Examples of Tensor Properties Using Matrix Fundamentals (A Physical Property)

Abstract

Properties of single crystals in particular often differ with different crystal directions as a consequence of constraints imposed by atomic packing and arrangements as well as the placement of substitutional impurities or other defects. For polycrystalline materials with more random orientations of crystal grains, these properties usually average out, and measurements represent the same values in any direction. Such multi-directional or multi-vector properties are intrinsically connected, and these connected properties are represented by tensors or arrays of vectors represented crystallographically (as crystal directions). These tensors or vector arrays have a rank characteristic of their complexity, which manifests itself in the scalar coefficients characteristic of a property. Such notations are conveniently represented by matrix algebra which in the case of tensor properties represents a powerful mathematical tool to study or predict properties of crystals, particularly crystal orientations.