We approximate the effect of exchange interactions among the conduction electrons if we assume that electrons with parallel spins interact with each other with energy $-V$, and $V$ is positive, while electrons with antiparallel spins do not interact with each other. (a) Show with the help of Problem 5 that the total energy of the spin-up band is
$$
E^{+}=E_{0}(1+\zeta)^{50}-\frac{1}{3} V N^{2}(1+\zeta)^{2}-\frac{1}{2} N \mu B(1+\zeta):
$$
find a similar expression for $E^{-}$, (b) Minimize the total energy and solve for $\zeta$ in the limit $\zeta<1$. Show that the magnetization is
$$
E^{+}=E_{0}(1+\zeta)^{53}-\frac{1}{5} V N^{4}(1+\zeta)^{2}-\frac{1}{2} N \mu B(1+\zeta):
$$
find a similar expression for $E^{-}$, (b) Minimize the total energy and solve for $\zeta$ in the limit $\zeta<1$. Show that the magnetization is
$$
M=\frac{3 N \mu^{2}}{2 \epsilon_{P}-\frac{l}{2} V N} B
$$
so that the exchange interaction enhances the susceptibility, (c) Show that with $B=0$ the total energy is unstable at $\zeta=0$ when $V>4 \epsilon_{P} / 3 N$, If this is satisfied, a ferromagnetic state $(\zeta \neq 0)$ will have a lower energy than the paramagnetic state. Because of the assumption $\zeta<1$, this is a sufficient condition for ferromagnetism, but it may not be a necessary condition. It is known as the Stoner condition.