Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 205

4.86 N Independent Spins, Revisited (Tennessee)

We use the expression where is the probability of the arrange-

ment of spins. For N spins we assume that are up and are down,

where The different arrangements are

Use Stirling’s approximation for the factorial to obtain

4.87 Ferromagnetism (Maryland, MIT)

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

the lattice as (see Problem 4.81)

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

where is a constant. We may rewrite (S.4.87.1) as

The susceptibility is given by