\sect{Baruch's  C07.}

Consider a ferromagnet with magnetic moments $m({\bf r})$ on a
simple cubic lattice interacting with their nearest neighbors.
[The symmetry is an Ising type, i.e. $m({\bf r})$ is the moment's
amplitude in a preferred direction]. The ferromagnetic coupling is
J and the lattice constant is a. Extend the mean field theory to
the situation that the magnetization is not uniform but is slowly
varying:
\begin{itemize}
\item [(a)] Find the mean field equation in terms of $m({\bf r})$, its
gradients (to lowest order) and an external magnetic $H({\bf r})$,
which in general can be a function of ${\bf r}$.

\item [(b)] Consider $T>T_c$ where $T_c$ is the critical temperature so that
only lowest order in $m({\bf r})$ is needed. For a small $H({\bf
r})$ find the response  $m({\bf r})$ and evaluate it explicitly in
two limits: (i) uniform $H$, i.e. find the susceptibility, and
(ii) $H({\bf r})\sim \delta^3({\bf r})$. Explain why in case (ii)
the response is the correlation function and identify the
correlation length.\\
\end{itemize}

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