Theory of Anharmonic Crystals

The anharmonic contributions to the energy, specific heat, frequency-wave-vector dispersion relations and damping of phonons in a crystal have been studied using the recent technique of thermodynamic Green's functions based on field-theoretic methods. General expressions for these quantities have been deduced. It has been shown that at absolute zero the number of density is finite for the anharmonic solid and it is of the same order as the square of the fractional change in the normal-mode frequencies. The cases of a linear chain and a simple model of a crystal have been studied in detail. The normal modes of a linear chain exhibit some unusual but interesting features for different amounts of anharmonicity. The half-width of the phonons of the anharmonic chain has been evaluated for all temperatures for both normal and umklapp processes. For a solid, the complicated integrals that occur because of anharmonicity are simplified by an approximation scheme suggested by us. Within the scope of this approximation it is found that the frequency-wave-vector dispersion curves for a solid show a dip at the maximum wave number, quite similar to that observed for solids like lead and copper. The width of phonons is proportional to the square of the wave number, and the thermal conductivity is seen to be finite at low temperatures and to vary inversely with temperature at high temperatures.