Behnke–Stein theorem

A Sommerfeld expansion is an approximation method developed by Arnold Sommerfeld for a certain class of integrals which are common in condensed matter and statistical physics. Physically, the integrals represent statistical averages using the Fermi–Dirac distribution.

When $\beta$ is a large quantity, it can be shown that

$\int _{-\infty }^{\infty }{\frac {H(\varepsilon )}{e^{\beta (\varepsilon -\mu )}+1}}\,\mathrm {d} \varepsilon =\int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +{\frac {\pi ^{2}}{6}}\left({\frac {1}{\beta }}\right)^{2}H^{\prime }(\mu )+\mathrm {O} \left({\frac {1}{\beta \mu }}\right)^{4}$

where $H^{\prime }(\mu )$ is used to denote the derivative of $H(\varepsilon )$ evaluated at $\varepsilon =\mu$ and where the $O(x^{n})$ notation refers to limiting behavior of order $x^{n}$. The expansion is only valid if $H(\varepsilon )$ vanishes as $\varepsilon \rightarrow -\infty$ and goes no faster than polynomially in $\varepsilon$ as $\varepsilon \rightarrow \infty$.

Application to the free electron model

Integrals of this type appear frequently when calculating electronic properties in the free electron model of solids. In these calculations the above integral expresses the expected value of the quantity $H(\varepsilon )$. For these integrals we can then identify $\beta$ as the inverse temperature and $\mu$ as the chemical potential. Therefore, the Sommerfeld expansion is valid for large $\beta$ (low temperature) systems.

References

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