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Phonons and the Dulong-Petit Law

  1. Apr 11, 2012 #1
    Hey,

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

    Quantum.png

    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
     
  2. jcsd
  3. Apr 11, 2012 #2
    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.
     
  4. Apr 11, 2012 #3
    Thanks for the quick reply!

    So I differentiate he partition function with respects to beta twice, then integrate, impose small beta then taylor expand?
     
  5. Apr 11, 2012 #4
    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!
     
  6. Apr 13, 2012 #5
    Is the series expansion for the logarithm


    [tex]\LARGE \frac{1}{\beta E}[/tex]
     
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