Einstein assumed that all of the 3N normal modes of a crystal containing N atoms have the same frequency ω_{0}. This is not a good model for the dispersion relation but it does a reasonable job in describing the specific heat. 
Debye used the long wavelength limit where the density of states increases like ω² up to a cutoff frequency where the density of states is assumed to abruptly go to zero. The cutoff is chosen so that the total number of states is 3N. 
body centered cubic 
face centered cubic 
hexagonal close pack pdf Mathematica notebook 

Eigenfunction solutions 



Dispersion relation  The dispersion relation can be expressed as the following determinant 
The dispersion relation can be expressed as the following determinant


Density of states D(k)  
Density of states D(ω)  
Energy spectra1 density 

Internal energy 

$\large u\approx \frac{3\pi^4}{5}nk_B\frac{T^4}{\Theta_D^3}\quad\left[\text{J/m}^3\right]$ 

Helmholtz free energy 
$\large f\approx \frac{\pi^4}{5}nk_B\frac{T^4}{\Theta_D^3}\quad\left[\text{J/m}^3\right]$ 

Entropy 
$\large s\approx \frac{4\pi^4}{5}nk_B\left(\frac{T}{\Theta_D}\right)^3\quad\left[\text{J K}^{1}\text{m}^{3}\right]$ 

Specific heat 
$\large c_v\approx \frac{12\pi^4}{5}nk_B\left(\frac{T}{\Theta_D}\right)^3\quad\left[\text{J K}^{1}\text{m}^{3}\right]$ 
