i S=.^> s

IC/83/216

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

STRUCTURE FACTOR OF LIQUID ALKALI METALS

USING A CLASSICAL-PLASMA REFERENCE SYSTEM

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

G. Pastore

and

M.P. Tosi

1983 MIRAMARE-TRIESTE

J • : t ; • * ' . ™ l m-jurmamm ••<!'•

IC/83/216

International Atomic Energy Agency

and

United nations Educational Scientific and Cultural Organization

INTERNATIONAL CEHTRE FOR THEORETICAL PHYSICS

STRUCTURE FACTOR 07 LIQUID ALKALI METALS

USING A CLASSICAL-PLASMA REFERENCE SYSTEM *

G. Pastore

International School for Advanced Studies, Trieste, Italy,

and

M.P. Tosi

International Centre for Theoretical Physics, Trieste, Italy,and

Istituto di Fisica Teorica dell'University, Trieste, Italy.

MIRAMARE - TRIESTE

November 1983

* Submitted for publication.

liill * .1

Al!2±Z££i' T n e pspni- presents ca"! yulatiuns of the l iquid :-t .-.ctuii::;

factor of the alkfill ivetals near freo^in,?:, s ta r t ing from the

c lass ica l plr'cma of bare ions as reference l iquid . The indirect

ion-ion intoras t ion ar is ing from elcct ror ic r.eroen.in;- i r t reated

by an optimised randon phn.fjc a.pru-oxi?<'Ltir>n (OHPA), imposing

physical r^nnirenenbfj as in tho ori^inr.1 ORPA scheme developed

by Weejcs, Chandler mid Andersen for l iquids with otrone.iy repulsive

core po ten t ia l s , A ooiiparison of the rec~o.lta with corr.vater simul-

ation data for a, model of l iquid rtvLidiuin shor.'s th;;t tv.e prevent

approach overcomes file well~!a>own d i f f i cu l t i e s net in applying to

th.see nstslr . the standard OTt'PA based on a reffranco l iquid of

neutral hard spheres, Ihs optimisation. sch<w:e i^ sO.so Hhovni to bo

equivalent to a reduction o±" the ri'.n^..: m:' ti;.c inflircct interaccicri

in monentur.i apace, as propoced eiTi.rir-xcrilly in ea r l i e r v;or!c, CoiTCiir-

itiofi with t.'periment for tli.e oilio.1' ;iX:;\ili6 hh<r.jis tji.;\t a good

overall reprerjentfition of the o.s.tn <.-.!>: 7;c- obc:"-i Tioci

potasEiuiji .'d-i;: osniura, "out; not for I i i •!. inrt, •vh

ci:nT;le for:" of th~ electron-ion "o tcjr'i-i'C1. rO.ii.

co;:vpro:;-i"LU.ity. The c.i:alA..-^i!fi? f:;: •••; i;or. •.•.:.•;: rt

expminau norc carG:;.Vil].y i,n tlio l,i.:'-,.n":; n." i-iH:.;:r;l

a v i f i " / t o j k - o : ; r 3 i ' b l c ; n f i ^ c r : : ! 1 ! : ; o : ' i.i.<- : T : ' " K " : . i - i .

one USES a very

i od to '-ho "i.inMiii.

Among perturbativc theories of the liquid structure factor

S(]t), the optimised random phf.use approxiriation (ORPA) developed "by1 2

V/coky, Chrmdlcr and Andersen * ha.s been very successful for l i

with strongly repulsive core pot entitle, includiiis sir-rplc liquid

mctfils such 3.3 I.lg and Al» After brep.liiiig the interatomic potential

into a purely repulsive term and a purely attractive term, this

method adopts the fluid of neutral hard spheres as reference liquid

end treats the difference between the true repulsive terra and the

hard core potentip.1, as well as the attractive term, by suitable

pcrturbative techniques. In particular, in dealing with the

attractive term this is modified in the excluded-volume region,

where tlie particles do not penetrate, in a way which entires "that

the pair distribution function g(r) continues to vanish in this

region.

A number of authors have shown, however, that this method

yields disappointing results for liquid alkali metals, essentially

independently of the detailed procedure followed in constructing

the interatomic potential by pseudopotential theory. This failure

is commonly ascribed to the fact that the core of the interatomic

potential in these systems ±3 too soft for the existing ORPA

approach to cope with. This conclusion has been confirmed for NaA

in the recent careful study of ncLGAighlin and Young,

In aJn entirely different approach, there have been various

attempts at evaluating the liquid structure factor of alkali inetols

on the basis of the one-component classical plasma (OOP) model.

