GENERAL DISCUSSIONMr. G. Mason (University of Bristol) said: I should like to report some workon random sphere packings using a computer to generate the co-ordinates of thesphere centres. This work is directed more towards a model of porous materialsthan to the problem of liquid structure, but the results are of interest in both fields.The method of generating the packing was first developed in two dimensions, packingcircles on a plane, and a short film has been made to illustrate this.* Three-dimensional packings have also been obtained, although these are as yet only of 100and 200 spheres.I20I008 8 0 g 1 8 60ru09c8 5 4 02 0 rr1o ! -+--=I I I I I0.3 0 - 4 0 . 5 0- 6 0*7 0- 8packing densityFIG. 1 .-Tetrahedron densities.bulk packing density = 0-63(4) ; number of spheres in packing = 200 ; total number of tetra-hedra = 727Initially a number of points are generated at random in a " box " in space.Thesepoints are then considered to be small spheres of a chosen radius. If two spheresoverlap, they are moved apart along the line of centres until they are just touching.The computer continues moving spheres apart until none overlap. The spheres arethen increased in size by a chosen increment and the process repeated. Spheres arefree to move out of the " box " during the process. The method may be regardedas a means of creating a random distribution of spheres in space, and bringing themtogether toward a point in the centre of the packing as if under the influence of a radialgravitational field.Packings of even these small numbers of spheres reach a limitingpacking density of 0.63-0.64 which is close to the density of experimentally produced* This film was shown during the discussion.75G. D. Scott, Nature, 1960, 188,90876 GENERAL DISCUSSIONpackings. The value cannot yet be determined precisely because of the small numberof spheres in the packings.The packings have been analyzed in terms of tetrahedral sub-units rather thanVoronoi polyhedra. There are several reasons for choosing tetrahedral sub-units.The analysis is simpler; there are more tetrahedra than Voronoi polyhedra in apacking of given size, so that the results are statistically more significant ; and it iseasier to correct for edge effects. Radial distribution functions have been calculatedover a wide range of packing density and detailed analyses of the tetrahedra in thedenser packings obtained.The packing density of each tetrahedron in one particularpacking was determined and fig. 1 is a histogram of the results. As would be expected,the histogram is spread over a wider range of packing density than the polyhedraof Bernal and Finney. It is hoped with more machine time to pack as many as 1000spheres and to vary their " hardness ".The present method has advantages over that of Bernal and his co-workers. Thepacking density may be varied in a more realistic manner than by the random creationof discrete holes, while the use in effect of a radial confining force avoids the asymmetrywhich may result from packing real macroscopic spheres under the influence of gravity.Mr.R. H. Beresford (University of Technology, Loughborough) said : Some of ourwork may contribute to the development of a statistical geometry of liquids and tothe study of Voronoi polyhedra. The work is concerned with the structure of randomheaps of random-sized hard spheres. Consequently it has the Bernal liquid modelas a limiting case ; it may allow a generalization by allowing an appropriate variationin size of molecules, with a resulting spread in closest approach distance. The basicapproach is to consider the Delauney graph (lines joining the centres of touching ornearly touching spheres) which includes the shortest total length of lines betweennon-touching spheres, and which still divides space into tetrahedral pieces.Lines__---(I XFIG. 1.corresponding to contact points are " edges ", and the others are "diagonals".Each edge represents a constraint-that the sphere centre separation equals the sumof the radii-and so the number of edges can be related to the number of degrees offreedom of the system. An attempt has been made to overcome boundary effects,and to tie up with crystal structures, by considering a heap as the limiting case ofa repeating primitive lattice. As the number of random spheres in this latticeapproaches a completely random heap.A 3-dimensional repeating lattice (fig. 1) can be divided into 6 tetrahedra per cellby 7 lines per cell (12 cell edges, each shared by 4 cells ; 6 face diagonals, each sharedby 2 cells; one body diagonal.The other two body diagonals are redundant).There is one lattice point per cell, and if a sphere is centred on this point, the numberof spheres per cell (= m) is 1. This system has 6 degrees of freedom, for if the sphereradius is taken as unity, the X vector has a 1 degree of freedom, the Y vector 2 andthe Z vector 3. Any additional spheres in this lattice are either " random "-witGENERAL DISCUSSION 77centres falling within an existing tetrahedron, or " ordered ", falling on a tetrahedronboundary. Ignoring the ordered case, an additional random sphere can be joinedby four lines to the four vertices of its surrounding tetrahedron (fig. 2). This addsFIG. 