Know that the motion of atoms in a crystal can be described in terms of the normal modes. There are 3 times as many normal modes as there are atoms in a crystal.
In a normal mode, all of the atoms in a crystal oscillate with the same frequency. Each of the normal modes has a wave-like solution that can be associated with one of the $\vec{k}$ vectors in the first Brillouin zone.
There are as many $\vec{k}$ vectors in the first Brillouin zone as there are primitive unit cells in the crystal.
Be able to describe how the phonon dispersion relation $\omega\text{ vs. }k$ is calculated: (1) Write the $3p$ differential equations for the motion of the atomic masses connected by linear springs, where p is the number of atoms in the primitive unit cell. (2) Insert normal mode solution into the equations, which result in $3p$ coupled algebraic equations. (3) Solve the algebraic equations for the normal mode frequencies.
Know that for a 3-D crystal, there are always three acoustic branches. and 3p - 3 optical branches. One third of the branches are longitudinal modes and two thirds are transverse modes.
The acoustic branches are linear near $\vec{k}=0$ while the optical branches have zero slope at $\vec{k}=0$. The phonon dispersion relation has zero slope at the Brillouin zone boundaries.
Know how to calculate the density states from the dispersion relation.
The integral of the density of states over all frequencies is 3 times the atomic density.
Know how to find a phonon dispersion relation or density of states for some material.
Be able to describe how the phonon dispersion relation can be measured with inelastic phonon scattering.
The total internal energy density stored in the phonons is the energy of a phonon mode $\hbar\omega$ times the density of states $D(\omega)$ times the Bose-Einstein factor integrated over all frequencies.
$$u =\int \limits_{0}^{\omega_{\text{max}}} \frac{\hbar\omega D(\omega)}{\exp\left( \frac{\hbar\omega}{k_BT} \right)-1}d\omega\quad[\text{J/m}^3].$$
From the internal energy density know how to calculate tthe specific heat $c_v=\frac{du}{dT}$, the entropy $s = \int\frac{C_v}{T}dT$, and the Helmholtz free energy $f = u - Ts$.
Be able to describe how to measure the specific heat and how the Helmholtz free energy determines the temperature of phase transitions.
