MANY BODY EFFECTS IN THE STRUCTURE OF SIMPLE FLUIDS

HAL Id: jpa-00225253

https://hal.archives-ouvertes.fr/jpa-00225253

Submitted on 1 Jan 1985

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

MANY BODY EFFECTS IN THE STRUCTURE OF SIMPLE FLUIDS

P. Egelstaff

To cite this version:

P. Egelstaff. MANY BODY EFFECTS IN THE STRUCTURE OF SIMPLE FLUIDS. Journal de

Physique Colloques, 1985, 46 (C9), pp.C9-1-C9-15. �10.1051/jphyscol:1985901�. �jpa-00225253�

JOURNAL DE PHYSIQUE

Colloque C9, supplément au n012, Tome 46, décembre 1985 page C9-1

MANY BODY EFFECTS IN THE STRUCTURE OF SIMPLE FLUIDS

P . A . Egelstaff

Physics Department, University of Guez&, tiueZph, Ontario, Canada NIG 2WI

Abstract - Classical radiation scattering experiments are related, via the first Born approximation, to the pair correlation function only. However statistical physics provides many inter-relationships between g(r) and higher correlation functions which can be exploited to study these functions. Some examples will be given and the limitations of this approach discussed.

Many physical phenomena which involve higher correlation functions also involve higher order forces.Thus a study to unravel the role of the higher correlation functions usually involves a study of higher order forces too. Because the theory of these forces is unsatisfactory, it is important that new experiments are attempted which highlight the principal features of these forces. Recent progress in this area will be reviewed. The examples used in this review will be taken from the work of the author and his colleagues over the past 10 to 15 years.

1. Introduction

The study of isotropic dense fluids, (gases, liquids and glasses) is centred around two sequences of functions: a sequence of positional correlation functions and a sequence of interatomic potentials. Whereas each member of the first sequence can be defined and will exist irrespective of what is assumed about the second, the existance and usefulness of members of the second sequence depends upon certain theoretical assumptions. For example it is assumed (in many cases) that in

calculating a property, the terms involving the pair potential are the most important of the potential terms and that terms involving higher order potentiala decrease rapidly in magnitude with increasing order. It is very difficult to test ~iiis assumption experimentally. One method is to evaluate the pair potential terms using reasonable approximations, and if this numerical value is a large fraction of the experimental value (eg. 75%) to assume that the series is convergent. Of course higher terms might oscillate in size and sign, or the series might be

conditionally convergent. An example of data of the former kind is given at figure la, showing the pressure of krypton at room temperature as a function of density.

Attempts havebeenmadetojustify theassumptionof c o n v e r g e n c e b y c a l c u l a t i n g the long range parts of the next term, but this is not convincing. There are difficult problems in evaluating the higher terms because insufficient data are available on either higher order correlation functions or higher order potentials. this reason it is usu.1 to make rather crude approximations. Meath and Arirfqf discuss some of these approximations and the effects produced hy cancellations between terms

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985901

JOURNAL DE PHYSIQUE

(a) The ratio P/pkT for krypton at room temperature as a function of p. The full line shows the experimental data, while the crosses are calculated from a Monte Car10 simulation using the Barker potential (ref. 1.).

(b) The differences between the full line and crosses of (a) are shown by crosses here and are compared to a simple extrapolation (full line) of the measured three body term at low densities (ref. 1.). The triangles show similar results for the P.Y. theory minus the simulation data.

2. A test of the existance of the convolution term in equation (12); the full line is the experimental data for liquid neon and the dotted line is the

prediction of equation 12, while the dot-dash curve is the p 1/3 mode1

on the predictions for dense systems. A simple example (in which g(r), g (r,~) denote correlation functions and u(r), u3(~,z) denote potential functionsj is provided by the series for the pressure, i.e.:

+ etc.

In order to evaluate the third term in this series, some approximation is needed.

