An equation of state for fully ionized hydrogen

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An equation of state for fully ionized hydrogen

G. Chabrier

To cite this version:

An

_equation

of

state

for

_fully

ionized

_hydrogen

G. Chabrier

Ecole Normale

Supérieure

de

_Lyon,

Laboratoire de

_{Physique (*),}

46 Allée d’Italie, 69364

_Lyon

Cedex 07, France

Institute for Theoretical

_{Physics, University}

of California, Santa Barbara, California

93106,

U.S.A.

(Reçu

le 2

_février

_1990,

_accepté

le 23 avril

1990)

Résumé. 2014 Nous

développons

un modèle

d’équation

d’état _pour des

plasmas complètement

ionisés et nous

_appliquons

ce modèle au cas de

_{l’hydrogène.}

Les effets de fort

_couplage

entre ions ainsi que les effets à

température

finie pour le gaz d’électrons sont inclus dans le modèle. Nous considérons deux différents modèles dans deux domaines de densité

correspondant

respective-ment aux limites d’interaction faible et forte entre ions et électrons.

_{L’équation}

d’état finale

couvre totalement le domaine de densité et de

_{température caractéristique}

des étoiles de faible masse, des étoiles de la

séquence principale,

des

_{planètes géantes}

et des

_expériences

de fusion par confinement inertiel. De

_plus

nous donnons une

_{expression analytique}

_pour la

règle

des sommes

pour la

compressibilité

du gaz d’électrons et _pour la

_longueur

d’écran

électronique

à

température

finie, ce

_qui

fournit un lien entre les théories

_quantique

de Thomas Fermi et

_classique

de

_Debye

Hückel.

Abstract. 2014 We

developed

a model for the

equation

of state of

fully

ionized

_plasmas

which we

apply

to

_{hydrogen. Strong coupling}

effects between ions as well as finite _temperature effects for

the electron gas are taken into account. Two different models are considered

corresponding

respectively

to weak and _strong limits for the correlation effects between ions and electrons. The final

_equation

of state covers the whole

_density

and _temperature domain characteristic of low-mass stars, main sequence stars,

giant planets

and

inertially

confined

plasmas.

An

_analytic

expression

for the electron gas

compressibility

sum rule and for the electronic

_{screening length}

at

finite _temperature are _derived,

_providing

a link between the quantum Thomas Fermi and the classical

Deby

Hückel theories.

Classification

Physics

Abstracts 52.25K - 05.70C

1. Introduction.

The

_properties

of dense ionized matter have been

_investigated

for

_long

not

_only

because of the intrinsic interest as a

subject

of statistical

_physics

but also because of the

important

applications

to a wide _array of

problems

such as inertial confinement fusion

_experiments

in

(*) Equipe

associée au CNRS.

plasma physics,

the

study

of

liquid

metals and other Coulombic

liquids

in condensed matter,

and the

study

of the interior of stars and

_giant

_planets

in

_{astrophysics.}

Under extreme

_conditions,

similar to those encountered in the center of white

dwarfs,

the ions

_(nuclei)

form a classical Coulombic fluid embedded in a

_{rigid background}

of

degenerate

electrons. Such a fluid can be modeled

by

the well known one _component

_{plasma (OCP),}

whose static and

_dynamic

_properties

have been studied

_{extensively [1].}

In

_{general however,}

the

_rigid

electron gas

approximation

is

rather

crude. In most cases of

_interest,

_{the electron gas}

is

_{polarized by}

the ionic

charge

distribution and the

_properties

of the

_plasma

are modified

_by

the response of the electron fluid. Various

investigations

have been made to

_analyze

the effect of the electronic

_screening

on the

_properties

of the

_{plasma, using}

Monte Carlo

_experiments

_[2,

3]

or variational calculations

[4].

The most extensive

_study

of this

problem

has been

performed by

Ichimaru and his collaborators

_[5],

who

_developed

a

_{complete theory}

of

interparticle

correlation in

_dense,

_high

_temperature

_plasmas.

Their

_theory,

however,

is restricted to a domain of

_relatively

weak

_coupling

for the

_{plasma ( T _ 2)}

and therefore is of limited

_application

for

_{astrophysical}

_{purposes. In} all these calculations the ion electron interaction is treated within the linear _response

_{theory (LRT).}

For

_{hydrogen plasmas,}

_attempts

have been made to extend these calculations

beyond

the

LRT,

either

_{by considering}

the electrons as semi-classical

_{particles [6]}

and

_using

a

pseudo-potential approximation

for the interactions in the

_plasma

_[7],

or

_{by using}

sophisticated

theories like the

_density

functional

_{theory (DFT) [8]}

or the

quantal hypernetted

chain

theory

[9]

to treat the full non-linear response of the electrons and ions

self-consistently.

Because of

its

_{approximation,}

the first method is confined to a low

_{density regime,}

and its domain of

application

is not

_yet

_clearly

delimited. This

_point

will be addressed in the

_present

_{paper. The} other methods

_present

a more

_complete

treatment of the

_problem

but have the

_disadvantage

of

_being

very cumbersome.

Consequently they

can not be used to

_generate

an

_equation

of

state

_(EOS)

over a wide

density

and

_temperature

_range.

_Moreover,

the pressure and the free

energy must be calculated

_by

numerical differentiation or

integration,

which

require

tremendous numerical efforts.

Such an EOS for a

_fully

ionized

_plasma

is of first

_necessity

for

_applications

in

_{plasma physics}

and

_astrophysics

where the

_{thermodynamics}

of the

plasma

must be known over a

_large

domain of

_density

and

temperature.

The purpose of the

present

paper is to describe a model

to

_generate

such an EOS. Because of its

_{astrophysical}

interest,

we

_computed

the EOS for a

pure

hydrogen plasma (Z

=

1 )

and compare it with

published

results. The formalism is

presented

for

_arbitrary

ionic

_charge

Z and can be

_applied

to _any

_fully

ionized fluid. The

_{theory developed by}

Hubbard and DeWitt

_[2]

is accurate

_only

in the

_{high density}

regime (rs

1 )

and assumes that the electron gas is

_{fully degenerate.}

Our model takes into

account the

_properties

_{of the electron gas} at finite

_temperature

and the final EOS table we

computed

covers

_essentially

the whole range of

coupling

and

_degeneracy

characteristic of a

fully

ionized

_{hydrogen plasma.}

We stress that _{the purpose of} our model is not to

_investigate

the onset of localization of bound states in the

_{hydrogen plasma}

but to

_generate

an accurate EOS for

_fully

ionized

hydrogen.

