1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Phonons and the Dulong-Petit Law

  1. Apr 11, 2012 #1

    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!
  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 :

    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]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook