Symmetry plays a crucial role in most formalisms of physics, and provides an elegant way to get a variety of results starting from a fully general mathematical basis. Such a principle of course applies to the realm of vibrational modes too. The structural symmetry of crystals, due to the ordered arrangement of atoms, causes particular properties of the vibrational modes, in terms of both the eigenvectors and eigenvalues of the dynamic matrix. The mathematical link that relates a space group, describing a geometrical symmetry, to the dynamics of a crystal, i.e. to a physical phenomenon, is provided by the theory of representations. Vibrational spectroscopy involved with lattice dynamics heavily benefits from the inferences attainable by combining the dynamics of the atomic motion in crystals with symmetry. Such a synergy has led to results of great practical use: selection rules, for instance, are a well-known powerful tool in interpreting experimental spectroscopic data.
This presentation aims at the following objectives:
illustrating the basic mathematical notions which underlie a rigorous approach to the relationships between lattice dynamics and symmetry, through the concept of group representations;
showing how the dynamic equations can be re-cast into a form leading straight to group representations;
providing the main links between lattice dynamics and crystal spectroscopy, in terms of theoretical inferences stemming from a combination of symmetry and dynamics, and experimental issues.
The points above have been developed with a particular care to the formal aspects, for three reasons: