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: Debye model and dispersion relation

  1. Mar 29, 2013 #1
    1. The problem statement, all variables and given/known data

    I have seen case studies of the 3D Debye model where the vibrational modes of a solid is taken to be harmonic with dispersion relation [itex]\omega = c_sk[/itex]. It is said that for temperatures much less than the Debye temperature, the heat capacity at constant volume [tex]C_V\sim T^3[/tex].

    Now I want to show that for bosons with dispersion relation [itex]\omega\sim A\sqrt k[/itex] has heat capacity [tex]C_V\sim T^4[/tex] for [itex]T\ll T_{Debye}[/itex].

    In the case studies I have read, I can't find where the dispersion relation comes into play. I have no idea how to see this. Please help!

    2. Relevant equations

    Debye Temperature is given by [tex]T_{Debye}k_B=\hbar \omega_{max}[/tex]

    3. The attempt at a solution

    Generally, I know I need to get the density of modes -- I have a suspicion that here is where the dispersion relation kicks in, but I don't know how.

    After that, I should find the ultraviolet cutoff frequency [itex]\omega_{max}[/itex].

    Then I should find the energy $$E=\int_0^\omega{max}d\omega {E(\omega)g(\omega)\over \exp(\beta(E-\mu))-1}$$ But what form does $E$ in the integrand take? I know that for photons with dispersion relation [tex]\omega = c_s k[/tex] we have [tex]E=\hbar \omega[/tex].

    After that, it's just a matter of taking limits and [tex]C_V=\left({\partial E\over \partial T}\right)_V[/tex] (should) give the required result...
  2. jcsd
  3. Mar 29, 2013 #2
    Please, somebody?

    OK, so I know that the density of state depends on the dispersion relation. What are the general definitions of [tex]E, p[/tex] in terms of [tex]\omega, k[/tex]? E.g. for the first case [tex]E=\hbar \omega[/tex] and [tex]p=\hbar k[/tex]. So the question is: what are the respective values for [tex]E,p[/tex] in general?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted