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On the transport properties of a dense fully-ionized hydrogen plasma. I. The semi-classical approach
V. Zehnlé, B. Bernu, J. Wallenborn
To cite this version:
V. Zehnlé, B. Bernu, J. Wallenborn. On the transport properties of a dense fully-ionized hydro- gen plasma. I. The semi-classical approach. Journal de Physique, 1988, 49 (7), pp.1147-1159.
�10.1051/jphys:019880049070114700�. �jpa-00210797�
On the transport properties of a dense fully-ionized hydrogen plasma. I.
The semi-classical approach
V. Zehnlé (1), B. Bernu (2) and J. Wallenborn (1)
(1) Chimie-Physique II, Association Euratom-Etat Belge, C.P. 231, Université Libre de Bruxelles,
1050 Bruxelles, Belgium
(2) Laboratoire de Physique Théorique des Liquides, Unité associée au C.N.R.S., Université Pierre et Marie
Curie, 75230 Paris Cedex 05, France
(Requ le 23 décembre 1987, accepté le 24 mars 1988)
Résumé. 2014 Dans le cadre d’une théorie cinétique classique on fait une étude des conductivités électrique et thermique d’un plasma d’hydrogène complètement ionisé valable dans un large domaine de températures et de
densités. Les effets quantiques sont pris en compte au moyen de potentiels effectifs. On montre que les résultats de la théorie cinétique sont en accord avec ceux de la dynamique moléculaire qui ont été obtenus pour des systèmes fortement couplés. On donne des formules analytiques valables dans le domaine du faible
couplage. Des propriétés inattendues du potentiel modèle qui tient compte des effets de symétrie quantique
sont discutées.
Abstract. 2014 A classical kinetic theory is used to study the electrical and thermal conductivities of a fully
ionized hydrogen plasma for a wide range of values of temperature and density. Quantum effects are taken
into account by effective potentials. It is shown that the kinetic theory results are in agreement with those of molecular dynamics which are available for strongly coupled systems. Analytical formulas valid in the weak
coupling domain are given and odd properties of the model potential which accounts for quantum symmetry effects are discussed.
Classification
Physics Abstracts
05.60-51.10-52.25F
1. Introduction.
During the last decade, much work has been devoted to study the properties of strongly coupled plasmas [1] in connection with the physics of inertial confine- ment fusion and of stellar interiors. Much is now
known about the equilibrium and dynamical proper- ties of plasmas for values of temperature and density covering a large domain of the fluid phase. In
absence of experiments, these results were obtained either by numerical simulations or by analytical or semi-analytical calculations. The agreement between these two very different kinds of approach have given confidence in both. Specially, in the case of the strongly-coupled classical one-component plasma (OCP), the values of the transport coefficients obtained by molecular dynamics [2] and by kinetic theory agree [3, 4] fairly well and have brought on
the extension of these methods to the study of hydrogen plasma properties. However, a hydrogen plasma is quantum mechanical even at high tempera-
ture and low density since the Heisenberg uncertain- ty principle is necessary to keep the electrons from
collapsing into the ions. The hydrogen plasma, and
other multicomponent plasmas, thus cannot be de-
scribed in the framework of classical mechanics unless the Coulomb potential is replaced by some
effective potential which accounts for quantum ef- fects.
The inclusion of quantum effects in the classical
partition function with the help of a temperature- dependent effective interaction potential was first suggested by Uhlenbeck and Gropper [5] in the early
1930’s. This procedure is clearly justified to study equilibrium properties [6, 7]. Recently, effective potentials have also been used to study the non- equilibrium properties of a strongly-coupled hydro-
gen as a classical system [8-11]. It is questionable
however that potentials constructed to evaluate static properties will be adequate for the calculation of dynamical properties. The justification can only
be given by a pure quantum kinetic theory [12].
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070114700
Few attempts have been made to elaborate a
kinetic theory valid for strongly coupled plasmas including all quantum effects [13]. Due to the complexity of the theory, some quantum effects
must be neglected or approximated [14, 15] in the quantitative evaluation of the transport coefficients.
