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Consider a lattice with sites, where each site can assume two
possible states , or spin ``up'' and spin ``down''. A
particular configuration or microstate of the lattice is specified by the
set of variables
for all lattice sites.
Now we need to know the dependence of the energy of a given microstate,
according to the configuration of spins. The total energy in the presence
of a uniform magnetic field is given by the ``Ising model'':
|
(114) |
where the first summation is over all nearest neighbor pairs and the
second summation is over all the spins of the lattice. The ``exchange
constant'' is a measure of the strength of the interaction between
nearest neighbor spins. If , the states with the spins
aligned
and
are energetically favored, while for the
configurations with the spins antiparallel
and
are the ones that are preferred. In the first case,
we expect that the state with lower energy is ``ferromagnetic'', while in
the second case, we expect it to be ``antiferromagnetic''. If we subject
the system to a uniform magnetic field directed upward, the spins
and possess and additional energy and
respectively. Note that we chose the units of such that the magnetic
moment per spin is unity.
Instead of obeying Newton's laws, the dynamics of the Ising model
corresponds to ``spin flip'' processes: a spin is chosen randomly, and the
trial change corresponds to a flip of the spin
or
.
Subsections
Next: Boundary conditions
Up: The Microcanonical Ensemble
Previous: Exercise 12.2: The Boltzmann
Adrian E. Feiguin
2004-06-01