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.
is a large quantity, the integral can be expanded[1][2] in terms of
is used to denote the derivative of
ε = μ
notation refers to limiting behavior of order
If the integral is from zero to infinity, then the integral in the first term of the expansion is from zero to
Integrals of this type appear frequently when calculating electronic properties, like the heat capacity, in the free electron model of solids.
In these calculations the above integral expresses the expected value of the quantity
Therefore, the Sommerfeld expansion is valid for large
(low temperature) systems.
We seek an expansion that is second order in temperature, i.e., to
is the product of temperature and the Boltzmann constant.
Begin with a change variables to
τ x = ε − μ
: Divide the range of integration,
using the change of variables
: Next, employ an algebraic 'trick' on the denominator of
, to obtain: Return to the original variables with
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: Hence, We can obtain higher order terms in the Sommerfeld expansion by use of a generating function for moments of the Fermi distribution.
and Heaviside step function
subtracts the divergent zero-temperature contribution.
gives, for example [4] A similar generating function for the odd moments of the Bose function is