Low-frequency electric microfield calculations by iterative methods

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Low-frequency electric microfield calculations by

iterative methods

B. Held, P. Pignolet

To cite this version:

B. Held, P. Pignolet. Low-frequency electric microfield calculations by iterative methods. Journal de

Physique, 1987, 48 (11), pp.1951-1961. �10.1051/jphys:0198700480110195100�. �jpa-00210638�

Low-frequency

electric microfield calculations

_by

iterative methods

B. Held and P.

_Pignolet

Laboratoire

_{d’Electronique}

des Gaz et des

_{Plasmas, I.U.R.S.,}

Université de

_Pau,

avenue de

_{l’Université,}

64000 Pau, France

(Reçu

le 15 décembre _1986, révisé le 29 mai

_{1987, accept6}

le 1er

_{juillet 1987)}

Résumé. - Dans cet

article,

une méthode itérative utilisant la

_règle

du second moment est

_proposée

_{pour le}

calcul de la composante basse

_fréquence

du

_microchamp

en un

_{point chargé.}

Les résultats, en bon accord avec ceux de travaux récents, sont déduits d’une simulation

_{numérique particulièrement simple.}

Cette méthode

permet

d’entreprendre

des calculs pour des

mélanges

d’ions avec des _temps de calcul non

_prohibitifs.

Abstract. - An iterative method

using

the second moment rule is

_proposed

for the calculation of the

low-frequency

component of the microfield at a

_{charged point.}

The

_results,

in

_good

_agreement with other recent

works,

are deduced from a useful and

_simple

numerical simulation. This method

_permits

extensive calculations for ionic mixtures without

_prohibitive

calculation times.

Classification

Physics

Abstracts 52.70

1. Introduction.

During

the last few years, because of the Stark

broadering diagnostic

of

_stripped

ions immersed in ionic

_perturbers,

_{many papers have} been devoted to

the microfield calculations

_[1]

_(and

references

_given

there), [2-15].

The

_density

and

_temperature

effects are in

_general

introduced into the

_microfield,

_using

the correlation

function

_{9 (y).}

In the

_past,

_{9 (y)}

was deduced from the

_Debye-Huckel

_potential

with or without

lineari-zation

_[1].

Recent studies

_improve

this

_method,

introducing

a more accurate function

_{9 (y) [6]}

or

propose an all-order resummation of the

Baranger-Mozer series

_[9-15].

In a

_previous

_paper

_[16]

hereafter referred to as

_I,

we

_proposed

a

_general

_{semi-empirical}

_expression

for

the correlation function

_{9 (y)}

in the

_{one-component}

and the two-ionic

_component

_plasma

cases.

The _purpose of this _paper is to

_present

an iterative method for the microfield

calculations,

taking

advan-tage

of correlation functions deduced from I.

The calculations are

_performed

at a

charged point

(Za e )

for the low

_frequency

_component

of the microfield

_{H({3), using}

the second moment rule as a

constraint for the resummation of the

Baranger-Mozer series from the second order term to

_infinity.

This numerical simulation

_permits

extensive

calcu-lations for ionic mixtures without

_prohibitive

calcula-tion time.

In section

2,

the formalism is

_presented

for the microfield and the second moment rule in the case of

two-component

plasma.

Section 3 is devoted to

_applications

of the

formal-ism where useful

_expressions

are

_given

for the

parameters,

the correlation

functions,

the low fre-quency

component

of the microfield and the second

moment rule used as the

_convergent

criterion for the

computational

method.

The results are

_presented

in section 4 and

com-pared

with other theories and

_finally,

section 5

_gives

general

conclusions.