In

_fact,

the EOS we

_present

in this paper is

_part

of a more

_general

calculation for fluid

_hydrogen,

in which _{the presence of atomic and molecular}

_hydrogen

is considered

[10,11].

For this reason, the determination of the conditions for pressure ionization and

temperature

ionization as

_given

in section 2 is

_{qualitative only}

and is meant to

_give

approximate

limits to the

_temperature

and

_density

range of

validity

of the

present

model. The final EOS table and the details of the model we

developed

for

atomic/molecular hydrogen,

2. Plasma _parameters.

We consider a

_plasma

of

_N;

ions of

_charge

Ze and mass M and

_Ne

electrons of

charge

- e and mass m in a volume il at a

_temperature

T. The concentrations and number densities are defined

_{respectively by xa -}

_{N a /N}

with a = i _{or e} and

n a - N a / Y = xa n

where

n =

NI V

is the total number

density

and N

= N;

₊

Ne

is the total number of

particles.

The

condition of

_{electroneutrality ne}

= _{ni Z} is assumed _over the whole

plasma.

The

_plasma

is characterized

_by

the usual

_coupling

_parameters,

T for the

ions,

Fe

for the

_electrons,

and the

density

parameter rs

for the

_degenerate

electron gas, defined

respectively

as :

Here a =

(3/4 -ff ni) 1/3

is the

ion-sphere

radius, ae

=

(3/4

7rn,)11’ =

aZ- 1/3

is the mean

interelectronic

distance,

and ao is the Bohr radius.

The

_degeneracy

_{parameter 0}

for the electrons is defined as :

where

_kTF

=

h2(3

TT’ 2 ne)2/3 /2

me is the Fermi

temperature

for the electron gas. The

temperature

T and the

_plasma

mass

_density

are related to these dimensionless

_parameters

through

the

_following

relations :

where A is the nuclear mass number in amu’s. We assume that the atomic nuclei in the

_plasma

are all in the

_fully

ionized state, which is achieved either

_through

the pressure ionization or

through

the

_temperature

_{ionization condition. The pressure ionization condition} can be

described

_{qualitatively by requiring}

the attractive Coulomb

_potential

_per electron

_{Ze2/ ae}

to

be smaller than the Fermi energy - 2

B2/maé i.e.

In the low

_density

limit

_(log

_{p m -}

_{4 )}

the

_temperature

ionization condition is

_given

_by

the Saha

_equation.

In the

_{p -T}

_diagram

the 99.9 % ionization

_boundary

for a one

_component

system

is

_{given by}

where _XH is the ionization energy of the considered

species.

At

_{high density}

this condition for a

_temperature

ionization is

_replaced

_by

the

requirement

for the

_temperature

to be

_greater

than the

_binding

energy of an orbital

_electron,

i.e. :

At very

high

density (rs 1 ),

relativistic effects become

_{non-negligible}

in the electron gas. Even

_though

such effects could be

easily

included in our model in this

limit,

we will

concentrate on the non-relativistic

domain,

defined

by

the

_requirement

that the kinetic energy

be smaller than the electron mass, i.e. :

Quantum

corrections to the

_{thermodynamic properties}

of the ions will be considered in our

calculations. We will define the ionic

_quantum

_{parameter A}

as :

In

_figure

1 we show the domain of

_application

of our

_study

in a

_{p -T}

_diagram.

The hatched area

in the lower left side of the

_diagram

is excluded because of

_{equation (4)}

and the

_{interpolation}

between

_{equations (5)}

and

_(6).

The hatched area in the _uper left side delimits the

_region

of

stability

for the solid

_phase.

The fluid-solid coexistence curve is the one obtained from

classical Monte-Carlo calculations

_[12],

i.e. T =

178,

corrected for

_quantum

effects when the

ionic

_{plasma frequency}

becomes of the order of the

_temperature

(ha 2: kT) [13].

Fig.

1. - Phase

diagram

of

_fully

ionized

_hydrogen.

Solid curves indicate constant values for the

plasma

parameter T and the

_degeneracy

_{parameter 8} defined in the text. Values of the

ion-sphere

radius in units of Bohr radius

_(r,)

are also indicated. The hatched

_rectangular

area at low temperature and low

_density

encloses the

_regime

dominated

_by

atoms, molecules and

_partial

ionization.

_Quantum

effects become

important

for the _protons above the line

_Ai

= 1,

indicating

that the

ion-sphere

radius is smaller than the

thermal de

_{Broglie wavelength.}

The hatched area located at low temperature and

_{high density}

is

essentially

defined

_by

r a 178, the value at which the classical OCP model freezes into a

_regular

lattice.

Because of

large

quantum effects

_{(rl > 1 ),}

_however, this

liquid-solid

transition may not occur in a

hydrogen plasma.

Dashed curves

_correspond

to interior models of

_{Jupiter (J),}

a brown dwarf

_(BD),

a

As the

_density

decreases

_along

an

_isotherm,

the electron gas becomes less and less

degenerate,

as seen from

_{equation (3),}

and more and more

_strongly

correlated.

_Finally,

at low

enough density

we end _up with a fluid in which the electrons behave almost as classical

particles.

For this reason, in our

_study

of the ion-electron

_plasma,

we will be

_using

twô

different models in the

_high

and low

_{density regimes,}

the

_{thermodynamic}

functions in the

region

of intermediate

_{coupling being interpolated}

between the values calculated in these two

models. The model used in the

_{high density region}

is based on the Two-Fluid

Model

initially

developed by

Ashcroft and Stroud

_[14],

solved in our case for a

_density-

and

temperature-dependent potential.

