\sect{Baruch's A13.}
A perfect lattice is composed of ${N}$ atoms on ${N}$ sites.
If ${M}$ of these atoms are shifted to interstitial sites (i.e.
between regular positions) we have an imperfect lattice with ${M}$
defects. The number of available interstitial sites is ${N'}$ and
is of order ${N}$. The energy needed to create a defect is
${\omega}$ .
\begin{itemize}
\item [(a)]
Evaluate the number of defects ${M}$ at a temperature ${T}$ (you
may assume that there is a dominant term in the partition sum).
Show that to first order in $e^{-\omega /2T}$ (i.e. ${\omega
>>T)}$
\begin{equation}
M=\sqrt{NN'} exp \left(-\omega /2T \right). \nonumber
\end{equation}
\item [(b)]
Evaluate the contribution of defects to the entropy and to the
specific heat to first order in exp ${\left(- \omega/2T\right)}$.
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\end{itemize}
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