Linear electronic transport in dense plasmas. I. Specific features of Boltzmann-Ziman formalism within the Lorentzian approximation

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Linear electronic transport in dense plasmas. I. Specific

features of Boltzmann-Ziman formalism within the

Lorentzian approximation

D. Léger, C. Deutsch

To cite this version:

Linear electronic

_transport

in dense

_plasmas.

I.

_Specific

features of

Boltzmann-Ziman formalism within the

Lorentzian

approximation

D.

_Léger

₍₁₎

and C. Deutsch

₍₂₎

(1)

Laboratoire des Matériaux Min6raux, Conservatoire National des Arts et _Métiers, 292 rue

Saint Martin, 75141 Paris Cedex 03, France

(2)

Laboratoire de

_Physique

des gaz et des

plasmas (*),

Université Paris XI, Bâtiment 212, 91405

Orsay

Cedex, France

(Received

on

_February

_{20, 1990,}

_accepted

in

_{final form}

on

_May

_29,

₁₉₉₀₎

Résumé. - Cet article est le

_premier

d’une s6rie consacrée à l’étude

_{systématique}

du _transport

électronique

linéaire dans les

_plasmas

denses fortement

_couples,

_comportant

_{plusieurs espèces}

ioniques classiques

plongées

dans un

_jellium

_{électronique}

fortement

_{dégénéré.}

Le formalisme retenu,

spécialisé

ici au cas des coefficients

_{thermoélectroniques}

et du _transport

_mécanique

(viscosité),

repose pour l’essentiel sur l’extension de la th6orie de _{Boltzmann-Ziman,} dans le cadre de

l’approximation

Lorentzienne. Les conditions de validité de cette dernière sont

largement développées,

et l’accent mis sur les corrections de

_température

finies et les corrections

inélastiques.

Les coefficients de transport sont

_{exprimés analytiquement}

sous forme de

_quantités

réduites, à la fois dans le cas

_élastique

et

_{inélastique, respectivement}

au moyen des solutions

exactes et variationnelles de

_{l’équation}

de _transport. Ceci _permet un calcul

_simplifié

des

corrections

précédemment

mentionnées, calcul

qui

sera détaillé dans l’article II de cette série.

Finalement, nous démontrons la validité de la méthode en

_{reproduisant,}

dans les limites

appropriées,

les formules de Ziman et d’Edwards pour la résistivité, ainsi que d’autres résultats

bien connus.

Abstract. - This is the first paper in a series devoted to a

_{systematic investigation}

of linear

electronic transport

properties

in

strongly coupled plasmas consisting

of a

_{multicomponent}

and classical ionic mixture embedded in a

highly degenerate

electron

_jellium.

The basic formalism

rests upon suitable extensions of the Boltzmann-Ziman

theory

as

_explained

in this work. It is hereafter

_specialized

in a

_{thorough investigation}

of thermoelectronic and mechanical _transport coefficients.

_Validity

conditions for the Lorentzian

_{approximation}

are first

carefully

examined.

High

temperature and inelastic corrections are

_emphasized.

Basic _transport

_quantities

are

expressed

under an

_analytic

and _compact form both in the elastic and inelastic cases,

through

exact and variational solutions of the transport

equation, respectively.

This allows for an easy

algebraic

treatment of finite

degeneracy

and inelastic _corrections, to be

_given

in the next _{paper II}

in this series.

_Finally,

the

_validity

of the method is demonstrated

_{by recovering}

Ziman’s and Edwards’

_resistivity

formula, and other well-known results, in the

_appropriate

limits.

Classification

Physics

Abstracts 05.60 - 05.30 - 52.25F

1. Introduction.

Equilibrium

and

_transport

_properties

in very dense and

strongly

coupled plasmas consisting

of

a

_fully

or

_partially

ionized

multicomponent

ionic mixture

_{interacting through}

the

_(screened)

Coulomb

_potential

and embedded in a

_{highly degenerate}

electron

_{jellium, play}

a central role

in several areas of Statistical Mechanics

[1],

Astrophysics

[2, 3]

and

Liquid

Metals

Theory [4].

As a

_particular

and very

important example,

let us mention the determination of

_equation

of

state and

_transport

_properties

of the

_{strongly coupled hydrogenic binary}

ionic mixture

H+-He2+

retained to model the Jovian

_planets

interior

_{[2, 3].}

_Recently,

we have

_already

investigated [1] the

critical

_{demixing properties}

of such

_systems,

which were assumed in this case to be in

_{thermodynamical equilibrium. Basically,}

the

_key

_parameters

in such

_analysis

are

the classical ion

_plasma

_parameter

T,

and the

_degenerate

electron

_jellium

_parameter

rs. This paper is devoted to a

systematic

investigation

of linear and electronic

_transport

properties

within a Lorentzian framework. This one is

_especially

well-suited to a

modelling

of

the interaction between a

nearly

fully degenerate

electron

jellium

and ion

_density

fluctuations.

This interaction constitutes the very basis for the evaluation of all

transport

quantities.

