Linear electronic transport in dense plasmas. II. Finite degeneracy contributions

(1)

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Linear electronic transport in dense plasmas. II. Finite degeneracy contributions

D. Léger, C. Deutsch

To cite this version:

D. Léger, C. Deutsch. Linear electronic transport in dense plasmas. II. Finite degeneracy con- tributions. Journal de Physique I, EDP Sciences, 1991, 1 (6), pp.837-853. �10.1051/jp1:1991172�.

�jpa-00246373�

(2)

J

Phys.

1 1

(1991)

837-853 JUIN 1991, PAGE 837

Classification

PhysicsAbsmc1s

05.@W-05.30d-5Z25Fi

Linear electronic transport ⁱⁿ ^dense plasmas. II.

Finite degeneracy contributions

D.

lAger(I)

and C.

Deutsch(2)

(1)

Laboratoire des Mat6riaux

Min6raux,

Conservatoire National des Arts et

M6tiers,

292 Rue saint Martin, 75141 Paris Cedex

03,

France

(2)

Laboratoire de

Physique

des gaz et des

plasmas(* ),

Universit6 Paris XI, Bfitiment 212, 91405

Orsay cedex,

France

(Received

9 Novetnber

199( accepted

in

final fonn

13

Febmaiy

1991)

R4sum4. Le formalisme

expos6

et d6tail16 dans le

premier

article de ceIte s6rie est ici

appliqu6

la d6termination des contributions de

d6g6n6rescence partielle

aux coefficents de transport thermo-

6lectroniques

et

m6canique (viscosit6),

coefficients

pr6alablement exprim6s

sous forme

d'expressions

rdduites. Los corrections de

temp6rature

finie sent

syst6matiquement analysfes

en relation avec (es

propri6t6s analytiques

de la fonction

d161ectrique

du

jelli

um. Alors que ^{cel le de} Thomas-Fermi fourni t

l'exemple

type de fonction

parfaitement r6gulidre

en q =

2kF,

celle de Lindhard et sa

g6n6ralisation

T finie sent au contraire caract6r1s6es par ^des ^d6riv6es

divergenIes

en ce

point.

^Des ^m6thodes

sp6- cifiques

sent

ddvelopp6es

_pour traiter correctement ces us

importants.

Nos r6sultats sent

pr6sent6s

sons forme de

d6veloppements analytiques

en

puissance

du

param~tre

de

d6g6n6rescence

_a, et des

expressions

exactes pour ^les corrections mentionndes sont ddriv6es

jusqu'A

l'ordre

a~.

_tour _traduc.

tion

num6rique

est finalement

appliqu6e

au cas des

m61anges

binaires

proton-helium complttement ion1s6s,

d'int6rlt

Astrophysique.

Le lien entre le

pr6sent

formalisme, et notamment ses

implications

num6riques,

et d'autres r6sultats

ant6rieurs,

fait aussi

l'objet

d'un _examen attentif.

Abstract. The formalism described in the first paper ^{in this} ^series ^is ^hereafter

specialized

to a thor-

ough investigation

of finite

degeneracy

contributions to thermoelectronic and mechanical transport

coefficients, conveniently expressed

_as reduced

quantities. 'lbmperature

corrections _are

systematically

discussed

through

the

analytical properties

of the

jellium

dielectric function, the Thomas-Fermi _one appears as a

paradigm

^of

regular

behavior _{at q} ₌

2kF

while the Lindhard and its

T-dependent

exten.

sion head a

singular

class characterized

by diverging

derivatives.

specific

^methods are

developed

for these

important

cases. Results _are

presented

in terms

ofanalytic expansions

in the

degeneracy

parameter a, and exact

expressions

for the above-mentioned corrections are derived up to order

a~. Finally

we

display

a number of numerical results

pertaining

to

fully

ionized

proton-helium binary

mixtures of

Astrophysical

interest. the connection of the

present

formalism and its numerical outputs ^with ^other

carefully

examined.

(*)

Assoc16 _au C.N.R.s.

(3)

838 JOURNAL DE PHYSIQUE I N°6

1. Intrtduction.

Thb

paper

^is ^devoted ^to ^the

analytical

derivation and numerical evaluation of finite

degeneracy

contrlutions

(FDC)

to electronic

transport

ⁱⁿ

strongly coupled plasmas consisting

of _a

fully

or

onlypanial~y

ionized

multicomponent

ionic mixture embedded in_a

highly degenerate

electron gas.

The classical ion

plasma

and the electron

jellium

are

parametrized

with r ₌

fle~ la; (fl

₌ I

/kBT)

and _rs ₌

aelao (ao,

Bohr

radius) respectively.

_a; or ae are related to the ion _or electron

density according

to ai,e =

[3/(4xni,e]1/3.

Finite

degeneracy

^effects are

actually

measured

by

a =

T/TF (TF,

Fermi

temperature).

Inelastic electron-ion

scattering

effects _are characterized

by

^a ₌

flhw.

In

previous _paper

I in this series

ill,

which we will hereafter refer to as

I,

the theoretical framework for

computing

all the

time-independent

electronic

transport

coefficients in such

plasmas

was laid

down

through

the Boltzmann-Ziman

equation

and the Lorentzian

approximation.

^The ^latter rests on a few number conditions _on the above-mentioned basic

parameters (see

Sect. 2 in

I),

which results in _a net

decoupling

of inelastic effects from finite

degeneracy

ones.

Therefore,

these two

types

^of corrections can be

safely

calculated

separately;

inelastic

scattering

will

investigated

in _a

forthcoming _paper

III in this series.

In this

work,

we now

emphasize

FDC to the standard _a

- 0 and D

- 0 results

dbplayed

in

Equations (4.20a-d )

of I. The relevant r _rs range ^where ^those contributions become

significant

is that of domain A delineated in

figure

I in I.

Presently,

_our

goal

is to derive the first

non-vanishing

a-corrections _to therrnoelectronic and mechanical

transport

coefficients. As _we shall see, those

are

only

of order

a~,

and

they

remain

numerically

accurate up ^to o. ~-

30ill;

this

explains

the

major

interest of _an

ana~yfical approach.

