GENERAL DISCUSSION Dr. M. La1 (Port Sunlight) said: The paper by Hoare and McInnes highlights only the minimum-energy configurations. From the statistical thermodynamic standpoint, however, the relevant configurational states would be those which correspond to the minimumfree energy of the system. It is only at 0 K that the minimum binding energy would coincide with the minimum free energy. At other temperatures entropic contributions, such as those arising from atomic vibrations and the relative flattening of the high-density regions in the configuration phase space, might be significant. How will the consideration of the entropic factors alter the picture as conveyed in the present work? Another important point concerns the presence of many-body interactions in condensed phases, particularly in the solid phase.It has been generally known that in such phases the assumption of the pair-wise additivity for interatomic interactions is inadequate and one niust consider at least three-body interactions in order to establish reliable relationship between the configurational state and the binding energy. Thus it would perhaps be worthwhile studying the influence of non-additive contributions on the configurational stability. The non-additive part for the three body interactions can be conveniently estimated using the Axilrod-Teller formula. Dr. M. R. Hoare (Bedford College) said: La1 is quite right that free-energy consider- ations are paramount in determining the actual occurrence of particular structures at a finite temperature. We do not, however, neglect this, as a reading of Section 7 of the paper will show.Further accounts of the relative contributions of energetic and entropic effects will be found in ref. (1 1) and (12). Although we have not yet pub- lished full details in terms of the complete isomer ensembles, our preliminary conclu- sion is as stated, namely that the relative insensitivity of vibrational frequency patterns to change from one minimum to another tends to minimize the contribution of en- tropic effects to the relative stability of different isomers, at least in the temperature- range for which the harmonic approximation is valid. The main exception to this is in the rotational contribution of clusters with very high symmetry, such as the icosahedron. We agree that the effect of many-body forces should be taken into account, though we have not found a way to do this in the computing time available.We suspect, in fact, that many-body contributions may be amplified in small particles, where surface effects are dominant, and particularly in the case of metals. Dr. J. F. Qgillvie (Newfoundland) said: In regard to the results of Hoare and Mc- Innes,l the great reduction of numbers of stable isomers according to the Morse potential, relative to those obtained with the Lennard-Jones potential, appears very striking. But if one compares the forms of the pair-potential functions, the reasons become evident. The figure contains three potential curves (in reduced form), for the Lennard-Jones function, V(r) = (1 -4-y- 1 M. R. Hoare and J.McInnes, this Discussion.64 GENERAL DISCUSSION where for the usual (6-12) function n = 6, and two Morse curves, V(r) = (1 -e--nr)2- 1 where n = 3 or 6. Examining the two curves with n = 6, we observe that the functions are quite similar, the significant difference being in the region of r - 2 where the Morse curve indicates a much smaller interaction energy than that of the Lennard-Jones function. The relation of these two curves to real molecular systems is that for two argon atoms, for instance, the true potential lies between these two model functions in the region r - 1.8 but rises more steeply than either in the repulsive region r < 0.9. The Morse function with n = 3 provides in contrast a much greater t . V 0 .o -0.5 - 1 .o FIG. 1,-Potential energy functions in reduced form for the Morse (M) or Lennard-Jones (L) models with the indicated exponents. binding energy at r < 2 than either of the n = 6 functions.The consequence is that the central atom can exert a much greater influence, under this potential, on second- nearest neighbours ; therefore, the less compact isomers are relatively much less stable. The Morse function with n = 3 is a good approximation to the true potential function of such strongly bound molecules as carbon monoxide,l and probably to diatomic molecules of metallic elements, but only in the lower region of the potential well. At large separations (r > 2.5), the space dependence tends to a limiting r-6 form. Thus this Morse function seems to have doubtful applicability as a model in the considera- tion of metal-atom clusters, in which long-range interactions are important.Dr. M. R. Hoare (Bedford College) said: While we obviously had metals in mind when studying the stability of the Morse-clusters, we were more concerned to demon- strate qualitatively the sensitivity of the number of minimum locations to the type of potential than to arrive at realistic values of the binding energy and thermodynamic properties. There are various estimates of the Morse exponential parameter in the metallurgical literature, some of which are surprisingly low, (e.g. a from about 1.18 for aluminium to 1.58 for gold).2 J. F. Ogilvie and R. W. Davis, Faraday Disc. Chem SOC., 1973,55, 189. Weaire, Ashby, Logan and Weins, Acta Metallurgica, 1971, 19, 779-88.GENERAL DISCUSSION 65 Dr.F. Abraham (San Jose) said: Until recently, the physical cluster has been pic- tured as a ‘‘ liquid droplet ”, a “ microcrystallite ”, or some other model construct, i.e., the statistical mechanical theories of a physical cluster have been heuristic, de- pending mainly upon the chosen model. The heuristic approach requires a successful choice of a suitable model which (l), represents reasonably well the system of interest, and (2), provides a relatively simple partition function from which numerical quantities can be obtained. While these models provide an insight into the origin of certain features of a physical cluster, they cannot serve to elucidate the cluster’s true molecular structure since this is assumed in one way or another.With the advent of large scale scientific computers, pursuing such an approach for the statistical thermodynamics of many-body systems has begun where only a form of the intermolecular potential function between two molecules is assumed. Two recent examples of this approach have been molecular dynamics and Monte Carlo simulations of clusters of Lennard- Jones atoms. In the Monte Carlo paper, we present a formal physical cluster theory for an im- perfect gas that is valid for an arbitrary definition of a “ physical cluster ”. The role of the deJnition of the physical cluster is stressed. For a particular definition of the physical cluster, which may be appropriate in nucleation theory, the Helmholtz free energy of 13-, 43-, 60-, 70-, 80-, 87- and 100-atom argon clusters is calculated in the classical limit for temperatures ranging from absolute zero to 100 K using Monte Carlo techniques.It is found that a cluster’s free energy is almost independent of its definition provided that the definition is reasonable and the temperature is sufficiently low (T < 75 K for an 87 atom cluster). The results are compared with the predic- tions using the harmonic approximation, deviations occurring for T 15 K. We make special note of the fact that the multiconfigurational states of the cluster arise as a direct consequence of the Monte Carlo computational procedure. Prof. H. Reiss (Los Angeles) said: Since the physical cluster is so prominent in nucleation theory, attempts are frequently made to define it rigorously. Abraham in his comments on physical clusters, described the work of himself and certain of his colleagues, and made the point that under certain conditions there may be no rigorous method of definition, and that in any event the definition should always be tailored to the measurement for which the theory is being developed.I thoroughly agree and would like to illustrate the point by reference to a physical cluster problem involving a one dimensional fluid.3 Here I will try to describe physical clusters which can be used in the development of the equilibrium equation of state of the fluid. Furthermore these clusters have a close connection to those which Abraham has studied and will probably be of value in nucleation theory. The most rudimentary physical cluster theory of the vapour phase assumes the clusters to be “ independent ” molecular species such that they constitute a mixture of ideal gases.Under this condition the configurational partition function for the entire gas may be expressed as , where {m,) indicates that the sum is over all terms such that 2 Iml = N. 1 D. J. McGinty, J. Chem. Phys., 1973, 58,4733. * J. K. Lee, J. A. Barker and F. F. Abraham, J. Chem. Phys., 1973,58,3166. H. P. Gillis, D. C. Marvin and H. Reiss, J . Chenz. Phys., 1977, in press.66 GENERAL DISCUSSION In these equations, N represents the total number molecules in the gas, ZN the total configurational partition function, zl, the configurational partition function for a cluster of molecules, and ml the number of clusters of l molecules. The immediate (rigorous) statistical mechanical consequence of eqn (1) and (2) is that the pressure of the vapour is given by where V is the volume of the vapour and viil the average of (if N + co) the most probable number of clusters of size 1.This is, of course, simply Dalton's law of partial pressures. Now, if the goal is to deJine the cluster, and therefore choose zl in eqn (l), such that the 2, are exact, this is impossible unless ZZ = Vbl (4) where bl is the reducible mathematical cluster integral involving l molecules. (Al- though implicitly obvious, this point has never been explicitly stated