2. Formalism.

2.1 GENERAL FORMALISM. - We consider

a

_plasma

with two ionic

_components

_{(a, b).}

If N is the total

number of ions

_(Na,

_{Nb )}

and E the total microfield

at the

_origin

_0,

it is convenient to consider the

Fourier transform of the

_probability

distribution

W(E)

[1] :

Introducing

the

_probability

of occurrence of a

given

configuration

of the N

_particles,

it is

_possible,

by

a cluster

_expansion

method

[17-18],

to _express

F (k)

with a

_development

of

_integral

functions of

correlation functions

_{g (r) [1].}

Finally,

the

_probability

distribution

_W(E)

is ob-tained

_by

_inverting

the Fourier transform:

The main difficulties which _appear with this method are the

_knowledge

of the correlation

func-tion g

(r)

and the resummation over all the orders for

the

_development

of

_{F (k) :}

these

_points

are

particu-larly

important

for

_increasing

values of the

_plasma

parameter,

i.e. for

_increasing

values of electronic

density

or

_{highly stripped}

ions

(fusion applications).

In

_I,

we introduced a

_{general semi-empirical}

expression

for the correlation function. We take

advantage

of these

_results,

_introducing

accurate

g (r)

functions in the

_general

formalism. The first

order terms of

_{F (k )}

are calculated

_classically

[1]

and we introduce a corrective function to serve as a

substitute for the resummation over all the other

orders.

_Assuming

that this correction is

_proportional

to the first order terms, the coefficient of

proportion-ality

is determined

_by

an iterative method

_using

the second moment rule

_{(Sect. 3).}

2.2 SECOND MOMENT RULE. - The second

moment

rule is a useful constraint for the microfield

calcula-tions

_[19-22].

If V is the total

_potential

_energy of the

system

and

_{Za e}

the

_charge

of the central

_ion,

the

second moment can be written in the form

_{[19] :}

where

_Vo

is the

_gradient

with

_respect

to the

_position

of the central ionic

_charge.

Using

the non-linearized Poisson-Boltzmann

equation,

it is

_possible

to introduce the correlation

function in the second moment rule

_{expression :}

where

_{ne, na,}

and

_nb

are the mean values for the

electronic and ionic

_densities,

_{uaa (r)}

and

_{uab (r)}

being

the

_two-body

_potential

functions.

In first

_{approximation,}

we assume that the

two-body

potential

functions

_{uaa (r )}

and

_{uab (r )}

are

_given

by

the

_Debye-Hfckel

screened

_potential,

this

non-linear formulation

_{(4) taking}

care of more

_coupling

effects.

_Finally,

the second moment rule can be

explicitly expressed using

the

_appropriate

correlation

function

_{g (r ).}

3. General results.

3.1 PARAMETERS. -

_Assuming

that the

_plasma

is

composed

of two ionic

_components

_{(a, b)}

and an

electronic one, the

description

of this

_plasma

can be

given

by

four

_parameters

_{[1] :}

- Correlation :

where re

is the mean inter-electronic

_length

and

Aj)

the

_Debye

_length

for the electrons

(Appen-dix A) ;

-

Proportion :

where

_{C a (C b)}

is the concentration of

_{ions ;}

- Number

of charges

on ions a and b :

_Za,

Zb.

Under these

conditions,

the two-ionic

_plasma

parameter

r

_[16]

can be written

_{(Appendix A) :}

where

Z

_{is the average ionic}

_charge,

Z

the root mean

square ionic

charge

(Appendix A)

and

F,

the

elec-tronic

_plasma

_{parameter :}

3.2 CORRELATION FUNCTION. - In

I,

the

correla-tion funccorrela-tion

_{g (r)}

is deduced from a

_{semi-empirical}

method,

using

physical

and numerical constraints.

These

results,

in

_good

_agreement

with Monte-Carlo

simulations,

are introduced in the

_general

_formalism,

the

_general

_expressions

for a two-ionic

_component

plasma

being given

by

[16] :

The

_screening

_{potentials Haa (y )}

and

_{Hab (y )}

are

deduced from the

_{screening potentials}

at the

_origin

[16] :

where y is the reduced

_length

_{(Appendix A)}

and

f(y)

the

_oscillating

function :

with

the

_{screening potentials}

at the

_origin

_Hij(o),

the

plasma

screening

length

d and the

_parameters

A’(T), B’(T), C’(T) being

defined in I.

