Sommerfeld expansion
Encyclopedia
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 we have used Big-O notation to describe the approximation's limiting behaviour. The expansion is only valid if 
vanishes as 
and goes no faster than polynomially in 
as 
.
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.
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...
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 thatwhere
is used to denote the derivative of evaluated at and we have used Big-O notation to describe the approximation's limiting behaviour. 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 modelFree electron model
In solid-state physics, the free electron model is a simple model for the behaviour of valence electrons in a crystal structure of a metallic solid. It was developed principally by Arnold Sommerfeld who combined the classical Drude model with quantum mechanical Fermi-Dirac statistics and hence it...
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 potentialChemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
. Therefore, the Sommerfeld expansion is valid for large
(low temperatureTemperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
) systems.