in regard to theories of physical clusters. It follows directly from the reversion of eqn (1) which yields 21 = 21 ( 5 ) L L The quantities on the right are just Vbl, Vb2, and Vb3, etc.) Since choosing thephysical cluster, such that its partition function is a mathematical cluster integral, is not very physical, especially since some of the GI will then have to be negative, this choice buys nothing in the way of a model, and so the strict goal of achieving (3) withphysical clusters is impossible.Thus, with a physical cluster approach, Dalton's law is strictly out of the question, and another formulation besides eqn (1) is necessary. The simplest adjustment is to include the so-called excluded volume contributions' in eqn (1). In reality this is a counting restriction associated with the combinatorics of expressions like eqn (l), i.e., it is not associated with exclusion due to physical potentials, but it has the same effect. For a three-dimensional system an approximate inclusion of excluded volume would replace eqn (1) with relations of the following sort for zl, z2, z3, etc.z, = z;e(i, o,o, 0 . . .) = z;v= z1 5 2! = 9 L ( 2 , 0 , 0 . . .)+z;e(o, 1 , o . . .) (6) - 2 3 - - A O ( 3 , @'I3 0, 0 . . .)+z;Z;o(l, 1, 0 . . .)+ZiO(O, 0, 1 . . .) 3! 3! etc. where z ' ~ = zl/Y and where, typically, O(0, i, j , 0 . . .) is an integral, much like a con- figuration integral, for a system containing no monomers, i dimers, j trimers, no tetramers, etc. The integral treats the clusters as each surrounded by a hard shell beyond which there is an arbitrarily defined minimal field of interaction with other F. H. Stillinger, J. Chein. Phys., 1963, 38, 1486.GENERAL DISCUSSION 67 clusters assumed to lie within their own shells.The shells are therefore like " hard " molecules and represent the configuration integrals of assemblies of such hard mole- cules. It can be shown that in one-dimension eqn (6) can be made exact. To illustrate this we choose linear molecules with pair interaction potentials where a, b, and E are constants and x is the distance between the centres of two mole- cules. Thus the molecules have hard-rod cores and attractive, square-well, outer potentials. A cluster of Z molecules is now defined as a group of I, such that every molecule is interactionwise connected. Put in another way, there must be no con- figuration in which the distance between two adjacent molecules exceeds 6. The maximum length of a cluster can be Zb, but, at that maximum length, not every internal configuration meets the cluster criterion.Tn fact, if we define Az as the length of an I cluster, in terms of the distance between its terminal molecules, an application of Markhoff's method, using eqn (7) shows that for In this equation, zl(Al) is the partition function of the physical cluster when its length is Az and L is the length (volume) of the entire system. The quantity r is an integer indexing terms in the sum, while Cf-' is the binomial coefficient. Thus the physical cluster must be defined not only by 2, but by its length &. In three dimensions its shape as well as its extent would have to be involved in its definition. With the definition of the physical cluster given above, it is possible to show that the following alteration of eqn (1) is exact. 5 2! = 0" 2! 6(2, 0)+rz;(A2)LdA2 a etc.Here the 8's have the same meaning as in eqn (6) except the hard-shelled molecules are replaced by hard rods of lengths determined by the various Al. The integers in eqn (9) are really sums over different cluster partition function products ; different values of ;Il corresponding to different clusters, even though I is the same. These integrals, which would be essentially impossible in three dimensions, have one redeeming feature which, fortunately, can be immediately investigated in one dimension, i.e., when Z is large enough the integrals have contributions only from a very small range of This is illustrated by fig. 1, 2, and 3 which are plots of z;(AI), from eqn (S), versus At in dimensionless units, for I = 3, 4 and 12, respectively. Notice how quickly z'(Al) moves toward &function behaviour; at I = 12, the half width is less than 5%68 GENERAL DISCUSSION I ' 00 0.4 0.8 I2 16 2.0 2.4 2.8 3.2 3.6 4.0 A 3 FIG.1. 