
\sect{Ferromagnetism for cubic crystal}

%Baruch's  C06

A cubic crystal which exhibits ferromagnetism at low temperature,
can be described near the critical temperature ${T_{c}}$ by an
expansion of a Gibbs free energy
\[\  G({\bf H},T)=G_{0} +\half  r{\bf M}^{2} + u{\bf M}^{4} +
v\sum_{i=1}^{3}M_{i}^{4} -   {\bf H}\cdot {\bf M}\]
where
${{\bf H}=\left(H_{1},H_{2},H_{3}\right)}$ is the external field
and ${{\bf M}=\left(M_{1},M_{2},M_{3}\right)}$ is the total
magnetization; ${r=a\left(T -Tc\right)}$ and ${G_{0}, a, u}$ and
${v}$ are independent of ${\bf H}$ and ${T, a>0, u>0}$. The
constant ${v}$ is called the cubic anisotropy and can be either
positive or negative.

\begin{itemize}
\item [(a)]
At ${{\bf H}=0}$, find the possible solutions of ${\bf M}$ which
minimize ${G}$ and the corresponding values of
${G\left(0,T\right)}$ (these solutions are characterized by the
magnitude and direction of ${\bf M}$. Show that the region of
stability of ${G}$ is ${u+v>0}$ and determine the stable
equilibrium phases when ${T<T_{c}}$ for the cases (i) ${v>0}$,
(ii) ${-u<v<0}$.
\item [(b)]
Show that there is a second order phase transition at ${T=T_{c}}$,
and determine the critical indices $\alpha,\,\beta$ and $\gamma$
for this transition, i.e. ${C_{V,H=0} \sim |T- T_{c}|^{-\alpha}}$
for both ${T>T_{c}}$ and ${T<T_{c}}$, ${|{\bf M}|_{H=0} \sim (Tc-
T)^{\beta}}$ for ${ T<T_{c}}$ and ${\chi _{ij} =
\partial M_{i}/\partial H_{j}\sim ~\delta_{
ij} |T-T_{c}|^{-\gamma}}$ for ${T>T_{c}}$.\\
\end{itemize}

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