Transport Properties of Equilibrium Argon Plasma in a Magnetic Field

D. Bruno *, A. Laricchiuta¥Istituto di Metodologie Inorganiche e Plasmi del CNR, Bari, 70126 Italy

M. Capitelli‡, C. Catalfamo§Dipartimento di Chimica dell'Università di Bari, 70126 Italy

A. Chikhaoui¶, E. V. Kustova#IUSTI - Université de Provence, Marseille, 13453 France

D. Giordano**ESA/ESTEC - Aerothermodynamics Section, Noordwijk, 2200 the Netherlands

Electron transport coefficients (electrical and thermal conductivity, electro-thermal conductivity) of equilibrium Argon plasma in a magnetic field are calculated up to the 12th Chapman-Enskog approximation at pressure of 1 atm and 0.1 atm for temperatures 500K-20000K; the magnetic Hall parameter spans from 0.01 to 100. The collision integrals used in the calculations are discussed. The convergence properties of the different approximations are assessed. The degree of anisotropy introduced by the presence of the magnetic field is evaluated. Differences with the isotropic case can be very substantial. The biggest effects are visible at high ionization degrees, i.e. high temperatures, and at strong magnetic fields.

I. Introduction

The transport processes in partially ionised gases moving in a magnetic field have been studied by the Chapman-Enskog method1 and by Grad’s 13-moment method2. The two approaches are mostly equivalent, but Grad’s 13-moment method corresponds to the second approximation of the Chapman-Enskog method in terms of Sonine polynomials so that its accuracy may turn out to be insufficient, as we shall see. Recent interest in MHD effects on hypersonic flows has brought renewed attention to this field. In a previous paper theoretical research on the subject has been reviewed3. There, only the calculation of the viscosity tensor was carried out for the equilibrium Argon plasma. In this work we extend the calculations to all other transport coefficients.

The Chapman-Enskog method allows to express the diffusion velocities, the heat flux, the stress tensor, the charge and the current density as linear functions of the spatial gradients of the macroscopic parameters (densities of the species, flow velocity, temperature, pressure) and of the electromagnetic field. The coefficients of these functions are the transport coefficients. If a magnetic field is present the transport coefficients must be expressed in tensorial form.

The calculation of the relevant coefficients entails the solution of systems of integro-differential equations. The coefficients are then expanded in a series of orthogonal polynomials (the Sonine polynomials, in fact) and the series truncated at the desired order of approximation. The calculation thus reduces to the solution of systems of linear algebraic equations of order N*K, K being the number of species and N the order of approximation of the expansion in Sonine polynomials. As a result, the transport coefficients are expressed as determinants whose coefficients are known functions of the macroscopic parameters, the fields and the usual collision integrals between particles.

With regard to the convergence of the above expansion, we can say that the transport coefficients that

* Researcher, IMIP, v. G. Amendola 122, 70126 Bari Italy, AIAA Member¥ Researcher, IMIP, v. G. Amendola 122, 70126 Bari Italy‡ Full professor, Dept. Chemistry, v. E. Orabona 4, 70126 Bari Italy, AIAA Fellow§ Student, Dept. Chemistry, v. E. Orabona 4, 70126 Bari Italy¶ Full professor, IUSTI, 13453 Marseille CEDEX 13 France# Researcher, IUSTI, 13453 Marseille CEDEX 13 France** Researcher, TOS-MPA, ESTEC, 2200 AG Noordwijk the Netherlands, AIAA Member

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35th AIAA Plasmadynamics and Lasers Conference28 June - 1 July 2004, Portland, Oregon

AIAA 2004-2161

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

involve collisions between heavy particles can be assumed to be converged at the second approximation; when the electrons make the highest contribution the convergence is much slower and can be dependent on the temperature and pressure of the plasma: it is the case of the thermal and electrical conductivity and of the electron-heavy particle diffusion coefficients4,5.