^At ^this

stage,

it

proves

use ful _to

get

a further

insight

into one of the basic

assumption

^of the Lorentzian

approximation, namely

the

neglect

^of ^electron-

electron collisions. As is well

known,

the latter

play

no role in the net evaluation of the electrical

conductivity,

due to the conservation of the electrical _current

density

in such

scattering

events.

lbrning

to the thermal

conductivity,

_we know from

Lampe's analysis

_{[2] that} electronic interactions

can

significantly

lower it when _a becomes

typically larger

than

4>e,

where

>e

₌

(me~ /xhpf)

^~~~

(kF

₌ _pF

/h,

Fermi

wavevector)

defines _a convenient

electron.jellium plasma parameter.

Thus

one can

safely disregard

e- e- collisions up ^to a m

1.283rj~~,

the "effective Fermi-Maxwell

transition",

_a bound which stands well

beyond

the scope of

present analysis (a _j~

0.30 and _rs

j~

^I

).

Linear

transport

^restricted to its elastic limit is

investigated

in section 2

through

an

analytic

a

-expansion

of basic

quadratures Ki(~

_over

generalized

relaxation limes.

[n

^these

calculations,

a

key

role is

played by

the derivatives of the effective _e- -ion interaction

V(q)

_c~

I/(q~e(q))

at twice the Fermi

kF-diameter.

two situations have then to be

considered, according

to the dielectric function

e(q)

within

I(q)

_is

_regular

_{at q}

=

2kF

_or not. In _a first

step, (Sect. 2.2),

_we

investigate o~-FDC

with _a Thomas-Fermi-like

~T~ jellium

^dielectric

function,

which

obviously

affords _a standard of

regular

behavior. It turns out however that FDC derived in this fashion _are also

fully

relevant for other dielectric

functions,

as

long

as the electron

jellium

remains

largely degenerate,

I. e.,

typically

_rs

_j~

^I. ^This

explains

the interest for

strongly focusing

_our attention to this

simple

case. Section 2.3 is devoted to the

opposite

^situation where the derivatives off

(q) diverge

at

=

2kF. Specific

methods have thus to be constructed in order to resum the

given

infinities in the

lhylor expansion

^of

quadratures Ki',~ (Ref. [3]).

^The

required "upper-bound"

formalhm is outlined in section 2.3.I and

Appendix

B. The

underlying

mathematical

developments

can be found elsewhere [3]

(and

D.

IAger

and C.

Deutsch,

to be

published).

A careful attention is

given

to the Lindhard

~L)

^dielectric ^function

[4],

^valid ⁱⁿ ^the a - 0

limit,

and its

finite-degeneracy

^extension

by

Gouedard-Deutsch

(GD) @.

The

corresponding transport

coefficients _are next derived _ana.

~yfical~y

up

^to ^order

a~,

which

completes previous

^results

pertaining

to the TF _case

(SecL 2.2).

A

straightforward

extension of the

present

^results ^to more involved dielectric functions is also

(4)

N°6 LINEAR ELECTRONIC TRANSPORT IN DENSE PLASMAS. II 839

investigated.

Finally _(SeCL 3),

we

analyse

the numerical

outputs

^of ^the

present

^formalism

implemented

in

strongly coupled

and

fully

ionized

H+ He~+

_mixtures. _Such

a

study displays

an obvious and twofold interest _:

first,

we shall be able to sort out all the

specific

features of _our finite

degeneracy

calculations without any additional and

obscuring

effect due to hard _core and short

ranged

ion- ion

interactions, although

our formalism

applies

to that _case as

well; second,

the results

given

and those

displayed

in the next

paper

^{III in} ^thb

series,

will be of obvious

utility

in the evolution

modelling

of

giant planets

such as

Jupiter

_or Saturn

[6,7j

^which are

essentially

fluids made of

strongly coupled proton-helium

mixtures

[8].

^In ^this

connection,

it is worthwhile to recall _some recent work

by

Hubbard

[6,7j

^and ^his

colleagues relating

the

planetary equation

of state to its observed

excentricity.

Our calculations are

performed

within the so-called

binary

ionic mixture

(BIM)

framework where the electron gas ^b taken as

mechanically rigid

and

non.responding.

Such

a model

proves

^to ^be

adequate

as

long

_as _rs remains

typically

smaller than

unity,

a feature which

prevails

in the

deep

fluid interior of

giant planets

_{[8]. It is} worth men

tioning

however that _a correct

description

of the

expected

critical

demixing

of such

binary plasmas

is

only possible

within the

polarized

version of the BIM

(PBIM) [3, 9] including

electron

screening

within the interionic

potentials.

^Ion-ion structure factors _are

computed through

the

hype

me tted -chain

(HNC) scheme, by neglecting brigde diagrams.

^Results ^for the thermoelectronic

transport

coefficients and the shear

viscosity

_are

investigated through

their r _rs

dependence. Finally,

the

present

^results ^for the

hydrogen plasma

are

compared

to those of other

authors, namely

Ichimaru and lhnaka

[10].

In the

sequel,

_any reference to an

equation (*)

in I will be denoted _as

((*)

_1). ^> ₌

(9~/4)1/3

will stand for a numerical _constanL

%ansport quantities

will be measured in atoinic units

(a.u.), namely

_ao ₊

pi

^I = e~

laoh

and _~o

= h

la(

for electrical

conductivity

and

viscosity respectively.

2. Linear

transport

at low

degeneracy (a £ 0.3).

2. I NomTIoNs. We _now intend to

apply

the dimensionless formulas

((4.12-4.13),

₁₎ to the calculation of FDC to thermoelectronic coefficients and the shear

viscosity.

Our

methodology

relies on the reduced variables

(See

Sect. 2.I in 1):

z =

q/2kF,

t ₌

k/kF,

_u

=

fl(Ek Fe),

_n

=

11/~,

a =

~a/2. (2.1)

and on

generalized

inverse relaxation times

r(I(k)

~-

tL;(1) (Eq. (4.10, 1)),

which

provide

er- act solutions of the linearized Boltzmann

transport equation

at any ^T ⁱⁿ ^the pure ^elastic ^limit.

