Lattice Vibrations

Abstract

In a crystal with N atoms or ions, there exist 3(N − 1) lattice vibrations. For an insulator they can be derived, in principle, from the full Hamiltonian by using a Born—Oppenheimer treatment which separates these low-energy excitations from the comparatively high-lying electronic excitations. For a metal an even more complicated treatment which describes the dynamics of electrons and ions in a self-consistent way has to be used. Until now, no calculations existed from first principles, only some semi-empirical methods starting, e.g., from plausible lattice potentials (screened coulomb potential, Born—Mayer potential, etc.) with some parameters adjustable to match experimental data. In a rigorous treatment, one uses a formal development of the lattice potential in powers of the ion displacements. The expansion coefficients are restricted only by conservation laws and by the symmetry of the crystal, and they must be determined from experimental information. The harmonic approximation leads to the picture of free quasi-particles, called “phonons,” each of which is characterized by its energy ħω,crystal momentum ħq,and branch index j. As can be seen from inelastic neutron spectroscopy, this picture is usually well satisfied since the lifetime of phonons is of the order of 100 vibration periods. The far-reaching analogy between quantum-mechanical and classical harmonic oscillators allows for a classical description of plane waves progressing in the direction of the wave vector q. Thus, each lattice ion moves with the frequency ω and has an elliptic polarization which is uniquely related to the branch index j, but which has a simple form only in certain symmetry directions (transverse or longitudinal). The cubic and higher-order parts of the lattice potential cause complicated phonon-phonon interactions. In the following, we separate the static effects from the dynamical ones (phonon lifetime, infrared absorption, etc.). We therefore use a quasi-harmonic approximation [²] in which all single phonon quantities (frequencies, force constants, etc.) are considered at a fixed temperature and depend on it as a parameter.