\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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%