Moreover the

_{superposition}

of a

_{fully degenerate}

_{electron gas} to a

_strongly

_coupled

ion fluid is also central to

_building

_up the so called

_Binary

Ionic Mixture model

_{(BIM) [5]}

where the

dynamic

and static

_properties

of the classical ion

_plasma

are

_decoupled

from a

_mechanically

rigid

and

_neutralizing

electron

_background.

_Focusing

attention on the electron

_component,

let us notice that its

_{complete degeneracy}

allows a safe

_neglect

of the electron-electron interaction. This allows a

_simplified

but accurate treatment of

_transport

_properties

within a

Lorentzian

_{approximation}

to the Boltzmann

equation [6]

which has

_already

been

extensively

applied

to the one

_component

_{plasma (OCP)}

model

[7a, b].

The main formal characteristics of this Boltzmann-Ziman

_(BZ)

formalism relies _upon

_{simple expressions}

for

time-indepen-dent

_transport

_quantities.

These

_{simple expressions essentially}

consist of

_{single quadratures}

of electron-ion effective interactions factorized out

_by

ion structure factors. The latter are

generally

deduced from the nonlinear

_hypernetted

chain

_(HNC)

scheme

_extensively

used in the

_study

of static and

_transport

_{properties [1,}

5,

7b].

Having

in mind a

_{thorough investigation}

of finite

_temperature

and inelastic contributions

(Paper II)

to the standard elastic

_scattering

at T =

0,

we detail at

_length

in section 2 the various

_parameters

used in this

_analysis.

Finite

_degeneracy

and inelastic

effects,

parametrized

respectively

with a =

T/TF

(TF,

Fermi

_temperature)

and ci

= 8 hw

(,B =

_{1 /kB}

T),

are

shown to act

_{significantly}

in

_{well-decoupled}

domains A and B of the

_{F-r. plane,}

which will allow a further

separate

analysis

of the

_previously

mentioned contributions. In section

3,

we

review the standard BZ formalism

[6],

hereafter

_specialized

to a _{Lorentz gas}

_interacting

weakly

with a

_{multicomponent}

ionic mixture. The

_investigated

transport

quantities

include electric and thermal

_{conductivities,}

as well as the

_{corresponding thermopower}

and the

(mechanical)

shear

_viscosity.

The bulk

_viscosity

is seen to have an

_identically

zero value in a

Lorentz gas.

Section 4 is devoted to a

_thorough

dimensional

_{investigation}

of the various

_transport

quantities

of

_present

concern. The latter are

_expressed

in terms of basic reduced

_quadratures,

both in the

_purely

elastic case

_{(Sect. 4.2)}

where exact solutions of the

_transport

_equation

are

available,

and in the inelastic case

_{(Sect. 4.3)}

where

_only

_approximate

solutions may be derived from the variational

_{principle [6].}

The

_significant

contribution

_{played by}

a number of suitable

_{r -rs-dependent}

_prefactors

is

_emphasized.

In both elastic and inelastic

_situations,

we

consider

_briefly

the

_limiting

a « 1 and J 1 cases, in view of

evaluating

finite

degeneracy

(ijn> 0 and (ij m>

1 explained

in this work. Our

_methodology

is

_applied

in section 4.2.2 to the simultaneous a --+ 0 and cv --+ 0

limits,

giving

back

straightforwardly

the well-known Ziman

resistivity

formula

_[6],

the standard Lorentz number as well as classical

_thermopower

and shear

_{viscosity expressions. Finally}

_{(Sect. 4.3.3),}

we intend to make a connection between the

variational

_approach

and exact elastic

_transport

calculations. For that _purpose, we focus our

attention on some variational

_transport

_quantities

_computed

with monomial trial functions in the

_appropriate

cv --+ 0 limit. We show that Edwards’

resistivity

formula

[8]

as well as some

other

_transport

_expressions,

belong

to that

_specific

class of solutions.

2. The Lorentz model.

2.1 NOTATIONS. - We consider a

_{multicomponent}

mixture built of

_degenerate

electrons and

Ps ion

species

subindexed with v, enclosed in the total volume Tf of a connex domain. Each ion

_species

is assumed to be

_fully

or

_{only partially}

ionized and endowed with atomic number

ZA,

effective

_{charge -}

_{Zv e}

(e

denotes the electron

_charge),

mass

_M1/

and

_particle

number

N 1/. Conventionally, v

= 1 refers _to the smallest

charge.

The

neutralizing background

is of

opposite sign

and contains

_{Ne nearly}

free electrons.

_{Electroneutrality}

thus reads as

Ne

=

ZN;

where

Ni = ¿

N 1/

and

Z

_c,

_Zv

is the mean valence. More

_generally,

the

v v

average

by

concentration number of a

_{given quantity X}

will be defined as

The ionic

_component

is characterized

_by

the densities _n; = lim

Ni/ v

and

n v = lim

N,,/ v.

v - m v - m

The

_partial

concentration numbers are _CJI =

N JI/Ni

and fulfill the

constraint L CJI = 1.

We

p

choose the ion

_{sphere radius a; =}

_{[3/ (4}

_{7Tni) ]1/3}

as a convenient unit

_{of length}

and we define

the classical

_plasma

_parameter

as T =

{3e2/ai.

