How Can Phonons, Einstein & Debye Models Explain the Dulong-Petit Law?

[Sekonda]

Hey,

My question is on Phonons, the Einstein & Debye models and the Dulong Petit Law. The question is displayed below:

I am told how to get to the heat capacity by using the logarithm of the partition function 'Z', and so I set about differentiating the logarithm of Z with respects to Beta twice.

However I'm unsure if I can manipulate the logarithm present in the integral (the one with the exponential functions with exponents BetaxE) to take on some other form which would allow the integration & differentiation to be simpler.

I'm not sure how I impose the normalizing integral condition and not sure how to use the normalizing integral with the integral displayed to the left of it. Also when to impose the 'large T' - I presume after differentiation and integrating?

Cheers guys!
S

 

[M Quack]

The high temperature limit would be beta very small compared to the energy scale of the problem, i.e. beta Lambda << 1, so that beta E << 1 for the entire integral. Then try a series expansion on the log.

 

[Sekonda]

Thanks for the quick reply!

So I differentiate he partition function with respects to beta twice, then integrate, impose small beta then taylor expand?

 

[Sekonda]

I've managed to attain that the term inside the logarithm can be approximated by 1/(betaE),
however substituting this into the lnZ equation gives me one term I want and another term like :
∫dEg(e)lnE

does this vanish? Or is my approximation wrong of 1/(betaE)?

Thanks again!

 

[Sekonda]

Is the series expansion for the logarithm $$\LARGE \frac{1}{\beta E}$$