Site symmetry and crystal symmetry: a spherical tensor analysis

TL;DR

This paper introduces a spherical tensor formalism to relate site-specific properties to overall crystal properties, improving computational efficiency in crystal property calculations using symmetry considerations.

Contribution

It develops a detailed spherical tensor analysis framework for crystal symmetry, enabling efficient property calculations and illustrating it with specific crystal examples.

Findings

Formalism reduces computation time in band structure calculations.

Application to chromium in spinel and vanadium in garnet demonstrates effectiveness.

Provides a systematic approach to relate site and crystal properties.

Abstract

The relation between the properties of a specific crystallographic site and the properties of the full crystal is discussed by using spherical tensors. The concept of spherical tensors is introduced and the way it transforms under the symmetry operations of the site and from site to site is described in detail. The law of spherical tensor coupling is given and illustrated with the example of the electric dipole and quadrupole transitions in x-ray absorption spectroscopy. The main application of the formalism is the reduction of computation time in the calculation of the properties of crystals by band structure methods. The general approach is illustrated by the examples of substitutional chromium in spinel and substitutional vanadium in garnet.