2.3 tetrahedra to the system by creating 4 new ones whilst destroying the original one.Each sphere will have (3 +a) degrees of freedom, where the 3 corresponds to the threeco-ordinates, and a is the number of degrees of freedom associated with the particleshape.If monosized spheres are employed a = 0. If any size of sphere is permiss-ible, a = 1. Then for a total of m spheres :no. of tetrahedra = T = 3m+3no. of lines = L = 4m+3no. of degrees of freedom = Ndf = 6+(m+ 1)(3+a)(The constants are obtained by reference to the case where m = 1.)This analysis gives the minimum number of lines needed to divide the cell intotetrahedra. It appears likely that the shortest set of lines can be obtained by anexchange system which replaces one long line by one or more shorter ones. Forinstance, consider two spheres which have n equatorial neighbours, and are joined bya " polar " line (fig.3). The resulting n tetrahedra can be replaced by (2n-4)~ _...... i I I ,_.,,,.......!FIG 3 . 4 1 2 = 5), . . . equatorial ; - . - . - . polar.tetrahedra if (n-3) equatorial lines replace the polar line. This increases both thenumber of tetrahedra and the number of lines by (n - 4). The original constructionensures that n >47 and a later analysis shows n+4+ as m-, 00. The net effect thereforeappears to be an increase in the number of lines. Let the shortest set of lines beL'>L = 4m-1-3; as EGN,,,D = L'-E>4m+3-6-(m-l)(3+a).and a large random assembly of equal spheres must have an average of at least onediagonal per sphere. For m = 1, this reduces to E<6, D 2 1 ; and the equalitieshold for crystalline close packing, which cannot have more than 12(= 2E) contactsper sphere.:.D>m-a(m- 1 ) = ( 1 -a)m+a78 GENERAL DISCUSSIONConsideration of a bounded cluster of spheres gives the approximation :L' 4m-6(m*- ; D (1 --ct)m--6(m*-- l)2.As both approaches giveas rn--+co,limit (D/m)>l-alimit (L'lm) 4FIG. 4.-bLATIVE FREQUENCIES OF TETRAHEDRAL TYPESstructures. . + ..face types .Dfrequency 2764-number012a20303a3c4a40562764-face nameoooo001 101 1211111122022211131223222222333333t,2 >964-frequency(L - 0)6L66(L-0)50L612(~-0)40*L63(L- 0 ) 4 PL612(~-00)303L64(L-0)303L64(L- 0)303L612(~-0)204L63(L- 0 ) 2 0 4L66(L- 0)D50 6L6-p.3 ; >*Y164-frequency ifL=4, D = 1 .7294096-145840969724096--24340963244096--108409610840961084096---27409618409614096--_IGENERAL DISCUSSION 79it would appear to be a useful hypothesis that a large random close-packed assemblyof equal spheres has 3 contacts and one near contact per sphere. This only appliesto spheres with an infinite range of sizes, Probably for equal spheres, L’/m--+7,0+4, the chances are 4/7 and 3/7, and there are tetrahedra round each line.Asecond hypothesis is that the elements of the Delaunay graph of such an assemblyhave a chance 4 of being a diagonal and $ of being an edge.This allows estimatesof the frequency of occurrence of closed tetrahedra in such an assembly, togetherwith the 10 other types of tetrahedra which can have two types of edge. These aregiven in fig. 4. Further statistical results should arise from the consideration of thesurface patterns on each sphere (fig. 4). One such result is that there are 4+ tetrahedraround each edge. This arises because the average number of points on each sphere( = 2L) is P = 8 and so the average number of triangles is T,. = 2P-4 = 12. Thusthere are 36 apices round 8 points. These correspond physically to tetrahedraround lines, and so the average must be 36/8 = 43 tetrahedra round each line.Edge.- _ _ _ - -DiagonalFIG. 