This is a classical fluid of identical point-like charger; on a

rifad uniform background, so that i t s direct use for real liquid

reetals includes the Coulomb repulsion between the bare ions but

o::ii.ts the indirect ion-ioa atbrp.ction arisin;; fx-om electronic

screening;, the properties of the sea of conduction electrons

- 1 - - 2 -

• ' - ' • . • ' i - y f l n p , b h f . 1 j - . i o d f ' L o j ' l l y '• . a : ; o i ' . . V .-:•.•* i • „ ' , • . - . . : : . i j :..:, .• i.:•:<.,• d j . i " ! • • ; >

Eieacaretl density of the metal. luinoo jrfc al vn& subsequently other

authors * have rieverthtO.e;.;^ shown that the :s Iructuro factor of

the OCP gives a good representation of S(k) .tor alkal:i s i.n the

region of i t s main peal;: raid higher momentum transfer:!. Electron::.•

screening i s crucial, however, in the small-angle scai/Lering region,

and. Chaturvedi et_ al ' have proposer! a v.'ay to include this through

a random phase approximation on the indirect ion-ion interact:!.on.

I t vjas found empirically that this interaction had to be cut off

in. momentum space in order to retain trio pacccas of the OCP model

at largo momentum transfer:;.

We present in th is paper a theory of S(k) for the alkali

metals which combines the above approaches by adopting the OCP ;-nodel

as the reference liquid and treating the electron-mediated attrf.ictive

term by nn optirriized random phase npproxiiiiation. Thfe craoial point

i s that , at lar. rc values of the co'J.-nling strength approaching

those of the direct ion-ion Couloiib rcpuj.;;,i.on in thorje liquid

metals near freezing, thn OGP rcodol ponccoccs a very well defined

region of excluded, volume around oneh p;-.rticlo. This i s very clear

from computer nimu3.ation d-.vt;i ' " for c(r) in the OC?i?, ejid indeed

the structure of the strongly coupled OGP c:n\ ho su.ccGSsful.ly

aroctellcd by a Ru.itahle hard-sphere liruict'."'" This f/tructural

property of the reference fluid ;O.lov:c;, tir. in the cri^insil OKPA

approach, an optirdKation of the povturb^.iivc treatr-icn-i; of the

imiirce-fc ion-ion attraction.

The layout of the paper i s ::.:• follovii1.. Aftor a presentation

of the theory raid of the rcfurcnoo: f'l>,i.\c! rjtrn.cburc in r;eotio:i 2,

the theory iy teritcd in eefiticm J r\'-;.iiriGt co:uputer si'iMlntion data

for a itLOdel of liqi^d Rbv 'I'.-AI :"ccoar:;i.' to fli-latifin data. "hn;;'3t\

on a particular l!TC-:-rcXo- \\-<: po\;i--v; i.YL '.?:vi:.il:-::-i\;:-;;u.d "h,; •J.T;c:'idO"ot^:itial

t h e o r y , c l ( ; a i - l y P 1 1 O V ; B ;•. t ' ~ : ; t o : i . " • • ' • i - .••! • ' i . • ; " ! • " i . c ; i i i r j i . - c h • i v . i M l

a p p r o a c h l.itiforc r a i d i n g I:ho r \ i e - ; l i.ijit c i ";;:•" ; c c u r s ' : ; o f t h e p o t o n L i a i

j ,;i dCirjori'^ir..-, t h e r ( ; a l l i o a i d ^ - J t n l . C o n t a c t vvit:i oui1 c a r j . i e rri Q

empirical treatment of electronic screening ' is mode in the

same spirit in section 4. A discussion of our results against

cxperirrontal data for the liquid alkali totals is presented in

section j and, with special attention to the swall-angle scattering

region, in section 6, Finally, a summary and some conclusions are

given in section 7.