The simplest is to assume that the density dependence is displayed in the second power of p in equation (1) and the implicit dependence on p in the 3rd order

correlation functions and the higher powers of p should be neglected. Although this is a poor approximation for the correlation functions and the higher terms over the range of density considered, it gives a good fit to the data (fig lb). This confirms that important cancellation effects occur, so that a much closer examination of these series is needed. This is a real challenge to experimentalists. Also shown in this figure are calculations using the Percus Yevick theory. By chance this fits the experimental data using the pair potential only, and therefore the error in it is about the size of the many-body force term.

In considering experimental cases we shall restrict atten n to noble gases, because the pair potentials are well known and it is believed "' that the higher order terms will be relatively small. Of particular interest will be the variation of some property with density - that is a traditional virial series approach to this problem will be examined. While there are many macroscopic properties which can be written as virial series, such as equation (l), they are not especially helpful because for each state only one point is measured. It is more informative to measure a functional property at each state, for example the structure factor, S(q), i.e.:-

where q may be varied over a wide range. It maybe measured approximately in a radiation scattering experiment interpreted through the series for the differential cross section, %,:-

ik r ikr

00 ⁼ b2s(q> + ^{p2b31 drL} 1 ^{d~ (e} -.- + ^eic.r)

dn g(r)

v v

+ p3b3111 drdsdte - - ik'&iq*+ ikt g3(r,s,t) + etc.

where b is the scattering amplitude, r , 5 and are intertomic vectors between three atoms in a triplet, and r, is a vector denoting the position of an atom chosen as one. and k' are the incsent and the scattered wave vectors respectively. The termpro- portional to b2 is the first Born approximation, while the two terms proportional to b3 give the second ~orndpproiimation. The first of these is the usual "multiple scattering"

term and it arises from scattering events which are far apart. Although it is one order in b/r less than the leading term, it can become important because it is multiplied by the volume (i.e. 1 drl) of the sample. On the other hand the second of

v

these terms arises from double scattering over a range in which g3(r,s) differs from

its assymtotic value. The integrals are al1 finite when V+-, so that this term

vanishes as b/r. Thus the standard procedure is to design the experiment in such a

way that the multiple scattering term is -10% of the total and so is readily

evaluated and subtracted leaving S(q). In this kind of experiment only the pair

correlation function g(r) may be studied. Even if the term in g3(r,ç) could be

C9-4 JOURNAL DE PHYSIQUE

isolated it would be difficult to extract information on g3(r,s) itself due to the complexity of this tenu. In order to study g3 it is necessary to scatter more than one coherent beam within the correlation range of g3. For atomic fluids this means within a few angstroms which is a very challenging task at present.

The pair correlation function is related to distances between the "centres" of atoms; effectively this means between their centres of mass or between nuclei. Thus equation (3) refers to any technique (e.g. neutron scattering) which determines nuclear positions, for which b/r is very small and for which other processes (e.g.

those leading to quantum changes in the atom or nucleus) contribute a negligible cross section. Of the possible techniques (neutron, X-ray, electron etc.) neutron scattering satisfies each of these criteria far better than the others and so is a much more accurate technique (in principle). Because of the relatively low neutron fluxes available for these experiments, there are practical difficulties which often prevent the full potential of the technique from being realised. Nevertheless for the dense noble gasf3)excellent experiments are possible if sufficient care and time are devoted to them . Since g(r) or S(q) can be measured accurately it is possible to determine small differences, and hence discuss either thermodynamic derivativesor the differences between computer simulations with accurate pair potentials and real data. In the former case effects due to higher order correlation functions may be studied, while in the latter case effects due to higher order potentials may be studied. In al1 cases the data will be incomplete because a multivariate function is being sought, while only a single variable (q) is being employed at each state.

Then, there is the question of measuring the frequency (w) change in the radiation which occurs during the scattering process, i.e. the frequency before v.s.

after scattering, and determining the scattering function S(q,w). This is related to the structure factor by:-

The dynamic structure factor S(q,w), contains further information on both kinds of higher order functions, and in some cases may depend more sensitively on them than does S(q). Again the two methods (i.e. derivatives or difference from simulations) are available and require high quality data. Fortunately again neutron inelastic scattering can provide suitable results.