The general formulation has been applied to the calculation of transport properties in Argon plasma in thermal equilibrium subject to magnetic field. The calculations are carried over for a wide range of temperature and pressure and magnetic field strength. The results provide a useful tool to all CFD researchers interested in MHD effects. Extensive calculations make it possible to assess the convergence properties of different transport coefficients at different plasma conditions. Also, the sensitivity to the uncertainties in the cross sections has been evaluated and prescriptions for future improvements have been derived. This point is of particular interest when electronically excited states with their enormous transport cross sections are taken into account6. Finally, and more important for practical applications, the relative importance of the contribution of the magnetic field to the transport properties of the plasma will be assessed.

II. Collision Integrals

The physical information on the interaction potential between colliding particles in the plasma is carried by the collision integrals. The collision integrals for the interaction between two colliding species (i,j) can be defined as follows7:

†

psRSW i, jl,s( )*

=2pm kTW i, j

l,s( )

12

s + 1( )! 1-12

1+ -1( )l

1+ l

È

Î

Í Í Í

˘

˚

˙ ˙ ˙

=4 1+ l( )

s +1( )! 2 l+ 1- -1( )lÈ

Î Í

˘

˚ ˙

e-g 2

g2s+3Qi, jl dg

0

•

Ú (1)

where g2 = µg2/2kT, sRS is rigid sphere cross section and

†

Ql is the transport cross section:

†

Qi, jl = 2p 1- cosl c

Ê Ë Á ˆ

¯ ˜ bdb

0

•

Ú = 2p 1- cosl cÊ Ë Á ˆ

¯ ˜ s g, c( )sin cdc

0

•

Ú (2)

s is the elastic collision differential cross section and c is the deflection angle.In this work we have concentrated on electron transport properties, thus the relevant collision integrals are

those pertaining to electron-heavy particle collisions.

As far as collisions with the neutral atom are concerned, transport cross sections,

†

Ql, have been calculated

by numerical integration of e-Ar elastic differential cross sections taken from the literature, following the scheme adopted by Rat8:

0.02 ≤ E < 1.0 eV: theoretical cross section calculated by Bell et al.9;1.0 ≤ E ≤ 10.0 eV: measured cross section by Gibson et al.10;10.0 ≤ E ≤ 100.3 eV: experimental results of Panajotovic et al.11 have been completed at low and high scattering angles with theoretical cross section calculated by Nahar and Wadehra12.

For collision with Ar+ we have used the following analytical approximation for the omega collision integrals of the screened Coulomb potential13. It is practically equivalent to the Kihara results14 except that it behaves better at low temperatures. The interaction potential is taken to be:

†

j r( ) = ±j0

r r0( )e-r r0 (3)

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where:

†

r0 = lD = e0RGTre

Me

Qe2

and

†

j0 =e2

4pe0

1lD

.

Now, let:

†

E* =E

j0

†

T * =KBTj0

†

Ql( )*

=N lQ

l( )

pr02

†

N l-1 = 1-

1+ -1( )l

2 l+ 1( )

†

lng = 0.57721 (Euler constant)

†

Cl = 1+13

+15

+ ... +1l

Ê

Ë Á

ˆ

¯ ˜ -

12l

, l odd

Cl = 1+13

+15

+ ...+1

l- 1, l even

†

A1 = 0, Am = 1+12

+13

+ ... +1

m - 1

The reduced omega collision integral is defined by:

†

Wl,m( )*

=1

m +1( )! T *Ê Ë Á ˆ

¯ ˜ m+2

Ql( )*

E*Ê Ë Á ˆ

¯ ˜ m+1

exp -E*

T*

Ê

Ë Á Á

ˆ

¯ ˜ ˜ dE*

0

•

Ú (4)

The following interpolating formula has been used13:

†

T*Ê Ë Á ˆ

¯ ˜ 2W

l,m( )*=

lNl

m m +1( )ln

4T*

g 2

Ê

Ë Á Á

ˆ

¯ ˜ ˜ exp Am - Cl( ) +1

È

Î

Í Í

˘

˚

˙ ˙

(5)

This formula interpolates the results obtained for the potential defined in eqs. (3) by neglecting any difference between repulsive and attractive potentials and any quantum correlation effect.