2, Optimized treatment of electron-mediated attraction and

representation of reference system

We write the interatomic potential q>(r) using a standard

pseudopotential formalism. This yields

2<f(r) = 2 + u(r) t2*1

wher^ the indirect ion-ion attraction u(r) is given in Fourier

transform by.2 2,,, .

in terms of a model electron-ion potential v(k) and of the dielectric

function £(k) of the homogeneous electron gas. Two properties of

the two terns of <f(r) in eqn (2.1) should be stroosed: (i) they

cancel each other exactly in the limit r -» oo , leading to the

Gmytein-Zernike formula for the structure factor S(k) in the

limit k -» 0, Eind (ii) the terin e"/r dominates over typical foras

of u(r) vrithin the excluded-volume rtsgion of the OCP for values of

the coupling ctrcngth typical of alkali votcJ.s not far above

freesr.ir'':. V,'e may for our present purpocer; define the radius <J of

thin region :is that value of r bei ov; v;hinh the pair distribution

function of the OCP i" unobservably O-I.T.11 in computer siimilation.

Property (ii) has boon explicitly iT lust rat r-d for instance in

ref. 14.

After addition and. subtraction of a rigid uniform background,

the Coulomb repulsive term in cqn (2.1) describes the OCP reference

system and the term u(r) is treated by perturbation theory, This16

yields for the Helir.holtz free energy density f of the liquid, to

within a stracbure-independent additive term f0, the expansion

f = f0 + |n2jdr go{r)u(r)

+ (higher-order terms) (2 13)

v/here n is the ion number density while go(r) ana S0(k) are the

pair distribution function and the structure factor of the OOP.

The terms that we hr.ve explicitly written in eqn (2.3) correspond

to the random phase approximation, which embodies the exact cancel-

lation of the ionic Coulomb tails mentioned in (i) above. Explicit

expressions for the firstfew terms of higher order can be found

in ref. 16: it ia important to notice here that they involve a

screened interaction and thus give only short-range corrections

to the KPkl

We can now proceed to optimize a truncated form of the

expansion (2.3), and in particular the RPA expression, by the same1 ?

procedure followed by Weeks _et al . ' To the extent that the

particles do not penetrate into the excluded-volume region, the

values of u(r) for r < CT are immaterial, 'l'he real potential u(r)

can therefore be replaced by an effective potential u(r), ,^ven byu(r) = u(r) for r>« (2.4)

end arbitrary otherwise. The optimal choice of u(r) for r<-& in

a truncated form of the free energy is based on the requirement

that the functional derivative of such a truncated form with

respect to u(r) should vrminh. for .v < u * This condition enr/uri.-;;

that the pair distribution function g(r) oslc-Jated for the; real

region remains the sama FLG in the reference liquid. Such a

physical requirement is badly violated in the EPA and must be

restored by the higher-order terms. V*'e stress that the replacement

of u(r) by u(r) for r < D' is not motivated by inaccuracies in the

determination of u(r) by pseu&opotontial theory, but by the need

to coiTect for the statistical mechanical errors introduced through

the xise of a truncated form of the free energy.

One finds in this way from the EPA expression for the free

energy that

g(r) = go(r) - (2u)~3n"1jdk p(k)So(k)[l + p(k)S0(k)] ~

1 (2.5)

and

S(k) = S0(k)[l + p(k}So(k)]-1 , (2l6)

where

p(k) = nu(lOABT . (2.7)

In the determination of v.(r) we have age.in followed Weeks _et si

in talcing an expansion in orthogonal polynomials, namely

u(r) = u(r) + 5(<y_r)[q - s(l-r/cr) + (l-r/crJ^^o^tZr/j-l)] (2.8)

v.'here §"(x) is the Heaviside step function and P (x) is a Legendrs

polynomial. The coefficients in eon (2.0) are determined from

the condition

r)l<tf . (2.9)f o r

Su(r)

as noted above. The inclusion of seven coefficients C was

sufficient to obtain convergence in S(lc) to within 1$ in the

calculations to be reported below.

In the i plemfinbr.tion of this programme one would ideally like

to have computer simulation data on the structure of the OCP

rei'erencc fluid at -nrGGx:;cJ.j: those values of the coupling strength

l iouici vaniiihcfj foi' nii ' icl" •Ull/ lJ tho irrc.'l VI Jud vol',i::ic

F= e /(ak T), with a = (4n.n/3) r for which siirrul-rSion data

and experimental data are available on liquid alkalia. Most of this

evidence refers to the alkalis close to the freezing point at

atmospheric pressure, with values of e /(ak T) which are somewhat

higher than the value To; 178 at which the OOP is "believed to17freeze. No simulation data on the structure of the supercooled

fluid OOP are as yet available, but a supercooled fluid state-11-7 ^

seems to have "been seen at 1 = 200, with an internal energy which

is well reproduced by an extrapolation of the internal energy

expression for the normal fluid state* V/e have consequently chosen,

in the calculations reported below, to extrapolate to higher f's

a theory of 5 0(k) and g o(r) which is known to be accurate up to

1 = 160 by comparison with computer simulation data! '