1 L - L 1.8 2.4 3.0 3.6 4.2 4.8 A 4 FIG. 2. of the entire range of iZl. For larger valuecof 2, ~'(1~): canlclearly be treated as a 8- function in the integrals of eqn (9). Now z; has been defined in term of an origin in some particular moZecu2e in the cluster, together with the overlap criterion enunciated above. The question arises as to whether it can be defined in terms of an origin at the cluster's centre of mass, as in Abraham's cluster, together with the overlap criterion and, if so, can the transformation between the two clusters be easily accomplished.The answer is affirmative, so that the same 8-function-like behaviour should occur.GENERAL DISCUSSION u 69 66 8 8 11.0 132 154 17.6 19.8 2 2 0 i, 12 FIG. 3. Another definition would involve dropping the overlap criterion but insisting that the cluster be confined within a given interval of length, centred on the cluster’s centre of mass, exactly as in Abraham’s cluster. For large enough clusters, configurations in which the overlap criterion is violated should carry very little weight, and the results should be essentially the same as with overlap. The significance of being able to dispense with the overlap criterion in this manner is not major, in one dimension, where it can be accounted for in a straightforward manner. However, in three dimensions a calculation including this criterion would be prohibitive.Thus as with Abraham’s cluster, the cluster in three dimensions can be defined in terms of confinement of the molecules to the interior of a sphere centred on the cluster’s centre of mass, and then even if with neglect of overlap its partition function converges to the same z ’ ~ obtained with both the requirement of overlap and the requirement that the origin be at a given molecule rather than at the centre of mass, the problem of evaluating z’~ becomes tractable. Furthermore eqn (6) becomes accurate under these circumstances. Dr. E. R. Buckle (Shefield) said: Hoare mentioned astrophysics, and I wonder whether small collections of atoms in configurations of absolutely minimal energy can be formed other than in an inter-stellar environment. With the computer one can choose them at will, but Briant’s conclusions pre-suppose the computer’s choice of configurations to be the same as the natural growth sequence.Might not the suc- cession be governed by kinetic mechanism? Is that what is meant by special entropic effects? Dr. M. R. Hoare (Bedford College) said : As you suggest, the mechanism of growth of small particles under realistic conditions might well exclude certain otherwise favoured configurations for essentially kinetic reasons, which may just as well operate in astrophysical conditions as in, say, nozzle-beam condensation. Put slightly differ- ently, one has no right to assume that solid-like clusters, formed under more or less70 GENERAL DISCUSSION rapid cooling, are effectively " annealed ", far less in true thermodynamic equilibrium.We have considered making an assumption of extreme quenching, assuming that particles are formed with equal probability in all accessible minima and computing the effective thermodynamic properties for this mix. However, this is just as likely to misrepresent kinetic and steric factors in collisions and would give weight even to chain-like configurations which are hardly likely to occur. Presumably the truth lies somewhere between the two extremes. So long as we are not trying to determine the morphology of the particles for its own sake, it would be satisfying if the nucleation parameters proved to be quite insensitive to which of the two extreme models we adopted.If' the growth process is kinetically determined, it might well be quite differ- ent according to conditions, presence or absence of carrier-gas, for example. Prof. J. Zarzycki (Montpellier) said: In calculating the interference functions of his model cluster Briant takes into account only the distances rlj between atoms in the same microcluster-and neglects distances between atoms in the adjoining clusters which could be of the same order of magnitude within a dense liquid. The problems discussed are similar to those already encountered in the study of glasses when the choice between the random network and microcrystallite theories appears now as a matter of degree. The difficulty lies in defining the " interstitial matrix " joining more or less locally ordered regions.Attempts at using pentagonal dodecahedra1 models for describing glass lattices go back to Tiltonl and Robinson.2 This model has recently been generalized to larger clusters by Ga~kell.~ Neither X-ray diffraction studies nor high-resolution electron microscopy, however, enable a choice to be made between these different hypotheses. Dr. C. L. Briant (Philadelphia) said: In our calculation of the interference function for the quenched 55 atom structure we include not only the distances within the 13 atom icosahedron but those in the disordered material around the isosahedron as well. This should consider the contribution of the interstitial matrix to the interference function. To completely answer this question, one must quench larger numbers of atoms which might then contain more than one ordered region.This would then allow a much more complete analysis of material interconnecting the ordered regions. Dr. M. La1 (Port Sunlight) said: The model for a microcluster considered in Briant's paper constitutes n(= 13) atoms suspended in vacuum, whereas in the paper by Abraham and Mruzik a microcluster is regarded as an assembly of particles (atoms, molecules or ions) enclosed