RADIAT DÍSTRIBUTION TUNCTIONS trOR ÄN

IIYDROGENOUS PLAS¡\'{A IN EQII IIIBRIU}/I

by

A.A. BARIGR. M.Sc. (laer. )

A thesis subnitted in aceord.ance with the requirements

of the Degree of Doctor of Philosophy.

l,tatherna tic al Physic s Departnent

University of .fidel-aide.

South A.r-rstralía.

Subnitted. September 1968.

t,

i

I

CONTT]I\TS

iIlTR0DU0lïON

1 .1 The Rad-ial Disi;ributi on tr\-lnc tl on

1.2 Eanly work in liquid.s

1.3 Extension to Plasmas

THE MOrfTE C,IRLO T.IETHOD

2.1 An Outline of the method-

2.2 Results

2.3 Discussi on ancl Concl-usi ons

THE ITfIEGR/ú EQU.¡ITTON I"{ETHOD

3.1 Introd-uction - The Percus-Yevick Equation

3.2 An /rsymptotic form of the Percus Yevickequa tion

3.3 Deficiencies in the Approach

A SUA}üTUII }{ECI]JI]']TCII,L CAICUL/iTION OF THE T!1iO ?ART]CI,E

DIS TRIBLTTION trUNCT]ON

4.1 Introd.uctl on

\.2 Formulation of the Eguation a¡d its evaluation

4.3 Results

4.4 Discussi on

SOLTJIION OF A T/IODIFIED PERCUS-YEVICI( ESUATION

5.1 A form sultable for solution on a computer

5.2 OutlÌne of the numerlcal pnoced-ure

5.3 Results and. discr-rssion

IT

TÏT

IV

v

CO]fIENIS

VI CONCT,USIO}i

6.1 Comlnrison of tqe two methods

6.2 Conclusion

APPENDIXA-''MonteCar]oc¿llculationsoftheradiald.istribution functions f or a proton-

el-ectron pltrsmarrr A.ii, Barker, Aust' J'

Phys. 13, 119. (1965) -

ttOnthePercus-Yevickequationrt¡'Ë\'À'Barker

Phys. Fr-uidse Pr 15go (1966)'

r';! o;uantum meci'ranica]- ca]-culatlon of the

rad.ial clistribution function for a plasmar'e

-A..rr!. Barl<ere -Ar-rst. J. Phys ' 27, 121 (tg68)'

rrtr[onte Carlo stud-y of a hyÔrogenous plasma

near the ionization tenperatirrett A'/tt' Barker

Phys. Rev. ß, 186 (tl68)'

APPENDIXB-Fortranprogramtoevaluatethequantummechanical d-istrlbution functions'

Fortran progran to sol-ve a mod'ified Percus-

Yevi ck eq:a'bi on.

SUT.Í}fiiRY

Radial d.istribution functl ons S"O (r) for a

d.ense hycli'ogenous plasma in equilibrium near the

icnization tempe rature are obtained. by two method.s.

The first is the lfonte Carlo (UC) method- origÍnal1y

applied. to fluid-s by läetropolis et a.L, and. recently

extended to plasmas by Brush, Sahlin ard- Tellerr and-

Barker. Although the technique seems reac11ly applica1cle

to high and- low temperatures, the MC results near the

ionization temperature shoir¡ that in thls regÍon the

Sro(n) obtained- are unusually sensitive to tv'ro parameters.

The firsi is the cut off imposed- at small. rad.ii on the

Coulomb potential between unlike particlese and- it is

found- that it 1s necessary to consider quantr:m effects

at these rad.ii. The second. 1s the maximum step length

A through v¡hich the particl-es are all-ovred. to move in

the IfC procedure. Near the ionization temperature the

plasrna behaves as a variable mixture of two phases, one

ionizeC, the other unionized.e and. the rnagnitud.e chosen

for A influences which phase d-ominatesr in the

relatively sma1l sample of configurations selected-

by the l,{onte Carlo procedure.

The second- technigue applÍed. is the solution of

integral equationsr and in particular the solution of a

mod.ified- Percus-Yevj.ck (Uftf ) equaticn. Eanly

investigatj-on of the Percus-Yevick (pV) equation

shov'¡ed. that j-n an asynptotic form for large rad-ii

it lvas inconsistent for systems interacting vr¡1th

attractive f orces, arrl to overcome this difficulty

terms suggested by Green Tvere includ-ed- to obtain

thre IÍIY equation"

The numerical solution of the tr,Ilf equation

immed-iately shov.¡ed. the inportance of ihe quantum

mechanical effects at small- rad-ii, and- 'uhat it would-

be necessary to calculate these accurately" An

ercpression is obtained. for the quantum mechanical

d-istribution f\rnction in a d-i1ute plasma, and from

this an effective quantum mechanical- potential is

d.efined., which merges lnto the Coulomb potential at

large rad-ii. Results are given for the temperature

range lxlOsoK - BxlO4oK for a neutral plasÍtâ.

Usi-rrg shield.ed. quantum meclranical Sun(") u"

input to the llf eguationrsolutlons are obtained- f or

tenperatures of \x'l04oK and JxlOaoK ¡rvith an el-ectron

b.ensity of 1çt u/ g". For temperaturæbel-ow thi s the

second-ord.er inconsistency mentioned. above causes

d-ivergence. Sol-utions of the MIlf eguation are also

presented.. These are foulnd- to improve on the Pi

results and- to also increase slightly the temperature

range over which the eguation can be applied-" The

percentage

encountered.

are d-iscus sed

ionization present,

the plasma tend.s

in d.etai1.

and. the d.i-fficulties

towards a neutral gasAS

@Iherebydeclarethatthisthesiscontains

no naterial which has been accepted. for the award

of any other degree or diploma in any universityt

and_ that to the best of my knovrledge ard- belief, the

thesis contains no rnaterial previously published- or

wrltten by any other persone except r¡¡here due referencc

is made in the text.

A" ir. Barker

ACKNOI¡ü-I,E DGEI,{ E T{i S

Throughout thj-s work the author has been greaily

stj-mulateC and. encouraged by Professcr I{.S" Greene v¡ith

r.,¡hom many valuabl-e discussions have been he1d. The autncr

is al-so particularly grateful to Dr. P"fü" Se¡rmour for

numerous hetpful- suggestlons, and- ind.eed- wishes to

acknowled-ge useful discussion rvith many visitors and meni¡ei"'

of the /id.elaid.e Universj-ty l'!athematical Physics Departlnen'c,

especially Dr. R, Storer and. l'fiss F' Tong.

F\rrther this v'¡ork couId. not have been completed-

without the generous grant from the /rustralian Instiiutl

of Nucl_ear science and. Ergineering for computing. Tht

computation was performed. at the Ad.elaide university

computing centre and. the cooperation of its staff l,"¡as

invaluable,

The author lvo¿ld also like to acknorivled.ge the scppor+u

of the Commonweal-th Governrnent in providing a Commonv¡eaiili

Post Grad.r.rate -frward. from 1965 to 1968.

Thanks are also given to }/llss M. Dodgson for her

cl-ose cooperation in the production of this thesis'

1.1

T INîRODUCTION

the ain of thls thesls is to d-etermine accurate

rad.ial d.istributlon functions for a neutral hydrogenous

plasma of electron number d.ensity 10t"e/ce in the ternpen-

ature region near ionization (i.u. 1o4"K to JxloaoK)'

So fan there has been relatively littl-e work on d'etermin-

ation of d-istribution functions for plasmas, especially

fon the density-ternperature region to be consid-ered'' ''üe

apply the tr[onte carlo and the lntegral eqi]ation techniquest

whieh have previously proved suecessful for non-ionized

gases, to obtain d-istribution functions 1n pIa$râs¡

1.1 The radial d.1s t ut

Before 19OO the theoretlcal work on flulds was mainly

based on the perfee-t gas 1aw of Boyle and- charles, and- 1ts

extension by clauslus and van d,er !Taa1s. In the early

19OOrs statistical mechanics was put on a much firner basis

bythesystematicapproachofGibbs,whosetheoryof

ensembles forms the basis of certain porrerful techniques

inusetod.ay.Then,iIrlg2ogtheapplicationofX-rayd.iffraction technigues to flIuid.s gave rlse to the eoncept

of the radial distribution function. The radial

distribution f\rnction S"O(r) between two prtiel-es of

types a and- b is defined as

\r.1(r)su¡ (r) (r.t )

%

1.2

where nb is the average nu¡nber density of partlcles of

type b in the fIuid. (* tacroscopic quantity)r "td %(")is thre mean nu¡nber d-ensity of particles of type b at a

d-istance r from the ath ¡article (a microscopic quantity).

the distance r is usual1y of micnoscopic orden, a¡d- 1n

thie work wilJ- be e4pressed. in units of Bohr rad-ii.

Thr¡.s Bu¡(t) i" a measure of the correlation in the positions

of partieles of type a and. br ârld- though not a stnict

probability, it is proportional to the probability of

find.ing a partlcle of type b at a d-istance r from a

particle of type âr About this time OrnsteÍn a¡rd-

Zernike [t ] also introd.uced. their concept of the llndirect

correlati on f\rnctiont nr.(r) (equal ¡e 9¿b(")-1 ) r which

is eomposed of a rd.irect correlation functiont cu.(r)

betlveen the two ¡nrticles a and b, plus a contribution

fron interactions with other prtlcles. The importance

of the concept of the raÔia] distribution function (and'

hence the correlation functions) was realized- r¿vhen 1t

ï/as gþewn that most thermodynamic variables can be

expressed 1n terms of g"O(r). Comprehensive treatments

of the properties of S.¡(r) and its relation to

thermod-ynamic properties are gilren by Green lzit H111 lll,ana Fisher [4].

1.3

1 .2 Early work on l,iquids

Several theories arose to predict g(r) for fluids'

Initially these ïuere associated partlcularly wlth crystal

lattlces ard. were knorirrr aS tcelI or latticer theories'

These have becorne guite refined. and lrave been extend'ed- to

tholet theoriese where the liguid. 1s imagined. to resemble

a crystal lattice from which some of the particles are

missing. such significant structure theories have been

particularly successful in pred.ictlng the properties of

d.ense f]-uids, and are d.iscussed- f\r]-Iy by Guggerùein [f ]'

prigogine t6] and Barker lll and. references therein.

In the 19,1+Ors interest revived in fluid theory when

I4ayer and l\{ayer tB] proposed a teluster modelr to

calcul-ate the virial coefflcients accurately. The radial

distribution funetion can be also elcpressed- as a power

series in the densltye and- for developments of this

approach see references [9] ana [to ].

In the late 194Orsr an integral equation for e(r)

was proposed by Born and" Green [tt], who closed. the sets

of equations obtained- previously by vvon [tl+] tv uslng

the superposition approximatlon of Klrkwood. llZl. Yvonrs

equations result from more general flynamical eguationse

by making substitutions appropriate to equllibriu¡n, and'

1.4

very similar dynarnical equations were al-so proposed by

Kirinvood lthr] an¿ Bogoliubov l13l abou-t thls tlme. The

lntegral eguation for eguilibrium 1s usr:a}ly neferred. to

as the BBGIC( on BGY equation, It was solvefl numerieal-]y

by a number of authors [t¿+¡] who obtained good agreement

with experiment for tenuous fIuid.s. An excelJ-ent review

of this field is given by Green [t4c].In the 195Or s the theorist received a setback when

the results of reliable machine ca]culations became

available for dense fl-uid.sr âs the results d-isagreed- with

those of the cell- theories, the BGY equation, and- the

virial expansione which d-oes not eor¡/erge at liguid- densit-

ies. The eomputing technigü€r developed- by I'ietropolis

et al. [t¡], is ca11ed. the l[onte Carlo method.. The approach

involves very few assumptions ard applies for a wid.e

tenrp,erature d.enslty rarrle¡ ând hence can be used to compare

other theories. It has been applied to hard- sphere

fluids by Rosenbluth and. Rosenbluth [16] and A]dere

Franlcel and- leu¡inson l17le and extended to particles

interaeting viith a Lennard.-Jones potential by Woocl and-

Parker [tA]. A simllar approach calIed. rmolecular

d.ynamlcst f¡ias been d.eveloped. by Ald.er and. ii'/ainwright Itg].

Reeent papers by Hoover and. Alder [zo], Verlet lzll and-

r¡Too0 lzZl give resul-ts which are in excellent agreement

1,5

with the ex¡rerimental d.ata available' the main dis-

ad-vantage of the method. is that it takes exeessive

eomputing time to obtain accurate rad.ial distributionfunctions for a given temperature and. d.ensity.

To improve upon this situation, in the last d-ecaôe

attentlon has reverted. to the integral equation approaeh.

In 1958 Percus and Yeviek l%) proposed- a new integral

eguation (nf ) based on a colleetive coord.inate proced.ure,

which has since been elegantly d.enived by Percus lZl+J

using a functional d.enivative technigue. The IY equation

rüas applied. to hard- spheres by Thlele¡ Ïlelfand.r Reisst

Fnisch and I,ebovtitz lZ¡l and- an exact solution fou¡d-

for hard. spheres by i¡rlerthein lZel and T,ebowitz lZ7l.ïVerthelm [Zg] has also obtained- an analytie solutlon for

a pair potential eonslsting of a hard core plus a short-

range taiI. A number of authors (see lzg) to llll) rrave

extended. the application of the PY eqi:ation to fluid.s

interacting wlth the Lennard.-Jones potentiaÌ, several using

the numerical solution procedure suggested by Broyles [¡h].Comprehensive calculations Trave been completed recently for

binary mixtures by Throop and. Bearman lSSl and Ashcroft

and. Langreth 116l.

About the same tine as the Pf eguation was proposed¡

another integral eguation ca11ed the tconvoluted- hyper-

1.6

netted- chaint (CHNC) was introd-uced almost simultaneously

by several- authors, see llll to [ltp ]" This equation

attempts to avoid- the c onvergence difficulties arising

from the series expansions in powers of denslty at high

d.enslties (see references [irt 1 to thJ]). It has been

applled_ to Lennard.-Jones fluid-s by verl-et and Ï,evesque

il+4]; and Klein and- Green [45] have also presented

extensive results for this case. There have been recent

papers by Helfand and- Kornegay it+01 and Hr:rst IUZ]

extendÍng the equatlon to take into account higher-ord-er

effeets. Baxter [l-¡B] nas numerically solved- the CHNC

equation involving thr: three ¡nrticle terrnr and. Khan

[l+g] gives extensive nesults.for 11quld Ar, Kr, Ne and Xe.

several approximate and- perturbatlon theorles have

been suggested-, most of which treat a region of the

interaction by one of the eguatlons mentioned abovet see

ref erenc es [ ¡o ] to l1ll, l,,{odern computer technlgues

have also enabled the BGY equation to be solved. more

accurately ([¡l+] and [¡¡]).

Reeent experimental d-ata has been published' by

ltichelsr et a1 lSel and. Mikola j and- Pings lsll' these

results being pnincipally fon Argon and Neon, though Khan

and. Broyles [58] have cons j-dererl 1igu1d Xenon.

Even though several of the theories for flui-ds

outlined. above are guite comprehensive, there 1s stil1 some

1.7

d.isagreement lvith e)q)eriment. This has leil to lapers

on the relationship betlveen pair potential-s ard

d.istribution functions (u.g. Strong and Kaplow lsgl) and

also to some work on j-nequalities that e(r) mr¡^st satisfy.

(see [60] to l6ZD. Diseussions of the d.eterminatlon

of intermolecular forces from macroscopic propenties are

glven by Row]-inson lØl and Hanley and' K1ein [64]'

f|henainhopefonfurtherimprovementinliquidtheory Seems to lie in the inclusion of three-bod'y forces'

There is consid.erabl-e work being d.one in thls fleld at

presente ând. recent papers by Rushbrooke and silbert

l6Sle Rorrvlinson 1661, Hend-erson 167l, Lee, Jackson and

Feenberg [68], Sinanog]-u 169)¡ ârrd Graben [Zo] are of

interest.

1.3 Ext ion to S

Prior to 1950, the work on electrolytes and

plasnas in equilibrium lvas dominated by the famous work

of Debye and ïfiÍcke1 in 1923 llll. In 1g5O Mayer llzl

showed that the divergence d.ue to the Coulomb lnteractlon

eould. be eliminated. from the eluster exlnnsion for the

equatlon of stater âfid shontly after thls several authors

d.eveloped.thlsapproachtohlgherordersofaccuracy(see l73l to lllD. There \¡as also at this tlme

consid.erable researeh, principally in Russia, directecl at

1.8

extension of the BGY equatlon to plasmasr and. this ls

d.iscussecl in d.etail in an excellent nevj-erv article by

Brush, De T4/itt and. Trulio [Zg]¡ which includ.es an

extens ive b1bli ography.

The diffleulty in extend.lng f1u1d. theories to

plasmas lies in the nature of the Coulomb f cr:eer beeause

firstly¡ it contains a singularity at the originr and'

second-Ìy it has long-range effects. The staìri11ty of

a system of particles interacting with such forccs has

been the sric ject of recent reviews and papers by Yang

llgl, Ter Haan [80], lncflieeney [81 ], Fisher and Ruclle fe?1,

and- Dyson and Lenard. [B¡ ]. The latter authors have Sl:'ot¡';n

that a necessary cond-ition for stabllity of the system

1s the inclusion of guantlrm statistics. It is also well

knolr¡n that the more obvious d-1fflcu1tie s associated with

the shont range of the Coulomb potentlal can be removed- by

taking into account guantum effects" The author at

first atternpted. to treat these very approximately in

extendinf fluld. theories to plasmas. However¡ they

proved to be so important that it became necessary to

make more exact ealculations.

ctrapter II presents the results of extend.ing the

Monte Carlo approach to plasmas. The theory of extending

the Monte Carlo approach to lorrg-range f cnces has been

d.eveloped ind.epend.ently by Brush, Sahlin and. Teller IAl]

1.9

for a one component plasnar âÐ.d- by Barker [S¡] for a tv¡o-

eomponent plasma, so only a brief descriptlon of the

rnethod. 1s given. The results are presented- in a serj.es

of tables and graphse and show that for temperatures

near icnizatlon 1t is very d.ifficult ard- expenslve to

obtain accu.rate di stribution flrncti ons.

CLrapter III deseribes the extension of the Hf

eguation to a hyd-rogenous system. It 1s shown tlat the

Hf equation is in fact inconsistent for such a systeme

and. to ensure consistency, higher-ord.er terms such as

those suggested by Green must be included.. This equation

1s referred. to as a nod.ifled. Percus-Yevie]6 equation(tUpy),

and. is ex¡pressed in a form sultable for solution on a

computer. An initial atternpt to solve the MPY equatlon

showed- the importance of quantal effectse a feature which

had alread.y been indieated. by the I'4C results.

To take aceount of the guantal effectsr â[ erc¡lression

for the two-lrartlcle d.istribution function is presented. in

Chapter IV. This erçression is then evaluated- numericallye

results are presented, and then d.iscussed in detail.

Refer,ences to research on the inclusion of qr:antal effects

1n fluids ard plasmas are given in the introductory

section 4.1 ,

In Ohapter V¡ using the results of Chapter IV,

1 .'lo

solutiorrs of the llf and- l,ilPY equations are obtaine d-.

Some emphasis is placeÔ on the numerical technigues used-,

for if a straightforward- iteration proced.ure is ad-opted-t

the method- d-iverges. Accurate d-istribution flrnctions for

l-¡xlOaoK are obtained-, a¡.d- somev¿hat less accurate results

for 3x1o4oK. At 2x1O4nK it is found- that, even v¡hen

appropriate stabilizing techniques are empì-oyed in the

computer progran, both the PY and MtrY equations d.iverge.

Before proceeding to the application of the MC

method and l4PY equation to plasmas, the author should-

mention in particular two other current lines of researeh

in this field. The first is the systematic development

of the diagram,'rnatic method.* to includ-e quantal effects'

Recent papers on this tec|.nlgue have been pu)rlished by

Bou¡ers and sal-peter tB6], De witt lell' Ifirt [Ae]'

Gaskell [89] and Diesend-orf and- Ninham [9O]. The second.

line of approach extend-s the tþeory of integral equations

to take into account long-range forces. The short-range

d-ivergenee f cn the BGY equation uras treated some tine

ago by Glauberman and. Yukhnovski [gt ]. The higher'ord.er

terms, horvever (".g. Green leZl¡, prove gulte important

as ind.icated. by o'NeiI and- Rostoker lgTl. Hirt [gt+]

an¿ Guernsey lg¡l have recently applied- the BGY eguation

to plasmas. the guantal effecis are inco orated-

* L{entioned. previously in a fl-uid context, this cl-usterexpansion ãpproach yields the DH result as a firstapproximati on.

1.11

into the BGY eqi-ratio:: 1n Bapers b¡r Mateudaira [9¡]. an¿

in an elegant paper by Matsud.a [Se 1. Extensions of the

Ilt and, CIINC method-s to a one eomponent plasma have been

made by Broylese Sahlin and. Carley lgll, ard- Carley [gA].

At the present tlme there are few plasma experimental

values for g(r), and even the thermodynamic fr-mctions

prove very d.lfficult to obtain. Most of the experimental

wonk completed so far has given e(r) for liquid metals

(see Johnson and. l,{areh [gg]: here theoretical results

are glven by Villars [tOO])' The experimental thermod'Fam-

1c properties of Hydrogen are discussed by Oppenheim and'

Hafemann [fof ] and. Theimer and Kepple [toZl' Numerlcal

nesults are also gi-ven by Rasaiah and. Friedman [to¡]

for the application of integral equations to ionic

solutiorls r

I .12

Refer ences to Chaoter I

tf ] ornstein, L.S and Zernike¡ F. (191+l+) Proc' Acad'

sci. (¡.rnst) !, 793.

lzl Green, H.s. (1952) tMolecular theory of Fluidst

(North-Holland Pub. Co. Amsterdam)'

13) HiIl¡ T.T,. (lgS6) Statistical llechanics (MeGraw-Hi1l'Lond.on)

(rg6o) statistlcal Thermod.ynamics (lad'ison-

\[/es1ey, London).

t4] Físher ' f .2. (1963) statistical Theory of lJiquids

(crricago Press¡ Lond-on) .

t¡] Guggenhelme E.A. (1952) Mixtures (Oxfora Uni. Press¡

Lond.on).

t6] prigoginee M. (lgSl) Mo1ecular theory of sol-utlons

(North-Ho11and- Ptlb. Co. e Amsterdam) 'l7l Barker¡ J.A. (196s) Lattice Theories of the Ligrrid

State (Pennagon Press Ltd'. e I,ondon) 'Ig ] Mayerr J.E. âfrd trilayere M.G, (tg¿+O) Statistical

Ivlechanies (,1. TViley and Sons, New York)'

tg] Ir/leixnere Jg editor (1961+) Proc. of fnt. Symp¡ orr

stat. l[echs. Germar¡y 1961+. (North-Ho1]-and Pub.

Co. Amsterd-am),

[f o ] De Boer, J. and Uhlenbeck, G.E. (t964) Studies in

St. 1{echs. Vol, I and II (North Holland Pub. Co"

Amsterd'am).

1 "13

[f f J Born, Mo and. Greene H.s. (19¿*6) Proc' Roy' Soe'

A1 BB, 1O.

[tz] rirkwood-e J.G. (lglS) Jo Chem. Phss' Jr 3oo

(1%9) J. chem. Phys. f r 919 . :

l13l Bcgoliubov, N,IT. rProblems of Dynamical Theory of

statistical Fhyslcs t (Otglish translation by

E.K.CoreAirForceCambridgeResearchCentre,

Publ ication TR-59 -23) .

i14] Yvon, J. (1937) ntuctuatlons en Densité; La

Propogationet]-aDiffusiond.e]-aLuminiére(Herrnan and Cie).

[tba] Klrkrvood., J.G. (tg46) ,r. chem. Phys' ü, 1Bo'

Klrkwood¡ J.G. ârrd Boggs, I{. (t94t )' J' Chem' Phys'

2, 511+'

s,, 394'

[14b ] noari g\rezr A.E. (tgUA) Proc. Roy. Soc ' T-,ondon'

Ser. A. EÉ; 73,

McLel-Iane A.G. (lg5Z) Proe. Roy. Soc. Lond.on 21pt 5O9

Cheng, K.C. (t950) Proc. Roy. Phys. Soc' Ser.Al 63t

428.

Kirklvoode J.G-e l\4aune E.K. and' A1der, B'J' (tg¡O)

J. Chem. PhYs. -1!, 1040.

Kirkr¡¡ood.e J.G. Lewinsone V.A. and Ald.er, B.J. USSZ)

J. Chem. PhYs " ?Q, 9L+9'

Itt+c]

Ir¡]

[16]

¡ttJ

[re ]

Irg]

Izo ]

121 l

lzzllztl

I zu]

lzs)

1.14

Green, H.S. (t960) rEncylopedia of Physiese

Vol.X. p.1l+t Ed. bY S. Flügge.

uletropolis, N.M. e Rosenbluth, A.Vli., Rosenbluth¡

M.N., Teller¡ .À.H. and- Tel-l-err E. (1953) Jour;

Chem. PhYs. 4, 1087.

Rosenbluth, I'Í.N. and- Rosenbluthe A.iI/. (f 954)

J. Chem. PhYs. 22t 881.

Alder, B.J', FranJrel, S.P. and' Lewinson, V'A'

(1955) J. Chem. PhYs - 23,, l+17.

Wood-e TV.,jT. and Parker, F.R. (1957) J, Chem' Phys'

4, 72o'

Ald.ere B.J, and. V/ainwrightr A.\¡'I. (1959) J' Chem'

Phys . il-, 459.

Hooverr V¡.G. and- Alder, B.J. (1967 ) 'l' Chem' Phys'

75J, 25o'

Venlet, L. (1967) pfrys. Rev. 159., 98

(rgea ) Jø, 2o1 .

Wood., 1,ü.il[. (1968) J. Chem. Phys. .[9,r 415.

Pereusr J.K. ârìd. Yevick¡ G.J. (lgSA) pfrys. Rev' ll!,t

Percuse J.K. (gAz) prrys. Rev. Letters Qe 462,

(1963) .r. Math- Phys. [r 116. see also

Lebowitz¡ J.T,, (1963) .1. Math. Phys. 4r 2l+8' :

Thiele, E. (1963) ,1. Chem. Phys . &., 1959 and- lPt

471+.

1

1 .15

lz6l

Reiss , H. g Frisch¡ H.T,. and Lebowitz e J'L' (1959)

J. Chem. Phys. 31-, 369.

Helfand., 8., tr'risch¡ I{.L. and' Lebo'ttitz¡ J'T" (t96t )

J. Chem' PhYs . 3L., 1037'

xrertheime M.s. (1963) pt'y"' Rev' Letters lQr

321, and E!o1.