5.Although there is a large number of possible surface patterns, the frequency ofoccurrence of those with a high density of diagonals must be low.Also the numberof possible patterns with a high density of edges is reduced by the geometrical require-ments for monosize spheres, and so it is hoped that enumeration of the likely patternswill be of value. The relationship of these patterns to Voronoi polyhedra has yetto be investigated. The number of redundant diagonals adjacent to an includeddiagonal can be evaluated for each section of a surface pattern ; the total number ofdiagonals and edges will be related to the number of faces on the Voronoi polyhedron.Adjacent Voronoi polyhedra will result in mirror image patterns round the line joiningtwo spheres, and will also set conditions for the surrounding polyhedra.This maybe relevant to the study of packing density. Finally, it is hoped that the cases whereD > m will shed light on less dense assemblies, and that the total length of diagonalswill be related to the separation of spheres, as discussed by Kohler and Springer(this Discussion).Prof. J. Walkley (Simon Fraser University) and Dr. I. H. Hillier (Universityof Manchester) said : Many of the problems associated with the prediction of thethermodynamic properties of a system at fluid densities arises from the conceptualdifficulty of describing a fluid. Bernal’s work lays particular emphasis on thestructural identity in a liquid. The concept of a “ coherent structure ” existing inthe high density fluid region agrees well with the molecular dynamics studies of hardsphere systems.2 We have made computer studies at the other end of the liquiddensity range where it might be expected that the coherent structure gives way to acompletely random structure.J.D. Bernal, Nature, 1960, 185, 68.B. J. Alder and T. E. Wainwright, J. Chem. Physics, 1960, 33, 143980 GENERAL DISCUSSIONThe generation of random configurations of hard-sphere molecules at a liquiddensity presents its own problems. By adding spheres to a “ box ” in a consecutivemanner in random positions (as simulated by the choice of random coordinates on acomputer) leads to maximum densities that are unexpectedly low. Any processinvolving the “ swelling ” of spheres already in the box at random positions has thedisadvantage that high densities can only be achieved at the expense of having only afew sample spheres in the box.A method of generating random configurations atthe required density is as follows.TABLE DISTRIBUTION OF FINAL PACKING DENSITY ; Np = 32, Ni = 80 ; PERIODICBOUNDARY CONDITIONnumber of boxes218507044142number of remaining Nj13141516171819The box is packed with Ni spheres by the assignment of random, x, y, z coordinatesto each molecule, irrespective of any overlap of any two or more spheres. The initialpacking number ATi may be greater or smaller than N,, where Np is the maximumnumber of spheres able to be placed in the unit cube ‘‘ box ” on a primitive cubiclattice. This defines the hard-sphere diameter.The coordinates of the centres ofeach sphere allows the scalar distance between centres to be found and by comparison26 c20 6 0 100 140 I00NiFIG. 1 .-Peak value of theNj distribution for Np = 32 ;curveA, without periodicboundaryconditions;curve B, with periodic boundary conditions.to the molecular diameter the number of spheres Nk which overlap the M,th spheremay be determined. Spheres are now removed from the box such that the sphereMq which incurs the greatest number of overlaps is removed. If spheres M,, M,and Mt each have the same (maximum) number of overlaps then a random choicebetween these is made to determine the one to be rejected. A re-evaluation of Nkfor each of the remaining (Ni- 1) spheres is now made and again the one incurring thegreatest overlap is removed. This procedure is repeated until there is no overlapbetween any two of the remaining N, spheres in the box.This process was carried out for Ni as small as 5 to values as large as 300 for GENERAL DISCUSSION 81typical box where Np = 64.In general, any ‘‘ run ”, i.e., the generation of a seriesof randomly packed boxes from a given Ni value, was continued until 200 sampleboxes were obtained. The distribution of the final N j population for any initial Nivalues was always narrow. A typical result for a ‘‘ box ” for which Np = 32 andfor Ni = 80 is given in table 1.In fig. 1 the peak value of the N j distribution for any given Ni value is plottedagainst Ni. Curves are given for a “ box ” with a torroidal (periodic) boundarycondition and for a ‘‘ box ” without such a condition.In each case the randompacking density approaches a maximum value, independent, of ATi. For a box size,Np = 32, this limiting random density is N j = 18 if periodic boundary conditionsare imposed and N j = 22 is they are not.The tentative result is that the random distribution of spheres can only occur to alimiting density. Bernal in an earlier paper comments upon the liquid-gas transitionand the implication of a A-point in the constant-pressure heat-capacity