For a molecular fluid composed of almost rigid molecules, the orientational correlations of neighbouring molecules is an important additional factor governing three particle and higher correlation functions. Correlations involving two atoms on one molecule and a third on a different molecule, as well as cases in which one takes three atoms each on a different molecule, will be of interest. If g(r,w ,w2) denotes the orientation dependent pair correlation function (wl and w2 being tke orientation variables for molecules 1 and 2). then the structure factor of a homonuclear diatomic molecular fluid is:-

where F(u ,q) = 1 ^{eiq'g'2, and} is the position vector joining molecular centres and d is the internuclear distance within one molecule.

This equation is equivalent to equation (2) for the atomic fluid. In the simple case where the orientations of neighbouring molecules are

uncorrelated the correlation function for molecular centres, gc(r), may be

extracted from the data since 2(a) becomes:-

Thus already we see from equations 2(a) and (b) that g(r,wl,wg) is "beyond gc(r)", in that to specify it requires higher order correlations between the atoms.

In section 2 we shall discuss models for g3(r,s) and how they may be tested through data on[ag(r)/ap] , while in section 3 we shall discuss the separation of the u3(r,s) term in the virial expansion of g(r). The extension of this work to higher densities, and the use of data on S(q,w) will be discussed in the following two sections. Finally some brief comments on correlations in simple molecular fluids will be made.

2. Models for g,(r,s) - and a simple test.

For very low density or when the atoms are far apart, the three body correlation function may be written as a sum of pair correlation functions, namely for 5 or 5 very large:-

where h(r) = g(r) - ^1. When this condition is not satisfied, g (5,s) can still be written in terms of g(r) if the pair theory is,+valid (1.9. by efiminating u(r) from the expressions for g(r) and gg(r,s) - see Abe or Stell ). The result rnay be expressed as a diagramatic series, and the first few terms are:-

+ p I ^{h(2) h(5-w)} h(w-5) dw + etc...

This series may be compared to the well known superposition approximation:-

If this approximation is expanded in terms of h(r), the expression (6) is obtained except that the integration over the position, w, of the fourth particle is replaced by a product, h(r) h(s) h(r-2). Thus the superposition approximation is valid at moderate densities only, where the four-particle terms are unimportant.

One way of testing expressions such as (5), (6) or (7) is to compare calculated and measured properties which involve g3(r,s). Perhaps the most convenient is the isothermal densit): derivative of g(r), which was first derived by Buff, and exploited by Buff and Brout to test the superposition approximation. It is:-

where xT is the isothermal compressibility. It has the advantage of being valid for

any system of potentials. Since S(q) - ^{equation 2} - can be measured it is convenient

to write

C9-6 JOURNAL DE PHYSIQUE

and after introducing a model g3(r,s-) into (81, evaluate aS(q)/ap from (9). If equation (6) is used, a simple result is obtained i.e.:-

p FIT ^{= ~ ( q ) (~(q) -1) + etc. .. .}

It should be remembered that this equation depends on the pair theory, and has included some four particle terms in the expansion. In contrast if the usual Ornstein-Zernike equation defining the direct correlation function, c(r), is differentiated with respect to p , the result is:-

Thus the higher terms in (10) are related to the derivative of c(r) in the pair theory, although (11) has a wider significance if a general theory for c(r) is available.

At low densities c(r) may be expanded as a virial series involving Mayer functions and then it will be seen that the leading term in ac(r)/ap is a three-body term, which illustrates that equation (6) is not a c3uster-type expansion. Also in this limit the next term in (10) is - ^{(~(~)-l} ,} and therefore (10) will be useful when S(q) is not too far from unity.

Egelstaff, page and ~eard"exp1oited equations (8) and (9) as a test of the superposition approximation (SA). In this case:-

and they showed that the final term is very important at high densities, in disagreement with experiment, so confirming that the S.A. is good at low densities only. An example of their method applied to liquid neon is given at figure 2. At high densities the convolution term in equation (12) generates a large peak in the calculated derivative at low q. This is not found experimentally, either for noble gases or for liquid metals. Since the convolution term arises from the product h(r) h(s) h(r,s), it is confirmed that the S.A. is in error at this level. On the other hand, txe-derivative can be f itted quantitatively by the uniform f luid model,

although the relationship of this model to the three body correlation function is not known.