The particular OOP theory that wo have adopted is the generalised

mean spherical approximation (GKSA) of Chnt.urvecli et al. The

main reason for this choice is that the G-T'SA incorporates the

free energy of the OOP from computer simulation work and therefore

also the low-k expansion of S 0 ( k ) , covering the sraall-angle region

up to ka 5L 1.5. This is the region where the results of the pro«ent

approach are most sensitive to inputs. ',7e shall point out explicitly

later on the errors in S(k) at large momenta and in g ( r ) , from

our approximate representation of the structure ox the reference

fluid.

Since the GMSA imposes through the virial theorem that g o(r)

should vanish sit some value of r, the value of a in cqns (2.8)

and (2.9) is also thereby knov.-n. In this respect, the main quali

difference betv.'oen our approach aaicl a standard OKPA is that go(*0

vanishes at r = a , instead of hewing a finite contact value an in

a reference system of neutral IKVE-J sphorcs.

3. Corny:r.rison vjith con-outer sirrnilr'tion of licuid rubidium

As we have already pointed out in the introduction, the main

test of the reasonableness of our approach lies in comparisons with

computer simulation work on models of liquid alkali metals with a

chosen potential <f(r). Liquid Rb, for which both molecular3 9 20

dynamics'" and Kont<j Carlo data are available, is the bast test

case. V/hile we are not unduly concerned at ' he present stage with

the reasonableness of the choice of u(r) in representing realliquid Rb, we may point out that it is based on the standard

21Ashcroft f oral for v(lc) ,

2v(k) = - |-cos(kr ) (3.D

with a core radius r determined fro:n phonon dispersion curves in22 c

crystalline Rb, and that it has also led to satisfactory results19

in computer simulation work on the dynamic structure factor and23

on the liquid-solid transition of Rb.

Figure 1 reports our results for S(k) at 319K, with a magnific-

ation of vertical scale by a factor 10 in the small-angle region,19

against the results reported by Rahman. A similar comparison with

data of ttiounu&in is given in Figure 2 for g(r) at 35OK. It is

clear that our statistical mechanical approach, which involves no

adjustable parameters since F ami or have been fixed a priori from

the true value of e /(ak^T) and from the theory of the OOP, is

working rather well! Indeed, the residual discrepancies with the

siinulatioii data in the short-range order of the liquid can be traced,

back to inaccuracies of the CKISA for the reference liquid. These

discrepancies arc a loss in phnse of the oscillations in 5(k)

around 4 A and inaccuracies in the first neighbour shell, in

particular in the height of the pain peak of g(r) and in the

re;;iclunl penetration into the correlation hole of each particle.

On corouv tin;? tV^o ^leor^tic;: 1 rBsu..-ta for u(k.) Yd th thone for

the corresponding OOP, one car: assess the effects of electronic

screening on the liquid structure factor, These effects arc, of

course, crucial for k -> 0 where S0(k) vanishes as k , and extend

over the nhole Binal3.-an';le region and the rising pert of S-'v)

towards it.'j wain peak, but are negligible at large rr.on.ent; .. The

°—1position of the main peak is shifted lay only about 0,02 A to

larger k and its height is little affected, since u{k) tends to

have a node in this proximity (cf Figure 3 'below). These qualitative

statements apply also to the results for other alkali metals,

reported in section 5 (cf Table 1 below).

The properties of ii(k) i''. 1"r.tro ted in Figure 3 have ai GO some

relevance in view of the replication of the present approach to

calculations of 3(k) for real liquid alkalis, to be reported below,

These results surest that the detailed shp.pe of u(k) at large

\.:>xentv. should not be unduly important after optimisation. This

is indeed fortunate, since large discrepancies exic>t in this region

between different pscudopotentials available in the literature.

In thin connection, it seems that the Ashcroft potential illustrated

in Figure 3 tendr, to exaggerate the structure of u(k) at large

momenta, tut one is encouraged to hope that the optimization

procedure helps to stabilize the theoretical results for S(k)

against variations of u(k) in this region.

