Relationship Between Physical Properties and Crystallographic Symmetry

RELATIONSHIP BETWEEN PHYSICAL PROPERTIES AND CRYSTALLOGRAPHIC SYMMETRY 

Second-rank tensors such as transport properties relate two first-rank tensors, or vectors. Thus, a second-rank tensor representing a physical property has nine components (32), usually written in 3 x 3 matrixlike notation. Each component is associated with two axes one from the set of some reference frame and one from the material frame. Three equations, each containing three terms on the right-hand side, are needed to describe a second-rank tensor exactly. For a general 

The tensor, with components Xy, is written in matrixlike notation as 

Note that each component of p is related to all three components of q. Thus, each component of the tensor is associated with a pair of axes. For example, X32 gives the component of p parallel to aa when q is parallel to aa. In general, the number of indices assigned to a tensor component is equal to the rank of the tensor. Tensors of all ranks, like vectors, are defined by their transformation laws. For our purposes, we need not consider these. 

Fortunately, several simplifications can be made (Nye, 1957). Transport phenomena, for example, are processes whereby systems transition from a state of nonequilibrium to a state of equilibrium. Thus, they fall within the realm of irreversible or nonequilibrium thermodynamics. Onsager s theorem, which is central to nonequilibrium thermodynamics, dictates that as a consequence of time-reversible symmetry, the off-diagonal elements of a transport property tensor are symmetrical (i.e., xy = X/,-). This is known as a reciprocal relation. The Norwegian physical chemist Lars Onsager (1903-1976) was awarded the 1968 Nobel Prize in Chemistry for reciprocal relations. Thus, the tensor above can be rewritten as 

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