They

involve _a "one-fluid" ion-ion _structure factor

S(z)

and the

square

ôf ân êffective ê- îon interaction

V(z) (Eqs. (3.4), (4.3)

^and

(4.8), _1).

For the sake of

convenience,

let us introduce now

the

following

notations _:

V(z)

₌

u(z)Vo(z)

with

Vo(z)

₌

~ ,

(x(~

₌

0.166rs), (2.2)

z +

z~pg(z) V~(z)S(z)

=

G~(z)i~(z)

_~

i~(z)

=

u~(£)S(z), _(2.3)

T =

<~(z

=

i)

₊

e(z

₌

i)

= i +

zlfg(i),

=

(T i)/T. (2.4)

Vo(z)

^stands for the screened Coulomb contribution within

V(x)

described in terms of an

appro-

riate

pseudopotenfial u(z)/z~.

Function

i2(z)

is assumed to be

regular (analytic)

at _z ₌ I.

e(z)

is the static

jellium

dielectric function endowed with

z§

₊

_{k(~ /(2kF} _)~,

normalized Thomas-Fermi

wavenumber,

times the

screening

^function

g(z).

In

present work,

we shall

mostly

concentrate on

(5)

the Thomas-Fermi-like

~TF~

^function

g(z)

=

Cste,

the Lindhard function

(L) (Ref. [4]) gL(z),

and its finite «-extension

by

Gouedard-Deutsch

(GD)

_[5] :

Where ₇ ₌

Pe/EF, Ao

₌

j ⁷

+

17~

+

4«~) _~/~l

^~~~ and

Bo(7)

₌

Ao(-7).

The whole GD function [5j is a bit _more involved than

(2.5)

which

represents only

_a first term within _an

asymptotic expansion. Nonetheless,

the latter b

already

sufficient to extend

correctly

the

Lindhard function at small _a

(Ref.

_[5j with the obvious result

limo-

ogGD

z)

₊

gL(

_z

).

^FDC to the

investigated transport

coefficients will be hereafter

analyzed through

a

straightforward

extension of basic

quadratures If

and

If, _parameter (° (Eqs. (4.19a,b), I),

and the Lorentz number

£°

=

(x~ /3)(kBle)~

,

rewritten in the

compact

^form :

p =

(16/9)r(

Z

poI), 1(~~

=

(16/9)r(

_Z

po(£°T)~~I) (2.6a)

Thus,

_the uantities II and I)

belong

to ^ectrical

esistivity and

inverse

thermalconductivity

spectively, /f to the

shear and (T

to

the

The

2.2 _~x~ coRREcrIoNs : cAsE oF REGULAR TF-LIKE

g(z)

FuNcrIoNs, In the

following,

_we fo-

cus our attention _on the

simple

functions

g(z

₌ Cste. Let us

emphasize

the relevance of finite a-corrections derived in this fashion _:

they provide

the main FDC contributions to

transport

^coef-

ficients, computed

with any ^other

regukr

_or

singular

function.

Following ^a2-

terms are at the very

most of order

z§a~Ina~. Recalling

the smallness of

z§

=

0.166rs

for _rs

j~ I,

we should thus be able to

neglect

them in _a first

approach.

As _a

provisional conclusion,

_one should state that results

derived in this section may ^be extended to

any

^other ^dielectric

Jknction provided

one makes _use of the TF-like

approximation

:

vo(z)

rz

(z~

+

zbg(z

=

1))

^-~

(27)

Our

approach

is based on

(t I)-expansion ((4.14),

₁₎ of basic

quadratures I(I(~ (Eq. (4.12), _1).

Making

use of standard ideal gas formulas

ii ii

for ₇

= pe

/EF just provides (see

notation

(2.1)):

t~"

= +

2u(& ^u&~

+

2u(u

_1)@~&~ ^~

u(u _I)fi&3

₊ ^~

u(u

I

)(u 2)(~&~

+

(2.8)

3 3 3

The first terms in

expansion ((2.14),

_{1) have}

already

been derived in I

~Eqs. (4.17)).

The calculation of the

following

ones is similar but very

lenghty, especially

for

Iii)).

It involves the net

evaluation of

quadratures (fin)

~

detailed in

Appendix

A~ When those _are

completed,

one obtains the final results

IiK'i/iTF)

- 1+ &~

(3

⁺

~

i~° _(-13 +1 Ill)

+

1(3

°)~) 12.9a)

(6)

N°6 LINEAR ELECrRONIC TRANSPORT IN DENSE PLASMAS. II Ml

IiK'i/ITF)

=

I ₁₊

&~

(l~

⁺ ^~

i~° _(-~l~

+ 7

Ill)

+

l~13

°)~)l

12.9b)

~'l/'~~

_"

i°~° _l~

_~

^°~ IS

_~

^{15)~ ~~~~}

_~

~l~

^~

Ill) (~~

~

II

/)

~ ~~

i~ ~

~

~~~~~)

_~~

_~~~~

'

'~

&

ig

io

₂

io

~

io

^~

I)K(~)

₌ l + &~

-2 -(3 (o) 3

+

3 (2.9d)

' 3 3

II

³

₁₎

³

₁₎

and next

(see

notation

(2.6):

IT/Ii(TF)

=

°~ 3

~

i~° ¹³

+

Ill)

+

1(3

°)~l (2.i°a)

IT/Ii(TF)

= 1-

°~ It

^~

_s~°

(~l~ 1+ 7%)

₊

_II ₍₃ ^°)~l _(2.i°b)

~~~~

~

5

^~

o~~ 1°3

~~~

+

~~

+

56 ~) ~"(~)

~

~l'l _1 _i'~ ^~°~ ^l~

~ ~~

Ill))

+ ~~

ii°~~

~~~~

(2.ioc)

@/2)

= 1- &2

35

¹⁹

^II

2 IO ~ ~o ²

~ ~

II ^3~~ ~°~i)

^~ ³

I (2,10d)

The first FDC _are thus shown _to be of order &~ in

agreement

^with

((" _)~

₌ 0 for _n odd

(Ap- pendix A).

Derivatives

i2'(1)

and

i2"(1)

in

equations (~9)-(2.10)

are taken with

respect

^to z. Vari-

able _T is defined

according

to

(2.4).

