Next: 6. Higher Order Effects Up: An Introduction to Crystal Physics (Description Previous: 4. Neumann's Principle
One frequently finds data in the literature which give the value of some physical property in a given direction. In this chapter the concept of the magnitude of a physical property in a given direction, and also the relationship between this value and the respective tensor components will be illustrated on the examples of the direction dependence of the electrical conductivity and Young's modulus respectively.
The specific conductivity in the direction of the electric field is defined as the ratio of the component parallel with the electric field of the current density () and the magnitude of the electric field (E), i.e. . Let the components of the electric field be Ei = Eni, where ni denotes the i-th component of the unit vector () pointing into the direction of the electric field. The component of the j current density parallel with is in tensor notation
Equation (5.3) may be applied in two ways. One possibility is to calculate the tensor components from the measured conductivity values and the corresponding direction cosines. For this purpose one should measure the electrical conductivity in different directions, which are not connected by symmetry, as many times as the number of the independent components. Another possibility of applying eq. (5.3) is quite opposite to the first one. With the aid of the already known tensor components the conductivity value can be computed for any direction.
Equation (5.3) becomes considerably simplified for crystals of the tetragonal, trigonal and hexagonal systems which have only two independent tensor-components ( and )
As another example we will study the direction dependence of Young's modulus. To begin with it should be stated that Young's modulus in the pulling direction is defined as the ratio of the longitudinal stress () and the longitudinal strain (. If the axis of the coordinate system is placed in the direction of the unit vector Young's modulus in this direction will apparently be
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