To test the domain of validity of the semi-classical
approach, we have thus chosen to proceed with weakly coupled systems of whose all quantum properties can be analyzed in detail. In particular,
for the first time, we have studied the effect of quantum statistics on electron-electron collisions from first principles, without simplifying assump-
tions, in the transition domain where the tempera-
ture is approximately the Fermi temperature. We argue that our criticisms of the semi-classical ap-
proach, established for a weakly-coupled plasma,
can be confidently extrapolated to strongly coupled systems. Our conclusions do not depend explicitely
on the value of the coupling parameter.
In the present paper (denoted as I) we will try to convince the reader that a semi-classical evaluation of the transport coefficients of a fully ionized hydro-
gen plasma exhibits unusual properties independent
of the method used, molecular dynamics or kinetic theory. We also shall show that these unusual
properties originate in the part of the effective potential which describes the quantum symmetry or quantum statistics. In the next paper (II) an explicit comparison between the present semi-classical and the pure-quantum approaches will be carried out.
We shall calculate the thermal conductivity of a weakly coupled electron gas, the simplest model exhibiting all quantum effects. This will also give the
contribution of electron-electron collisions to the heat transport in a multi-component plasma which
has generally been neglected in the study of the transport properties of quantum plasmas.
The contents of I are organized as follows. In
paragraph 2, we define two semi-classical models of the hydrogen plasma corresponding to two different
effective potentials and discuss their domain of
validity. The classical kinetic theory of strongly coupled plasmas is briefly discussed in paragraph 3
and is applied in paragraph 4 to the semi-classical models in order to evaluate the transport coef- ficients. These are compared to the molecular
dynamics results. In paragraph 5, we show how to
recover the previous results with Landau theory [16]
in the weak coupling domain. This allows us to
analyse the unusual behaviour of the transport properties. ,with. simple analytical expressions. We
finally comment briefly in paragraph 6 on tem-
perature-dependent potentials.
2. The effective interaction potential.
The thermodynamic state of a fully ionized hydrogen plasma made up of equal numbers of ions and
electrons of density ni = ne = n at temperature T - (kB 9 )- 1 can be characterized by two dimen-
sionless parameters which we choose to be the coupling parameter :
and a density parameter :
In these definitions e is the ionic charge, a = (3/4 rrn)"3 is the ion-sphere radius and ao =
h2/Me e2 is the Bohr radius.
The effective interaction potential vab (r) between
two particles of species a and b (a, b = e, i :
electrons or ions) distant of r is defined as [5-7] :
where gab (r) is the radial distribution function when the pair of interacting particles is removed from the surrounding plasma. It is known that a classical estimate of gei(r) based on the Coulomb potential diverges when the electron-ion interdistance van-
ishes. Quantum effects only are able to avoid this
unrealistic tendancy towards the collapse of the system. However, when gab (r) is quantum mechani- cally evaluated, the effective potential (2.3) can be
used in the classical partition function in order to obtain the thermodynamics of the system.
Since the pioneering work of Uhlenbeck and
Gropper [5] a lot of work was devoted to the computation of the effective potential [6, 7]. Re- cently, Minoo et al. [7] have proposed an analytic interpolation between the exact effective potential at
short distance and the Coulomb potential at large
distance which compares well with numerical evalua-
tions ; their expression of the effective interaction
potential between species a and b can be split into
two parts : one which accounts for the quantum diffraction effects and the bare Coulomb interac- tion :
and one which accounts for the quantum symmetry
(exchange) effects of electrons
In expressions (2.4) and (2.5), iB ab is the thermal de
Broglie wavelength of the pair (a, b ) :
In what follows we shall consider two models of the
hydrogen plasma characterized by interaction poten- tials constructed with (2.4) and (2.5) :
As was expected, the effective potentials (2.7)
remain finite as r --> 0 and differ notably from the
Coulomb potential on a distance of the order of the de Broglie wavelength. Such simple models however,
are not valid in any domain of the parameters rand rs as they were obtained from assumptions we now
summarize.
