Cubic crystal system

About: Cubic crystal system is a research topic. Over the lifetime, 4039 publications have been published within this topic receiving 85286 citations. The topic is also known as: isometric crystal system.

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.

Anisotropy of the Electronic Work Function of Metals

Abstract: Work function is experimentally known to be different for different faces of a crystal by amounts ranging from one-tenth to half a volt. For tungsten the faces can be arranged according to decreasing work function as follows: 110, 211, 100 and finally 111. The explanations so far suggested for the differences of the work function are discussed and shown to give either an incorrect sequence or a wrong order of magnitude of the observed differences. The author uses the picture of Wigner and Bardeen according to which the work function is a sum of a volume contribution and a contribution due to a double layer on the surface of the metal. The origin of the latter can be described in the following manner. With every atom one can associate a polyhedron ("$s$-polyhedron") with the atom at its center, such that it contains all points nearer to the atom under consideration than to any other atom. If the distribution of the electron density within these polyhedra of the surface atoms was the same as for the inside atoms then there would be no double layer on the surface. However, this is not the case since the total energy is lowered by a redistribution of the electron cloud on the surface. There are two effects: the first is a partial spread of the charge out of the $s$-polyhedra and the second is a tendency to smooth out the surface of the polyhedra. In consequence of the second effect the surfaces of equal charge density are more nearly plane than in the original picture. The two effects have opposite influences and since they are comparable in magnitude, it is not possible to predict the sign of the total double layer without numerical computations. Some general formulae for the double layers are derived and discussed more fully in the case of a simple cubic and a body-centered cubic lattice. The minimum problem of the surface energy is solved for four faces of a body-centered crystal and the results are applied to the case of tungsten. One obtains the differences between the work functions for different directions. The results agree satisfactorily with the experimental data: assuming a reasonable density of the free electrons, one obtains the correct sequence of faces and the correct differences of the work function. The surface energies are calculated an d found in agreement with the observed stability of certain crystal faces.

Single Band Motion of Conduction Electrons in a Uniform Magnetic Field

Abstract: The effect of a uniform magnetic field on the conduction band of metal is investigated, using as model the tight-binding approximation for a simple cubic crystal. The normally discrete magnetic levels pertaining to free electrons are shown to be non-uniformly spaced and broadened as a result of the lattice forces.

Infrared Absorption at Longitudinal Optic Frequency in Cubic Crystal Films

Abstract: Thin flat films of cubic ionic crystals have sharp, strong infrared absorption bands at the frequencies characteristic of polar longitudinal optic modes of long wavelength in infinite crystals The bands occur only when the incident radiation beam is not normal to the surface and only in the $p$-polarized component of the radiation They occur not only in transmission through the films, but also in reflection when the films are deposited on thick metallic or dielectric substrates Such a band was observed in films of LiF and was found to be in good qualitative agreement with, but somewhat stronger than, the band predicted from a dielectric dispersion curve obtained from other types of measurements by Bilz, Genzel, and Happ