Crystals are classified by their symmetries. Every crystal can be associated with one of the 32 point groups. A point group is the collection of symmetries (rotations, reflections, inversion) that a crystal has where at least one point in the crystal does not move during the transformation. Point group symmetries can be represented by $3\times 3$ matrices. Rotational symmetries are described by the following matrices where the main symmetery axis has been taken to be the $z$-axis.

\begin{equation} C_1=\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]\hspace{1cm} C_2=\left[\begin{matrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]\hspace{1cm} C_3=\left[\begin{matrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 1 \end{matrix}\right]\hspace{1cm} C_4=\left[\begin{matrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}\right]\hspace{1cm} C_6=\left[\begin{matrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{matrix}\right]. \end{equation}

All of the symmetries of a point group can be generated from at most three generating matrices. Multiplying the generating matrices with themselves and with each other results in other symmetries of that point group. The resulting matrices are multiplied with themselves and with the generating matrices until no new symmetries appear. For instance, $C_4$ is the generating matrix of the point group 4. All of the elements of point group 4 are given below.

\begin{equation} C_4=\left[\begin{matrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}\right]\hspace{1cm} C_4^2=C_2=\left[\begin{matrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]\hspace{1cm} C_4^3=C_4^{-1}=\left[\begin{matrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}\right]\hspace{1cm} C_4^4=C_1=E=\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]. \end{equation}

Multiplying any of these matrices with any other, results in one of these four matrices. These are the four elements of point group 4.

Matrix multiplication $A=$ -1 0 0 0 -1 0 0 0 1   $B=$ -1 0 0 0 -1 0 0 0 1   $C=$

Every equilibrium property of a crystal can be expressed as a derivative of an energy. A system where all of the extrinsic variables are held fixed goes to the minimum of the internal energy. Extrinsic variables are those that scale with the size of the system like the number $N$, the volume $V$, the entropy $S$, the magnetization $\vec{M}$, and the electric polarization $\vec{P}$. For a solid, the internal energy is a sum of the electron contribution and a phonon contribution.

\[ \begin{equation} u=\int\limits_{-\infty}^{\infty}ED_e(E)f_{FD}(E)dE+\int\limits_{-\infty}^{\infty}ED_{ph}(E)f_{BE}(E)dE \end{equation} \]

Here $E$ is integrated over all energies, $D_e$ is the electron density of states, $D_{ph}$ is the phonon density of states, $f_{FD}$ is the Fermi-Dirac function and $f_{BE}$ is the Bose-Einstein function. Often in an experiment it is easier to hold the temperature constant than the entropy. If the temperature is held constant then the system will go to a minimum of the Helmholtz free energy $F$ which is constructed from the internal energy $U$ by a Legendre transformation,

\[ \begin{equation} F=U-TS. \end{equation} \]

Further Legendre transformations are typically made to construct a Gibbs free energy $G$ where the independent variables are temperature $T$, electric field $\vec{E}$, magnetic field $\vec{H}$, and the stress tensor $\sigma$. The conjugate variables to these independent variables are,

\[ \begin{equation} S = -\frac{\partial G}{\partial T}\qquad P_i = -\frac{\partial G}{\partial E_i}\qquad M_i = -\frac{\partial G}{\partial H_i}\qquad\epsilon_{ij} = -\frac{\partial G}{\partial \sigma_{ij}} \end{equation} \]

Here $\epsilon$ is the strain tensor and Einstein notation is used for the tensors. Taking further derivatives of these quantities results in many measureable quantities such as:

Stiffness tensor	$s_{ijkl} = -\frac{\partial\epsilon_{ij}}{\partial\sigma_{kl}}=-\frac{\partial^2 G}{\partial \sigma_{ij}\partial\sigma_{kl}}$
Thermal expansion	$\alpha_{ij} = \frac{\partial \epsilon_{ij}}{\partial T}= -\frac{\partial^2 G}{\partial \sigma_{ij}\partial T}$
Piezoelectric effect	$d_{ijk} = \frac{\partial P_i}{\partial \sigma_{jk}}=-\frac{\partial^2 G}{\partial E_i \partial \sigma_{jk}}$
Electric susceptibility	$\chi^E_{ij} = \frac{\partial P_{i}}{\partial E_j} =-\frac{\partial^2 G}{\partial E_{i}\partial E_j}$
Magnetic susceptibility	$\chi^M_{ij} = \frac{\partial M_{i}}{\partial H_j}=-\frac{\partial^2 G}{\partial H_{i}\partial H_j}$
Pyroelectric effect	$\pi_{i} =\frac{\partial P_{i}}{\partial T}= -\frac{\partial^2 G}{\partial E_{i}\partial T}$

It is also possible to further Taylor expand the physical quantities to include nonlinear effects such as,

\[ \begin{equation} P_i=\chi^{(1)}_{ij}E_j+\chi^{(2)}_{ijk}E_jE_k+\cdots \end{equation} \]

Here $\chi^{(2)}_{ijk}$ is the nonlinear electric susceptibility that is used in nonlinear optics. The strain can also be expanded to higher order in electric field,

\[ \begin{equation} \epsilon_{ij}=d_{ijk}E_k+Q_{ijkl}E_kE_l+\cdots. \end{equation} \]

Here $d_{ijk}$ are the piezoelectric coefficients and $Q_{ijkl}$ are the electrostriction coefficients.

Tensor quantities like the electric susceptibility must be unchanged by the application of a symmetry element of the point group. This restricts the form that tensors can have for every crystal class. Some of these restrictions are listed in the table of crystal classes. All odd-rank tensors are zero if inversion is an element of the point group. Thus, a crystal does not exhibit pyroelectricity, piezoelectricity, piezomagnetism, or a nonlinear optical response if there is inversion in the point group. Symmetry also requires all second rank tensors for cubic crystals to be proportional to the identity matrix.

The form of tensors that describe nonequilibrium properties such as the electrical conductivity or the thermal conductivity are also restricted by the symmetry of the crystal.

Electrical conductivity	$\sigma_{ij} = \frac{\partial j^e_{i}}{\partial E_{j}}$		$\vec{j}$ is the current density and $\vec{E}$ is the electric field.
Thermal conductivity	$\kappa_{ij} = -\frac{\partial j^{th}_{i}}{\partial (\nabla T)_j}$		$\vec{j^{th}}$ is the heat current and $T$ is the temperature.