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Many of the physical properties of solids are described by tensors. For instance, when force is applied to a piece of rubber in the *x* direction, compresses in the *x* direction but expands in the *y* and *z* directions. Crystal physics is the study of the tensor properties of crystals and how these properties are related to the symmetries of the crystals.

**Reading**

Kittel chapter 3: Elastic constants and Elastic waves

An Introduction to Crystal Physics Ervin Hartmann

*Thermodynamische Eigenschaften von Festkörpern:* Appendix F in Festkörperphysik, R. Gross und A. Marx (Available to TU Graz students through the TU Graz library).

- Be familiar with Einstein notation for tensors. In this notation, you sum over repeated indices. A scalar product $\vec{P}\cdot\vec{E}$ would be written $P_iE_i$ and the matrix equation, $\vec{P}=\chi\vec{E}$ would be written as $P_i=\chi_{ij}E_j$.
- 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. Some examples are:

Rotation about the $x$-axis by

Rotation about the $y$-axis by

Rotation about the $z$-axis by

Rotoinversion about the $x$-axis by an angle

Rotoinversion about the $y$-axis by an angle

Rotoinversion about the $z$-axis by an angle

$$\text{Identity }E= \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$ - Be able to generate all the elements of a point group from the generating matrices.
- Know how to calculate the internal energy, specific heat, Helmholtz free energy, entropy, electric susceptibility, magnetic susceptibilty, piezoelectric coefficients, pyroelectric coefficients, and the stiffness tensor from the microscopic electron and phonon states. Solutions of the Schrödinger equation → density of states → free energy → thermodynamic properties.
- Know how symmetries can restrict the values that the coefficients of physical properties can have. For instance, a crystal with inversion symmetery cannot exhibit pyroelectricity, pyromagnetism, piezoelectricity or and piezomagnetism. For cubic crystals, all properties that are described by matrices have the form of a constant times the identity matrix.

**Resources**

Table of crystal classes in 3-D

Kristallsymmetrie und Punktgruppen - Erklärung - Wir versuchen in diesem Video die grundlegenden Elemente der Kristallsymmetrie, deren Nomenklatur und die Mathematik dahinter zu erklären.

Table of crystal classes in 2-D (pdf)

Crystal physics, Johann Potoschnig 2011

Matlab code: Input elements of generating matrices, Generates group elements from generating matrices, Transform rank 2 tensors (matrices), Transform rank 3 tensors, View saved matrices, Download matlab files (zip).

SGTE data for pure elements - The Gibbs energy as a function of temperature for many elements.