in an impenetrable " cell ". Will the author please comment on the apparent contrast between the models. Dr. C. L. Briant (Philadelphia) said: The difference in these two models is basically as follows. In the model we used, an atom boiling off the surface of the cluster is allowed to do so and could only return to the cluster if the potential energy drew it back.In Abraham's model an atom passing beyond some constraining radius is directed back at the cluster rather than being allowed to escape. In the solid and low temperature liquid clusters examined in this paper, one never finds any atoms boiling off the surface so the result would be the same for either model. Only if one wanted to study high temperature liquid clusters would the model critically affect the results. L. W. Tilton, J. Res. Nat. Bur. Stand., 1957,59, 139. H. A. Robinson, J. Phys. Chem. SoIids, 1965,26,209. P. H. Gaskell, Phil. Mag., 1973,32,211.G E N ERA L D IS CU S S I 0.N 71 Dr. F. F. Abraham (Sun Jose) said: You noted that in examining the 55 atom quenched cluster you found that a 13 atom icosahecron had formed which included atoms near the surface, in contrast to forming at the centre of the cluster.Don’t you feel that it would be more appropriate rapidly to quench the bulk phase simulated by the use of periodic boundary conditions so as to eliminate any “ surface ” effects in the creation of the amorphous state? (e.g., see A. Rahman, J. J. Mandell and J. P. McTague, Molecular dynamics study of an amorphous Lennard-Jones system at low temperature, J. Chem. Phys., 1976, 64, 1564.) Dr. C. L. Briant (Philadelphia) said: It is true that one would like to quench more bulk-like systems in order to establish a model for amorphous metals. However, the use of periodic boundary conditions presents problems, as they can induce rather artificial structures and also give misleading ideas as to the frequency of various seed structures.I feel the best approach to this problem would be to quench large numbers of atoms using free boundary conditions. One could then examine the structure of the centres of these quenched aggregates where there should be little surface influence. Also such a method should give more information about the inter- connecting tissue between the structural units. Prof. H. Reiss (Los Angeles) said: There is some work,l* employing molecular dynamics, which relates closely to research described by both Hoare and Briant. This work generally confirms the conclusions of both Hoare and Briant. I I I I 0.5 I .o 1.5 2.0 2.5 3.0 l/u FIG. 1. First an extremely supercooled (“ glassy ”) Lennard-Jones fluid. A 500-particle liquid sample was compressed and rapidly cooled to a reduced temperature (kT’/c) of 0.11 and a reduced density (pa3) of 0.95. The radial distribution function, shown in fig.1, is similar to that found by other investigators for random-close-packed systems, The doubled second peak in the radial distribution corresponds to the A. Rahman, M. J. Mandell and J. P. McTague, J. Chein. Phys., 1976, 64, 1564. M. J. Mandell, J. P. McTague and A. Rahman, J. Chein. Phys., 1976, in press.72 GENERAL DISCUSSION shoulder in the second peak of the structure factor, which was described by Briant. The diffusion constant was zero within the limits of their measurement. The dynamic structure factor (left, fig. 2) shows that the amorphous system has propagating longitudinal phonons. The longitudinal sound speed is roughly equal to that for solid argon.The transverse current correlation function (right, fig. 2) 0 10 20 cd.I 0 10 20 r3.z FXG. 2. indicates propagating transverse phonons at long wavelength, with a sound speed about 25% less than that for solid argon. In the course of beginning the study of the " glass transition " these investigators found that by using sufficiently slow cooling rates the Lennard-Jones fluid would spontaneously crystallize. Fig. 3 records the history of a run made with 108 particles at a density of 0.91 with constant energy. The nucleation event is characterized by simultaneous vanishing of diffusion, increase in temperature and decrease in pressure. Similar results have been achieved for 256 and 500 particles. For the larger systems, crystallization requires higher densities and/or lower temperatures.Mandell developed the technique of searching for the absolute maximum (away from the origin) of the structure factor in reciprocal space. The history of this quantity for the nucleation event illustrated in the previous figure is shown in fig. 4. In the super- cooled fluid one infers partially ordered structures which are fairly long-lived. The interpretation that these structures are sub-critical nuclei is tempting. The appearance of the Bragg peaks at crystallization is indicative. Fig. 5 shows the projection of a 256-particle crystal. The sample has been quenched after its formation at high temperature. Close-packed (1 1 1) planes