Effect of anharmonicity on the phonon density of states and specific heat of a monoatomic, one-dimensional crystal lattice

Abstract

We analyze the lattice equation of motion involving terms up to third order in lattice displacement. The phenomenological arguments suggest that the force constant D 1 of the quadratic term must always be positive and the force constant B 1 of the cubic term may take either positive or negative value. The criterion for stability of the lattice provides constraint on the relative magnitudes of the three force constants. We solve the equation of motion using root mean-square spatial fluctuation approximation and obtain the seminonperturbative dispersion relation both for positive and negative B 1 . The nature of phonon density of states curves for positive B 1 show some close resemblance with the experimental observations. At very low temperature, the specific heat of this system to leading order in large positive B 1 varies as square root of temperature and it obeys Debye's T law in one dimension for small negative B 1 . At very high temperature, the specific heat may fall below or above its classical value depending on the relative magnitudes of B 1 and D 1 for B 1 Ͼ 0 and it always falls above its classical value for B 1 Ͻ 0. The lattice model with positive B 1 emerges as a good candidate for description of a monoatomic crystal.