N Independent Spins, Revisited

Consider a system of N independent spin-1/2 particles. In a magnetic field H, in the ẑ direction, they can point either up or down with energy s_iμH, where μ is the magnetic moment and s_i = ±1. Derive expressions for the entropy S( f ) in the case of a microcanonical ensemble, where the number of particles N and the magnetization M = μf N (f = Σ s_i/N) are fixed.

SOLUTION

We use the expression S = ln ω where ω is the probability of the arrangement of spins. For N spins we assume that N_u are up and N_d are down, where N = N_u + N_d. The different arrangements are

(1)

(2)

(3)

Use Stirling’s approximation for the factorial to obtain

(4)