Statistical Mechanics - Physics at Oregon State University

158 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.
this model to show many of the techniques employed in statistical mechanical
calculations of solids. In this sense the Ising model is a standard workhorse or
toy-problem. More complicated and realistic lattice models can be solved along
similar lines, although the calculations are more complicated and are often only
feasible numerically.
Basis for the Ising model.
The Ising model is a simple description of a magnetic solid. We assume that
each atom labelled with i has a total angular moment Si. This total angular
moment is a combination of the total spin and the total orbital angular momentum
of all the electrons. In a real system, both the direction and magnitude of
Si are a function of time. The magnitude fluctuates around an average value,
which could be zero. The directions are able to change. For example, in iron
above the Curie temperature the individual atoms still have a local spin associated
with them, but the orientation of these spins is random. Only below the
Curie temperature do we find an ordered, ferromagnetic state.
A simple model for the internal energy of a collection of atomic spin-moments
is the Heisenberg model (the traditional semantics uses spin moment here, but
it can easily include orbital effects):
H = −
i