Application of symmetry groups of four-dimensional space in spectroscopy of crystals

Abstract

A method of classification of many-electron states that involves symmetry groups of four-dimensional space is developed. Taking into account the spatial and temporal symmetry, the selection rules for electric and magnetic dipole transitions are investigated. The reduction relations for the irreducible representations of the orthogonal group O4 and the group R4 of pure rotations of the four-dimensional space on the groups and Oh4 and O4 of the four-dimensional cube and of the octahedron on the three-dimensional groups Oh and D6h of the cubic and hexagonal systems, respectively, are obtained. The four-dimensional classification of the levels of the spin-orbit interaction of rare earth ions for the intermediate reduction on the group of the four-dimensional cube is performed. With the help of the irreducible representations of the O4 group, the selection rules of magnetic and forbidden electric dipole transitions, as well as of intercombination transitions, are investigated.