Analytical and Numerical (Monte Carlo) Studies of Point and Space Group Symmetry-Breaking in the Liquid–Solid Phase Transition

Abstract

We present a complete classification of symmetry-breaking terms in the one- and two-body densities for crystallographic space group P63/mmc according to irreducible representations (IR) of the extended Euclidean space group E(|R3) in three dimensions. We show that these IRs of E(|R3) are the order parameter symmetries of the liquid–solid phase transition from a spatially uniform liquid, with space group E(|R3), to a crystal of space group P63/mmc. The analytical group-theoretical methods derived in this paper furnish mathematically rigorous criteria at the level of the microscopic spatial structure of materials for identifying the occurrence of a spontaneous solidification of a liquid and the low-symmetry crystallographic subgroup of E(|R3) in which the liquid crystallizes. We prove in particular that all of the irreducible order parameter symmetries of the liquid–solid phase transition appear in the two-body density, whereas the vast majority of symmetry-breaking IRs of E(|R3) is absent from the one body density. The novel aspects arising from this finding for analyzing phase transitions are discussed. We present numerical results of an exact Fourier path integral Monte Carlo simulation of spontaneous crystallization of liquid natural argon at the experimental triple point temperature into a hexagonal close-packed crystal, with space group P63/mmc. The experimental relevance of this system in particular is addressed. We mention the relation of the theory presented here to experimental X-ray diffraction and neutron scattering analysis of the microstructure of materials. We outline the applicability of formal and/or computational techniques derived and employed in this research to (envisaged) Monte Carlo investigations of adsorbed solid and liquid films, phase transitions in liquid crystals, computation of the local electron number density for electronic energy band structure calculations, and variational shadow wave functions Monte Carlo studies of spontaneous crystallization of superfluid 4He.