^Notation "TF' refers

explicitely

to the Thomas-Fermi-like

approximation (~7)

but

expression (2.10d)

for the shear

viscosity

holds for all

e(z) _up

to order

&~,

ⁱⁿ

agreement

^with ^the term

z3(1 z~)

ⁱⁿ

L2(t) (Eq. (4.I16), _1).

The

given

&2 -contributions have been

expanded according

to

increasing

_powers in

(3 (°

=

i'2(1)S(1) /2 If (Eq. (4.19b, _1).

Thb _assert the crucial role

played by _parameter (°

in

establishing

the first finite T corrections to

transport quantities:

in _a way,

(°

allows _us to

compare

^the ^width ^of ^Ziman's

integrant VI (z)i2(z)

to that of the Fermi distribution derivative

(-3 f(/3Ek _)k=k~

when _o increases.

2.3 CASE OF DIELECTRIC FUNCTIONS WITH DIVERGING DERIVATIVBS AT z = 1.

2.3. I

Upper

bound

formalikm.

In lieu of

dealing

with the derivatives

of12+m IL; (t)

in the calculation of series

((4.14), 1),

an alternative

approach

would consist

by making

a

pansion

of functions

L;(t)

around _t

= I. More

specifically,

thermoelectric coefficients may ^be deduced from

K(~(

first written _as

1(~~(

₌

dq ^~~~ q~

~

with

t~Li(t)

=

dzr~l/(z)i2(z), (2,ll)

~

~( 3~

_t

~(t)

(7)

&2 JOURNAL DE PHYSIQUE I N°6

which leaves _us with the burden of

evaluating

the

upper

^bound contribution in

expressions

such as

i+J

I ₌

/ _dzG(z)

with

G(z)

₌

V/(z)16(z)

and

4l(z)

_e

x~i2(z). (2.12)

16(z)

and all its derivatives _are assumed _to be

regular

at z = I.

In the _case of _a

non-analytical g(z)

function _at

z =

I,

thin second method is

preferable

as it allows for _an easier

quantitative handling

of

singularities

within the

b-expansion

^of I. Let _us consider for instance GD'S

expression (2.5)

_: for _n >

I, g(~)(l) _gets

_a

logarithmic singularity

as well _as

poles ix I)-P

with _p < _n I. Lindhard's

g(z)

behaves in _a similar fashion. In both

cases,

lhylor's expansion1

₌

£~>~ b~G~-~(l)/nl

breaks

down,

which may ^be circumvented

upon replacing

it

by

an

equivalent

one with derivatives

G(~) (z computed

at _z ₌ I + b

(Append

ix

B).

It remains _on us to evaluate this series with the

underlying problem

^of the above-mentioned

g(~)

l +

bl's singularities.

Such _an

analysis

is outlined in

Append

ixB for the GD

(and L)

dielectric function. This

provides

the final

output

:

1 =

1(TF)

+

R(Ci

&~b ₊

C2b~

₊

C3b~), (2.13)

with

Coefficients ^Cl, C2

and

^C3

in

_2,13)_are unctionsof the variable

A

₌ _b/& (Eqs.

(B6a-c)),

lb

estimate

the

thermopower

Q

up to &~, I

has been

_expanded

(2.13) split

into a regular contribution I(TF) _which arises from F-like proximation (2.7), and

additional terms ^~-

R

which

come

from the xtra-contribution linear

in

_x§ within the

potential

Vo(z)(see definitions (2.2) and

(2.4)).For the _{sake of} _we

and _its ^rivatives when _to

be taken at z

1.

2.3. 2 a2

coJTecfions

: case of

ingular

g(x)

^kncfions. - The

foregoing

^results

estimate corrections _f

III<~~i

=

dn (- II _n~t~ _Ii

~~i~

⁺

(~~l~')

~

(~~i~)

^~

-~ (Ci&~b

⁺

^C2b~

⁺

C3b~]

+

~

~~~(~ l, ^(2,15)

(Ii _)"

~~

Expression (2.15),

with lower bound

-ripe pushed

down to _-cc

(See previous

footnote 2 in

I),

is further estimated

by replacing

b

by (t I)

and

16(z) by x~i2(x). ^The1~

term is derived from

expansion (2.8).

The

corresponding

calculations _are based _on _a

systematic

recollection of the

expansion

terms in &° and &~ for

I(j)/

_or

_K)~/

,

and also in and &~ for

Ill _{)/ (Ref.} _[3]).

It is

easily

checked out that the TF contribution

withir~ (2.15)) yields previous

results back

exactly (2.9a-c).

The calculation of R&~ corrections is

quite straightforward

but also

pretty lengthy, especially

for

Ill _)/.

_The

corresponding algebraic expressions

are

displayed

in

Appendix

^C both for L and GD'S functions. In contradistinction to the TF

situation,

the numerical estimate for R&2 corrections

require

the evaluation of

quadratures f(@))~

^for several distinct functions which do not

pertain

to

monomials,

the value ofwhich is available in the first column of table II

(Appendix A). Finally,

(8)

N°6 LINEAR ELECTRONIC TRANSPORT IN DENSE PLASMAS. II &3

we

get

^the

required

enact numerical results for basic

IT quadratures

and

parameter (T (for

L and GD'S

g(z))

under the form _:

1)/1)(L)

=

I)/If(TF) _R&~(3 (°) ((1/3)In(&~/4) 0.39917)

,

(2.16a)

1)/1)(GD)

=

I)/I)(TF) _R&~(3 (°) ((1/3)In(&~/4)

+

0.14362)

,

(2.16b)

1)/1)(L)

=

I)/I)(TF) _R&~(3 (°) ((7/5)In(&~/4) 0.28913)

,

(2,16c)

1)/1)(GD)

=

I)/I)(TF) _R&~(3 (°) ((7/5)In(~~/4)

+

1.08424)

,

(2.16d)

(T/(°(L)

=

(T/(°(TF)

_{+ R&~}^~~

^~~~ (((7/2)In(A~/4) 0.sll12) ((14/5)In(&~/4)

+

0.35507)

+ ~"~~~

((7/15)In(h~/4)

+

0.05918]]

T

i~(I)

R«~ 1°

i(37/15)1n(«~/4) _o.179111

,

12.16e)

(T/(°(GD)

=

(~/(°(TF)

₊ _R&~^~~

^~~~ (((7/2)In(a~ /4)

₊

2.31314) ((14/5)In(&~/4)

+

2;50332)

+

')~~

((7/15)In(&~/4)

+

0A1722])

T i~

I)

R«~

^~~

j1°~~ i(37/15)1n(«2/4)

+

2.024841 (2.16f)

Other dielectric functions

may

^{also be}

investigated. Usually they

are

only

defined at T

=

0,

and

they approximate exchange

and correlation effects in various ways. Hubbard

[12]

^for ^instance has

proposed

a local field correction

g(z)under

the form

g(x

= gL

(z) Ii

I

x§gL lx )(g(z) lx

^~

_)].

Therefore,

_even for a

regular G(z), previous analysis

about

gl")

_{I + b} derivatives has to be done

again.

However in the

foregoing expression,

the

z§

term will

appear

as

xi

in the

expansion

^of

Vo(z)

at z = I + b.

Thus,

for small _rs values

(I.e.,

_rs

_j~ I)

the

given

contributions

~-

(0.166rs)~

can be

safely neglected. So, by

reference to

Tfi

_one _may introduce _a "Lindhard

approximation"

through