Lebowitz, J.L. (196\) prrys. Rev' E, AB95'

\flertheime M.s. (196+) J. ulath. Phys' Jr 643'

Broyles, A.A. UgA ) ,1. Chem' Phys' fu., 1068 and'

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verlet, T,. (1961+) PhYsiea flt 95'

Broylese A.A. Chung, S.U' and Sah11n' H'L' (lgGZ)

J. Chem. PhYs. il-, 2462.

Verlet, L. and Levesquee D' (lgøZ) pfrysica 29' 1121+'

Rowlinson, J.s . (1963) lrlole c. Phys ' lr 349 'Broyles¡ A./r. (t960) J. Chem. Phys' 33,, l+56'

fu' 359'

Throope G.J. and- Bearilâh¡ R.J. (tgge) J' Chem' Phys'

44' 1l+23 '(1967) J. Chem. Phys.

(l+Z), 3036.

.Asbcrofte N.1,T. and Langreth, D.C. (1967) fnys. Rev.

156c 685.

i\leeron, E. (lgSg) prrys. Fluids lr 246'

127 l

Iza ]

lzel

[¡o ]

ltt l

ltzll33l

t3tÈ l

lssl

t]e1

1 .16

lttl

[¡e ]

ltg l

Meeron, E. (t ge o) J. l{ath. Fhys . !r 192.

Iviorita, T. (lgSA) erog. Theor. Phys. Kyoto. 4c-a2O.

I{oritae 1. and. Hiroike, K. (tgeO) Prog. theor.

Phys. Kyoto þt 385' 829 and 1oo3i 4¡ 317;

Æp 537.

Van Leeuwen, J.If .J.¡ Groenrrueld-, Jo and- De Boerr J'

(tgSg) Physica !þt 792.

Green, M.So (tgeo) J. Chem. Plqrs. 51., 1403,

Rushbrooke, G.S. (lg6o) physic^ Æt 259.

Verlet, L. (gAo) lduovo Cimento lpr 77,

Salpetere EnE. (lgf1) A.*. FWs' 5¡ 183.

Verlet, T,. and. Levesquee D. (lg6z) prlysiea ZBc

1124.

Klein, M. and. Green, };1.S. (19Ø) .1. Chem' Phys.

22, 493 , 1367 t 1388 .

Helfand., E and. Kornegaye R.L. (196+) physlea 30,

1481 ,

Hr:rst, C. (lgøZ) eroc. Phys, Soc. $r 193,

Baxter¡ R.J. (lg6l) Preprint.

Khan, A.A. Ug6+) errys . Rev. 1A, A367 i 1J6-, A1260

(1966) lreprintsCarley¡ DnD. and Lad.oe F. (1965) pfiys. Rev. Ef,,

N+2.

Lado, F. (19fu) prtv". Rev. 115,, Ã1013.

(lg6l) ¡. chem. Phys. l+7 r 4828.

It+o 1

[t+t 1

It+21

It+tl

tl+t+ l

t45l

Iue 1

t47l

[l+B ]

[4e ]

[¡o ]

[¡t ]

lszl

lstl

Il+]

lrsllrel

157 l

[æ1

lsg )

[6o ]

161 I

1.17

Wertheim, M.S. (1965) ¡. Chern. Phys. l+3, 1370.

llS6t) J. chem. Phys . !É., 2551 .

(1567) .1. l,{aths. Phys. 9r 927.

Barker, JoÄ. and. Hend-ersonr D. (1967) ,1. Chem. Phys.

l+7, 2856 and. l+711+.

Kazak, J.J. and. Ricee SoA. (geg) ,1. Chen. Phys.

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Young, D.A. and. Ricee S.¡1. (1967) ,1. Chem. Phys.

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Physica fut 769.

Miehels¡ A.., T,evelte J.1i,[.H. and- De Graaf, TlI. (1958)

Phys ica Zt+e 659.

Plngse C.J. Qg67) Di-sc. Farad.. Soc. Lãr 89.

Molec. PhYs. 12, 5O1.

Mlkoloj, P.G. and- Pingse C.J. (lg6l ) ,1. Chem' Phys.

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Khan, A./\. and- Broyles, rr.A. (1965) J- Chem. Phys.

Eâ' 43 'Stronge S.f,. and Kaplow¡ R. (lgeö J' Chem. Phys.

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Yamad.a¡ l.{. UgA) erog. Theor. Phys. Ilyot'o 25, 579.

Garrod-, C. and- Percus¡ J.K. (lg6+) ;. iVIath. Phys.

5, 1756"

1 .18

166)

167 l

[68 ]

l6e)

Izo]

lzt I

16zl

¡at)le ul

lesl

ltzl173l

tzl+l

l75l

ltø1

lttl

Feenberg, E. (1965) J. lltath. Phys. Sr 658.

Rorvlinson¡ J. S. Ug6S ) Dis c. Farad. Soc. lpr 19.

Hanleye I{.J.lVTo ârÌd K1ein, M. (lg6l) N'B.s.

technical note 360.

Rushbrooke, G.S. and. Silbert, M. (1967) Molec. Pþys.

E, 5o5' 1

Rowlinsonr J.S. (1967) Molee. Phys, 9, 513.

Ilenderson¡ Dn (1967) .t. Chem. Phys. M-, l+306-

Leee D.K., Jaeìrsone II.ïV. ard- Feenberg, E. (1967)

Ann. Phys. À4, B¿+.

Sinanoglu, f,. (1567) Adv. in Chem. Phys. i2, 283.

Graben, H.It. (tggg) efrys. Rev. Letters 2Ot 529.

Debye, P. and. Hü'ckele E. (1923) pnys. Zelts. 4,185, 3O5. tThe collected papers of P.J.U'[. Debyer

(Interscience Pub., New York, 1954).

Mayerr J.E. (t950) ¡. Chem. Phys. 13, 1L+26.

Montroll-, E,lltr. and. rälard.t J.Q, (195A) ffrys. Fl-ulds 1,

55.

Friednane H.L. (lgSg) ulo]-ec. Phys. !r 23t 19o and 436

Meeron, E. (lgSl) J. Chem. Phys. 26, Boh.

(1958 ) errys . Flulds fr 139 .

Abe, R. (195Ð Pnogr. Theor. Phys. Kyoto 4t 475

anô. 4, 213.

Land-au, L.D. and- Lifs]nitze E.lú. (lgSA) tstatistical

Physlcst p.229 (Permagon Press. New York).

1.19

Iza]

ltglIeo ]

ler l

[82 ]

¡e:1

¡atJ

Ie¡]

I B6]

[87 ]

IBB ]

¡asJ

Igo]

[gt ]

Brush, S.G., De rtrUittr HoE. and Trul-io¡ J'G'

(1963) wuc. Fusi-on Jt 5.

Yang, C.N. (lg6z) Rev. trfod-. Phys - fu,, 691+.

Ter Haar, D. UgA ) nept. Prog. Phys . A, 3U.

l,lc vrleeney, R. (1960) Rev. L{od-. Phys. 1?, 335.

Fisherr trÍ,g' and Rue11e, D. ?gee) ,1. lilath. Phys'

Z, 260 'Dyson¡ F.Jo and- Lenard, A. (1967) ,1. l/lath' Phys' 9r

423, 1538.

Brushe S nG. e Sahlinr H.T,. ârrd Te11er, E' (1%6)

J. Chem. PhYs. M-t 2102.

Barker¡ A.A. (1065) Aust. J. Phys. 19, 119 - see

.A.ppendix A.

Bowersr D.T,. and. Salpetere E'8. (t960) Phys' Rev'

11g¡ 1180. ,!

De tüittr H. (lg6Z) J. ûlath. Phys. fr 1OO3e 1216.i'

(1965) Prrvs. Rev. ß!, ah66. '

(1966) J. Irath. PhYs. lr 616.

Hlrte C.''ü. (lg0>) ftrys. tr'luid-s Br 693, 1109.

Gaskel1, T. (tgeg) Proc. Phys. Soc. Ser 2 lt 213î.

Diesendorf , I[.0 . and. I{inTram, B.ri¡" (t ge g ) '1. n[aths.

Phys. (to be Pr:bllshed).

Glauberman, A.E. (lgSl) ootraay Akad' Nauk SSR Jgt 88.

GlaubeFûâ1e .¿!.8. and- Yukhnovski (lgSZ) Zfrur. Ekop.

Teor. Fiz. 22. 562.

1.20

lgzl Green, H.S. Qgû) i;iuclear F\rsion fr 69.

lgll oNielr Tn and. Rostoker (lg6S) prrys. Fluids $r 11o9.

t194] Hlrte c.rilt. (196i) errys. F1u1ds Qr 693.

(i967) l_Q.,565.

lgSl l,,latsudalra, IT. (1g66) Fhys. Fluids lr 539.

tSe 1 l,latsuda, K. (lgøe) Ðrys. tr'l-uids !!, 328.

lgl) Broyles, A.A.¡ Sahllnr I{.L. âlld- Carleyr D.D' (1963)

PhYs. Rev, Lett. -!:Qe 319.

tga] carleye D.D. (lg6l) Èrys. Rev' 717-, 1406.

(lg6S) ,1. chem. Phys . [!, 3489

(1s67) Éc 3783

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3, 313'

[too] v1ltars, D.s. (19Ø) prrys' Fluids !r 7)+5'

[f Of ] Oppenheim" f . arrd Hafenânne D.R. (1963) ,1. Chem. Phys.

Z2c 101 'lloz] Thelmer, o. and- Keppre¡ P. (l%l) prrys. Rev' 165,

168.

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PhYs. 49, 27l.+2.

2.1

TI TTIE MOT\TTE CARLO }/ETTIOD

The extension of the l,{onte Carlo (UC) method

to plasmas 1s described. 1n d.etail by Brushr Sahlin

and- Te1ler [1 ]' and Barker lzl- Emphasls 1n this

ehapten is placed. on the results of thls authorrs

recent extensive eomputer calculatj-ons f or a h¡nlrogenous

plasma of density lolBionsr/ce at, a temperature of 1o4oK.

2.1 An outline of the meth

Asystemcomposed.ofNindividualpartie}esie

conflned- in a volume V at a temperatr:re T. The particles

are assumed- to obey el-assicaf statistics and- to interact

in accord.ance wlth the Coulonb potentlal. The problem

1s redueed. to a feasible size by consldering only a

finite nr¡mber of particles N¡ and. in this case N-32c

a Value which proves eOnvenient anC. gives reasonable

accuracy (see Alder and- i,Vainri-ght t¡]). In the j.nltial

configuration the particles can be either placed randomly

or in an ordered fashion in the unlt ce1l 0f volume

V=LJ9 ârd. in the case of an electriCally neutral plasma

they can also be in1tlalIy paired as neutral ¡articles

or dissoclated- as i-ons. The unit ce}I is surround'ed-

by a network of ldentical ce]lsr th¡s enabling the enerry

of a configuration to be calculated conveniently as

2.2

d.escribe6. in detail in [t ] and lZl. Another configurat-

ion is obtalned. by displacing a parti.cle by a rand-ont

amount, which can have a maximum value A' The energy

of the new configuration is calculated-, and. the Mc

procedure d-ecides if the move 1s acceptable or not, (see

[2]). In the present calculation each parti-c1e is

displaced systematieal.ly (although they can be rnoved

randomly) in this manner, until the energy of the system

approaches its equilibrium value. The cniterion for the

choice of A 1s such that the rate of approach of the

system to eguilibrium is minimlsed. In previous lfc

calculations Ir ], lzl, [l], 1t has been found' convenient

to use A:r,r/úln), where L is the unit cell lergtht and

N is the nr:mber of partlcles in the eeIl for then a has

the rlght order of magnitud-e to secure near-optimun con-

vergence.Inthisworkthislmpliesava]-ueofAof

order 9 Bohr rad.ii.

Another paraneter vrhich proves important in the

calcul-ation 1s the cut-off AO imposed- on the attractive

coulomb potential at short radi1. This l-irnits the

cl_oseness of approactr of tvrro particlesr ârld hence the

potential energy between them. The value given to Ao is

tv¡ice the Bohr rad-ius, for at this rad-ius (Uv Bohrrs

orbit theory) a partiele has its lov¡est potential energy

possible without any kinetic energy, and the value of the

2.3

potential energy is the sane as the ionization energy

for the particle. Here pairing 1s consid'ered' to occur

between two particles when they are closer than AO' i'e'

in tlrelr grourd. state¡ âfi.d. particles are not considered'

palred. when they are in excited- states'

Thecomputerprogramused-fortheea}culationisgiven in lzl, although several nodifications were

necessary to ad_apt i-t to the c.D.c. 64oo computer on

which the present calculatlons xrere conpleted. The

eomputlng time involved was four:d to be excessiver fooo

iteratlons taklng one hour on the 6400 computer (each

iteration gives every partlcle in the unit cel1 a chance

to move up to the maximum d'istance A)' For this

reason the stud-y has been confined. to a slngle temperature

of 1o4oK, and d-ens1ty, l ol Bions /ec'

2.2 Results

ThreenaincomputerrunsTyerenad.e.Theflrst

started. ttrre particles from a random configunatj-on of

protons and electrons and- 1nitially put A - 12'5 aot

and_ proceed.ed. for 5orooo lteratioIlS. Then the energy of

the system appeared- to h,ave settled' d'own to the

equlllbrium value, ard. to check thj-sr a second run Tvas

earried through¡ starting the particles as pairs of

2.4

protons anÔ electrons approximately eguid.istant from

each other in the cel-J-. Again the maximum

dlsplacement was ehosen as a = 12.5aoe but here the

run covered 12.OOO iteratiorlso This run seemed

to approach a slightly dlfferent energy equllibrium

valuer ând- so a third- run was eomplæted-, again

starting the partlcles as pairs, distributed evenly

throughout the cell, but now allc¡rning them to move

with A = 5Oao for 'lO¡OOO iteratiorrs¡

Fig2..|showsthevariationofthenorma]ised.

cel_1 energy per particfe E/UXî vrÍth the m:mber of

iterati-ons completed- for the three runs mentioned above'

The energy is averaged. over IOOO iteratlons for each

point pl-otted., and this smooths out mar¡y of the

extreme energy fluctuations which oceur from iteration

to iteration. It can be seen that I'rith such fluctuations

1t becomes d.iffleult to obtaln comprehenoive sanpling

of all states r.Lsing the MC proced-ure unless a very long

run is taken. on the rlght-hand- side of the figure

Ievels are presented. whlch show approximately the number

of pairs in the unit ee1l eorrespond.ing to a selected-

value of E/NkT. In Fig 2.2 the like ard unlike

d.istribution functions are d-rawn (on a 1og scale) ¡V

consldering lterations 5orooo to 5or0oo of run 1, in

which regi-on the system seemed- to be near equilibriun.

2.5

The corresponding classical distribution f\rnctiorrs are

al_so drawn; to incorpor.ate the required. cut-off at

srnall_ rad.ii, they Ïrave been given a constant vafue

be10v¡ 2.O Bohr Rad1i. The d.ata for these characteristics

are given to the nearest three figures in Table 2"1 t wþere

the subscript U refers io the unllke easer L the like case

and- c refers to the cl-assical d.istribution function'

Fig 2.3 shows rrnl-ike distribution functions taken

from selected- sectlons of Flg 2.1. The results to 2

places of d.ec1nals are given nl-unericatly in Table 2"2

with the correspond-ing Debye Huckel d-istribution f\rnctioir.

For a plasma of this tenperature and d-ensity the Debye

shield.ing C.istance is 92'l+ Bohr radii, and- is denoted' o;L

the graphs bv ÀD" Fig 2"1+ and. Tabl-e 2.3 present the

d-lstribution functions betrreen the llke particles for tlle

corresponding sections of Fig 2o1 o The non-integral

values of r appearing 1n the Table arise because the

program (see lZl, Appendix B) i\ias run in mesh uni-tse

and. 1 Bohr rad-lus = 2"1OO mesh units. The energy has

also been expressed.a. a d.imensionless number, lvhere

the energy in cel-l units E is converted- into tire d-ime::sion-

less E/NkT by multlpl-ylng it by 2"072 x rc-3 for thet,-

Temp of 1o4oK and electron d-ensity of lotue/ce'

Fis 2.5 isolates the unl1ke d-lstribution functiors

at smal1 rad.ii, and. thre values are obtained- by taking

2"6

logs to the base ten of the figures given in lable 2.2"

Again the cl-assical- curve guc = "Pøu(r) is glven nibir

the ÉU(r) u.sed- in the L{onte Carlo cal-culation (i.""

Coulom-o for r>Zao, and const for r<2ao")

Distribution functions taken from i-r,erations "ljOOO to

Z3OOO of run (t) are d-enoted in the Tables as 13OOO-23C4':

(r ) "

2"7

TABT,E 2. 1

RADIAL DISÎRIBUÎION FUNCTIO]'íS fOT LiKC ANd. UNI1KE

cases from iterations JOTOOO to 50' l0O of Run 'l '

r Log¡o Loglo Logro Lo$to(norrr Rad ii) (e"(") ) (so (r) ) (su(") ) (eu. (r) )

1.19

2.OO

3 "57

5.95

8.33

1O "71

13 "1O

15'48

17 "85

20 "24

22 "6

25 "O

27 "429.B

32.2

34"5

36.9

39 "3

- .576 - 6 "857

-6"857

-3,841

-2.305

-1 "646

-1 . 280

1 "Ol+7

-o " 886

-o.786

-o "678''o"607

-o " 549

- o. 5oo

- o "¿+60

_o.426

- o"397

- o "372

- o '3119

l+.740 6.857

6"857

3,841a 7^ç:Lc_rv)

1 .6j-t6

1,28O

1 .O'LI

O. BB5

o "758

o "678

o "607

o "51-19

0"5o0

0"460

o.426

o "397

O.-

o.349

"339

"139

"21ì+

,1 50

"072

" OBJ

" oo7

"o59

"o2B

.o55

no4,

"065

^o7

"103

"o51

.047

3.2940

2 "32',1

1.glB

1.659

1 .5014

1 ,365

1.244

1 "166

1 . O¿+1

.959

. 8116

"778

"730

.61+0

"579

"517

2"8

r

41 "7

4¿+"0

46.4

48 .8

51 "2

53 "5

55.9

58 "3

60 "7

63 "1

65 "5

67 "B

70 "2

72 "6

75.O

17 "\79 "B

82 "1

84" 5

86 "g

Bg "3

91 "6

94" 0

96.4

Log¡ o

(s"(r) )

"o51

.088

" 140

"'i34

" 113

" 140

. oE7

.06I¡

" 05B

"o20

,01B

" 061

"036

"017

.o20

.01 0

"o16

- .o09

"012

.036

" o20

"o12

"o'.'.3

1c44

-o"329

- o,312

-o "295

-o "281

-Ð "268

- o "zD6

-o "245'- o "235

-o "226

-o "217

-o "2o9

-o "202

-o "195

-o"1BB

-0,1 BJ

-o "177)-^

-Uolll

- o "167

- o.162

-0.158

- o "154

- 0"150

- O.1 116

- o.1\2

"\59.l+3o

"1+o7

.3-i2

"334

"297

"268

"221+

"179

.139

"127

,11J

,o97

"072

.045

.045

. 046

.01 5

.011

"o33

.o'12

.0'1h

"o17

"o31

Logr ct

(su.(");

o "329ô 7-i,)

O "29/o

O "28"'

o "268o "256

O "21+5

v"¿¿o

Uoél t

O.2OC)

U"¿U¿

rl 4 0:

o,-i 88

o"18j

o "i7-io "172

O "'i67

o "162

0.158

o "i 5\-

0.-l 50

o..1)-".5

O "1L¡2

Log¡o Log¡o

(s".(r) ) (eu(r) )

o "23'

rLogro

(s"(r) )

Logr o Logto(s".(r) ) (eu(r) )

2"9

o,139

o "136.

o "132

o "129

o.127

o.124

o "121

o"119

o "116g

"1..15

o"112

o.110

0.108

0"106

O"1Oir

o.102

o "'1

00

0"099

o"o97

o,og5

o" og4

o "o92

0,091

o. 0Bg

Logr o

(sur(") )

98. B

101 "1

103.6

105 "g

1O8.3

11O.7

113 "1

llt 5 "5

117 "9

119 "1

122.6

125

127 "4129.8

132.1

134"5

136 "g

139 "3

141 "7

144.0

146 "4

148 ,8

151 .2

153"6

"o27

. o34

.o3g

"o22

.015

- "o20

- "oog

- "o32

- "o22

- .056

- "o52

-,051

- .o31

- "o47

- .o47

- "046

- "050

- "o47

-.o48

- "o34

- "o43

-.o38

-.o53

-.o32

- o "139

- o,136

-o "132

- o.129

- o.127

- O.12L,.

- o "121

-o "119- o "116

-o "115

-o "112

-o"'l 1 0

-0,1 08

-0. 1 06

-0"104

-o"102

-o.1 00

-0 " ogg

-o "og7

-o" 045

-o. og4

-o "og2

-o "og1

- o. o8g

- .006

,017

"o12

.oo2

-.01 4

- .026

- "o43

- .056

- "o44

- ,o77

- "o77

-.078

- .061

- "oB2

- .067

- "o75

- "o75

- "061

- "067

- "o6c

-"068

-"ubÕ

- "068

-.064

r

156.O

1 58.4

160.7

165'1

165.5

167.9

17O.2

172.6

175.O

.o35

. o¿J+

.o4g

.o24

.oJ8

.o16

.o16

. oo5

"012

-0.088

- 0.086

- o.oB5

- 0.084

- o.o83

- o.oB2

- o.o81

- o.o79

- 0.078

.O53

.076

,o72

- .oh4

- .065

- . olJ+

- . ol+7

- .o34

- .o37

2.10

O¡ O8B

0. 086

o. o85

O. OBI+

o. oB5

o. o82

o. o81

o.o7g

0.078

Logro l,oglo Loglo Loglo(s"(r) ) (s". (r) ) (eu(r) ) (su.(r) )

2.11

TÂBLE 2.2

Un11ke radial distribution functions taken from various

n-Lns as shown and- companed with the correspond.ing Debye

Huckel d-is tributl on tr'\,lncti on.

1.19

3.57

5"95

8.33

1O.71

13.1o

15.48

17.85

20.24

ZZ.O

25.O

27 "4

29 "B

32 "2

34.5

36.9

39.3

2"35x10r 1

4.94 *10"

1.45x1O2

3.19x1O

1.3Bx1O

8.1 0

5.61

4"29

3.50

2.gB

2 "62

2.36

2.16

2 "OO

1 .88

1.77

1.69

r(ao) D.Ho1 fooo-ryoo (1)

2"95x104

1 .21-¡x1 03

1.JOx1O2

5.tt4x1O

3$7xto

2 "63x1o

2 .O1 x1O

1.58x1O

1 "2J x1O

1.O7x1O

8.64

6.73

5 .54

l+- 84

4"O3

3 j,37

2 ,81+

JOOOO-4oooo(1 )

5 "51x1Oa'l . !Bx1 03

2.1 5x102

B "31x1O

4.59x1O

3 .27 x1 I'

2.33x1O

1.J)x1O

1 "48x1O

1 "OBxl O

9.10

6"BD

5. BB

5 "32'l+"45

3.82

3.11+

4oooo-5oooo ( 1 )

5,47x104

1 "96x103

2"O4x102

B.25x1O

\"J2x1O

3 "11 x1O

2,30x1O

1 "72x1O

1 .45x1 O

1 "19x1O

g.oB

7 "34

6.12

5"41

Lt.26

3.76

3.U)1

40ooo-5oooo (i)

6.75x1os

2 "OOx1 02

2 "JBxiOg "98

6.01

5"36

5"01

3 "85

3 "\\2 "Bg

2.64

2 "62D <C)Lo ,) t

2 "23

2 "O9

i "gB

1"99

2.12

r (ao) D.H.1 SOOO-23ooo(1)

JOOOO-4oooo ( 1 )

40o0o-5oooo(1)

4oooo-5oooo (3 )

41 .7

44. o

46 r4

48. B

51 .2

53.5

55.9

58.3

60.7

63.1

c.5.5

67.8

70.2

72 "6

75.O

77.3

79.B

82.1 0

84.5

86 "g

89.3

g1 .6

1.62

1 ¿56

1 .51

1.46

1.42

1 .39

1"36

1 .33

1 .31

1 .29

1.27

1.25

1.23

I .22

I .21

1 .19

1.18

1.17

I .16

1 .15

1 .1¿+

1 .14

2 "43

2.15

1.86

1.76

1 .58

1.53

1 .44

1 . J+'1

1.38

1.47

1.45

1 .55

1.34

1.32

1.27

1.29

1,30

1.29

1.30

1.22

1.18

1 .17

3.10

2.91

2.81

2.56

2.33

2.10

2.11

1 .83

1.64

1.53

1.42

1"llt1.37

1.32

1"22

1.32

1.23

1 .12

'l .08

I .14

1.13

1"08

2.66

2 "47

2.31

2 "16

1 .99

1 .86

1 "59

1 .51

I "38

1"23

1.26

1 .18

1 "13

1 ,04

.gB

"90oo

.96

,gB

1 .O2

"93

.gB

1.95

1.BO

I .75

I "7O

1 "67

1"64

1"r+

1"fl+

1"51

1"49

1 ,41

1" 38

1 .l¡1

1"35

1 "31

1.31

I "29

1 "26

1"31

1 "25

1 .26

I .21+

r (ao)1 Sooo-zJooo ( t )

50000-Loooo(1)

¿+OOOO-

5oooo(1)

2.1/,+

4oooo-50ooo (3 )D.li.

149.8

151 .2

153t6

156.O

1 58.4

160.7

163.1

165.5

167 .g

170.2

172.6

175.O

177.1+

179.8

182.1

1 84.5

196.9

189.3

191 .7

191+-l

196.11

1gg.g

201 .2

t¡d+

1.d+

1.04

1 .04

1 .04

1.d+

1.03

1.03

1.03

1.03

1.O3

1.O3

1.03

1.03

1.02

1.02

1.12

1.02

1.02

1.02

1.O2

1.02

1.O2

1¡ 05

1 i07

1.04

1.03

1..O5

1 .08

1.01

1"O2

1.OO

.gB

.95

.97

.92

.93

.90

.gB

,Bg

.87

.87

.87

.90

.Bg

.88

.97

.99

1 .01

.99

.95

.98

1.O1

.gg

1 .05

1. 0'l

1.04

1.O1

.98

.gB

-97

.97

.92

.90

.87

.93

.98

.gt

.90

.73

.7O

.7'l

.77

.73

.72

"79

.73

.75

.79

.81

.83

.84

.83

.81

-70

.82

.83

.84

.85

.83

.85

.87

1.05

1.05

1. 04

1 .06

1.O5

1 .0¿+

1"O3

1 .04

1. 04

1 .01+

1.0¿+

1.03

1.02

1.)2

1.OO

1.O2

1 .01

1,O1

1.OO

1.01

1.OO

,99

1 .00

2.15

r (ao) D. H.1 fooo-23000 ( 1 )

.89

.92

.92

.91

.90

.92

.93

.93

.94

.93

.92

,93

,92

.95

fooo0-4oooo(1 )

.89

.Bg

.91

.85

.96

.91+

.BB

.go

.87

.89

.90

.91

.95

.95

4oooo-5oooo(1 )

l+OOOo-5oooo(5 )

oo

.99

.gg

.99

"99

.97

.97

.97

.99

.98

:97.96

.96

.gB

2O3.6

206.o

208.3

21O.7

213.1

215.5

217.9

220.2

222.6

225.O

227 .4

229.8

232.1

231+.5

1.q2

1.02

1.02

1.02

1.O1

1 .01

1.01

1 .01

1.01

1 .01

1 .01

1 .01

1 .01

L01

.89

.96

.88

.88

.97

.9¿+

.95

.96

1.O1

1.O3

.99

1.OO

1.OO

1 .04

2.16

TT+BLtr' 2.3

T,ike rad-iaI distribution functions taken from lterati-ons

aS shorrvn, and- compared. with the correspond-lng Debye-

Huckel d.istrlbution functi on.

