The specific heat is the derivative of the internal energy with respect to the temperature.
$$c_v = \left(\frac{\partial u}{\partial T}\right)_{N,V}.$$This can be expressed in terms of an integral over the frequency ω.
$$c_v = \frac{\partial }{\partial T}\int\hbar\omega D(\omega)\frac{1}{e^{\frac{\hbar\omega}{k_BT}}-1}d\omega.$$The Leibniz integral rule can be used to bring the differentiation inside the integral. If the phonon density of states D(ω) is temperature independent, the result is,
$$c_v = \int\hbar\omega D(\omega)\frac{\partial }{\partial T}\left(\frac{1}{e^{\frac{\hbar\omega}{k_BT}}-1}\right)d\omega.$$Since only the Bose-Einstein factor depends on temperature, the differentiation can be performed analytically and the expression for the specific heat is,
$$c_v = \int\left(\frac{\hbar\omega}{T}\right)^2 \frac{D(\omega)e^{\frac{\hbar\omega}{k_BT}}}{k_B\left( e^{\frac{\hbar\omega}{k_BT}}-1\right)^2}d\omega.$$The form below uses this formuula to calculate the temperature dependence of the specific heat from tabulated data for the density of states. The density of states data is input as two columns in the textbox at the lower left. The first column is the angular-frequency ω in rad/s. The second column is the density of states in units of s rad-1m-3.
After the 'DoS → cv(T)' button is pressed, the density of states is plotted on the left and cv(T) is plotted from temperature Tmin to temperature Tmax on the right. The data for the cv(T) plot also appears in tabular form in the lower right textbox. The first column is the temperature in Kelvin and the second column is the specific heat in units of J K-1 m-3.
| D(ω) [1015 s rad-1m-3] | cv(T) [106 J K-1 m-3] | |||
ω [1012 rad/s] | T [K] | |||
Input: ω [rad/s] D(ω) [s rad-1m-3] | Output: T [K] cv(T) [J K-1 m-3] | |||
Load a phonon density of states:
Nearest-neighbor mass-spring models: