In the Einstein model, all of the modes have the same frequency ω0. The density of states is a delta function D(ω) = 3nδ(ω - ω0). Here n is the density of atoms.
The form below generates a table of where the first column is the angular frequency ω in rad/s and the second column is the density of states D(ω) in units of s/(rad m³). The delta function is approximated by a square pulse that is ω0/1000 wide.
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The internal energy density in the Einstein model is,
$$u(T) = \int\limits_{0}^{\infty}\frac{\hbar\omega D(\omega)}{\exp\left(\frac{\hbar\omega}{k_BT}\right)-1}d\omega=\frac{3n\hbar\omega_0 }{\exp\left(\frac{\hbar\omega_0}{k_BT}\right)-1}.$$The specific heat is the derivative of the internal energy,
$$c_v = \frac{du}{dT} = \frac{3nk_B\left(\frac{\hbar\omega_0}{k_BT}\right)^2\exp\left(\frac{\hbar\omega_0}{k_BT}\right)}{\left(\exp\left(\frac{\hbar\omega_0}{k_BT}\right)-1\right)^2}.$$This has the correct high temperature behavior, $c_v\rightarrow 3nk_b,$ and it goes to zero at $T=0$ as it should, but the temperature dependence near $T=0$ is not correct. Debye improved on Einstein's model and calculated the correct temperature dependence at low temperatures.