International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 2.1, p. 266
Section 2.1.1. Introduction^{a}Institut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D-37077 Göttingen, Germany |
Interatomic interactions in crystalline solids not only determine the equilibrium atomic structure but also the possible excitations of the lattice: the motions of atoms, molecules or ions. The investigation of dynamical processes provides us with detailed information about the interatomic forces. Obviously, there are a huge variety of possible collective motions within a solid, which represents a multiparticle system with 10^{23} degrees of freedom. As long as the solid may be described by an equilibrium structure and atomic displacements from the average positions are small compared with interatomic distances, the dynamical behaviour of the lattice is essentially determined by well defined lattice vibrations or phonons. These elementary excitations of a solid are described by eigenvectors and eigenfrequencies reflecting the strength of interatomic interactions. Owing to the symmetry (space group) of the equilibrium structure there are constraints for the individual eigenvectors. In special high-symmetry cases, phonon eigenvectors can even be predicted merely on the basis of group-theoretical considerations.
This chapter is devoted to the implications of lattice symmetry on the form, i.e. on the eigenvectors, of lattice vibrations. We restrict ourselves to the consideration of perfect crystals and harmonic vibrations. In addition, some aspects of anharmonicity are discussed in terms of a quasi-harmonic model, yielding the connection between microscopic dynamics and macroscopic thermodynamic quantities such as thermal expansion. However, intrinsic anharmonic effects associated with the interaction of phonons, phonon damping or localized vibrations due to defects, for example, are beyond the scope of this article. In Section 2.1.2 we present the fundamentals of lattice dynamics with special emphasis on the role of the dynamical matrix. Section 2.1.3 deals with the symmetry properties of this matrix along with its eigenvectors and eigenfrequencies. Symmetry-induced degeneracies will be considered in some detail as well as compatibility relations for phonon wavevectors corresponding to points of higher symmetry within the reciprocal space. Finally, the optical selection rules for long wavelength vibrations are presented. Some examples are included in order to illustrate the theoretical results.
For a further discussion of other phenomena associated with lattice vibrations the reader is referred to the monographs of Leibfried (1955), Maradudin et al. (1971), Reissland (1973), Srivastava (1990) or Dove (1993).
References
Dove, M. T. (1993). Introduction to lattice dynamics. Cambridge University Press. (ISBN 0–521–39293–4.)Leibfried, G. (1955). Gittertheorie der mechanischen und thermischen Eigenschaften der Kristalle. In Handbuch der Physik VII/1, edited by S. Flügge, pp. 105–324. Berlin: Springer.
Maradudin, A. A., Montroll, E. W., Weiss, G. H. & Ipatova, I. P. (1971). Theory of lattice dynamics in the harmonic approximation. In Solid state physics, Suppl. 3, edited by H. Ehrenreich, F. Seitz & D. Turnbull. New York: Academic Press.
Reissland, J. A. (1973). The physics of phonons. London: Wiley. (ISBN 0–471–71585–9.)
Srivastava, G. P. (1990). The physics of phonons. Bristol: Adam Hilger. (ISBN 0–85274–153–7.)