Debye model questions

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Hi all, I have trouble understanding some ideas relating to the Debye model.

In my text (Oxford Solid State Basics by Steven Simon, page 11), it was stated that Debye wrote the following expression
⟨E⟩=3∑→kℏω(→k) [nB(βℏω(→k))+12]
What was not stated was the meaning of this expression. The only mention was that it was completely analogous to Einstein's expression for the averaged energy of a quantum harmonic oscillator in 1D.
⟨E⟩=∑kℏω [nB(βℏω)+12]
However, I can't seem to draw the link between the 2 expressions. Could someone explain to me
1) the interpretation of Debye's expression
2) how Debye's expression arises from a partition function (and how the partition function comes about),
3) and also the link between the 2 equations?

Many thanks in advance!
 

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dRic2
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The derivation of both model is nicely carried out in Zimann's book "principles of theory of solid" (I think in chapter 2). Although I don't really recognize the formulae you wrote...

This should answer question 1 and 2. About your question 3, if I remember correctly Einstein model was derived assuming all ions vibrate at the same frequency. You can think of the Debye model as a "correction" of the Einstein model which introduces a "wight" for different frequencies of oscillations (phonons).
 

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