Einstein assumed that all of the 3N normal modes of a crystal containing N atoms have the same frequency ω0. 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 cut-off frequency where the density of states is assumed to abruptly go to zero. The cut-off is chosen so that the total number of states is 3N. |
body centered cubic |
face centered cubic |
hexagonal close pack pdf Mathematica notebook |
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Eigenfunction solutions |
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Dispersion relation | The dispersion relation can be expressed as the following determinant |
The dispersion relation can be expressed as the following determinant
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Density of states D(k) | ||||||||||||||||||||
Density of states D(ω) | ||||||||||||||||||||
Energy spectra1 density |
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Internal energy |
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$\large u\approx \frac{3\pi^4}{5}nk_B\frac{T^4}{\Theta_D^3}\quad\left[\text{J/m}^3\right]$ |
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Helmholtz free energy |
$\large f\approx -\frac{\pi^4}{5}nk_B\frac{T^4}{\Theta_D^3}\quad\left[\text{J/m}^3\right]$ |
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Entropy |
$\large s\approx \frac{4\pi^4}{5}nk_B\left(\frac{T}{\Theta_D}\right)^3\quad\left[\text{J K}^{-1}\text{m}^{-3}\right]$ |
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Specific heat |
$\large c_v\approx \frac{12\pi^4}{5}nk_B\left(\frac{T}{\Theta_D}\right)^3\quad\left[\text{J K}^{-1}\text{m}^{-3}\right]$ |
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