Towards fast converging lattice sums : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand - Burrows, Antony (Massey University, 2023)
In the field of solid state physics there are many open questions surrounding the best configuration of packing spheres to calculating binding energies to J/mol accuracy. Many of these problems have attracted attention from individuals in many faculties from mathematics, physics and chemistry over the course of the last four centuries. A significant amount of work has been done modernizing interaction potentials from the early twentieth century by the use of modern computers and quantum chemical software programs extending versions of the most common two-body potential. The historical survey of the methods leading up until the late nineteen eighties serves as the basis for where we step off for much of the analytic techniques for evaluating lattice sums and their use in answering these open questions. Investigations in to the stability of certain packing configurations compared to others in the solid state can be made with the use of fast techniques to evaluate the properties of such systems, many of which are developed here and used throughout the work in the various projects seen below. The aim of this work is to show that the evaluation of lattice constants and the formulae to calculate them can be given in a concise and efficient form with the use of mathematical and numerical methods. Analytical expressions can be found that are given in terms of real exponents and these expressions can be evaluated to arbitrary precision within a satisfactory amount of computer time. In contrast to the infinite structure that forms the lattice in the physical world, the techniques to calculate its sum have evolved from an infinite direct summation to methods that treat the sum associated with the quadratic form of the lattice re-expressing it as a sum of simple functions using number theoretic techniques and treating sums in terms of fast converging series or sums of hyperbolic functions. The results of this investigation are multiple new formulae for the cubic lattice systems, including expressions for the simple cubic lattice and famous Madelung constant in N--dimensions. A new expression was found for the hexagonal close packed structure that is computationally elegant and allowed the examination of the behaviour of the two-body Lennard--Jones potential in terms of the lattice parameters. A single parameter sum was found for the simple cubic system that was used to investigate the effect of pressure on body centered cubic system compared to the face centered cubic system.
Keywords: Lattice theory; Solid state physics; 490407 Mathematical logic, set theory, lattices and universal algebra