MECHANICS AND ENERGETICS OF THE BAIN TRANSFORMATION

The general mechanics and energetics of the Bain transformation are pr esented. The Bain transformation takes a crystal from its b.c.c. confi guration into its f.c.c. structure, or vice versa, by means of homogen eous axial deformations. The crystal remains b.c.t. on the transformat ion path, and different types of Bain transformation may be distinguis hed by the response of the transverse lattice parameters to incrementa l changes in the longitudinal lattice parameter. A rational means of c omparing the various types is made possible by defining the longitudin al stretch as the independent variable or 'degree of transformation'. It is shown that, among possible Bain transformations, the one that oc curs under a uniaxial-loading environment has the lowest binding energ y at any given stage of transformation. In addition, the lowest possib le 'barrier energy' for any Bain transformation occurs when the crysta l passes through a special unstressed tetragonal state that resides at a local energy maximum on the uniaxial-loading Bain transformation pa th. A set of simple inequalities among the crystal's elastic moduli (a t any stage of transformation) is developed to determine whether or no t any incremental departure (including those that break tetragonal sym metry) from a Bain path can result in a lower binding energy at the sa me degree of transformation. In order to illustrate the general princi ples, pseudopotential model calculations are made for a sodium crystal undergoing Bain transformations on three distinct paths, namely uniax ial loading, constant volume and uniaxial deformation. The computation s include the energy and stress 'barriers' for the transformations, as well as the binding energy, atomic volume, longitudinal and transvers e stresses, and elastic moduli. The pseudopotential model and computat ional techniques are those of Rasky and Milstein. The computed elastic moduli are used to show that, if the sodium crystal is in a current e quilibrium state on the uniaxial-loading Bain transformation path, the n any nearby state that is reached by an 'arbitrary' incremental latti ce distortion, at the same degree of transformation, will have a highe r binding energy than that of the current state. There are, however, o ther uniaxial- or shear-loading transformation paths that are not of t etragonal 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, where in this configuration appears as a special state, differently oriented on the non-tetragonal paths. Finally, it is hypothesized that the min imum energy barrier for any homogeneous b.c.c. <-- --> fc.c. transform ation, on an equilibrium path between unstressed and elastically stabl e initial and final states, is that associated with the same unstresse d tetragonal configuration that occurs on the uniaxial-loading Bain tr ansformation path.