How can crystal symmetry operations be used to reduce the number of unique properties of a solid?

Question

Can anyone please give an example or a reference which shows how crystal point groups and symmetry operations can be used to reduce the number of parameters describing the property of a crystal, leaving only the unique parameters?

Answer 1

I think a classical example is electrical conductivity and resistivity (see Wikipedia), or any physical quantity which is described in the anisotropic case by a tensor (see also elasticity tensor as suggested in the comments by Jon Custer). Consider the Ohm's law in the anisotropic case $$ J_{i}=\sigma_{ij} E_j, \qquad E_i=\rho_{ij} J_j $$ The conductivity $\sigma$ and resistivity $\rho$ are 3x3 matrix (tensors), for example $$ \sigma=\begin{bmatrix} \sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\\ \end{bmatrix}, $$ and where $\rho=\sigma^{-1}$ is the matrix inverse of the conductivity $\sigma$.

If the crystal lattice is cubic, directions $x$, $y$, and $z$ are equivalent, and therefore the physical properties are isotropic. In this case one has $\sigma_{xx}=\sigma_{yy}=\sigma_{zz}$, and $\sigma_{ij}=0$ for $i\neq j$, and therefore one has $$ \sigma=\sigma_0\mathbf{1},\qquad \rho=\rho_0\mathbf{1}, $$ where $\mathbf{1}$ is the identity matrix and $\rho_0=1/{\sigma_0}$. In this case the Ohm's law reduces to the standard vector form $$\mathbf{J}=\sigma_0 \mathbf{E},\qquad \mathbf{E}=\rho_0 \mathbf{J}$$ The physical description in the anisotropic case needs 3x3 parameters (the components of the conductivity tensor). The cubic symmetry reduces this parameters to just one, $\sigma_0$.

PS: I've found this book Symmetry, Group Theory, and the Physical Properties of Crystals by R C Powell.