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. Begin with a change variables to
:
![{\displaystyle I=\int _{-\infty }^{\infty }{\frac {H(\varepsilon )}{e^{\beta (\varepsilon -\mu )}+1))\,\mathrm {d} \varepsilon =\tau \int _{-\infty }^{\infty }{\frac {H(\mu +\tau x)}{e^{x}+1))\,\mathrm {d} x\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7fd72e0bf8e8e9c946d0859d51ad59e6bf5b8d)
Divide the range of integration,
, and rewrite
using the change of variables
:
![{\displaystyle I=\underbrace {\tau \int _{-\infty }^{0}{\frac {H(\mu +\tau x)}{e^{x}+1))\,\mathrm {d} x} _{I_{1))+\underbrace {\tau \int _{0}^{\infty }{\frac {H(\mu +\tau x)}{e^{x}+1))\,\mathrm {d} x} _{I_{2))\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/300eb0420614ce00ae78eb5800e31b989a8b5b41)
![{\displaystyle I_{1}=\tau \int _{-\infty }^{0}{\frac {H(\mu +\tau x)}{e^{x}+1))\,\mathrm {d} x=\tau \int _{0}^{\infty }{\frac {H(\mu -\tau x)}{e^{-x}+1))\,\mathrm {d} x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c446d6e21bb52ecb8dfb2e6e430fd9fa9231b22d)
Next, employ an algebraic 'trick' on the denominator of
,
![{\displaystyle {\frac {1}{e^{-x}+1))=1-{\frac {1}{e^{x}+1))\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f532aa437ec686b81dae0b0727c69bb24f05d489)
to obtain:
![{\displaystyle I_{1}=\tau \int _{0}^{\infty }H(\mu -\tau x)\,\mathrm {d} x-\tau \int _{0}^{\infty }{\frac {H(\mu -\tau x)}{e^{x}+1))\,\mathrm {d} x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb7e3223d027e0929c1de6dd095838638a42591)
Return to the original variables with
in the first term of
. Combine
to obtain:
![{\displaystyle I=\int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +\tau \int _{0}^{\infty }{\frac {H(\mu +\tau x)-H(\mu -\tau x)}{e^{x}+1))\,\mathrm {d} x\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8d10947fbca99e9c5f9c8ba6e12f1ef598d952)
The numerator in the second term can be expressed as an approximation to the first derivative, provided
is sufficiently small and
is sufficiently smooth:
![{\displaystyle \Delta H=H(\mu +\tau x)-H(\mu -\tau x)\approx 2\tau xH'(\mu )+\cdots \,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fae98dfb0cded51e475ff09ff57d4b9dafdd59)
to obtain,
![{\displaystyle I=\int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +2\tau ^{2}H'(\mu )\int _{0}^{\infty }{\frac {x\mathrm {d} x}{e^{x}+1))\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92607fc98eed201cf9ce556b45d6d91b8f29d908)
The definite integral is known[3] to be:
.
Hence,
![{\displaystyle I=\int _{-\infty }^{\infty }{\frac {H(\varepsilon )}{e^{\beta (\varepsilon -\mu )}+1))\,\mathrm {d} \varepsilon \approx \int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +{\frac {\pi ^{2)){6\beta ^{2))}H'(\mu )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8bb47dc5c8caa4b62a99e1cd2c3bb5610d8793b)
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
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi ))e^{\tau \epsilon /2\pi }\left\((\frac {1}{1+e^{\beta (\epsilon -\mu )))}-\theta (-\epsilon )\right\}={\frac {1}{\tau ))\left\((\frac {({\frac {\tau T}{2)))}{\sin({\frac {\tau T}{2)))))e^{\tau \mu /2\pi }-1\right\},\quad 0<\tau T/2\pi <1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5284e2e04749c0926b29c48be39ab41d589d8964)
Here
and Heaviside step function
subtracts the divergent zero-temperature contribution.
Expanding in powers of
gives, for example [4]
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi ))\left\((\frac {1}{1+e^{\beta (\epsilon -\mu )))}-\theta (-\epsilon )\right\}=\left({\frac {\mu }{2\pi ))\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f87d5cb601e027145fd5d96538a9f63885a2c7)
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi ))\left({\frac {\epsilon }{2\pi ))\right)\left\((\frac {1}{1+e^{\beta (\epsilon -\mu )))}-\theta (-\epsilon )\right\}={\frac {1}{2!))\left({\frac {\mu }{2\pi ))\right)^{2}+{\frac {T^{2)){4!)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63889301fa874652fb1e78c1bb3da85c5b63ed97)
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi )){\frac {1}{2!))\left({\frac {\epsilon }{2\pi ))\right)^{2}\left\((\frac {1}{1+e^{\beta (\epsilon -\mu )))}-\theta (-\epsilon )\right\}={\frac {1}{3!))\left({\frac {\mu }{2\pi ))\right)^{3}+\left({\frac {\mu }{2\pi ))\right){\frac {T^{2)){4!)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0c3b33b04517dff8a7f98705377e0508771a33)
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi )){\frac {1}{3!))\left({\frac {\epsilon }{2\pi ))\right)^{3}\left\((\frac {1}{1+e^{\beta (\epsilon -\mu )))}-\theta (-\epsilon )\right\}={\frac {1}{4!))\left({\frac {\mu }{2\pi ))\right)^{4}+{\frac {1}{2!))\left({\frac {\mu }{2\pi ))\right)^{2}{\frac {T^{2)){4!))+{\frac {7}{8)){\frac {T^{4)){6!)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ffd6b9d1a636fde5c99e13ccc7fa43f5e7495e9)
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi )){\frac {1}{4!))\left({\frac {\epsilon }{2\pi ))\right)^{4}\left\((\frac {1}{1+e^{\beta (\epsilon -\mu )))}-\theta (-\epsilon )\right\}={\frac {1}{5!))\left({\frac {\mu }{2\pi ))\right)^{5}+{\frac {1}{3!))\left({\frac {\mu }{2\pi ))\right)^{3}{\frac {T^{2)){4!))+\left({\frac {\mu }{2\pi ))\right){\frac {7}{8)){\frac {T^{4)){6!)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c526b7a329ee5fbe3e7fac2f4f588a303bccf5d8)
![{\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi )){\frac {1}{5!))\left({\frac {\epsilon }{2\pi ))\right)^{5}\left\((\frac {1}{1+e^{\beta (\epsilon -\mu )))}-\theta (-\epsilon )\right\}={\frac {1}{6!))\left({\frac {\mu }{2\pi ))\right)^{6}+{\frac {1}{4!))\left({\frac {\mu }{2\pi ))\right)^{4}{\frac {T^{2)){4!))+{\frac {1}{2!))\left({\frac {\mu }{2\pi ))\right)^{2}{\frac {7}{8)){\frac {T^{4)){6!))+{\frac {31}{24)){\frac {T^{6)){8!)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b3dd4499f1d5bbf459edfaa3821fa28d59e4629)
A similar generating function for the odd moments of the Bose function is