~~~~

~

l

bgL~)~ )G(z

=

i1' ^'~

It h worth

mentioning

that such an

approxirrtation

also

permits

_us to extend GD'S results

(2.16) (with gL(z)

_-

goD(z))

to the _case of _more

sophisticated

dielectric functions endowed with _an

«-dependent G(z) ^function,

such as that

proposed

_very

recently by

^Chabrier

[13].

^In each _case, the

parameters

T and R, have to be

correctly

evaluated

according

to

(2.4)

and

(2,17).

(9)

3. A number of salient numerical results

(H+ ^-He~+ mixtures).

3. I NUMERICAL _ANALYSIS _oF

m~-coRREcTIoNs

_IN _HYDROGENIC _MixTuREs. We _now im-

plement

^numerical ^results

(2.16)

^vithin

strongly coupled

and

fully

ionized

H+ He~+ binary

mixtures. In the

sequel

_we shall

mostly

concentrate on domain A delineated in the

(r _r~) plane

in

figure

I of I with

typically

_rs

_~

l and r < 10. In the

pure

^Coulombic case,

V(z)

reduces _{to Vo}

lx (Eq. (2.2))

with

u(z)

_e I and

S(z)

identifies with the static

charge density

structure factor

Szz (z) (Eqs. (3.6,1)

and

(4.3, 1)).

^Parthl ^ion-ion structure factors

S~v(z)

have been derived within the BIM model

through

the HNC

approximation.

As _a rule of

thumb,

_we know that

discrepancies

between

Szz (z)

values

computed

^within a BIM _or _a PBIM framework [9] ^increase with _rs _or

(and )

r. Therefore the BIM

model,

with bare Coulomb interionic

potentials,

_proves

fairly appropriate

to

transport

calculations in domain A~

Numerical data for basic

I~ quadratures

and the

parameter (T

have been

computed

for L and GD'S

functions,

with _rs ₌

0.5,

< r < 9 and _c2

increasing

from 0 to 1009b

He,

thus

covering

the range ^2.5 x

10-~ j~

a

j~

^0.25. Quadratures

If

and

If,

and

parameter (°,

which

pertain

_to the

&

- 0 and £l _- 0

limits,

have also been

systematically derived,

_as

they provide

standard

quantities

to be

compared with,

when

evaluating

the _net effect of FDC. As

already

stressed in

I,

the r _rs

dependence

^of

transport

coefficients

((4.20), ₁₎

^and

(2.6)

is

mainly

^located within their

pre.factors, namely (rf

Z for _p,

(r(

Z for

q/~, (r)

Z

r')

for

K~~

with T _~-

(i>sr')-~,

and _a

~-

(rs/r')

for

Q,

where r'

= r Z 1/3

(see

notation

((2.I), 1)).

~

Bin Lindhard ~ ^Bin ^Lindhard

8

_r

" 60 cz = 0.75

/

r~ m i cz = 0.75

~

_l

(

₂

~ ^m

'*

la ) 161

a

« o

_ ~

~ _

z ~

P°

)

_K° _r/20

- ~

tl g 1

« .

~ o

« ~

"

u ~

- «

z «

~ ~

j m

w ~

o

0 0.5 1 1.5 0 40 60 100

r, p

Rg.

I. Variations of

(a)

electrical

resistivity p°

in terms of _rs and

(b)

thermal

conductivity K°

in terms of

r, featuring

the

prefactor

influence in transport calculation.

p°

and

I[°

_are

computed

in the elastic limit at T

= 0 within the

H+ He~+

_mixture. _Those _results

pertain

to BIM model with HNC structure factors and Lindhard's

fig)

_{[4] within} electron-ion interaction.

(10)

This feature is

clearly exemplified

in

figure

I which

again explains

the interest of

directly

_an-

alyzing

the backbone

quantities I), If, (°

and their finite T

generalization II, I), if

_and

_(T.

_A

number of numerical datas are

given

in table I. A more exhaustive tabulation is available in

[3].

lhble I. Numerical datas

for

reduced

quantities I), II

_and

(° (Eqs. (4.19),

₁₎ and their

finite degen-

eraqy «tension

II, If, if

_and

_(T _(Eqs. _(2.10))

_and

_(2.16))

_within

_H+ He~+ binary

ionic

mhmres,

in terms

ofthe plasma _parameter

r at rs = 0.5

for five

distinct helium concentration numbers. Present results

pertain

to domain A delineated in

figure

I in I

(Ref [I])

Calcularions _were

pe~fiormed

within the HIM model with HNC ion-ion structure

factors

and Gouedard-Deursch

(Eq. (2.5)

electron- ion

screening

r _a

If I( I( l~

_I~ ~°

~'

X*

l 0.270 0.5912 0.4413 0.4688 0.3064 0.1879 2.2668 1.8064 2 0.135 0.4934 0.4667 0.4668 0.2364 0.2077 2.1261 2.0315 3 0.090 0.4438 0.4343 0.4336 0.2020 0.1914 2.0231 1.9873 4 0.068 0.4115 0.4070 0.4063 0.1800 0.1749 1.9370 1.9195 5 0.054 0.3879 0.3854 0.3848 0.1641 0.1614 1.8610 1.8512

12.5 He

0.260 0.6343 0.4881 0.5094 0,3224 0.2057 2.2429 1.8306

2 0.130 0.5427 0.5154 0.5129 0.2543 0.2263 2.1038 2.0183

3 0.087 0,4983 0.48#1 0.4855 0.2216 0.2111 2.0067 1.9732 4 0.065 0.4703 0.4652 0.4631 0.2009 0.1958 1.9284 1.9112 5 0.052 0.4503 0.4473 0.4455 0.1861 0.1833 1.8614 1.8511

50 He

0.236 0.6319 0.5150 0.5208 0.3090 0.2141 2.1943 1.8740 2 0.l18 0.5614 0.5376 0.5302 0.2517 0.2296 2.0584 1.9893 3 0.079 0.5306 0.5208 0.5150 0.2251 0.2166 1.9714 1.9419 4 0.059 0.5126 0.5073 0.5027 0.2086 0.2044 1.9062 1.8892 5 0.047 0.5006 0.4972 0.4935 0.1969 0.1945 1.8537 1.8422

75 He

0.224 0.5963 0.4977 0.4972 0.2864 0.2061 2.1711 1.8896

2 0,l12 0.5373 0.5165 0.5075 0.2359 0.2175 2.0361 1.9735 3 0.075 0.5131 0.5042 0,4974 0.2127 0.2057 1.9534 1.9252 4 0.056 0.4997 0.4947 0.4894 0.1985 0.1949 1.8940 1.8769

5 0.045 0.4913 0.4880 0.4838 0.1885 0.1864 1.8483 1.8360

xwz+

0.214 0.5565 0.4730 0.4684 0.2635 0.1953 2.1519 1.8995 2 0.107 0.5069 0,4885 0.4788 0.2187 0.2034 2.0175 1.9595 3 0.071 0.4876 0.4796 0.4724 0.1984 0.1924 1.9382 1.9108 4 0.054 0.4777 0.4730 0.4675 0.1859 0.1829 1.8839 1.8664 0.043 0.4719 0.4687 0.4643 0.1773 0.1755 1.8440 1.8311

(11)

-j

Go

~

He He

~ ~

~

~- a Q

- ( )

L Go---

~~

Go

L ^----j r

Go

---

~2 0.37

1

I I

,24

_~~

~

Go L

~~

Go

^~~

L ^--jjr L

--j

Go

--

-

I

o

~~

~

÷e C

~

C _

o ~~'

'

÷e

~

l~ '

~

a H

0.

_I (~)

1

I I

Fig,

2~ Finite temperature corrections. Plot of

(a) quadratures II

_and

II

_contrasted _to

II, _(b)

_normalized

Lorentz number

CT /C°

=

II /I)

,

(c) quadrature if

_contrasted _to

It

_and

_(d)

_parameter

(T

_contrasted _to

(°,

_in _terms _{of r} _at

rs = 0.5 for

H+

_and

He~+ _plasmas.

_Quantities _with

a

superscrip

T refer _to electron

transport

taken in the elastic limit and

including

finite

temperature

contributions

according

to results

(2,10)

and

(2.16).

Actual

plotted

values derive from BIM model with HNC structure factors and both the Lindhard [4] and Gouedard-Deutsch

(Eq. (2.5))

e~ -ion

screening.

Figure

2 ekhibits the behavior of the above-mentioned

quantities,

as well _as that of the normalized Lorentz number

£T /£°

=

II /I).

Finite

temperature

corrections

significantly

reduce

quadratures I)

and

It, already

for _a

=

0.I,

which amounts to a simultaneous increase of thermal and elec- tric

conductivities,

and also of shear

viscosity.

This result arises from the

widening

of the Fermi

(12)

N°6 LINEAR ELECTRONIC TRANSPORT IN DENSE PI,ASMAS. II 847

function derivative

(3 f) /3Ek

when the

degeneracy parameter

a increases. This is in

qualitative agreement

^with ^the

asymptotic Spitzer's

limits

[14]

: at

large

_a, a and Ii behave _as

a3/~/lrt

_a and

Ta3/~ /ln

_a

respectively;

_{qs b}

expected

to behave as

a5/~ /lrt

_a in accord with

equation (4.13d, 1).

^A

systematic

derivation of the

high temperature

^limit ^of

expressions (4.13,

₁₎ will be

given

elsewhere

(D. L6ger

and C.

Deutsch,

to be

published).

Finite

degeneracy

corrections

pictured

in

figure

2 stem

essentially

from TF-like

expressions (2.10),

characterized

by

absolute

&~-arguments larger

than I. This

explains

the relative

impor-

tance of observed FDC

(lhb. I)

_: for r ₌ I and _rs

=

0.5,

one

gets

a = 0.27 in

hydrogen

with

(I) II

_< 0,15 but

already ((I) II) Iii

m259b.

Actually,

our

analytical &~-corrections

should thus be restricted to a "relative" _narrow _a range,

typically

_a

_~

0.3. In this

connection,

it would be also instructive _to

compute

^&4 corrections with the TF-like

potential (2.7)

in order to

extend these results to a

larger

domain.

3.2 COMPARISON _wTH _OTHER _RESULTS. As

already

mentioncd in

I,

Ichimaru and lhnaka

(IT~ [10]

^have

already computed

electrical and thermal conductivities for _a

partially degenerate hydrogen plasma,

within the range ^r ^< ^{2 and} a < 10.

They

used Edwards' formula for p

(equation (4.33,

_{1) and} another _one

analogous

to

equation (4.34,

₁₎ for Ii. Their results

rely

on a HNC-like

approximation

for the ion-ion _structure

factors,

while electron

screening

is retained both within

e~ -ion and ion-ion interactions

through

_a full GD function

complemented by

a local field correc-

tion function

G(q). Comparison

of IT's results with _ours

~llqs. (2.16a-d))

is

pictured

in

figure 3,

where we

plot

the

a-dependence 8f quadratures II

and

If

within the range ^0.05 ^< a < 0.5 at r = I for the

hydrogen plasma.

