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Conductance Theory of Nonideal Plasmas.

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Annalen der Physik. 7. Folge, Band 36, Heft 6, 1979, S. 429-437
J. A. Barth, Leipzig
Conductance Theory of Nonideal Plasmas
By W. EBELING
and G. ROPKE
Sektion Physik der Humboldt-Universitiit Berlin und Sektion Physik der Wilhelm-Pieck-Universitiit Rostock
Abstract. The conductivity of nonideal plasmas is investigated by using the kinetic theory as
well as the Kubo-type correlation function theory. The effects of dynamical screening, short range
forces and the Debye-Onsager relaxation effect on the conductance problem are considered without
and with quantum mechanical corrections. For some special plasmas it is found that deviations from
Coulomb’s law and quantum effects have only a small effect on the conductivity, whereas the numerically most important effects are connected with the dynamical screening and with the DebyeOnsager relaxation force.
Leitfiihigkeit fur nichtideale Plasmen
Zusammenf assung. Die Leitfiihigkeit fur nichtideale Plasmen wird unter Verwendung sowohl
der kinetischen Theorie als auch der Methode der Korrelationsfunktionen untersucht. Es werden die
Effekte von dynamischer Abschirmung, kurzreichweitigen Kriiften und dem Debye-Omgerschen
Relaxationseffekt auf die Leitfiihigkeit ohne und mit Beriicksichtigung quantenmechanischer Korrekturen betrachtet. Fiir einige spezielle Plasmen wird erhalten, daS Abweichungen vom Coulombschen Gesetz und Quanteneffekte nur einen geringen EinfluS auf die Leitfiihigkeit besitzen. Die numerisch am wichtigsten Effekte sind mit der dynamischen Abschirmung und der Debye-Onsagerschen
Relaxationskraft verbunden.
1. Introduction
In the last time the experimental and theoretical investigation of nonideal plasmas
has been the subject of many investigations [l-51. The results of the classical conductance
theory based on Boltzmann equations [6--81 are in part not in satisfactory agreement
with the experimental findings. The aim of this investigation is the application of recent
concepts and results of the kinetic theory [9-111 as well as of the Kubo-type transport
theory [12-161 to the conductance problem. Especially we are interested in the following effects which are not contained in the classical conductance theory: 1. Dynamical
screening: The influence of deviations from the static Debye screening on the conductance
is investigated.
2. Short range forces: The influence of devjations from the Coulomb’s law on the
conductance is studied. This effect could be important especially for a h l i plasmas. A
possible model for the short range interactions is [17]:
1
V ( r )= T [l - Ae-ar].
3. Debye-Onsager relaxation effect: The induced relaxation forces opposite to the
external field and their influence on conductance is calculated.
Other effects which are connected with the interaction with neutral particles or with
short range order of the ion distribution function are outside the scope of this paper.
w. EBELINQaria G.ROPKE
430
2. The Method of Kinetic Equations
The kinetic equations for the distribution function f a ( v ) of species a for a homogeneous
plasma in an external static field E read after BALESCU-LENARD-KLIMONTOVICH
[lo]
OD
-&T‘
e
+ i(u - k - w ’ ) t’+ i b
k
Ez’~]
(3)
2mb
0
where E ( W , k) is the dielectric function [lo], V ( k ) the Fourier transform of V(r), and
> 0 a small damping factor E -+ 0. From eq. (2) we obtain by multiplication with v
and integration
E
and the evaluation of the resistance R defined by
E=Rj,
j=2ea$dwvfa(w),
(5)
a
reduces to the calculation of the r.h.s. of eq. (4). After partial integration and substitution
we obtain
* kk JE(o, k)I2 {%aau
8
--n
0
fb
fa
?fb)
ma
mb
aV’
+ i ( o - k .V ) z - i
2ma
k . E$]
Eq. (6) may be written in the form ( E parallel t o j in isotropic systems)
E = R@)j+ QE
431
Conductance Theory of Pl'onideal Plasmas
with
m
* 1dt' e--Er'cos [(o- k . u') t'],
n
* ( k . u' - k
m
u)fAo)(w) fio)(w') Re
1dt exp
+ i (o- k . u ) t - i
--~t
0
Herc fAo)(v) is the Maxwell distribution.
Let us first calculate R(O). The most difficult problem is the treatment of the factor
/ a ( o , k ) 1-2. Following a proposition of KLIMONTOVICH
[lo] we replace this part by its
mean valiie with respect to w
C
x2
=
2 rice; 4nP.