'on (10) may be tested in experiments on low density gases. Winfield and Egelst%PP showed that it had the right shape near the critical point of krypton, and Teitsma and ~~elstaff('~)measured S(q) as a function of p for krypton at room temperature, confirming the correctness of (193 at denssties less that half-critical.

An exampie of their results (at p = 2.42 x 10 atoms/m ) is given in figure 3. For

q > ¹ A the agreement between equation (10) and the data is good, while at low q

significant discrepancies occur. These are probably associated with the three body

forces discussed in the next section, and which lie outside the derivation of

equation (6). Thus g3(,r, is known only approximately for densities up to about

critical. Over thia rangeffh)the measured ac(q)/ar riil give an estimate of the error

in equation (6) where the three body forces do not matter.

One defect of the above analysis is that only an isotropic average over gg(r,s) is measured. Another, angular dependent average is offered by the BBGKY equation.

Again this equation depends on pair theory. It is:- du(r) Sin q.r

S(q) -1 = & J ^{g l r )} -a;; - ^{COS^^}

9

2 Sin q.(r-s) - ^{Sin q.s}

+ -& Y ^[g3(~,s) - ^g(r)] _dr ⁹ 1 ^Cos 0 d r d ~ where 0 is the polar angle of r. Unfortunately the use of this equation involves a good knowledge of u(r), and densities such that the second term on the right hand side is as large as the first. In the form of the Born-Green equation it has been widely used with the S.A. to calculate g(r) or S(q) and compared toeither computer simulated or experimental data. The weakness of the S.A. at high densities was demonstrated clearly.

Isothermal density derivatives of g(r) higher than the first are related to g4 or higher functions, but al1 distances except one are integrated out. This means that only the simplest information about the higher fu stions may be obtained this way. Some work has been reported by Egelstaff and Wanz !

3. Three Body Potential

The virial expansion for the direct correlation function is well known, and if S(q) is measured accurately at many States along an isotherm it can be exploited to test models for the three body potential. To do this it is necessary to separate the term C2(q,T) in the expansion:-

where:

B(~,T) = f f(r)eig.% and f(r) = expi-fiu(r)} -1.

These definitions are consistant with those used in the conventional virial expansion of the pressure.

Teitsma and ~~elstaff'O'measured c(q) along the room temperature isotherm of

krypton, and deduced the function C2(q,T). These data are shown in figure 40,

compared to predictions using the Axilrod-Teller (A.T.) triple dipole potential ie:-

JOURNAL DE PHYSIQUE

3. A comparison of the prediction of quation $10) with experimental data on krypton gas at T = 297K and p = 2.42 x IO2' atoms/m . The discrepancies at low q reflect the need for terms depending on 3-body forces in equation (6).

5. Structure factors fo dense krypton gas at room temperature and the densities

shown in units of 1of7 atoms/m . Çolid circles are the experimental results and

the full line is a computer simulation using the known pair potential for krypton

(from ref. 12).

1.0

CI

\ ^0.5,

I 3

V

(V

O

rc)

I 0.0

4.(a) The contribution to the third virial coefficient of the direct correlation function, arising from the three body potential. Experimental data are showi by circles, and a prediction using the Axilrod-Teller triple dipole potential is shown by crosses (from ref. 10).

-

(3,

$ ^EXPT + ^AT

--+++

*+ P

+++ + p

+ -

l I I I

g ($1

(b) The size of the three-body force term at the peak g(r). The maximum reduction is about 10%.