curve. Thedensity limit observed in the present study agrees well with the limit suggested byBernal for the break up of the liquid “ coherent structure ” and the inherent implica-tion in the heat-capacity curve.Mr. R. Collins (University of Salford) said: Bernal and King (I) and Bernal andFinney (11) use the term “polyhedron” in two different contexts, which perhapsought to be distinguished.Polyhedron in I1 means Voronoi polyhedron (VP),but polyhedron in I does not. The lattice of polyhedra in I is a subset of the latticeof tetrahedra topologically inverse to the lattice of VP’s. The definition of neighbourin I1 is much more precise than the J2 criterion in I. If the I1 definition of neighbouris adopted, then the average number of neighbours of a given atom in a real liquidincreases with decreasing density since it is 14 for (slightly perturbed) close-packing,14.28 at the intermediate density described in I1 and for vanishingly small densitiesapproaches the random distribution (perfect gas) figure of 2+48n2/35 = 15.54.lThe basic problems in using a realistic ‘‘ soft ” potential #(Z) appropriate to aninteratomic distance I are to calculate the energy and the entropy of any irregularconfiguration.Both are easier to express in terms of the lattice of tetrahedra inverseto that of the Voronoi polyhedra rather than the VP lattice itself. In this connectionthe work described by Beresford also breaks the structure down into tetrahedra only.In addition to the entropy contribution arising from the local atomic co-ordinationnumbers 8, discussed by Everett, there is also the contribution arising from theprobability function $(I) of the neighbour distances 1. We have estimated thisusing information theory arguments, and even if the entropy contribution from the8, is completely neglected, there results an equation of state for a general #(I) whichgives reasonable qualitative agreement with experiment.2 Contrary to a widely heldview, the geometric theory is capable of yielding thermodynamic results for a realistic4(0.The histogram of what are effectively atomic co-ordination numbers 6, in fig.6of I1 is valuable since it provides a starting point for evaluating the correspondingentropy contribution, which has so far proved an intractable analytic problem.Real progress would become possible in this direction if fig. 6 could be repeated forvarious densities, and extended to cover configuration generated by a soft (e.g.,Lennard-Jones) potential. I would expect that the other statistics (such as illustratedin fig.7) will prove of much less thermodynamic importance, although of sometheoretical interest.J. L. Meijering, Philips Res. Rep., 1953, 8, 270.R. Collins, Proc. Physic. Soc., 1965, 86, 199. D. C. S. Allison and R. Collins, to be published82 GENERAL DISCUSSIONIn an attempt to develop a realistic physical formalism, I would suggest that inthe first instance it is easier to consider initially a two-dimensional liquid. Here thetopology is much easier ; the Voronoi polygons have an inverse lattice which consistsof triangles only, and the mean co-ordination number lV is a strict topological constantequal to 6 at all densities. Also, one can see directly how the long-range orderappears in the crystalline state on freezing.If an adequate mathematical formalismcould be developed on this basis, it would not have the defect, inherent to the lattice-gas model of a liquid, that a long-range ordered lattice of sites is used to describe anessentially disordered physical state. If the problem cannot be solved in two-dimensions, there seems little hope of a three-dimensional solution.Prof. D. H. Everett (University of Bristol) said : Although work on the randompacking of spheres gives a valuable insight into the structural properties of liquids,it is not immediately apparent how the results of such studies can be applied to thecalculation of thermodynamic properties. In introducing these papers, Finneyhas mentioned that the results can be used successfully to calculate the energy of aliquid and hence the heat of fusion: but it is also necessary to be able to use themodel to calculate the entropy. An approximate approach to this problem seemspossible along the following lines.Simple cell theories of liquids predict too low an entropy for the liquid state.One feature of these theories which makes a major contribution to this discrepancyis the assumption that each molecule can be regarded as moving in the mean field ofits neighbours (the average potential model): the energy of a captive molecule at thecentre of its cell is the same for all