1.19

3.57

5.95

B,33

10.71

13.1O

15.48

17 .85

20.24

22.6

25.O

27.ì+

29 .8

32.2

3l+.5

36.9

39 .3

41 .7

44. O

46.4

r (ao) D.H.1 looo-4oo(1)

50000-4oooo(1 )

o.53

3.06

1 .44

1.60

1.03

1.11

1.21

1 .O9

1 .10

1.10

1.O7

1.O3

1.13

1.31

1 .28

1.22

1.17

1.32

1 .l+1

1.74

4oooo-5oooo(1 )

4oooo-5oooo (3 )

o. oo

.18

.08

.19

.35

zo

.38

.61+

.58

.55

.61

.62

.69

.63

.77

.67

.68

.75

.77

.83

I¡.2Jx1O-1 2 o.oo

2.O2x1O-a .oo

6.gzx1o-3 .oo

3.11+x1O- 2 . 01

J.2Jx1O-2 .o5

;12 .13

.18 .18

.23 .13

.29 .14

.34 .11+

.38 .15

.l+2 .16

,46 .2O

.5O .25

.53 .29

.56 .37

.59 .32

.62 .39

.64 .38

.66 .l+3

o. oo

1.30

1.32

1.66

1.79

1.25

1.21

.95

1.18

1.O2

1.20

1.1+5

1.19

1 .19

1.2/+

1.ñ2

'1 .05

.92

1.O5

1.35

2 !t7

r (ao)1 1000-2tooo(1 )

30000-4oooo(r )

4oooo-5oooo(1 )

40ooo5oooo(l)D.H.

l+8.8

51.2

53.5

55.9

58.3

60.7

63.1

65.5

67.8

70.2

72.6

75.O

77.31+

79.8

82.10

84.5

86,9

89.3

91.6

9l.o96.4

98. B

.68

.7O

.72

.74

.75

.76

.78

.79

.Bo

.81

.82

.83

.84

.85

.85

.86

.87

.87

.88

.89

.89

.90

.40

.49

ç,2.)J

.65

.69

.71+

.91

1 .O2

I .14

1 .09

1.O3

1.OO

1.18

1.16

1.15

1.13

1.11

1.13

1.O9

1 .09

'l .13

1.56

1.51

1.49

1.46

1.30

1.28

1.22

1.20

1.31

1.28

1.23

1.18

1.23

1 .19

1 .06

1.14

1 .15

1.15

1.12

1 .10

1.20

1.14

1.16

1.O9

1 .24

.98

1.O0

.90

.BB

.88

.gg

.80

.Bj

.72

.82

.89

.90

.90

1.O2

.94

.92

1 .01

1.01

.gB

.84

.85

.81

.96

.81+

.85

.85

.Bg

.90

.90

.91

.88

.go

.93

.93

oz

.92

.90

.95

.97

.95

.97

1.11

101.2

1O3'.6

1 05.9

1 OB.3

11O.7

113.1

115.5

117.9

120.2

122.6

125

127.4

129..8

132.1

134.5

136.9

139.3

1'l+1 .7

144. O

146.4

148.8

151 .2

153 "6

.go

.91

.91

.91

.92

.92

.92

.93

.93

.93

-94

.94

.91+

.94

.95

.95

.95

.95

.96

.96

.96

.96

.96

r (ao) D.H.1 SOOO-23ooo(1)

'l ¡o9

1 ¡O7

1 .06

1.O7

.99

1.00

1 .01

.96

.99

.gB

1.O3

1 .0¿+

1.O1

1 .O5

I .OB

1rO7

1 .06

1 .04

1 ,OB

1 ,08

1 .08

1.06

SOOOO-4oooo(1 )

1¡11+

1.22

1 .16

1.16

1.O5

1.10

1 .09

1.15

1 .08

1.O5

1.O3

1.OB

1.O2

1.06

'l .01

.gg

1.O1

I .O3

1.10

1 .06

1.04

1.03

1.OB

2.18.

lr.OOOO-

5oooo(1)

1.62

.99

.94

.90

.85

.96

.77

.75

.68

.72

.75

.78

.78

.73

,79

.79

-79

,75

.74

.7l.+

, .80

.73

,78

40ooo-50ooo (5)

.97

.96

.99

.99

1 .O3

1 .00

.97

.99

.gg

.97

1.OO

.gg

1.O2

1.OO

1.O3

1.OO

1.OO

1.O3

1 .01

1 .01

1.OO

1 .01

1 .01

1 .11

2.19

r (ae) D.H.1 fooo-2looo(1 )

1.O5

1.OB

1 .06

1 iO3

1 .04

1.O2

.98

.97

.98

.96

.95

.95

.92

.91

.87

.89

.92

.91

.92

.93

.95

.95

foooo-4oooo(1 )

1.O4

1 .04

1.O3

1 .07

1.O5

1.O9

1.O9

1 ,06

1 .06

1.06

'1 .04

1.O5

'1 .OO

.97

.95

.99

1.O5

1 .00

.97

.97

,95

l+OOOCI-

5oooo(1 )

.80

.76

"75

,83

.77

.81

.83

.89

.BB

.90

.88

.BB

.85

.86

.87

.89

.Bg

.89

.90

.91

.93

oz

l+OOOO-

5oooo (5 )

1.O2

1 ,O2

1.O2

1.O2

1 .04

1.O3

1 .O3

1.02

'1 .O1

1.O2

1.C1

1.O2

1.O2

1 .O2

1. OO

1"O2

1 .O2

1.ù1

1.O3

1.O3

1"01

1 .01

156.O

158.4

160.7

163.1

165.5

167 .9

170.2

172.6

17 5.O

177.4

179.8

182.1

1 84.5

186.9

189.3

191 .7

191+.1

196.4

1gB.B

201 .2

2O3.6

206.o

.96

.96

.97

.97

.97

.97

.97

.97

.97

.97

.98

.gB

.98

.gB

.gB

.gB

.98

.gB

,gB

.98

.gB

.98

1 .11

n (ao) D ¿H.l rooo-25ooo ( 1 )

.96

.98

.96

1 .OO

1.OO

.99

,99

1.OO

.gB

1.OO

1.01

1.Q1

foooo-4oooo(1 )

¿95

.90

.93

,gB

.94

.96

.93

.94

.94

.97

.gg

1.01

2.20

4oooo- :

5ooo0(1 )

193

.95

.99'

.99

1.OO

1.OO

1.06

'l .o7

1 .06

1.07

1.06

1.10

4oooo-5oooo(t )

1¡01

1 ,01

1.OO

.98

1 .01

1.OO

1 .01

.99

1 .00

.gg

1 .00

.99

2O8¿3

zlot7

213.1

215.5

217.9

220.2

222.6

225.O

227.4

229.8

232.1

234.5

¿98

¡98

.99

.99

.99

.99

.gg

.99

.99

.gg

.99

.99

-5(1) from a randorn conf;guration

witn ô : 12.5aovNKT

-15

-25

-35

2

redwittr

ao

10000 30000

nu m ber of it erqt ion s

50000 I

rFIG. 2.I . APPROACH TO EQUILIBRIUM

*+* + +++

1+lcn

oCNo

-l

+5

-5

CLASSICAL (urtlhc)

MONTE CARLO (untike)ooa tta

oaooaoa

O¡

l.tONTE CARLO (t¡þ,

.-CLASS|CAL (tikc I

FlG 2.2 RADIAL OISTRIBUTION

FOR LIKE AND UNLIKE

CLASSlCAL GRAPHS,

FUNCTIONS ( drawn on a log scate )

CASES, WITI{ THEIR CORRESPONDING

TAKEN FROM ITERATIONS 3OOOO TO 5OOOO

OF RUN 1.

r25

r (Bohr Ro d li )

| )to : sz.c

oc

+

6

9utrl

o

o

o+

DEBYE HUCKEL

a

a

oo

ol9

oo

o

oo

Ào

10 120r (Bohr Rqdii)

UNLIKE OISTRIBUTfON FUNCTIONS COMPAREOCORRESPONDING DEBYE HÜCKEL OISTRIBUTlON

FoR lo4o x AND 1018 elcc.

from iterations 13000 to 23000 of run I30000 " 400æ

" ¿0000 " 50000 .' 3

200

Wf TH THEFUNCTION

AoBoC+

o

o

1o

a2

FlG. 2 .3

0

o

AB

c

oo

o trom iteration 13000 to 23000 of run 1

o 30000 " ¿0000 "+""3

o

o

DEBYE HÜCKEL

200

ooo o

o

o

eOo

o

o

o

o o1.2 o

ooo@

o

o o

oo

.+++++Þ+

0.8

0.4

+++

attt'

+++

a

oa

a

o

++

+

ì++

+

O

g(r) a

a+

++

o

a

aa

+

+o

0

FrG. 2.'4

D

t0 120r (Bohr Rodii)

LIKE D¡STRIBUTION FUNCTlONS COMPAREO WITH

CORRESPONDING DEBYE HÜCKEL DISTRÍBUTION¡ AND 1018 e/cc.

THEFOR 104

oK

6

cut otf at 2oo

A o from iteration 13000 to 230ffi d run'lBo .. " ó0ÛÛ0"5000C+ .. .. .. t' " 3

clqssicol

+

510r (Bohr REdii)

FtG. Z.5 . LOGS OF UNLIKE DISTRIBUTION FUNCT]ONS AT SMALL RADII COMPARED

wtTH THE CLASSICAL DISTRIBUTION FUNCTION pm 1040r lruo 1018 e/cc.

o

o

+L

L

çn

c,cn.o

oo

ao2

+

oa

8+

++

0

I,

2.21

2.3 SSl Conc icn

tr'romtr'ig2.litcanbeseenthattheapproach

of the system to its equil-lbrium energy value is

influenced. d.rastically by both tbe parametar Àr and- the

inltia] configuration chosen. At the present

temperature and- d-ensity, nrn 2 shows that equilibrium

isobtained.muchfasterfromaconfigurationwitha]lparticlespaired.ItcanalsobeseenthatwithRunJ

AnotonJ-yinf]uencestherateofapproachtoeguilibrium¡

but the eguil-lbrir:m energy value attained'' This

contrasts with the results of Wood. and Parker [4] wnot

workingr¡¡ithafluidinteractingvrithaT,ennard-Jonespotential¡ noted that thelr results lvere ind'epend'ent of a'

0n cl-oser examination of the results presented it was

found- that this difficulty occurs in the temperature

range 1o4olt to 3x1o4oK, where the plasma appears to

behaveasavariablemlxtureoftrrophases,ionized-arf1

unionizedr which phase dominate is influenced rather

sensitive]ybytheparamcterAforthere]-ativelysnall

sarnple of configurations considered' in this work'

The levels (4) on the right of Flg 2'1 give approximately

the number of pairs (i,". unlike particles closer than

zao), in the unit cell for correspondj-ng E/NkT. A

d.eta1led. stud.y of pârticle movements shorvs that it

lstLrenumberofpartic]espairedthatissod.epend-ant

2 .22

on the size of A.

The distributlon f\rnctions taken from run 1 aS

1t approaehes a corìstant enerry value, i'e' from

iterations SOTOOO onwards as in Fig 2.2e sholir a marked-

d-lfference from the classiea] and- the Debye-Huckel CâS€S'

The rapid_ rise of g¡(r) to a value abor¡e unity is due to

a proton or an electron collid'ing with a pair' this

occurs particularly lvhen there is a relatively large

number of palrs presente âIId- is evident in B when A is

less than fj-f teen Bohr radil. Fig 2.1+ lllustrates this

situation very cJ-ean1y. In ru¡ 5 (A = 5oto) whene from

the energy graphe there are usually no palrse âñd

occasionally one pair is f annd., S¡(r) is simil-ar to the

Debye-Huckel case, but approaches unity much faster.

In run 1¡ where the particles were started. as rand-om]y

distrlbuted. ions¡ ârrd iiriere in the process of approaching

egllilibriunr but initially there were f ew collisions

between pairs and. lonsr S¡(r) lvas srnall for r<60.

Howevere alread.y the tend-ency of ions to collide i¡vith

pairs is ind-icated. by the appearanee of a sharp peak at

r = BO ao. In run 2 taken from near eguillbrium

there are peaks at f = 5ao ard r = l+Oao¡ ârrd' from a

stucly of the particle movementsr these peaks appear

following a collislon between a pair and. an ion.

2.23

The number of pairs present a1.so has a marked'

effectone¡(r)'RunJinFlg2'3shcn'vsth¡atfora

Iarge, Sff (r) 1s quite similar to the Debye-Huckel cllrvet

but falls appreciably bel-ow it at small rad.i ir while in

the range r = 4Oao to P = BOao 1t lies above. For

A = 12.5 âo as in 2 there is a tend-ency for rrnlike

particles to stay about lOao to 6oao apart. Examination

of particle movements conflrm that two unlike ions d.o

tend. to ward.er arourd. the cel] togethere Sorlêtimes coming

close as a pair¡ and- sometimes straying 4Oao or 5Oao

apart, but rarely escaping fully the otherts influence.

However, if a is increased. they do escape fu11yr and- if a

is decreased- to 5aoe they tend. to falI lnto palring

comple te1 y.

Fig 2.5 il-I¡strates the lmportance of the choiee of

another para¡neter used. 1n the calculation, the cut-off

AO inposed on the coulomb potential- at smal1 rad.ii. Tf

the cut-off were applled aL one instead. of at two Bohr

radli SâV r then an unlike ion on moving to 2 tsohr radil

apart would. be sub ject to a considerable change in potent-

ial enersrr âDd. this move would. be most improbable by

the l[c procedure. on the other hand., if the inter-

particle potential wag cut off to make the r¡re11 shallcne

then unlike particles could escape each other?s

influence quite easily. To rlgorously determine the

2.2/'+

form of the lnterparticle potential at smalI raoii 1t

would be necessary to treat the close interactions

guantun mechanical-ly. The present choice ¿9 = 2ao

is based on cl-asslcal consid-erations only' ft was

alsonotieed.frontr'lg2.SthataSAbecamesmal}erthe d.istrlbution f\rnction at small radii tend-s to

approach the classical curve' Furthere êxâil1nation of

partì-elemovementsshowed-thatifAwerelarge,then

almosteverytria]-movementtookoneir-rIri¡ve11aÏVay

fnonanotherrandalthougþthismeantalargepotentialenerry change, eventually a nove was allowed; whereas

fon srnal-l Ar the particles tend- to move apart and-

togethen frequentfy, but rarely to escape very far

from each other.

Tables 2.2 and' 2'3 present ttre d'istribution

furrctions obtalned from lterations 30'oOO to 4OtoOO and

4OTOOO to 5OrOO0 of run 1, to show the variance that

occurslvithlnar'trnoflo'oooiteratioflsoltcanbeseen that this variance is guite large' being reguJ-arl-y

greater than O-2 and- although a long run would tend

to smooth out these f luctr¡ations ¡ 1t seems unlikely that

sucharunwouldinprovethed.istributlonfunctionsmuchbeyond. the first place of decimals '

fn concluslon then, it appears that the I'{onte

Carlo approaclr is 'not partlcularly successful in

2.25

obtaining aecurate distribution functlons for a two

eonponent plasma in the temperaùrre region near

ionization. In this reglon the maximrrm step length

parameter A is analagous to a l-imit of the energy of

quanta absorbed. or emitted- from the radiation fieÌdt

and if A is smaIl one particle may move slightly awai¡

from the interactì-¡g particl-e, but rarely escapes fu11y;

lvhereas ]f A is relatively large the partieles completely

separate. This behaviour is peculian to the temperature

range near ionlzation as at low and hÍgh temperatìlres

the results are ird.epend.ent of a for a long enough

lìllrr. It is in this respect that the plasna appears

to behiave aS a vaniable mixture of two phases 1n the

region of ionization, lu.ith the choice of A determining

lr'hich phase d-orninates. It also appears highly d'esirable

to include quantum mechanical interactions between

protons and_ electrons at small radii in the l{.c.

calculation. Because of the excessive eomputlng time

involved (IOOO iterations taking one hour on a C.D.C.64OO

courputer), d-istribution functj-ons should be obtained

more economically in thls negion by solving a nodified-

Percus-Yevick eE¡ation. using this latter approach

1t is hoped. by comparing results to resolve the

d.ilemrna of the choice of a and hence improve the K

result s.

2 .26

Referencep t,o Cha'pter II

It ] Brush, S.G.¡ Sa]r1inr II.L., and Teller, E.

J. Chem. Phys. 45¡ 2102 (1966).

lZl Barker¡ A.A. l{.Sc. thesis¡ University of Adelaid.e,

s./t. (196i+); Aust. J. Phys. 13, 119 (965).

lZl Ald.err B.J.e ând. Ïilalnwrlghtr T. J. Chem. Phys.

31, 5 (t960).

tt+] t'llood.r VÍ.IV.¡ ârÌd Parken, T.R-, J. Chem. Phys. 4,

72o (1957),

t¡] l{etropolis¡ N.r Rosenbluth, A.\V., Rosenbluthr I{.N.¡

îeI1er, .A.W.r âDd- Tel-lerr E. J. Chern' PlSIs'

4, 1oB7 (953) .

t6] Barker, .4.4. 1o be pubtisherl- in Phys. Rev'

JuIY 1968.

3.1

TTT THE II\]'IEGRAL EOJJATTON METTTOD

3.1 od.ucti- on - ercus-Yevi E tion

In Chapter I the extension of lntegral equations

to d-eal with plasmas was discussed generally. the

three main integral equations which have been applied.

to fluid.s are the Born-Green-Yvon (gCV) eguati on, the

percrrs-Yeviek (fV) equation, and the Convoluted- Hyper-

netted Chain (CSUC) equation. In ¿eelcling that the

Fercus-Yevick eguation could. be app11ed. best to a plasma

of lOrae/cc at temperatures near 1O4oK, the author was

infl-uenced- by a number of factors. tr'irstly Green [1 ], and-

Stell lZl ha¿ d.eveloped- higher order eguatlons which

contained- the Hf eg¿atj-on ás a first approximation"

Subsequently Verlet lll, Verlet and- Levesque [¿+]t

Allnat [f ], TVertheirn t6] and. Hend.erson lll have all

proposed higþer order terms to improve the PY equation.

Second-ly several eomparisons of the three approaches wlth

the \{onte Carlo method. for fluids [B], and- for the electron

gas [g]r ind-icated. the superlority of the H{ equatlon

in most cases. A recent paper by \4/atts [tO] confirms

that the Hf equation is superior near the critical region

for a Lennard-Jones fl-uid. Ilorvever, l-ittle 1s known about

the merits of ,9f equation where attractive forces are

involved.

3.2

3.2 tic forn of the Percus-Yevick equation

for larse r

The Percus-Yevick equationr generallsed for a

fluid- mixturer has the form

gab eab = I [{.^"-r) *"" (eo"-t) d"*" 'Ðn^nv

(j.t)

where eab = exp(,8pro), this ean f\rrther be written

in the form

s"¡(r) e.o(r) 12rr ne

s+r

3-r[*¿O

["r"( ")-t ]

(3. z)

eu"(e) [*o"(t)-t ]tat sd.s,

where n" is the number density of partlcles of type c

per unit volume, Ðc sums over all types of particles in

the rnixture, and. d-3xe ranges over the volume of particles

of the cth type. Let the integral term be funagined. to

physical-Iy correspond to a particle of type a at xrr a

particle of type b at ëb, and. a ¡nrtlcle of type c at

zoi and further, Iet r - lx, - =¡l

s = l¿" - ="1

and-

t = l¿U - *"1. Because of the spherical symrnetryr the

integration over the range O to 2t due to the rotation of

the g! plane about r can be d.one immediately. Further,

3.3

since I = ! + ! we can use the siræ ruJ-e to change the

variable and- r¡n"ite the integrations over lengths only

as 1n (3.2). Broyles [tt ]r by d.ifferentiating over rt

rewrote this equation in the form

(s+r)*o"(l=l)"oo

-oo

[*o"(I "*tl-1) ] [1-eac(s) ] sds, (s.t)

which is much easier to hand-1e computationally'

To obtain an asymptotic form of equation (3.2) ror

large ¡.¡ we make the follov¡ing assumptionst (i) That fon

large r, and the integer n>Or ÞþaA O("-^); and for attract-

ive forces at small r, pþ^a is finite. These assumptions

exclude gravitational forcese and. require a cut off at

sma1l r for coul_omb forces. They imply that we can

express ga¡(r) = 1 + er¡(r), trvhere er.(r) will- be finite

for snal-l r¡ and wtII -be snall- for large ri ancl without

these cond.itions statistical mechanics is probably

inapplicab].e here. (li) Triât euo(r) t*->o fot large r

and. for sufficiently smal-l m. (iii.) trrat I I-.rrrractiverco 'O

(") drl> I / €repulsive (o) drl for mixtures, whì-ch inlo

a plasma is a Conseguence of screening between particles.

Now by (i) above i_t is posslble to expand. in porvers

of / for large I.e and with retention of terms involving

3.1+

only small poïvers of þ, equation Ø.2) becomes

Ir + euo (r) 1

ï"nc

It+taø.o(*) + LrB' Ér5"(r) + ...

s+r

oo

-r

1+l

2rr [1-"ue(") ]. It+er"( s) J f r¡" (t) td.t scLs.

l*"1(s.t+)

Changing the variable to y = s-Fe and neglecting e.O(r)

by assumptions (i) and- (ii)r eguatlon (l-4) red-uces to

oo

Fóu¡G) + +P" Øro(r)'2¡r [1-erc(v**) ]+ Ðn

cvraaa t

Y+2r[t + eu"(v+r) I eo"(t) tdt (v+r) dy oa

i, vl(1. s)

Usirrg assumption (ii) it can be seen that the most

important contributions to the right-hand. sicl-e integral 1n

equetlon (l.S) arj_se when y is smallr and- hencee a cut-

off parameter rrarr is introd-ueed., v¡here a(r for large re

beyond- r*'hlch contributions to the integral are assunred-

negligible. F¿rther by assumption (i) the ri$rt-hand-

slde is fj-nite, and. since r is large and- y small , ,u.(V**¡

can be neglected-; so the right-hand- side can be expand-ed-

in powers of É(Y+r) r to give

3.5

9Øu6(r) + LP" ør¡'(n) + ..o = 2ff Ð rtU

t-P d""(v+r)ï

a

-a

+p'øac'(v*") . . . l. (+) I ,'r)Í" eo"(t) tdt dv (t.6)

As there are no gerreral existence theorens for

solutions of non-linear integral equationsr even 1f we

obtain agreement from a eomparison of the left-hand and.

right-hand- sid-es of. eguation (5.6) c ure cannot be sure

an exact solution to eqr:at1on (3.2) exis-bs' Howevert if

we a1e able to satisfy the asymptotic equation (3.6), there

may exist an accurate solution to equation (3,2), whereas

if (3.6) nas no solution, no exact solution of (3.2) can

exist.For a system of partieles involving attractive

forces on1y, it is evid.ent from equation (5.6) that a

solution is irnpossible; f or the. Ér" is always negative¡r Y+2rI r, r -.J , , eattractj-ve (t)at is posltive for smalI y by

tvtphysical considerationse âILC þuø is negatlve. Thtrsr to

first ord.er the left-hand side is negative' while the

right-hand. sid-e is positive; and- to socond- ord-er the

left hand. sid.e is positive, w[i1e the right-hand- s jde 1s

negativer both orders being mathematlcally inconsistent.

Hovuever, bV nod.ifyi-ng the above reasoning for a system

3.6

of repul-sive f orces only, xve see that þ

¡t+2rpositive, while f 6¡"(t)d't becomes

' lvl

become sac

negatlve; SO

that now both f jrst- and second--order agreement in ø

can be obtalrred. For a s)tstem of mi-xed forcese several

cases arise, for þuø can nou'¡ be positive or negative, and

if þaA is positivee so that a particle rrarr repels a

partlcle rr¡n, then particle rrarr can attract a particle

ncr? while particle rr¡tr may repel particle ttctt. BeCause

many of these cases are unphysical, rffe shal-lt fcn

d.efiniteness, consifler mixtures of charged- partlcl-es'

Then, 1f particle rrarr repels particle rbr and- attracts

particle ,,cilr particle rrbr¡ will attract particle rrcrr alsoo

Fon these eharged. particle mi-xtures, f irst-ord-er agreement

follows by the same reasoning as above and. using

assumption (i-ii), ¡ut second order cons j-derati ons lead'

to fllsagreement. The above d.iscussion is Summarised' in

Table 3.1 .

3.7

Ti,BrE 3.1

Surnmary of the consi.stency of the asymptotie eguation

(2.6) for various cases

lype of force Present

Order in

þ^.

Wtre ther (3.6) i"eonsistent to thisord.er

No

No

Yes

Yes

Yes

No

lilttractive only

Repulsive onlY

Ilixtures of Ch,arges

For a T,ennard--Jones type of interpartlcle potentialr which

is repulsive at short d-istances, and which falls off

rapid.ly, the PY equation applies welJ-¡ ard- not only is

a solution to the asymptotic equation (3.6) possiblee

but solutions to the Hf equation (l.Z) have been found.

Howevere for mixed. Coulornb interparticle potentlals an

exact solution of (3.2) is clearly not possible due to

the inconsistency in the second. order tenns of equation

(3.6).

Because of thls d-lfficufty, the ad-d.itional terms

proposed. by Green [t ] were consl¿ered. The lntegral

FirstSecord-

FirstSeeond.