The detailed calculations are

_presented

in section 3. The models free

energy in the low and intermediate

_{density regimes}

are

_given

_respectively

in sections 4 and 5

and the final results for the EOS of the H

_plasma

are

_presented

in section 6.

"

3. The

_{high density regime.}

3.1 SCREENING LENGTH AND EFFECTIVE POTENTIAL. - As

long

as the Coulomb

_potential

energy

e2la

is smaller than the Fermi energy of the electrons EF, we can assume that the ion-electron interaction is weak

_compared

to the kinetic contribution of the electrons and can be

treated as a

_{perturbation, retaining}

the linear contribution

_only.

This condition leads to

equation (4)

and is

_{supported by}

results of the

_inherently

non-linear

Density

Functional

Theory

in which bound states _{appear in the}

_plasma

_{for rs}

=

2,

indicating

the onset of

strong

non-linear effects

_[8].

Under this

_condition,

the exact Hamiltonian of the ion-electron

_plasma

can be rewritten as

_{[14] :}

Here

_He

is the familiar

_{« jellium »}

Hamiltonian

[15]

for the

_electrons,

and

Heff

is the Hamiltonian of the screened ionic

_fluid,

which takes into account _{the response of the}

polarizable

electron

_background

to _any variation to the ionic

_{charge density.}

It is

_expressed

as :

where K;

is the ionic kinetic

_{contribution,}

_{p K} is the Fourier

_component

of the

_microscopic

density, and ECK, 0)

is the

_screening

function of the electron fluid in the adiabatic

approximation,

to be discussed below. The fluid is

_equivalent

to the

_{superposition}

of two

uncoupled

fluids,

i.e. a

_jellium

electron gas and a fluid of

_{pseudo-ions interacting through}

the

short range screened

potential :

which is the sum of the external ionic

_potential

and the induced

_{polarizable potential.}

We define a dimensionless temperature- and

density-dependent screening

_parameter

Qe

as :

Here

_KF

=

(3

7T2ne)1/3

is the Fermi wavevector,

Q

=

KKF

and X is

the screened

density

temperature

static Lindhard function which

_yields

to the

_following

_expression

for the

screening

parameter :

Here

_KTF

=

(6

rne

e2/kTp)1/2

is the Thomas-Fermi wavevector, a =

J.L o/kT

and

,

° is the

chemical

_potential

of the

_{noninteracting}

electron _gas at finite

_temperature,

defined

_by

the condition :

roc

where

_{In ( a )}

=

Jo

dx xn/ (ex - a

+

1 )

denote the Fermi

_integrals.

The chemical

_{potentials a}

0

are calculated with the fit derived

_by

Dharma-Wardana and

Taylor [16]

and for the Fermi

integrals

we use

_{polynomial expressions,}

of which values agree to within one

_part

in

10- 6

with the numerical estimations

_[17].

The

_screening

wavevector, or inverse

screening

length, K,,

defined as

_Ke

=

KF Qe (Q )

is a

generalization of

the Thomas Fermi and

Debye

Hückel wavevectors recovered

_respectively

in the zero

_temperature

limit

(0

1 )

and in the

classical limit

(0

>

1 ) at Q

= 0. In the limit of

large wavelengths (6-0), equation (13)

yields :

We note that the

_screening

wavevector is

_independent

of the

_wavelength

in this

limit,

which is

a direct consequence of the

_perfect

_screening

condition.

_Using

the results for the

compressibility

of the

_{noninteracting}

electron gas at finite and zero

_temperature,

equations

(12)

and

_{(15) yield :}

where the _xoe denote the

_{respective compressibilities}

of the

_{noninteracting}

_{electron gas} at

zero and finite

_{temperatures.}

The

_{equation (16)}

is the

_{generalization}

at

_finite

_temperature

_of

the zero _temperature

_{compressibility}

sum

_{rule for}

the electron _gas

_[15].

We will see in the next

section how to

_modify

this relation for an

_interacting

electron _gas.

In

_figure

_2,

we show the behavior of the dimensionless

_screening

wavevector

_aKe

as a function of

_Q

for different values of the

_density

and

_degeneracy

_parameters.

A

_large

value of

(aKe )

corresponds

to a

_relatively

short range effective interaction between the ions

_compared

to the bare Coulomb

_potential.

At intermediate densities

_(rs

=

1 )

we note the

_strong

wavelength dependence

of

_Ke

and

_consequently

the

_strong

_departure

from the Thomas Fermi model

_(aKTF

= 1.56

rs 2)

recovered in the

long wavelength

limit for the

strongly degenerate

case. At short

_wavelength,

the

_screening

wavevector vanishes and we recover the unscreened

bare Coulomb

_potential.

At constant

_{density, Ke}

decreases with

_increasing

_temperature,

indicating

that the

_screening

of the ionic interactions becomes less and less efficient. This is a

consequence of the fact that the electron gas

spreads

out as the

temperature

increases at constant

_density.

The same conclusion holds

_obviously

as the

_density

increases at constant

degeneracy

since _{the electron gas becomes less}

_{polarizable, becoming ultimately}

a

_rigid

Fig.

2. - Electron

_screening

wave vector as a function of K for two different densities and two different

temperatures. The dotted line and the short dashed line indicated

_respectively

the Thomas Fermi limit and the

Debye

Hückel limit. The dashed lines are RPA calculations whereas the solid line indicates that

a LFC has been included in the calculation of the

_{polarisability.}

The calculation of the distribution functions and then the

_{thermodynamic}

functions of the

system

necessitates the

_knowledge

of the screened

_potential

_{Veff (r)}

_{(or screening}

function

rheff(r)/(Ze)2)

in the r _space. In order to avoid

_long

range oscillations in the Fourier

transform of

_{equation (11),}

it is more convenient to calculate the Fourier transform of the non-Coulombic

_part

of the effective

_potential,

defined

by f/1 (K)

=

Veff (K) -

4 7r

(Ze)2 IK 2

which behaves like

_{Q - 6 at}

_large

Q.

