Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

# Introduction

Some physical properties, such as the density or heat capacity of a material, have values independent of direction; they are scalar properties. However, in contrast, you will see that many properties vary with direction within a material. For example, thermal conductivity relates heat flow to temperature gradient, both of which need to be specified by direction as well as magnitude - they are vector quantities. Therefore thermal conductivity must be defined in relation to a direction in a crystal, and the magnitude of the thermal conductivity may be different in different directions.

A perfect crystal has long-range order in the arrangement of its atoms. A solid with no long-range order, such as a glass, is said to be amorphous. Macroscopically, every direction in an amorphous structure is equivalent to every other, due to the randomness of the long-range atomic arrangement. If a physical property relating two vectors were measured, it would not vary with orientation within the glass; i.e. an amorphous solid is isotropic. In contrast, crystalline materials are generally anisotropic, so the magnitude of many physical properties depends on direction in the crystal. For example, in an isotropic material, the heat flow will be in the same direction as the temperature gradient and the thermal conductivity is independent of direction. However, as will be demonstrated in this TLP, in an anisotropic material heat flow is no longer necessarily parallel to the temperature gradient, and as a result the thermal conductivity may be different in different directions.

The occurrence of anisotropy depends on the symmetry of the crystal structure. Cubic crystals are isotropic for many properties, including thermal and electrical conductivity, but crystals with lower symmetry (such as tetragonal or monoclinic) are anisotropic for those properties.

Many (but not all) physical properties can be described by mathematical quantities called tensors. A non-directional property, such as density or heat capacity, can be specified by a single number. This is a scalar, or zero rank tensor. Vector quantities, for which both magnitude and direction are required, such as temperature gradient, are first rank tensors. Properties relating two vectors, such as thermal conductivity, are second rank tensors. Third and higher rank tensor properties also exist, but will not be considered here, since the mathematical descriptions are more difficult.