- #1

Othin

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## Homework Statement

We are asked to derive the expression for the internal energy of an ideal Fermi degenerate gas using Sommerfeld expansions, writing out terms up to the fourth order in ##(\frac{T}{T_F} )## , that is, we must determine ## \alpha ## in the following expression: $$ U= \frac{3}{5}N\varepsilon_F \left\{ 1 + \frac{5\pi ^2}{12} {\left( \frac{T}{T_F}\right)}^2 + \alpha {\left( \frac{T}{T_F}\right)}^4 \right\}$$

The method is basically the one used in the text of Kubo on Statistical Mechanics.

## Homework Equations

We're treating the 3-D "particle in a box" model with periodic boundary conditions, so that the energy spectrum is given by: $$\varepsilon_k = \left( \frac{\hbar^2 k^2}{2m} \right) , k=\left( 2n\pi /L \right)$$

We are also supposing that the number of particles is sufficiently large for we to approximate the summations characteristic of a discrete spectrum by integrals. We then have:

$$

N= \gamma V \int_0^\infty D(\varepsilon)f(\varepsilon) d\varepsilon, \\

U= \gamma V \int_0^\infty \varepsilon D(\varepsilon)f(\varepsilon) d\varepsilon \\

$$

where ## \gamma ## takes the spin degeneracy into account and:

$$

D( \varepsilon ) = \frac{1}{4 \pi^2} {\left( \frac{2m}{\hbar ^2} \right)}^{3/2} \varepsilon ^{1/2} \equiv C \varepsilon ^{1/2} ,\ \ f(\varepsilon) = \frac{1}{exp[\frac{\varepsilon - \mu}{k_bT}] + 1}.

$$

The Sommerfeld expansion is also given, whereby we can solve integrals of the form ## I = \int_0^\infty f(\varepsilon) \psi(\varepsilon) d{\varepsilon}## :

$$

I = \int_0^{\mu}\psi(\varepsilon) d\varepsilon + \frac{1}{2!}I_2 {\left( \frac{d \psi}{ d \varepsilon} \right)}_{\varepsilon= \mu} + \frac{1}{4!}I_4 {\left( \frac{d^3 \psi}{ d\varepsilon^3} \right)}_{\varepsilon= \mu} + ...

$$

Within approximations, the integral ##I_2## and ##I_4## are taken to equal, respectively, ## \frac{\pi^2 (k_bT)^2}{3} ## and ## \frac{7 \pi^4}{15}(k_bT)^4##

## The Attempt at a Solution

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Since U must be given in powers of ##T/T_F ##, we must first use the equation for N in order to write #/mu# in terms of T and #T_F#. Therefore, for N:

$$

\psi(\varepsilon) = \varepsilon^{1/2} \implies N = \gamma VC \left \{ \frac{2}{3}\mu^{3/2} + \frac{\pi ^2}{12} (k_b T)^2 \mu^{1/2} - \frac{3}{8} \frac{1}{4!} \mu^{\frac{-5}{2}}I_4 + ... \right \} \\

$$

$$

\frac{N}{\gamma VC}= \frac{2}{3} \mu^{3/2} \left[ 1 + \frac{\pi^2}{8}\left (\frac{k_b T}{\mu} \right)^2 + \left( \frac{7 \pi^4}{640} \right) \left (\frac{k_bT}{\mu} \right)^4 \right]

$$

Now, after this expression Kubo (and other books I've checked) make some kind of expansion around ## \mu_o ## and arrive at the following expression:

$$

\mu = \mu_o \left[1- \frac{\pi^2}{12} \left( \frac{k_b T}{\mu_o}\right)^2 - \frac{\pi^4}{80} \left( \frac{k_b T}{\mu_o}\right)^4 \right]

$$

and , no matter how I try, I can't get anything like that last expression (I managed to get the first term by truncating the series and using a binomial expansion to the first order, but when I include the next power I can't make it work)