The

_problem

of the

_{discontinuity of f/1 (K)}

at K = 0 can be circumvented

_{by rewriting Q}

as the sum of two

_{potentials :}

one

_being

the difference between a Coulomb and a Yukawa

_potential,

of which the Fourier transform is calculated

_{analytically,}

the other one

_being

the

_{complementary}

_part

_f/12(K).

Both have a finite value at K = 0. The final result

reads,

in dimensionless

quantities (x

=

rlae ; q

= _ae

K) :

where A o is a

screening length

defined as :

and f/12(X)

is

_{given by :}

with qe(q) = ae Ke(K).

decreases

rapidly

as the

_{temperature increases,}

at fixed

density

[3].

The function

03C82(x)

represents

the

_departure

of the effective

_potential

from a Yukawa

potential.

In

_figure

3 0393

we _compare the effective interionic

_potential

Veff(X)

with the Yukawa

potential

e

x

(see

Eq.

(17))

for different _{values of 0 and rs. For distances smaller than the} mean interionic

distance,

the two

_potentials

are very similar and become

essentially

the Coulomb

_potential

as

r _goes to zero. As the

_density

decreases

_(i.e.

_rs

_increases)

the

_departure

from a

_Yukawa

potential

is very

important

in the domain of

_strong

_degeneracy,

because of the

_strong

inhomogeneity

of the electron gas

but,

as the

_temperature

increases,

the two

_potentials

become very similar

(for

r, = 1 the

highest

discrepancy

is of the order of 10 % at 0

_{= 0.5)}

and become

_{essentially indistinguishable}

at 0 - 1. Below a certain

_{density and/or}

a

certain

_temperature,

the effective

_potential,

via the function

_03C82’

exhibits an

_oscillatory

behavior in the

_region

_{of short range order. This}

_corresponds

to the onset of the well known Friedel oscillations. It defines a transition between a

Thomas-Fermi-type regime,

at

_high

density (rs 1 )

and finite

_temperature

_{(0 > 0.1 )}

in which the effective

_potential

_VefT(x)

can

be

_{approximated by}

a Yukawa

_potential

of which the

_{screening length}

is

_{given analytically by}

equation (18),

and a

_{Friedel-type regime}

at lower densities

_and/or

_{temperatures.}

This

_point

has been discussed in detail

_by

Gouedard and Deutsch

_[18].

It is

_important

to stress that

Ao

in

_{equation (18)}

is a

general density-

and

_{temperature-dependent screening length,}

the

density

and the

_temperature

_{dependence being clearly separated (note}

that

_Ao/qTF

is a

universal function of the

_density

_parameter

_0).

It is a

_{generalization}

of the Thomas Fermi

(Ào

=

qTF)

and

Debye

Hückel

(Ào

1= qDH

=

J"3f:)

screening lengths

recovered

respecti-vely

in the

_{fully degenerate ( 0 --> 0 high density}

_{(rs 1 )}

limit and in the classical

(0

>

1 )

low-density (rs

>

_{1 )}

limit.

The main

_{approximation}

in our model free energy is the linear

approximation

in the

treatment of the electron-ion interaction. The most

_{powerful, although extremely}

cumber-some,

theory

available so far to include non-linear effects in the calculation of this

interaction,

is the

_Density

Functional

_{Theory (DFT) [8].}

_However,

because of the

_complexity

of the

method,

no

_equation

of state has been

_computed

so far within the framework of the DFT.

Therefore it is not

_possible

to

_appreciate

the relative contribution of non-linear effects on the

Fig.

3. -

(a)

Effective interionic

_{potential (Eq. (17) (-)}

and Yukawa

_{potential (---)}

for two different densities at 0 = 0.0543.

(b)

Same _as

thermodynamic

functions of the

_plasma.

We must resort to a

_comparison

of the effective

ion-ion

_potentials

of the

_respective

models. In

_figure

4 we _compare the effective interionic

potential (15)

with and without the inclusion of a local field correction

_(LFC)

in the dielectric

function,

with the result of

_Density

Functional

_{THeory (DFT)}

calculations

_[8]

for

rs = 1. As

already pointed

_out

by

Perrot

_[19]

and Dharma-Wardana et al.

_[20],

one of the main consequences of

using

a linear response

_theory

is to overestimate the extent of the

screening, leading

to a too

_{repulsive potential}

and an underestimation of the electron

density

at the nucleus. This

_stronger

_screening

inhibits the formations of bound states. The two

potentials

differ also in both the

_amplitude

and the

phase

of the Friedel oscillations

_although

this misbehavior is

_{improved slightly by}

the inclusion of a LFC in the RPA dielectric function. However the most

_interesting

feature is that the

_discrepancy

between the screened

_potential

and the DFT

_potentials

diminishes _very

_rapidly

as the

_temperature

increases,

indicating

that a

linear

_{approximation}

becomes more accurate in the domain of low and intermediate

degeneracy (

2:

_{0.2 ),}

the electron cloud

_becoming

more and more

spread

out. We also note

the

_{rapid damping}

of the Friedel oscillations as the

_temperatue

increases,

consequence of the

smearing

of the Fermi

_sphere.

As we

_already

_mentioned,

the

_potential

becomes a Yukawa

type

potential

in this

_region.

In view of this

_{comparison, given}

the

_good

_agreement

found for

rs = 1 at 0 - 0.3 between the two

potentials,

we conclude that non-linear effects will

_modify

appreciably

an EOS

_only

_{for rs 2:}

1 in the

_region

of

_strong

_{degeneracy ( B 0.2 ).}

In this

region,

however,

bound states are

_likely

to _{appear and} a more

_{complete theory}

must be

developed [ 10, 11 ] .

Fig. 4.

- Effective interionic

potential given by equation (17) (-)

or

_by

DFT

_{(---) (Ref. [20])}

at

rs = 1 for _two different

temperatures. The dotted line indicates the

_{potential (17)}

within the RPA

approximation

for the dielectric function.

