416 13 Monte Carlo Methods in Statistical Physics
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Fig. 13.1Example of a cubic lattice with atoms at each corner. Each atom has a finite magnetic moment
which points in a particular direction.
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mains range from a 0.1 mm to a few mm. When an external magneticfield is applied, the
domains already aligned in the direction of this grow at the expense of their neighbors. For a
given ferromagnetic material the long range order abruptlydisappears at a certain temper-
ature which is called the Curie temperature for the material. The Curie temperature of iron
is about 1043 K while metals like cobalt and nickel have a Curie temperature of 1388 K and
627 K, respectively, and some of the rare earth metals like gadolinium and dysprosium have
293 K and 85 K, respectively. We could think of an actual metalas composed of for example a
cubic lattice with atoms at each corner with a resulting magnetic moment pointing in a par-
ticular direction, as portrayed in Fig. 13.1. In many respects, these atomic magnets are like
ordinary magnets and can be thought of in terms of little magnet vectors pointing from south
to north poles. The Ising model provides a simple way of describing how a magnetic material
responds to thermal energy and an external magnetic field. Inthis model, each domain has
a corresponding spin of north or south. The spins can be thought of as the poles of a bar
magnet. The model assigns a value of +1 or -1 to the spins northand south respectively. The
direction of the spins influences the total potential energyof the system.
Another physical case where the application of the Ising model enjoys considerable success
is the description of antiferromagnetism. This is a type of magnetism where adjacent ions
spontaneously align themselves at relatively low temperatures into opposite, or antiparallel,
arrangements throughout the material so that it exhibits almost no gross external magnetism.
In antiferromagnetic materials, which include certain metals and alloys in addition to some
ionic solids, the magnetism from magnetic atoms or ions oriented in one direction is canceled
out by the set of magnetic atoms or ions that are aligned in thereverse direction.
This spontaneous antiparallel coupling of atomic magnets is disrupted by heating and dis-
appears entirely above a certain temperature, called the Néel temperature, characteristic
of each antiferromagnetic material. (The Néel temperatureis named for Louis Néel, French