III. Electron Transport Properties

If the large mass ratio between the electrons and the heavy particles is taken into account, the Boltzmann equation for the electron component can be effectively decoupled from the others15. Since this component is mainly responsible for the thermal and electrical conductivity in the plasma we have reduced our calculations to the simplified system obtained in the limit of vanishing mass ratio.

In the N-th order of approximation, this entails the solution of linear systems of order N (N+1, for the thermal and electro-thermal conductivity). At this stage, a comment is in order. The nomenclature on this subject is not unique in the literature. Some authors count the order of approximation in terms of the order of expansion in Sonine polynomials, others refer to the first nonvanishing solution. We here stick to the latter convention. Therefore, for the electrical conductivity (and for the diffusion coefficients) the first approximation comes from retaining only the lowest order term in the expansion in Sonine polynomials; for the thermal and electro-thermal conductivity(and for the thermal diffusion coefficients), instead, for which the lowest order term gives vanishing contribution, the first approximation is obtained by retaining the first two lowest order terms. The solutions give the following transport coefficients:

†

DTe : thermal diffusion;

†

fe : electro-thermal conductivity;

†

Die i = 1, ...,k : diffusion;

†

se : electrical conductivity;

†

le : thermal conductivity;

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When a magnetic field is present, for each of these coefficients, three independent components exist. This is caused by the anisotropy introduced by the magnetic field. In vector form, the linear dependence of the flux (of mass, of charge, of energy) on the driving gradient (of pressure, of concentration, of temperature, etc.) can be written:

†

J = a ⋅ —A (6)

now, in the isotropic case, the matrix a has only one scalar value:

†

a =

a 0 0

0 a 0

0 0 a

(7)

In the presence of the magnetic field, we choose a reference frame with the x-axis along the magnetic field and obtain:

†

a =

a / / 0 0

0 ao a t

0 -a t ao

(8)

Explicit expressions for the macroscopic fluxes used in CFD calculations in terms of the transport coefficients used here can be found elsewhere1. In the following we turn to analysing the results for the electrical, thermal and electro-thermal conductivities for equilibrium Argon plasma. The first issue to be addressed is the convergence of the expansion in Sonine polynomials. This is what we discuss in the next section.

IV. Convergence

It has been recognised long ago4,16 that the convergence properties of the electron transport coefficients are much worse than those of a neutral gas. This is a consequence of the particular features of the electron scattering cross sections. Other studies report that the convergence may depend on the magnetic field strength17. We have therefore analysed the convergence of the different transport coefficients up to the 12th approximation for an equilibrium plasma at p=1 atm and at several values of the temperature, from 500K to 20000K, and for several values of the magnetic field strength. The magnetic field strength is here reported in terms of the electron Hall parameter. This adimensional number is the product of the Larmor gyrofrequency and the mean free time for Coulomb collisions. It is given by:

†

be = wet e (9)

where the electron Coulomb mean collision time is:

†

t e = 316

me

reWe- ion1,1( ) (10)

and the electron Larmor gyrofrequency:

†

w e =e

meB (11)

We start this analysis by looking at the convergence properties of those transport coefficients that do not depend on the magnetic field (in the following we shall omit the subscript e where it is not essential) namely

†

l/ / ,

†

s / / ,

†

f// .Figs. 1a-1c show the relative error of each approximation, with respect to the 12th approximation, for

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different values of the temperature. At the lowest temperature, the convergence appears to be very slow: this is a consequence of the particular behaviour of the electron atom cross section that shows a very deep minimum at low energy (the Ramsauer minimum); however, recall that, at this temperature, the (equilibrium) ionization degree is very small so that the absolute value of the transport coefficients is also extremely small. As the plasma temperature rises, the convergence is much faster. It is still evident, however, that the presence of electron-atom collisions slows down the convergence. For example, in the case of thermal conductivity, in order to reach 1% precision, at 20000 K the 2nd approximation is sufficient, at 10000 K the 3rd is needed and at 7000 K at least the 5th approximation should be used.