cells. A more realistic model-which might becalled the lzeterogevleotis cell model-would taken account of the fact that at anyinstant a molecule finds itself in a field determined by a particular arrangement ofneighbours.We might therefore regard the liquid as divided into cells, each of whichdetermines a characteristic potential energy controlling the motion of the captivemolecule. The depth of the potential energy minimum in a cell will differ from cellto cell, and through time a given molecule will sample cells of all kinds with a pro-bability proportional to the frequency with which each type of cell occurs in the liquid.We could use this consideration to justify an intuitive evaluation of the configurationalentropy. A rigorous calculation of the configurational entropy arising from thedifferent arrangements of these heterogeneous cells would, however, presumably needa detailed knowledge of the number of ways of dividing space into cells.Studies ofthe statistical geometry of sphere packing, such as those described by Bernal and hisco-workers, are, however, based on an analysis of only one (or a very few) specificrandom packings. Thus, only one particular division of space is examined, fromwhich, when some arbitrary method of distinguishing cells of different kinds hasbeen chosen, enables us to evaluate the number of cells of different kinds in therandom packing. If a large enough number of spheres is considered, then the dis-tribution function derived for one particular random packing will approximate tothe mean distribution function derived from a large number of different packings.It seems reasonable to suppose that an approximation to the configurational entropycan be derived by equating the number of ways of dividing space to the number ofways of arranging the cells of a given packing among themselves.It may be arguedthat not all arrangements will completely fill space. The error so introduced willdepend on the precision with which we define a " kind " of cell. If we were to specifythe precise shape of each cell, then the error would be larger than if we specified, forR. Collins, Proc. Battelle Coll. Phase Changes in Metals (Geneva-Villars, March 1966), to bepublished by McGraw-HillGENERAL DISCUSSION 83example, the potential energy of a molecule at the centre of the cell : minor changesin cell shape needed to fit a cell into its surroundings can be made without seriouslyaffecting the energy.For the purposes of a preliminary estimate of the configurationalentropy corresponding to a random packing of spheres, we may classify cells accordingto the number of nearest neighbours which define the cell without specifying theprecise geometry of the cell. If a fraction ei of the cells are formed from i nearestneighbours, then the configurational entropy is simply - Rx6, In Bi. The intuitiveargument mentioned above would lead to the same result. The histogram given byBernal and King for close-packed spheres leads to a value of 2.77 cal deg.-l mole-l.This is certainly of the correct order of magnitude for the configurational entropy ofmelting of a substance with freely rotating spherical molecules (e.g., CH4, 2.5;CC14, 2.4).Other contributions to the entropy of fusion will arise from the looseningof the vibrational degrees of freedom in cells with smaller numbers of nearest neigh-bours; and from the fact that Bernal’s analysis refers to spheres in closest randompacking, to which liquids will tend only at the lowest temperatures. At the meltingpoint of most substances the liquid will probably have a significantly less densestructure so that we shall expect the figure of 2.77 cal deg.-l mole-1 to represent aminimum value for the entropy of fusion : in fact, only metals have substantiallylower values, while the noble gases (e.g., Ar, 3.4) have rather higher values.It will be of considerable interest to see how the configurational entropy of randomsphere packs varies with the density.However, we should not expect the methodof reducing the density by random removal of single spheres used by Bernal and Kingto be realistic : random sphere packs containing discrete holes of “ molecular ”size are highly improbable distributions and calculations based on them are unlikelyto be reliable. The method employed by Mason in which a continuous series ofrandom packings of varying density is generated should (provided that surface effectscan be eliminated) lead to much more realistic values.To carry the theory further it would be necessary to show how the density variedwith temperature : this is clearly a more complex problem and