)

Hnα(t)

Snα(t).

(36)

The next two terms in the expression forΛ†_{are the following}

ik 2

⊥

k2

sin2ψ

(1 +iζαt)

|n|T

+

n

Tn−

I|n|+1

µ _S

nα(t)

1 +iζαt

¶

×(iχnαit+ναcosψ−iSnναsinψ) (χnαit)

+ik 2

⊥

k2

sin2ψ

(1 +iζαt)

|n|T −

n

Tn+

I|n|+1

µ _S

nα(t)

1 +iζαt

¶

×(iχnαit−ναcosψ−iSnναsinψ) (χnαit)

=ik 2

⊥

k2

sin2ψ

(1 +iζαt)

|n|I|n|+1

µ

Snα(t)

1 +iζαt

¶

(χnαit)

×

·

iχnα(it)

µ_T+

n

Tn−

+T −

n

Tn+

¶

+ναcosψ

µ_T+

n

Tn−

−T −

n

Tn+

¶

−iSnναsinψ

µ_T+

n

Tn−

+T −

n

Tn+

¶¸

= 2k

2

⊥

k2

sin2ψ

(1 +iζαt)

|n|I|n|+1

µ

Snα(t)

1 +iζαt

¶

(χnαit)

×Hnα(t)

Snα(t)

[χnα(it) +Snναsinψ]. (37)

The last two terms inΛ†_{can be written as follows,} −kkk⊥

k2

sinψNk (1 +iζαt)

µ1α/2χnαHnα(t)T

−

n

Tn+

I|n|+1

µ

Snα(t)

1 +iζαt

¶

(it)2

−kkk⊥

k2

sinψNk (1 +iζαt)

µ1α/2χnαHnα(t)

T+

n

Tn−

I|n|+1

µ _S

nα(t)

1 +iζαt

¶

(it)2

=−kkk⊥

k2

sinψNk (1 +iζαt)

µ1α/2χnαHnα(t)I|n|+1

µ _S

nα(t)

1 +iζαt

¶

(it)2

×

µ_T+

n

Tn−

+T −

n

Tn+

¶

=−2kkk⊥

k2

sinψNk (1 +iζαt)

µ1/2

α χnαI|n|+1

µ _S

nα(t)

1 +iζαt

¶

(it)2

×Hnα(t)

Snα(t)

£

να2−χ2nα(it)2

¤

. (38)

Using Eqs. (36), (37) and (38),

Λ†== 2χnαsinψ (1 +iζαt)

I|n|+1

µ _S

nα(t)

1 +iζαt

¶_H

nα(t)

Snα(t)

½_k2

⊥

k2nνα(it)

+k

2

⊥

k2|n|sinψχnα(it) 2

−kkk⊥

k2 Nkµ 1/2

α (it)2

£

να2−χ2nα(it)2

¤ ¾

(16)

Using this expression in Eq. (28),

εl= 1−

X

α

ω2

α

ω2µα

∞

X

n→−∞

½_k2

⊥

k2 n 2_G

0,1,|n|−1,|n|

+k

2

k

k2

³

G0,1,|n|,|n|+µαNk2G2,1,|n|,|n|

´

−2k

2

⊥

+2k

2

⊥

k2χ 2

nαsin2ψ

£

G2,3,|n|,|n|+|n|G2,2,|n|−1,|n|

−|n|G2,2,|n|,|n|+1−να2G2,3,|n|,|n|+1+χ2nαG4,3,|n|,|n|+1

¤

−2kkk⊥

k2 Nk|n|µ 1/2

α

£

SnναG1,1,|n|−1,|n|−sinψχnαG2,1,|n|−1,|n|

¤

+2kkk⊥

k2 Nkχnαµ 1/2

¾

+iX

α

ω2

α

ω2µ 2

α

∞

X

n→−∞

Z ∞

0

dt

e

2 kt

2

/2

×

e

−(ν2

α+χ 2 nαt

2

[Hnα(t)](|n|−1)

(Snα(t))|n|

×2χnαsinψ (1 +iζαt)

I|n|+1

µ _S

nα(t)

1 +iζαt

¶_H

nα(t)

Snα(t)

½_k2

⊥

k2nνα(it)

+k

2

⊥

k2|n|sinψχnα(it) 2

−kkk⊥

k2 Nkµ 1/2

α (it)2

£

να2−χ2nα(it)2

¤ ¾

.