C
Then, in eq. (8) the integrals may be performed with exception of the k-integral which
yields a factor
where a cutoff value km is introduced to make the integral finite. For the special case
of Coulomb interaction eq. (11)yields the so-called Coulomb-logarithm
L
= Inl'b
+ Lz/x2 w In (kmfx),
(12)
and usually for km the approxiniation
km
= 3kBT/e2
is adopted.
The value of RCo)as given by
(13)
W. EBELINU
and G. ROPKE
432
<
in the case e, = -ei = e, mJmi
1 (e electron, i ion) depends on the special approximation used for fa which determines the factor f. Using a shifted Maxwell distribution
we would obtain f = 4 1/%/3 M 3.34, the CHAPMAN-ENSKOQ
method yields f m 1.69.
Improvements by choosing GRAD’S 13-moment approximation give after KLIMONTOVICH and EBELINQ
[ l l , 51
f M 1.73.
(15)
The only difference t o the earlier calculation [ll, 51 of R(O)(14) consists in the new value
for L which depends now on the parameters of the pseudopotential (1).
Let us consider now the term Q which is given by eq. (9). Using again the KLIMONTOVICH approximation (10) we get after some calculations the expression
Por the special case of Coulomb interactions follows
1
Q = - Be2%.
6
The expressions ( l l ) , (16) which determine the resistance according to
R=f
m~~2e2
L
(kaT,)3J2 1 - Q
take into account modifications of the Coulomb interaction which may occur, f.i., for
a,lkaliplasmas at small distances. Following an earlier work [16], in the case of Cs-plasmas we propose to use eq. (1) with the values
A = 1.371,
0.964 [Al-l,
(19)
and A = 0 for H-plasmas e.g. Numerical evaluations of L and Q are given in section 4.
OL =
3. The Method of Correlation Functions
The aim of this chapter is to give a quantum mechanical generalization of the Coulomb
logarithm L and the relaxation function Q which in principle is valid for any order of
the perturbation theory with respect to e2. Using ZUBAREV’Smethod [12] the resistivity
of a n electron system imbedded in scatters can be expressed by correlation functions as
shown in [13] :
R=- Q
3N2 e2 1
(F;F)
+ (3Nnt)-l < P ; F)‘
Here i2 i s the volume, N the total number of electrons, P
the electron system and F
=
2 pi
the total impuls of
i
= i/li [ H , PI the total force acting on the electrons.
the small positive parameter
E
again acts as a convergence parameter.
Furtheron,
433
Conductance Theory of Nonideal Plasmas
The generalization to a two-component system consist,ing of electrons (index e)
and ions (index i) reads, see [14, 151:
where
Practically the whole current in a plasma is transported by the electrons, Re
we have
< Ri, and
R w Re,
(25)
The obvious result (25) is obtained if the force-force correlation functions { F i ; Fi),
( F , ; Fi) are omitted in comparison with ( F , ; F,). This can be justified for a plasma with
a large mass ratio between electrons and ions if the coupling of the system to further
degrees of freedom is taken into account, see [14].
Since the Zubarev nonequilibrium operator corresponds to a shifted equilibrium
distribution eqs. (24), (25) have t o be compared with eq. (18) withf = 4 1/E/3.
I n this way we obtain for the Coulomb integral L the general expression
F,
=
z
- [ H , P,] = i
R.
2 ke2V(k)af&
u;;;& a$? at:.
(28)
PI Psk
Furtheron, comparison of (24) with (18) yields for the relaxation function
=C-
eb
3eNernb0
b
r d t f d A e-'I Sp{e,Fb(--t - &A) P e l .
0
To evaluate the expressions (26), (29) for L and Q, respectively, we use the perturbation
theory with respect t o e2. I n the lowest approximation in eq. (26), (22) H is replaced by
H",and by using WICK'Stheorem we obtain
=
( F , ; F,)
=
J
B
J" dt
0
0
2
lc2e41 V ( k )l2
Pip&
* exp [ (L - f -t)
( ~ g $ k
+ EZ?-~- E$? - ~ g ? ) ]
fe(pl+
k ) f i ( A - k)
434
W. EBELWG
and G . ROPKE
The t,1integrals may be performed immediately, and by using Dirac's identity a n energy
conserving delta function occurs. Furtheron, in the liniit me/mi 1 the p,, p2 integrals
may be perfornied wjth the result
<
The corresponding expression for R(O)coincides in the liniit T + 0 with the well known
ZIMANformula for the resistivity, see also [17].