In _our

calculations,

_we have retained both the BIM and PBIM

models

(in

the _c2

- 0

limit)

^and the GD function

(2.5)

to

proceed

to a

meaningful comparhon.

In both cases, one

gets

a

good _agreement

~vith a better _one for the PBIM than for the

BIM,

as

expected.

Nonetheless _our results

stay

^below ^those ^{of IT} ^who ^have

computed

_p ^and Ii

through

variational solutions of the

transport equation.

ll~is

implies _1)~"~~

>

I)l~~~~~, ii

₌ 1,

2),

_as _can be

readily

demonstrated from the basic content of the variational

principle

itself

[15]. Moreover,

it should also be

appreciated

that IT do not make _use of

exactly

the _same

fig

as that in the

present

^work For _a

= 0.

I,

their

If

and

II

values

stay

^above our

If, despite they

should

approach

^it

asymptot- ically

in the _a

- 0 limit. Such _a

discrepancy clearly points

out the qua ntitative relevance of the various

approximations

involved in

modelling _processes.

4,

Summar%

Finite

degeneracy

contributions to

time-independent transport

coefficients have been worked out in the elastic

scattering regime through

the theoretical framework

explained

in the

[I],

which is based on a Boltzmann-2iman

approach

within _a Lorentzian

approxi-

mation. The _use of

generalized

relaxation times allows for _a

systematic presentation

of

transport

coefficients in terms of reduced

quantities.

Finite

temperature

^effects are derived from various _ex-

pressions

for the

jellium

dielectric function

e(q),

^two situations have to be

distinguished according

to the

analytic

behavior of

e(q)

at q =

2kF.

In _a first

step

we consider the _case of

perfectly regular

Thomas-fermi-like

(Tn

dielectric functions. Exact _a

-expansions

_are ~erived in this _case

up

to order

a~,

both for thermoelectronic and mechanical

transport

coefficients

expressed

as reduced

quadratures.

As

demonstrated,

TF-like results

provide

the very ^bash ^for our finite

degeneracy analysis,

even for _more

sophisticated e(q)

^functions as

long

as the electronic

parameter

rs ^re-

mains smaller than

unity. Next,

we

investigate

the case of the Lindhard and Gouedard-Deutsch dielectric functions. Their

non-analytic

character _at _q

=

2kF brings

to additional terms of _or-

(13)

II BIfi( l~

pBjfi jT

j^T

)

_l 2

', _Ii

Ii

,BIfi

_1(~

BIfi

---1(~

'

jT _~

---.

I(

₂

~

~~ ~m

~ ~

,

,,

i , , I

,,

m

,,

,, ^m

,,

°<

(a) '",

^'' _°<

_b "),

~ '~),

l- Present results I - Present results

2» Ichlmaru and 2 - Ichimaru and Tanaka

HYdrooen r

= I Hydrogen r

= 1

0.05 0.20 0.35 0.50 0.05 0.20 0.35 0.50

a «

T/T~

^a =

Tli~

Fig-

3. Finite temperature corrections. Present results for

(a) quadratures If

_and

II

and

(b) quadratures

If

_and

II

contrasted to those of Ichimaru and lbnaka

(II~ [10]40r

_a

hydrogen plasma

in terms of the

degenaracy

parameter a at r

= I. Our results _are based _on BIM and PBIM models

(taken

in the _c2

-

0

limit)

with HNC structure factors and also the Gouedard-Deutsch

(GD) (Eq. (2.5)) fig)

function. IT results

depend

on HNC-like ion structure factors within _a

polarized

one component

plasma (OCP) through

a modified GD function.

der

+~

a~In(Cste

_x

a~)

times

z~p

=

0.166rs

,

which have been derived

exactly.

Extension of the

present

^results ^to ^other

e(q)

is next

investigated. Finally,

the numerical

outputs

^of our formalism are

implemented

within the

strongly coupled ^H+ ^He~+ binary

^mixtures.

Comparison

to

pre-

vious results

by

Ichimaru and lhnaka

[10]

^for ^the

partially degenerate hydrogen plasma

shows _a

good agreement

_up to _a _+~

30il.

Thermoelectronic and mechanical

transport

coefficients have been tabulated in _a reduced

form,

convenient for _a numerical

exploration.

Acknowledgements.

We wish _to thank H. Minor and D.

Levesque

for useful discussions as well as P

Fromy

for efficient

help

Mith tile numerical calculations.

(14)

Appendix

A

Properties

of

quadratures f(())~.

Those are defined

acccording

to

((4.15),1) ((

=

q/x)

_:

if(n)io

₌

C

dn (- II _f(ni

=

i~°'dn~~l~llln~~ ^(Ail

so

tlley obviously

^vanish ^for odd

f(().

First _we consider monomials _:

~i~~~0

^~

/~~ _~Q

~~

~j2 i /~~ _~Q)~) (P

~

l). (2~~)

The

right

hand side in

(A2)

^is

expressed

ⁱⁿ terms of Bernoulli's numbers

[lfl,

thus _one obtains the rational data

((°)~

₌

l, (i~)o

₌

1/3, ((4)~

=

7/15,

and _so on, of

frequent

use in this work.

In the

general

_case, for _a

given f(@),

one has to

compute

^the

integral (Al) numerically.