From the definition (2.3), it is already clear that
the effective potential although temperature-depen-
dent does not take into account any collective effect : on one hand, this means that the screening at large distance must be treated in the same way as in the case of the bare Coulomb potential (1), on the
other hand, the potential is pairwise additive i. e.
three-body (or more) interactions due to the overlap- ping of three (or more) wave packets are neglected.
This assumption is valid if the density is such that there are no more than two particles in a de Broglie sphere or equivalently if the system is not strongly degenerate :
where TF is the electron Fermi degeneracy tempera-
ture. Note that this condition is compatible with a density-independent potential.
When obtaining the simple expressions (2.4) and (2.5), s-scattering states only have been considered
and the bound states have been neglected. This last assumption is valid for temperature higher than the
ionization temperature (kB T > 1 Ry ) or
(1) Klimontovich and Kraeft [17] however have derived
an effective potential for an electron-ion pair which takes into account static screening.
Fig. 1. - Domain of validity of the effective potential as given with conditions (2.8) and (2.9).
The symmetry part of the effective potential (2.5)
comes from an average radial distribution function for electrons with parallel and antiparallel spins. A
more refined model which treats separately spin up and spin down electrons has also been proposed and
tested [11, 18]. We shall not consider this model here
as it gives qualitatively the same doubtful results we want to discuss. We rather shall compare kinetic
theory results obtained from the use of the diffrac- tion potential v d alone (model I) and those obtained
from the full potential va6 + Uab (model II).. It is expected that model I will be valid as long as the
electrons are not degenerate i.e. T > TF or r Irs $
0.5 which restricts the ( T, rs ) domain to more
classical values than in equation (2.8).
3. Classical kinetic theory.
’
The transport properties of a classical fluid can be
obtained from the kinetic equation obeyed by the two-point equilibrium correlation function [19] :
Sal a2 (k, z ; pl, p2) is the Fourier-Laplace transform of the equilibrium correlation function of the phase-
space density-fluctuation :
where (p,,,(pl) is the Maxwell distribution function and n «1 the number density of species a 1.
S« 1 «2 (kl, pl, PZ ) = S«, C’2 (k, t = 0 ; Pi? p2 ) is the initial condition of the two-point correlation function :
where we have introduced the static structure factor Sal a2(k). The exact memory kernel can be split into
three parts,
a free-streaming term
a mean-field term :
with C al a2 the direct correlation function and a collisional term which can be written in the form (coming
back to ordinary space and time variables) :
where LI (12 ) is the two-body interaction operator while C (11, 22) is a contracted four-point corre-
lation function which is constructed by using three-
and four-point generalisations of S(12) [19].
In order to study the dynamical properties of the moderately coupled hydrogen plasma we shall ap-
proximate the collisional memory kernel (3.6) in full analogy with the case of the OCP [4]. In the expression (3.6) we replace C (11, 22 ; t) by its
disconnected part :
and renormalize one of the interaction operators by
means of equilibrium correlation functions in such a
way that the initial condition Xc , 1 a2 ( 12, t = 0 ) of the
collisional memory kernel is preserved (asymmetric renormalization). The collision is then nearly exactly
treated at short time while the approximation (3.7) is equivalent to neglecting close collisions for a longer
time. The resulting memory kernel generalizes the
linearized Balescu-Guernsey-Lenard collision operator for finite values of k, z and r [20]. This procedure of asymmetric renormalization has been shown to be valid for a moderately coupled plasma [4, 9, 21].
In this approximation scheme, the collisional memory kernel of the OCP is straightforwardly generalized to a multicomponent plasma :
where V,b is the Fourier transform of the interaction
potential between species a and b. Aal a2 is a part of
the memory kernel proportional to k. pl, which does not contribute to the evaluation of the desired transport coefficients.