10-4

10-3

10-2

10-1

100

0 2 4 6 8 10 12N

|l//(N

)-l //(1

2)|/l

//(12)

(a)

10-4

10-3

10-2

10-1

100

0 2 4 6 8 10 12N

|s//(N

)-s //(1

2)|/s

//(12)

(b)

10-4

10-3

10-2

10-1

100

0 2 4 6 8 10 12N

|f//(N

)-f //(1

2)|/f

//(12)

(c)T=1000

T=7000

T=10000

T=20000

Figure 1. Convergence properties of the parallel component of electron transport coefficients for p=1 atm at different plasma temperatures. a:

†

l/ / ; b:

†

s / / ; c:

†

f// .

We now turn to analysing the effect of the magnetic field. In this case, it is seen that the convergence of the transport coefficients depends on magnetic field strength in a nontrivial way. Fig. 2 illustrates results for the orthogonal components of the electron transport coefficients of a plasma at T=10000 K, p=1 atm. Different curves refer to different values of the electron Hall parameter. It can be noticed that the convergence is generally reduced at moderate field strengths (

†

be ~ 10 ) and then considerably enhanced at strong fields (

†

be ~ 100 ). The effect is much smaller at the magnetic fields values most interesting for applications (

†

be ˜ < 1). However, recall that, for fixed value of the Hall parameter, the magnetic field decreases as the pressure is decreased. Last, we show the differences in the convergence properties of the orthogonal and transversal components of the thermal conductivity coefficients. This plot is somewhat representative of the general trend: the transversal component converges more quickly at moderate and high values of the Hall parameter.

From the results presented it is difficult to draw conclusions with general validity. The precision of each order of approximation depends on the degree of ionization (worse for low ionization) and on the magnetic field strength (worse for

†

be ˜ > 1). The 6th approximation appears to be converged to within 1% in most cases.

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10-4

10-3

10-2

10-1

100

0 2 4 6 8 10 12N

|lo(N

)-l o(1

2)|/l

o(12)

(a)

10-4

10-3

10-2

10-1

100

0 2 4 6 8 10 12N

|so(N

)-s o(1

2)|/s

o(12)

(b)

10-4

10-3

10-2

10-1

100

0 2 4 6 8 10 12N

|fo(N

)-f o(1

2)|/f

o(12)

(c)b

e=0

be=1

be=10

be=100

Figure 2. Convergence properties of the orthogonal component of electron transport coefficients for p=1 atm, T=10000 K at different values of the electron Hall parameter. a:

†

lo ; b:

†

so ; c:

†

fo

10-4

10-3

10-2

10-1

0 2 4 6 8 10 12N

|l(N

)-l(

12)|/

l(12

)

Figure 3. Convergence properties of the orthogonal (full symbols) and transversal (open symbols) components of electron thermal conductivity for p=1 atm, T=10000 K at different values of the electron Hall parameter (circles:

†

be = 10 ; squares:

†

be = 100 ).

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V. Anisotropy

The transport coefficients explicit the proportionality between weak deviations of the plasma parameters from equilibrium (i.e. spatial gradients of temperature, flow velocity, pressure, composition) and the resulting fluxes of mass and electric charge, momentum, energy. The presence of the magnetic field vector breaks the spatial symmetry of the plasma so that the transport coefficients become 2nd rank tensors. In the direction parallel to the magnetic field, the particles suffer no influence by the Lorenz force and the relative transport coefficient is unaffected by the presence of the field. Not so in other directions. In the direction normal to the magnetic field, but where the driving gradient still has a component, the response of the plasma medium (the orthogonal transport coefficient) approaches the parallel value as the magnetic field vanishes. When the magnetic field is increased, the particle trajectories are governed by the Lorenz force and this tends to mask the effect of collisions; the transport coefficient decreases and the plasma tends to behave like a nonviscous fluid. On the other hand, the deviation of the particle trajectories creates a flux in a direction normal both to the magnetic field and the driving gradient. This flux is described by the transversal transport coefficient. The latter vanishes when there is no field, then rises and becomes comparable to the other components. Eventually, for high values of the Hall parameter, the particle trajectories are completely dominated by the Lorenz force, the plasma behaves as a Eulerian fluid and the transversal transport coefficient vanish in this limit. Given this general behaviour, we now go on to examine the results of calculations of the electron thermal, electrical and electro-thermal conductivity for a equilibrium Argon plasma at pressures of 1 atm and 0.1 atm and for several values of the temperature and of the magnetic field strength.

