Conduction electron ferromagnetism 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 anti parallel spins do not interact with each other.

(a) Show with the help of Problem 5 that the total energy of the spin-up hand is E⁺ = E₀ (1 + ζ) ^5/3 – 1/8VN² (1 + ζ)² – ½NμB (1 + ζ); (find a similar expression for E-. 

(b) Minimize the total energy and solve for ζ in the limit ζ << 1. Show that the magnetization is so that the exchange interaction enhances the susceptibility. 

(c) Show that with B = 0 the total energy is unstable at ζ = 0 when V > 4ε_F/3N. If this is satisfied, a ferromagnetic state (ζ ≠ 0) will have a lower energy than the paramagnetic static. Because of the assumption ζ <<1, this is a sufficient condition for ferromagnetism, but it may not be a necessary condition. It is known as the Stoner condition.

3N 2r - VN