With the expression (3.8) of the memory kernel,
the kinetic equation (3.1) becomes a closed set of equations for S(12, t) which can be solved by
iteration in terms of the initial condition i.e. the
equilibrium correlation functions. In that way the
simplest approximation for S(12, t ) is obtained by canceling in equation (3.1) Z’ and Xc ; it can be
called the free-streaming (FS) approximation :
However, SFS doesn’t possess all the symmetries of
the exact solution. In particular Sa a2(k, z ; pl,
PZ) #: Sa al(k, z ; p2. pi). Only the weak-coupling
limit SWO of SFS is not ambiguous :
As we want preserve the exact initial condition of all the dynamical quantities in the case of the strongly coupled plasma, we will consider the mean-field (MF) approximation of S ( 12 ) which is obtained by canceling ’¡C in equation (3.1) and which possesses all the needed symmetries :
where we have introduced the dielectric function matrix :
In the next section, we shall use the collisional memory kernel (3.8) with the mean-field approxi-
mation (3.11) to evaluate the thermal and electrical conductivities of the hydrogen plasma. It should be pointed out that the weak-coupling limit -Vwc of expressions (3.8) is obtained by approximating S (12, t ) by equation (3.10) and C ai aj (l ) by
-13V"i"j. It can then easily be shown that
Ewc a, -Of2 (k = 0, z - + i0 ; pl, P2) is the linearized
Landau collision operator used in Braginskii’s paper
[16].
4. Thermal and electrical conductivities of the semi- classical hydrogen plasma.
The long-wavelength modes of a two-component plasma have been obtained on the basis of the kinetic equation (3.1) [22] and of macroscopic hydro- dynamic equations [23]. The comparison between
the two approaches gives a microscopic definition of the transport coefficients in terms of matrix elements in momentum space of the memory kernel.
4.1 THERMAL CONDUCTIVITY. - Using the stan-
dard notation, summarized in the appendix A, the thermal conductivity in absence of electrical current
can be written as (2) :
where
) c) is the kinetic energy state and Q projects onto
the subspace orthogonal to the multicomponent hydrodynamical states (see appendix A). The ex- pressions (4.1)-(4.4) are quite general and are in fact
valid for any fluid (as far as the memory kernel is not
specified). For a moderately coupled hydrogen plas-
ma, molecular dynamics simulations have shown
[10] that the kinetic contribution to the thermal
conductivity dominates. This contribution corres-
ponds in the Green-Kubo integrand to the autocorre-
lation of the kinetic heat flux alone. With the present formalism of the memory kernel, the kinetic part of the thermal conductivity is expressed with n ind only
and takes the form :
In what follows, we shall consider equation (4.5) as
the starting point for the evaluation of the thermal
conductivity K of the hydrogen plasma (we thus shall drop the supercropt kin from K kin).
To proceed, we must invert the collisional memory kernel X(0, 0 ; p, p ‘) in the space orthogonal to the hydrodynamical variables. This can be achieved
using a limited polynomial expansion of X’. In the weak-coupling limit Baginskii [16] has shown that the two-Sonine-polynomial (or equivalently the 21 moments) approximation satisfactorily reproduces
the exact value of the plasma thermal conductivity (2) We have restored the correct factor in front of the
right-hand side of equation (4.1). It differs from equation (4.1) of reference [22b] by a factor COICV, the ratio of heat capacity of a perfect gas to that of the non-ideal
plasma.
[24]. Therefore we restrict our calculation to the
same approximation ; for symmetry reasons it is then sufficient to consider the space spanned by the following polynomials (k = klx ) :
in the one-Sonine-polynomial approximation. In the two-Sonine-polynomial approximation one has also
to consider :
Taking into account that ESlEfs l2a=
(2a 1 -VFs 1 £) = 0 as can be easily shown, and intro- ducing into equation (3.8) the following dimension-
less quantities >
equation (4.5) becomes :
where the functions S§f§ are defined in appendix A
as matrix elements of S(12, t ).
We have numerically computed the matrix ele- ments (4.11) of 2 within the two supplementary approximations : (i) the dynamical functions
S§f§ (l , t ) are replaced by their mean field approxi-
mation deduced from equation (3.11) ; as was al- ready pointed out this approximation satisfies unam- biguously all the needed properties of I. Moreover it avoids any divergence problem since it takes into
account the dynamic screening at large distance ; however, to avoid numerical problems some algeb-
raic transformation of the matrix elements is necess-
ary (see appendix B) ; (ii) the equilibrium corre-
lation functions are evaluated in the hypemetted
chain (HNC) approximation which is known to reproduce rather well the equilibrium properties of
the hydrogen plasma [8]. When the interaction
potential is chosen, as model I or model II equation (2.7), the equilibrium functions as well as the matrix
elements of 2 are then determined.