The

_quantum

jellium,

endowed with electron

density ne

= lim

Ne/V,

is

parametrized

with rs

=

ae/ao

where

ao is the Bohr radius and

v - 00

ae =

[3/(4 7Tne) ]1/3

=

ail ZI/3

is the electron

sphere

radius. This unit

of length

allows to define

r’ =

_{3e2/ae

as another

_plasma

_parameter.

The effective

_strength

of the

_{plasma coupling}

is

measured

_{by rerr}

=

rZ;rr.

By

reference to the

ion-sphere

model of

Salpeter

[9],

the effective

quadratic charge

Z ff

can be chosen as

_{(Ref. [10])}

Z;rr

=

ZI/3 ZS3.

Throughout

this paper, A will stand for the numerical constant

₍₉

7T /4 )1/3.

We will dénote

by ZJI

=

ZJI/Z

a reduced

_charge

number.

_Also,

it will _prove convenient to deal with the variables :

which will be detailed in the

_sequel.

_Transport

_{quantities investigated}

in this work will be measured in atomic unit

_{(a.u.), namely}

_oo =

e 2/ao h

= 4.60 _x

104 n-1 cm-l

and

710 =

À la)

= 7.12 x

10- 3

_poise,

for electrical

_conductivity

and

_{viscosity respectively.}

2.2 VALIDITY CRITERIA FOR THE LORENTZIAN APPROXIMATION. - As

_previously

stated,

interacting weakly

with a

_{highly degenerate}

electron

jellium.

The latter

_assumption

is written

in terms of the

_degeneracy

_{parameter a}

as

so that T

_{(K) =}

6 x

105 rs 2.

_EF

stands for the Fermi energy of the

jellium

which can be

treated

_{nonrelativistically}

as

long

as

EF/m,

c2

1, i.e.,

_rs >

10- 2

_(me,

electron

mass).

The weak

_{coupling hypothesis (WCH)}

is ensured when

_EF

remains

_larger

than the

(absolute)

coulombic e - -ion energy. This can be

readily expressed by evaluating

the latter for

each

_ion-species

in a neutral

_sphere

of radius

a’

=

[3

Zv/(4 -ff n)]1/3

and next

performing

the

c, average

according

to

_(2.1).

This

_{yields :}

which

_typically

_{restricts rs}

to be lower than

_unity

in the

_{hydrogen-helium}

mixture.

Alternatively,

the WCH can be

_implemented

as an

_adequate

criterion

_validating

the first Born

approximation

in the calculation of the e--ion

_scattering

cross-section. For our _{purpose, and}

following

Stevenson and Ashcroft

[11],

we write it in the form

_{u app/ 4}

7Ta2 ::;;

1,

where the

«

_{geometrical »}

cross-section 4

7Ta2

is evaluated with the characteristic Thomas-Fermi

screening length

À TF

=

(6

7tune

e2/EF)- 1/2

while

0-app,

the

apparent

cross-section per

ion,

is

defined

by

the

_{identity :}

An obvious estimate for the time relaxation T is

provided by

the standard

expression

a =

ne

e2 T /me

for the electric

conductivity.

We thus arrive

at (T / (T 0

Z/ 7r 2,

,

i.e.,

just

a

lower bound for o,. A more suitable conclusion can however be

reached,

_by

making

use of the

following

reduced

_expression

for the well-known Ziman

resistivity

formula

including

the Born

approximation,

detailed in section 4 :

This

_yields

which formally corresponds

to

_{inequality (2.4).}

As we shall _see, this assertion is corroborated

by

the fact that the basic dimensionless

_quadrature

_Io

_entering

in Ziman’s formula

_(2.6),

a)

remains close to

_{unity, b) only}

involves ion

_charges

under their reduced form _za and

_c)

also exhibits a smooth rs variation

( 1).

Moreover,

additional constraints such as

_a)

the electron

(1)

An

_upper-bound

for

₁₀

can be

_crudely

derived with

V (x) =

(x2

+ 0.166

_rs)-1,

the Thomas-Fermi

potential,

and

S(x) =0,

the

_asymptotic

limit of structure factor

_S(x).

This

provides

jellium

remains

_only

weakly

polarized

by

ion

density

fluctuations,

namely À TF a’, v

and

b)

the electron mean free

_{path A e}

= _{VFr is}

large

in

comparison

with _ai, also _agree with

(2.4)

or

_(2.7)

endowed with a

z-

2/3 factor. We are thus led to conclude that in addition to the

previous (2.3), inequality (2.7) provides a

correct criterion for the Lorentzian

approximation,

including

the Born one.

Looking

at the

_{thermodynamical}

coherence of the Lorentz model

through

the

_requirement

p =..e p, + pi -- 0

also

_provides

another

_{upper-bound for rs (Ref.[lb]),}

which amounts to

r, 5

(0.206

+ 0.405

Z/Z)’

i.e.,

rs 5 1.6 for pure H

and rs s

1.2 for pure He.

Negative

pressure for

non-negligible jellium

correlations

(typically rs - 1.5)

can thus arise at

_large

He concentration. This is a

_warning

that a

polarized background

should be introduced within the

so-called

_polarized

BIM model

_{(PBIM) (Ref. [1])}

in lieu of the

_rigid

BIM one.

