\sect{Baruch's C04.}
Model of ferroelectricity: Consider electric dipoles ${\bf p}$
on sites of
a simple cubic lattice which point along one of the crystal axes,
${\pm \langle 100\rangle}$. The interaction between dipoles is
\[\ U =\frac{ {\bf p}_{1}\cdot{\bf p}_{2}- 3({\bf p}_{1}\cdot {\bf r})
({\bf p}_{2}\cdot {\bf r})/r^{2}} {4 \pi \epsilon_{ 0}r^{3}}\]
where ${\bf r}$ is the distance between the dipoles, $r=|{\bf r}|$
and $ \epsilon_{ 0}$ is the dielectric constant.
\begin{itemize}
\item [(a)]
Assume nearest neighbor interactions and find the ground state
configuration. Consider either ferroelectric (parallel dipoles) or
anti-ferroelectric alignment (anti-parallel) between neighbors in
various directions.
\item [(b)]
Develop a mean field theory for the ordering in (a) for the
average polarization $P$ at a given site at temperature T: Write a
mean field equation for $P(T)$ and find the critical temperature
$T_c$.
\item[(c)] Find the susceptibility
$\chi =\left(\frac{\partial P}{\partial E}\right)_{E=0}$ at
$T>T_c$ for an electric field $E||\langle 100\rangle$, using the
mean field
theory. \\
\end{itemize}
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