3.2 THE THERMODYNAMIC FUNCTIONS. - The free _energy related to the Hamiltonian

₍₉₎

is

written

where the trace is taken over the states of the

coupled

electron-ion _system. The

_superscript

(id)

denotes the

_{non-interacting}

contribution to the free _energy

_given

either

by

the standard

where a is the chemical

potential

of the

_{noninteracting}

electron gas defined

by equation (14).

The last two terms on the

_right

hand side of

_{equation (20)}

denote

_respectively

the non-ideal ionic and electronic

_{contribution,}

to be discussed below.

3.2.1 Screened

_{ionic fluid.}

- Because of the _temperature

_dependence

of the effective

_potential

veff,

the excess internal energy is no

_{longer equal}

to the canonical average of the

_potential,

but includes an extra kinetic term. On the other

hand,

the excess pressure includes a term due to

the

_{density dependence}

of the

_{potential :}

where

v ( K _ ) 4’T (Ze)2 2

and

_{S’ ( K - ) N-1}

_{Px Px * %}

is the ionic structure factor of the

K 2

screened ionic fluid.

_{The Q}

and p derivatives are

_trivially

rewritten in terms of the dimensionless variables defined in

_{equations (1)-(3).}

The _{interaction energy eint is}

_equal

to the

sum of the two first terms on the

_right

hand side of

_{equation (22a).}

It is a _very well known

feature of the HNC

_theory

to allow the calculation of the chemical

_potential,

and then of the free energy,

by

direct

_integration

of the distribution functions

[21].

The _{free energy in the} HNC scheme is thus obtained with an _accuracy

_comparable

to those of the internal energy and pressure

contrarily

to methods like Monte Carlo and DFT where numerical

_integration

of the interaction energy with

respect

to the

temperature

has to be carried out. This method has been used for

_temperature

and

_{density independent potentials only}

but it has been extended

recently

to

_temperature

and

_dependent

screened

potentials

[22].

More

_recently,

DeWitt and

Rogers [23]

derived a more

_{simple expression}

for the same free energy, in the HNC scheme :

where

_è(K)

is the dimensionless Fourier transform of the direct correlation function and

h(r)

the

_pair

distribution function of the screened ionic fluid.

_è,(K)

is defined as

cs (K) - c (K)

+

_{P 16}

V eff (K) / Z2

and cp is

given by :

where

_Ao

and

_’P2(x)

have been defined in

_{equations (18), (19).}

In order to assess the

_validity

of the HNC scheme for the calculation of the

_{thermodynamics}

of a

_plasma,

we

_compared

out results for the excess _pressure

p ex,

internal _energy

ue",

interaction _{energy eint, and free energy}

_{f eX (Eqs. (22), (23))}

with

_existing

Monte Carlo

(MC)

calculations

_(see

Tab.

I).

Such simulations have been carried out for a

rigid

electron

Table 1.

_{- Comparison}

between MC results

_(left

column)

and HNC results

_(right

column)

(Eqs.

(22), (23)) for

the ionic excess

thermodynamic functions.

The

quantities

are dimensionless

(

1 Ni kT).

The MC results are taken

respectively from : (a) reference [12] for

rs =

0,

(b)

reference [3] for

calculations with the

_finite

_temperature Lindhard

_{function, (c) reference [2a]}

and

_{(d) reference [2b] for}

the calculations with the zero-temperature

Lindhard function.

equation (11) [2],

and a finite

_temperature

dielectric function

_[3].

In most of the cases the

discrepancy

between MC and HNC results in less than 1 % for _any value of T from 1 to 160. The

_{largest disagreement (-}

1-2

_%)

found for F 2 in the

_comparison

with the results of

Totsuji

and Tokami

_[3]

can

probably

be attributed to the small number of

_{particles (64)}

involved in the MC calculation in this case, since the

disagreement

diminishes as T increases.

The

_discrepancy

on the free energy in the

comparison

with the results of Hubbard and DeWitt

[2]

at small T is due to _{the poor} accuracy of the MC method in this

region,

where the

_Debye

radius becomes

larger

than the unit cell

dimension,

and to the

_approximated

mean _square fit

in this

_{region [24].}

Dharma-Wardana

_[20]

derived two

_separate

_expressions

for the

exchange

part

and the correlation

_part

of the free energy and the chemical

potential.

The correlation

part,

however,

was

approximated by

the

_ring

sum

_{contribution,}

i.e. the RPA

expression,

of which _{range of}

validity

is

_unknown,

as

_pointed

out

_by

these authors. It is

known,

moreover, that the RPA scheme

_always

underestimates _{the interaction energy.}

_Recently

Tanaka,

Mitake and Ichimaru

[5,

26]

calculated the interaction energy of an electron gas at

_finite

_temperature

_using

the

Singwi-Tosi-Land-Sjolander (STLS) approximation [27]

to treat the

_strong

_coupling

effects between electrons

_beyond

the RPA.

They

parametrized

their results as a function of rand 0 and

_performed

a

_coupling

constant

_integration

to obtain a

_general

fit for the

_exchange

and correlation free energy

Fxx(ne, T). During

this

_{procedure, they}

took into consideration the known violation of the

_{compressibility}

sum rule in the STLS

_scheme,

and the

_consequent

departure

from the exact Monte Carlo calculations in the classical

_[12]

and

_{fully degenerate}

[28]

limits as the

_coupling

constant

_{T or rs}

_increases,

_{by anticipating}

similar derivations in their

parametrization.

Their free _energy and its

_temperature

and

_density

derivations

_reproduce

the

classical results

_{[12, 29]}

and the results for the

_ground

state

_{[28, 30]}

with

_digressions

of less than 0.6 %. In the

_region

of intermediate

_{degeneracy (0 ~ 1),}

their result is in

perfect

agreement

with variational calculations

_[31].

Because of the

_{integration procedure,}

the

authors claim a total error of - 5 % on

_Fxc.

_{In any} case this result seems to be the most accurate treatment of the

_{thermodynamic properties}

of an

_interacting

electron _gas at finite

temperature

published

so far.