The conditions derived above form the very basis for the Lorentz model under

investigation.

It should be

_appreciated

however that

they

do not match the condition that atomic nuclei

get

completely stripped

of their orbital

electrons,

especially

at

_large

Z values. The latter

_requirement

can be

_roughly

derived

by demanding

the electronic pressure

p, -

2/5 ne

EF

to be

_{large enough}

for pressure ionization to hold :

Even in the

_deep

fluid interior of

_{giant planets}

such _{as, say}

_Jupiter,

where

_{(Ref. [12])}

rs .--

0.85,

Feff -.

45 and cl -.

0.9,

condition

(2.8)

indicates that the heaviest

(helium)

component

may be

only partially

ionized.

Turning

to the ion

_component

_properties,

we

_inquire

for ion

_departure

from their classical

behaviour,

a feature which

plays

an

_important

role within the BZ

_transport

_formalism,

due to

the enhancement of inelastic electron-ion

scattering

processes at the Fermi surface. The latter

are

_usually

measured

_by

where

_kF

=

(2

_me

EF) 1/2/h

is the Fermi wavevector while 8 =

A/ai

defines a « one-fluid »

parameter

with A =

_[{3h2/{2

_{1TM)] 1/2,}

_{the average thermal De}

_Broglie

_wavelength.

A detailed

_{analysis (Paper II)}

shows that inelastic

_scattering

contributions to the BZ formalism

are to be measured in

_FyIlO

units. For our _purpose, the classical ion _range is thus

well-characterized

_{by y «}

1 for r « 10

_{and y «}

_IOIF

for r 2: 10.

_Finally,

the OCP fluid-solid transition

_[13]

also

_provides

the upper bound

Feff --

171,

i.e.,

T _ 171 for pure H and

F -- 42.75 for pure He.

The various

_F-r,

domains

_featuring

the overall

_inequalities

described above are

_pictured

in

figure

1,

in the case of a 90 % H-10 % He mixture

_{corresponding}

to

_{Jupiter’s deep}

interior. Inelastic contributions are

_expected

to act

_{significantly}

in domain B.

_Hopefully,

this latter appears

well-decoupled

from domain A where finite

_{degeneracy (a-dependent)}

corrections to

the T = 0

jellium play

an

important

role. This

decoupling

is the most

_{conspicuous by-product}

of the

_present

analysis.

It allows us to treat the

_{corresponding}

_transport

formalism

_separately,

with a

_{significant simplification}

of the

_{required algebras.}

Those are

_explained

in section 4 of

Fig.

1.

_{T-rs plane}

for the H+

-He2+

mixture at _C2 = 0.1

(10

%

He), corresponding

to the

deep-fluid

interior of

Jupiter.

Finite

degeneracy

and inelastic electron-ion

scattering

effects get

significantly

enhanced in domains A and B

_{respectively.}

3. Boltzmann-Ziman

_transport

_theory.

Let us consider a

jellium

fluid submitted to an external electric field 9 and a thermal

gradient

VT. We also introduce the local and

_{macroscopic velocity v (r, t )}

of the ionic

_component

with

respect

to the observer in order to account for

_viscosity

effects - Vv. The v --+ 0 limit taken at

the end will ensure that we

_get

the correct limit for

_transport

coefficients in a frame where the ions are at rest. In

_agreement

with the Lorentzian

_{approach, nearly-free}

electron states are

represented by plane

waves of energy

_Ek

=

h2 k 2 / (2 m,)

and

velocity

vk =

kk/m,,.

Then,

electron

_transport

can be worked out

_through

the linearized Boltzmann

_{equation :}

Function CP (k)

within the collision term

_provides

a standard linear estimate

f, - f e 0 - 0 (k) (8feo/ aE k)

for the true electronic distribution

_{f e}

in terms of the Fermi function

_{f °.}

The e--ion transition

_probability

per unit time

F (k --+ p ) computed

in the first Born

Here we define

V(q)

as an obvious linear combination

of the screened electron-ion

_{pseudopotentials}

_{V,,(q) ==}

_Û,,(q)/s(q)

_{(Ref. [4]), including}

a

static limit of the

_jellium

dielectric function

_{s (q)}

= lim

s (q, ltJ).

Hence the «one-fluid»

w - 0

structure factor

_{S(q, £ù )}

reads as :

The

_S,,,’s

stand for

partial

ion-ion structure factors which are double Fourier transforms of

t-dependent

correlation functions

[10].

The

_{physical meaning}

of definitions

(3.3)

and

_(3.4)

becomes clear in the coulombic case where the

_i-dependence

is

_only

contained within the

charge

number

_Z,.

This

_yields

and

Szz(q, úJ ) corresponds

to the reduced

_{charge density-charge density}

structure factor with the

emergence of reduced valences z 11 =

Z,, 12,

previously

defined in section 2.1.

