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## Main Question or Discussion Point

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

$$\langle E \rangle = 3\sum_{\vec{k}} \hbar \omega (\vec{k}) \ [ n_B (\beta \hbar \omega (\vec{k})) + \frac{1}{2}]$$

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.

$$\langle E \rangle = \sum_k \hbar \omega \ [ n_B (\beta \hbar \omega ) + \frac{1}{2}]$$

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!

In my text (Oxford Solid State Basics by Steven Simon, page 11), it was stated that Debye wrote the following expression

$$\langle E \rangle = 3\sum_{\vec{k}} \hbar \omega (\vec{k}) \ [ n_B (\beta \hbar \omega (\vec{k})) + \frac{1}{2}]$$

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.

$$\langle E \rangle = \sum_k \hbar \omega \ [ n_B (\beta \hbar \omega ) + \frac{1}{2}]$$

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!