Mechanics and energetics of the Bain transformation

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Abstract
The general mechanics and energetics of the Bain transformation are presented. The Bain transformation takes a crystal from its b.c.c. configuration into its f.c.c. structure, or vice versa, by means of homogeneous axial deformations. The crystal remains b.c.t. on the transformation path, and different types of Bain transformation may be distinguished by the response of the transverse lattice parameters to incremental changes in the longitudinal lattice parameter. A rational means of comparing the various types is made possible by defining the longitudinal stretch as the independent variable or degree of transformation. It is shown that, among possible Bain transformations, the one that occurs under a uniaxial-loading environment has the lowest binding energy at any given stage of transformation. In addition, the lowest possible barrier energy for any Bain transformation occurs when the crystal passes through a special unstresssd tetragonal state that resides at a local energy maximum on the uniaxial-loading Bain transformation path. A set of simple inequalities among the crystal's elastic moduli (at any stage of transformation) is developed to determine whether or not any incremental departure (including those that break tetragonal symmetry) from a Bain path can result in a lower binding energy at the same degree of transformation. In order to illustrate the general principles, pseudopotential model calculations are made for a sodium crystal undergoing Bain transformations on three distinct paths, namely uniaxial loading, constant volume and uniaxial deformation. The computations include the energy and stress ‘barriers’ for the transformations, as well as the binding energy, atomic volume, longitudinal and transverse stresses, and elastic moduli. The pseudopotential model and computational techniques are those of Rasky and Milstein. The computed elastic moduli are used to show that, if the sodium crystal is in a current equilibrium state on the uniaxial-loading Bain transformation path, then any nearby state that is reached by an ‘arbitrary’ incremental lattice distortion, at the same degree of transformation, will have a higher binding energy than that of the current state. There are, however, other uniaxial- or shear-loading transformation paths that are not of tetragonal crystal symmetry, in general, and not ‘in the neighbourhood’ of the uniaxial-loading Bain path, which have the same minimum barrier energy at the same unstressed tetragonal crystal configuration, wherein this configuration appears as a special state, differently oriented on the non-tetragonal paths. Finally, it is hypothesized that the minimum energy barrier for any homogeneous b.c.c. ↔f.c.c. transformation, on an equilibrium path between unstressed and elastically stable initial and final states, is that associated with the same unstressed tetragonal configuration that occurs on the uniaxial-loading Bain transformation path.