The

neglect

of e--e- collisions within

_{F(k --* p ) (Eq. (3.2))}

has

_obviously

no

_practical

incidence in the calculation of electric

_{conductivity.}

However,

turning

to the thermal

conductivity,

we

_already

know from

_Lampe’s

_{analysis [ 15]}

that electronic interactions within a

partially degenerate electron-jellium

could lower it

_by

an amount of 25-50 % ! This is a

warning

that linear

_transport

_theory

within a Lorentzian framework has to be

strictly

restricted to its range of

applicability,

delineated

_{by inequalities (2.3)}

and

_(2.7).

Linearized

_transport

_{equation (3.1) plays}

a fundamental role in

assessing

thermoelectronic

and

viscosity

transport.

E = 6 -

_{V J.L}

_e/e

behaves like an effective electric field. Another

possible

choice,

among

others,

reads as

_{(Ref. [14b],}

Sect.

3.2)

E’ = 6 -

_(Vil,,)Tle

where the

gradient

operator

is taken at constant

_temperature.

_{Then J.Le}

within

_{equation (3.1 a)}

has to be

replaced by he, enthalpy

per electron.

li

corresponds

to the

_symmetrical

and traceless

_part

in

Vv. As in the classical case

[14],

the

viscosity

term does not exhibit an

independent

contribution - div v.

Consequently,

the bulk

viscosity [16]

is seen to have an

identically

zero

value to all orders in the

_degeneracy

_parameter.

This result can be

proved

on a

_rigorous

basis

[1b] through

a

_{Chapman-Enskog-like}

treatment of the

_transport

_{equation (3.1) belonging}

to

quantum

Lorentz gases. It extends a

_previous

and similar conclusion

_already

obtained

_by

Abrikosov and Khalatnikov

_[17]

at order a = 0 within the _context of the Landau

theory

for

quantum

liquids,

and it agrees with the

neglect

of e--e- interactions.

with

Equations (3.8) provide

the

_{required microscopic expressions}

for electric and heat-flux

currents and the stress _tensor,

_{respectively. Supplementing}

_transport

_{equation (3.1 a), they}

will allow us to determine thermoelectronic coefficients and the shear

_viscosity.

This is achieved

_{by comparing}

them with their

_{phenomenological homologues}

detailed in the next

section.

4. Basic

_transport

_quantities.

4.1 GENERAL. - Linear

transport

coefficients are defined

_{by [6, 16]}

L,, =- a- corresponds

to the electric

_conductivity

at constant

_temperature

while the thermal

conductivity

is defined

_by

U == - K V T with

_{J = 0,}

which

_implies

K =

-

(Lrr - LTE

LET/LEE),

Crossed

_quantities

_LTE

and

_LET

fulfill

_Onsager’s

relation

_[14b],

LTE

= -

TLET.

The Lorentz number C relates the thermal and electrical conductivities

through

K = £ Ta-.

Finally,

the

_thermopower

is defined as

_Q

= -

L ET /LEE

-Phenomenological equations (4.1 a-b)

hold for the effective electric field E

_{= F, - V» e e}

retained

_throughout

this work. With the alternative definition E’=82013

_(VU,)Tle,

the thermal current

_{(4.1 b)}

contains an additional term

_(Jle)(Tse)

_{with se,}

_entropy

_per

_{electron ;}

thus a and K remain

_unchanged

but

_Q

must be

_{replaced by Q’ -}

Q -

(se/e).

4.2 LINEAR TRANSPORT IN THE ELASTIC CASE. - We intend to solve linearized

transport

equation (3. la)

in the standard limit where electrons are

_elastically

scattered

_by

ionic

density

fluctuations. The actual content of elastic diffusion is the

_{following : through}

successive

collisions,

the electron absorbs as much _energy as it reemits. It is then sufficient to

_put

i =

_{6 hw --* 0,}

in

agreement

with the electron-ion cross section

_proportional

to

_{S(q, w )}

(Eq. (3.4)).

Also,

it should be

_kept

in mind that in the é - 0

limit,

S(q, to )

as well as the

S,,,, (q, w

)’s

remain

_symmetric

under w - - w, in

agreement

with the

_principle

of detailed

balancing.

As a _consequence, emission or

_absorption

of a

_quantum

-hW is

equally likely

to

for the basic

_{Fo (k --+}

_{p ) expression}

which

actually depends

on the static

_(classical)

« one

fluid » structure factor :

Equation (3.1 a)

also shows how intertwined the vectorial thermoelectronic coefficients are

with the order 2 tensorial shear

_viscosity.

_So,

one is entitled to

_{write 0 (k) = P 1 (k) + 0 2 (k)}

where

_{functions c/J t (k) and c/J 2 (k) correspond}

to

_separate

solutions for thermoelectric and mechanical

_transport

_{respectively.}

Within the elastic

limit,

those can be derived

_exactly

because collision term

_(4.2)

relates

_only

p states located on k

_sphere.

Therefore

c/Jt (k)

and 0 2 (k) may be

written as

with

_generalized

relaxation times

_Tl(k)

and

_{T2(k) depending only}

on k. There is no need for

two distinct relaxation times in E and

_{V T respectively,}

because

_{(Ek -}

_{,u, e )}

remains constant on

k

_sphere.