O O. 5 1 .O 1.5 2 .O

2 . 0

I I I I

- ^I

(b > ^-

1.9 -

- V I R 1 A L

-0- MONTE C A R L O

-0- E X P E R I M E N T a -

e--

C9-10 JOURNAL DE PHYSIQUE

where A is a constant depending on the polarisability and el, e2 and 8 , , are t h e interna1 angles of a triangle formed by the three atomic nuclei. This term corresponds to the long range behaviour of ~ ~ ( 5 , s ) and is not expected to work at short range, even though many theoretical papers have employed it for the whole range of distances because of the mathematical difficulties involved in making a more appropriate choice at short range. The discrepancy shown in figure U i s consistant with the failure of the A.T. term in the range 3 to 108.

The A.T. term can be used in computer simulations1' of g(r), and it is found that the principal modification to the pair potential prediction for g(r) occurs at its major peak. This is due to the strong r dependence in (15), and normally the peak height is reduced (fig 1 of ref 11). However the peak of g(r) is just the region over which the A.T. term fails and other effects must be included, and hence over the r-range for which equation (15) is applicable it has a negligible effect on g(r). In addition if a densiFr of between two and three times the critical density is used in these calculations , even the effect at the peak of g(r) almost

disappears. Thus, although widely used, equation (15) is unsatisfactory. An improved mode1 incorporating short range effects should be parametrised and the parameters determined by fitting to data such as that in figure U.Figure 4b shows

the real change in the peak height of g(r) due to many body forces.

4. The Dense Fluid

The advantage of low density experiments is that they can be analysed using density expansions, and meaningful potentialstf,or other data) extracted. If we adopt an expansion in a series of terms involving n order potentials, there is little hope of a term y term analysis when many terms are important. Also the'work of Meath and ~ z i d 2P suggests that strong cancellations occur, so that a term by term analysis would be inappropriate. Instead, at higher densities we can look for the magnitude of a total many body potential effect.

Since the Monte Carlo workl"using the triple-dipole potential has indicated that its effect is reduced as the density is increased, it is useful to enquire whether the total many body effect does likewise? To answer this question we need to compare the measured structure f2ctor with a computer simulation using the known pair potential. Figure 5 shows data 'taken at densities of about twice the critical density on the room temperature isotherm of krypton. The full line is the simulated result and differences between it and the data points are evident. The order of magnitufg and approximate position on the q-scale are the same as found at lower density . This confirms that the higher order potential effect does not vani@ with increasing density, and leaves open the question of whether the expansion in n order potentials is satisfactory or not?

Studies of this kind are dependent upon a good knowledge of the pair potential.

Fortunately by extracting the term B(q,T) - equation 14 - from the experimental data on c(q), and then Fourier transforming, a direct measuremefs of the pair potential rnay be accomplished. This has been done by Barocchi et al' with results that agree closely with the earlier work using a parametric analysis of bulk data.

5. Dynamical Structure Factor (S(q,wx

The equal time correlation functions (g(r), g3(r,s) etc) for a fluid are a thermal average over the positions of atoms taking part in many body collisions. In part this rnay be expressed by writing g(r) as the integral over time of the imaginary part of the Van Hove space time correlation function, and noting that the imaginary part reflects the relaxation processes in the system. Thus we rnay expect that a study of the dynamical behaviour rnay throw light on the various contributions to the static correlation functions. This rnay be especially true for the study of effects produced by many body forces as these rnay influence many body collisions,

particularly the ring collisions which are important at high density. If we compare

the dynamic structure factor obtained from a computer simulation based on the known

pair potential with that measured experimentally, the total contribution due to many

- Experiment; or x Simulation.

6. Examples of comparisons of cornputer simulations of S ( q , w ) - using the krypton

pair potential - with the experimental results of ref. 12. Note how the effect

increases with increasing density (from ref. 15 & 16) for low q.

C9-12 JOURNAL DE PHYSIQUE

body forces can be determined. Because (fig. 5 ) the structure factor shows an important differeece for q < ^la- , we would expect S(q,w) to show differences there also. schommers' ' has argued that such differences should be larger relatively than those in S(q). This might arise if the difference is large but changes sign as a function of frequency so that the integral in equation (4) shows only a small effect.

