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 is a large quantity, it can be shown that
where is used to denote the derivative of evaluated at and where the notation refers to limiting behavior of order . The expansion is only valid if vanishes as and goes no faster than polynomially in as .
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 . For these integrals we can then identify as the inverse temperature and as the chemical potential. Therefore, the Sommerfeld expansion is valid for large (low temperature) systems.
- Ashcroft & Mermin 1976, p. 760.
- Sommerfeld, A. (1928). "Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik". Zeitschrift für Physik 47: 1–3. Bibcode:1928ZPhy...47....1S. doi:10.1007/BF01391052.
- Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics. Thomson Learning. p. 760. ISBN 978-0-03-083993-1
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