Ferromagnetism

The spins of a regular Ising lattice interact by the energy

(i)

where B is an external field, μ is the magnetic moment, and the prime indicates that the summation is only over the nearest neighbors. Each spin σ_i has z nearest neighbors. The spins are restricted to equal ±1. The coupling constant J is positive. Following Weiss, represent the effect on σ_i of the spin–spin interaction in (i) by the mean field set up by the neighboring spins σ_j. Calculate the linear spin susceptibility χ(τ) using this mean field approximation. Your expression should diverge at some temperature τ = τ_c. What is the physical significance of this divergence? What is happening to the spin lattice at τ = τ_c?

SOLUTION

Using the mean field approximation, we may write the magnetization M of the lattice as

(1)

where n is the density of the spins and B_eff is the sum of the imposed field and the field at spin σ_i produced by the neighboring spins:

(2)

where λ is a constant. We may rewrite (1) as

(3)

The susceptibility χ is given by

(4)

For B and M small we may rearrange (4), yielding

(5)

where τ_c ≡ λnμ². The divergence of χ at τ = τ_c indicates the onset of ferromagnetism. The spins will align spontaneously in the absence of an applied magnetic field at this temperature.