On the Material Symmetry Group of Elastic Crystals and the Born Rule

The aim of this paper is to investigate the extent to which non-linear elasticity theory can be used for describing the behavior of crystalline solids. The results we obtain show that some strict and definite boundaries must be set to the possibility of actually doing so. Specifically, we address here a two-fold problem: On one hand, we pose the question of the validity of the so-called "Born Rule", a fundamental hypothesis due to CAUCHY and, in a weaker form, to BORN, by means of which continuum theories of crystal mechanics are formulated. On the other hand, we explore the possibility of developing an effective elastic model independently of the Rule, in those cases in which the Rule itself does not work. Our results are based on a close study of the implications of the phenomenon of mechanical twinning with regard to the symmetry properties of the energy function of an elastic crystal. These are summarized by the choice of a "material symmetry group" G: The main experimental features of twinning lead one to consider a class of "twinning subgroups" of G that are particular "reflection groups", in fact, particular linear representations, possibly unfaithful, of abstract "Coxeter groups". In the "generic" case, the properties of such groups prevent the Born Rule from holding. Only when some very special non-generic conditions are met by the twinning modes of a crystalline substance is the Rule valid; the development of an elastic model is then possible by following a well-known procedure. The analysis of relevant experimental data confirms that, while basically all crystals exhibit twins, most of them do exhibit generic twinning modes for which the hypothesis of BORN is violated. We also show that in such generic cases any tentative thermoelastic approach developed independently of the Rule does not give physically sound results and thus cannot be usefully adopted, because some quite undesirable conclusions regarding the symmetry of the energy can be drawn that definitely make elasticity inadequate for our purposes. Experimental data point out nonetheless two quite remarkable classes of "non-generic" materials for which the Born Rule is never violated, and to which an elastic model safely applies.