is not easily solublewithout a much more detailed analysis requiring the introduction of an assumedintermolecular force law between pairs of spheres.The real value of Bernal’s modelis that it brings out clearly what is implied by the term randomness as applied to thestructure of liquids, and emphasizes the fundamental geometrical distinction betweensolids and liquids.Dr. J. Finney, Birkbeck College, London, (communicated). The advantages of atetrahedral description of a random packing have been stressed by Beresford, Collinsand Mason. Beresford and Mason are particularly interested in porosity and hencethis approach is of greater use than polyhedra, while Collins’ approach via neighbourdistances implies the convenience of a tetrahedral lattice. We are interested in thegeometry of the array and the packing problem itself and therefore the Voronoipolyhedron is the obvious unit as it defines completely unambiguously a region ofspace associated with one point only.The polyhedron shapes may be complex,but they do give us information which could be used to elucidate the thermodynamics.For instance, Everett’s use of the co-ordination polyhedra to estimate the con-figurational entropy shows an approach which could yield fruitful results, althoughuse of the Voronoi geometrical co-ordination might be more realistic. To justifysuch an approach to configurational entropy, we must establish a relationship betweenthe particular histogram (here the coordination number) and the “ site energy ” of acentre. Further work on this has shown the site energy to be effectively unrelatedto geometrical coordination, but suggests a remarkably high correlation of 0.83-0.9084 GENERAL DISCUSSIONbetween the Voronoi volume and the centre energy.The physical reason for this isobscure-it may be connected to the packing restrictions-but the existence of sucha high correlation shows promise for evaluating an improved approximation to theentropy and facilitates the use of a large number of small samples to this end, whererealistic energy calculations could not normally be made.The general problem of calculating the entropy of a random array is fraught withuncertainties. The approach of Collins and Allison via information theory entailsthe difficulty of sorting out what information is relevant, and how much is needed togive an unambiguous minimum description of the system.In this connection the useof either coordination number or geometric neighbour histograms is an insufficientdescription, for the information contained about each site is incomplete in energyterms.Regarding the use of soft potentials suggested by Collins, we have in fact carriedout analyses on some high density Monte Carlo runs kindly made available to us byDr. Singer and Dr. Mcdonald of Royal Holloway College, in an attempt to comparea random packing with the structure of a “ liquid like ” arrangement generated in thecomputer. Initial results show a remarkable similarity between fig. 6 for the hardsphere model and the total histogram for several configurations at different timesin the same Monte Carlo run.Moreover, there are interesting variations betweenthe different configurations, suggesting complete inadequacy of single samples ofabout 100 centres for approaching the problem via geometrical neighbours.As for the two-dimensional liquid, we think it is essentially different in kind fromthe three-dimensional system. For example, = 6 exactly in two dimensions, whileiV is a function of density in three; moreover, the generation programme of Masoncrystallizes in two dimensions but not in three, suggesting differences in the basicnatures of the two problems.Beresford’s attempts to develop a statistical geometry in terms of a tetrahedraldescription are interesting, but entail an ambiguity in the choice of “ diagonals ”,or near contacts.This ambiguity could be removed simply by defining tetrahedradirectly from the Voronoi set-i.e. by choosing the inter-centre links to be theVoronoi face normals. With each Voronoi vertex are associated four centres, thusdefining uniquely a set of tetrahedra completely filling space. After removing thisambiguity, it would be interesting to see what this approach to the statistical geometryyielded.Mason’s random model generation programme could be a step forward inproducing data for analysis on a large scale as similar programmes constructedpreviously have not converged to a limiting density. We cannot see a theoreticalargument to show that Mason’s programme would converge. Moreover, it is notcertain what the characteristics of the packing are, how