3.3 MICROFIELD. - General

_expressions

for the

microfield are deduced from the

_general

formalism

introducing :

- the reduced microfield :

with

-

a new variable for the

_expression

of

_{F (k) :}

with

With this

notation,

if

_{H(13 )}

is the

_probability

distribution of the scalar-microfield

_/3,

it is

_possible

to write :

where

_{F (u )}

is

_given

_{by :}

with

1/’ fa

and

_{1/’ fb}

_being

_given

_{by [1] :}

where

_jo

is the

_spherical

Bessel function of order

zero and

x

_being

a reduced

_length

defined in

_Appendix

A.

The

_asymptotic

behaviour of

_{H(Q )}

can be

_given

with an

_{analytical expression [1] :}

Finally,

using

parameters

introduced in section

3.1,

we find

_{(Appendix A) :}

and the

_asymptotic

form

_(for

_high

values of

_{(3) :}

where ya and Yb are

_given

in

Appendix

A as :

3.4 SECOND MOMENT RULE. - The second moment

rule can be

_expressed

_using

the definition of the

reduced microfield and the

_parameters

(Appen-dix

_A).

For a

_{two-component}

_{plasma :}

the

_{one-component}

_plasma

result

_being

obtained in the _{limit p} = 0.

Taking,

for the

_two-body

_potential

function

u(r)

the screened

_{Debye-Huckel potential:}

it _{is easy} to see that :

with

It is of interest to note

_that,

for a

_{one-component}

plasma

without correlation

_{(Ae --+ 0),}

the limit is

given by [19] :

3.5 ITERATIVE METHOD FOR THE MICROFIELD CAL-CULATIONS. - The microfield calculation

is per-formed

_using

the

_general

formalism results for the

first order terms in the

_development

of

_{F (u )}

(Eqs. (33)

and

_(34)).

The

_higher

order terms are

calculated with the second moment rule

_assuming

that

_{(22) :}

where

_k1/l’1

1 is the corrective term.

In

_practice,

the microfield is calculated in four

steps :

- calculation of the correlation function

g (y )

with a

_{semi-empirical}

method

[16] :

the first order

terms

_41,

₁

₍₂₃₎

to

₍₂₅₎

and the second moment rule

«(3 . 2)

r

(38)

are deduced from

g (y)

- for k =

0,

determination of the microfield

distribution

_{H (/3 )}

_{(Appendix A)}

- verification of the conditions of normalization :

- calculation of the second moment :

and

_comparison

with the second moment rule. After an incrementation

on k,

the last three

_steps

are still

_performed

and so on, until the convergence

is reached.

_Finally,

the final

_ko

value is determined

by

a linear

_{interpolation}

_using

the last two values of

4.

_{Applications.}

In tables I and

_II,

the results for the test of

converg-ence are

_presented

for a

_{one-component}

and a

two-component

plasma

for various values of the

_plasma

parameter

r

_{(7) ;}

the second moment

_{rule J3 2)}

r

(38)

is

_compared

with the second

_{moment (/3}

_{2) c}

_{(N )}

calculated from the microfield

_{distribution,}

N

_being

the number of iteration on k.

Table I. -

Comparison

between the second moment

_rule

_/3

_{2) r}

and the second

_{moment (}

_{f3 2) c}

_{(N )}

calculated

for

several values

_of

_{k(N) (one}

_component

_plasma).

Table II. -

Comparison

between the second moment

_{rule f3}

_{2) r and}

the second moment

f3 2) c

_{(N )}

calculated

_for

several values

_of

_{k (N ) (two-component plasma).}

Fig.

1. - Microfield distributions of

H(¡3)

for a

one-component

plasma (test

of convergence for various values of

_k)

with v = 0.173, _p = 1,

Za

=

_Zb

= 1 and r = 0.01.

Fig.

2. - Same as

_figure

1 with v =

0.2, p

= 1,

Za

=

Fig.

3. - Same as

_figure

1 with v =

0.4, p

= 1,

Za

=

Zb

= 35 and r = 20.03.

Fig.

