Symmetry Constraints on Properties

Abstract

This chapter introduces the distinction between isotropic and anisotropic properties as well as the application of coordinate transformations to solve problems involving anisotropic properties. Neumann’s Principle and the Curie Principle are also discussed.

The beginning of wisdom is the definition of terms.

—Socrates

Review Questions

• 1. Briefly explain the Curie Principle.

• 2. An object with point symmetry $m\overline{3}m$ (O_h) distorts under uniaxial stress along [001]. What is the point group of the distorted object? What is it when the stress is along [111]? If an applied electric field removes the centre of symmetry in each case, to which point groups would the distorted objects now belong?

• 3. Carbon atoms in graphite are bonded within (001) via strong triangular (120°) sp² hybrid σ bonding. The remaining unbonded 2p electrons form delocalized π bonds, and the adjacent sheets are held together by weak Van der Waals forces. For this reason, both the electrical and thermal conductivities of graphite are higher in the (001) than normal to it.

  The thermal conductivity k_ij is a tensor that relates the heat flux h to the temperature gradient dT/dx according to:

$$ {h}_i={k}_{ij}\frac{dT}{dx_j} $$

The thermal conductivity tensor of graphite is:

$$ {k}_{ij}=\left|\begin{array}{ccc}355& 0& 0\\ {}0& 355& 0\\ {}0& 0& 89\end{array}\right|\kern0.5em {\mathrm{Wm}}^{-1}\;{\mathrm{K}}^{-1} $$

• (a) If a gradient of 100 K/m is applied along the y direction, what is the resulting heat flow (in kW/m²)?

• (b) If a gradient of 100 K/m is applied along the z direction, what is the resulting heat flow (in kW/m²)?

• 4. Explain how Fourier’s Law, Ohm’s Law, and Fick’s Laws of Diffusion are related.