The effective longitudinal dielectric constant for plasmas in inhomogeneous magnetic fields

The effective longitudinal dielectric constant for plasmas in inhomogeneous magnetic fields

R. Gaelzer^I; L. F. Ziebell^II; R. S. Schneider^II

^IInstituto de Física e Matemática, UFPel, Caixa Postal 354, 96010-900 Pelotas, RS, Brazil

^IIInstituto de Física, UFRGS, Caixa Postal 15051, 91501-970 Porto Alegre, RS, Brazil

ABSTRACT

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 dispersion relation for electrostatic waves in inhomogeneous plasmas, considering for simplicity the particular case in which the magnetic field is homogeneous and other plasma parameters can be inhomogeneous, and deriving a general expression 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 because it features a relatively simple geometry, which has been useful for the discussion of basic features, like the symmetry properties of the effective dielectric tensor. More general situations which also feature inhomogeneity in the magnetic field introduce considerable difficulty to the derivation of the effective dielectric tensor and to the effective dielectric 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 written in a general form which is well known from the literature [6],

where the is the dielectric tensor and e_l is the longitudinal dielectric constant,

It is also important to remember here that it has been demonstrated that for an inhomogeneous plasma the dielectric properties are given by the so-called effective 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 appearing 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 formulation 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 relation 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 numerical results to forthcoming publications.

The structure of the paper is the following: In section the formulation employed in Ref. [5] is adopted and applied to the derivation of a general expression for the effective dielectric 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 dielectric constant

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

where

with

In these expressions we have

The geometry utilized has been the following. The magnetic field has been considered pointing in the z direction, and inhomogeneous in the x direction, B₀ = B₀(x) e_z. The waves were assumed propagating in arbitrary directions, with k_|| and k_^ as the components of the wave vector respectively parallel and perpendicular to the magnetic field. The wave angular frequency has been denoted as w. y denotes the angle between the vector k_^ and the direction of the inhomogeneity. The inhomogeneity has been assumed to be weak, such that the cyclotron frequency has been written as

where W_a(x) = (q_aB₀(x)/m_ac) is the particle cyclotron angular frequency, m_a is the particle rest mass, q_a is the particle charge, c is the velocity of light, , and is the unperturbed position of particle a.

We also used

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

where we did not use the indexes na in the components P_i, for simplicity. We see that it is necessary to evaluate the following product,

Therefore, the quantity of interest is given by

After some straightforward algebra, using G⁺ G^– = –1 and F^1/2F^1/2 = F, we arrive to the following

Using now the expression for F^1/2G^±, and separating the terms containing S_n cos y from those with i sin y, we obtain

This expression must be used along with Eq. (5), which constitutes a general form of the longitudinal dielectric constant. An alternative form can be obtained by use of the following property of Bessel functions,

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

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, n ® 0, ® b_a, and F_na ® . In that case it is easy to show that Eq. (7) is reduced to the following expression,

and therefore,

where we have performed the integration over the time variable,

Equation (8) corresponds to the homogeneous contribution 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, b_a ® 0, where

The Bessel functions become all independent of k_^. Therefore, for vanishing k_^, all terms which have combinations of Bessel functions multiplied by k_^ should vanish. After a small amount of algebra, it is easy to show that Eq. (5) becomes

For propagation parallel to the direction of the inhomogeneity,

and

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

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

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

3 The dielectric constant for the case of a Maxwellian distribution function

Let us assume a Maxwellian distribution function for particles of type a,

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

The quantities and F_na do not depend on the parallel momentum, which only appears in the product . Therefore we are left with the following integrals over parallel momentum,

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

Using these results,

where

In order to simplify the ensuing calculations, we introduce the following notation,

where

Moreover, we define

and

Using this notation,

and the dielectric constant becomes the following

where

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

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

where

Using the symbols introduced by Eqs. (19), and the notation introduced by Eqs. (21), we can write

f_na = m_aH_na(t),

where

and

The dielectric constant therefore can be written as follows,

where

Using the integrals given by Eqs. (21),

where

where we have taken into account the following

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

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

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

where

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

Let us examine the terms appearing in the expression for L^†. Initially we consider the following terms,

Now we may consider the following,

This expression can be inverted, and we obtain,

Multiplying numerator and denominator by ,

Consider now Eq. (30),

Taking into account the following property,

we obtain,

In analogous way,

Collecting together these results,

Therefore, we have,

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

The next two terms in the expression for L^† are the following

The last two terms in L^† can be written as follows,

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

Using this expression in Eq. (28),

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

which, as expected, corresponds to the expression for the component e₂₂ of the effective dielectric tensor, for a Maxwellian distribution function for particles of type a, in the non-relativistic approximation [5].

Acknowledgments

The present research was supported by Brazilian agencies CNPq and FAPERGS. L.F.Z. acknowledges useful discussions with M. Bornatici, and thanks the hospitality of Instituto di Fisica A. Volta, Universitá di Pavia, Italy, during part of the North Hemisphere Winter of 2003.

Received on 26 January, 2004

• [1] L. F. Ziebell and R. S. Schneider, in X Latin American Workshop on Plasma Physics combined with VII Encontro Brasileiro de Física de Plasmas, SBF (SBF, Săo Pedro, SP, Brasil, 2003), pp.93-, oral presentation.
• [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 in X Latin American Workshop on Plasma Physics combined with VII Encontro Brasileiro de Física de Plasmas, SBF (SBF, Săo Pedro, SP, Brasil, 2003), pp.43-, poster presentation.
• [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. JETP 65, 715 (1987).
• [8] R. A. Caldela Filho, R. S. Schneider, and L. F. Ziebell, J. Plasma Phys. 42, 165 (1989).
• [9] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970).

Publication Dates

• Publication in this collection
  04 Nov 2004
• Date of issue
  Sept 2004

History

• Received
  26 Jan 2004