4. -

Microfield distributions of

_{H(13 )}

for a

two-component

plasma

(test

of convergence for various values of

_k)

with v =

0.2,

_p =

0.5, Za

=

9,

Zb

= 1 and r = 0.32.

The

_{corresponding}

microfield distributions

_{H(13 )}

are

_plotted

in

_figures

1 to 3 for a

_{one-component}

plasma

and in

_figures

4 to 6 for a

_{two-component}

plasma.

For

_increasing

values

_{of k,}

the microfield

_{H(13 )}

is

shifted towards smaller values of the reduced mic-rofield

₁₃

with a

_pinch

effect on the distribution and

on enhancement of the maximum.

Comparing

_{(13 2) rand (13 2) c (N)}

(Tab.

I and

_II),

it is easy to determine the

_convergent

_ko

values

Fig.

5. -

Same as

_figure

4 with v =

0.8, p

=

0.5, Za

= 9,

Zb = 1 and r = 5.13.

Fig.

6. - Same

as

_figure

4 with v =

0.6, p

=

0.5, Za

= 17,

Zb

= 1 and r = 8.39.

(Appendix A)

and to deduce the final microfields distribution

_{H (f3).}

These results

_(Fig.

7 to

₁₂₎

are

compared

with the more recent results

_given

_by

other theories

_using

the

_Debye-Huckel

screened

potential approximation

[2-8] :

next-nearest-neigh-bour

_(NNN)

_{approximation, Hooper}

_(H)

or

Tighe-Hooper

(TH)

calculations,

adjustable

parameter

exponential (APEX) approximation,

Monte Carlo

(MC)

simulations.

Fig.

7. -

Definitive microfield distributions of

_H(J3)

for

a _{one-component}

_plasma

_compared

with the results of

Iglesias

et al.

_[2-8]

with

_{ko =}

0.138,

v =

0.173,

p = 1 9

Za = Zb = 1

and r = 0.01.

Fig.

8. - Same as

_{figure 7}

with

_{ko = 0.415,}

v = 0.2,

p = 1, Za=Zb=9

and F=0.52.

with H and APEX. In the other cases, the

discrepan-cy is in

general

small,

our results

_being

inside the

dispersion given

by

NNN, TH,

APEX or MC.

5. Conclusion.

In _{this paper,} a

_general

iterative method for the

microfield

calculations has been

_presented

in view of

Fig.

9. - Same

as

_figure

7 with

_ko

=

0.456,

v =

0.4,

P = 1, Za=Zb=35

and F = 20.03.

Fig.

10. -

Definitive microfield distributions of

_{H(,B )}

for

a _{two-component}

_plasma

_compared

with the results of

Iglesias

et al.

_[2-8]

with

_{ko =}

0.304, v =

0.2,

p =

0.5,

Za = 9, Zb =1

and r=0.32.

applications

to the Stark

_diagnostic

of

_{highly stripped}

ions.

The results are in

_good

_agreement

with APEX and the calculation times are _{very short}

_compared

with

Monte-Carlo simulations or more usual theories

_[1].

With this iterative

_method,

it is

_possible

to

con-sider extensive

_{applications,}

_taking

care of the

Fig.

11. - Same as

_figure

10 with

_ko

=

0.171,

v =

0.8,

p

= 0.5, Za = 9, Zb = 1

and r = 5.13.

Fig.

12. - Same as

_figure

10 with

_ko

=

0.423,

v =

0.6,

p=0.5, Za=17, Zb=l

and T=8.39.

Zb =1,

i.e.

_{v =1.732 )}

and the Holtsmark limit

(v = 0).

Finally,

it is of interest to note _{that the accuracy of}

these numerical results could

_certainly

be

_improved

by

introducing

the electronic

_degeneracy

in the electronic

_screening

function

_[23-26]

or

_using

effec-tive

_{two-body potential}

function solution of the

non-linearized Poisson-Boltzmann

_equation.

Other cases were studied for different values of the

_parameters

v, p,

Za, Zb

and these results are

available on

_request.

Appendix

A. General results.

1. PARAMETERS. - We consider a two-ionic

com-ponent

plasma

with a

neutralizing

background.

