The Royal Society

Parts I and II deal with the theory of crystal growth, parts III and IV with the form (on the atomic scale) of a crystal surface in equilibrium with the vapour. In part I we calculate the rate of advance of monomolecular steps (i.e. the edges of incomplete monomolecular layers of the crystal) as a function of supersaturation in the vapour and the mean concentration of kinks in the steps. We show that in most cases of growth from the vapour the rate of advance of monomolecular steps will be independent of their crystallographic orientation, so that a growing closed step will be circular. We also find the rate of advance for parallel sequences of steps. In part II we find the resulting rate of growth and the steepness of the growth cones or growth pyramids when the persistence of steps is due to the presence of dislocations. The cases in which several or many dislocations are involved are analysed in some detail; it is shown that they will commonly differ little from the case of a single dislocation. The rate of growth of a surface containing dislocations is shown to be proportional to the square of the supersaturation for low values and to the first power for high values of the latter. Volmer & Schultze’s (1931) observations on the rate of growth of iodine crystals from the vapour can be explained in this way. The application of the same ideas to growth of crystals from solution is briefly discussed. Part III deals with the equilibrium structure of steps, especially the statistics of kinks in steps, as dependent on temperature, binding energy parameters, and crystallographic orientation. The shape and size of a two-dimensional nucleus (i.e. an ‘island* of new monolayer of crystal on a completed layer) in unstable equilibrium with a given supersaturation at a given temperature is obtained, whence a corrected activation energy for two-dimensional nucleation is evaluated. At moderately low supersaturations this is so large that a crystal would have no observable growth rate. For a crystal face containing two screw dislocations of opposite sense, joined by a step, the activation energy is still very large when their distance apart is less than the diameter of the corresponding critical nucleus; but for any greater separation it is zero. Part IV treats as a ‘co-operative phenomenon’ the temperature dependence of the structure of the surface of a perfect crystal, free from steps at absolute zero. It is shown that such a surface remains practically flat (save for single adsorbed molecules and vacant surface sites) until a transition temperature is reached, at which the roughness of the surface increases very rapidly (‘ surface melting ’). Assuming that the molecules in the surface are all in one or other of two levels, the results of Onsager (1944) for two-dimensional ferromagnets can be applied with little change. The transition temperature is of the order of, or higher than, the melting-point for crystal faces with nearest neighbour interactions in both directions (e.g. (100) faces of simple cubic or (111) or (100) faces of face-centred cubic crystals). When the interactions are of second nearest neighbour type in one direction (e.g. (110) faces of s.c. or f.c.c. crystals), the transition temperature is lower and corresponds to a surface melting of second nearest neighbour bonds. The error introduced by the assumed restriction to two available levels is investigated by a generalization of Bethe’s method (1935) to larger numbers of levels. This method gives an anomalous result for the two-level problem. The calculated transition temperature decreases substantially on going from two to three levels, but remains practically the same for larger numbers.

The Becker-Kersten treatment of domain boundary movements is widely applicable in the interpretation of magnetization curves, but it does not account satisfactorily for the higher coercivities obtained, for example, in permanent magnet alloys. It is suggested that in many ferromagnetic materials there may occur ‘particles’ (this term including atomic segregates or ‘islands’ in alloys), distinct in magnetic character from the general matrix, and below the critical size, depending on shape, for which domain boundary formation is energetically possible. For such single-domain particles, change of magnetization can take place only by rotation of the magnetization vector, I _O . As the field changes continuously, the resolved magnetization, I _H , may change discontinuously at critical values, H _O , of the field. The character of the magnetization curves depends on the degree of magnetic anisotropy of the particle, and on the orientation of ‘easy axes’ with respect to the field. The magnetic anisotropy may arise from the shape of the particle, from magneto-crystalline effects, and from strain. A detailed quantitative treatment is given of the effect of shape anisotropy when the particles have the form of ellipsoids of revolution (§§ 2, 3, 4), and a less detailed treatment for the general ellipsoidal form (§ 5). For the first it is convenient to use the non-dimensional parameter such that h = H /(| N _a - N _b |) I _O , N _a and N _b being the demagnetization coefficients along the polar and equatorial axes. The results are presented in tables and diagrams giving the variation with h of I _H / I _O . For the special limiting form of the oblate spheroid there is no hysteresis. For the prolate spheroid, as the orientation angle, θ , varies from 0 to 90°, the cyclic magnetization curves change from a rectangular form with | h _O | = 1, to a linear non-hysteretic form, with an interesting sequence of intermediate forms. Exact expressions are obtained for the dependence of h _θ on θ , and curves for random distribution are computed. All the numerical results are applicable when the anisotropy is due to longitudinal stress, when h = HI ₀ /3λδ, where λ is the saturation magnetostriction coefficient, and δ the stress. The results also apply to magneto-crystalline anisotropy in the important and representative case in which there is a unique axis of easy magnetization as for hexagonal cobalt. Estimates are made of the magnitude of the effect of the various types of anisotropy. For iron the maximum coercivities, for the most favourable orientation, due to the magneto-crystalline and strain effects are about 400 and 600 respectively. These values are exceeded by those due to the shape effect in prolate spheroids if the dimensional ratio, m , is greater than 1·1; for m = 10, the corresponding value would be about 10,000 (§7). A fairly precise estimate is made of the lower limit for the equatorial diameter of a particle in the form of a prolate spheroid below which boundary formation cannot occur. As m varies from 1 (the sphere) to 10, this varies from 1·5 to 6·1 x 10 ^-6 for iron, and from 6·2 to 25 x 10 ^-6 for nickel (§ 6). A discussion is given (§ 7) of the application of these results to ( a ) non-ferromagnetic metals and alloys containing ferromagnetic ‘impurities’, ( b ) powder magnets, ( e ) high coeravity alloys of the dispersion hardening type. In connexion with ( c ) the possible bearing on the effects of cooling in a magnetic field is indicated.