they depend upon the exactmechanisms written into the programme, and how they compare with physicallybuilt models.For example, the final result may depend on exactly how we removean overlap, and what system we use for picking out centres during the search foroverlaps. The construction of a random model under a central gravitational fieldis an improvement over the physical models, but out present inability to describethe essential characteristics of a random model make it difficult to compare explicitlythe results of the two modes of construction.One of the features of such a generating programme is that it gives random packingwith a large density variation up to the maximum of a random close packing andcould give the data necessary for investigating thermodynamic properties with tempera-ture variation.There are serious difficulties, however, in the long machine timesnecessary even for 200 spheres, and the rate of increase of time with sample size seemGENERAL DISCUSSION 85to prohibit the generation of much larger ones. Thus physical models still have animportant place. However, if the thermodynamic significance of the Voronoivolume can be firmly established, a large number of smaller packings will be invaluable.Prof. C. Domb (King’s Cottege, London) said: I wish to speak about the behaviourof the specific heat of a fluid in the critical region, and more particularly about com-paring experimental results with the predictions of the lattice gas model. Followingthe accurate experimental work of Voronel and his collaborators, various analyseswere attempted to establish the experimental values of the critical indices for thisspecific heat.There were differences of opinion as to whether the data could best befitted by a logarithm or a power law.2 Since the data cover a limited range withappreciable experimental errors, a formula of the typeinvolving several disposable parameters was used, and it is not surprising that the datacould be fitted by a range of values of a. (The in the formula refers to the regionjust above and just below the critical point, and the value a = 0 corresponds to alogarithm.) In fact, for a theoretical model there are no disposable parameters,and to test the validity of the model, its predictions should be compared directly withexperimental results.This has not been possible previously because of the lack ofsufficiently precise theoretical information on the three-dimensional lattice gas model.A. V. Voronel, V. G. Gorbunova, Yu. R. Chashkin and V. V. Schekochikhina, Soviet PhysicsJETP, 1966,23, 597. A. V. Voronel, Yu. R. Chashkin, V. A. Popov and V. G. Simkin, SovietPhysics JETP, 1964, 18, 568. M. I. Bagatskii, A. V. Voronel and V. G. Gusak, Soviet PhysicsJETP, 1963,16, 517.M. E. Fisher, Physic. Rev. 1964, 126, 159986 GENERAL DISCUSSIONRecent calculations by members of the theoretical research group at King’s Collegehowever, greatly improved the accuracy of the theoretical calculations.and his collaborators havesucceeded in adding four new terms to the high temperature series expansion for thespecific heat.This means, that series are available as far as l/TI3. (The terml/TI4 should be available shortly.) The result of plotting the ratio of successivecoefficients as a function of 1/N is shown in fig. 1 ; the upper points refer to thesusceptibility and the lower points to the specific heat. The critical index is deter-mined by the limiting slope of this ratio for large N and the evidence is convincingthat the specific heat index is &. It is then possible with so many exact terms andBy a remarkable co-ordination of techniques, Sykes654e --- 832Io_ - 5 - 4 - 3 - 2 - I 0log,, (1 -Tc/T)FIG. 2.a limiting asymptotic form to calculate the specific heat accurately, and the result ofsuch calculations for a number of three-dimensional lattices is presented in fig.2.It will be seen that the specific heat depends little on crystal structure. At lowtemperatures, the position is much less satisfactory since the series behave less regularly.However, Gaunt has examined other critical indices and has thereby producedindirect evidence that the critical index has the same value of $. For the diamondlattice, the low-temperature series are all positive, and Gaunt has made an estimateof the specific heat.When we compare with experiment, we follow the approach of F i ~ h e r , ~ remember-ing at the same time that a lattice gas is a crude model and we should not expect toomuch. In fact, we wish to determine how adequate it is in accounting for criticalM.F. Sykes, J. L. Martin and D. L. Hunter, Proc. Physic. SOC., in