A similar

_reasoning

would have

_easily

convinced us that a

hypothetical

bulk

viscosity term - f (Ek -

/ e )

div v,

where f

is a

given

scalar

function,

should lead to a

vanishing

elastic contribution.

Upon

replacing

formal solutions

(4.4)

in

_{equation (3. la)}

_yields

after

_{straightforward}

identifications :

Ti l(k)

is

_nothing

but the well-known Ziman-like

_{[6] integrant extending}

to a

_running

_upper

bound,

while T2

1 (k)

taken in

_{the k --> kF}

limit does the same for

_Baym’s

_viscosity

formula

_[18]

in the elastic case.

4.2.1 Reduced

_quantities

and

_{methodology. - Bearing}

in mind a

_{thorough investigation}

of

a-dependent

contributions to the electronic

_transport

(domain

A of

_{Fig. 1),}

it appears useful at

this

_stage

to _re-express

_(4.4)

and

_(4.5)

in a reduced form

_especially

well suited for further

algebraic analysis

and numerical

_integration.

This will also _prove convenient for

_{investigating}

inelastic

_corrections,

as is detailed in section 4.3.

_First,

let us introduce the dimensionless variables

t has to be taken as a function

_{of q}

and a =

T/ TF

With the

_help

of the dimensionless variable x, the effective interaction

V(q)

(Eqs. (3.3)

and

The

_jellium

dielectric

_{function e (x)}

within

_{V (x) depends solely}

on x.

Also,

the structure

factor

_S(q

=

2 kF x)

is now

_simply

denoted as

_S(x).

Electric

_conductivity

and shear

_viscosity

are

_respectively

measured in a.u.

_(see

Sect.

2.1).

We will also introduce the

_{corresponding}

time unit

Generalized relaxation times

_rl(k)

and

_{T 2 ( k )}

thus read as

with

Next,

one introduces

_quadratures

Kn, m

such as

from which one derives

and

Equations (4.13) pertain

to exact solutions of the

_transport

_{equation at}

_{any temperature,} in the elastic limit. Numerical

_exploration

would

_require

a tabulation of

_{L 1 ( t )}

and

_{L2 (t)}

deduced from

_{V (x),}

for a

_{given E (x) suitably}

chosen in the

_{temperature-density}

domain of interest. In a

_forthcoming

_work,

we shall

_{give specific}

attention to the

_temperature

extension of the Lindhard dielectric function

[19],

worked out

_by

Gouedard-Deutsch

_[20].

These functions hereafter referred to as L and GD

_{respectively,}

will

_play

a central role in

_evaluating

Anticipating

the main

_{analysis displayed}

_there,

let us notice that an

_algebraic

a

_expansion

of

solutions

_{(4.13) obviously}

necessitates an

underlying expansion

of the

t-dependent

_part

in the

integrant

of

_quadrature

_K(i)

-This in turns

_brings

about a simultaneous

_expansion

of

_{t(,q, a )}

with

_respect

to q

leaving

us

with the evaluation of

_quadratures

such as

(2)

where notation

(2.2)

has been used. Those

_quadratures

are

_identically

zero for n odd or

expressed analytically

in terms of Bernoulli numbers in other cases

_[21],

with the

_simplest

results :

The scheme described above constitutes the very basis of our

_methodology

for

_evaluating

finite

_degeneracy

corrections. It is

_{straightforwardly applied}

in the

_sequel

to the

_simple

a -.> 0

_limiting

case. Let us

_emphasis

however that the calculation of

_higher

order terms in

expansion (4.14)

is not an obvious matter. When

_dealing

with L and GD dielectric functions for

_exemple,

this

_requires

first a

_highly

non trivial

analysis

of the

singularities

contained in the

derivatives of these functions at x = 1

(i.e., q

= 2

kF).

The net contributions of these

singularities

have next to be resumed in order to

_get

the correct

_T-dependence

of

_transport

quantities,

even at lowest

a2

order.

4.2.2 Linear _transport in the a --+ 0 limit. - At the lowest

order,

a

expansion

of

K(’)

yields

at once

Upon

introducing

these results into

_{equations (4.13a-d),}

we obtain the

_{required expressions}

for the linear

_transport

coefficients in the T - 0 limit. In order to avoid the

_repetition

of

lengthy expressions,

it appears useful to introduce the

_following

notations

(2)

For a « 1,

quadratures (4.15)

are

_{simplified, replacing}

their lower

_{bound - 6}

_{JL e}

_{by -}

oo . This is

and

(5 =

Dirac

_{distribution) .}

Then,

one

_finally

_gets

which

provide

standard

expressions

for thermoelectronic

coefficients 0- (),

Ko

and

_{Q 0}

as well as

for the shear

_{viscosity °,}

_expressed

in a reduce form. It should be noticed that

_superscript

0 in our notation refers

_specifically

to the simultaneous cv - 0 and a -+ 0 limits.

(Lowerscript

0

on the other

hand,

refers to

_a.u.).

It should also be

_appreciated

that

_{equation (4.20a) yields}

back the well-known Ziman

_expression

for electrical

_{resistivity :}

Q’o

is at least of order a, in

agreement

with a term of

order q

arising

from

Kl,IJ.