areGENERAL DISCUSSION 73 I R2 1 .o 0.5 (3.0 500 250 0 .80 .70 .60 .50 I .... press u re tempera! ure c vertical and perpendicular to the plane of the figure. Unlike the 108-particle system, the larger samples invariably form imperfect crystals. Nonetheless, these samples are large enough that their crystalline structures can be unambiguously recognized; defects such as vacancies and dislocations can be located. The final figure (fig. 6) is the projection of a spherical region taken from a 256-particle sample shortly before a crystallization event. The region contains 33 particles (28 shown and 5 hidden). The distances are consistent with having vertical close-packed planes as in the previous slide, and the orientation coincides with that of the not-yet-formed crystal. Optimistically, this might be regarded as the first observation of a spontaneously-formed crystal nucleus.74 a 1 GENERAL DISCUSSION I .I I I J 1 o4 c p ,of Y 1 02 0 I 0 */ 0 0 A -(1.2, -1.2, 6.4) o -14.8, 3.7, 2.6) -(1.2, 1.2, 6.4) A -(-3.7.3.7,3.71 FIG. 4. 4 , 1 , - 1 0.1.1 5 r A I I I - 3 - 3 - 2 - 1 - 0 - -1 -2 -3 - - - + + 4 + + + * * 4 b + * + * + + ' * ; 4 : -4 + + -7 -3 FIG. 5.GENERAL DISCUSSION 75 + + + +f + 4- t f ++ + + + + + t + + ti t i- +. + +I + FIG. G. Dr. M. La1 (Port Sunlight) said: Monte Carlo technique based upon Metropolis sampling is most suitable when the phase space contains a single minimum energy " well ',. The method produces a sample in which most of the points (configurations) lie in the " well " in the vicinity of the minimum. If, however, the configurational phase space is composed of multitude of minima separated by energy barriers, realiza- tion of a truly converged sample might be difficult within the present computational limitations.In this context the question concerning the magnitude of the energy barriers separating the various stable configurations of the clusters, as posed by Everett, is most relevant. If such barriers are high, then the suitability of the method to the cluster problem is questionable. Dr. F. F. Abraham (Sun Jose) said: For finite temperatures, the exact structure of the water-ion clusters corresponding to the lowest energy (T = 0 K) is of marginal chemical interest since many conjigurations with nearly equal stabilization energy exist (€3. Kistenmacher, H. Popkie and E. Clementi, J.Clzem. Phys., 1974, 61, 799). For this reason, we use the Monte Carlo method to generate the Boltzmann-weighted configuration space and, hence, to get thermodynamic and structural properties, especially for the high temperature of 298 K. It should be noted that the Monte Carlo method has been proved successful for calculating the equilibrium properties of dense systems (e.g., liquids) where there exists no single minimum energy well. Prof. J. M. Thomas (Aberystwyth) said: My comment refers to the optimum cation-water configuration envisaged for ion-water pairs (fig. 13(a)). There are a number of monovalent cations which possess substantially similar ionic radii (Rb+ = 1.48, NH+ = 1.43, T1+ = 1.40A) but which, on the basis of their differing polarizabilities, give rise to quite widely varying properties in, for example, ease of cation migration, particularly in the solid state. One wonders whether the heuristic approach outlined in Abraham's paper is capable of taking into account these relatively subtle differences between ostensibly similar cations. Dr. F. F. Abraham (Sun Jose) said: If given an accurate potential energy surface for the cation-water system, the variation of physical properties due to differing polarizabilities would be taken in account when calculating the equilibrium properties using the Monte Carlo method. Dr. E. R. Buckle (Shefield) said: In regard to the discrepancies in fig. 8 and 9, the vapours of the alkali-metal salts contain polymeric species which increase in abun- dance in the order K, Na, Li. This could complicate the interpretation of the mass spectra.76 GENERAL DISCUSSION Dr. F. F. Abraham (Sun Jose) said: Such possible errors were examined by Kebarle (ref. (9) and (lo)), and he concluded that they probably contributed little to the total experimental error. Dr. E. R. Buckle (Shefield) said: Turning to the work of Kahlweit, what is being described as ageing is analogous to the course of a homogeneous chemical reaction. It keeps things simple if the constituents of the mixture can be chosen so as to give the correct predictions with the ideal laws of kinetics and thermodynamics. If the description of the growing particle and the way it behaves towards its neighbours were correct the course of reaction would follow from the laws of motion. The trouble, and here the problem is quite general in liquids,l lies in distinguishing the motion of a coherent group of atoms from the motions of the atoms themselves. E. R. Buckle, The Chemical MetaZZwgy ofIron and Steel (Iron and Steel Institute, London, 1973), p. 52.