Behnke–Stein theorem: Difference between revisions

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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 statistics|Fermi–Dirac distribution]]. 


When <math>\beta</math> is a large quantity, it can be shown<ref>{{Harvnb|Ashcroft & Mermin|1976|p=760}}.</ref>  that


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:<math>\int_{-\infty}^\infty \frac{H(\varepsilon)}{e^{\beta(\varepsilon - \mu)} + 1}\,\mathrm{d}\varepsilon = \int_{-\infty}^\mu H(\varepsilon)\,\mathrm{d}\varepsilon + \frac{\pi^2}{6}\left(\frac{1}{\beta}\right)^2H^\prime(\mu) + \mathrm{O} \left(\frac{1}{\beta\mu}\right)^4</math>
 
where <math>H^\prime(\mu)</math> is used to denote the derivative of <math>H(\varepsilon)</math> evaluated at <math>\varepsilon = \mu</math> and where the <math>O(x^n)</math> notation refers to limiting behavior of order <math>x^n</math>. The expansion is only valid if <math>H(\varepsilon)</math> vanishes as <math>\varepsilon \rightarrow  -\infty</math> and  goes no faster than polynomially in <math>\varepsilon</math> as <math>\varepsilon \rightarrow \infty</math>.
 
== 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 <math>H(\varepsilon)</math>.   For these integrals we can then identify <math>\beta</math> as the [[thermodynamic beta|inverse temperature]] and <math>\mu</math> as the [[chemical potential]]. Therefore, the Sommerfeld expansion is valid for large <math>\beta</math> (low [[temperature]]) systems.
 
==Notes==
<references/>
 
==References==
*{{cite doi|10.1007/BF01391052}}
*{{Cite document| 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| ref = harv| postscript = <!--None-->}}
 
[[Category:Concepts in physics]]
[[Category:Statistical mechanics]]
[[Category:Quantum field theory]]
[[Category:Particle statistics]]
 
{{condensedmatter-stub}}

Latest revision as of 08:29, 20 August 2013

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 β is a large quantity, it can be shown[1] that

H(ε)eβ(εμ)+1dε=μH(ε)dε+π26(1β)2H(μ)+O(1βμ)4

where H(μ) is used to denote the derivative of H(ε) evaluated at ε=μ and where the O(xn) notation refers to limiting behavior of order xn. The expansion is only valid if H(ε) vanishes as ε and goes no faster than polynomially in ε as ε.

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 H(ε). For these integrals we can then identify β as the inverse temperature and μ as the chemical potential. Therefore, the Sommerfeld expansion is valid for large β (low temperature) systems.

Notes

References

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