For bosons, the energy spectral density u(ω) is the energy of a boson $\hbar\omega$ times the density of states D(ω) times the probability that the state is occupied. This probability is given by the Bose-Einstein factor,
$\Large u(\omega) = \frac{\hbar\omega D(\omega)}{\exp\left(\frac{\hbar\omega}{k_BT}\right)-1}.$
The form below can be used to calculate the energy spectral density from the density of states. The density of states is input as two columns of text in the lower left textbox. 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(ω)' button is pressed, the density of states is plotted on the left and u(ω) is plotted on the right. The data for the u(ω) plot also appears as eleven columns of text in the lower right textbox. The first column is the angular frequency ω in rad/s. Columns 2-11 are the energy spectral density u(ω) for ten temperatures 0.1Tmax, 0.2Tmax...Tmax in units of J s rad-1 m-3.
| D(ω) [1015 s rad-1m-3] | u(ω) [μJ s rad-1 m-3] | |||
ω [1012 rad/s] | ω [1012 rad/s] | |||
Input: ω [rad/s] D(ω) [s rad-1m-3] | Output: ω [rad/s] u(ω) [J s rad-1 m-3] | |||
Load a phonon density of states:
Nearest-neighbor mass-spring models: