Crystallographic Classes Research Articles

Laser-beam intensity dependent, optical circular birefringence in crystals

Using a phenomenological approach, the optical circular birefringence of crystals in the presence of intense laser light is shown to be the sum of natural gyration, self-induced optical rotation (exhibited by optically inactive bodies), and a self-induced nonlinear variation in the natural gyration. These effects are discussed for all crystallographic classes and the conditions for the separate observation of each effect are specified.

Crystal symmetry groups and their representations II orbits, strata and crystallographic classes of space groups

Crystal symmetry groups and their representations II orbits, strata and crystallographic classes of space groups

Polarization dependence of the two-photon absorption coefficient in crystals

The theory of two-photon absorption with provision for exciton effects is built on the basis of general principles without using any model of electronic states of the crystal. Electron-phonon interaction is neglected. The splitting of exciton levels into levels of transverse and longitudinal excitons is taken into account. The polarization dependence of two-photon absorption coefficient of allowed two-photon dipole transitions from the ground state of crystal for all 32 crystallographic classes is calculated. The results obtained are compared with the experimental data available.

Experimental study of nuclear electric quadrupole splittings of 27Al in Topaz

The nuclear magnetic resonance spectrum of 27Al in single crystals of natural topaz (Al 2[F,OH] 2SiO 4) has been investigated at room temperature at 10.500 MHz. Observed quadrupolar perturbations have been analyzed by the method of Volkoff, for three orthogonal planes, to yield eight nuclear quadrupole coupling and electric field gradient tensors. Satellite splitting indicated that the crystallographic class of topaz is holohedral. Eigenvalues and electric field gradient asymmetry parameters of the tensors fall into two groups of four, the members of each group having identical parameters. Eigenvectors of all tensors are different. The eight equivalent Al 3+ sites given by the X-ray space group are thus divided into two types of magnetically nonequivalent sites, each type consisting of four magnetically equivalent positions and the term magnetic being used to describe site symmetries as revealed by magnetic resonance. Quadrupole coupling constants and field asymmetry parameters for the two types of sites are: Type α sites Type β sites C z= eQφ zz h 1.5918±0.008MHz 1.6506±0.008MHz ν=(φ χχ−φ yy)φ! zz 0.704±0.007 0.501±0.005

Pattern of mutual relation between crystallographic lattice point groups and crystal systems

The suggested scheme is based on the principle of importance among the holohedral groups and the lattice points in individual crystallographic classes. The holohedral groups coincide with groups of symmetry characterizing the corresponding coordinate systems. These groups differ from the conventionally accepted systems by the nonpolar nature of axes and by the fact that their coordinates have no + or - signs. -- E. A. Alexandrov.

Wave Propagation in an Infinite Piezoelectric Plate

The problem of wave propagation in an infinite piezoelectric plate belonging to crystallographic class C6v, situated between shorted electrodes, is rigorously analyzed for two special orientations of the sixfold axis. The solution is derived using the linear piezoelectric equations. The analysis shows that for a given frequency and wave number in the propagation direction there are three independent solutions of the differential equations, and, furthermore, that these three solutions couple at the traction-free boundaries of the plate. The dispersion curves can be calculated from the resulting transcendental equations. It has previously been shown that the resonant frequencies at infinite wavelength of a plate, in which the electromechanical coupling factors are high, deviate considerably from those of the purely elastic solution. Thus the dispersion spectrum calculated from this analysis will deviate considerably from that of the purely elastic analysis. It is shown in an appendix that the solution for the plate of infinite width is also applicable in the case of the plate of infinitesimal width.

On the Symmetries of Spherical Harmonics

Letbe a finite group of transformations of three-dimensional Euclidean space, such that the distance between any two points is preserved by all transformations of the group. Such a group is a group of orthogonal linear transformations of three variables, or, geometrically speaking, a group of rotations and rotatory inversions. Thirty-two groups of this type are important in crystallography and are known as the crystallographic classes.

Axes of symmetry and the crystallographic classes

It is usual to divide axes of rotatory symmetry into those of the first and those of the second sort. In the former a crystal is brought into coincidence by a rotation through a certain angle round the axis and in the latter by such a rotation followed by a reflection about (or, as I prefer to say, a reversal relatively to) the plane at right angles to the axis. In 1906 Mr. H. Hilton proposed to define an axis of rotatory symmetry of the second sort as one in which the rotation was followed by an inversion about (reversal relatively to) a point, and he showed that his proposal would, if adopted, simplify the definition of the crystallographic classes.

Societies and Academies

LONDON. Mineralogical Society, January 29.—Prof. H. A. Miers, F.R.S., president, in the chair.—Experiments bearing on the order of crystallisation of rock-constituents: Prof. H. A. Miers. The general results of experiments made by Miss F. Isaac and Prof. Miers with mixtures of salol and betol in all proportions were described; the experiments have established the supersolubility curves even beyopd the points where they cross below the eutectic temperature. Owing to the fact that the main separation of crystals in the cooling mixtures takes place only when the liquids have been supercooled to temperatures given by the supersolubility curves, it has been found (i) that in general the mixtures do not solidify as a eutectic mixture; (2) that, according to the conditions of supercooling, either substance could' be made to crystallise before the other in mixtures approaching the eutectic in composition. It was suggested that these results are applicable to the solidification of many rocks and alloys.—Serpentine rock from the Tarnthaler Köpfe, Tyrol: Dr. A. P. Young. The minerals found in the serpentine are diopside, tremolite, clinochiore, picotite, magnetite, iron pyrites, and a fibrous mineral referred to antigorite. The latter mineral is regarded as holding a place between the micas and chiorites. On the surface of the serpentine are projecting bastite pseudomorphs coated with films of silvery lustre. The serpentine is a non-foliated intrusive core which on the borders is highly foliated and passes into talc-bearing phyllites.—A simple tabular arrangement of the thirty-two crystallographic classes: Dr. J. W. Evans. The table is based on the character of the symmetry of the principal zone axis or zone axes. Each column contains classes with the same rotational symmetry round the axis, and each row thoe which agree in the other symmetrical characters of the axis.—A new model of crystal refractometer: Dr. G. F. 1-lerbert Smith. This instrument is identical in principle with that previously described, and is intended for use with large mineral specimens and mounted gem-stones. No part extends above the level of the plate holding the dense glass hemisphere. Further, the optical combination has double the focal length of the earlier form, and provides, consequently, greater refinement.—Isomorphism as illustrated by certain varieties of magnetite: Prof. B. J. Harrington. Analyses are given of specimens of magnetite from St. Joseph du Lac, Canada, and from Magnet Cove, Arkansas, both showing the unusual combination of octahedron and trapezohedron {311}. The Canadian specimen contained about 5 per cent. of TiO2, 8 per cent. of MnO, and 3 per cent. of MgO, and the specimen from Arkansas about 10 per cent. of Al2O3, 2 per cent. of TiO2, 2 per cent. of MnO, and 9 per cent. of MgO.—Mr. Fleischmann exhibited a collection of olites from Japan, Dr. Evans an objective giving a flat field with convergent light, and Prof. Miers a goniometer to be used for the measurement of the refractive indices of cooling solutions, for which purpose it is provided with means for maintaining a constant temperature for any desired period.