\sect{Mean field approximation for ferroelectricity}

Consider electric dipoles ${\bf p}$ that are situated on sites of a simple cubic lattice, 
which point along the crystal axes ${\pm \langle 100\rangle}$. 
The interaction between dipoles is
%
\[
U = \frac{ {\bf p}_{1}\cdot{\bf p}_{2}-   3({\bf p}_{1}\cdot {\bf r})
({\bf p}_{2}\cdot {\bf r})/r^{2}} {4 \pi r^{3}}
\]
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where ${\bf r}$ is the distance between the dipoles, and $r=|{\bf r}|$.

\begin{itemize}

\item [(a)]
Assume nearest neighbour interactions and find the ground state
configuration. Consider either ferroelectric (parallel dipoles) or
anti-ferroelectric alignment (anti-parallel) between neighbours in
various directions.

\item [(b)]
Develop a mean field theory for the ordering in (a) for the
average polarization~$P$ at temperature $T$. 
Write the mean field equation for $P(T)$, 
and find the critical temperature~$T_c$.

\item[(c)] 
Within the mean filed approximation find the 
susceptibility $\chi =\left({\partial P}/{\partial E}\right)_{E=0}$  
for ${T>T_c}$ with respect to the electric field $E||\langle 100\rangle$. 

\end{itemize}



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