FirstSecond-

3.8

eguation resulting from the inclusion of the first

additlonal term is referred- to as a nrodified Percus-

yevick eguation (Upy) and is d.iseussed in d.etail in

chapter v' tr'or the present we shall observe that by

includ.ing this add-ltional- term the inconsistency 1n the

second-ord-er asymptotic egr:ation for cLrarged- mixtures

is removed. The main d-isadvantage of thls ad-dltional

termisthattheeqrrationcannolongerbeexpressedin

the convenient form of (3.3), ard- so computational solution

of the equation will be correspondingly more d-iffieult'

3.3 oach

Before atternpting to solve the MPY equationt

theauthord.ecid.ed.towriteaFortranprogramtoevaluatethe first-order, or Percus-Yevick termr âs in equatlon

(3.2). This form of the equati on was preferred- t o the

form (l.l)r âs using eguation (3'z) the program could'

be easily expand-ed to incorporate the ad-ditlonal term

at a later stage.

The progran used- to solve the PY eguation is

incorporated. in the final program used t o solve the IIPY'

whichlatterprogramispresented.in.Êr'ppend.lxBgand

thusmostofthepresentd.iscussionalsoappliestothe f lnal- program. It urasdecid.eil- to initially store the

Br¡(*) and- É.¡(r) in intervals of 'l Bohr rad'ius for

70

val-ues of r from zero to three tinres the Debye shielding

d.istance. It v,ras further d.ecid.ed- to t:se the Debye-Huckel

e"¡(r), (i.". BDH(*)), and the Coulomb É"¡(r), (i.e. É"(r))'

in the following form

Bat(r) = gon(") and- /uo(r) = Ó.(r) ror r>2ao

cr¡(r) = BDH(zao) and /.o(r) = þe (zao) for r<2ao

as input data to evaluate the integral on the right-hand

slde of eguation (3.2). This choice of the cut-off value

at 2uo ts identical with that used- for the ItlC calculation

(see 2.1g where the cholce of the cut-off was discussed

fu11y, and ,,vas ref erred- to as ÂO), and t o al1ow a eonplete

eom¡nrlson with the MC resultse the initial data is chosen

for the same temperature (tO4'f ) and. d.ensity (1O'ee/ee).

At first an attempt vras mad-e to evaluate the lntegral

by uslng a proced-r:re suggested by T,yness ltZ1, whlch corrld-

easily be extend.ed. to integrations of higher dircension

without an excessive inerea.se in the number of eval-uation

points. Hovuever, it was found- that this technlEre could'

not be applied. to the PY integral because of (1 ) the

ar¡¡kvuard. range of integration in the inner integnalt ary]

(ff) the non-smoothness of the e'n(r) and. É"(r) that are

used as input d-ata, Iience f inally 1t was d.ecid.ed. to

use a slmple trapezoidal- rule to evaluate the lntegral¡

and to increase the mesh ratio unti] tfp desired' aceuracy

3.10

Tyas attained.

The initial interpolation proeed.ure ad.opted. fon

obtaining S.O(r) and- Ø"O(r) at the mesh points,

from the gab(t) and Ør5(r) stored- at set values of rr l^tas

the usr:al linear lnterpolation. Honrevere it was found.

that this eaused. large errors, espeeial-Iy for sma11 r,

unless a very flne mesh was used, and this proved time

consuming, To avoid. this d.ifficulty the gra(r) and.

dr¡(*) stored. were converted. into logarithns and a 11near

interpolatlon mad-e betvreen the logarithmic val-ues. This

procedure gave reasonabl-e accuracy without taking an

excessive nunber of rnesh points.

.Nr fìrrther d-evice employed in the evaluatlon of

the integral was to divide it into several regiofirio This

al-lovirs the use of d.lfferent mesh ratios in the different

regions, and lt is then possible to see which regions are

most important. The integral of equation (3.2) is also

terrninated- by an upper l1nrit (feQfif ) ot the variable B, so

that the lntegral becomes

T,BCUT 6+rr (r) t.c [eu"(s) 1] cu"(s)s (uo"(t)-1 )t¿tas

{ s-r

It is d.ivid.ed. into reglorrs as shovn: in Fig.3,1 .

LB

t

LA CUÏ

LA CUT

FrG. 3.t SHoW¡NG

r+s

lr-sl

f S LB CUT

THE SEPARATE REGIONS OF ¡NTEGRAT¡ON

OF T DTM.

r

3.11

In rAppendix Bt region a is referred to as the

ttinner regiolltt, region b the "large rtt region, and region

ce the rmain regioïtt. The mesh ratios r:sed. in each

regio' have brackets after them, with (Zn) to ind-icate

that they refer to the two dimensional integral abovee and'

(¡n) to indicate they refer to mesh ratios for the five

d.imensional- term considered' in Chapter V'

Theevaluationoftheintegralfora¡nrticular

value of r shOvr¡ed- that the results are extremely sensitive

to the form of the d.istribution and potential used' as

input (tet us d-end e these oU Ur* and' f,r*)' especiall-y

at snal1 radii, and hence are extremely sensitive to the

cut-off val-ue AO. .lis ttre choice of AO is based- on semi-

classical crlteria, the validity of which has been thrown

into Some dorrbt by the Mc results, it appears necessary

to d.etermine the lnput accurately by taking into accorrnt

guant*m effects 1n d.etai1. Further evaluations of I(r)

show that this is so for al-l values of r from ze?o to LBCUTT

though the de¡nndence is nrost marked. f or r smaIl. The

next ctr;apter will take lnto aecount the quantal effeets at

snall- rad.ii.

3.12

References to Chapter J

tr ] Green, H.s. Ug6Z) prrys. Fluld-s Er1.

lzl stell, G, ?gØ) rrw" ica 29e 517 .

lSl verlet, L. (tg64) Fhysiea 39, 95.

(1965) " L, 959

(tg66) " æ., 3o4.

t4] Verlete T,. and. Levese'rl€¡ D. (lg6l) prrysica &;251+. See also

Levesque, D. (l gø0) rirys ica 32, 1985 "

(i967) " ZÉ, 254"

t ¡ ] t r:-nat, .ii.R . (1966) Phys iea 32 t 133.

t6 ] lTertheim, tr{.s . (1967) J. I\[ath. PL5is. pr 927 "

tZ] Hendersone D. (t167) oisc. Farad-. soe" (ee) !2126"

t8] Broyles, lr,..A-., Chung, S.Uo and- Sairlin, H.I-,. (geZ)

J. Chem. PhYs. if-, 2l+62.

t9] carley' D.D" (1963) prrys. Revn Å-, 1406'

[to ] ïvatts, R.c . (t 968 ) ,r. chem. Fhys . 49, rc .

[11] Broyles¡ .1i..[+. (t960) J. Chem. Phys. 11, L+56, 1068"

l12l T,ynessr J.N. Preprint

l{ustard., D.¡ l,ynesse J.N. and- B}att, J'lvl' (1963)

Comput. Joi:r. !r 75

r,yness, J.N . (1965) uatrrs. comput, 79, 260.

¿+'1

IV A QUANTUI,Í I\,{ECILANICAL CALCUTAT]ON OF TTTE TIIVO PARTICLE

D]STRIBUTION FUNCTION

L.1 Introduction

It 1s necessary to distinguish between two klnd-s

of effects which are commonly referred to aS quantum

effects:

A - The effect d-ue to quan@ which gives rise

to the texchanget terns¡ oF terms arising from the

pauli rexclusiont principle; and which can Jead- to

correlations even 1n the absence of interactions.

B-Theeffectduetotheg@:d-vnamicswhichisd-irectl-y associated with .'t{eisenbergt s uncentainty

princiPle.

As early as 1930 quantum mechanical expressions for g(r)

vïere proposeö by Born and Oppenheimer [t ], Slater lZl,

London lll, Kirkwood. [f+], Uhlenbeck and- Gropper Il],

irligner t6] and- others [Z]. These $Íere applled' to real

fluids i,e. see [B], to obtain the guantum eorrections

to their eguation of state. Experimental evid.ence

eonflrmed. that quantum correcti-ons for fluid-s vrere

lmportant (especlally for Hyd.rogen and Heliun) at low

temperatufes. Extensive reviews of later work have

been mad-e by Ðe Boer [g], Coleman [tO], and- Mayer and

4.2

Band ltt ], to narne but a felv, an¿ recently many papers

d.iscussing guanial effects in fluids have been published-

ltzl"Recently holvever, lvith the increasing interest

1n plasma physlcs, it has been found that quantum

effects are partieularly important at 1ow temperatureso

where for an hyd.rogenous plasma "1ow" temperature

is O(1O4oK), As the quantal effects are also derrsitSr

d.epend_ent, for the case of atomic nurnber z=1 c many

authors use the Debye shield-ing d-istance

KT,|

12\ r where the neutral-ity cond-ition hi=De

Btm^e2(t

appiles i or some plasma length parameter which d.epend-s

on both density and temperature, to obtain a criterion

for the range of irnportance of the guantal effects"

The d.ifferences in the quantal effects for interactions

betlveen various types of trarticles are very large, and

¡D =(

frequentiy reference +

wavelengthÀ=ñ/+\"- " \2r/

s nade to the thermal De Broglie

, It is because of the Presence

of particles with relatlvely small- mass m (eIectrore)t

and- al-so the change of the d.omirrant interparticle

potential from the Lennard-Jones to the coul-ombic type.

that the quantal effects become so important for plasmas

at lovr temperaturesc In thls thesis ïYe are concerned-

4.3

lvith a rather d-ense hyd-rogenous plasmar having an

electron numben density De = 1olge/ec (=tp, the proton

nunber d.ensity), and temperatures ranging from 1O4oK

(pre-ionization) to JxlOaoK 'uvhene the gas is f\-rl1y ionlzed'

and ls a true plasma.

The fact that the classical Coulomb potentlal

É"(r) = uÊ*

has a divergence at the origin, and- that

this d.ivergence can be removed by taklng guantal

consid_erations into account, has also increased the

interest in this field. The three main approaches taken

were: -(") Using tbound.-unboundr state considerations f cnr

interaetions between un11ke larticles. This has

been d-eveloped. reeently by a large number of

authors (see lßl to lzl)), and various criteria

have been proposed for the transitlon from the

bound reglon to the unbound. region. Most of these

references also refer to the degree of ionization

present¡ ând- several (lll (") ], [19 ]' l2Jl) offer

improvements to the Saha egìJation.

(¡) Extending the l/lontroll-Ward. l24l pertu::bation

expansion for plasmas by including guantal terms

in the formurae (see l25l to [zg]).

4.4

(") By applying recent mathematical technigues to

the expressions obtalned for lncludlng guantal

effects in fluids (i.".[t] to [s]). At that

time d.ireet evaluation of the sums involved was

inpossiblee but sophisticated. mathematics arr1

the ad,vent of the computer has novT made a d.irect

evaluation feasible (see references [æ1 to ISS]).

In this ctrapter a quantal expressi on is f ormulated.

for a trro particle d.istribution function which specificallyd.oes not includ.e the effect of other partlcles. The

method takes into aecount the Heisenberg effect only¡

and- is an extension of the Slater sr:m for B.¡(o) [Z].The ex¡rression does not include guantum statistical effects,

as even at 1O4oK the number of el-ectrons approachi-ng

eaeh other closely is expected to be smal-l' Holüevere

there is some evidence from references lzel, lz8] an¿

[¡o] that these are not negllgible for the eleetron-

el-ectron (e-") interactions at short distanees. Quantum

statistical effects fal1 au¡ay as I,4 and. lf2 for the electron-

proton ("-p) and proton-proton (p-p) interactions, where

Iú is the proton to electron mass ratior and so the

effect of statistics should be negligible in the

calculation of gep and. gpp.

The two-partlcle distribution fi:¡rction is

L+.5

evaluated on a CDC 6¿+00 computer to obtait gp" and- g""

over the range of temperatures 1O4 to ix1Oa. Because

of the diffieulties encountered- in evaluating the

Coulomb vvave functionsr gpp could. not be evaluatedt

but due to the large reduced- mass for this systemr the

quantal effect should. be gui-te negligible in this case,

Results are presented 1n Tables 4.1 to 4.11¡ and

correspondirrg graphs are drawn in Figs. 4.1 to 11.8.

These are discussed. 1n section 4.i*r and concluslons madeo

l+.2 A formulation of the e)qcression for g^- (*)

the raÖial- d.istribution functiorr BaU(r) iu

usually defÍned by gab(r) = Dab G)/\ where o"o(n)

is the number d.ensity of prticles of type b at a distance

r fnom a particle of type às urrd % the average nurjben

d.ensi ty of type b throughout the fluid . Hou¡ever,

gru(") "^t also be d.efined. as the ratio of tthe

eond.itional- probability of flnd.ing ¡nrticle b in the

volume element dx(2) given particle a in volume element

ax(1 )r to 'the ind.epend.ent probabillty of find.irrg

particle b in volume ax(Z) and. also particle a in

volume element ax(1)t, i.€.

ga¡(r, - "r¡(') ¿*(t ) ¿T!?) , (4.1)

n, ax(1) % dx(2)

'l+.6

Uslng the usual probability interpretatlon of the

Ìvave f\rnction and assurning that Boltzmanrr statisticsapply to the system, the equation (4.1 ) can be

written for a proton-electron system as:

Ðr, exp(-ÉErr) /lrrrro/fir*+) exp (-Æ'n" /z^) þEþL ,

8pu (") =ì "*p

(-Æ'n"/zn)ú oþ o*

E

(4.2)

where the summation over n sums over the bound states

of energy E' of the h¡nlrogen atom, and. /n1m tre the

lvave fr-mctions normal

The sumnatlon over k

hyd-rogen aton and. the

ised so that I ,",-* úfim dv = { '

sums over scattered states of the

lt- are the wave functions,k

normalised so that úuþl dv=1 . ,l) is the v,¿ave function

of an electron without a proton presentr that ise a

prane urave, but normalised. so that I p^PoËdV=1 " PuttingJ"

in the wave functions as given by Paullng and- Wl1son lsel

for the bound- states¡ and. by Sehiff l3ll tor the

scattered states; and- replacirrg the summation over k

by an integral and removlng the angular depend.ence

we have:

I o

sn" (r ) = [=i, ""n (-PEr')

n-1

1=o

2I+1

L+rr

ú

")

/22\",

417

(n-1-1 ) 3

2n[(n+1)lJ3

exp(-¡3t<,n2 /2m)a2at

exp( -p)pzrr"ili (ù1" . k,lo- "*n çßk2n2/2n)

oo 2L+1

1=o k2r2 ¡ l ["

oo4r(zr)"

(l+.3)

where p - 2rZ/nao (ao being the first Bohr rad-iu^se Z

the atomic number, and n the prlncipal quantum number),

-2I+1"ili'

are associated- T,aguer,re polynomialse and. F1(arkr)

are the Coulomb TÌ¡ave functions wlth cx - Zme2/-n2k fork = wr/lt and m beirrg the reduced. mass of the particles,Evaluating the d.cnominator directl-y gives (zrfgnz/n)-3/2 e^-3

which can be eonveniently e:çressed. j_n units of (nofrr

raali)-5, and. we have

3/2 /z\%

/2trÊh2\î:

3/2 exp (-purr-R)

Tm4

top ( )r oo

fl=1e

n-1 (zt+t)(n+1+1 )l1=o [(^ * 1) 3 j"

"'7?:.1 ror] +

"n(- ) L"(zr+t )

l+.8

Irr(ørt<r) ]2or<I)

+ 2m2ttr2

(4.4)

\nie can interpret eguation (4.4) as being

composed of a normalisation constant (zrr\n'/n)3/',

which nultiplÍes a bound. state contribution (tfre first

term) ad.ded to a scattered. state eontrlbution (second.

term) to give the rad.ial d-lstribution fu¡ction between

two partlcles" The eguation can be applied to e-e and-

p-p interactions equally well but 1n these cases there

are no bound. states. A1so, the Coulonb lvave functions

now refer to the e-e or p-p interactionse ard the

normalisation constant alters due to the change 1n the

reduced. nass m of the system' It should be emphasize'-

that eguation (4,4) refers to the distribution function

between two particles only, and- does not a1low for the

presence of other ¡nrticles.

4.28 Evaluation of E tlon (lr"L)

The bound, state contribution to gpu is evaluated-

by generatlng the associated laguerre polynomials by

two methodse one usirìg recurrence relations oþtained-

from the d-ifferential equation, and. the other using t'he

poïver series prorrided by Paullng and iVilson lSel, these

4.9

provlde a consistency check. The summation over n 1s

terminated- when the contribution from the last nth state

is less than one ten thousandth of the sum to ensure

accuraey to three figr:res, The program was written

to evaluate Sn"(t) for r in intervals of hal-f Bohr radii"Trre parameter p is d.imensionless, "E is the term -Fßn

/2,\whlch i-s expressed. as l3l . 0n nultiplying

\n/the bound. state contributlon by the normalising constant

(totfr expressed in units of Bohr rad-ii) tfre d-imensionless

bound state contribution to Cn"(r) i" obtained-.

The scattering contribution proves a litt1e more

d-ifficult to evaluate¡ ard. differs for the three cases of

proton-proton (p-p) proton-electron(p-e.) an¿ eleetron-

electron ("-")interactions, tr'or the p-e and. e-e eases

the first few Coulomb ï\iave f\rnctions can be generated by

t'wo method-s, one using a power seriesr and. the other using

an asymptotic expansion, d.epend.ing on the range of cr and-

kr; the well known resurrence relatlon technique of

Abramowitz [58] was then used- to generate the functions of

higher ord.er. The summation over 1 is again terminated.

'vr¡hen the last term becomes smallr ârrd- this occurs when

I becomes much larger than t<r+lcrl , for then the Coul-omb

wave functlons falL off very sharply. The integral

is evaluated using a tratrnzoidal rule with upper and lotver

4.',l0

l1mits¡ whichr when d-oubled. and. iralved respeetivelye

failed. to alter the value of the integral by more than

one ten thousand.th of the value of the integral. Initially

an attempt vras mad.e to express the Coulomb wave functlons

in their integral represen-r,ationsre and. to then take the

summation insid.e the integrals evaluate it analyticallyt

and. complete the integration, Unfortunately 1t was

impossible to evaluate tire sum ana1ytica1ly, and although

a eomputer program rras lvritten using this approach, the

final d.ouble lntegration over a sumrnation proved. tlme

consumlng¡ and. thls progran u/as only used. to check the

seattered. c ont nibuti on.

Several points need. to be noted. concernlng the

scattered. term in equation (¿+.4). As cx-)O¡ i.e. forsmall charge or large k, trren i (Zf+t )

1=o

and. the integral can be evaluated. analyticalIy. It can

be eas1ly shor,vn that 1f the sun is of ord.er 1 then the1

maximum val-ue for the integrand. occurs f or k ¡ ({Sn,¡Zo¡-2,

and- the integrand. fal-Is reasonably sharply for smaller or

largen values of k. Even soe 1t is necessary to integrate

cnrer a fair range of k to obtain an accurate result.This in turn means the parameter kr in the Coulomb ftnetions

varieg consid.erably. tr'or the p-e and e-e case the

Ft(arkr) /k"r'-) 1 ,

4.11

reduced. nass m (used. in a and- tÌre weighting term

exp[-Pk'n'/ (z*) ]) l" of the ord.er of the electnon ilâsse

ø remains smal-le a1.Id the range of k is not excessive'

Hovrever, in the p-p case, cr becomes large, the maximum

value for the integrand 1s large and accurate evaluation

of the integral requires a large range of k to be considered"

This in turn reguires the evafuation of Coulomb functions

over an extensive range of ¿ and. kr¡ and so would' reguire

many d.ifferent generating techr:1ques, as shown by

Froberg llgl, Abramowitz [1S1, Slater [Uo] and- others [¿+t ].

Fon this reason the sane method- for calculating the p-p

guantum meehanical d.lstribution f\rnction is not appropriate

here. ft is possible to evaluate the p-p d-istributlon

using simplì-fied lvave functions of the W.K.B. approximation;

but thls simply sholvs that the quantal effects are not

important. On physical grounds, and also from the high

temperature results of references lzîl, Lz}] and lfo1o itis el-ear that the guan'r,um nechanlcal eorrections for the

p-p d.istribution function are relatively smallr ârid. Bpp

lies close to the corresponding classical S"(r).As stated. in Section 1.5¡ it 1s planned to use the

two-panticl-e e-^(r) and. its associated effective poten.tiale"Be'

as inputs to a modified Percus-Yevick equation, to take

into account the effects of other particles. However it

4"12

lvas d-ecid-ed. to obtai-n an approximate estimate of the

shleldirrg of other particles on g(r) inmed.iately by

includ-ing a Debye-Hücke1 shielding factor in the charl{c

so that z in equation (4,4) is replaced. by ze-r/)tn where

b is the Debye sh1eId1ng length. This lnvolves the

approxinatlon that the wave functions obtaj-ned- by

solving the Schroedinger eçluatlon for a Debye-Huckel

shield.ed- potential ane egual to the wave func-ri ons

obtained. by solving the usual hydrogen atom waYe equa',,ir--:,,

with the charge in these $¡ave functions mod.lfied b5t a

Debye-Huckel shield.ing factor, This quasi-classical

approximation seems reasonable and. is supported by receni;

results of Rouse lzll, Harris [16] and- Storer [42].The computer program is given in Appendix Be ûitc-r:

is reasonably economicale one run to obtain g(r) (* going

from zero to 2OO ao in steps of uo) taking approximatel!:

90 second-s on a CDC 6400 computer"

4.3 Results

The results are presented. for five tenperatures

1O4"K to JxlOa"g in intervals of 1O4"K. Tabl-es I¡.1,

4.3r 4.5r 4.7r 1¡.! give logro(Cpe) for the five tenperr

tures, and. for 1Oa"K and. 5x1O4"7{ the f j-rst bound staie

contribution is also given" Tables 4.2, 4.4, 4"6, L!-" B,

4.13

l+.'1 O, are the correspond-ing logl o (9"") values ' f n

the tables on1y, J-ogro (e(t)) l" presented, and'

conseguentty the follolving abbreviations are made in

the tables:

r The rad-ius in Bohr radii

S"(r) Í,ogro of the correspond.ing classical d.istribution

func ti on.

S1"(r) Log I o of the first b ound. state contribution"

Sg(r) - T,oglo of the total bound- state contribution'

N the number of bound. states contributing to the total

bound. state contribution before contribution of a

further state ad.d-s less than one ten thousand-th of

the total bound- state contribution"

E (r) LoBlo of the proton-electron d-istributionoPe

functi on.

s"u(r) LoEro of the electron-electron d-istribution

frrncti on.

e'n(r) Log I o of the corres'pond-ing Debye-Huckel

d.Ístr lbution func ti on.

gr(r) - Logro of the appropriate d-istribution functio;r

includ.ing shielding effects.

sometimes the deseription t guantum mechanicalr is

ad.d.ed. to the d.istribution functions, but largely this is

assumed- understood. The figures were ealeul-ated' with an

4.11t

estinated. error of !5 in the fourth figure.

The Tables 4.1 to l¡.1O are represented graphica]-l-y in

Flgs. 4.1 to 4.5¡ although most emphasis is placed. on

the temperature of 1Q4o7< where the quantum effects are

most apparent. Table l+.11 presents (f or the proton-

electron case) for varj-ous temperatures, three parameters

d.efined. as f ol1ows: -

rJ - the rad.ir:s 1n Bohr rad-ii such that (Sn"(") e"G))/

c (r) is less than.o! for r>r-. This in the tex.b isé3r- z J

referred- to as the t¡oining radiusre as for r>r" the

quantum mechanical curve is within 17" of the

correspond-ing classical curve. Values are not

given fcr. the e-e case as they are very similar to

the p-e values'

I the percentage ionization =R ï

I" Esctrr (")4tt"'¿*x 1OO

where SraOrr(r) is the scattering contributlon to

sn"(r) and- r, is the rad.ius chosen such that

L 2 = electron number densityr (i.". 4nrr"/3 isl+tre.-t r

the volume which on the average contains one electron")'

rRr.i "

gBe(r)4trczdr

4.15

^D

For the present cal_culations r:.sing a numben density

of lOl\e/cc rI has the value of 117.2 Bohr rad'ii.

