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Homework Help: Ripplons quantum stat mech problem

  1. Aug 29, 2012 #1
    1. The problem statement, all variables and given/known data


    2. Relevant equations

    3. The attempt at a solution

    I have a vague idea of how to do this problem, but I'm not sure. Here's the plan I have for solving it:

    1. Find the dispersion relationship [itex]\epsilon(k)[/itex]
    2. Find the 2D density of states for wave vectors k: g(k)dk
    3. Using the dispersion relation, find the 2D density of states [itex]g(\epsilon)d\epsilon[/itex]
    4. Integrate this and the Bose Einstein occupation number and the energy to find the average energy...?
    5. Use this to find the heat capacity

    The dispersion relationship should be:

    [tex]\epsilon = \hbar \omega = \hbar(\alpha_0 k^3/\rho)^{1/2} \rightarrow k = (\rho/\alpha_0 \hbar^2)^{1/3}\epsilon^{2/3} \rightarrow dk = (2/3)(\rho/\alpha_0 \hbar^2)^{1/3} \epsilon^{-1/3} d\epsilon[/tex]

    Because the number of states is a circle in 2D, the number of states between k and k + dk should be:

    [tex]g(k) dk = (A/(2\pi)^2)2\pi k dk[/tex]

    Plugging the above in:

    [tex]g(\epsilon) d\epsilon = (A/2\pi)(2/3)(\rho/\alpha_0 \hbar^2)^{2/3} \epsilon^{1/3} d\epsilon[/tex]

    Now, we plug this into the integral with the B.E. occupation number:

    [tex]E = \int_0 ^\infty \epsilon n(\epsilon) g(\epsilon) d\epsilon = (A/2\pi)(2/3)(\rho/\alpha_0 \hbar^2)^{2/3} \int_0 ^\infty \frac{\epsilon^{4/3} d\epsilon}{e^{\beta(\epsilon - \mu)} - 1}[/tex]

    But here's where I'm stuck. First of all, I don't know how to analytically do this integral. Second, we were never given the chemical potential...am I supposed to figure it out from the first line of the problem, using α? I don't see how, though...

    Can anyone help me out? Thanks!
    Last edited: Aug 29, 2012
  2. jcsd
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