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%TCIDATA{Created=Thu Sep 18 08:04:32 1997}
%TCIDATA{LastRevised=Thu Sep 14 13:33:17 2000}
%TCIDATA{Language=American English}

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\hfill {\Large Phys 861, Fall 2000}

\noindent {\huge Specific heat of electrons in metals}

\bigskip 

Use $c=T\partial s/\partial T$, where $s$ is the entropy per unit volume.
Start from 
\[
s=-k_{B}\int g({\cal E})\left( f\,\ln f+(1-f)\,\ln (1-f)\right) d{\cal E}
\]
where $g({\cal E})$ is the density of levels and $f$ is the Fermi function 
\[
f(x)=\frac{1}{e^{x}+1}\quad \quad \quad {\rm with\quad }x=\frac{{\cal E-\mu }%
}{k_{B}T}
\]
Note in passing that $s$ vanishes for $T\rightarrow 0$, as it should,
because $f$ is either $0$ or 1 in that limit. Then 
\[
\left( \frac{\partial s}{\partial T}\right) _{\mu }=-k_{B}\int g({\cal E}%
)\ln \left( \frac{f}{1-f}\right) \,\frac{\partial f}{\partial T}\,d{\cal E}
\]
Use $\dfrac{f}{1-f}=e^{-x}$ and $\dfrac{\partial f}{\partial T}=\dfrac{%
\partial f}{\partial x}\dfrac{\partial x}{\partial T}=-\dfrac{\partial f}{%
\partial x}\dfrac{x}{T}$ to obtain 
\begin{equation}
\left( \frac{\partial s}{\partial T}\right) _{\mu }=-k_{B}\int g({\cal E}%
)\,x^{2}\,\frac{\partial f}{\partial x}\,\frac{d{\cal E}}{T}  \label{exact}
\end{equation}
This formula is still exact. Now note that 
\[
-\frac{\partial f}{\partial x}=\frac{e^{-x}}{\left( e^{-x}+1\right) ^{2}}
\]
is strongly peaked at ${\cal E}=\mu $ when $k_{B}T\ll \mu $. Then, in this
limit,
\[
\frac{\partial s}{\partial T}=k_{B}^{2}\,g(\mu )\int_{-\infty }^{\infty }\,%
\frac{x^{2}e^{-x}\,dx}{\left( e^{-x}+1\right) ^{2}}\,dx
\]
The exact value of the integral is $\pi ^{2}/3$. At low $T$, one can replace 
$\,g(\mu )$ with $g({\cal E}_{F})$, where ${\cal E}_{F}$ is the value of $%
\mu $ at $T=0$. Then $c=\gamma T$ (at constant $\mu $) and also $s=\gamma T$%
, with 
\[
\gamma =\left( \frac{\pi ^{2}}{3}\right) k_{B}^{2}\,g({\cal E}_{F})
\]
For a gas of non-relativistic free electrons in 3d, substitute from AM's eq.
(2.65), 
\[
g({\cal E}_{F})=\frac{3}{2}\frac{n}{{\cal E}_{F}}
\]

To get higher order terms from Eq. (\ref{exact}), insert the Taylor series $g(%
{\cal E})=\sum_{p}g^{(p)}(\mu )({\cal E}-\mu )^{p}/p!=$ $\sum_{p}\left(
k_{B}T\right) ^{p}g^{(p)}(\mu )x^{p}/p!$ and look up in Arfken and Weber the
values of
\begin{equation}
\int_{-\infty }^{\infty }\,\frac{x^{2+p}e^{-x}\,dx}{\left( e^{-x}+1\right)
^{2}}
\end{equation}
for even $p$ (the odd values of $p$ give 0). 

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