A. Thermal conductivityIn fig. 4 we report the comparison on the parallel component of the electron thermal conductivity as calculated in this work and in the classic work of Devoto15: the agreement is very good, thus justifying the set of collision integrals used in this work. Also, the total thermal conductivity is reported16 for comparison.

0

1

2

3

4

0 1 104 2 104

T (K)

l // (W

/m*K

)

Figure 4. Parallel thermal conductivity of partially ionised Argon at p=1 atm. Full line: electron contribution (ref. 15); dashed line: electron contribution (this work); symbols: total thermal conductivity (ref. 16).

In fig. 5 we report the ratio of the orthogonal and transversal components of the electron thermal conductivity to the parallel value. The ratio is plotted against the electron Hall parameter that, at fixed pressure and temperature, is a measure of the magnetic field strength. Different curves refer to different plasma temperatures. Fig. 5a,c refer to the p=1 atm case whereas fig. 5b,d refer to the p=0.1 atm case.It is evident that the effect, depending on the plasma temperature and the magnetic field strength, can be very substantial. The effect of the pressure is twofold. As the pressure is decreased, the ionization equilibrium is naturally shifted towards higher ionization. This is seen in the small shift of the curves towards lower values of the Hall parameter. The biggest effect, however, is connected with the dependance of the transport coefficients on the magnetic field strength that is, eventually, the quantity of technological interest. Given a fixed value of the Hall parameter, and therefore a fixed degree of anisotropy, the magnetic field necessary to obtain such a result is inversely proportional to the plasma pressure. So, for example, if at T=10000 K, p=1 atm, the value

†

be = 0.1 is obtained for

†

B = 0.09 T , the same conditions at p=0.1 atm are reached at

†

B = 0.03 T .

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0

0,2

0,4

0,6

0,8

1

1,2

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

(a)

†

lol/ /

0

0,2

0,4

0,6

0,8

1

1,2

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

(b)

†

lol/ /

-0,1

0

0,1

0,2

0,3

0,4

0,5

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

(c)

†

ltl/ /

-0,1

0

0,1

0,2

0,3

0,4

0,5

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

(d)

†

ltl/ /

Figure 5. Ratio of orthogonal (a, b) and transversal (c, d) electron thermal conductivity to the parallel component for Argon plasma at different temperatures and pressures of 1 atm (a, c) and 0.1 atm (b, d).

0

5000

10000

15000

0 1 104 2 104

T (K)

s // (m

ho/m

)

Figure 6. Parallel electrical conductivity of partially ionised Argon at p=1 atm. Full line: ref. 15; dashed line: this work.

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B. Electrical conductivityIn fig. 6 we plot the parallel component of the electrical conductivity of partially ionised Argon at p=1 atm as calculated in this work and in ref. 15. Again, the agreement is very good at all temperatures. We can then go on and estimate the effect of the magnetic field on the other components of the electrical conductivity tensor. Results are shown in fig. 7.

0

0,2

0,4

0,6

0,8

1

1,2

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

†

s os/ /

(a)

0

0,2

0,4

0,6

0,8

1

1,2

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

†

s os/ /

(b)

-0,1

0

0,1

0,2

0,3

0,4

0,5

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

†

sts/ /

(c)

-0,1

0

0,1

0,2

0,3

0,4

0,5

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

†

sts/ /

(d)

Figure 7. Ratio of orthogonal (a, b) and transversal (c, d) electron electrical conductivity to the parallel component for Argon plasma at different temperatures and pressures of 1 atm (a, c) and 0.1 atm (b, d).