4. Optimized -potential

Figure 3 compares, for the same model af liquid Rb at 319K,

the effective J.IPA potential u(k) obtained from our opti.Tiization

procedure with, the ft:aie' potential u(k), over a relevant region

of momentum transfer. Clearly the optiTdif.-T.tion is leading to a

strong reduction in thu range of the indirect ion-ion interaction

in noi.ieatan space, to the point that u(k) ±; encentially ;:,ero for

momenta somewhat above thr> position of the nnln perfc of S(k)

(indicated j.n the Figure by a double arrov?).

These :rcircuits provide a ctatiotical mech;!.nict;l justification

for our earlier empiric;.1-! approach",' wriert u(k) v/.'-.s taken ecual to

u(k) up to its first node (marked in t-ie Figure L\" an arrov;) and

equal to zero thoroaftcr- Indeed, tlii» air--'.<-•_>. rcc/ipp yields values

of "00 whicli are dose to the thcorotitvi.! rc.-ultf; reported in

Pigure 1 above, v.dt]- dir crop.-.uicinr; of O.o\r.?..l bscor.ing visible in

the mriiii pcOv cuicl in tho r;;na].l-angle region.

5. Structure factor of alkali Hiet

In the following ca3.culations, v;hich are aimed at comparisons

v.dth measurementrs of S(k) by diffraction from liquid alkalis near

freezing, we have adopted two alternative choices for u(k), both

being based on the Ashcroft form for v(k) in eqn (3,1). The first22

choice is the potential adjusted by Price e_t al to phonon

dispersion curves, which has already been used for Eb in section

3 (model I). In the second, choice (model II) v:e have used the24

refrultr. of W'shif;hta and SiriyvrL for the electronic dielectric

function in eqn (2.2.) and have at the same time readjusted the core

radius r to fit experimental values ' of S(0), related to the

isothermal compressibility of the liquid notal through the Ornstein-

Zernike relation. A tabulation of the input data and of some results

is presented in Table 1, The thennodjmnmic state for the various

metals corresponds to rt teimcraturo of 319K' for Rn and to the

te::'Lperr:,turo of diffraction experiments for the other metalc (see

i-igares 4-7 bcJov.')j the corrcsjjoriding density iioiiv; laxen i'ro.a

Smithel l s^

The resu l t s for S{k) of Ha at 100C, K at 65G end Cs at 30C are

compared in Figures 4, 5 and 6 v.lth the X-ray diffraction cicta of

Greenfield e_t al and Huijben and van der Lugt . The ver t ical°—1scale has again been enlarged by a factor 10 for k below 1 A . I t

i s evident tha t , on the scale used in these Figures, i t is sufficient

to include information on the compressibility in the electron-ion

coupling to obtain already a rather good description of the

diffraction data for these metals.

The main regions of momentum apace v.'here there i s sensitivity

to the details of the model for the liquid metal are the snail-angle

scattering region and the region of the main perfr. V'e shall examine

more carefully the former region in the next section, making use also29of the recent diffraction data of Wasedn., Concerning the height

of the main peak, i t i s clear from i t s theoretical values in Table

1 that i t can s t i l l be aoaev/hat sensitive to the details of the

electron-ion coupling, although the comparison with the corresponding

peak height for the OCP in the same Table shows that the effect of

this coupling is rather small (excluding for the moment the case of

M from consideration).

qIn our earlier vrork i t v/as suggested, by comparison of the OOP

peak height with the data of Greenfield et_ al for Ha and K, that

a reduction of the bare-ion coupling strength in tha reference OCP,

to a value somewhat lower than the trae value of e /(ak '£) for the

liquid metal, vms indicated. It in evident from Figures 4-S that

such a reduction is not necessary, at lenst with the present choice

for u(r ) , if preference; is given to the data of Iluijbam and van der

lugt. On the other hand, i t has also been pointed out' that X-ray

diffraction experiment;1:; tend to ovtT^.rl'inate ftoin(iv,'Vi,".t the pen1/.

] • ; . ; ' ; • ' i r : t ' n ' :-.oi)-:-. o n :-'l T V C . 1 ! ' " :•;• j ••'••!••....r.- o ; 1 . ) . ''.'h •••'.•.••• f , u c ; ! . i w r ; r ' r v i . i n

interesting ones, in view also of similitudes in freezing' of the

Various alkali metals and of the OCP, but difficult to M G O S S more

precisely on the baoirs of presently available data.

Fiv. lly, our re cults for Li at igOC are compared in Figure 7

vdth diffraction data," Simple pceudopotentiala models have generally

had less success for this liquid metal than for the other alkalis,

and this remains true in our calculation of the liquid structure

factor.