3.3 LOCAL FIELD CORRECTION. - The dielectric function defined in

_{equations (10)}

and

₍₁₁₎

is the Lindhard dielectric function calculated within the RPA

_{approximation.}

It is well known that the RPA

_{approximation}

fails as the

_density

of the electron gas decreases because it does

not take into account _{the short range correlations in the motion of the electrons. This}

approximation

can be

_{improved by taking}

into account the

_polarization

effects in the electron-electron interaction in the

_{density-density}

_response function in term of a mean-field

approach :

where

_{Xo (K)}

is the

_{density-density}

response function for the

non-interacting

gas,

v (K)

=

47T2 is

the Hartree

potential

which characterized the

long

_{range correlations} of the

K2

electron

_system

and G

_{(K, w )}

is the so-called Local Field Correction

_{(LFC) describing}

all the

exchange

and short range correlation effects

_beyond

the RPA. The dielectric function is then written :

where EL

denotes the Lindhard dielectric function

_{(Eqs. (12), (13)).}

In the formulation of a static

_theory,

the most

_important

effect of the LFC stems from its static evaluation at w =

0,

i.e.

G (K) -

G

(K, w

=

0 ).

Moreover,

in our

_{calculations,}

we will assume the adiabatic

_{approximation}

to be valid and then we will use the static limit

(w

= 0 )

of the dielectric function. The

_validity

of this

_{approximation}

has been assessed

The

_{long wavelength}

behavior of the LFC is

_{governed by}

the

_{compressibility}

sum rule

_{[ 15].}

with

Here

_Xo,

=

3 / (2

nEF)

is the

_{zero-temperature}

_{compressibility}

of the

_{non-interacting}

electron

gas and X

xc

=

1

a

2

F xc

T is the

exchange

and correlation contribution.

gas and d -

xxc

1 =

1/2 a n2(

FXC )

T is the

exchange

and correlation contribution.

1/n2

ô2

(

F xc )

T 0

h h d 1 0

Ob 0

n2 an2

/

The small

_wavelength

behavior of the LFC is related to the short range behavior of the

electronic

_pair

distribution

_{g (r) through}

the exact relation

[33, 34] :

lim

The finite value of

_{g (0)}

is a _consequence of

_{quantal tunneling.}

Although equations (27)

and

(28)

have been derived

_originally

for the

_ground

state

(0 = 0 ), they

are still valid at finite

temperatures

[35].

We can

_easily

extend our

_previous

result

(16)

for the

_{compressibility}

sum rule for the

non-interacting

gas to the

_interacting

electron _gas at finite

_temperature.

_{Using equations (16), (26)}

and

_(27),

we

_{get :}

-lim

where

_KD

=

(4

_7Tne

f3

e 2)1/2

is the inverse

Debye screening length,

_{X o} =

(ne kT)-1

1 is the

compressibility

of the classical

_perfect

gas and

X ( 0 )

denotes now the total

_{compressibility}

of the

_interacting

_{electron gas} at finite

_temperature

_{(X ( (o )- 1 =}

_{x - 1 ( o )}

+

X xc-1 ( (0 )). The

second

equality

in

_{equation (29)}

can be obtained

_{by using}

the classical version of the Fluctuation

Dissipation

Theorem,

i.e. e(K)-1 =

1 -

K51 K2 Szz(K)

and the

_{long wavelength}

limit of the

charge-charge

structure factor

_{Szz(K --> 0) =}

_{(K51 K + X o/X )-1}

1

_[36].

The

_{equation (29)}

provides

a link between the zero

_temperature

_{( 0}

=

0)

and

high

_temperature

(0

>

_{1 )}

limits of

the

_{long wavelength}

behavior of the dielectric function for an

_interacting

electron gas.

A new

_screening

length A 0 xc(ne,

_T)

associated with the finite

temperature

interacting

electron gas, can then be derived from

_{equations (26), (27), 29) :}

where A F 1 = ae

KF

is the Fermi

_{screening length}

and A _o is the

_{screening length}

of the

non-interacting

electron gas

(Eq.

(18)).

The

_{ratio k F/k 0}

is

equal

to the

_screening

_parameter

Qe(O)

defined in

equation (15).

The second term in the bracket arises from the

_exchange

and correlation contribution and can be evaluated

_analytically

from the second derivative of the free _energy

_Fxc

discussed in the

_previous

section.

Various

_expressions

for G

_(K)

have been

_proposed

in the

_past

_by

_many

_{investigators.}

In the

zero-temperature case, the most

_interesting

solution seems to be the one derived

_by

Utsumi and Ichimaru

_[37]

hereafter referred to as

_GUI.

Not

_only

the LFC derived

_by

these authors

compressibility

derives from a

_fitting

formula

[30]

in

_perfect

_agreement

with the

_quantum

Monte-Carlo simulation

_[28],

and the

expression

for

_{g (0 )}

is the one calculated

_through

a

diagrammatic

calculation of the electron-electron ladder

_diagrams

_[38].

In the case of a

finite

_temperature electron

fluid,

the situation is not so

_simple

and no

analytical expression

has been derived

yet.

For

_large

values of the

_degeneracy

parameter 0

the electron fluid becomes

_essentially

classical. Thus the

_principal

effect that _goes into the determination of the LFC is the

interparticle

correlation

_brought

about

_by

the Coulomb

repulsion,

no

_exchange

effects need to be taken into consideration. A self-consistent formulation of such a LFC can be obtained

_using

the linear

_{approximation}

and the classical version of the Fluctuation

_Dissipation

_Theorem,

which

_{yields :}

where

_{Socp (K)}

is the structure factor of the well known One

_Component

Plasma

_(OCP).

Because of the

_{long-wavelength}

behavior

_{of Socp (K)}

_[36],

the

_{compressibility}

sum rule

(27)

is

satisfied,

whereas the condition

(28)

is

_{automatically}

fulfilled since

_{g (0)}

= 0 for a

_fully

classical

_system.

However the

expression (31)

for the LFC

_corresponds

to an electron

OCP,

i.e. a

_system

of

_point

charges

embedded in a

_{uniform positive}

_neutralizing

_background,

without interaction with this

_background.

