a Study of the Fundamentals of Modern Lattice Dynamics with AN Application to Quantum Crystals

We describe some advances made in the effective potential Monte Carlo (EPMC) method. In particular, the effective potential has previously been evaluated by expanding the smeared potential up to finite order; we show that the infinite series can be summed analytically. Previous work also included various forms of an isotropic approximation and the low-coupling approximation; we discard the former but initially retain the latter. We find significant differences with earlier work and argue that the finite-order EPMC method is now obsolete and that the new formalism supersedes it. A deficiency in the EPMC method is that the Gaussian averaging process neglects odd terms in the Taylor expansion of the potential. We describe an improved effective potential theory (IEP) in which this is partially removed by incorporating into the EPMC method, using perturbation theory, a cubic contribution from the potential. We show that IEP theory leads to a marked improvement over the results obtained from the EPMC theory while the EPMC method's speed and computational facility are fully preserved. These fundamental advances in formulating a reliable all-temperature lattice dynamics contain an uncontrolled approximation--the low-coupling approximation (LCA)--routinely used in the implementation of the EPMC theory. We examine the validity of the LCA, by making calculations that do not use it, and find that the LCA is not a good approximation when used with EPMC. We show that the errors caused by making the LCA are properly compensated for only when it is used in conjunction with the IEP theory. We find that at the very lowest temperatures there are small anomalies in the thermal properties. While these are not significant for the heavier rare-gas solids, they probably will be important when IEP is applied to helium. Finally, we examine the validity of a variety of potentials to account for the experimental thermal and elastic properties of solid neon, using IEP theory. We show that so-called realistic potentials are currently not able to account for the experimental results, even when many-body effects are explicitly included. A simple -minded nearest neighbor Lennard-Jones model is shown to give the best account of the thermodynamics of solid neon.