6. "cattcring

The present situation for the liquid stn.icturo factor of Na, K,°—1 °—1

Hb and Cs in the small-angle region, for 1c between 0.1 A and 1 A ,

is illustrated in Figure 8 on a further enlarged vertical scale. The

theoretical results reported in this Figure are based on model II.

Severn! observations are immediately evident:

(i) the agreement of the data for Na from different authors seems

very good, if one keeps in mind that T/nneda'a data refer to a

slightly higher tempera-tare (1050 instead of 100C), and the numerical

results of the present theory need little improvement;

(ii) there ;-.re aome discrepancies for K between the Huijbcn-vpjri

der Lu£t data and the theoretical results on the one hand, and the

data of Greenfield et rl and of V.'ascda (the latter referring to 70C

rather than 65C) on the other;

(iii) there are clear discropruiclcs in Rb and Ca bctvroen the

theorcticr;! remits and V/aseda's data.

- J . l -

m»••• « . M

*_ 5 «•

Of course, the prflsent th^or;* should be .:ios^ ;i!:prcT;ri:;r':

precisely for liciuid Na, the Eei?':. vi'fcii the veal: set olcetroa-iorL

interaction. On the other hand, 7;a::;ed::s data :-:o-'.-:j the most Ei'C:;:1.,:

ac yet available in this region of ranerrtvia ir~~-i.:;i<:i , =;JIO c.nt. i r j

hope to obtain fro::i the;n sore detailed e'^ii'lcal ini'jjmation or: \\it>,

interionie potential in those rr-yLn:.-... ", "'Iiis ^us;::tion :-.s carrenfly

under inveaii£fitj on, usinj t:vj injirh-j rained in this "o:''J!:o

A related, question oo.r csnis uhe pr'&rericr= of a linear t e n

S(k) = 3(0) + a_ k + . . . in the small-.;.; expansion of the structure

factor, firr^t suggested by IJntthni am! 'Ife-rcli; Kaseds" '' lias

convincingly demonstrated that such a linear tsr:s i s present in his

X-ray diffraction jiaUemn, On the other hc-.n-i, no such term appears

in our Bimnle treatment of the electron-ion cotuilirv.

7» Sumrfiary r-.nd c qncluEin^ remarks

All the evidence jji-^sently available iudios.tsn that the OCP

model Cfin be uraeful !?.£3 FI ciartin^ point for tho theory of liquid

alkalis. I t would DO rr.ther hard, hoivover, to give a physical

Justification for adop'ting the zv.^.e starting point in polyvalent

simple metals such as K and Al. Without going into a detailed

discussion, i t may oe cu^fieient to point out here that ono v;ould

have to adopt a Coulombic; coupling nbrcn.fjth for a reference OGP

fluid which is much smaller than the value of (Ze) /(al<:_T) apnvo-or: ate

to the real metal. In the calculations reported above for alkalis

we have not had to troat the coupling; strength 1 as n. disposable

parameter, this attitude beiuj undoubtedly correct vis-a-vis tho

sintAlation data of Eali .-iri pad Mountain rind at least approximately

correct when confronted with experimental structural data.

?I-:iving adopted an OOP reference system, a aiinple RPA treatment

cf the Glectr^n-Tiodifjt^d ion--ion attraction correrjpondr. procisGly

to a treatment of tho oiectron-ieri coupling by linear response

theory. I t i s clear frorii our vrork that higher-order temits are very

irr.jiorisnt in the calculation of contributions from close collisions

to the free cracky and ine ;:rL--'ici;ure. For tun.it o] y, th-j effect of

such irrr.ns tiirnn out to fco largely e< •.;:.valnivt to a ruppretision of

electronic c-;cre-';nin;<v in cloye aoll:i BiLons, These coiiflusionn have

boon reached by our direct comparison with the sirxilation dr.ta on

Rb, but are confirmed by our study of the structure of liquid Na,

K and 0se

We have also seen that for these liquid alkalis, but not for

Li, one can obtain already a fairly satisfactory account of the

over""!.l structural data by a simple pscudoyiotential approach

adjusted to S(0), after including the higher-order terms approximately

through an opti'niportion of the PJ?A. 0?he insejieitivity of many

structural details to the details of tho effective interionic

potential emphasizes once core that a refined asrjessment of the

interactions fron liruid structure in these iietals CEUI only rely

on very accurate data in the small-angle scattering region.

