| ||||

| ||||

At finite temperatures, the atoms in a crystal vibrate. In the simulation below, the atoms move randomly around their equilibrium positions.

If the atoms only make small displacements from their equilibrium positions, the minima of the bond potentials can be approximated as parabolas and the forces between the atoms can be described by linear springs. Any possible motion of a linear mass-spring system can be described in terms of its normal modes. In a normal mode, all of the atoms move with the same frequency. There are as many normal modes as there are degrees of freedom in the problem. In the simulation above, there are 64 atoms each of which can move in 2 dimensions. This system has $2\times 64=128$ normal modes. The simulation below can display all of the normal modes of this system.

$k_x:\,$

$k_y:\,$

A three-dimensional crystal containing $N$ atoms has $3N$ normal modes.

The atoms in a normal mode oscillate sinusoidally with an angular frequency $\omega$ like a harmonic oscillator. When the normal modes are quantized, the allowed energies have the same form as for a harmonic oscillator, $E=\hbar\omega \left( n+ \frac{1}{2}\right)$, where $n$ is an integer. Every energy quanta $\hbar\omega$ in a normal mode is called a **phonon**. The integer $n$ is the number of phonons in that normal mode. The phonon has the same $\omega$ and $\vec{k}$ as the normal mode it belongs to.

In thermal equilibrium, the average number of phonons in a normal mode is given by the Bose-Einstein factor,

\[ \begin{equation} f_{BE}=\frac{1}{\exp\left(\frac{\hbar\omega}{k_BT}\right)-1}. \end{equation} \]Phonons are bosons so there can be an arbitrary number in each normal mode.