Group theory and physical properties of crystals

A new approach to the application of group theory in the study of the physical properties of crystals, which is more powerful than the direct method described in Section 15.2, has been developed by Nowick and is described fully in his book Crystal Properties via Group Theory (Nowick (1995)). A brief outline of Nowick s method will be given here. The equilibrium physical properties of crystals are described by constitutive relations which are Taylor expansions of some thermodynamic quantity Yt in terms of a set of thermodynamic variables Xj. Usually, only the first term is retained giving the linear relations 

Additional symmetries arise when the tensors Xj and/or Y, are symmetric, and from crystal symmetry in accordance with Neumann s principle, as seen in Section 15.2. These symmetries are properties of the tensor and the crystal point group, and, if different physical properties may be represented by the same kind of tensor, it will exhibit the same structure, irrespective of the actual physical property under consideration. 

In the linear-response region the fluxes Jt of matter and heat are related to the thermodynamic forces Fk by linear phenomenological relations 

The forces Fk involve gradients of intensive properties (temperature, electrochemical potential). The Ljk are called phenomenological coefficients and the fundamental theorem of the thermodynamics of irreversible processes, due originally to Onsager (1931a, b), is that when the fluxes and forces are chosen to satisfy the equation 

See the text for an explanation of the determination of the orientation of the polar and axial vectors in the doubly degenerate E representation. 

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