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C. Electro-thermal conductivityFinally, we show the anisotropy effect, produced by the presence of the magnetic field on the electro-thermal conductivity coefficients. These transport coefficients describe the transport of electric charge due to the thermal diffusion process. They can be positive or negative, depending on the temperature and pressure of the plasma. The curves in fig. 8 show that also the magnetic field can change the sign of these coefficients.

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

†

fof//

(a)

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

†

fof//

(b)

-0,1

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

(c)

†

ftf//

-0,1

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

10-3 10-2 10-1 100 101 102

T=4000T=5000T=6000T=7000T=10000T=20000

be

(d)

†

ftf//

Figure 8. Ratio of orthogonal (a, b) and transversal (c, d) electron electro-thermal conductivity to the parallel component for Argon plasma at different temperatures and pressures of 1 atm (a, c) and 0.1 atm (b, d).

VI. Conclusions

Electron transport properties of equilibrium Argon plasma have been calculated for a wide range of temperatures and magnetic field strengths up to the 12th Chapman-Enskog approximation. Results show that the convergence of the different approximations can be very slow at low ionization degrees and that it can furthermore depend in a nontrivial way on the magnetic field. More important for practical applications, we have shown that anisotropy induced by the presence of the magnetic field can alter strongly the transport coefficients thus making the isotropy assumption untenable. This is particularly relevant at low pressures and at high temperatures, i.e. regimes of interest for hypersonic applications, where even moderate magnetic fields of technological interest, can produce deviations from the isotropic behaviour in the order of tens of percent.

VII. Acknowledgments

This work has been supported by ESA/ESTEC under contract 16745/02/NL/PA.

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VIII. References

[1] J. H. Ferziger, H. G. Kaper, “Mathematical Theory of Transport Processes in Gases”, North-Holland, Amsterdam, 1972.[2] V. M. Zhdanov, “Transport Phenomena in a Multicomponent Plasma” (in russian), Energoizdat, Moscow, 1982. [3] D. Bruno, M. Capitelli, A. Dangola, "Transport coefficients of partially ionised gases: a revisitation", AIAA 2003-4039 (2003).[4] R. S. Devoto, “Transport Properties of Ionised Monatomic Gases”, Phys. Fluids 9 (1966) 1230.[5] M. Capitelli, "Transport Properties of Partially Ionized Plasmas", J. de Physique 38 (1977) C327.[6] M. Capitelli et al., "Electronically excited states and transport properties of thermal plasmas: the reactive thermal conductivity", Phys. Rev. 66 (2002) 16403.[7] J. O. Hirschfelder, C. F. Curtiss, R. Byron Bird, Molecular Theory of Gases and Liquids, John Wiley & Sons, Inc. (1966), p. 525.[8] V. Rat, P. André, J. Aubreton, M. F. Elchinger, P. Fauchais, D. Vacher, J. Phys. D 35 (2002) 981.[9] K. L. Bell, N. S. Scott, M. A. Lennon, J. Phys. B 17 (1984) 4757.[10] J. C. Gibson, R. J. Gulley, J. P. Sullivan, S. J. Buckman, V. Chan, P. D. Burrow, J. Phys. B 29 (1996) 3177.[11] R. Panajotovic, D. Filipovic, B. Marinkovic, V. Pejcev, M. Kurepa, L. Vuskovic, J. Phys. B 30 (1997) 5877.[12] S. N. Nahar, J. M. Wadehra, Phys. Rev. A 35 (1987) 2051.[13] H. S. Hahn, E. A. Mason, F. J. Smith, Quantum transport cross sections in a completely ionized gas, Phys. Fluids 14 (1971) 278.[14] T. Kihara, O. Aono, J. Phys. Soc. Jpn. 18 (1963) 837.[15] R. S. Devoto, Phys. Fluids 10 (1967) 2105.[16] R. S. Devoto, Phys. Fluids 10 (1967) 354.[17] U. Daybelge, SUIPR Report 263, Institute for Plasma Research, Stanford University 1968.

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