Week 8: Faraday’s Law and Induction 293
a) antiparallel spins b) parallel spinsenergy outFigure 110: A cluster of five magnetic moments (spins) is illustrated with the central spins in two
possible configurations. When the central spin is antiparallel to thefour surrounding spins, it has
potential energyUa= +4Jm^2 in a suitable system of units. When it lines up parallel to the four
surrounding spins, it’s energy isUp=− 4 Jm^2.
study ofmagnetic phase transitionsbetween paramagnetic and ferromagnetic states (the latter with
permanent magnetic dipole moments) and is called theTwo Dimensional Ising Model^81.
In this figure two spin configurations are presented – the first withfour neighboring spins (all
in the same direction) surrounding a spin that points in the opposite direction. The energy of the
centralantiparallelspin in this case isUa= +4Jm^2. In the second, the central spin is parallel to
the surrounding spins and the energy is nownegative:Up=− 4 Jm^2. The energy difference between
these two configurations is hence ∆U= 8Jm^2.
At high temperatures, both configurations are nearly equally probable in a given lattice of spins,
with the parallel configuration only slightly favored, and the systemwould behave like a paramagnet
or even a diamagnet if the diamagnetic response was larger than theparamagnetic alignment to an
external field (this is controlled with a different coupling constant in the case of the Ising model
between the spins and an external field).
As one cools the system, one removes heat energy from it. That energy comes from (among
other places) the potential energy of interaction between the spins. In very rough terms, as soon as
the energykBT(wherekBis Boltzmann’s constant) is smaller than the energy difference between
parallel and antiparallel configurations, the parallel configurationstarts being much more likely to
be found in the lattice and the spins in the lattice start to “order” in small clumps of locally parallel
spins that grow (and compete) as the system further cools. At acritical temperature, the size of one
of the clumps spans the lattice and the system develops a macroscopic magnetization characterized
by a permanent magnetic dipole moment. Notallof the spins point in the same direction (until one
reaches absolute zero) but themajoritydo, with a fraction that increases to unity as one approaches
zero temperature.
One last time we resort to our magnetization picture, this time (in??) to illustrate thepermanent
macroscopic magnetization of a bar magnet in theabsenceof an external field.
The Curie Temperature and Neel Temperature
The critical temperature for the paramagnetic-ferromagnetic transition is called theCurie Tempera-
tureafter Madame Curie, who discovered it. The critical temperature for the related antiferromag-
netic transition is called theNeel Temperaturefor similar reasons (no, not because Curie discovered
it, think harder).
Physicists find the classic ferromagnetic phase transition to be very interesting because it is an(^81) Wikipedia: http://www.wikipedia.org/wiki/Ising Model.Note well the other links at the end of this article to an
(as promised!) dizzying array of magnetic models and theories. Magnetism in matter isinteresting and importantand
a simple Ising model computation/simulation is well withinthe reach of a student looking for a project who knows a
programming language or how to use e.g. Matlab or Mathematica.