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Temperature vs phonon

  1. Dec 14, 2008 #1

    I was looking at the high- and low-temperature limits of the specific heat in the quantum theory of cristals (Ashcrof&Mermin, Chap. 23).

    To get the behavior under these limits, one consider first the case where T is large compared with all the phonon frequencies and second, when T is low compared to these frequencies.

    But, the temperature shouldnt be (in some way) proportionnal to the phonon frequency? If this was right, then the low limit [tex]\omega\gg T[/tex] would be a non-sense.

    So I realize that I dont really understand the relation between temperature and phonons. Sure, I know that the number of phonon of each type will come to play, but I cant make a whole picture of all that in my head.

    Can someone try to explain, or give some refs where this is clearly explained?

    Thanks a lot,

    Last edited: Dec 14, 2008
  2. jcsd
  3. Dec 16, 2008 #2
    The phonon dispersion relation (the ω-k relationship) is determined only by the lattice properties of the solid, and is not a strong function of temperature. Each point on the ω-k diagram corresponds to some vibrational mode of the system, and since phonons are bosons, the probability that a phonon exists in any given mode is given by Bose-Einstein statistics. Put another way, at every frequency/phonon energy, you have some density-of-states determined by the dispersion relation. But only a fraction of those states are filled, and Bose-Einstein statistics tell you how many are filled at a certain temperature.
  4. Dec 17, 2008 #3
    Hi Manchot,
    Thanks for your answer!

    To make my question more precise, thit is the answer that satisfied my curiosity :

    There is two quantities that link phonon and temperature : the frequency of the phonon and its probability in the overall distribution. If you isolate T in the distribution, you get :

    where [tex]\omega[/tex] is the frequency and [tex]n[/tex] its associated probability. So for a given probability, the temperature is proportional to the frequency but for a given frequency the more the probability is small, lower is the temperature.

    My question was something like : How can you obtain small temperature from hign phonon frequencies. The answer is simply that these frequencies must have low probability.

    Your comments are welcome,

  5. Dec 17, 2008 #4
    ^ Yes, that is correct. At phonon energies considerably higher than the temperature, the occupation fraction is small and so there simply aren't many phonons present.
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