\sect{Stoner ferromagnetism}

Consider Fermi gas of $N$ spin $1/2$ electrons, at temperature $T=0$.
Define $N_{+}$ and $N_{-}$ as the number of "up" and "down" electrons respectively, 
such that $N=N_{+} + N_{-}$. Due to the antisymmetry of the total wave function 
the energy of the system is ${U = \alpha N_{+}N_{-}/\mathsf{V}}$, where $\mathsf{V}$ is the volume. 
Note that this interaction favors parallel spin states. 
Define the magnetization as ${M=(N_{+} - N_{-})/\mathsf{V}}$.

\begin{itemize}

\item [(a)] Write the total energy $E(M)$, 
including both the kinetic energy and the interaction,
and expand up to 4th order in~$M$.

\item[(b)] Find the critical value $\alpha_c$, 
such that for ${\alpha>\alpha_c}$ the electron gas can lower its
total energy by spontaneously developing magnetization. 
This is known as the Stoner instability. 

\item[(c)] Explain the instability qualitatively,
and sketch the behavior of the spontaneous magnetization versus~$\alpha$.

\item [(d)] Repeat (a) at finite but low temperatures~$T$,
and find $\alpha_c(T)$ to second order in $T$.

\end{itemize}


{\bf Guidance:} In the last item explain why the energy $E(M)$ 
should be replaced by the $M$-constrained "free energy" $F(M)$.
Use know results [Patria] for the free energy of electrons 
at finite temperature.  


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