Crystal physics

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.

Outline
• Stress and strain
• Einstein notation for tensors     W
• Review of statistical physics
• Intrinsic symmetries
• Maxwell relations     W
• Thermodynamic properties
• Pyroelectricity     W Pyroelektrizität.pdf
• Ferroelectricity     W Ferroelektrizität.pdf
• Pyromagnetism
• Piezoelectricty     W
• Piezomagnetism     W
• Electrocaloric effect     W
• Electrostriction     W
• Magnetostriction     W
• Thermal expansion     W
• Groups and symmetry
• Examples of how symmetries affect the properties of solids
• Piezoelectricity     W
• Nonlinear optics     W
• Birefringence
• Table of crystal classes and their associated point groups

Reading
Kittel chapter 3: Elastic constants and Elastic waves
An Introduction to Crystal Physics Ervin Hartmann

For the exam you should
• be familiar with the Einstein notaion for tensors.
• 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
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.