In this case, the first non-zero contribution in

_expansion

_(4.14)

behaves as

(t -

1).

Moreover,

in the a 1

_limit,

the Lorentz number retrieves its Wiedmann-Franz value

CO =

_{( 7r 2/3) (k]Ble )2 .}

Finally,

transport

coefficients in the elastic limit at T = 0 are

derived from the structure factor

S(q)

and the mean

_potential

y ( q )

_through

It,

f

and the local

_{parameter (°.}

_Physics

thus lies in those dimensionless

_quantities.

As we will

show in a

_forthcoming

_work,

a salient feature is that

_pre-factors

contained in

_expressions

(4.20)

contribute

_mainly

to the net behavior of the

_{corresponding}

_transport

_{coefficients,}

through

their T

_{and rs}

_{dependence (see previous}

footnote

_1).

This

_explains

the interest of

introducing

scaled

_expressions

like

_(4.20)

when

_{studying analytical properties}

of those linear

transport

quantities.

4.3 LINEAR TRANSPORT IN THE INELASTIC CASE. - We now turn to solve

_{equation (3. la)}

in the more

_general

case where inelastic collisions between electrons and the ionic

_component

cannot be

_neglected.

As

_already

_stressed,

this feature arises in domain B of

_figure

1. In contradistinction with the elastic case,

_{only approximate}

solutions to the linear

_transport

equation

can

_actually

be

_derived,

with the

_help

of a well known variational method

_[6]

which

functions. A few well known

_transport

_expressions

such as Edwards’

resistivity

formula

_[8]

are

shown to

_belong

to this class of solutions.

.4.3.1 Standard variational method. - Let us consider

_transport

equation (3.1a)

written in the

compact

form

X (k)

= P 0

(k)

where

P e (k) = - 3 lin (0 (k»

defines a

_linear,

definite

_positive

and

_self-adjoint

operator

acting

on a class of real trial

_{functions e (k) = E}

_y

i e i (k). Entropy

balance

(3.7)

reads as : 1

where

_{brackets (... )}

refer here to the scalar

_product

_{( 1 / V)}

_E

(... ).

The variational

_principle

k

states that

_amongst

all

_{0 (k) fulfilling equation (4.22),}

the exact solution of

_transport

_equation

optimizes

the

_system

_{entropy ( 0}

_(k),

P fl

_(k)

whereas the

applied

external

_fields

are

_kept

constant. The scalar

_parameters

_’Yi have to be

_{variationally adjusted}

which

provide

the

variational solutions :

--Pii

and

_P

stand for matrix

_{elements ( oj, P oj >}

_{and (cPi’ P cPj)}

_{respectively.}

The tilde refers hereafter to trial functions and other related

_quantities

_specialized

to

_viscosity.

.l¡, Ui

and

_fIi

_correspond

to the rotational invariant

_part

of currents

(3.8), computed

with trial

functions ei (k)

which have to be introduced in the form

in order to

_produce

a nonzero

_output.

Unit vector û and tensor

Û,

which have

_arbitrary

directions in the elastic case, are removed from the variational currents

_{(4.23) through}

an ad

hoc

_projection.

For

_anisotropic

_systems,

or when

_addressing

the calculation of more

sophisticated

transport

quantities

such as the Hall

_coefficient,

a tensorial extension of the

variational method

_{(D. Léger}

and C.

_Deutsch,

to be

_published)

can be worked _out, which

allows us to

_completely

circumvent the

_dependence

of the trial functions on

_unphysical

û or û

_quantities.

4.3.2 Variational reduced

_expressions.

- As

was done for the elastic case, the

_required

variational

_transport

coefficients _may be

given

a more convenient form

_involving

reduced

Functions

_{’P i ( ij)}

and

_{ei(i7) correspond respectively}

to

_{’P i (k)}

and

_ej(k),

the latter

_being

implicit

functions of

_{(Ek - .¿ e)}

or also

_/3

_{(E k - U E)IIT - = -4.}

From a

_general

_point

of

_view,

çoe ;

(77 )

and

è

(?7 )

may be taken as

_{monomials 7jn,}

or chosen in any set

_{of linearly independent}

functions.

Let us now set

_to

_{= Ip - k}

_/2

_kF

_{and tl}

=

(p

+ k )/2

kF

and introduce dimensionless matrix elements

_{P ij}

and

_Èii

as

where

_{S(x, w )}

stands for

_{S(q = 2 kF x, W ).}

We also define dimensionless

_integrands

1JI’ij

and

_Îj

in terms of x and the new variables ii and

_{fi ’}

previously

defined in

(2.2)

so that

and

_similarly

for

_gii.

_Upon

_{introducing :}

and

_using

similar definitions for

_{À,, A 2}

and

_A3

with _{’Pi --+}

_Pi’

one arrives after a number of

lengthy algebraic manipulations

at

Variable fi’

is useful in

_computing

_transport

coefficients at a low

temperature,

i.e.,

a « 1. When

_adressing

the

_computation

_{of P;,}

and

_Èij,

we should in fact consider

_quadratures

similar to

extending

to the inelastic case

_{integrals (7jn)o}

_already

considered in

_(4.15).

