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,

Since only the Bose-Einstein factor depends on temperature, the differentiation can be performed analytically and the expression for the specific heat is,

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^{-1}m^{-3}.

After the 'DoS → cv(T)' button is pressed, the density of states is plotted on the left and c_{v}(T) is plotted from temperature T_{min} to temperature T_{max} on the right. The data for the c_{v}(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) [10^{15} s rad^{-1}m^{-3}]

c_{v}(T) [10^{6} J K^{-1} m^{-3}]

ω [10^{12} rad/s]

T [K]

Input: ω [rad/s] D(ω) [s rad^{-1}m^{-3}]

Output: T [K] c_{v}(T) [J K^{-1} m^{-3}]

Load a phonon density of states:
Nearest-neighbor mass-spring models: