Pauli spin susceptibility the spin susceptibility of a conduction electron gas at alsolute zero may be approached by another method. Let N⁺ = ½N (1 + ζ); N^– = ½ N (1 - ζ) be the concentrations of spin-up and spin-down electrons. 

(a) Show that in a magnetic field B the total energy of the spin-up band in a free electron gas is E¹ = E₀ (1 + ζ)^5/3 – 1/2NμB (1 + ζ) where E₀, = 1/10Nε_F, in terms of the Fermi energy ε_F in zero magnetic field. Find a similar expression for E^–. 

(b) Minimize E _total = E⁺ + E^– with respect to ζ and solve for the equilibrium value of ζ in the approximation ζ << 1. Go on to show that the magnetization is M = 3Nμ²B/2ε_p, in agreement with Eq. (45).