Ackrio'.vlefirfemen11;;, V/e are very grateful to Dm A, Rahman and R.D.

Itcuntaln for sending us tabulations of their F3in.ilration results, to

Profof3,-or V.'. van rler Lugt for tabulations of experimental stn.ictv.re

factors, and to Dr H. !•;. De'Vitt for simlction data ^n the OOP. It

is also a pler-r/are to thank l)r 5. Senaboro for Ufnoful discussions.

This v;ork v-as supported by the Kinir.tero della Pubblica Istnisione

and by tho Conf;i,";lio N:\r.ion.ple delle Hicerche.

KEr'EHLNCES

1. J.D. Waeks, D. Chandler and H.C.- Andersen, J. Chcm-Phys. 34, 5237 (1971).

2. H.C. Andersen, D. Chandler and J.D. Weeks, J. Chem, Phys. _5jj, 3812 (1972).

3. Se.e, e.g. N.K. Ailawadi, Phys. Repts. 5_7_, 241 (1980) ,,nd references given

therein.

4. I.L. McLaughlin and W.H. Young, J. Phys. F_lj7, 245 (1982).

5. H. Minoo, C. Deutsch and J.P. Hansen, J. Physique LeUres 38, L191 (1977).

6. M. Ross, H.E. DeWitt and W.E. Hubbard, Phys. Rev, A2_4, 1016 (1931).

7. K.K. Mon, R. Cann and D. Stroud, Phys. Rev, A24, 2145 (1981).

8. O.K. Chaturvedi, G. Senatore and M.P. Tosi, Lett. N. Cimep.to 30, 47 (1981).

9. D.K. Chaturvedi, H. Severe, G. Senatore and M.P. Tosi, Physica IUJ_1, 11 (1981).

10. S.G. Brush, H.L. Sahlin and E. Teller, J. Chens. Phys. 45, 2102 (1966).

11. J.P. Hanscn, Phys, REV. A8, 3096 (1973).

12. M.J. Gillan, J. Phys. C_7_, LI (1974).

13. D.K. Chaturvedi, G. Senatore and M.P. Tosi, N. Cimento B^2, 375 (198L).

14. G. Senatore and M.P. Tosi, Phys. Chem. Liquids U , 365 (1982).

15. See, e.g. N.W. Ashcroft and D. Stroud, Solid State Phys. 3_3, 1 (Academic

Press, New York 1978).

16. H.C. Andersen and D. Chandler, J. Chem. Phys. 53^ 547 (1970) and

55, 1497 (1971).

17. W.L. Slattery, G.D. lloolen and U.K. DeWitt, Phys. Rev. A2jj, 2255 (1982).

18. S- Galam and J.P. llansen, Phys. Rev. A_14, 816 (1976).

19. A. Rahman, Phys. Rev. Lett. 1?_, 52 (1974) and Phys. Rev. A9, 1667 (1974).

20. R.A. McDonald, R.D. Mountain and K.C. Shukia, Phyf. Rev. B20, 4012 (1979);

R.D. Mountain, Phyr,, Rev. A?_6, 2BS9 (1982).

21. N.W. As!u:voft, J. Phv;. Cl. 232 (I'll)1').

•11. D.L. Price, K.S. Singwt and M.P. Toai, Phys. Rev. %2_, 2983 (1970);

H. Shyu, K.S. Singwi and M.P. Tosi, Phys. Rev. E3, 237 (1971).

23. M. Parrinello and k. Rahman, Phys. Rev. Lett. 4j>, 1196 (1980).

24. P. Vashishta and K.S. Singwi, Phys. Rev. B6, 875 and 4883 (1972).

25. M.J. Huijben and W. van der Lugt, Acta Cryst. A3_5> 431 (1979).

26. Y. Wasecia, The Structure of Npn-Cryst a Hne^MaLer^a^s (McGraw-Hill, New York

1980).

27. C.J. Smithclls, Metals Reference Handbook, 5" Edition (Butterworth,

London 1978).

28. A.J. Greenfield, J. Wellendorf and N. Wiser, Phys. Rev. A4, 1607 (1971).

29. Y. Waseda, Zs.Naturforsch. 3^a, 509 (1983).

30. P.A. Egclstaff, N.H. March and N.C. HcCill, Canad. J. Phys. 5_2, 1651 (1974).

31. A. Ferraz and N.H. March, Solid State Commun. 3_6, 977 (1980).

32. C.C. Mat thai and N.H. March, Phys. Chcm. Liquids U_, 207 (1982).

TV,El r i

<I- CA;'T0':; ;

Metal

Li

Na

'K

Rb

Cs

3.