However, to avoid divergencies which are due to the Coidoinb potential, partial
summations of the power series expansion of (26) with respect t o e2 have to be performed.
I n the non-degenerate limit, n(2nn+,/,9P)-3f2 1, by expanding the density matrix with
respect t o e2 one of the potentials V ( k )in (30) will be substituted by a statically screened
one.
<
Furtheron, we introduce a cutoff value km for the k-integral in (31) to exclude the contributions of collisions with small collision parameter. To make a coniparision of (31) with
(11) possible we choose km is accordance with eq. (13). Then, after substitution of
exp (,8pe)by n(E2,8/2nme)3'2/2
a generalization of eq. (19) is
which accounts for quantum mechanical corrections to eq. (11). Whereas the distribution
function in (31), (33) is introduced in a correct manner, the introduction of a statically
screened potential and of the cutoff parameter km should be improved by a fill1 quantuniinechanically treatment of the higher order terms with respect t o e2 in eq. (26).
I n order t o evaluate Q within the frame of the perturbation theory a t least the first
order with respect to e2 must be considered:
Conductance Theory of Nonideal Plasmas
435
with
AE = a2(k. pl/me - k .pz/nzi
+ (m;l + m r l ) k2/2).
The further evaluation of (34) may be performed in the nondegenerate limit. Similar
t o the evaluation of L, the expansion of the density matrix with respect to ea leads in the
non-degenerate limit to the substitution (32) so that the divergencies of the Coulomb
potential are avoided. After performing the integrations in (34) we obtain
* exp (-k2t2/2pme)
sin (tik2t/2me)
(35)
which accounts for quantum mechanical corrections to (16). The expression (35) was oband EBELING
[ll]and in the
tained also from the kinetic theory by KLIMONTOVICH
classical case by KADOMTSEV
[18].
The expressions (33), (35) for L and Q coincide in the classical limit with the expressions (11)and (16), respectively. The quantummechanical effects decrease L and Q in
comparision to their classical values.
Discussion
To show how the conductance of a plasma is influenced by the different effects considered in chapter 1, numerical values of the Coulomb logarithm L tind of the relaxation
function Q are summarized in Table 1 for several plasmas at T = 10000 K. In dependence on the density characterized by the plasma paramet,er p , the value In (k,,,/x)
for L in the classical kinetic theory is compared with the values of L for H- and Cs-plasmas (19), respectively, without (11) and with (33) quantum mechanical corrections.
Similar, Q is evaluated for H- and Cs-plasmas, respectively, without (16) and with (35)
quantum mechanical corrections.
Table 1 Numerical values of the Coulomb-logarithm L and the relaxation function Q
for several classical and quantum-mechanical plasmas at T = 10000 K
LL = Be% :
0.1
0.5
1.0
1.5
2.0
~~
In (kmlx)
L : class. H-plasmas
L : qum. H-plasmas
L : class. Cs-plasmas
L : qum. Cs-plasmas
Q : class. H-plasmas
Q : gum. H-plasmas
Q : class. Cs-plasmas
Q: qum. Cs-plasmas
R/Rkin gum. H-plasmas
R/Rk, qum. Cs-plasmas
3.401
3.402
3.385
3.374
3.351
0.017
0.017
0.017
0.017
1.012
1.002
1.792
1.805
1.789
1.777
1.756
0.083
0.080
0.079
0.076
1.085
1.061
1.099
1.151
1.138
1.129
1.112
1.167
0.153
0.151
0.138
1.201
1.174
0.693
0.805
0.794
0.788
0.775
0.250
0.219
0.216
0.189
1.467
1.379
~
0.405
0.589
0.581
0.577
0.567
0.333
0.278
0.274
0.228
1.987
1.813
It is to be seen that the effects of deviations from Coulomb's law and of quantum
effects are small and do not exceed a few percent in the case of L and twenty percent
in the case of Q within the region ,u 5 2. The numericdy most important effects are
28'
W. EBELINQand G. ROPKP~
436
connected with the dynamical screening and with the Debye-Onsager relaxation force.