The Debye shield.ing d.istance in Bohrrradii d.efined.

in the usual manner as Ào = ( kr-)z

'' \Bzrn "Q=

/

Figs. l+.6 and- 4.7 present these results graphically'

In tr'ig l+.7 a compari-son is mad-e wi th the percentage

ionization pred-icted by the Sa-lra equation [4¡]'

Fig. 4.8 shoï,¡s effective potentials v, (multip1ied. by

P to make them dimensionless), for e-e and- e-p

interactions f or 104"11 defined- from the QIlf distribution

function as follows:

s""(r) = exp(-P v""(r))

e"n(r) = exp(-P v"n(r)) a

Hence the effective potential values can be obtained from

the logs of the correspond-ing d.istribution functions by

multiplying then by -2.30259. the classical eurves

S"(r) using the Coulomb potential É"(r)

using s"(r) = exP(-P ø"(r)) 'r,vhere P"(r) = gtfo* e-e case

e2 for p-e case

are also drawn

ar

IABIE 4.1A

Proton-electron d.istribution functions at 104"7<

showi-ng first bound- state and- totat bound- state

contrlb uti ons ,

B"(r) *r"(*) N %(r) *o"(r)r

oo

27.4382

13.7141

9.1427

6.8571

5.4856

4.571)+

3.9183

3.4285

3.O475

2.7428

2.4935

2.2857

2.1Ogg

1.9592

1.8285

1.711+3

1.613l.+

1.5238

g.Bo23

9.3681 2

8.9338 2

B.l+995 2

8.0652 2

7.6309 2

7.1966 2

6.7623 3

6.3280 3

5.8937 3

5.4594 3

5.0251 4

4"5911 4

4.1565 5

3.7222 6

3.2879 B

2.8536 11

2.)+194 14

1.9851 1 B

9.3681

B.g33B

B.4gg5

8.0652

7.6309

7.1966

6.7623

6.3281

5.8939

5.4599

5.0262

4.5931

4.1611

3 .7315

3.3068

2.8919

2.1+960

2.13ù,6

4.16

9.8023

9.3681

8.9338

B.4gg5

8.0652

7.6309

7.1966

6,7623

6.3281

5.Bg3g

5.1+599

5.0262

4"5931

)+.1612

3.7317

3.3074

2.8933

2.4994

2.1421

o.o

O.5

1.0

1.5

2.O

2.5

3.o

3.5

4.o

l+.5

5.O

5.5

6.o

6.5

7.O

7.5

B.o

8.5

9.O

4.1V

g"(") *.,u(r) N s"(r) un.(r)r

9.5

10.0

10.5

11 .O

11 .5

12.O

1z.D

13.4

13.5

14. O

14.5

15.O

1.41+36

1.3711+

1 Õo61

1.2)+67

1.1925

1.1428

1.0971

1.0549

1 .01 59

.9796

.9458

.9143

1.5508

I .1160

o.6822

o.2478

-.1 864

-.62o7

1.8271

1.5877

1.4137

1.2888

1.1947

'1 .11 BB

1 "0539

.9961

.9437

.8955

.8509

. Bogl+

1.8417

1.612o

1.1À89

1.3346

1.2505

1.1842

1.1286

'l .0800

1.0363

.9967

,9602

.9265

22

z6

29

32

311

35

37

39

4o

42

l+3

4l+

4. 1B

TABLE 4.18

Proton-ele ctron distributlon functions including

shleld.ing at 1O4oK and. showing flrst bound state

and. total bound. state contnibutions,

son(r)) (s.'r(')) N (sr(r)) (c*('))r

o.o

0.5

1.0

1.5

2.O

2.5

3.O

3.5

¿+. o

4.5

5.0

5.5

6.0

6.5

7.O

7.5

B.o

9.5

9.0

oo g.Bo23

9.2871 2

8.7726 2

8.2589 2

7.7460 2

7.2339 2

6.7220 2

6,2119 3

5.7O2O 3

5.1929 3

4.6845 4

)+.1768 4

3.6698 5

3.1636 7

2.6580 g

2.1531 12

1 .61+90 15

1.1455 1g

r 6u26 23

g.8o2j

g.2Bg4

8.782O

B.27gg

7.7833

7.2920

6.8o59

6.3252

5.8497

5.3796

4.9149

4.4561

l+.OO39

3.5600

3.1279

2.711+O

2.3297

1.9914

1.]150_

27.2799

13.5662

B.gg52

6.7099

5.3389

4.4250

3.7724

3.2830

2.9024

2.5981

2.3491

2.1],+17

1.9663

1.8159

1.68j7

I .5718

1.4714

1.j821

9.2894

8.782O

B.27gg

7.7833

7.2920

6. Bo5g

6.3252

5.8¡+97

5.3796

4.9149

l+.4560

4. ooSB

3.5597

3.127O

2.7118

2.321+6

1.g\o5

1.69119

4.19

(so"(r)) (s.'r(r)) (s"(r)) s*(r))Nr

9.5

1 0.0

10.5

11 .O

11.5

12.O

12.5

13.O

13.5

th.O

14.5

15.'O

1.3023

1.2305

1.1655

1.1065

1.A527

1.OO3l+

. g58o

.9162

.8775

.841 6

. BOB2

.7770

.1 405

- "3610

-.9619

1 .471+3

1.3108

1 .1 885

1.0926

1.0128

.9432

.88ol+

.8226

.7687

.7181

.67o5

.6255

1.Do6o

1.3548

1.21JJa.3

1.1596

1.O9O7

1.0321

.9806

.93U+

.8924

.9540

.81 86

.7859

27

30

3z

33

35

)o

3B

39

¿+o

¿+t

43

lr4

4.n

TÁBLE ¿+.2

The el-ectron-electron distribution furatlons at 104"K

r sc (")

-oo

8""(t) *on(r) sr(r)

0.0

o.5

1.0

1.5

2.O

2.5

3.o

3.5

4.0

\.55.O

5.5

6.0

b.5

7.O

7.5

8.0

8.5

9.O

-27.1+282

-13.71/+1

9.11+27

6.8i71

5.1+8ffi

- 4.5714

- 3.9183

- 3.4285

- 3.0l+76

- 2.71+28

2.4935

2.2857

2.1 O9B

1.9591

1.8285

1.7143

1.6131+

1.5238

-3.1+176

-3.2116

-3.O192

-2.8411

-2.6769

-2.5255

-2.3858

-2.2569

-2.1377

-2.O27'l+

-1.9252

-1.BJ04

-1 .71+23

-1.6605

-1.5845

-1.5137

-1 .L+l+77

-1.3862

-27.2799

-13 Õ566

8.9952

6.7099

,.3389

- 4.'l+25o

- 3.7724

3.2830

2.9O2ù,

- 2.5981

- 2.3l+91

2,'1417

1.9663

1 .8'1 60

1.6857

1 .5718

1.1+714

1 .3821

-3.4017

-3.182O

-2,9776

-2.lB9O

-2.6155

-2J+559

-2.3O90

-2.1731

-2.O479

-1.9322

-1.8251

-1 .7263

-1 .631i+

-1.5493

-1 .4703

-1 "3967

-1.3284

-1.2648

4.ä

ec (r) 8".(") *on(r) sr(r)r

10.5

11 .O

11.5

12.O

12.5

13.O

13.5

14.O

1l+.5

15.O

1 ,3061

1"21+67

1.1925

1.1428

1.0971

1.0549

1 .O159

o.9796

0.9458

o.9143

-1 .225l.1

-1.1787

-1.1346

-1 .0935

-1 .O55O

-1.01BB

-o. g84g

-o.9530

-O.922l-+

-0. Bg4o

1.1655

1.1065

1.0527

1.OO3l+

o. g58o

o"9162

o.8775

0. B¿+16

o. Bo82

o "7770

-1 .0992

,1.O513

-1 .0066

-O,961+7

-o.9252

-o.8885

-0. B54o

-o.821 5

-o.791O

-o.7622

l+.22

TABLE 4.J

Proton-electron d.istribu.tion functions at 2x1O4"K

r s.(r) ep"(") son(r) sr(r)

o.o

o.5

1.O

1.5

2.O

2.5

3.O

3"5

4.0

4"5

5.O

5.5

6.o

6.5

7.o

7.5

8.0

8.5

9.0

9.5

10.o

13 "7141

6,8071

4.5714

3.4285

2.7428

2,2857

1.9592

1.7143

1 .5238

1.3714

1.2467

1.11+28

1.0549

"9796

.9143

.8571

. B067

.7619

.7218

.6857

5.92lJL

5.4901

5.O559

4.6218

4. t BB4

3 "7563

3.3275

2.9057

2.4978

2 .116l.+

1.7791

1.5033

1.2947

1"1436

1.0331

.9486

. 8806

.8236

.7745

"7315

.6934

oo

13.6616

6.8047

4.5191

3.3764

2.6907

2.2337

I "9073

1 "6625

1 "4721

1.3198

I "1953

1.O915

1.0036

.9281+

"8632

. 8061

"7558

.7111

"6711

.6351

5.92U+

D "4606

5. OOO4

r+.5439

4.0915

3.6U1+3

3.201+3

2.7760

2.3677

1.9931

1.6698

1 .4115

1 .2187

1 . O7B4

.9741

"8928

"8264

,7704

.7219

.6794

.6Ì+16

oo

r

10.5

11,O

11 .5

12.O

12.0

13 "O

"6531

.6234

.0963

.5711+

.5486

,5275

.6592

"6284.6003

.57'l+B

.5511+

.5298

.6o25

.5729

.51+59

.5212

.498¿r

.l+771+

4"23

"6077

.5171

.51+93

.5239

.5OOl+

,4792

gc(") en"(r) so(t) s.(t)

4.24

TABLE 4,4

El-ectron-electron d.istribution functiorrs at 2x1Oa"I<

B"(*) e""(t) *on(r) s.(r)r

o.0

O.5

1.0

1.5

2.O

2.5

3.O

ztr). )

4"0

4.5

5.o

5.5

6.0

6"5

7.o-7trl.)

B.o

8.5

9.0

-oo

-13.711+1

6.Bj7l

- 4.5714

- 3.4285

2.7428

2.2857

1 .9592

1.7143

I "5238

1.3714

1.2+67

1 .1428

1.0549

o.9796

O.911+3

- 0.8571

- o"B067

- 0.7619

-2.4641+

-2.2613

-2 "O75

-1.9065

-1"7546

-1 "6183

-1 .4959

-1 Õ861

-1 .2875

-1.1g\g

-1.1192

-1 "0475

-o.g\2g

-o.9247

-o.8719

-O.821J,2

-o " TBog

-o.7415

-oo

-13.6616

6. Bo47

- )+"5191

- 3.376+

2.6907

2.2337

1"9073

1 "6625

1.4721

I .3198

1.1952

1.O915

1.0036

o.9284

o.8632

- o.8061

- o.7558

- 0.7111

-2.)+560

-2.2461

-2. O54

-1.B8Og

-1 .7251

-1 "5854

-1 .l+601

-1 "3479

-1.2480

-1.157O

-1 "0759

-1.OOJO

-o,9374

-o.8783

-0.8260

-o.7689

-o "7329

-o.6932

r

4,25

g"(*) e""(t) son(t) sr(r)

9,5

10. o

10.5

',l1 ,o

11 .5

12. O

12 "5

13.O

13.5

14.0

14.5

15.O

-o,7218

-o"6857

-o.6j3o

-o "6234

-o.5962

-o.5714

-0.5485

-o.5275

-o.5079

-O.l+898

-O"/,1729

-O,1+571

-o "7055

-o.6727

-o.6426

-o, 61 49

-o "5892

-o "j656-O "51+38

-o "5236

-o " 5049

-o.4875

-C.1+729

-O.l+714

-o.6711

-o "6351

-o "6025

-o "5729

-o "5459

-o.5212

-o,4gB4

-o "4771+

-o.45Bo

-A")aJi)

-O "l+231

-o.4075

-o.6j69

-o "6238

-o.5935

-o.5657

-o.53gB

-o "5164

-0.4946

-o,4743

-o"4554

-o.4379

-o,4215

-o.4063

l+.26

TABT,E ¿+.5

Proton-electron distribution functions at 3xlOaoK

g"(*) er(r) (son (r) ) (c.(r) )

l+,5189

4.0682

3.6203

3 "1771

2.7421

2.3221

I .9303

1 .5855

1 .3061

1"0971

,9470

" BJBO

.7551

.6892

.631+6

.5884

.5485

"5137

.4830

.4557

"4312

.4092

r

o.o

0.5

1.O

1.5

2.O

2.5

3.O

3.5

4.O

4.5

5.0trÊ-)o)

6"o

6.5

7.O

7.5

8.0

8.5

9.O

9.5

10 .0

10.5

9.1427

4.5714

3 "0476

2 "2857

1.8286

I "5238

1.3061

1 .1428

1 .O159

"9143

.8312

"7619

"7033

.6531

.6o95

.5711+

.5378

.5079

.4812

.\571

.4354

4.5189

4.0851

3.6511+

3.2194

2"7925

2.3771

1.gB5B

1.6374

1.3514

1.1358

.gB1 1

.869¿+

"7852

.7185

.6635

.6i70

"5770

.5420

.5112

.4838

.4593

"4371

9.111+2

4"5428

3.O191

2.2572

'l . Booo

1 .l+95/.+

I .2778

I .1146

o.9876

o. 8861

o. BoSo

o.7538

o.6752

o "6251

0.581 6

o .1+35

o.5099

o.48ol

o "4534

o.4294

o "t+o77

oo oo

r

4 "27

s"(r) e*(r) (son(") ) (s*(r) )

"4156 "4170 0.3879 "389211 .O

l+"23

TABI,E 4.6

Electron-electron d-istributlon functi ons at 3x1 04oK

B"(*) e""(t) eon(*) er(r)r

0.0

O.5

1.0f,-l.)

2.O

2,5

3.O

3.5

h.o

4.5

5.O

5"5

6.0

6.5

7"O

7.5

B.o

8.5

9.0

9¡5

10 .0

-oo

-g "1427

-4" 5713

4. 0476

-2 "2857

-1 "8285

-1 "5238

-1 Õ061

-1 .1428

-1 .O159

-o "9143

-o.8312

-o.7619

-o "7033

-o.6531

-o.6095

-o.5714

-o.5378

-o.5079

-o.4812

-o.4571

-2" 01 90

-1 ,81 85

-1 .6376

-1 .l+77O

-1 .3354

-1 "2112

-1,1024

-1 "OO71

-o "9238

-0" B506

-o.7864

-o.7299

-o.6801

-o.6361

-o "5968

-o "5621

-o.5305

-o .5024

-O "l+769

-o.4538

-9.1142

-)+.5428

-3. 01 91

-2.2572

-1.800

-1 .\954

-1 "2778

-1 .1146

-o.9876

-0. BB61

-o " SoJo

-o.7338

-o "6752

-o.6251

-o.5816

-o " 5435

-o.5099

-o.4801

-o.4531+

-o "4294

-2 "0134

-1 " B0B¿+

-1.6239

-1 ,l+603

-1 "3166

-1 "1906

-1.OBO2

-o "9839

-o.ggg5

-o.8257

-o.7609

-o.7039

-o "6538

-0.6094

-o.5701

-o "5322

-o " 5011

-o.4752

-O " l+496

-o.)+265

4.zg

r s"(r) s""(") so'(") s*(r)

10,5

11 .O

11 .5

12.O

12.5

13.O

13.5

14.0

1l+"5

15.O

-o "4354

-o "4156

-o "3975

-o.3Bo9

-o.3657

-o "3516

-o.3386

-o.3265

-o "3153

-O,5Ol+B

-o.4328

-0.41 58

-o "3959

-o "3797

-o. j648

-o.3509

-o.3382

-o"3263

-o "31 52

-O " JOI+B

-o,4077

-o "3879

-o "3699

-o.3533

-o "3382

-O,321+2

-o "3112

-o "2991

-o "2879

-o "2774

-o,4054

-o.3862

-o"3686

-O.JiZLt,

-o"3376

-O "32-t8

-o "3111

-"o "2993

-o " 2BBJ

-o "2777

4.n

TÁBLE 4.7

Proton-el-ectron distribution fÏnctions at l.rx1O4oK

s"(r) (e(r) ) (son(r) ) (er(") )r

0.0

O.5

1.O

1.5

2"O

2.5

3.O

ztr

4.0

4.5

5.0

5.5

6.o

6.0

7"O

7.5

8.0

8.5

9.0

9.5

6.8571

3.4285

2.2857

I .7143

1 .3711'+

1.1428

"9796

.8571

.7619

.6857

.6234

.5714

"5275

.4898

.l+571

.l+286

.4054

.3809

.3609

3.-7622

3.3300

2.8975

2.)1709

2.0594

1 .6800

1 .3559

1.1O44

.9241

.7975

"7055

.6351

,5787

.5320

.4925

"4587

.4293

.4036

.JBOB

.3605

6. BfB4

3.1+1OO

2.2671

1 .6958

1.3529

1"12U+

o.9612

o.8387

o.7l+35

o.6674

o.6051

o"5531

o.5092

O.l+715

o.4JB9

O.l+1 04

o "3852

o "3628

O.3l+28

3.7622

3.3184

2.8768

2.1+436

2.0284

1 .61+58

1.3268

1 "0791

.go2o

.7774

.6865

.6168

.5608

.5145

.l+753

.A+17

.41'11+

.3867

J640

.3437

co oo

4.31

TABr.Ð 4. B

Electron-electron d-istribution functions at /ax1 O4oK

r s"(n) 8""(*) con(r) s*(r)

o,o

O.5

1.0

1.5

2.O

2.5

3.O

3.5

4.0

\.55.O

5.5

6.0

6.5

7.O

7.5

B.o

8.5

9'0

9.5

-oo -oo

-6 "8571

-3 "l+285

-2.2857

-1 .7143

-1 .37111

-1 .1428

-o.9796

-o.8571

-o.7619:o.6857

-o .6234

-o.5714

-o.5275

-o.48gB

-o.4571

-o.4286

-0.40J4

-0. JBog

-o "3609

-1 .7457

-1.5475

-1.3714

-1 .2179

-1 .0852

-o "9713

-o "8737

-o.7901

-o.7185

-o,6569

-o,6oJB

-o.5579

-o.5179

-o.4B5o

-o.4522

-O.421+9

-o,4005

-o.J78B

-o,3592

-6. BJBh

-3"41OO

-2.2671

-1.6958

-1 .3529

-1 .1211+

-o.9612

-o.B3B7

-o.7435

-o.6674

-0.6o51

-o.5531

-o.5092

-o "4715

-o.43Bg

-0.4'104

-o.3852

-o.3628

-O,31+28

-1 .7414

-1 .5400

-1 .361',1+

-1 .2059

-1 .O718

-o.9568

-0. B5BJ

-o.7740

-o.7o19

-0. 6400

-o.5866

-0.5406

-o.5oo7

-o.\662

-o.4345

-0.4071

-o.3B2B

-o.3610

-O.31+15

r

4.32

s" (r) s""(r) son(") sr(r)

'lo.o

10.5

11 .O

11 .5

12.O

12.5

13.O

13.5

1l+.0

1l+.5

15.O

-O.31+28

-o.3265

-o.3117

-o.2981

-o.2857

-O.271+3

-o.2637

-o.2540

-o.2\\g-o.2365

-o.2286

-o "3417

-o.3257

-o.3111

-o.2978

-o.2855

-O.271+1

-o.2636

-o.2539

-a.2U19

-o.2366

-o.22Bg

-o,3428

-o.3085

-o.2936

-o,2801

-o.2677

-o.2563

-o.2458

-o.2360

-o.2270

-o.2186

-o.2107

-o.3238

-o.3o7B

-o.2931

-o.2797

-o.2675

-o.2562

-o.2457

-o.2361

-o.2271

-o "2187

-0.2108

4.33

TABLE l+.9A

Proton-electron distributi on furrctiorrs al 5x1O4oK

showing first bound- state and. tota] boi:nd. stateeontributions.

(s"(*)) @.,"(r)) IT (su(*)) (s"-n(r))r

o.o

O.5

1.0

1.5

2.O

2.5

3.Q

3"5

4.0

4.5

5.o

5.5

6.0

6.5

7.O

7.5

8.0

8.5

9.0

,.1+856

2.7428

1.8286

113711+

1.O97

o.9143

o.7837

o.6857

o.6095

0.5486

o "4987

o.4571

o.4220

o .3918

o.3657

o.3429

o.3227

o. Jo4B

3.2761+

2.8371

2.1+O2B

1,9685

1 .5342

'1 .1000

o.6656

o.2313

-.2O3O

-.637i

2. B4¿+0

2.4120

1.gB7B

1 .5838

1.2217

o.9252

o.7a2o

o.5363

o.4056

o.2g4B

o.1973

0.1114

o "0369

-.0271

-. OB2l+

-.1312

-.1756

-.2177

3.2760

2.8470

2.4166

I .9984

1.6099

1.2766

1 .O197

o.B3B5

o.7137

o.621+8

o.5576

o .5045

o. h6'1 1

O.42,l+8

o.3g3B

o.3672

O.5i+40

o.3235

o.3054

oo

B

9

12

16

20

2l+

2B

3o

3z

311

36

39

41

l+3

44

45

46

47

l+.31+

TABLE 4.98

Proton-electron distributi on fu¡ctions inehd.ing

shiel-dlng at JxlOa"K andt showing the total bound. state

contribution.

r

0.o

0.5

'l .o

1.5

2.O

2.5

3.O

3.5

4.0

4.5

5.O

5.5

6.0

b.5

7.O

1trl.t

B"o

8.5

9.O

son(r) sr(r)

3.2761+

2.835)+

2.3969

1.9683

1.5621

1.1995

o.9035

o.6Bo7

o.5152

o.3845

o.2734

o.1750

0.0875

0 .01 06

-.0563

- .1143

-.1654

-,2116

-.2545

s.(r)

3.2760

2.8384

2.4O16

1.9793

1.5892

1.2569

1.OO21

.8228

.6992

.6109

.5¿y'+o

.l+912

.U179

.4115

.5806

.3540

.3308

.31C5

.2921+

N

oo I

I

9

12

16

20

24

28

3o

32

34

36

38

4o

42

¿+4

\5

46

t+7

5 "l+724

2.7296

1 .8153

1 .3582

1.oBJo

o"go11

o.7705

o.6725

o .5964

o ,5354

o.4856

O.¿+440

o.4o8g

o. JTBB

o.3526

o.32gB

o.3097

o.2917

l+.3 5

ÎÁBLE 4.10

El-ectron-electron distribution f\rnctions at 5x1O4"K

s"(r) 8""(*) *on(r) s.(r)r

0.o

o.5

'l ,o

1.5

2.O

2.5

3.O

3.5

4.0

l+"5

5.O

5.5

6.0

o.)

7.O

7.5

B.o

8.5

9.o

9.5

-oo -oo

-5.¿+864

-2.7428

-1.8285

-1 .3714

-1 .0971

-O.91i.+3

-o.7837

-o.6857

-o.6095

-o.5486

-o.4gB7

-O.ì+571

-o.4220

-o.3918

-o.3657

-O.31+29

-o.3227

-0. Jo4B

-o.2BB7

_4 ÃtrÃÃt . J2-/-/

-1 .3596

-1.1879

-1.O4OB

-o.9161

-o. 81 '1 1

-o.7229

-o.54Bg

-o.5865

-o "5338

-g.la889

-o "4505

-o.4174

-o.3BB7

-o.3635

-O.31+13

-o.3214

-o.3o3B

-o.28Bo

-5.1+724

-2.7296

-1.8153

-1 .3582

-1.0839

-0.901 1

-o.7705

-o.6725

-O.5961+

-o "5351+

-0.4856

-o.4440

-o.4o8g

-o. JTBB

-o.3526

-o.3298

-o.3097

-o.2917

-o "2757

-1.5521

-1 J536

-'1 .1 BOO

-1 .0314

-0. 9058

-0. Bool

-o.7113

-o.6369

-o .5742

-o.5213

-O.l+763

-o.4379

-O "l+046

-o.3758

-o.3507

-o.3283

-o "3087

-o .291 1

-o 1753

4.ß

s"s(r) son(r) sr(r)r

10 .0

10.5

1'l .o

11 .5

12.O

12.5

13.O

13.5

14.0

14.5

15.O

s" (r)

-o.2743

-o.2612

-o.2493

-o.2385

-o.2285

-o.21 94

-o .2110

-o.2032

-o.1 959

-o.1 892

-o.1 B2g

-o.2739

-o.261o

-o.2493

-o.2385

-o.2285

-o.2195

-o.2111

-o.2033

-o. 1 960

-o.1893

-o.1 B3O

-o.2613

-ç,.21+83

-o"2364

-o.2z56

-Q.2156

-o.zo65

-o. 'l 98'1

-o.1go3

-o.1831

-o "1763

-o.1 7OO

-o.2611

-o.2481

-o.2363

-o.2255

-o.2155

-o.zo66

-o .1982

-0. 1 go4

-o.1832

-o.1765

-o.17O2

l+.37

TABT,E ,l+.11

The varlation in the Debye shieldlng length' the

jolning rad-ius and. the percentage ionization with

temperature for shield.ed. and non-shield-ed. calculati ons o

TE}[POK

104

SHIELDING

NO

YES

NO

YES

ITO

YES

ITO

YES

NO

YES

NO

YES

NO

YES

NO

YES

NO

YES

's13.O

13.5

10 "5

10 .0

9.5g.o

8.5

8.0

7.O

7.O

6"5

6.5

5.O

5.O

4.5

l+"5

I

.04

.06

12.1+3

14.55

42 "76

46. BO

71 .O4

74.6',1+

go"BB

93 "06

95 "16

96.68

97.U+

98.34

gB "25

98. 85

ID

1.Jx1Oa

1 .7jx1Oa

2.Ox1Oa

2.5x1 oa

JxlOa

llx1Oa

jxlOa

Bx1 Oa

92.21

11 2.94

121 .99

13O.)+1

1 45.80

159.72

18\.L¡J

206.20

260 "82 3.O 99.16

6

2xl}a

ctassrcal(4 r lo1)

classlcal

t-gl

oENo

( 101)I

FrG. 4. 1

rcal

o1

lst bound state(lo1 ) f

\

(z )

2D

tate

4r Iú

510r (Bohr Rodii )

PROTON ELECTRON RAOIAL DISTRIBUTION FUNCTIONSSHOWING CONTRIBUÎIONS FROM THE FIRST BOUNOSTAT E ANO TOTAL BOUNO STAT E S FOR VAR IOUSTEMPERATURES. THE CORRESPONOING CLASSICAL RAOIALOISTRISUTION FUNCTIONS ARE ORAWN FOR COMPARISON.

Fr (Bohr Rcd ii)

((\f

ru

o$ o$. Ic1lL 9c

OAO K

vARlotJs

CORRESPONDING

1

úr

oolo

-2

I

-6

¡G. 1-2F Q.M. ee OISTRIBUTION FUNCTIONS FOR

TEMPERATURES COMPARED W]TH THEIR

CLASSICAL O¡STRIBUTION FUNCTIONS.

t

IEBYE HUCKE

THEWITHWTH

9pe(r)ì

lst bound statecontlì

\

6

l¡gì

oElo

L

2

\

9s(r)

\

boundcont0

5 l0

FrG. 1.3

r (Bohr Rodíi)PROTON ELECTRON RADIAL DISTRIBUTION FUNCT]ON

A SHIELOING FACTOR TNCLUDE0, S5(r), S COMPAREO

9ps (rl ANO THE OEBYE XÜCXEI- 0ISTR]BUTI0NFUNcTloNs FOR 104

ox.

7

5

¡-(,¡

o)

ï¡cC'

q,o-

cn

tllltr

3

Itl\

\\

¡l9p.(r )

9' (r)

Ip

\I

\\

\\ \

\\\

shietded bountt--\state cmtl

\ qon- sh ietd ed Þound state cont4\./

T¡

0

FtG. ¿.5

50 100r ( Bohr Rodíi)

THE SHIELOED ANO NON. SHIELOEO OISTRIBUTIONFUNCTIONS, WITH THEIR CORRESPONDING BOUNDSTATE CoNTRIBUTIoNS, TO LARGE RAOil AT 104

oK

TempOK

5r10¿

3¡ lOa

D¿

sh ietd ed {

-o

5 r.¡ (Bohr Rqdii)

JolNlNG RAOIUS vs TEMPERATURE FoR SHIEL0EO AND

NON - SHIELDEO CALCULATIONS.

l0

FrG. 4. 6

I

O

I

@I

lI

r00

c.ocl

.Nc.9

(ugìE

cout¡,,CL

60

20

FlG. t,.7

Saha

s h ietd ed

3'10 Itemp. (degree Ketvin )

PERCENTAGE IONIZATION GRAPHS FOR

NON - SHIELOED CASES WITH THE

SAHA CURVE FOR A PLASMA OF

SHIELDEO ANDCORRESPONDTNG

DENSTTY 10l8 e/cc.

I

l+'!8

4.4 Dlscussion

The Tables 4.1 - 4.10 with Flgs. l+'1 - 4.5 show

that for both p-e and e-e pairings, the S*(r) n'ns

smoothly onto g"(r) at a certain joining rad.ir¡s rJ.

