Low Temperature Expansion of Chemical Potential

[QuasiParticle]

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):

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

where, in the Hartree-Fock approximation, we have

<br /> \epsilon(p) = \frac{p^2}{2m} + n V(0) - \frac{1}{(2\pi \hbar)^3} \int d^3p&#039; V(\textbf{p} - \textbf{p}&#039; ) f(p&#039;).<br />

Here, m is the effective mass, n is the particle density, V(0) is the interaction potential V(q) at zero momentum transfer. The potential may be assumed to depend only on the momentum transfer V(\textbf{p} - \textbf{p}&#039; ) = V(| \textbf{p} - \textbf{p}&#039; | ) = V(q). The F-D distribution f(p&#039;) in the exchange term may be approximated with the non-interacting one. The chemical potential is determined by the condition (spin-1/2):

<br /> 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<br />

Now, the right-hand side should somehow be expanded as a series in ( k_B T/ \mu)^2, which can then be inverted to give \mu as a function of T. 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:

<br /> \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] ,<br />

where
<br /> 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 .<br />
and
<br /> 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) .<br />

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

Has anyone come across this problem or know any good references? I would really appreciate any assistance.

 

[QuasiParticle]

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:

<br /> 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) .<br />

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