This result is

_obviously

different from the case of a

lattice of electrons

_interacting

with a lattice of

_point

ions,

which is the case for the

hydrogen

plasma

in the-classical limit for the electrons. This difference is illustrated in

_figure

5 where we

show the electro-electron structure factors obtained in the two situations.

Consequently

we can

_expect

the LFC to be different in the two cases.

Fig.

5. - Electron-electron

structure factor calculated within the framework of the HNC

_theory

for the OCP model

(-)

and for the TCP model

_{(---) (Eq. (38))}

for r = 1

(and

e = 20 for the

TCP).

Another

_approach

for the finite

_temperature

LFC has been

_{proposed recently by}

Tanake and Ichimaru

_[35],

who extended to finite

_degeneracies

the STLS scheme

_{initially developed}

by Singwi et

al. for the

_ground

state

_[27].

The STLS functional form for G

_(K)

satisfies the

(27).

This

_point

has been

_partially

corrected

by

Vashista and

_Singwi

who

_approximated

the LFC

_by

the

_following

form

_{[39] :}

where

_Q

=

KIKF

and the

_parameters

A and B are fixed

_by

the conditions

₍₂₇₎

and

_(28).

Finally

the last

_{possible parametrization}

scheme for

_{G ( K)}

would be a finite

_temperature

extension of the formula first

_{proposed by}

Hubbard

_[40],

i.e. :

where A and B are the coefficients used in

_(32).

The Hubbard

_{approximation corresponds}

simply

to

_using

the Hartree-Fock value of the

_pair

correlation function in the STLS scheme. A common feature of the forms for

_{G (K) given by equations (31)-(33)}

is that

they

do not

exhibit a

_peak

structure around K = 2

KF,

associated with the

discontinuity

of the electron

density

at the Fermi surface. In the zero

_temperature

limit,

the presence of this

peak

in the

LFC,

associated with a

_{logarithmic singularity}

in its

_slope

has been demonstrated

rigorously

[41, 42].

Using

the

_Density

Functional

_{Theory (DFT),}

Dharma-Wardana and Perrot

_[20]

showed

that,

for values of _rs of the order of

1,

the

_{exchange potential}

is still dominant

compared

to the correlation

_potential

up to 0

of the

order

of 1. Consequently,

since the LFC is

related to the second derivative of the

_{exchange-correlation potential,}

we can assume the

exchange

part

of the

LFC,

and hence the

_peak

structure, to be still

_important

at intermediate

degeneracy.

For this reason the form we _{propose for the LFC} is a finite

_temperature

extension

of the form

_{given by}

Ichimaru and

_Utsumi,

that we will refer to as

_GUI.

This is achieved

_by

using

the finite

_temperature

_{compressibility}

in

_{equation (27b)}

instead of the one for the

ground

state. For that we differentiated the

_parametrized

form of the

_exchange

and

correlation free energy at finite

_temperature

mentioned in section 2.2

_[5].

This insures

consistency

between the

_screening

function

_entering

into the calculation of the effective interionic

potential,

and the

_{thermodynamic properties}

_{of the electron gas} at finite

temperature.

The

_term

_{yo in}

_{equation (27b)}

is then

_{given by :}

where

_Fxc

is the

_fitting

formula

_given

_{by equation (3.83)}

of reference

[5].

The

zero-temperature

limit and the classical limit

_{(0 > 0)}

of the

_{compressibility}

calculated from this

differentiation

_reproduce

the Monte-Carlo results

_{[28, 12]}

within less than 3 %. Because of the lack of

_parametrized

form for

_{g (r}

=

0)

_at finite

_temperature,

we

_kept

the zero

temperature

value. It is thus clear that the condition

₍₂₈₎

is violated. However we found out

that the final

_{thermodynamic}

functions of the screened ionic fluid are

_quite

insensitive to _any

important change

in

g (0).

This comes from the

fact,

as seen from

_figure

_2,

that the

polarizability quickly

vanishes

_beyond

K ~ 2

_KF

and that the effect of the LFC is dominant

only

in the

_region

K 2

_KF,

so that the short

_wavelength

behavior of the LFC does not have

any

quantitative

effect on the

_{thermodynamics}

of the

_plasma.

The

advantage

of the form

GUI

is to recover the accurate form in the

_{zero-temperature}

limit and to

_give

an accurate

temperature

dependence

in the

_{long wavelength}

_limit,

where correlation effects are

dominant.

The behavior of the different form of G

_(K)

discussed above is shown in

_figure

6 for

0 = 1. We

explored

the

consequence of these different forms for the finite

temperature

LFC

on the

_{thermodynamic}

functions of the H

_plasma.

In the

_region

of intermediate and low

region

of intermediate and _strong

_degeneracy,

the results differ

_{significantly.}

In this

_region,

the

_quantum

effects between

_electrons,

_resulting

in a

_peak

structure at K - 2

_KF,

become

important

and must be taken into account in a _{proper calculation of} an LFC. We conclude

that as

_long

as we are interested with the calculations of static

_properties,

a

_quantum

treatment of the LFC is necessary in the

region

of

_strong

and intermediate

_degeneracy

(0 1 )

whereas for lower

_degeneracies

the

_{thermodynamics depends essentially}

on the

correlation,

i.e. the

_{long-wavelength (K « KF)}

_part

of the LFC.

The enhancement of the

_screening

due to the inclusion of a LFC

_(GÚI)

can be seen in

figures

2 and 3. The effect on the

_{thermodynamics}

of the

_plasma

is

_quantified

in table II and

shown in

_figure

7

_{by comparison}

with RPA results. The difference can reach a few

_percents

on

the pressure, the internal energy and the free energy for rs a 1 at

_large

_T,

and can be more

than 10 % on the

_entropy.

Fig. 6.

Fig. 7.

Fig.

6. - Different forms of the LFC at 0 = _1, T = 1. The solid line

(-)

represents

GUI,

the dashed line

(---) G H (Eq. (33)),

the dashed-dotted line

_{(-.-) Gvs (Eq. (32)),}

the dashed-double dotted line

(-..-)

Gocp (Eq. (31)),

and the dotted line

(...) G STLS (Tab.

IV of Ref.

[5]).

Fig.