This paper is concerned with the evaluation and tabulation of certain integrals of the type (* 00 I(p, v; A) = J J fa t) ) e~cttxdt. In part I of this paper, a formula is derived for the integrals in terms of an integral of a hypergeometric function. This new integral is evaluated in the particular cases which are of most frequent use in mathematical physics. By means of these results, approximate expansions are obtained for cases in which the ratio b/a is small or in which b~a and is small. In part II, recurrence relations are developed between integrals with integral values of the parameters pt, v and A. Tables are given by means of which 7(0, 0; 1), 7(0, 1; 1), 7(1, 0; 1), 7(1,1; 1), 7(0, 0 ;0), 7(1, 0;'0), 7(0, 1; 0), 7(1, 1; 0), 7(0,1; - 1 ), 7(1,0; - 1 ) and 7(1,1; - 1 ) may be evaluated for 0 < b/a ^ 2, 0 ^ c/a ^ 2.

The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect' functional provided due account is taken of its gauge symmetry. This enables topological conclusions to be drawn about the critical sets and leads eventually to information about the moduli space of algebraic bundles over the Riemann surface. This in turn depends on the interplay between the holomorphic and unitary structures, which is analysed in detail.

The equations of motion, boundary conditions and stress-strain relations for a highly elastic material can be expressed in terms of the stored-energy function. This has been done in part I of this series (Rivlin 1948 a ), for both the cases of compressible and incompressible materials, following the methods given by E. & F. Cosserat for compressible materials. The stored-energy function may be defined for a particular material in terms of the invariants of strain. The form in which the equations of motion, etc., are deduced, in the previous paper, does not permit the evaluation of the forces necessary to produce a specified deformation unless the actual expression for the stored-energy function in terms of the scalar invariants of the strain is introduced. In the present paper, the equations are transformed into forms more suitable for carrying out such an explicit evaluation. As examples, the surface forces necessary to produce simple shear in a cuboid of either compressible or incompressible material and those required to produce simple torsion in a right-circular cylinder of incompressible material are derived.

The study of phase equilibria is historically one of the most important sources of information about the nature of intermolecular forces in non-electrolyte liquids and their mixtures. Many of the main features of vapour-liquid and liquid-liquid phase behaviour were already well characterized experimentally during the early part of this century, but the theoretical explanation of phase equilibria for a wide variety of substances and over a large range of pressures and temperatures has lagged far behind. This paper presents theoretical studies of phase equilibria in binary mixtures obeying the van der Waals equation, especially liquid-liquid equilibria that can occur at high pressures. The variety of fluid phase behaviour that occurs in binary mixtures can be qualitatively discussed in terms of the changes in thermodynamic properties near critical points. Upper critical solution temperatures (UCSTs) occur when a heterogeneous (two-phase) system becomes a homogeneous (one-phase) system when the temperature is raised. The maximum temperature along the temperature-mole fraction (T, x) coexistence curve for constant pressure is the UCST at this pressure. Lower critical solution temperatures (LCSTs) occur when a homogeneous system becomes a two-phase system when the temperature is increased. The LCST is at the minimum of theT, xcoexistence curve. Thermodynamic considerations of critical points yield requirements for the curvature of the mixing functions plotted againstx.

It is shown that in calculating transition integrals it is permissible to neglect the departure of the potential of an atom or ion from its asymptotic Coulomb form. This enables a general analytical expression for the transition integral to be derived. Tables are compiled from which the absolute strengths of large numbers of spectral lines can at once be obtained if the term values of the upper and lower levels are known. s-p , p-d and d-f transitions are all treated. Comparison with experimental data shows that for the simpler systems (i. e. systems with a single electron outside closed shells) the method gives remarkably accurate results; indeed, it appears superior to the normal rather laborious procedure involving the computation of the necessary wave functions, in each individual case, by solution of the appropriate Hartree or Fock differential equation. The method (in its most elementary form) may not be so satisfactory for complex systems (i. e. systems with unclosed shells) owing to difficulties associated with the identification of certain energy parameters. However, the rather scanty comparison data available suggest that even for such systems it yields useful (and in some cases precise) information on the line strengths. Incidentally, in the course of the work the accuracy of a few wave functions based on the self consistent field approximation (including exchange) was tested by using them to evaluate line strengths from both the dipole moment and the dipole velocity formulae. Appreciable defects were revealed.