The

_analytic

detailed in the

_following

_{paper II.}

For our _purpose, let us

_just

note here that

_quadratures

(ij ")

1 are

identically

zero for

odd n,

while in other cases

they

express as

where

_n(cJ)

=

(e e - 1 ) -

1 refers to the Planck

function,

while

pp(w2)

is a

_polynomial

of

degree

p.

Expressions (4.31)

are thus

especially

useful in

computing

inelastic corrections

obtained as

expansions

in

(à 2which again explains

the basic

methodology

described above.

Finally,

one

_gets

the

_required

_transport

coefficients in the reduced form

Once

_{cP i ( 1j)}

and

_{Pi ( 1j)}

are

selected,

these

expressions

allow for a variational calculation of

transport

coefficients at _{any temperature.} As is

_readily

demonstrated from the basic content of the variational

_principle

itself,

variational

_{quantities a "1}

Kvar and

7J

v a ’have

to be smaller than their exact

_counterpart.

4.3.3 Variational transport

coefficients

in the elastic limit. - Before

_proceeding

further with the estimation of inelastic contributions

_(Paper

II),

it proves useful to

_briefly

examine the connection between variational

_techniques

and the elastic

transport

calculations detailed in

previous

sections. For that purpose, it is sufficient to derive variational results from the

simplest

monomial trial

_{functions n}

which

_produce

a non-zero

_output

for

uvar,

, Kvar and

7J ;ar.

The elastic limit is characterized

_by

lim

_to

=

0,

lim

tl

= t and also

Estimating

_{p a, _ p 11 IJ,2}

with

_{’P 1 ( 7j)}

= 1

provides straightforwardly

since

_JI

- 1 in this _{case ;}

expression (4.33)

can be shown to be identical to the well-known Edwards formula for

_resistivity.

in

_agreement

with an

_expression

_recently

derived within a different theoretical framework

_by

Ichimaru and Tanaka

_[22]

for thermal

_conductivity

in dense

_plasmas.

However,

it should be

appreciated

that these authors have

_replaced

the chemical

_potential

_g, in variable

fi =

8 (Ek - / 7r

by

he,

enthalpy

per electron. This is a

_simple

consequence of the definition retained for the effective electric

_field,

as

_{explained previously.}

Let us also notice that a

_complete

calculation of Kvar with the correction term -

_{TL ET 2 IL EE}

_requires

at least two

trial functions in order to

_produce

a

_{non-vanishing}

_output

for the latter. The

_complete

expression

is then much more involved that

_{equation (4.34),}

even at a level of a two functions

approximation.

Finally

we

_{compute 7J:ar}

with

_{§S, ( fi )}

=

1,

and obtain

Equations (4.33)-(4.35)

derive from the variational

_principle

with one

_single

trial function in each case. In the a - 0

limit, they

reduce to exact results

(4.20a)

and

(4.20c-d)

respectively.

Another

_interesting

_problem

would consist in

_{demonstrating}

that a

_complete

basis of

monomials r,n,

or more involved

functions,

yield

back the exact solutions at _any

T

_provided

in

_{equations (4.13).}

A more immediate _program would consist in

selecting

out

trial functions within a limit a «

1,

giving

back exact

_expansions

of

_transport

coefficients up

to a finite order

_{a . By}

_restricting

to thermoelectronic

coefficients,

we have been able to

check out that the function

_{pairs (1, 77)}

and

_{(r" r,2)}

_yield

back

_{respectively 0,}

and

K _up to

order a 2.

_So,

one may

conjecture

that the

thermopower Q

should be retrieved at

order

a2

with

(1, fi,

r, 2)..

Once we have chosen the

_given

trial

functions,

we can thus

apply

the above formalism to

include inelastic effects. In this

fashion,

we shall be able to

_{compute u}

at order

a 2,

the inelastic contributions

_being

included both into the zero-order term and in the

a 2 correction.

This will be done in the next

_{paper II.}

5. Conclusion.

The theoretical framework for

_computing

all the

_{time-independent}

electronic

transport

coefficients in

_{strongly coupled}

and

_{multicomponent}

ionic mixtures is laid down

_through

the Boltzmann-Ziman

_approach

and the Lorentzian

_{approximation.}

The latter rests on a few

number conditions on the basic

_parameters

r

_{and rs,}

which have been examined

_carefully.

The

_present

_analysis

results in a crucial

simplification arising

from the

decoupling depicted

in

figure

1 of inelastic effects from

_{temperature-dependent}

_{corrections. This remark paves the}

way for a

_separate

_{analysis given}

in the

_following

_{paper II}

in this

series,

in terms of exact

a 2 and

_{(j 2expansions.}

_Specific

features of Boltzmann-Ziman

transport

formalism within the

context _{of Lorentz gases have been reviewed with}

_specific

attention to the

Chapman-Enskog-like treatment of the

transport

equation. Finally,

thermoelectronic and mechanical

_transport

coefficients have been

_rexpressed

under a suitable reduced

_form,

both in elastic and inelastic cases,

ready

for further

algebraic

and numerical

developments.

Acknowledgments.

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