Belor r" there is a marked. d.ifference betweet ge¡,f(*)

and. s"(r). At r=o the Ql.[ eurve tends to a constant

(approximately equal to the f jrst bound. state contributio

of (zrrpnr/n¡3/z exp( 15.78o/rxto-4)/rr tor temperatures i :

below l.rx1O4"K in the p-e case) while the cl-assical cr¡rve

approaches infinity. For small r and lovr temperatures

the quantr:rn rnechanical p-e curve lies close to the first

bound state contribution, i.€. (zrr¡n' /n)3/'.*p[ (t ¡. 78o/

(trtO-4) )-Zrl/r, (where r is in Bohr radli), whilst the

correcponding classical curve

s (o) = exp [ $.156/ (rx1o-4x2r) ]Þc\-

falls away much more sharply. As the rad.ii increase,

other bound states and scattered- states stant making an

appreciable contribution to 8p" (") t until at r" it

effectively joins the classical curve' As can be Seen

from the graphs the e-e case is essentially slmilar, but

in this ease no bound states exist, and- the log e"(r)goes to minus lnfinitY,

Figs. l+,'1 and 4"5 show ttrat for tfìe p-e ease the

bound state eontribution 1s guite 1arge, especially at

l+.39

Iew tenperatures, and for 1O4oK even at 50 Bohr rad.ii

the bound. states contribute 1Bf, of Sn"(r) and' stil1

contribute 11% at 1OO Bohr radii. the value of n at

which the bound--states sum terminates is also of intereste

and for 1040K at,'10 Bohr rad-i ir26 terms Ïvere needede -

at !O Bohr radii, 84 terms, and at 1OO Bohn rad-iie 11O

terms contributed. In a simil-ar fashion the number otr

terms contnlbuti-ng to the scattering States rose as the

rad.ii increased.

The temperature d'epend-ence of the both the g""(r)

and. gn"(r) is also evident from Figs. h.1 and 4.2" As

the temperature is increased. the guantal curve becomes much

closer to the classical curve¡ and. for the p-e ease the

bound. state contributlon fa1ls off nueh fasterr âId

the contributlon of the first bound state is less important'

Comparison of gp. (") and. g""(r) with recent results

obtained- by Sto:..er tl¿+] and- Sto:er and- Davies [æ] give

agreement to 15 in the fourth figr:ree v¡hich is less thafl

the estlnated. ernor for these calculations. It should

benoted-thatathlgþertemtreraturestharithosecalculated.henee i.e. >5xl OooK, gp" (O) contains important contribut-

ions from n>1 statesr a feature shown clearly by the

results of Storer.

Fig 4,6 shows that the joining radlus r" fa11s off

sharply as the temperature increases from 9x1 03 oK to

4.1+o

JxlOa"K but at higher ternperatures varies only sllghtly.

there 1s no obvious analytlcal d-epend-ence of rJ on

ternperature.

The inclusion of the approximate shield.ing factor

1n the results of Figs " 4.3 arxl 4.1+ show there is little

effect on the general shape of the curvee but that it

causes an appreciable change in values, So that now the

shiel¿e¿ Sg(r) ¡oins its respective tD.H.t curve above

a certain rad-ius. Frorn Flg 4.6 it can be seen that

this joining rad.ius (aefined as before, except now the

eriterion is that er(r) approaches within 57o of g'n(r),

not g"(r) as before) only dlffers from the non-shiel-d.ed.

case at temperatures belc,r/It 3x1O4"K. The effect of the

shield.ing 1s more pronounced. on the total and first

bourd- state contributions, and. tr'ig. l+.5 shows that these

fa1] off appreciably faster than the non-shielded- câ'see

especially at large rad-ii. In Flg. 4.5, because the

classical curve is nearly id-entical with 8--(n), and-Pv

slmilarly since son(r) remains so close to sg(r)r the

B"(r) and. 5o(r) are not drawn.

As the quantum rnechanical erçression for gp"

cllvldes it into bound. and scattered state contributions,

1t is possible to obtain the percentage lonizat'ion present

in the hyd.rogen gasy and Fig. 4.7 shows that this d'iffers

only slightly f or the non-shie1d.ed. and- shield-ed casest

4'l+1

The effect of shield.irrg is to increase the ionization

two or tlrree per cent, which is to be expected.r âs the

shield.ing preclud.es some of the bound. states. The

results agree guite closely with those of Saha tl+l], the

main d.isagreement being iust above 1.5x1O4"K where the

non-shield.ed. ionization value is only half Sahars

value and even the shield.ed. value is 15î4 bel-ow. Also

by Sahars theory between 'l .5x1o4"K and 2xlO4oK, la9'7 of

the the ionization occurs, while the quantun mechanical

calculation gives 59/" without shielding, ana 6Oíi with

shie1d.1ng. A,t 2.5x104oK the non-shie1d.ed, theory implles

there are twiee as rlany neutral partlcles as pred.icted.

by Sahat s results. Sahat s origirral- theory, see [¿+l+]

aIl-ovr¡ed only for lov/er bound- states in an approximate

manner, and- has since been improved- by a number of

authors [tZ(c) ] and- lSnl, to include hlgher bound states,

and. some attempt has been mad.e to also a11ow for shield.ing

effects lZl(c)]. The d-egree of ionization obtalned- fron

these refinements is sti11 surprizlngly close to the

val-ues obtained by Saha.

Fig 4.8 shows that the effective potentials obtained

by al.lowing for guantal effects d-iffer from the

classical Coulomb potential at small rad-ii, and are

finite at the origin. As the temperature increases

the effective potentials become eloser to their

4.42

corresponding coulomb potentials ¡ while rernaining

finite at the origin, the Vr"" merge wlth their

corresponding coulomb potentlals at smal-1 rad.ii. The

d.if ference between tfre v""(r) arrl the V", (r) is most

marked, In the e-e case the lnclusion of the q''anta1

effect eonsiderably reduces the repulsive Goulomb

potential; tvhereas for the p-e case there is an

increase 1n the attractive potential frofl r=P" to

ll=1.5ao, then a reduction of the attractive potential

f or r=1 .JÐ.o f,6 ¡=0.

Inconelusiontheresultsareofimportancebecause they show the rather large deviations of the

two ¡n.rt1cle quantal d.lstrlbution functions from the

classlcal- Coulomb theory at short lnterpartlcle d'istances,'

and. because they ind.lcate that belorn¡ r¡' guanturn mechanical

effects become lrnportant f æ the p-e ard e-e câsese

especially at low fempenatures. unfortr:nately the

complexity of evaluating Coulomb $'ave functions over

large ranges pr"eclud.es the cal-culation of gnn(*), lut

the guantum effects should- be small for this case. An

approxirnate al-lowance for shield.ing indicates that

results are gualitatively the sane, both for e-e and.

p-e câsêse as for no shieldingr but the gr(r) tend' to

the correspond.ing go'(") instead- of joining e"(r) "sfor the two particle (i.e. non-shielded.) case. The

4.113

incl-usion of shielding causes changes j-n the d'egree ol

lonization present¡ but the d.egree of ionization remair''s

remarkably cl-ose to values obtained- from the saha and

improved Saha equatiorlso Al-so from the p-e results

one can see some iustification for consid.erir:g the

f irst bourd. state as the ma jor contrilcution to the

bound. states, especially at ternperatures befcu¡ 3x1O4oIl

(ror the density lotse/cc). The d.ecrease in the

repulsive coulomb potential for the e-e interactions is

most marked., and in contrast to the p-e case' Ve-e

neverenlrancestheCoulornbpotential'Theinc]-usion

ofqrrantumstatistieswou]-dhavemosteffectonthee-eintenactlorrs (see 133D, but should be smal1 relative

to the guantum effect calculated'

l+,l+l+

References to Chapter l+

tr l

lzl

13l

t4l

t¡l

t6ltzl

tBl

tglIto]Irt ]ltzJ

Born, M. and. Opperrheimer, E. (1927) Aott. Physik

&, t+57.

Sl-ater, J.C. (lgSl ) prrys' Rev. fr-, 237.

(1933) J. chem. Phys. fr 687

lond.on, F. (lglo) z. Physlk. 63, 245.

Kirkwoode J.G, (lg3z) Phys. zeits 33t 39

(1933) Phys. Revo 44, 31.

Uhlenbeck, G.E. and. Crropper, L. (lglZ) fnys' Revo

41' 79

Wigner, E. (lWZ) Pnys. Rev. 40, 7l+9

Bethee E. and. Uhlenbeck, G.E. (1937) pfrysica &'

91 5.

Margenau, H. r (glo) Phys . Rev. &, 1782.

(lgSl ) prrys . Rev. il' I o14 and 1l+25;

&., 747 and 1785 "

DeBoere J. (tg¿rg) Report Progr. PtrSrs. 72, 3o5.

Colemane A.J. (1963) nev. Mod. Phys. fr, 668.

Mayer, J,E. ârld Band., IÛ. (lgt+l) Jo Chem. Phys. L2r141 .

In the last ten years papers have appeared. by

Isihara, A.iMontro11, E.VY. i Lee -, T.D. and- YangrC.N';

tr'ujitae S. and. Hirota; Imre ¡ K.i Ozizmir, E.i

Rosenbaum, M. and- Zweifel , P.i Lee, J.C. and-

Broyles, A.A, ; Hiroike g K. i Jaen, J.K. and

Khan¡ A.A,; A1len¡ K.R. and. Dunnr T.i Lad.or F.!

4.1+5

and. T,arsen, S.Yn and. manY otters.

l13l Fermi, E. (192ì+) z. Phys. ?É, 54. Assumes bor¡¡rd.

states d.o not exist if their energy is >

potential energy of tt¡¡o ions at an average

distance apart.

[14] Feix, i\l.R. in 'Proceedirrgs of the Sixth International

Corrference on Ionlzation Phenomena in Gases'r

T. Hubert and E. Cnemin, Ed.s. (Bureau d-es Ed-itlonsr

Centre d.'Etudes Nucleaires de Saclay, Parisr 1964)"

Vol. I p.185. Treats the classical electron gas

by a random phase approtimationt this hou¡ever

fails at sma11 interparticle d-istances.

l15l Von HagenoTr, K. and Koppe, H. In the conference

above Votr. I P.221 ,

[16] Harrisy G.lü. (1959) J. Chem. Phys. il-, 211..

(lg6z) enys. Rev. 125e1131

(1964) Æ., I\)127.

Calculates nelv bour:d. state leve]s by using shield:¿

potentials.

It7](a) Ecker, G. and lÏeizele !'f. (gf6) Attt. Physj.k.

!, 126. Consid.ers shield-ing on the s states

to deternine the bound- state cut-off.(¡) Ecke::', G' and. Krol1¡ TiI. (1963) pnys. tr'luids 6' ta

(") Ecker, G. anÖ Krol-l-r 1,v. (geø) z. Naturforsch

21a 2O12r2O23.

4.46

[tg] Kelbg, G. (lgSl+) Anrro Phys. 12, 351+. Proposes

an effective potentlal to cut-off the bound- states.

[ 1 9 ] Dekeyser, R . (1965) Ptrvs ica Lg 1 4o5 .

[zo ] T,arbe G.T,. (t96+) Phys. Rev. Letters. 72¡ 683.

(1965) errys. Rev. l-4Q, L1529"

Using the expansion technique of Gold'berger and-

Ad.ams obtainsa eonvergent correlation functlon

for r> the therrnal De Broglie vravel-ength.

lzl l Guernsey ¡ R.L. (lg6S) prty" . Fluids Z 792.

lZz) Oppenheim, I. and. Hafemanne D.R . (1963 ) ,1. Chemo

Phys . 2 1O1. Using the relative size of the

bound. and unbourrd terms they eliminate the QIt,f

d.ivergenceso

lzllþ) Rousee c.A' (gq ) prtv". Rev. E, 41 '(¡) þ, 62.

(") Solving the Schroedinger eguation for a Yukawa

(screened.) potential¡applies his solutions to

eguations of state and- lonizalion ealculatior:s

(preprint ) .

[24] Montro11, E.ïf. and- ',vard.p J.C. (t959) fnys. Fluids.

!,55'l25l B1och, c and- d.e Dominieus¡ c. (lgSA) l. Nuc1. Phys.

Z, 459^

C. and. d.e Ðominiorse C.Blocht (1e5e)

!, 181

Jn Nue1. Phys.

and 5o9

lz6)

lztlIzs ]

lzgl

[:o 1

4.47

De lÏltt, H. (1962) ¡. Math' Phys. fr 1216.

(1966) J. Math. Prgrs. /r 616.

Ni-nhame B.W. Preprint.

Diesendorf, I,,{.0. Private Communlcation.

oNeile T. and Rostoker, N. (1965) pfrys. Fluids Sr

11 09.

Trubnikov¡ B.A. and. E]-esin, V'F. (lg6S) ¡'n.T.P.

20 866. Jnclud.es exchange A¡/l effects, but makes

no al-lcruvance for shielding.

Dekeyser, R, (1967 ) erryslca 22, 506.

Kremp, P, and Kraeft, -vlloD. ?geg) Atn. Phys, 20, J!¡O.

n{atsuda, K. UgøA) prrys. Fluids !!r 328.

storer R.G, ?geï) J. llaths. PhJrs. 9¡ 964.

Ebeling, in/. (lgøg) lrrysica 38, 378.

Pauling, L., and. \r1/ilsone E.B. "f ntrod-uction to

er¡anturn Mechanicstt Ch.Vr (1,[c6raw-Hi1]-: New York 1935)"

Schiff, L,f . tQuantu.m Meclranicsr Ch.V (MeGraw-H111:

New York.1955).

Abramowitze M. and Stegun, I.A. rrHandbook of

I\4athematieal Funeti ons rr r Ch. 1 4. (wati onal Bureau

of Standards: trtlashington) .

Fröberge C.E. (1955) Rev. }'[od. Phys. 27t 399.

Slatere L.J. tConf1uent Hypergeometric Funetionsl

ltt I

ltzllttl3\l|.¡nl

ltø1

lttl

[58 ]

l3e)

It+o 1

[41 ]

It+21

Iu¡ ]

t44l

4.1+8

N.B.S. Tables of Coulomb liilave tr*unctions AMS 17

Abrarnowitz, M. Phys. Rev. 99, 1187 ?955).

Abramowitz, Iú. and Stegun, I.A. Phys. Rev. 99t

1851 (1955).

Storere R.G. Private Communicatlon

Storer, R.G. and Davies, B. (1%8) pnys. Rev. 7-f.7t

150.

Sahae LÍ.N. and Saha, N'K. (1934) ttA Treatise in

Mod.enn Physicsrr p.795 (tfre fndian Press: Calcutta).

Thompsonr i4¡.8. (1962) Att Introd.uction to plasma phys-

ics'p.1 9 (Pergarnon Press: oxford) .

5.1

V SOLUTION OF THE MODI}'IED PERCT]S-rcVICK EQUATTON

5.1 A form table for solution on a computer

In Chr,apter III we expressed the Hf equation in a

form suitable for solution on a computerr ârld gave an

outl-ine of the computer proeed-ure to d-etermine S"O(r)

from this eguation, TVe also d.erived an asynptotic

form of the llf egr:ation for large Fe which was found- to

be inconsistent in the secord-order terms. It was

further shown that the integral term was extremely

sensitive to the inBut potentÍa1CIand- distribution f\rnctions

at sma11 r. In this section we sha11 apply the same

reasoning to the MPY equation which, in its asymptotic

form for large Y2 has the advantage of self-consistency

to all ord.ers. The input potential-sand- distribution

functlons wil-1 be taken from the accu.rate quantum

mechanical calculations of CLrapter IV'

The mod-ified- Percus-Yevlck equation as proposed by

Green t1 ] has the form

8ab eab 1 + Dc

(t-e.u) d"*" d"*d.

/ t*o"-.. I 8"" (t-e"") d"*"tc

+ L " '" ä "d /f {uo"-r¡ (sou-t) u"u Bae Ba.(t-er")

(r t¡

whene e"O = exp(Þó"b), the summations are over the

types of particles in the mixture, anÖ the higher

ord.er terms have been neglected. In thls Ctrapter,

uslng computer notation, shal1 refer to the PY term

(i.e. the first integration term), as TDTMe ard- the

last lntegration term as FDTM. Since TDTM was considered

in detail in Chapte" 3r here emphasis will be placed on

fDTiVI whlch effectively d.eseribes the f our-particle

interactions" Using an approach similar to that in the

3-Wrticle caser tr'DTM takes the form

512

[*o"(t)-1 ]M,k2

(r+s)

r-s I

(r+u)

r-u I

[*'on-

ctì)d.

oo

t t,ltI

Tn

otJOc

[*ou(v)-1 J s""(s) sru(u). [1-"""(s) ] [t -"ra(.r) ]

su"(w) d-g vdv td-t ud.u sd.s, (5.2)

where w is deflned bY the equation

2r2wz - 2r2 (s2+u2) - (r2+s2-t2) (r2+u2-vr)

[4r's' (r2+s2 -1'¡"1t . ( 5.2a)

Il+""o'-(r2+u2 -u'¡" 1L cos â .

The guantities rysrtrürv and w refer to the interparticl-e

distances between four particles vrhlch are placed. at the

5.3

vertices of a tetrahedronr viz.

lË^ - ë¡1, l¿, - asl, t l*o - *"1, u -Þ-

l¿" - å¿1, ìr = l*¡ - xdl and w - l¿u - Ãsl, vrhere 5, 'Ãb, ëc, and Su d_eflne the positions of the 4 particles

of types àcbcc and d, respectively. 0 is the angle

betr,veen the pfane Ie E¡ ! and. q,3r Ip and is related-

to the length w by (5.2u). It can be seen that w

aehieves its maximum value (r,\ilmx) when particle c and d-

are d.irectly opposite each other on either sid.e of $r

and this occurs when O='tr. Correspondinglyr the rnininum

value (WUfU) j-s obtained uuhen 0 - O"

An asymptotic form of FDTI,{ akin to (3'Z) can be

obtained from the same assumptiofJSe using similar

reasonir:g. Eguation (5.2) tiren becomes

((p*y) + ... ] 1

Ujø^6(q**)+. o. ) (t+a/r)

r

a.#) t,t ,! _ã,

>n^cv

2rr"à)h^ I tpó.d o- l '"ac'-a

a

IB+2r

pl

q+2r,o"(t) t eou(v) tdo

îr[l+eu"(*) ] d-g vd-v tdt

dq d-p (5.3)lql

5.1+

where11

w2 = t2 + v2 - 2pq. + 2(t2 p')z (n' g")z ces 0o

This e)cpression can also be obtained by letting r

become large in the geometrical interpretation of the

1ntegra]" \Mhen it is ad.d.ed to the asymptotic forn of

the Pereus-Yevick eguation, it makes the resulting

asymptotic eguation consistent to second ord.er for

charged ni-xtures, aS can be seen from argUments analagous

to those of section 3.2"

The numerical evaluation of fDTll as given in (5"2).

is based. to a large extent on the technigues mentioned-

in secti on 3.3, This 1s to be expected', as FÐTtr'q 1s

essentiatly composed. of two parts which are identical

to TDTi\{, but which are modlfi-ed. by the inner integralrTft

" su"(w) d.0. rn the program (see append-ix B) thisioirurer integral is evaluated. in terms of w. [[is is d'one

by using (5.2a) to obtain d.0 in terms of dw" The inner

integral then becomes

su" (w)2r2w 0r¡¡

DEi\T)

2r2wDENwhere we have used the substitutlon d0 - dÏt¡ "

5¿5

Holvever, f or tlæ trD[}[, although the integrations can

be d-ivid.ed- into regions and. evaluated- using a

trapezoidal rule aS for TDTl'f , the mesh ratiots that

can be u,sed. are much smaller d.ue to the higher dimension.

Further d.etails of the program for calculation of

trDTI/i are given in the notes with Appendix B'

The application of the MPY eguatj-on to a two

component (p-") plasma encounters ¿ifficulty in the

choice of input, for it can be seen from equation (5.1)

that there exist four distribution functi-ons to be

ca1culated., gee, gep, gp" "tU Unn' As it is intended-

to u^se an iterative technique to solve the l,4Pf eguatÍon,

this would mean solving f our linked integral eguations'

Classicall-y, the following j-d-entitles hold-: É"" = Ønn e

g"e = gpp, Øpe = ø"n ttu gp" = gep. These reduce

(l.t ) to two linke¿ equatlons. Holvever, from the guantal

considerations of Chapter I\I it t¡r¡as shovrn for smal1 r,

that although the effective potential-s and d-istributlon

functlons for interactions betr,veen un11ke particles

i¡/ere equal (meaning v"p = un" "tu g"p 9p" respectively),

this was not the case for like particles" For I1ke

particle" V"" and- 9"u d-1ffer very appreciably for smal1

r fnom the correspond-ing Vpp "^d gpn' Furthermore

fr"om Chrapter IV we eould not obtain accurate values for

5.6

gpp ( and- hencu unn) by the same method- used' for 8ee,

though ii lvas d-ed-uced- that for the p-p case the guantal

effects should- be small, ard- "o

gpp is expected- to

remain cl-ose to its corresponding el-assical curve and.

V- -- cl-ose to the comespond-ing Coulornb potentj-al-.pp

To resol_ve the d.ifficulty of obtaining accurate

input data for l-1ke particl-es at small r, it is f ound.

convenient to assume that the combined effect of the

e-e and p-p potentlals can be represented. by reflection

of the e-p potential from below to above the r-axise ioo"

vL = - vu, lvhere v" refers to the combined effect of the

e-e and- p-p potentials, and Vg refers to the interaction

potential betlveen unlike partlcles. This assumptiont

besic.es alleviating the need- for accurate p-pinput d-atat

reduces the number of linked integral eguations obtained'

via (5.t) from three to t'vo, and. so greatly reduces

cornputational- d.ifficulties. From Fig. 4.8 it can be

seen that reflection of v"n about the r axj-s to obtain

the ccrnbined- effect of th€ O-e and- p-p potentials results

in a v" characteristic ';vhi-eh differs narkedly from the

V"" curver ârd from the c1asslcal curve to whi"n Unn

closely approximates. HOlvever, since the forces are

repulsive, the number of partleles of the same charge

approachlng one another very closely is expected. to be

small, and. the error involved unimportant, at least aL

5¿7

temperatures above 2x1O4"K. Below this temperature

the number of pairs present may encol-rage the fonmation

of complex ions for rrhich a more rigorous treatment of

quantal effects between like particles (includ-ing

quantum statistics) would- be desirable

5.2 Outline of the rical procedure

The evaluation of the Hf term ard. the ilIPY term

has been discussed in some d.etail 1n secti ons 3.3c 5"1 t

arrd Append-ix B. ]n this seetion lve shall Ôiscuss the

iterative procedure adopted in solving the MPY equation.

Tn 1960 Broyles l2l proposed an iterative procedure

i¡¡here an initial trial *Í;) (") is inserted- in the

right-hand. sid.e of equation (5.1), the integrations

are then performed to give a first-lmproved- triat- s!l) f tl,

This ean be used. to obtain a third triaI, and so ofi¡

simple iteration in this fashion d.id- not lead. to a

convergent Sequence and it was f o:.nd. necessary to lnclud-e

a mixing parameter a to Secure convergellCe. Conseguently

the (n+1 )tfr input was bu1lt up using tÌæ rule

srN(n+1) - * *o,il) + (r-a) *o,lî-') (5'4)

It bas been found. by Throop and Bearman lll that the mixing

constant ø is inversely proportional to t]1e d-ensity for

LJ fluidse and. as the d.ensity increasese c d.ecreag€S¡

5.8

and, hence the rate of eonvergence becomes appreciably

Sloffer. Broyles lZl al-so pointed out that convergence1.")

could be improved. if 1t was assumed that the g' '

result is approached. exponentiallyr for then

(*( i)*) -( i*1 )_-(i)

6ó

1-R

( ¡)

$.0)úÞ +

tö(J-1 )

t

vr¡here (¡*t)cft)

D :rlTt)

I and- thus as the so-]-uti onsg

( )ooapproached. the f inal- resulto the g could be pred'icted"

It is unfortunate that a technigue recently

proposed by Baxter t4] for the solution of the Iry

equation does not apply for long-range forces' His

method rel-ies on the interparticl-e potential- vanishing

beyond. some range M, for then the PY equation can be

written in a forn vyhich d-epends on the d'irect correlation

fìrnction and t he radial distribution function over the

range (OrU) on1y. Watts t¡] Lras applied- thls teehnigue

suceessf\r1ly to a Lennard.-Jones fluid. near the critical

regi on.

Tn the caleulation presented. here, the j-terative

method- d.ue to Broyles vì/as used., but several mod-ifications

vt¡ere neeessary to obtain convergence, and. these will

be discussed in the relevant sections. It rras decided- to

lnltial_ly attempt to solve the ]\/TPY eguation for a

5.9

temperature of 1O4oK and. d.ensity 101 8e/cee and- to

graôua1l-y increase the temperature to cbtain results 1n

the region where the DH approximation is valid-. The

first difficulty is choice of the initial 8t* and' VIN"

It was polnted out in section (5.1) that, by assuming

the combined- effective potential for like particles

\ilas equal to minus the effective potential f cn unlj-ke

partielesr the problem 1s reduced to solving two llnked

integral equationsr âÍrd so we chose

where the l_oglo(en"(")) are presented. in Table 4.14"

Sinee in Chapter l+ we also determined a dlstribution

function to appnoximately take lnto account shieldingt

it was d.ecided. to use those results f or EINI i'eo

Losro (s"(r)) - -Log,o(sU(r)) = -Losro (s"(t)), which

can be obtained. from Table 4.18. Thus input w1l-l be

frequently referred to as the guantum mechanical

Debye-Huekel (amn¡ d-istributlon function'

Theprogrami-nAppend.ixBd-oesnotcalcu}atethe

d.lstribution function at r=O. To calculate gab(O) it

is necessary to take the limits of the integrals in the

MHf eguation as r->Oe aIId eval-uate the resulting equation

LBCUT

s 2 d-s+

dc

5.10

$.1)

s.o(o) "a¡(o) = i+,1+ø E n" I [1 -"ac(") ]erc(") [s¡"(s)-t ]

o

LBCIJTBn : "" ä % [" (1 -"ac(s) )sac(") [s¡"(s)-t ]s"

. LBCUT - I| --- - [t-e "(") ] suu(u)[*ou(u)-1 ]u2 |/'ao,i o o

îrao (on) d odud s a

where w = (s' + u2 Zsu cosO)å . From this eguation

1t can be seen that the presence of other ¡articles

effects the di-stribution between tvvo particles, even

at zero interparticle d.istance. Horvever, beeause the

effect of the other particles 1s expected. to be smaIl

(i.e. the integrals in eguation (5'7) are small)¡ it

uras d.ecided to fix g(O) to the q¡antal value determined'

1n Chapter IV until the last few iteratioh's. This step

should. help stabilise the iterati-ve ^r'echnigue' Theoreticall-y

of coursee g(O) should have no influenee on the value of

trE integrals; howeverr âs g(+) was obtained as the

geometrlc mean of g(o) and- g(1 ), its value d.oes affect the

calcula ti on.