The geometrical fit of the continents now separated by oceans has long been discussed in relation to continental drift. This paper describes fits made by numerical methods, with a ‘least squares’ criterion of fit, for the continents around the Atlantic ocean. The best fit is found to be at the 500 fm. contour which lies on the steep part of the continental edge. The root-mean-square errors for fitting Africa to South America, Greenland to Europe and North America to Greenland and Europe are 30 to 90 km. These fits are thought not to be due to chance, though no reliable statistical criteria are available. The fit of the block assembled from South America and Africa to that formed from Europe, North America and Greenland is much poorer. The root-mean-square misfit is about 130 km. These geometrical fits are regarded as a preliminary to a comparison of the stratigraphy, structures, ages and palaeomagnetic results across the joins.

Physical processing or pretreatment of lignocellulosics concerns the ultrastructural modification of materials such as wood, straw and bagasse. The substrates produced can be subsequently converted by chemicals. The various pretreatment options will be discussed in the light of the ultrastructural, polymeric and chemical modifications that are obtained. The processes can be classified as follows: (i) steam; (ii) aqueous; and (iii) organosolvolysis treatments. All of these have their antecedents in the thermomechanical processes developed by the pulp and paper or fibreboard industries. Sequential application of thermomechanical technology leads to fractionation of the substrate into the major polymeric fractions: cellulose, hemicellulose and lignin in varying degrees of modification. A number of pretreatment concepts are now at a commercial scale and are being applied to produce foodstuffs from lignocellulosics for use by ruminant animals. The same techniques are being piloted in the energy and chemicals from lignocellulosics field.

Lead isotopic compositions of young volcanic rocks from different tectonic environments have distinctive characteristics. Their differences are evaluated within the framework of global tectonics and mantle differentiation. Ocean island leads are in general more radiogenic than mid-ocean ridge basalt (m.o.r.b.) leads. They form linear trends on lead isotopic ratio plots. Many of the trends extend toward the field of m.o.r.b. On plots of ²⁰⁷ P b / ²⁰⁴ Pb against ²⁰⁶ Pb / ²⁰⁴ Pb, their slopes are generally close to 0.1. Island arc leads in general are confined between sediment and m.o.r.b. type leads with slopes of ca . 0.30 on a plot of ²⁰⁷ P b / ²⁰⁴ Pb against ²⁰⁶ Pb / ²⁰⁴ Pb. Pb, Sr and Nd isotopic data of Hawaiian volcanics are closely examined. Data from each island support a two-component mixing model. However, there is a lack of full range correlation between islands, indicating heterogeneity in the end members. This mixing model could also be extended to explain data from the Iceland-Reykjanes ridge, and from 45° N on the Atlantic Ridge. The observed chemical and isotopic heterogeneity in young volcanic rocks is considered to be a result of long-term as well as short-term mantle differentiation and mixing. Lead isotopic data from ocean islands are interpreted in terms of mantle evolution models that involve long-term (more than 2 Ga) mantle chemical and isotopic heterogeneity. Incompatible element enriched ‘plume’-type m.o.r.b. have Th/U ratios ca . 3.0 too low and Rb/Sr ratios ca . 0.04 too high to generate the observed ²⁰⁸ Pb and ⁸⁷ Sr respectively for long periods of time. Elemental fractionation in the mantle must have occurred very recently. This conclusion also applies to mantle sources for ocean island alkali basalts and nephelinites. Depletion of incompatible elements in m.o.r.b. sources is most probably due to continuous extraction of silicate melt and/or fluid phase from the low-velocity zone throughout geological time. Data on Pb isotopes, Sr isotopes and trace elements on volcanic rocks from island arcs are evaluated in terms of mixing models involving three components derived from (1) sub-arc mantle wedge, (2) dehydration or partial melting of subducted ocean crust, and (3) continental crust contamination. In contrast to the relation between ⁸⁷ Sr/ ⁸⁶ Sr and ¹⁴³ Nd / ¹⁴⁴ Nd ratios of ocean volcanics, there is a general lack of correlation between Pb and Sr isotopic ratios except that samples with very radiogenic Pb ( ²⁰⁶ Pb / ²⁰⁴ Pb > 19.5) have low ⁸⁷ Sr/ ⁸⁷ Sr ratios (0.7028- 0.7035). These samples also have inferred source Th/U ratios (3.0-3.5) not high enough to support long-term growth of ²⁰⁸ Pb. Data suggest that their mantle sources have long-term integrated depletion in Rb, Th, U and light r.e.e. High ²³⁸ U / ²⁰⁴ Pb (y a)values required by the Pb isotopic data are most probably due to depletion of Pb by separation of a sulphide phase. Relations between Pb, Sr and Nd isotopic ratios of young volcanic rocks could be explained by simultaneous upward migration of silicate and/or fluid phase and downward migration of a sulphide phase in a differentiating mantle.ration of a sulphide phase in a differentiating mantle.