Crystals and Melts

Crystalline phases consist of regular (periodic) space-filling assemblages of atoms, ions, or molecules that, more or less, compactly fill 3-dimensional space at an average mass density, ρ s . Crystalline phases are subject to the rigorous constraint imposed by long-range order (LRO) and satisfy the spatial invariance determined by the lattice’s point-group symmetry. That is, LRO assures the presence of special symmetries and provides long-range correlations of the particle positions, even over macroscopic distances. Thermodynamic equilibrium selects the particular crystalline phase that will minimize a system’s free energy at some fixed pressure for all temperatures below the phase diagram’s co-existence curve with the melt phase. Crystalline phases, as defined here, also broadly constitute many of the common ‘engineering materials’, including metals and alloys, crystalline ceramics, semiconductors, and many important polymeric systems.