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\Large
{Problem set for chapter 4, Due Wed.\ Feb.\ 1}
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\noindent Useful references for this problem set include:
\begin{itemize}
\item Callaway, chapter 1.
\item Ashcroft \& Mermin, chapter 22.
\end{itemize}

\noindent{\bf{1.}} Argon is a Van der Waals FCC crystal, for which the
pair potential is of the Lennard-Jones type:
\[
V(r)=4\epsilon\left[
\left(\frac{\sigma}{r}\right)^{12}
-
\left(\frac{\sigma}{r}\right)^{6}\,,
\right]
\]
where $\epsilon=1.05\times10^{-2}\;\;{\rm{eV}}$ and $\sigma=3.4\AA$.
Neglecting all be nearest-neighbor forces, calculate the dispersion
$\omega({\bf{k}})$ for the longitudinal phonons in the (1,0,0) direction.
\medskip

\noindent{\bf{2.}} Read about the theory of elasticity in Ashcroft and 
Mermin.  Then, consider a crystal of Ag, and look up its atomic mass 
and elastic constants.
\begin{itemize}
\item Obtain an expression, in terms of the lattice frequency
spectrum, for the mean-squared excursion $\left|{\bf{s}}(0)\right|^2$
of a crystal atom about its equilibrium position.
\item Look up the atomic mass, elastic constants, and melting
temperature $T_M$ of Ag, and estimate $\left|{\bf{s}}(0)\right|^2$
at $T=0$ and $T=T_M$.
\end{itemize}
\medskip

\noindent{\bf{3.}} Calculate the dispersion of a simple linear chain of
atoms each of mass $M$, bound together with springs of spring constant
$f$.  Now, reconsider the example of a diatomic linear chain
discussed in Ibach and L\"uth Sec.~4.3.  Show that when $M=m$ for the
latter example, the two results are consistent ( hint remember
that when $M=m$ that the principle translation distance halves).
\medskip


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