Section 4.1.5.3. Interpretation of dispersion relations

The usual procedure for analysing dispersion relations is to set up the Born–von Kármán formalism with interatomic force constants . The calculated frequencies are then derived from the eigenvalues of the dynamical matrix D (Section 4.1.2.1) and the force constants fitted, by least squares, to the observed frequencies. Several sets of force constants may describe the frequencies equally well, and to decide which set is preferable it is necessary to compare eigenvectors as well as eigenvalues (Cochran, 1971).

The main interest in the curves is in testing different models of interatomic potentials, whose derivatives are related to the measured force constants. For the solid inert gases the curves are reproduced reasonably well using a two-parameter Lennard–Jones 6–12 potential, although calculated frequencies are systematically higher than the experimental points near the Brillouin-zone boundary (Fujii et al., 1974). To reproduce the dispersion relations in metals it is necessary to use a large number of interatomic force constants, extending to at least fifth neighbours. The number of independent constants is then too large for a meaningful analysis with the Born–von Kármán theory, but in the pseudo-potential approximation (Harrison, 1966) only two parameters are required to give good agreement between calculated and observed frequencies of simple metals such as aluminium. In the rigid-ion model for ionic crystals, the ions are treated as point charges centred on the nuclei and polarization of the outermost electrons is ignored. This is unsatisfactory at high frequencies. In the shell model, polarization is accounted for by representing the ion as a rigid core connected by a flexible spring to a polarizable shell of outermost electrons. There are many variants of this model – extended shell, overlap shell, deformation dipole, breathing shell … (Bilz & Kress, 1979). For molecular crystals the contributions to the force constants from the intermolecular forces can be derived from the non-bonded atomic pair potential of, say, the 6-exponential type: Here, i, j label atoms in different molecules. The values of the parameters A, B, C depend on the pair of atomic species i, j only. For hydrocarbons they have been tabulated for different atom pairs by Kitaigorodskii (1966) and Williams (1967). The 6-exponential potential is applicable to molecular crystals that are stabilized mainly by London–van der Waals interactions; it is likely to fail when hydrogen bonds are present.