Thermodynamics and lattice vibrations of minerals: 3. Lattice dynamics and an approximation for minerals with application to simple substances and framework silicates

The principles of lattice dynamics are briefly reviewed in this paper, and from these principles a simple, generally applicable lattice vibrational spectrum is proposed for minerals. The spectrum can be used to calculate the thermodynamic functions in the harmonic approximation. The model proposed is consistent with lattice dynamics and is sufficiently detailed in its assumptions about the distribution of modes that the thermodynamic functions are closely specified. The primitive unit cell, containing s atoms, is chosen as the fundamental vibrating unit; it has 3s degrees of freedom, three of which are acoustic modes. Anisotropy of these modes is included by use of anisotropic shear velocities for the two shear branches: dispersion is included by use of a sinusoidal dispersion relation between frequency and wave vector for all three acoustic branches. Optic modes, which comprise 3s–3 degrees of freedom, are represented by a uniform continuum, except for ‘intramolecular’ stretching modes (such as Si-O stretching modes) which can be enumerated and identified as being isolated from the optic continuum. Parameters for the model are obtained from elastic and spectroscopic data; the model is independent of any calorimetric data. It is applied to give the temperature dependence of Cv over the range 0°–1000°K of halite, periclase, brucite, corundum, spinel, quartz, cristobalite, silica glass, coesite, stishovite, rutile, albite, and microcline. The specific heat of the simple Debyelike substances, halite and periclase, is reproduced well by the model. The influence of the additional formula unit of H2O on the vibrational and thermodynamic properties of brucite, as compared to periclase, is discussed. The heat capacities of the relatively simple minerals, spinel and corundum, are given accurately by the model. The heat capacities of quartz, cristobalite, and coesite are accurately predicted from spectroscopic data through the model. The heat capacity of silica glass is discussed in terms of the classic continuous random network (CRN) model and a paracrystalline model, the pentagonal dodecahedral (PD) model of Robinson (1965). The PD model appears to be more consistent with measured Cv data than the CRN model. The heat capacity data of rutile are reasonably reproduced by the model as are the data for stishovite at temperatures above 50°K. Measured data for stishovite below 50°K appear to contain an excess heat capacity relative to the model; this excess may arise from surface energy contributions, as was suggested by Holm et al. (1967), and it is suggested that the model provides a better estimate of the low-temperature vibrational heat capacity of stishovite than the measured data. The heat capacities of albite and microcline are reproduced well by the model. The model is only a simple approximation to real lattice vibrational spectra, but it appears to work well for a large number of minerals and is therefore useful in correlating structural, compositional, elastic, spectroscopic, and thermodynamic properties for purposes of extrapolation and prediction of these properties.