These effects give rise to a factor L(l - Q)-l in the conductance formula which may
deviate from the value of the classical kinetic theory by about +lOOyo at y = 2 and
by about +20:(, at ,u = 1. Therefore the resistivity values calculated here are remarcable
higher than those predicted by the classical kinetic theory. Introducing the following correction factor
to the kinetic theory, this factor can read values up to 2 as shown in Table 1.
A comparision between experimental values and different theoretical approaches
to the conductivity of a plasma has been performed in [l, 191 where it is shown that the
experimental values of the resistivity are higher than the values obtained from the
Spitzer theory and improvements of it.
The theoretical values calculated here are in the middle between the predictions of
the classical kinetic theory and the experimental values for the resistivity which are
still higher. Therefore there must be additional effects which increase the resistivity of
real plasmas which are not jet taken into account here.
The authors would like to thank R. BALESCW,
Yrr. L. KLIMONTOVICH
and W. D.
K R A Efor~ fruitful discussions.
References
[l] S. G. BAROLSIUI,
N. V. Y E R M O P.
~ P.
, KULIK,and V. A. RUBIT,Teplofiz. Vys. Temp. 114,
702 (1976);
S. G. BAROLSIUI,N. V. Y E R Y O ~B., M. KOVALIOV,
P. P. KULIK,and V. A. RIABII, Proc.
XII ICPIG Eindhoven I. 3., 181 (1975).
[2] M. SKOWRONEP,
J. Rous, A. GOLDSTEIN,and F. CABANNES,
Phys. Fluids 18, 378 (1970).
[3] R. RADTKEand K. GUNTBER,
Beitr. Plaemaphysik 16, 299 (1976); J. Phys. Lond. D 9, 1131
(1976).
[4] G. E. NORMAN
and A. A. VALUEV,
Proc. XI1 ICPIG Eindhoven 11. 1. a 257 (1975); Teplofiz.
V p . Temp. 16,191 (1977).
[5] W. EBELING,
W. D. KRAEFT,and D. KREMP,in: Phenomena in Ionized Gases. Invited Papers.
Berlin 1977.
[6] L. SPITZER,
Physics of Fully Ionized Gases. New York;
L. SPITZERand R. H ~ R MPhys.
,
Rev. 89, 977 (1953).
[7] S. G. BRAGINSKI, Zh. exp. teor. Fiz. 88, 459 (1957).
and B. S. LILEY,Rev. Mod. Phys. 82, 731 (1960).
[8] R. HERDAN
Statistical Mechanics of Charged Particles, London 1963.
[9] R. BALESCU,
[lo] Yu. L. KLIMONTO~CH,
Statisticeskaja teorija neobratimych prozessov, Moscow, Nauka, 1975.
[ll] Yu. L. KLLWONTOVICH
and W. EBELINQ,Wiss. Z. Univ. Rostock MNR 11,355 (1962); Zh. exp.
teor. Fiz. 48, 146 (1962); 68, 905 (1972).
[12] D. N. SUBAREW,
Statistische Thermodynamik des Nichtgleichgewichts. Berlin 1976.
[13] G. ROPKEand V. CHRTSTOPH,J. Phys. C 8, 3615 (1975).
[14] G. ROPKE,Physica 88 A, 144 (1977).
[15] V. CHRISTOPII and W. SCHILLER,
phys. stat. sol. (b) 86, 231 (1978).
Conductance Theory of Nonideal Plasmas
437
[16] G. ROPKE,C.-V. MEISTER, K. KOLLBIORQEN,
and W.-D. KRAEFT,
Ann. Physik (Leipz.) 86,377
(1979).
[17] W. EBELINQ,
C.-V. MEWTER,
R. SANDIG, and W.-D. KBAEFT,Ann. Phpik (Leipz.) 86, 321
(1979).
[18] B. B. KADOBITSEV,
2%. exp. teor. Fiz. 33, 151 (1957).
[19] Yo. V. IVANOV,
V. B. MINZEV,V. E.FORTOV,
and A. N. DEEMIN, Zh. exp. teor. Fiz. 71, 216
(1976).
Bei der Redaktion eingegangen am 25. April 1979.
Anachr. d. Verf.: F’rof. W. EBELINQ
Sektion Physik der
Humboldt-Universitit Berlin
DDR-1086 Berlin, August-Bebel-Platz 2
Dr. G. ROPKE
Sektion Physik der Wilhelm-Pieck-Univ.
DDR-25 Rostock, Universitiitsplatz 3
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