\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 % $U = \frac{ {\bf p}_{1}\cdot{\bf p}_{2}- 3({\bf p}_{1}\cdot {\bf r}) ({\bf p}_{2}\cdot {\bf r})/r^{2}} {4 \pi r^{3}}$ % 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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%