A Sommerfeld expansion is an approximation method developed by
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
for a certain class of
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s which are common in
condensed matter
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
and
statistical physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations ...
. Physically, the integrals represent statistical averages using the
Fermi–Dirac distribution Fermi–Dirac may refer to:
* Fermi–Dirac statistics or Fermi–Dirac distribution
* Fermi–Dirac integral (disambiguation)
** Complete Fermi–Dirac integral
** Incomplete Fermi–Dirac integral
See also
* Fermi (disambiguation)
Enrico Fermi ...
.
When the
inverse temperature
In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:\beta = \frac (where is the temperature and is Boltzmann constant).J. Meixner (1975) "Coldness and Tempe ...
is a large quantity, the integral can be expanded 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
.
If the integral is from zero to infinity, then the integral in the first term of the expansion is from zero to
and the second term is unchanged.
Application to the free electron model
Integrals of this type appear frequently when calculating electronic properties, like the
heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
, in the
free electron model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quant ...
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
In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:\beta = \frac (where is the temperature and is Boltzmann constant).J. Meixner (1975) "Coldness and Tempe ...
and
as the
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
. Therefore, the Sommerfeld expansion is valid for large
(low
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
) systems.
Derivation to second order in temperature
We seek an expansion that is second order in temperature, i.e., to
, where
is the product of temperature and
Boltzmann's constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas 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
to be:
:
.
Hence,
:
Higher order terms and a generating function
We can obtain higher order terms in the Sommerfeld expansion by use of a
generating function for moments of the Fermi distribution. This is given by
:
Here
and Heaviside step function
subtracts the divergent zero-temperature contribution.
Expanding in powers of
gives, for example
:
:
:
:
:
:
A similar generating function for the odd moments of the Bose function is
Notes
References
*
*{{Cite book , last1 = Ashcroft , first1 = Neil W. , last2 = Mermin , first2 = N. David , authorlink2 = David Mermin , title = Solid State Physics , page
760, publisher = Thomson Learning , date = 1976 , isbn = 978-0-03-083993-1 , url-access = registration , url = https://archive.org/details/solidstatephysic00ashc/page/760
Equations of physics
Statistical mechanics
Particle statistics