Chapter 2 Lattice dynamical applications of variational effective potentials in the feynman path-integral formulation of statistical mechanics

Abstract

This chapter discusses several formalisms to calculate the lattice dynamical properties of crystals. The original harmonic, or quasi-harmonic, formalism is valid only if the amplitudes of the atomic vibrations are small. For most materials, this restricts the method to low temperatures. For quantum crystals, the harmonic approximation is never satisfactory as the zero-point motion of the atoms is too large. Anharmonicity can be included through a perturbation expansion and this extends the temperature range over which the theory can be applied. However, the successive terms in the perturbation expansion become more and more complicated, and in some cases, the expansion appears to be only asymptotically convergent. Self-consistent phonon theories were originally devised to deal with the quantum crystals. The harmonic force constants are replaced by their averages calculated self-consistently over a trial set of harmonic oscillator functions. The theory can be shown to be equivalent to a summation of an infinite subset of the perturbation theory terms. A great triumph of the first order self-consistent phonon theory (SCI) is that it provides a physically reasonable starting point for the description of quantum crystals, such as helium.