4

5

5

5

rS

28

05

.02

.42

.77

r

212

211

187

185

18!

or/a

1.463

1.462

1.450

1.449

1.447

r (A)c

0.74

0.59

0.95

0.86

1.26

1.21

1.33

1.35

1.44

1.43

S(0)

0.016

(0.03t)b

0.017

(O.O256)a

0.020

(O.O241)a

0.021

(O,O2A5)b

0.021

(0.0256)3

s —

3.11

2.38

3.21

2.98

3.113.11

3.10

3.18

3.14

3.14

0

3.

3

3

2

2

eak

21

20

.00

.99

.95

* For each metal, the two rows refer to models I and II discussed in the

text. Measured values of S(0) are in parentheses (a, from M.J. Huijben

and W. van der Lugt, Eef. 25; b, from Y. Waseda, ref. 26) and have been

fitted to determine the values reported for r in model II. S arcc

the calculated values for the height of the main peak, the corresponding

theoretical value for the OCF being given in the last column. In the

case of iNa , we have also seen some sensitivity of the precise values ofpeak peakS

17,18peak , 17,18and S to the input data on the free energy of the OCP.

20

theory (fi i l l curve) and from molecular dynnnWcs resin Ls of19

A. Rahman ( d o t s ) .

F_u;<ire_2_ Pa Lr distribution function of a model for liquid Rb at 350 K from the

present theory (full curve) and from Monte Carlo data of R.D. Mountain

(dots).

Fi fturc 3 Comparison between the optimized RPA potential ~u(k) (broken curve) and

the true potential u(k) (full cuj-ve) for a model of liquid Rb at 319 K.

The double arrow denotes the position of the main peak in S(k) , and

the single arrow narks the first node in LI(JC), The potentials are in

units of k T/n .B

JFi_£ure_4 Strur.Lure [actor of liquid Na at 100 C from model I (broken curve) and

model II (full curve). The experimental data are from X~ray diffraction

of Huijben ant! van der Lugt (clots) and of Greenfield et <•? 1.

(circles).

Figure 5 Structure factor of liquid K at 65 C from model 1 (broken curve ) and

model II (full curve). The experimental data are from X-ray diffraction

25 28of Huijben and van der Lugt (dots) and of Greenfield et al.

(circles).

Figure 6 Structure factor of liquid Cs at 30 C from model I (broken curve) and

model II (full curve). The experimental data (dots) arc from X-ray

diffraction of Huijben and van der Lugt.

Figure 7 Structure factor of liquid Li at 190 C from model I (broken curve) and

model II (full curve). The experimental data (dots) are taken from

26Waseda.

F i g u r e 8 S t r u c t u r e f a c t o r o f l i q u i d N'a , K , Rb a n d Cs i n t h e s m a l l - a n g l e s c a t t e r i n g

• - 1 « - l 2 9r e g i o n ( 0 . 1 A f k i H ) . The e x p e r i m e n t a l d a t a a r e f r o m W a s e d a

25 28

( b r o k e n c u r v e s ) , H u i j b e n a n d v a n d e r L u g t ( d o t s ) a n d G r e e n f i e l d v\. a l .

( c i r c l e s ) . The r h e o r e ! i c a J r e s u l t s ( f u l l c u r v e s ) a r e f rom m o d e l M , at:

L c n i p e m L u r e r . e tum 1 t o t i t o s o o f W.nscil.-i' .•-• da t f ! ( 1 0 5 C f u r N.i , 70 C for K ,

40 C f o r Rb a n d 30 C f o r C s ) . The o t h e r e x p e r i m e n t a l J..it ,! f o r N\i ,md

K a ' O .IL r, 1 i ;>ti i. I y l o w e r : . M i r c r . i i i ; r c : ; ( 1 0 0 <; l u r M.-i a n d d'i C f o r K ) .

Fig.2

-19--20-

Fie-3

- : ; • ! . -

3

to'

2

1

x1O <j

ift\

Fig. 6

3

^-^

co

2

1

x1O

Ifft

r '

r 1

\[\l\1 X

Fis.T

0.090.080.070.060.050.040.030.02

Fig. 8

- 2 6 -