We also notice that

f(())~

is the first term in a class of functionals

lhble II. Functional vah~es

So(f), Si(f), 52( f)

,

do(f), d2( f)

and

d4 f) for

various

f(@).

f((

_S~ _e

f(() §

S, Sz d~ dz

d,

In((~)

_-2.5407257 _-1.1310942 _-0.5138560 _-5.4444122 _2.1035996 _-4.0180951

(~In((~)

_-0.0658396 _-0.0529693 _-0.0185312 _-0.1410848 _0.2296372 _0.7011999

I'm((~)

_0.3702898 _-0.0074970 _-0.0027380 _0.7934781 _0.9691479 _1.6259410

In(1+~~)

0.2318630 0.0366897 0.0095516 0.4968494 0.3005142 -0.1296160

(~In(1+~~)

0.2450928 0.0151025 0.0023798 0.5251999 0.4929503 0.3650778

I'm(1+~~)

_0.5829876 _0.0123032 _0.0011105 _1.2492592 _1.3473934 _1.7523704

(~/(l+~~)

_0.1775330 _0.0317621 0.0086578 0.3804278 0.2029356 -0.1589599

(~/(l+~~)~

0.1107193 0.0243485 0.0071954 0.2372556 0.0929562 -0.1568519

(~/(l+~~)3

_0.0764691 _0.0193771 _0.0060969 _0.1638624 _0.0452470 _-0.1269614

1'/(1+l~~)

_0.1558004 _0.0118016 _0.0020129 _0.3338579 _0.2970644 _0.1589599

1'/(1+l~~)~

0.0668137 0.0074136 0.0014623 0.1431722 0.1099795 -0.0021079

1'/(1+~~)3

_0.0342502 0.0049715 0.0010986 0.0733932 0.0477092 -0.0298906

( tan"1(

0.2606614 0.0388104 0.0099064 0.5585602 0.3558739 -0.0986306

(15)

lhble II

provides Sn

values

(n

₌ 0 _-

2)

^for the main functions

entering

the

quadratures 1()~(

dbplayed

in

Appendix

C. For _our purpose ^we ^have âlso încluded ^{in it the} ^values ôf some

lineir

combinations

do

₌

) _So, _d2

=

[So 3Si], d4

₌

[So 15Si

+

3052] (A4)

which will be used in the next paper ^{III of} this series.

Appendix

B

Upper

bound calculation.

Here _we outline the derivation of

expansion (2.13) displayed

in Section 2.3.I of main text. The basic

parameters

r and defined in

(2.4)

'vdl be

extensively

used. We concentrate on the GD

dielectric function

(2.5),

Lindhard's results

being

recovered in the & _- 0 limit.

We first

reexpress lhylors's expansion

of

(2.12)

as1 ₌

£~> ~(- l)"~ ^lb" G"~~(l+b) In! (Ref. [3]),

which _we first _we

conveniently split

into a so-called

trapez&dal

contribution T and _a

remaining

series R _:

The

trapezoidal

contribution is next derived

by expanding G(

I +

b)

_up to

b~,

which

yields

T values accurate up ^to

53

~ ~

~~2 12b2 2z~p l2b~iP

(gjl

+

b) 9( ill G(i

+

b)

m

~~~

^~~

^~ _i

r

~ T~ _~ ~~

On the other

hand,

a direct calculation for

Ag

₌

g(I

+

b) g(I) yields

_:

fi /~2 /~2 _{~ fi2} /~2 /~2 fi2 &2 ~ fi2 /~2fi2

~~ 4

~

4~

~~'

4

24~~

₄ ₄

~

2~~

4

~

6 &2 ₊ b2 ^~

~~~~~

whence

~~ ~~ ~~ ~~ ~,, ~~, ~ ~~

~~~~~

T2 ^~ 2T2

~'

T

~ T2 4 _r _T ~

T2

9b~lb &~

+ b~

9b34l &~

₊ b~ 2 &~

2T2 ^~'~ 4 ~

2T2 ^~ 2~~

4 ~

3 &2 + b2

Next _we turn to the

non-singular _{part R~s}

in the series

R,

which reads _as

R~s

=

-(b3/12)

_lim

G((I

₊

_b),

J-o

G( denoting

the

holomorphic part

^of

G"(z)

with

logarithmic

term included. This

provides

b3

_16" 216' 16 21b

9b34l 5

&~ + b~

~~~~~~

r2 12 ~

3 _T ~

3 _T r2 2r2 ₆ ^~

6~~

₄

(16)

~

l

~A~ ^~ _(l ^~A2)~

^~

_(l ^~A2)~

'~ ~

~ ~~

~~

l

~/~~

(82)

It remains to

complete

the

analysis

with the calculation of

Rs,

the

singular part

^{in R}

expanded up

^to

b3.

_The _worst

singularities

in

G(")(I

₊

b), I-e-, b-("-

_{II and}

S-("-~)

_arise _from

g(")(I

₊

b)

and

g("- _~)(l

₊

_S), respectively. Looking

for successive

G(z) derivatives,

which are linear in

z~,and using

an inductive

argument gives

G(")(1

₊

_b)

+~

e(n 3) (z~fng(")(1+ _S)) ) ~~')

⁺

⁸⁽ⁿ ²⁾ (z~fg(")(1

⁺

b))

^x

T T

and also

by

recursion

~~~~~~ _{~ ~~}

~~~

_~~

4-

~2) ₂₎ ~-

_~

_~n

_~

^~~

_~

_~~~

~~

4-

~

where

X

= A(A -

I)

^/( I

+

_A~ ) _stands

for a _convenient _complex _throughA = b16

and

8(p) denotes the

Heaviside step

unction :

fl(p)

= I

for

p >

Finally,

the _of _complex

conjugates

(cc)

within ₍₈₄₎_and the

resulting

evaluation of

the G(")(I

+ )

singular ntributions

~B3)

s

us to

derive ^he

9b~4l (~ ^lrt(I _+A~)j

9534l

j ₅

T2 T2

18

^3A2

A

_ _{I I} _ I

I

l~

953

(61b

~,) j2 4 _tan~lA _ ~j 2

1 +

A2

3

_{(1 +}

A2j2 ^r2

_r

3A2

1+

A2

~ (85)

Uponperfoming the

sum

of

(Bl), (82)

and (85) yields back

the

for

GD'S

unction ^layed _in quation _2.13)

of

main ext

ith

ecifically

r

C2(GD)

₌

$

(I lrt~

^~~

((~~ ^ln(I

+

~))

,

(B6b)

T

C3(GD)

₌

$

((

⁺

In~

+

In(I

₊

A~) j

l' ^~ l)

T

JOURNAL DE PHYSIQUE i T I, M 6,JUIN 1991 34