Symmorphic Space Groups Research Articles

Reduction of Crystallographic Point Groups to Subgroups by Homogeneous Stress.

It is assumed that the symmetry operations possessed by a homogeneously stressed crystal will be those common to the crystal and to the stress. Application of stress either leaves a point group unaltered or lowers it to a subgroup. Any stress-induced minimum step of symmetry lowering can be caused by unaxial stress but three of the possible stress-induced compound steps of symmetry lowering require biaxial stress. The uniaxial or biaxial stress required for any stress-induced symmetry lowering is tabulated and stereograms are given for each of the minimum steps showing the relation of the remaining symmetry operations to the initial symmetry and showing the splitting of a set of initially equivalent general directions into inequivalent subsets. These stereograms provide a direct representation of the splitting of a general position in a crystal into inequivalent subsets for the symmorphic space groups and the latter are listed; the effect of glide planes and screw axes must be considered in the remaining space groups yet the stereograms still provide the correct pattern of splitting. Uses of the stereograms, for instance, to obtain the splitting of a special position, are described.

The determination of irreducible representations of symmorphic space groups using semidirect product groups

The determination of irreducible representations of symmorphic space groups using semidirect product groups

Application of Space-Group Theory to the Vibrational Problem of di-Tetramethyl Ammonium Uranium Hexachloride

The absorption spectrum of [(CH3)4N]2UCl6 (abbr. TMA) is discussed. The spectrum is vibronic in nature, i.e., the electronic transitions between levels belonging to the 5f2 configuration of the U4+ ion are coupled with the vibrations of the TMA lattice. The number and symmetries of the normal modes of vibration of a unit cell are calculated in the factor group approximation. The Winston—Halford decomposition formula is used to reduce the representation of motions among the irreducible representations of the symmetry group of TMA. Calculations are compared with the experimental results. The gross features of the spectrum are found to be in agreement with the theoretical predictions. The method of obtaining the character tables for finite symmorphic space groups is presented in summary form. A general vibrational problem in crystals is discussed.