The data on krypton give reference 12, has b compared to computer simulations by Egelstaff et a?l3'and by Salacuse et aJfp! Some exam esof thejr results are given 'in figures6(a)-pnd (b). At a density of iP.6 x 1 ' ' 0 atoms/m there are small differences for q < ^1A (~fpecially for q = 0 . 8 1 1 . ), which are larger than those observed at q = 1.35 or 1.78 A . It should be noted that there are no

adjustment constants used in these comparisons. These di erences yt q < ^{IR-' become}

much more substantial for the higher density of 13.8 n ' ' 0 1 atoms/m , and this is to be expected if the expansion if terms of high order forces does not converge easily.

In this event, the use of an n order force expansion to describe properties at high density may not be realistic, although more work should be done on the interpretation of the S(q,w) data in order to test this possibility.

6. Orientational Correlations in Diatomic Molecular Fluids

In many cases the interna1 vibrational energy levels of a molecule are widely spaced compared to kT, and then the molecule may be treated as a rigid body. It is usual, in this situation, to specify the positions of atoms in the molecule by the molecular structure, the position vector rl of the molecular 'centre' and a set of angles (denoted by w ) which determine the orientation of a molecular axis.

Therefore it is possible to consider a general pair correlation function for two molecules, g(r12, wl, w2). We may also consider a partial pair correlation function between an atom denoted by a on molecule 1 and by.$ on moleCule 2, i.e. gaB (ru$).

This function contains less inforgation'than the orientation dependent function since it is an average over the angular !variables w and w2. herefore, if available, g(r11,,7, u2) would yield more idformation i bout many b dy correlations than the part a unction g a$' However onjy the set of go$ may be measured in a diffraction experiment using isotopic substitution to Vary nuclear scattering amplitudes.

Properties of the orientation dependent function must be infered indirectly e.g. by the r dependence or state dependence of g

I aB'

The simplest example is that of the homonuclear diatomic molecular fluid, and since an elongated molecule leads to rger effect the case of chlorine is interesting. Sullivan and Bgelstaf#h'f'~ have studied liquid chlorine at several states, and the pair correlation function for liquid chlorine at room temperature is shown in figure 7. The molecular bond at 2 A is well separated from the liquid structure peak aC 3.8 A. The latter is broad, compared to an atomic liquid, and has a double maximum. This arises because the end to end and tee configurations of neighbouring molecules generate two distinct distances. Thus the presence of this distinctive ff7ture is related to the population of these configurations. Sullivan and Egelstaff studied three states and compared the relative heights of these two peaks with those calculated for a pair potential model at low density and at al1 the experimental densities. With increasing density the height of the peak at - ^{5.6 A}

falls relative to the peak at - ^{3.8 A} in the model, indicating that the open configurations (end to end and/or tee) are less favourable at the higher density.

Since the ordering of the peaks in the model agreed with that observed

experimentally, they concluded that this effect likely occured in the real liquid.

Their results are thus related to changes in the many body correlation functions.

Possibly the most reasonable approach is to use the experimental data to test

pair potential models, and then via computer simulations to predict the simplest many

body correlation functions. In the example above, molecular size and shape were

shown to be important factors in nearest neighbour correlations, and if such data are

measured accurately and fitted well a good basis for predictions is obtained. In

this case multiple correlations such as two atoms on one molecule with one atom on a

second molecule would be the easiest to justify, while the correlation of three atoms

one each on three different molecules would be poorly understood.

7. The pair correlation for liquid chlorine at 296 K and 11.9 x lo2' molecules/m 3

Structure factor data taken from reference 18 and transformed to r-space by

information theory. [John Root is tkgnked for thif figure]. The experimental

data were extrapolated from q = 11A to q = 22A- by information theory, and

transformed to r-space similarly.

JOURNAL DE PHYSIQUE

An alternative techniqud19' is to analyse the data in terms of the spherical harmonics of g(r12,~1,~2). Some data on the higher order functions can then be infered £rom the relative size of different harmonics and £rom the molecular

structure. The availability of two independent approximations to these functions is useful as a test for large errors. Detailed work on this topic is not available yet.

