Magnetization Fluctuation

Consider N moments μ with two allowed orientations ±μ in an external field H at temperature τ. Calculate the fluctuation of magnetization M, i.e.,

SOLUTION

The energy of a dipole in a magnetic field ε_m may be written

The partition function Z is simply

(1)

Since the moments are all independent, we may express the average magnetization ⟨M⟩ as

(2)

On the other hand,

(3)

For i ≠ j the ensemble averages are independent, and

We are left with i = j so (2) and (3) give

(4)

We then obtain

(5)