The electronic

_plasma

_parameter

_Te

is defined

by :

where re

is the inter-electronic

_{length :}

Introducing

the average ionic

_charge

Z

_{[16] :}

the root mean _square ionic

_charge

Z :

and the electronic

_{Debye length :}

the two-ionic

_plasma

_parameter

r can be written

[16] :

In order to _{express the} microfield and the second

moment rule

_explicitly,

it is

_interesting

to introduce

the relation for the concentrations and the conditions

of

_{neutrality :}

In these

conditions,

we obtain the rates

_{[1] :}

Finally,

for the correlation function

_{applications,}

it is useful to define two reduced

_lengths

x and y :

where r is the

_position

of the

_{charge particle,}

and

with

the relation between x and y

being given by :

2. MICROFIELD. - The

asymptotic

behaviour of the microfield at a

_charged

_point

(ion a)

is

_{given by}

[1] :

1

with

and the notation

For the ionic

_part

of the

_microfield,

_p and

_/3

are

connected

_{by equations (31)}

and

_(32).

Using

the definitions for x _{and y, it is easy} to show that

For

_high

values of

_f3,

the

_{exponential integral}

term

vanishes and the

_{asymptotic expression}

becomes

fairly

simple :

where

with k = a or b.

3. SECOND MOMENT RULE. - It is

possible

to

_give

explicit expressions

for the second moment

rule,

using

the reduced microfield and the

_parameters.

For a

_{one-component}

_{plasma :}

it is easy to show that :

and we find :

Noticing

that :

with

the second moment rule

_becomes,

_using

the reduced

length x :

For a

_{two-component}

_{plasma :}

4. CONVERGENT METHOD. - For the microfield

calculations,

we consider three

regions.

For small

values of the reduced microfield

_f3,

_H(13)

is

com-puted starting

from the

_general

formalism

_{(21), (24),}

(25), (45)

and

_using

the

_{Krylov-Skoblya}

approxima-tions for the Fourier transform

_[27].

These

calcula-tions are

_performed

until numerical noise appears,

criterium on the

_slope

stops

the numerical pro-cedure. For

_large

values of

_f3,

the microfield is

_easily

deduced from the

_{asymptotic development (28)}

to

(30)

and

_(35).

In the intermediate

_region,

we use an

appropriate

function for the

_{interpolation}

of

_{H(f3 ).}

This function is introduced from the

_general

asymp-totic

_{development, assuming}

that :

The four

_parameters

are deduced from constraints on the value of

_{H(i3 )}

and the first derivative of this

term for the last

_{computed point (Hl}

=

H({3I),

and the first

_{asymptotic point}

The numerical values of

_H({3)

can be

_adjusted

after verification of the conditions of normalization :

Noticing

that

_{(45) :}

the microfield distribution calculation is

_performed

for every iterative value of k.

N

_being

the iteration

number,

_{(f3 2) r the}

second

moment rule

_(38),

_{(f3 2) c (N)}

the calculation of the

second moment

₍₄₇₎

from the N-th

_{H (/3 )}

microfield

distribution,

the

_discrepancy

_{A (N )}

can be written :

Assuming

that:

the definitive

_ko

value is calculated when the

con-vergence is reached.

k(N ) being

the N-th value of

k,

ko

is

_given

_{by :}

where

_{(N - 1 )}

and N are the last two iterations

before and after the convergence.

References

[1]

HELD, B., J.

_Physique

45

₍₁₉₈₄₎

1731.

[2]

IGLESIAS, C. A., HOOPER C. F. Jr., DE WITT, H. _E.,

Phys.

Rev. A 28

₍₁₉₈₃₎

361.

[3]

IGLESIAS,

C. A.,

LEBOWITZ, J. L.,

Phys.

Rev. A 30

(1984)

2001.

[4]

ALASTUEY, A., IGLESIAS,

C. A.,

LEBOWITZ, J. L., LEVESQUE, D.,

Phys.

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GRECO ILM

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[6]

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PERROT,

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₍₁₉₈₅₎

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