εl= 1−

X

α

ω2

α

ω2µα

∞

X

n→−∞

½

k2

⊥

k2 n 2_G

0,1,|n|−1,|n|

+k

2

k

k2

³

G0,1,|n|,|n|+µαNk2G2,1,|n|,|n|

´

−2k

2

⊥

+2k

2

⊥

k2χ 2

nαsin2ψ

£

G2,3,|n|,|n|+|n|G2,2,|n|−1,|n|

−|n|G2,2,|n|,|n|+1−να2G2,3,|n|,|n|+1+χ2nαG4,3,|n|,|n|+1¤

−2kkk⊥

k2 Nk|n|µ 1/2

α

£

SnναG1,1,|n|−1,|n|−sinψχnαG2,1,|n|−1,|n|¤

+2kkk⊥

k2 Nkχnαµ 1/2

+2k

2

⊥

k2nναχnαsinψG1,2,|n|,|n|+1

+2k

2

⊥

k2|n|χ 2

nαsin2ψG2,2,|n|,|n|+1−2 kkk⊥

k2 Nkµ 1/2

α χnαsinψ

×£

να2G2,2,|n|,|n|+1−χ2nαG4,2,|n|,|n|+1

¤ª

.

εl= 1−

X

α

ω2

α

ω2µα

∞

X

n→−∞

½

k2

⊥

k2 n 2_G

(17)

+k

2

k

k2

³

G0,1,|n|,|n|+µαNk2G2,1,|n|,|n|

´

−2k

2

⊥

k2nναχnαsinψ

£

G1,2,|n|−1,|n|− G1,2,|n|,|n|+1¤

+2k

2

⊥

k2χ 2

nαsin2ψ

£

G2,3,|n|,|n|+|n|G2,2,|n|−1,|n|

−να2G2,3,|n|,|n|+1+χ2nαG4,3,|n|,|n|+1

¤

−2kkk⊥

k2 Nknµ 1/2

α ναG1,1,|n|−1,|n|

−2kkk⊥

k2 Nkµ 1/2

α χnαsinψ£−G2,2,|n|,|n|− |n|G2,1,|n|−1,|n| +να2G2,2,|n|,|n|+1−χ2nαG4,2,|n|,|n|+1

¤ª

. (39)

⌈

An interesting limiting case to be considered is the case of propagation perpendicular to the direction of the mag-netic field and of the inhomogeneity. In this case from Eq. (39) we obtain,

εl= 1−

X

α

ω2

α

ω2µα

∞

X

n→−∞

©

n2G0,1,|n|−1,|n|

−2nναχnα£G1,2,|n|−1,|n|− G1,2,|n|,|n|+1¤

+2χ2

nα

£

|n|G2,2,|n|−1,|n|+G2,3,|n|,|n|

−να2G2,3,|n|,|n|+1+χ2nαG4,3,|n|,|n|+1

¤ª

, (40) which, as expected, corresponds to the expression for the component ε22 of the effective dielectric tensor, for a Maxwellian distribution function for particles of typeα, in the non-relativistic approximation [5].

Acknowledgments

The present research was supported by Brazilian agen-cies CNPq and FAPERGS. L.F.Z. acknowledges useful dis-cussions with M. Bornatici, and thanks the hospitality of In-stituto di Fisica A. Volta, Universit´a di Pavia, Italy, during part of the North Hemisphere Winter of 2003.

References

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[2] L. F. Ziebell and R. S. Schneider, accepted for publication in the Braz. J. Phys., (2004).

[3] O. J. G. Silveira, L. F. Ziebell, R. S. Schneider, and R. Gaelzer, in inX Latin American Workshop on Plasma Physics combined with VII Encontro Brasileiro de F´ısica de Plasmas, SBF (SBF, S˜ao Pedro, SP, Brasil, 2003), pp.43-, poster pre-sentation.

[4] R. Gaelzer, R. S. Schneider, and L. F. Ziebell, Phys. Rev. E

51, 2407 (1995).

[5] R.Gaelzer, R. S. Schneider, and L. F. Ziebell, Phys. Rev. E

55, 5859 (1997).

[6] T. H. Stix, The theory of plasma waves (AIP, New York, 1992).

[7] V. S. Beskin, A. V. Gurevich, and Y. I. Istomin, Sov. Phys. JETP65, 715 (1987).

[8] R. A. Caldela Filho, R. S. Schneider, and L. F. Ziebell, J. Plasma Phys.42, 165 (1989).