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\sect{Defects in Lattice}

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 ${w}$. Assume that
${T<<w}$, and show that the leading order estimate for the typical
number of defects is ${M=NN' exp\left(\frac{-w}{2T}\right)}$.
Evaluate also the contribution of the defects to the entropy, and to
the specific heat in the same level of approximation.




