Find the equilibrium value at temperature of the fractional magnetization M Nm = 2 < s

Find the equilibrium value at temperature τ of the fractional magnetization

M Nm = 2< s >/N

Of the system of N spins each of magnetic moment m in a magnetic field B. The spin excess is 2s. take the entopy as the logarthithm of the multiplicity g(N,s) as given in

σ(s) ≈ log g(N, 0) – 2s2/N (40)

for |s| << N.

σ(U) = σ0 – U2/2m2B2N (41)

with σ0 = log g(N, 0). Further, show that 1/τ = –U/m2B2N, where U denotes < U >, the thermal average energy.