Ordering in Close-Packed Structures

Abstract

Landau theory predicts significant differences between the ordering transitions in hexago­ nal close packed (hcp) and face centered cubic (fcc), based on symmetry arguments alone, even though it is possible with pairwise first and second neighbor interaction energies to assign identical energies to corresponding tetrahedral (T) and octahedral (0) clusters. High precision cluster variation method (CVM) calculations on the fcc structure with first and second neighbor interactions in the T-0 approximation are compared with corresponding calculations on the hcp structure with identical interaction energies. Using a high tempera­ ture CVM expansion we prove that CVM is able to calculated the small, but significant, differences that were missed in previous CVM and Monte Carlo (MC) studies. The results are also compared with results from exact ground state calculations and low temperature expansions. Initially we judged these differences to be too small to convert second order transitions in one structure into first order transitions in a corresponding structure when required by Landau theory, but later calculations*) showed that the tiny differences are indeed sufficient to give agreement with Lifshitz's predictions from Landau theory by giving weak first order transitions in these cases. However, among degenerate structures within fcc and hcp CVM does on occasion predict an incorrect hierarchy of stability.