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Low Temperature Expansion of Chemical Potential

  1. Jun 16, 2008 #1
    I'm trying to derive a low temperature series expansion for the chemical potential of a weakly interacting Fermi gas. The starting point is, of course, the Fermi-Dirac distribution function (p is the particle momentum):

    f(p) = \frac{1}{e^{\beta(\epsilon(p) - \mu)}+1} ,

    where, in the Hartree-Fock approximation, we have

    \epsilon(p) = \frac{p^2}{2m} + n V(0) - \frac{1}{(2\pi \hbar)^3} \int d^3p' V(\textbf{p} - \textbf{p}' ) f(p').

    Here, [tex]m[/tex] is the effective mass, [tex]n[/tex] is the particle density, [tex]V(0)[/tex] is the interaction potential [tex]V(q)[/tex] at zero momentum transfer. The potential may be assumed to depend only on the momentum transfer [tex]V(\textbf{p} - \textbf{p}' ) = V(| \textbf{p} - \textbf{p}' | ) = V(q)[/tex]. The F-D distribution [tex]f(p')[/tex] in the exchange term may be approximated with the non-interacting one. The chemical potential is determined by the condition (spin-1/2):

    n = \frac{2}{(2\pi \hbar)^3} \int d^3p f(p) = \frac{1}{\pi^2 \hbar^3} \int_0^\infty p^2 f(p) dp

    Now, the right-hand side should somehow be expanded as a series in [tex]( k_B T/ \mu)^2[/tex], which can then be inverted to give [tex]\mu[/tex] as a function of [tex]T[/tex]. It seems that the Sommerfeld method used for a non-interacting system is not easy to use in this case. I know the result should be the following:

    \mu (T) = \mu_F (T) + n V(0) - \frac{1}{2} n \left[ F + G \frac{\pi^2}{12} \left( \frac{T}{T_F} \right)^2 \right] ,

    F = \frac{3}{2 k_F^3} \int_0^{2 k_F} k^2 \left( 1 - \frac{k}{2 k_F} \right) V(k) dk .
    G = 3 \left( V(2 k_F) - \frac{1}{4} \int_0^{2 k_F} \frac{k^3}{k_F^4} V(k) dk \right) .

    The potential is now written as a function of the Fermi wave vector ([tex]p = \hbar k[/tex]). [tex]\mu_F (T)[/tex] is the chemical potential of a non-interacting Fermi gas. The zero temperature limit, i.e. [tex]F[/tex], is rather simple to derive.

    Has anyone come across this problem or know any good references? I would really appreciate any assistance.
  2. jcsd
  3. Dec 29, 2008 #2
    I guess I could update this thread a little bit. I was able to derive the requested expansion somewhat after posting the above message. My approach was, however, slightly different. The result also had an extra term and reads:

    G = 3 \left( V(2 k_F) + \frac{1}{4} \int_0^{2 k_F} \left( \frac{k}{k_F^2} - \frac{3}{2} \frac{k^3}{k_F^4} \right) V(k) dk \right) .

    The reason for this small discrepancy is unclear. By comparing the two results to numerical calculations using the exact equations, I find that my [tex]G[/tex] is a better approximation.
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