The value of the thermal conductivity is then
obtained by inverting in the two-Sonine-polyno-
mial approximation (3). The results for the strong- coupling domain ( T > 0.1 ) are summarized by
table I, where the dimensionless quantity K * _ K / (kB Cope na 2) is displayed (w pe 2 = 4 7Te2 n/me) .
K ll (3) is little sensitive to the potential model ; its
common value for both model I and model II is thus entered in the table. On the other hand, as
K ei is everywhere less than 1 % of the total thermal
conductivity K, we don’t indicate its values.
Inspection of table I shows that our results are
systematically but slightly, higher than those ob- tained by molecular-dynamics simulations of model II [10]. Since the size of the molecular dynamics
error bars is probably optimistic both methods are in
reasonable agreement. The main point is that the
value of the electronic part of the thermal conduc-
tivity is lowered by the symmetry effects included in
(3) The one-Sonine-polynomial approximation for
model II is given in reference [11].
Table I. - The reduced thermal conductivity K * = K/(kB lùpe na2) in the strong coupling domain.
K *’, is the molecular dynamics result obtained for model II [10]. K *1 and K ee*II are the electronic part obtained
from kinetic theory in the 2d-Sonine-polynomial approximation for model I and model II, respectively.
Kif is the ionic part. K * corresponds to the results of reference [25]. Columns A and B give the ratio of the
second- to the first-Sonine-polynomial approximation for models I and II, respectively.
model II. This will become even more evident in the weak coupling limit.
The results of Cauble and Rozmus [25] are nearer
to those of model I than to those of model II possibly
due to the use of the full potential (2.4) to evaluate equilibrium functions while the potential which
appears explicitely in equation (3.8), is taken from
equation (2.5) and thus symmetry effects are neg- lected. But it is also possible that the agreement
comes from the artificial symmetrization of y which
was made in connection with the FS approximation (3.9) of the dynamics. There are too many unjus-
tified approximations in this calculation.
4.2 ELECTRICAL CONDUCTIVITY. - A lot of work has been concerned with the electrical conductivity
of a strongly coupled hydrogen plasma [9, 14, 15, 25] ; we thus shall be rather concise on that subject.
Nevertheless we shall point out some peculiar properties that appear in connection with the use of the effective potentials.
The electrical conductivity takes the form [22c] :
where IL ei is the reduced mass of the electron-ion
pair and
with It e) the electron longitudinal momentum state defined in appendix A. Using the approximate mem-
ory kernel (3.8) the expression (4.15) becomes :
Retaining only this contribution (f2,,)dir in the
evaluation of the electrical conductivity (4.12) is equivalent to the so-called one-Sonine-polynomial (or five moments) approximation. It is interesting to
remark that this approximation does not depend explicitly on the electron-electron interaction poten- tial. The quantum symmetry effects can only appear
through the correlation functions. It is thus expected
that these symmetry effects will more influence the two-Sonine-polynomial approximation. This one
needs the consideration of ( f2 ee )ind which is written
as :
where i is defined by (4.11) and where
As in the case of thermal conductivity, we have
evaluated the electrical conductivity by approximat- ing the dynamical functions by their mean field limit and all the static correlation functions by the HNC
scheme. Our results in the strong coupling domain
are displayed on table II with those of references [8,
9, 10, 25].
Once more, our values for the electrical conduc-
tivity of the hydrogen plasma are in good agreement Table II. - Reduced electrical conductivity u * = u / f2 p (f2 p = 4 7re’ n/ 11 ei) in the strong coupling domain
for model I and model 11. cr * is the molecular dynamics result [8, 9, 10] ; a * is our result in the second-
Sonine-polynomial approximation ; QSHp is the theoretical result of reference [9]. A and B give the ratio of the
second- to the first-Sonine-polynomial approximation for models I and II, respectively. u:c is the theoretical result of reference [25].