press.C. Domb and M. F. Sykes, J. Math. Physics, 1961, 2, 63.D. S. Gaunt, Proc. Physic. Soc., in press.M. E. Fisher, Physic. Rev. 1964, 126, 1599GENERAL DISCUSSION 87behaviour. Fig. 3 (prepared by D. L. Hunter) shows such a comparison; it willbe seen that, when the critical point is closely approached, the experimental resultsshow small but definite deviations from the theoretical calculations. The low-temperature comparison is shown in Fig. 4. Gaunt points out that in the range shownv-5 - 4 - 3 - 2 - Ilog,o (1 - TdT)FIG. 3.Points : Date for N2 (Voronel et al.)Curves : king model (a : F.C.C. ; b : S.C.)the theoretical curve could easily be mistaken for a logarithm since it is almost linearon the logarithmic scale.The disagreement between experiment and theory is moremarked than on the high-temperature side; this was already noted by Fisher whoregarded it as a serious defect of the model. We may conclude that, although theM. E. Fisher, Physic. Rev. 1964, 126, 169988 GENERAL DISCUSSIONlattice gas model has considerable success in accounting for critical behaviour, thereis room for improvement which could perhaps be obtained by relaxing the rigidityof the model and allowing holes of different shapes and sizes.Dr. B. L. Smith (University of Sussex) (communicated): In connection with thecritical point properties of simple fluids, I report the results of some preliminarymeasurements of the refractive index, surface tension and density of xenon in thisregion.The surface tension was measured in the temperature range 189 to 286°Kby a capillary rise method. The results may be represented by y = yo(l -T/289.74)*with yo = 54.6f0.1 dyne cm-I and p = 1.287+0.017. According to the law ofcorresponding states, p should be a universal constant ( N 1.22). Previous resultsfor simple molecules 2 p indicate that for argon p : 1.28, and for nitrogen, p = 1.24.It seems unlikely that the lack of sphericity of the nitrogen molecule (acentric factorl ~ ) = 0.04) is sufficient to account for the difference between the value ofp for nitrogenand those for argon and xenon. A more likely explanation would appear to be thatthe (pz - p,) data used to compute the surface tension are unreliable, and that moreaccurate values would result in better agreement.It is noted thatp N 1.28 is in excellentagreement with a calculation by Widom for a lattice gas model, based on a reformul-ated van der Waals, Cahn-Hilliard theory of surface tension.Refractive index measurements were carried out on xenon liquid and vapour incoexistence and also on fluid xenon at temperatures above the critical point. Studies,e.g., by Abbiss et aZ.,6 have suggested that an anomalously large deviation from theLorentz-Lorenz function might occur in this region, much greater than that predictedby * The results of over 500 measurements lead us to the conclusion thatthe Lorentz-Lorenz function (n2 - l)/(n +2)p remains constant to within f 3 %over the whole range studied (0~002-0~024mole~m-~) and that at the critical pointn, = 1.1383fO-0008, L.L. = 10.5f0.1 cm3 mole-l. The variation may be muchsmaller, since most of the possible error in the Lorentz-Lorenz function arises fromuncertainty in density.The lack of accurate density data, in particular, reliable values for (pz - p,), haslead us to develop a direct experimental method for obtaining this information.According to Guggenheim,l the difference in density between a liquid in coexistencewith its vapour is given by pz - p , = A( 1 - T/T,)P, where fl N 0.33. The results of ourpreliminary observations suggest that /3 = 0.343+0*010. This agrees well with thevalue = 0.345f0.015 obtained by Fisher lo from an analysis of the results ofWeinberger and Schneider,ll and also with /3 = 0.341 obtained from our refractiveindex measurements by assuming that the Lorentz-Lorenz function is independent ofdensity, i.e.,[(n2 - l)/(n2 + 2)p]z - [(n2 - l)/(n2 + 2)pJ, = K( 1 - T/Tc)tE. A. Guggenheim, J. Chem. Physics, 1945,13,253.F. B. Sprow and J. M. Prausitz, Trans. Faraday Soc., 1966, 62, 1097.D. Stansfield, Proc. Physic. Soc. A, 1958,72,854.G. N. Lewis, M. Randall, K. S. Pitzer and L. Brewer, Thermodynamics (McGraw-Hill, NewYork, 1961).B. Widom, J. Chem. Physics, 1965,43, 3892.C. P. Abiss, C. M. Knobler, R. K. Teague and C. J. Pings, J. Chem. Physics, 1965,42,4145.L. S . Taylor, J. Math. Physics, 1963, 4, 824.S. Y. Larsen, R. D. Mountain and R. Zwanzig, J. Chem. Physics, 1965, 42, 2187.B. L. Smith, J. Sci. Instr., 1966, 43, 958.lo M. E. Fisher, J. Math. Physics, 1964,5,944.l1 M. A. Weinberger and W. G. Schneider, Can. J. Chem., 1952,30,422