Much of the energy stored in a crystal is in the form of the vibrations of the atoms. For crystals which are not electrically conducting, the phonons are the dominant contribution to the energy. Phonons also play a dominant role in the thermal conductivity of electrical insulators. For electrical conductors, the conduction electrons also contribute to the energy stored in the crystal and to the thermal conductivity.

The total internal energy density stored in the phonons is the energy of a phonon mode $\hbar\omega$ times the denisty of states $D(\omega)$ times the Bose-Einstein factor integrated over all frequencies.

$$ u =\int \limits_{0}^{\omega_{\text{max}}} \frac{\hbar\omega D(\omega)}{\exp\left( \frac{\hbar\omega}{k_BT} \right)-1}d\omega. $$

The density of states is the number of phonon modes per unit volume per unit frequency. $D(\omega)d\omega$ is the number of phonon modes in the interval bewteen $\omega$ and $\omega + d\omega$ per cubic meter. The maximum frequency in the density of states is called $\omega_{\text{max}}$. The Bose-Einstein factor $1/(\exp (\hbar\omega/(k_BT)-1))$ is the mean number of phonons in a mode at temperature $T$.

The specific heat at constant volume is the derivative of the internal energy density with respect to temperature.

$$ c_v = \int\limits_{0}^{\omega_{\text{max}}}\left(\frac{\hbar\omega}{T}\right)^2 \frac{D(\omega)e^{\frac{\hbar\omega}{k_BT}}}{k_B\left( e^{\frac{\hbar\omega}{k_BT}}-1\right)^2}d\omega.$$

The specific heat tells us how much energy is needed to raise the temperature of a cubic meter of the crystal by 1 Kelvin. This is a relatively simple experiment that can be used to check if the calculation of the phonon modes was done correctly. The specific heat goes to zero as $T\rightarrow 0$ and rises like $T^3$ at low temperatures. At high temperatures the specific heat is constant and obeys the Dulong-Petit law $c_v\approx 3nk_B$ where $n$ is the atomic density.

The entropy density is,

$$ s = \int\frac{C_v}{T}dT=k_B\int \limits_{0}^{\omega_{\text{max}}}D(\omega)\left[ -\ln \left( 1-\exp\left( \frac{-\hbar\omega}{k_BT} \right) \right)+\frac{\hbar\omega}{k_BT\left(\exp\left( \frac{\hbar\omega}{k_BT} \right)-1\right)} \right]d\omega$$

The entropy is proportional to the logarithm of the number of ways that the phonon states can be occupied for a given temperature. At low temperatures, $s\rightarrow 0$. This is known as the third law of thermodynamics. At high temperatures the entropy increases like $\ln(T)$.

By combining the internal energy density and the entropy we have an expression for the Helmholtz free energy density,

$$ f=u -Ts = k_BT\int \limits_{0}^{\omega_{\text{max}}} D(\omega) \ln\left[ 1-\exp\left( \frac{-\hbar\omega}{k_BT} \right)\right] d\omega$$

A system held at constant temperature will go to a minimum of the Helmholtz free energy. Experiments are usually performed at constant temperture so this is the relevant energy to consider.

The phonon density $n_{\text{ph}}$ is the density of states times the Bose-Einstein factor integrated over all frequencies,

$$n_{\text{ph}} = \int \limits_{0}^{\omega_{\text{max}}}D(\omega) \frac{1}{\exp \left( \frac{\hbar\omega}{k_BT} \right)-1}d\omega$$

The buttons below lead to programs that will numerically calculate the phonon contribution to a thermodynamic property from the phonon density of states.