Philosophical transactions - Royal Society. Mathematical, physical and engineering sciences 

About: Philosophical transactions - Royal Society. Mathematical, physical and engineering sciences is an academic journal. The journal publishes majorly in the area(s): Nonlinear system & Laser. Over the lifetime, 1886 publications have been published receiving 109846 citations.

The Growth of Crystals and the Equilibrium Structure of their Surfaces

Abstract: 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.

A Mechanism of Magnetic Hysteresis in Heterogeneous Alloys

Abstract: 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.

On the Mathematical Foundations of Theoretical Statistics

Abstract: Centre of Location. That abscissa of a frequency curve for which the sampling errors of optimum location are uncorrelated with those of optimum scaling. (9.)

Sound Generation by Turbulence and Surfaces in Arbitrary Motion

Abstract: Monograph on sound generation by turbulence and surfaces in arbitrary motion, discussing sound and multipole fields and governing equations

Multiple-relaxation-time lattice Boltzmann models in three dimensions.

Abstract: This article provides a concise exposition of the multiple-relaxation-time lattice Boltzmann equation, with examples of 15-velocity and 19-velocity models in three dimensions. Simulation of a diagonally lid-driven cavity flow in three dimensions at Re = 500 and 2000 is performed. The results clearly demonstrate the superior numerical stability of the multiple-relaxation-time lattice Boltzmann equation over the popular lattice Bhatnagar-Gross-Krook equation.