University of Adelaide · 2017. 5. 9. · t, i I CONTT]I\TS iIlTR0DU0lïON 1 .1 The Rad-ial...
Transcript of University of Adelaide · 2017. 5. 9. · t, i I CONTT]I\TS iIlTR0DU0lïON 1 .1 The Rad-ial...
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 .
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Pereusr J.K. ârìd. Yevick¡ G.J. (lgSA) pfrys. Rev' ll!,t
Percuse J.K. (gAz) prrys. Rev. Letters Qe 462,
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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
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Lebowitz, J.L. (196\) prrys. Rev' E, AB95'
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verlet, T,. (1961+) PhYsiea flt 95'
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Throope G.J. and- Bearilâh¡ R.J. (tgge) J' Chem' Phys'
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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.
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Van Leeuwen, J.If .J.¡ Groenrrueld-, Jo and- De Boerr J'
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Klein, M. and. Green, };1.S. (19Ø) .1. Chem' Phys.
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Khan, A.A. Ug6+) errys . Rev. 1A, A367 i 1J6-, A1260
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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.
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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.
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55.
Friednane H.L. (lgSg) ulo]-ec. Phys. !r 23t 19o and 436
Meeron, E. (lgSl) J. Chem. Phys. 26, Boh.
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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
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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.
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PhYs. Rev, Lett. -!:Qe 319.
tga] carleye D.D. (lg6l) Èrys. Rev' 717-, 1406.
(lg6S) ,1. chem. Phys . [!, 3489
(1s67) Éc 3783
igg] Johnson, trf .D. and }.ilarche N.H. (1963) ffrys' Lett'
3, 313'
[too] v1ltars, D.s. (19Ø) prrys' Fluids !r 7)+5'
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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..