Beforecommencingalongcomputerrun,extensive

hand. checlcs and trial runs to opti-mlze the integration

variablesÏVerecompleted-'Theoptimumvaluechosen

5 .11

for LACIfl, whlch d.ecid.es the size of the regions in

the integration procedurer Tuas found to be appoximately

equal to the joining rad-ii.Ls mentioned- in section 4.4.

This means that the reglons lvhere the quantal effects

be.come important are treated- in greater d-etail-' The

mesh ratios chosen for the various regions ìffere

d-etermined by accuracy eonslderations. Graphs of the

integral- value versus mesh ratj-o show that the integrals

attain a nearly constant val-ue when the mesh ratio

becomes sufficiently 1arge. Althor.rgh it 1s possible

to ad.opt large mesh ratios for the PY tern (fmU), this

is not feasibi-e for the ad.d.itional l,lPY term (¡'¡tU)

because in this case the fi-ve- d.lmensional integration

becomes too tine consuming. Thus the choice of the

mesh ratios for the regions in FDTI{ are d.eternined- by

time limitatiorÌs. The optimun size of LBCUT for

termination of the range of integration has to increase

with temperature. This is because it necessarily

lntrod-uces an error in the calculation of g(n) as r

approaches LBCUTT ârd is conseguently chosen to yield-

accurate values for the integral for r.3b. Hence

LBCUT is of O(4"b)" Analytical- cheeks for realistic

input data proved- imposslble¡ though an analytical check

for S(r) = coilst was completed.. Several hand.

5.12

cal-culations lvere made for realistic d-ata to ponf irm

that threre were no errors in the integration procedure"

The iterative proced.ure of Broyles \irias applled- to

the MHf eguation in the f æm

1+TDT\4+FDTM(5. e)si(") = I

eo(r)

where subscript u refers to unl-ike particles and the

Superscript j refers to the jth :'-teration, vrith a similar¡

equatlon for 11ke particles. At 104oK the iterative

process dlverged. on the second. iteration, und-ergoing

extreme f -uctuati ons.ç especially at small- rad'i i " It

lvas further noticed- i,hat the Seguence of terms in the

numerator on the right-hand side of eguation (5'B) formea

a dlverging seguence for small F. This implies that

the improved. Percus-Yevick equatiorÌ cannot be applied- at

thls temperature since it forms a diverging seguenceo

To d-etermine 1f thls i,ras the case at higher te'nperaturesn

the temperature 'ffas raised- 1n small steps. At 2x1O4"I<

the j-terations also d.lverge, even if u¡e Ìlse a large

mixing constant d.e and. after four iterations S"(r)tt1 ,

so that FDTM becomes negative, and- this results in

inad.nis sibJ-e negative d-istribution functions. At

2"Jx1O4oK. divergence d.oes not occur until the eighth

i tera ti on.

5,13

At 3x104oK several neTv moclifications Yrere

introd.uced to help secure convergence of the iterative

procedure. The mixing constant was removed, and a

seguence *(rn) , e(1 ) , rQ) ob tained-o From these threeI "")values a g' ' can be cal-culated using equation (5"5)

"

This *(-) is then used as input to generate another

sequence ",(rl-) ,e(3) , g(l+), and another e(-) can be

d-etermined-. In this way it 1s hoped- to obtain a

quence of g(*)"" Holvever it proved

necessary to overcome tlo d-ifficul-ties" The f irst

occurs when the R calculated- for equation (S.Z) is

xl g fcr then g(-) may become excessively large" This

is overcome by testing the values of R obtained-e and

if ln-t I is less than O,! the val-ue of R is replaced- by

o.5 (ir n is <1 ), or 1.5 (i-r n is >1 ). The second'

d-ifflcutty is that the f irst u("') cal-cul-ated seems to

overshoot the f inat *(*) ¡ ârd- causes the sequence of *(*) t "

to oscill_ate. This is overcome by using the mixing

constant technique to includ-e some of the previous g(-);

thus a ne\M input u(ru) is obtained. frorn BrN = o eÍ-)*

(r-a) *(i].' e where uÍ*) is the nth *(-) that has been

calcul-ated.. It is f ound- that the choj.ce a = 3 secures

reasonable convergeoCe o A trial run lïas also made v'¡here

the input was comPosed as fol-l-ows:

5 "14

roe(sr') = a log{u[-)) * (r-o) roe(*f:]l. rhis

mixing or- the logarithmic vafues inproves convergence

of the like distribution functions, but has an ad-verse

effect in the unlike case" ft d.oes, hol'rever, prevent/\

the g(*/ t s obtained- from becoming negative, which

occasionally occur.i.ecl f cn S"(r) when r is small'

The iterative process proves quite time

consuninge one iteration taking approximately t hor:r

on the CDC 6400 computere and- for this reason it lvas

decid.ed- to move to Ì;he temperature of l-¡x'1Oa oKu

rather than continue the run at 3x1O4oY,, where the

results, although conr¡ergent f or large r valuest

fluctuated for r<10 Bohr radii, even after 20 iteratiorç'

At the higher temperature the iterative technique

converges rapidly to give d-istribution fhnctions

id_entical- to four places of d.ecimals af ter only f our

iteratio'Sa If g(O) is al-l-o,¡,.¡ed- to vary, and not

fixed_ at its quantum mechanlcal value, thls rnerel¡'

al-ters the results belou¡ ! Bohr radii, and' convergence

to four decinal- places again occurs within four itera'c-

ions o

By removing the MPY term the program i¡/as

rearranged. to solve the Hf equation, and' this was

applied. to a range of temperaiures' At 4x1 O4o7<

5.15

convergenee to four decimal places was obtained. after

six iterations. At 3x1O4"K hov¿ever, the Iry equation

ran into similar, and probably more fu¡damental

d-iff1cuIties, than the l'4PY equati-on. The TDTtr,l

became relativel-y large at sma11 radiir but renained-

less than unity, and. after 36 iterations the g(*)ts

obtained- were reasonably conslstent. If holvever this

*(") is used as input for the nth iterati on, then

*(n+t) urrr""" slightlyr ând- *(n+z) urrr""" considerably,

atthough by using equation (5.5) with u(n+t ) utt¿ *(n+2)

r g(*) very simi]-ar to the *(*) used as input is

obtained. This means that the final *(*) generates a

non-convergent sequence on simple iteration. Such

behaviour d,iffers from the IITPY equation, i,'¡here the

iteratlons tend to remain fairly stable for large r

values, but become erratic at small radii. On further

s1mp1e iteration of the l.,lPY equation the erratic

behaviour at smal-l- r gradually effects the r¡¡hol-e CrO(r)"

At temperatures bel-o¡¡ 3x1o4"K, the PY equation prod-uces

negative d.lstribution functÍons. This is a d.irect

result of the inconsistency of the second-order terms

when attractive forces are present; for with sLlch

forces TDTII is negative, and. at these low temperatures

the second. order TDTI'Í has rnod-uh¡.s greater th¿-n unity,

5.16

and. this lead.s to à negative distrlbution function.

This inconsistency d.oes not occur r,rith thre tr/lPY equation¡

for if TDTII becomes l¿rge and negative, FDTI\,{ is

invariably larger and positive, and so equatlon (5'g)

yietds a positive distrlbution function. Ho"'¡ever¡

the divergence of the series 1e TDTlf, trÐTl[ soon caugeg

d-ivergence of the iterat ive technique r and- the MPY

equation applies over only a sllghtly greater temperature

range than the PY equation.

5.3 Results and. d.is sion

The results obtained by solvirrg the IrY and LIPY

equations for an hydrogenous plasma at 3x1OaoL< are

presented_ in Tabl-e 5.1 . They are compared with the

inltial input d-ata which is labelled QI'{DH¡ âs it is

composed of the Debye-Huckel d.istrlbution f¡nction at

large rad-ii but includ.es guantal effects at small radii.

The QI{DH resul-ts may contain errors for Y<15 Bohr radii

of less than t5 in the for-t'th d-ecinal p1ace, and. for r>'15ao

they are correct to the f ourth d-ecimal place. the PY

results are obtained from the final *('") d.erived- from

iterations 35 and 36, and- only d.iffer from the previous

g\-/ bv !5 in the last figure given in the table'

The resul-ts of the [,{PY are simllar]y obtainedr but in/ -_\this case g\*/ 1s derlved. after only 16 iteratiorlse

5.17

It can be seen the errors increase rapidly at small-

radii, where g(r) can only be given to two d-ecimal

places. The results are sho¡rn graphically 1n

Fig. 5,1 for like distribution functions r ârld |n Fig. j.2

for rrnlike d is tribut ion fb.nc t ions 'For [x1 O4oI< the results are given in Table 5'2

and. sh.oiun graphically in Figs. 5.3 and- 5.4. At th js

temperature each u(-) tabulated is accurate to four

d-ecimal places, and further reproduces itsel-f on

simple iteration. Tt vrrill be recalled- that the

calculation of g(O) was not used in solvlng the PY

equation, and g(O) remained. fixed. at 1ts quantal value;

this particularly proved- a stabil-izing factor at the

lower temperature of 3x1 04oK.

It should- be noted- however, that although the

results at i+x1O4oK converge to four decimal plaeese

there is an estirnated. error of approxlmately t! in

the fourth d.ec1rnal p1ace. At srnaIl r this is malnly

caused. by inaccuracies 1n the evaluation of the integral

especially f or the trDflf, where a reasonably smal1 mesh

ratio must be used. At l_arger values of r an error

in the fourth decimal figure 1s caused by the cut-off

LBCUI imposed. on the integral. At 3x1O4"7< these

enrors become qutte large at sma11 rad-iir for in

particular FDTII beeomes ]arge, and this term is subject

:

5.18

to errors of up to JO;,. To improve the accuracy a

large mesh ratio 1s need.ed in FÐTMr ârld this would-

involve a considerable increase in computing time.

From Fig. 5.1 1t can be seen that the nfPY results

are very erratic bel-ov¡ ten Bohr rad-ii; they become

relatively large near the origin, but then fal] sharply

away at 2-3 Bohr rad_ii, before returning to quite large

val_ues at ! Bohr radii. For r>1O the ]'1tPY cal-cul-ation

of e"(r) remains signlficantly larger than its PY

eguivalent, a feature which might be predicted- from

equation (5.8), where as FDTII is always positive the MPY

results wj-l-1 lnvariably be greater than the correspond'ing

pY results. The Iìf results in turn 1ie above the QI/IDH

results for r<15Oao, but then they gradually fa11 slightly

be]ov'l the 0,1DH results' The MIY results, hol'¡ever, remain

abovetheQl\./iDHresultsforal-lradii.lhismeansthe

DH d.istribution function is between the PY and. x[PY results

for e>15Oaoy and even allolving for an error of 5 too l-ow

in the fourth decimal place in the Hf results, this

imp]-ies s 1s sunprisingly good. The inclusion of- -QMDH

the additionaÌ term in the MPY calculation makes an

appreeiable difference to the d.istribution functions.

rQnilDH

Llke Un]-ike

5,2C

PY i',IPY

Like Unl-ike Like Unlike

25

3o

35

40

1+5

5o

6o

7o

BO

9o

100

120

140

i60

lBO

200

220

240

260

2BO

joo

.6977

.7\77

"7854

.Bi48

.8382

.8573

.9865

.9076

" 9234

.9356

.9'l+53

.9595

"9692

.9761

"9812

"9851

" gBBo

.9903

"9921

"9935

"991+7

1.433ì+

1 .3375

1"2732

1 "2273

1.193O

1.1664

1|1281

1.1019

1 "OBJO

'1 . 0688

1.0579

1 .Oì+22

I "O318

1"0245

1.O191

1 "O152

1 "A121'l . oog8

1.OOBO

1 "0065

1,0054

"7060

.7535

.7896

"8191

"8410

"8591

, BBTO

.9087

"921+1

"9350

.)+57

.9595

"9692

.9759

" g8og

" 9846

"9874

.9913

.9914

.9933

.991+4

1.4160

1"3245

1 "2611

1"2217

1.1B5ro

1,1621

1"1220

1 .0938

1,0784

1"0654

1 "0540

1 , O¿+20

1 ^ Or-;O

1"0229

1 ,01 BO

L0147

1 .0-1 14

1.OO93

1 "OO7B

1"OO59

1 ,0047

"7126

,7588

"7983

" B24O

,8475

.8665

"8970

"9162

"9323

.9U+2

,9540

" 96BB

"ggo5

"9870

,g\gl

"9899

"9926

"9925

"9933

" gg44

oo Ãz

1 "4315

1"3361t

1"2727

1"2270

1,192/,+

1 .1684

1"13O8

1 .1056

1 "OB85

1 .0755

1 .0650

1 "0506

1 "0400

1 . 0341

1 .0276

1 "0201

I "0143

1 .0'1 10

1"OO92

1 . OOTO

1 "0059

rQiT;DH

Like Unlike

HT

Like Unlike

5 "21

MTry

Un]ikeLike

34o

380

420

46o

500

540

5Bo

620

66o

700

740

"9963

"9974,9982

,9987

"9991

,9993

"9995

"9997

"9997

.ggg\

.9999

1 ,OO37

1 " 0026

1 .OO1 B

1.OO13

1,O0Og

1.OOO7

1 "O0O5

1,OOO4

1.OOO5

1.OOO2

1 . OO01

"9960

"9972

.9981+

.9986

.9988

"9992

,9996

.ggg6

1.OOO

.ggg\

.9995

1"OO33

1 "OO22

'1 .OOl1

1,OO12

-1

" OOO9

1 ,0oo4

1 "

OOO1

'1 " OOO2

"99971.000

1 ,0oo4

"9966

.9976

"ggg1

"9996

1,OOO2

.999\

"9996

"gg9g

1.OOO1

oooz. ,,/,/ ))

L O0O7

1.OO4o

1 "OO2g

1 .OO21

1 "001 1

1 "OO12

1,0006

1 "OOO5

1 , 0006

1.OO1l

'1 "OOl

j

1 "001 5

(Notes Errors came in lfPY about 5OO)

5.23

DH PY T.{HT

r Like Un11ke Like Unlike Like Unlike

25

30

35

4o

45

50

6o

7o

BO

9O

100

110

120

130

140

150

170

1go

210

230

250

350

1,3175

1 .2506

1 .2051

1.1772

1.1473

1.1279

1,0997

1.OBO2

1.0660

1.0553

1 .0470

1.O4O3

1.0349

1.O3O5

1.0267

1.0236

1 .01 86

1.0149

1 .O121

1.OO99

1 ,0082

1.OO3l+

o.7607

0. BooB

o "8307

o. B5J8

o.8722

o.8869

o,9092

o.9256

o.g3B1

o.9475

o.9550

o.9609

o.9658

o. 97OO

o.9737

o.9767

o. gB1 4

o. gB4B

o. g87B

o.gÙgg

o.gg15

o.9963

1.3130

1 "2471

1.2023

1.1697

1.1449

1.1261

1 "OgB5

1.0791

1.0647

1.0542

1.0459

1.0396

1 .0344

1.0299

1.0260

1.O23O

1.O182

1 .O1l+7

1 .0117

1.OO95

1.O0BO

1 . OOJLI

.7620

, Bo20

.8319

.8550

.8735

.8881

.9105

.9268

,9394

.9488

.9564

.9623

.9672

.9715

.9752

.9782

.9830

.9867

.9896

.9911+

.9926

.9966

1 .311+9

1.2489

1 .2039

I " 1713

1.1465

1.1276

1 .1 000

'1 .OBO5

1.0662

1.0557

1 .0474

1 .041 1

1.0359

1 .O315

1.0276

1.0246

1.01gg

1 .O1 66

1.O135

1.0111

1 . OO91

1.OO37

o.75go

o.7996

o. 82gB

a.8531

o.8716

0. 8866

o.gog3

o.9257

o.g3\o

o.9476

o.9551

o.9612

o.9663

o. g7o4h

o.g73g

o.9769

o.9817

o,9853

o. gBBo

o.9902

o.gg1g

o.9966

DHï,ike Unlike Like

5,24

MPYUn11ke Llke Unliker

450

550

6jo

750

850

950

o.gg85

a.ggg3

o.9996

o.9998

o.9999

I . OOOO

1 .OO1 5

1 .OOO7

1 .0OOl+

1 .OOO2

1 . OOOI

I .OOOO

HT

o.9981

o.9989

o.9991+

o.9997

o.9996

o. ggg8

1.OOl6

1.OO1g

1.OO15

1.OOO2

1.OOO3

1.OOO2

.9983

.9990

.ggg4

"9997

.9997

.9999

1.OO17

1.O01 0

'l .0006

1.OOO3

'l .ooo4

1.OOO3

L

cn

3.0

1.0

2.0

PY

MPY

QM0H

Dl{

l0r (Bohr

FIG. 5.2 COMPARISON OF UNLIKE

20Rqdii)

otsTRtBuIoN FuNCT¡0NS lt 3 r106,

c,cno

0

r (Bohr Rqdii )l0 20

MPY

QMOH

PY

OH

FtG. 5 . 3 COMPARISoN OF L|KE OIST RTBUT|ON F UNCTTONS Ar lr r 10¿ o r

- 1.0

- 2.0

EN

e,úlo I

lI

- 3.0

EN

clolo

2.0

3.0

0

IIPY

PY

QMDH

t.0

OH

lo r Fohr

FIG.5.¿ COMPAR¡SON OF UNLIKE FUNCT|q.|S ¡l 4¡lol cX

20Rodii)OISTRIBUTþN

5. 25

In I'ig. 5.2 (ancl Tabfe 5.1) for the unl-ike

d.lstri-bution functiorrs it can be seen that the PY

results lie l'¿ell- belor¡¡ the QIIDH results. The I'[PY

equation, which should- improve on the PY results, lies

guite cl-ose to the QI{DH results, fyirg below them for

r<5Oaoe except for g(o), and- then remaining slightly

above then for large r¡

An almost ld.entical analysis occurs for the three

sets of results given in Table 5.2 at L,rx1 O4"Ko For

the like case the PY results remain above the QL{DIí results

for rc8Oaoe and. the lûFY results l-ie above them. Beyond-

r = BOao the 8L4DH valrres lie between the I\,{P'Y and PY

values. In the unlike case the Ilf results remain bel-ow

the QI,IDH results, while the TIIPY results remain smal],er

for yaJOaot but beyond. that become greater than the

Ql/iDH results.

Perhaps the most signiflcant feature of the results

is the fairly large increase in the value of gt(r) indicated-

at snal1 radii. In the Monte Carlo results obtained- at

1O4oK it was noted. that the peak 1n g"(r) at smal1 r was

probably due to col_l_isions between ions and. pairs.

At the higher temperature of 3x1O4"K the guantum

nechanical results lnd.icated. that the pl-asma was

approximately 9Oi; ionized-, and henee there is still a

5.26

reasonable chance of a collision between an ion and

a pair. As the temperature is lotvered the mrmber of

pairs increases, at the sane time the PY and. MPY results

start to d.iverge, lvhich ind.icates that it 1s again the

fornation of pairs which causes the difficulties"

Hovr¡ever, in the integral equation approach this divengence

appears in the folloirlng manner. Firstly S¡(r) increases

sharply for small r in the evalr:ation of g{1 )(*), then

on using g{1 ) 1r¡ as input this causes sÍ2) (*) to

increase sharply at smat-l re this in turn increases S{3) {"),

and, the serj-es d-iverges unless extrapolated- baek to g(-).

This d-iffieulty is also associated. wlth the concept of

the ccrnbined- effectlve potentlal vL = -vur whieh was

introduced- in ord.er to red.uce the mrmber of linked

integral eguations in the MPY eguation. For the divengeneet

which initially starts in g"(r)r is very closely

connccted with the Vtr chosen. From such consid-erations

it appears that to rigorously improve upon the results

presented- at JxlOa"K or to proceed to lovren ternperatures

it is necessary to treat the e-e and. p-p interactions

separately. In such a procedure the quantal Vpp

needs to be accurately determined-, and for completeness

quantrm statls tical effects should. be lncl-ud.ed- in the

5.27

caleuJ-ation of V"u.

In concl-uslon then the Percus-Yevick integral

equation approach can be successf\r1ly applied- to

plasmas when the second- order terms renai-n smal1e

and. this occurs when the plasma is ful1y ionized-.

The inclusion of hlgher ord-en termsr âs 1n the MPY

eguation, alters the results appreciably. Holvever

the solution to this equation also becomes unstable

as the pairing present in the plasma becomes slgniflcant.

To extend. the region of applicability of the MPY eguation

1t j-s neeessary to obtain accurate quantal effectlve

potentials between the particles, and. to solve three

linked integral equatlons. At tenperatures above

the ionization temperaürre the nethod ylel-ds accurate

d.lstributi-on functions r,vith-in a fevr iterations. the

d_istrlbution f\rnctions obtalned for an hyd.rogenous

plasma of 1018e/cc at 3x1O4oK are rernarkably slmilar

to those obtained- by the DH theorye except at smal-l

radi i.

5.28

References to Chapter V

[t ] Green, H.S. hg6S) errys. Flulds Qr 1.

lzl Broyles r A.An (t geo) J. Chem. phys . ä., 406.

lSl îhroop¡ G.J, and Bearman, R.J, ?gee) .1. Chem.

Phys . .!4, 1423.

t¿+] Baxter e R. J. (1g67) Phys. Revo 151+t 17o.

t¡] r,4lattsr R.o. (t964) ,1. chem. phys. .!9, n.

6.1

VI CONCLUSION

6.1 Compaqison qf the two me thods

It is unfortur:ate that the main I\4C results Tvere

obtalned. for the temperature of 1OaoK, for it was

subsequently found. that the PY and LÍPY eguations could-

not be applled at thls temperaturer and. a d.irect

comparison of results became impossible. Another d.iff erence

1n the cal-culations is that the I\,lC computations l'rere mad.e

by taking quantal effects into account only in a rather

crud.e mannerr whereas such effects were treated. more

exaetly in the Hf and- I\IPY caleulations. For these reasons

only a broad. comparison of the two nethods is mader and

conclusions pertinent to the presented results are

contained- in section 6.2. The I''tC approach has the

ad.vantage that the derivation of tie nethod. is relatively

free of assumptions compared, to the PY approach. Howevert

the l,rIC method also exhibits its usual d.isad-vantage, namelyt

that eal-eulations of accurate d.istribution functiorrs are

very tinre consu:ning; and- in this tern¡erature range the

presence of long-range forces in conj¿nction with pairing

of unlike charges aggravates this situation. 0n the

other hand., rvhile the lff eguation can be solved. numerically

reasonably quickly, to obtain aecurate distribution

functions; when higher ord.er terms are includ.ed- to obtain

6.2

an improved. equation (i.e. MPY), thls approach also

becomes very time consu-niing. The imprwed- aeeuracy

of the results is perhaps the main advantage'

6.2 Conclusion

In this u¡ork lve have applied- two of the vreIl-

establj-shed- liqrid. theoriese the i\Íonte Carlo (tøC) method.

and. the Percus-Yevick (PY) eguation, to a d.ense tgrdrogeneous

plasma (.u = 1Otle/cc) near the ionlzation ternperature.

The tr{C method was app}ied. at 1O4oKr and extensive results

of these calculations are presented in the Tables arx1

Flgures in chapter II. From these results it lvas

conclud-ed- that guantum mechanical consid-erations are

important at smal-I radii for this temperaturet and the

e;t-off AO used- lvith the coulcmb potential should. be

replaced- by an aecurate qr:antal effective potentlal at

smal_l rad-ii. The resul'r,s also lndicated- that the maxi¡nurn

step length A used. in the UiC procedure mus t be carefully

ehosen when consid.ering the temperature nange corresponding

to the transition from the neutral gas to the ionized

plasma. For in this region the plasma appears to

þehave aS a mixture of turo phasese wlth the choice of

A deterrnining whlch phase d.oninates in the relatively

srnall sample of conflguratlor:s selected- by the MC

procedure.

6.3

The initlal appllcation of the Hf equatione in an

asymptotic form fcn large r, indieated. that the IY

equation can be successfully applied. to systems composed-

of particles irith repulsive interactions at short

d.istances. Hovrreverr if attraetive forces are present,

an ineonsistency arises in the asymptotic equation.

This inconsistency is remor¡ed- b5t consid-ening ad.Citional+u€rms to the eguation such as those suggested by Green.

The resulting eguation has been termed a modified Percus-

Yeviek equation (unf). Further initial investigations

into solving the lr.{PY equatlon fon a trvo-component plasna

showed that the integrations involved. rvere hlehly sensitive

to the form of the interparticle potentials and. inter-

particle d.istribution functions at small radii. 1o

obtain such accurate two-¡nrticl-e potentials and

d.istributlon firnction^s f or an hydrogenous plasma it 1s

necessary to inel-ud-e cluantal effects. TheÞr by using

accurate two particle potentials in the MPY equationt

1t should- be possible to obtain aecurate d.istribution

functior:s f or the many particle system.

The calculation of aecurate quantum mechanical

two-partiele d.istribution functions has been presented

in detail in Cþapter IV. The expression obtained. for

the ti,.,'o-lnrticle distributi on func tion, equation (4'4) ,

6.4

takes the important Heisenberg effect into account,

but neglects the smaller quantal effect due to

statlstics. The cornputer prognam written to

evaluate (4.4) 1s listed in Appendlx B¡ ând proves

extremely efficient for calculations ot gp" and. 9""

over a range of temperatures. However 1n the p-p

ease the large inerease in the red-uced. mass of the

two-partiel-e system eauses computational difficultiest

and this particular prograTn 1s inapplicable. Fortunately

the semi-classieal- \MKB appnoxirnation nay be used- there.

The quantal cal-culations cf Bu" and- *"n t"" pnesented.

for the range of tenperatures 1O4oK to lx1 o4oK. Because

of the eonvenient form of eguation (4.t+), the cornputer

Brograrn gives the flrst bound.-state eontrlbution, the

nr:mber of bound- states contributlng to g^-(r) to obtainep' ' ,

a flxed. accuracy¡ âIId- the total bound.-state contnibution

for that accuracy, It al-so calculates the percentage

ionizatlon present, and. the radiuse PJ, belcnnr lvhich

guantal effects become important. The program is

furttrer eas1ly mod.ifled to includ.e shield-ing effects in

an approximate nanner¡ âIld hence ind.icates the form of

the distributlon function for a nany-particle system.

the quantal results sholv that there are rather

large d.eviations from the efassical theory at shont

6.5

interparticle d.istances, and- that bel-ou¡ r" guantum

mechanical- effects become important, especial-ly at

lolv tem¡eratures. The approximate allowance for

shield.ing in the calcul-ation of e"(r) inaicates the

results are qualitatively the same for this casee but

aL large radii g"(") merges r,vith the Debye-Huckel qra(r),in contrast to the two-particle Sn"(") casee whlch merges

with the cJ-assical S"(r). The inclusion of shield-ir,rg

a1-so slightl-y increases the degree of ionlzation presentt

'rvhich is to be expected-, as the shielding precl-udes

some of the bound states. Nevertheless the results showed.

that elther by fu11y taking account of the bound. stateqt

or by attemptlng to allo,rv for shlelding, the d-egree of

lonization rvas surprisingly close to the values obtained.

by Saha. Alsor the results ind.icate that there is some

justification for considering the first bound. state to

provld-e the major portlon of the total bound- state

contributione especiall-y at temperatures bel-ow 3x1Oa"Kq

The aceurate two-particle g-^(r), and an"pe'associated- effective potential- Vn"(r), tvere then used- '

as input to the I4PY eguation. In order to reduce the

computer progran to a feasible size the input d.ata u^sed for

the combined. effective potential between like partlcles

firas taken to be the reflectlon of the interparticle

potential between unlike particles. This procedure

6.6

al-so avoid.ed the need. for a separate calcul-ation of

vnn(r).It rn¡as f o.nd that the PY and. I,{PY equations could

both not be solved. at lovv temperaturesr where the

integrations on the rlght-hrand side of eaeh of the

equatiorrs formed. a d-ivergent senies for smã1l r. Tn

the I¡f case thls also 1ed- to negative d.istribution

functions d-ue to the inconsistency of the seeond-ord-en

terms. At temperatures above JxlOa 1t became possible

to obtain solutions to both the IIf and L,1llf equations by

employing the iteration techni.que described in Chapter'Vi

Accunate results are presented for l-¡xl OaoK and some*f.l

less accurate figur.es for 3x104"71. these results sholi¡

that tl.e g.(r) characteristlc, obtained- by includ-lng

a Debye-Huckel shielding factor 1n the two-partlcl-e

guantal calculation, ties between the Hf and. llflf curves

in the l1ke case for 1âTl$ê Ito The L{PY results are

aÌways larger than the PY results.

A close analysls of the divergence in the integrations

on the night-hand sid.e of the i\fiIlf r shot¡'¡ed that it vras

physically related to the formation of pairs in the

plasma. To proceed. to lower temperatures it wouÌd be

necessary to solve three linked. integral equatlons for

gee, gnn and- gp". Fr:rther, it rivorld' æpear d-esirabl-e

6.7

to includ.e quantum statistical effects for the e-e

interactions" Because of the sensltive dependence ofthe integral--eguation proeed.we to the initial input atthe 1-ovu temperatures, in thre future it may ice preferable

to o-btain input d-ata by extnapolation of soluticn athighen temperatrJ.r €s o

In conclusion, botLrthe ì-ntegral equation approaeh

and. the l,4C method encounter difficul-ties as the plasma

becomes only partialJ-y ionized. At temperatures 1n

the regicn of the ionizatlon temperaturee quantal- effeets

play an inpontant role, and should be incorporated. into

both approaches in a rigorous fashion. For temperatunes

jrrst below the ionízaüon temperature, ind-ications are

that it lvi1l be necessary to includ.e three-body interact-

ions. For tem¡nratures above the ionization temperature,

and al-l-owing for quantum effects, the IY equation yields

approximate distributlon fl-rnctions very eeonomically.

They ean be improved. by solving the lfPY equation, as

d.emons trated. 1n Chapter V j

ÂPPENDïX li

The articles Ín this append.ix vüere published

by the author during the course of this rrvork,

Barker, A. A. (1965). Monte Carlo calculations of the radial distribution

functions for a proton-electron plasma. Australian Journal of Physics 18(2),

119-134.

NOTE:

This publication is included in the print copy

of the thesis held in the University of Adelaide Library.

It is also available online to authorised users at:

http://dx.doi.org/10.1071/PH650119

THE PHYSICS OF FLUIDS VOLUME 9, NUMBER 8

On the Percus-Yevick Equation

Ä. A. B¡'msn(Jniaersity o! Adelaide, Ailelaüle, South.Auslralia .

(Received 10 Èõïemd, í965; firrui*uouscript received 14 February 1966)

using an asYmPtotic form ofattraòtive forces are removedassumes an inconsistent form

I. INTRODUCÎION

] Gre"tt' are included.

tr. AN ASYMPTOTIC FORM OF"TIIE- PbRcus-YEvIcK EQUATIoN

The Percus-Yevick equation, generalized for a

fluid mixture, has the form

g"ta"t :1 - I n" | {r"" - l)g*(gu" - l) d'"r", (1)

where

edb : exp (0óò,

which can be written

o"oçr)e",(r): | -'++n" Ï"- 1,".-",[¿""(s) - 1]

's""(s)[So"(Ú) - 1]t dÚ s ds, (2)

where of tYPe c

per un Particles

in the olume of

r J. K. PhYs. Rev. rl-0,, 1 (1958)'r A. A. 37;2462 (ts62).! D. D. 1406 (1963).. H. S. (1965).

particles of the cth type. Broyleso tewrote this equa-

tion in a form

fi,V0",{r)",,{r).1 - 1 : r* Ðn" [' -(s * r)s""(lsl)

'[so"(ls + rl - 1)]11 - e""(s)ls ds, (3)

which is much easier to handle computationally'To obtain an asymptotic form of the equation for

large r, we make the following assumptions: (i) That

Pþ:rØ is O(r-") for large r, and for attractive forces

it is finite for small r. This assumption excludes

gravitational forces, and requires a cutoff at small rfor Coulomb forces. It implies that we cân express

g"r(r) : 1 f e'6(r), where e"6(r) wiII be frnite for

sÃall r, and will be small for large r; and without itstatistical mechanics is probably impossible' (ii) That

Now by assumption (i) we can expand in powers

of ó for lirge r, anã with retention of terms involving

only small powers of þ, Eq. (2) becomes

[1 * ..,(r)l[1 * Éó.0(r) 1- Lþ"ó'"'@) + "']

: 1 + t+ E"n" ["' Lt - e'"(s)][l * .'"(s)]

' f "*' ro"çt¡t d't s d,s " ' (4)IJ l¡-r I

Changing the variable t'o E : s - r, and neglecting

..u(r) by assumptions (i) and (ii), Eq' (4) reduces to

Aþ"0@)*Ë9'ó2,@) + "'

/l tt - e""(a * r)ltl * e""(a t r))

' 1,,.,* e¿"(t)t d't (a * r) d'v " ' ' (5)

.4,UGUST T966

2zr -:; L"n"

1590

5 À. A. Broyles, J. Chem. Phys. 33, 1068 (1961)'

ON TIIE PERCUS-YEVICK EQUATION 1591

asymptotic Eq.consrstent to

this orderType of force

present

Attractive only

Repulsive only

Mixtures(of charges)

Using assumption (ü) it can be seen that the mostimportant contributions to the right-hand side inte-gral in Eq. (5) arise when E is small, and hence, acutoff parameter "a" is introduced, where a 1 r forlarge r, beyond which contributions to the integralare assumed negligible. Further by assumption (i)the úght-hand side is finite, and since r is large and.y small, ,""(U I r) can neglected; so the right-handside can be expanded in powers of ó(A + r), to give

ßó*(r)+àA'ó:'(r) + "': 2" 2,"n" L" l-pþ".(a +r) - Èg'02"(a *") ...1

(*) 1,",*,"'

u"{')'o'or' (6)

As there are no general existence theorems for solu-tions of nonl.inear integral equations, even if weobtain agreement in Eq. (6), we cannot, be sure anexact solution to Eq. (2) exists. However, if we areable to satisfy the asymptotic equation (6) theremay exist a solution to Eq. (2), whereas if (6) hasno solution, no exact solution of (2) can exist.

For a system of pariicles involving attractiveforces only, it is evident from Eq. (6) that a solutionis impossible, since ó." will always be negative. Also,Jilï' ,^,r,".r,,..(t) dt is positive for small g by physicalconsiderations, and þ"¡ is negative. Thus, to firstorder the left-hand side is negative, while the right-hand side is positive; and to second order the left-hand side is positive, while the right-hand side isnegative, both orders being mathematically incon-sistent. However, by applying the above reasoningto a system of repulsive forces only, we see that 9""becomes positive, while Jifi" eb"(t) dt becomes nega-tive; so now both flrst- and second-order agreementin { can be obtained. For a system of mixed forces,several cases arise, for þ"0 can now be positive ornegative, and if @"u is positive, so a particle ,, a,,repels a particle ttö," then particle "a" ear_ atfuact aparticle "ct' while particle "b" rnay repel particle"c." (See Table I.) Because many of these casesare unphysical, this paper will be concerned withmixtures of charged particles. Then, if particle ,,a,,

repels particle t'ö" and attracts particle "c,,, particle"b" will altract particle "c" àlso. For these chargedparticle mixtures, first-order agreement follows bythe same reasoning as above, but second-order con-siderations lead to disagreement-on using âssump-tion (iü).

For a Lennard-Jones type of interparticle poten-tial, which is repulsive at short distances, and falls

T¡sLn f. Summary of the consistency of Eq. (6) forvanous cases.

NoNoYesYesYesNo

Q.&"a :1 + I" n" [ {n," - r)s""G - e"") d,"r"

+ + >,"rr" 2,o"" Il (su" - l)(soo - t)Q"og""Q"o

.(L - e"")(t - e"o) d,"r. d,"ro. (Z)

Rewriting the "Iast tem" above in terms of inter-particle distances, with the same notation as inEq. (2), we obtain

'f >"n" Eon, I"' I"- I,'.':"",' I,'.':.",'

J" \s,,"(t) - 7llsu"(a) - rls""ß)s""(ø)11 - e,"(s)l

'll - t:"¿(u)lga"(w) dO u du t dt u d,u s ds,

where

2r"w' : 2r'(s" I u") - (u" t r' - u")(r, + s, - tr)

I lhr's" - (r" + s2 - t"¡'11

'lLr'u" - (r' + u" - u')')r cos o.

Order inó""

FirstSecondFirstSecondFi¡stSecond

Whether(6) is

off rapidly, the Percus-Yevick equation applies well,and, in acdition to a solution to Eq. (6) being pos-sible, a solution to Eq. (2) has been found. However,for mixed Coulomb interparticle potentials an exactsolution isr clearly not possible as there is an incon-sistency irL the second-order terms of Eq. (6).

III. ÄDDITIONALTERMS FOR MIXED FORCES

Greenn has proposed an integral equation which,compared with the Percus-Yevick equation, con-tains additional terms. As each term can be ex-pressed in powers of ô"b(r), which have been assumedto fall off with r, only the first additional term wilIbe considered. The equation for the radial distribu-tion function now becomes

5921 .A.. A. BA.RKER

2o E"n" lon, I" "ßö".(p * r) . '.1(t r p/r)

'l""wø",{o*r)"'l(1 + oh) [,',*," r0"(ì) 1,",',"',,u@)

.f " U + €,"(,ro)l d'o a du t d't d'q d'p,

If this term is treated in a manner analogous tothat adopted in the previous case, an expansion interrns of 6(r) reduces, for large r, to the followingform:

rV. CONCLIIDING REMARKS

The Percus-Yevick equation can be successfullyapplied to systems composed of particles with repul-sive interactions at short distances. However, iÏattractive forces are present, for the existence of anasymptotic solution it is necessa,ry that correctionsof the type suggested by Green should be included.The main disadvantage of this additional term isthat the equation can. no longer be expressed in theconvenient form of (3), and so computational solu-tion of the equation will be correspondingly-moredifrcult. In the future the author hopes to publishcomputed radial distribution functions for a proton-electron plasma for various densities and tempera-tures.

ACKNOWLEDGMENTS

The author is indebted to Professor H. S. Greenfor many constructive discussions on the abovematerial and to Dr. P. W. Seymour for his usefulcomments.

where

7t)" : t' I u" - 2pq J-2(f - p1'ø' - q')å cos d.

When this term is added to the second-order termof the Percus-Yevick equation, it makes the asymp-totic equation consistent to second order for chargedparticle mixtures, by adopting arguments analogous

to those used previousþ.

Barker, A. A. (1968). A quantum mechanical calculation of the radial

distribution function for a plasma. Australian Journal of Physics, 21(2), 121-

128.

NOTE:

This publication is included in the print copy

of the thesis held in the University of Adelaide Library.

It is also available online to authorised users at:

http://dx.doi.org/10.1071/PH680121

iyionte CaiÍo Study oi a l-I5,irgf,enous Piasma near theJonization lempeiature

;,:. 1

', ' .: A. A. Bo,*¡r : ,: r r,,1.¡l..,,:,r i ,, ,rij t,,r

'' paþarlmcnlolMalhcnolù;atPhysìcs,tlnhersityo!Adelaide,AdelaìdaSOOO',Soøf AuryoËXt,"':: :. . ': ;'

. (Received 18 December 19ó7)t I ,. ,,,; .. llJ i:, '¿ | .);i,. l

Results of a recent lllonte Ca¡lo study of a hydrogenous plaqma nesr the ioniz¿tion temperature'sþgry )¡l rtriii '

_ posed at small radü oa the Coulomb potential bctween unl r;,,i.,,r,,,,:i;. ,: I )r,rlj j, ,

,'l:t: ij .

,/ jr Íll ,r i,, ., . \ r/,,,,1i1

' 1,,,,;,'; i,r-F;E probiem of obtaining distribution functions rounded by a network of identical ceils, thus enabüqg4 íoi long-range forces has been considered by the energy of a configura'uion to be óalc-.rlated con]

,ffi':l¿;:i !;l'gi,l,xi:äl',il'Í, äîi.,Ti[ ï.'"ff:'å'iååîi;:,it'i;"oi:"'."i'*::study of a one-component plasma imum value A. The energ-y of tbe new configuration is- has been completed by Brush, Sahlin, ãnd Teliìr.3 In calculated, and the MC f,rocedure d,ecid.es if-the move

tiis paper tåe author presents the results of extending is acceptable or not. EacL particle is consid.e¡ed it this',åe MC procedure, described in detail by Barker,{ to a manner until the system approaches an equiübriurntwo-component plasma,

- and

_ particularly considers energy level. The criterion foi-the choice oi Aìs usuaily

iemperatures in the region where ionízation occurs. based on minimizing the rate of approach of the systenrThis region is of considerable interest, but is also the to equilibrium; howlver, as is shônn below, the iesulsmost di6,cult to deal with from the mathematical of ttris calculation indicate that ottrer considerationsstandpoint. It is found that for a plasma of density|Ðrïef cc a'u a temperature of 105oK, acceptable radialrüsl¡ibution functions a¡e obtaiued using the MC'rechnique, and below 9X 103'K the particles becomepairei, forming'¡-he neutral gas. However, in the range10{-5X1üoK, the plasma appears to behave as amixture of two pbases, ionized and un-ionized. Whichphase dominates is influenced rather sensitiveiy by aparameter A.

As the parameter A assumes some importance in ttrefollowing discussion, the manner in which it arises willbe briefly discussed. In MC calculations of this t¡æe, anumber of particles-ló protons and 1ó electrons in thiscase-aie placed in a unit cell. This unit cell is sur-

should also be taken into accouat.Another important choice is the cutofi imposed oa

tåe aitractive Coulomb potential at short radü. Theneed for this choice is aiso c¡.cor¡ntered when trying tasolve integral equations with attractive Coulomb io;c:.,,;present. It can be overcome'oy treating tàe close hic¡-actions quantum mechanicall.y (QM), and Barker6 andStore¡o have independentiy calculated, wi'rh cioseagreement, efiective potentials ó. which should be uscdwhen unlike particles approach closer than a certaildistance r¡, which depends on temperature. However,the MC calculations were compleied previous io tirecalculation of 6u, and the results are presented in Fig. 1,

which shows the unlike racüal, dist¡ibution functiorguo obtained from itèrations 30 000 to 50 C00 rvithA:12.5øo (øe is the Bohr radius), and u'sing the usi;aiCoulomb potential {a but with a constant value below

¡ A. A. Barker

_ ¡ A s, FJ, I,, g¿hlin, and D. D. Carley, Phys. Rev,Lette 19ó3).

â D , Phys. Rev. 131, 140ó (1963).I S. H. L. Sa¡ri¡D, and B. Teller, J. Chem. Pbys. 45,

2LC2

'4, A. Barker, Aust. J, ShyrÍcs 18, 119 (1965).

r=2ø0, A classical distribution function go:epd" isd¡awn for comparíson, and the equivaient quautummeclianicai casÇ gqu:eró' is also shon'n. The guc faiis'uo reproduce gu at small radii, and this is almost cer-tainly due to too large a choíce of the step size A, withthe result that not enough sample points are consideredat short interparticle distances. At higher radii, grurc isweiì above 8", which implies that two unlike particlesprefer to remain some 25 cs apart. This implication isconñrmed by fclose study of particle movements, fromwhich it is found that two unlike particles tend to move

¡

around the cell together. - -_ __ Ì ,;. _ , .i, - i

r R. G, Storer,, Aust, J. Physics 21 121 (i9ó8).J, Math, Phys. 9, 9É¡ (i968).

a pair (effectiveiy neutrai) and an íon or elcciron. Thechoice of A also infiuences markedly both the rate ofapproach of the system to equiiibrium, and even theequilibrium energy levei attaincd.

Figure 2 shows the variation ot the ceii encrg'y peiparticle (16,protons and i6 eiectrons ir^ the unir ceil),with the number of iierations com¡lctcd (each itcratioa .

i¡t tho unit ceiì o cliadco tû ¡novc B.-..*-.-'':,.;:..'.'..-'J*

givcs every particle

t7t MONTE CARLO STUDY OF:f .. .

PLAS M A i¿i I

9

i '.,'. ,

.'iì |

',,,'i

r

!

I

It, 'i':¡.,,¡,';.):,r' .t !',, .,rl:'

' ": .". i'tt.,'.,¡ij tti :i. i:r,'ì l.ì ,., j l,'.1 . :,)r!.jl ii',i ii ',. l,,u.i:,,1 ' ;

C,;1,, - :rl',jr,.,!t -,.! r.i.

,,r.t,rrl.i,;,,¡l','..,t, . ;., . ì

' -,,,ì.i,,j¡¡l rì..1',: ;.,.1... r :i

, ".'.i .,"'; ':.í. 1(1i ,: ;, .' , t ..

È

NKT

qgÌ

oao

. , 'Jj i,ì,r;"3I I i':

I'

, ìi

i rl; l' l' ii1', rir i

tttt.ìt it. I'

i ¡- I i

I i l'"ii i; lll,i ¡. ri

',i))

-L0

50 100 EO

20000 40000 60000

NO OF ITERATIONS

approximately the number of pairs,existing. at this ene¡gy.

: (BoHR RAull) i

F¡o. ons, (drawn on dlogscsle),for ¿ h i0t8 efcc and temperêtutelüoK, 50 ooo.

maximum distance Å). In (b) and (c)'il can bc seen that the temperature range nea¡ ionization, as at low an<i ., the equiiibrium energy is extremely scnsitivc to A, and high temperatures the resuits are independent of A for :

(a) and (b) show that approximate equilibriuin energy , a trong enough run. trt is ia this iespect that the plasmais reached faster by approaching fro¡h an ordercd appears to behave as a rrlrture of two pbases in ihe :

configurâ'Lion.Pairingissaidtooccurwhenthepotential, region of ionization, wiih lhe choice of A deterrniningenergy betwcen a proton and electron is greater than which phase dominates.the ionization energy (this occurs when they are iess In conclusion, then, the potential g. shouid be usedtíar^ 2øo apart), and (d) shows approximately the in MC calculations for attractive Coulomb i¡teractionsnumber of pairs in the unit cell at that energy. The where r(r¡; and the maximum step iength A musi.'c,estudy of particie movements mentioned conflrms that carefully chosen when in

"he range near the ioniza'úon

the degree of pairing depcnds on À. Previous MC calcu- temperature. It might be preferabie (though uroieiations3,l'7 have emphasizcd that results should be expensive computationally) io use a step size with aLrdependent of A, and the magnitude of A has been, Gaussian distribution, corresponding to the Boltzmannchosãn empirically by consideriig the rate of converg-' distribution of energy in the radiátion ûeld, iu this

I ence oi the system to equilibrium, It has been found range. Howeyer, because of the excessive coroputingthat L=L/(31/) =$.S¿o in this case, where,Z is the dme involved (3000 iterations tal¡inø iä on a CÐC,

: unil-ceil iength, and Iy' is the number of particles in the ó400 copputer), dist¡ibution functous shourd 6ecell, has the right order of magnit'rde to _secure ne¿r' obtained'more ácono¡:oícagy in this region by sol,vlng a

r optimum convergence. However, in the region neâr. th.e - .oain.¿ percus-yevick áquations ãnC caicujationsI io¡ization poiniìt appears as íf  is anaiogous to a iimit using this approach are at p;ese't being caried out. Itoi the encrgy of quanta absorbed or emitted from the Ís hãped by lomparirrg thåse results w'i¡h the MC re-raciiationfield,an<iif Áissmall,.oneparticier_naymoye ,uftrîr.rolve the diiemma of the choice of A, andsiightly away from rhe. inreracting parricie, lyt rarel¡ h;;;" i,"p."ve the MC resulg.escapes fully;whereas if Á is¡eiatively large the,parti- - ifr.-

"itf,or wishes to acknorvledge the heÌpful

cies completely separate. This behavior is peculiar to ,ugJ.rtiã", of p¡ofessor II. S. Green ãnd Dr. p.-W.

, t w.w.wood andr. R. parker, J. chem, phys,Z1,??Q (iqsÐ. Ï{i"ir on this work. The MC resuits we;e compieted

r , A. A. Barker, Phys. Fluids 9, 1590 (1966)'

I(d)

-L

il : :r':: I 3

,,i ,.,ir.

l:H___.

8.1

APPENDIX B

FORTRAN PROG}ìAM}IES

1 To eval-uate the Q.ln di stribution functionsa

The program listing is for the calculation of

snu(r)r for r varying from .5ao to 117ao in steps

of one half Bohr rad.ii. The prefix or suffix HBR

signifies the variable expressecL in hal-f Bohr

rad-ii¡ and. BR signifies Bohr rad.ii. The program

only need-s slight mod-ifications to calculate gu"(r).The charge term SHZ becomes -1 e the reduced. mass

REDùÏ becomes O.5, and- the bound. state cal-culation

(statements 20 to 16) 1s removede v¿i-th several minor

pro gram ehanges.

A separate fortran program TVas v/ritten to

calculate the first bound state contribution and

eval-uate the d-j-stribution functions at zero rad.ii,

but as it is mainly a simplification of the program

listed., 1t 1s not includ.ed-. Tlro ad.dltional programmes

were run to check the values of the bound and-

scattered- contributions by using alternati-ve techniques

as d-lscussed- in section 4.2. These were run only

for sel-ect r val-ues, and. are also not listed..

8.2

2. 1o solve a L{od.if1ed Percgg@

The prograln reads the results of the l-ast run

from tapee transfers them to another tape using thetFather-daughtert p*oced.i:re, and. begins this ca1eu1-

ation using the gro(r) d-erived- in the previous iteration.Suppose for d-efiniteness the jth iteration (fmn) f,as ,

been read- in, i-oe. g(i)(r), where n(j)çr) refers to both

I1ke and. unlike câseso the variable LA is given the

value 1 to d.enote interactions between like particles,

and- 2 to d-enote interactions betvreen unlike particles.

LC and- LD are used in a similar role d-r:ring ¡þs ). and.

Ðd. The equation is progranmed. f or the coinputer in

the f orm: -

e0 (LAer,R+1 ) = (t +rott'rr+FDTlil)/nxp(ez(myr,R+1 )

where LR 1s the rad.ius 1n Bohr rad.il (1 is add.ed 1n

ind-ices to avoid. storage d.ifficulties when r-O).

TDTI'4 is the PY tern:LRF s+LR

[1 -"."(s) J.Bac(-) [s¡"(t)-t ]tdt.sdsTs-I,R 1

for = erc¡) lpz(ucrLR+1 ) ]

s""(r) = explçz(t¿c,rn+1 )l

=+ ) D- tLav!vo

"r" (o )

t

8.3

FDTM is the ad.d.itional term suggested by Green:

I,RT' IRF LR+S LR+U T2r fJ"l"

ndd

nctor

I rn-s I lm-ul

[*o"(t)-1 J Ieo¿(")-t J sr"(s) e"u(u) [i-"ra(,r) ]s."(*)

d dvd.v td.t ud.u sd.s for

2r2w2 = 2r'(s2+u2) - (u2+r2-v') (r2+s2-t2)

+ 4[r'"2-(¡z+s2-t2 )']+ [4*"o'-(*" +rf -v2)"]+ cos0 .

The TDTM and FDTI\t are d-ecomposed. into terms obtained_ by

completing the respective summations to give

TDII,Í = TDCON*IIO(r-,nrt) + TD(r¿,2)]

FDTM = FDCON*[¡'¡ (r¿, 1 ,1) + FD (t*t,l ,Z)

FD(tA 12r1) + trÐ(rt,z12)l

+

a

The two d.imensional integration contained. in TD(f¿rf,C) ,

is carried- out in the D0 loop from statement 1O9 to 1O2e

ard. the five dimensional integration contained. inIÐ(LATLCTLD) is completed in the DO loop from statement'

1O2 to J2.

As descrlbed ln section 3.3¡ the integrationproced.ure flnally adopted. was a sirnple trapezoid_al nu1e,

The mesh ratio used (UttSe I,IIHU and tr[HZ for the DD case,

and UIHT and- MHF for the 2D case) were altered. aecording to

the region of integration, i.e. to MHI (fn¡ and- IIITT (ZO)

B'4

for the "inner" regione to lrH(5D) and. MIfl'(2D) fon

the ltmainil region and to Um,a(5O) anA IÍrf,n(ZO) forthe t'large rtt reglon. The procedure used_ to obtaing(r) for non-integer values of LR was to d.o a llneanlnterpolation betvrreen GZ(LAeLR+1 ) and. GZ(T,t\eLR+2),

i.ê. on the logs of g(r).

The final GO(tArtn+1 ) obtained_ is stored. (statement

1BB), and the GO(lartn+1 ) i" returned. to the start ofthe program as gj*t. A simil-ar iteration is mad_e but

the G0(r-.AeLR+1 ) d-erlved. are noïï stored. as g j*zL (statement

195) and. then r:.sed. to obtain a goo as described insection þ.2 (see statements 190 to j73). The val_r;e ofg_ may be mixed. ("u tn 199) with the gj read in initially

to obtaln the next gIN, or may be used. directly as input

for the next series of iterations, gïN, sj*j, gj*4, to

obtain the next estimate of g*. 0n each iteration the

main results are printed. out (ZO¿+) and written on tape

(t7t) .

The progran used. for the llf calculation 1s

essentially the sane as the one listed. ( nut d.oes not

contain the time consuming 5D calculation) and so isnot presented. Sna1l progralnmes to calculate g(o)

from equation (a ) and the therrnod.yrramic integrals

I and. J referred to in Chapter J are also omitted..