2 - Symmetry considerations

Summary

Introduction

The distinguishing feature of crystalline solids is their symmetry, manifest microscopically in their X-ray diffraction patterns and macroscopically in crystal morphology. An ideal crystal is an infinite regular repetition in space of identical structural units. The symmetry of a particular ideal crystal is specified by the set of symmetry elements, comprising rotations, reflections and translations, which leave it invariant. Real crystals are not only of finite extent, but also contain a variety of imperfections such as inclusions of minority phases, grain boundaries, dislocations, impurities and point defects; the latter two are especially relevant in the present context. An isolated impurity or point defect in an otherwise ideal crystal obviously removes translational symmetry. It may also reduce the residual point symmetry in some circumstances, exemplified by ions with degenerate electronic states (the Jahn–Teller effect), impurities in the form of small molecules, small substitutional cations which move off-centre, vacancy pairs, bipolarons, etc. A principal theme of the present monograph is exploration of the ways in which the properties of a laser-active centre are controlled or affected by its crystalline surroundings, including their residual point symmetry. We shall find that it is often useful to distinguish a dominant site symmetry, determined by coordination alone, which is somewhat higher than the actual point symmetry.

The formal description of symmetry exploits a branch of mathematics called ‘Group Theory’, which is not a physical theory in the sense of ‘Quantum Theory’, but is rather a collection of principles deduced from a chosen set of axioms.