Density of states → Internal energy density u(T)

The internal energy density u(T) is the integral of the energy spectral density over all frequencies,

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

Here $\hbar\omega$ is the energy of phonons with frequency ω, D(ω) is the density of states, and $1/(\exp(\hbar\omega/k_BT)-1))$ is the Bose-Einstein factor.

The form below uses this formula to calculate the temperature dependence of the internal energy density 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 → u(T)' button is pressed, the density of states is plotted on the left and u(T) is plotted from temperature Tmin to temperature Tmax on the right. The data for the u(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 internal energy density in units of J m-3 .

[1015 s rad-1m-3]
[108 J m-3]

ω [1012 rad/s]

T [K]

Tmin:  K  Tmax:  K

Input: ω [rad/s] D(ω) [s rad-1m-3]

Output: T [K] u(T) [J m-3]

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