Sommerfeld expansion

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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 the inverse temperature is a large quantity, the integral can be expanded[1][2] in terms of as

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[edit]

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.

Derivation to second order in temperature[edit]

We seek an expansion that is second order in temperature, i.e., to , where is the product of temperature and Boltzmann's constant. Begin with a change variables to :

Divide the range of integration, , and rewrite using the change of variables :

Next, employ an algebraic 'trick' on the denominator of ,

to obtain:

Return to the original variables with in the first term of . Combine to obtain:

The numerator in the second term can be expressed as an approximation to the first derivative, provided is sufficiently small and is sufficiently smooth:

to obtain,

The definite integral is known[3] to be:



Generating functions[edit]

A generating function for moments of the Fermi distribution function is

Here and Heaviside step function subtracts the divergent zero-temperature contribution. Expanding in powers of gives, for example [4]

A similar generating function for the odd moments of the Bose function is


  1. ^ Ashcroft & Mermin 1976, p. 760.
  2. ^ Fabian, J. "Sommerfeld's expansion" (PDF). Universitaet Regensburg. Retrieved 2016-02-08. 
  3. ^ "Definite integrals containing exponential functions". SOS Math. Retrieved 2016-02-08. 
  4. ^ R. Loganayagam, P. Surówka (2012). "Anomaly/Transport in an Ideal Weyl gas". JHEP. 04: 2012:97. arXiv:1201.2812free to read. Bibcode:2012JHEP...04..097L. doi:10.1007/JHEP04(2012)097.