The quantum theory of elastic constants

Abstract

A recent rigorous and rotationally invariant treatment of the theory of a homogeneously deformed material is discussed. It is shown that it leads to different strain perturbation terms in the Hamiltonian from those arrived at in the standard treatment of Born and Huang. These differences are discussed. It is shown also that the Hamiltonian of the deformed system may be directly expressed in terms of the Lagrangian finite strain tensor, with the result, for example, that the correct symmetries for the thermodynamic quantities are apparent at all times. The advantages and simplicity of the finite strain treatment are discussed. It is also shown that the Born-Huang treatment fails a simple test of rotational invariance and so cannot be correct. This accounts for the difference in the two treatments. As an example, it is shown how the soft mode of beta quartz, following the method of Axe and Shirane (1970), contributes a logarithmic divergence to the elastic constants in addition to the (T-Tc)-1 divergence discussed by these authors. It is weaker than this latter divergence, but it may be of interest further away from the transition.