Linear Chain
Einstein Model
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 Model
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.
Linear chain 2 masses
simple cubic
body centered cubic
And similar expressions for the y and z directions
face centered cubic
And similar expressions for the y and z directions
hexagonal close pack pdf Mathematica notebook
And similar expressions for the y and z directions
And similar expressions for the y and z directions
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Nearest neighbors and next nearest neighbors
From Theoretische Festkörperphysik, Czycholl
The dispersion relation can be expressed as the following determinant
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Nearest neighbors and next nearest neighbors
From Theoretische Festkörperphysik, Czycholl
The internal energy increases like T² at low temperatures and is linear in T at high temperatures