7. - Relative

_importance

of the LFC in _percent for the excess ionic free energy

(e, 0),

internal energy

(A, A)

and pressure

(~, ~)

as a function of T for _{r, = 2}

_(solid

line

_{( ))}

and _rs = 1

_(dashed

line

(--- )).

3.4

_QUANTUM

CORRECTIONS. - In the low

_temperature,

_{high density region}

of the

_diagram

shown in

_figure

_1,

we can

_{expect quantum}

effects on the ions to become

_important.

As

long

as

the

_parameter

ll

_{given by equation (8)}

is smaller or of the order of

unity,

we can consider

these

_quantum

effects to be a small correction to the

_{thermodynamics}

of the ions and we use

the

_{Wigner expansion}

to the lowest order

_(-

_{h 2).}

Since the

_quantum

corrections occur in the

high

density regime ( rs

«

_{1 ) (cf.}

_Eq.

_(8)),

we can

_approximate

the effective

potential

in this

Table II. - Excess ionic

_{thermodynamic}

_functions

within the RPA

_{approximation}

and with a

Local Field Correction. The

_quantities

are dimensionless

(

1 Ni

kT).

For each value

of

rand

rs, the

free

energy on the

first

line has been calculated

_from

the

_expression

_of

Iyetomi

and Ichimaru

_(Ref.

_[5])

whereas the value on the second line has been calculated

from

equation

(23).

where

_Ao

is the

_{screening length}

defined in

_{equation (18)}

or

_(30).

Then the

Wigner

expansion

for the

_quantum

correction to the ionic free _energy

_{yields :}

The first term is the

_quantum

correction for the OCP recovered at _very

_{high density,}

whereas the second term is the

_(negative)

contribution due to the electron

_screening.

The correction to

the internal energy

uqu

and pressure

pqu

are

_{given by}

the

_temperature

and

_density

derivatives. The contribution of the V and T derivatives of the last term in the bracket in

_equation

₍₃₆₎

are

found to be

_negligible

so that

_{ui qu}

and

_pqu

can be

_{approximated by :}

In table III we show the relative contributions of the different terms of the total excess free

energy

(20)

and the related pressure and internal energy for different values of T and rs. The relative

importance

of the

screening

contribution on the

_{thermodynamic}

functions of

the ionic fluid is shown in

_figure

8 as a function of

_r,

and then

0,

for different values of the

Table III.

_{- Different}

contributions to the

_{excess free}

_energy, internal _energy and pressure

of

the

plasma for

different

densities and temperatures. All the

_quantities

are

_given

_per

_{particle (IN)}

in unit

_of

kT. We recall that

_u¡qu

= 2

p 9u .

2

f¡qu

(Eq. (36)).

Fig.

8. - Relative

importance

of the

screening

contribution in percent for the excess ionic free energy

(e,

0,

+),

internal _energy

_(A,

_A,

_x)

and pressure

(U,

D,

*)

as a function of T

for rs

= 0.1

(solid

line

(-)), rs

= 1

4. The low

_{density regime.}

In the

_{low-density regime (rs}

>

_{1 ),}

the ion-electron and electron-electron Coulomb _energy

become

important

compared

to the Fermi energy of the electrons and the dielectric

formulation

_developed

in section 3 can no

_longer

be used.

On the other

_hand,

in the

_low-density,

_high

_temperature

limit

(0

>

_1),

_{the electron gas is}

weakly degenerate

and the

_plasma

_{may be treated} as an almost classical

two-component

plasma (TCP).

Quantum

diffraction and

_symmetry

_effects,

which

_prevent

the

_collapse

of

purely

classical

_systems

of

_particles

of

_{opposite charge,}

can be handled

through

pseudopoten-tials,

derived

_{by expressing}

the

_{quantum-mechanical}

Slater sum in a form reminiscent of the classical Boltzmann factor. For the

_{hydrogen plasma}

effective

_pair

potentials

have been

derived from exact numerical

_computation

of the

_two-particle

Slater sum for electron-electron

and

_{electron-proton pairs}

_[7],

and

reâd :

where

_Àaf3

=

h/(2

71’maf3

kB

T)1/2 and maf3

is the reduced mass of an

a-f3

pair (f3 = 2

denotes

the electron

_component).

In

_{equation (38)}

the first term arises from

_quantum

diffraction

effects,

while the second takes care of

_symmetry.

Because the contribution of bound states to the

_{electron-proton}

Slater sum as well as the

density dependence

of the

_potential

are

_neglected,

this model Hamiltonian can be used

only

in the low

_{density, high}

_temperature

_{( T >}

1

_{Ry ) region.}

Bernu et al.

_[44],

_considering

the influence of the electron

_spin

on the structure and the

thermodynamics

of this

model,

restrained its domain of

_reliability

to the

_{region ’g 2:}

5.4,

so

that densities are low

_enough

for the

_temperature

T to

_greatly

exceed the electron

degeneracy

temperature

TF.

We will discuss the

_validity

of this limit as well as the

_reliability

of the TCP model, The

pair

distribution functions and the

_{thermodynamics}

of the TCP were calculated in

the

_regime

of interest within the framework of the HNC

_{approximation.}

On the other

hand,

in the limit of weak

_coupling

_{(r «}

₁₎

and low

_degeneracy

₍₀

>

1),

the

_{thermodynamics}

of the ion-electron

_plasma

can be evaluated in term of the

_quantum

generalization

of the

ring

expansion [45].

The free energy then reads :

where

_{(Z2) = L Xa Z;, a}

_denoting

the ionic and electronic

_species.

The first term in

03B1 03B1

equation (39)

is the

_leading

term of the classical cluster

_expansion,

i.e. the

_Debye

term,

whereas the second term in the bracket

_represents

the correction for the electron

_quantum

effects. The

_quantum

diffraction

_parameter

_{ye is}

_{given by :}

where Ae

is the electron

_equivalent

of the ionic _quantum

_parameter

defined in

_{equation (8).}

0393

For Coulombic

system

(- 1/r),

the

_potential

_/3

V - T

decreases as

_1/kT

as the

_temperature

a