7. Conclusions

-

At first sight the overal success of the pair theory of liquids suggests that it is a good starting point for the discussion of al1 fluids. For low density gases, where virial expansions are successful, this ts certainly true. However at high density, roughly corresponding to triple point densities, the status of the pair theory is not clear even for the 'noble gas' liquids. Most of the claims made in the literature are not based on sound experimental evidence about the magnitude and behaviour of higher terms. This is simply because such evidence is very hard to obtain, and so very little exists. Higher order data correspond to a mixture of terms involving higher order correlation functions, higher order potentials, and may involve quantum corrections before they are compared to the classical expressions.

Because of these difficulties most of the published work has concentrated on the triplet correlation function and the triplet potential. Probably the data on both functions now justify further theoretical effort at sxplanation at the level of a parametric mode1 based on fundamental theory.

The question of whether terms involving higher potentials are small compared to the pair term when the density is large, is unresolved. Recently large many body potential effects have been observed in the dynamic structure factor. Because many particles occupy a larger volume than one or two, these effects will be seen

primarily at low q. This is amply confirmed by experiment. It is possible that some higher order correlation functions play a more important role than (e.g.) g3 or g4, at high densities. To study this matter experimentally is one of the challenging future tasks.

Orientation dependent effects can be seen in g(r) for simple molecular fluids, and one can infer some information the relative likelyhood of different many body correlations £rom the shape of g(r). But perhaps the best use of such data at present is to test models of the pair potential, which could then be used in computer simulations of these and other effects.

While radiation scattering experiments have provided a great deal of useful data, there are major limitations on the extent of the information that can be extracted. However this technique has the advantages of quantttativeness andalack of ambiguity concerning the quantity being measured. When other methods, using for example atomic beams containing clusters or multiple beam experiments with synchotron radiation, are brought to fruition it may be that, by combining these diverse data, a proper understanding of many body effects could be achieved.

Ref erences

1. J. Ram and P.A. Egelstaff. Phys. Chem. Liq. 14, 29, (1984).

2. W. Meath and R. Aziz. Molec. Phys. 2 1 , 225, 71984).

3. P.A. Egelstaff. Ch. 14 in "Neutron Scattering" Ed. K. Ski5ld and D.L. Price Academic Press. This chapter describes the procedures necessary for good quality neutron data.

4. R. Abe. Prog. Theor. Phys. 2, 421, (1959).

5. G. Stell. Physica 2, 517, (1963).

6. F.P. Buff and R. Brout. J. Chem. Phys. 2, 1417, (1960).

7. P.A. Egelstaff, D.I. Page and C.R.T. Heard. Phys. Lett. A s , 376,

( 1969).

8. D.J. Winfield and P.A. Egelstaff. Can. J. Phys. 51, 1965, (1973).

9. P.A. Egelstaff and S.S. Wang. Can. J. Phys. 2, 684, (1972).

10. A. Teitsma and P.A. Egelstaff. Phys. Rev. AL!, 367, (1980).

P.A. Egelstaff, A. Teitsma and S.S. Wang. Phys. Rev. A 2, ¹⁷⁰²

( 1980).

P.A. Egelstaff, W. Glaser, D. Litchinsky, E. Schneider and J.B. Suck.

Phys. Rev. A z, 1106, (1983).

F. Barocchi, M. Zoppi and P.A. Egelstaff. Phys. Rev. A s , 2732 (1985).

W. Schommers. Phys. Rev. A 2 2855, (1980) and A 3, 2241, (1983).

P.A. Egelstaff, J.J. Salacuse and W. Schommers. Phys. Rev. A z, ^374,

(1984).

J.J. Salacuse, W. Schommers and P.A. Egelstaff. To be published.

J.D. Sullivan and P.A. Egelstaff. J. Chem. Phys. 76, 4631, (1982).

J.D. Sullivan and P.A. Egelstaff. Chemical Physics 89. 167, (1984).

J.R. Sweet and W.A. Steele. J. Chem. Phys. c. 3022:(1967).