\sect{Baruch's C20.}
Stoner ferromagnetism:
The conduction electrons in a metal can be treated as a gas of
fermions of spin $\half$ (with up/down degeneracy), and density $n
= N/V$ . The Coulomb repulsion favors wave functions which are
antisymmetric in position coordinates, thus keeping the electrons
apart. Because of the full (position and spin) antisymmetry of
fermionic wave functions, this interaction may be approximated by
an effective spin-spin coupling which favors states with parallel
spins. In this simple approximation, the net effect is described
by an interaction energy \[U = \alpha \frac{N_+N_-}{V}\] where
$N_+$ and $N_- = N-N_+$ are the numbers of electrons with up and
down spins, and V is the volume.
\begin{itemize}
\item [(a)] Define $n_{\pm}=N_{\pm}/V=n/2\pm \delta$ and assume
$\delta\ll n$. Expand the total energy at temperature $T=0$ (kinetic and interaction)
to 4th order in $\delta$ and find the critical value $\alpha_c$
such that for $\alpha>\alpha_c$ the electron gas can lower its
total energy by spontaneously developing a magnetization.
(This is known as the Stoner
instability.) \item[(b)] Explain the instability qualitatively,
and sketch the behavior of the spontaneous magnetization as a
function of $\alpha$.
\item [(c)] Reconsider (a) at finite but low temperatures $T$,
and find $\alpha_c(T)$ (consider the effect to the lowest nonzero
order of $T$).
\end{itemize}
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