W9_parallel_resonance.eps

292 Week 8: Faraday’s Law and Induction

Internal current cancels

Net surface current

Surface current does not

Applied External Magnetic Field into page)

Figure 109: Just as was the case for a diamagnet, the internal currents of aligned magnetic moments
cancel (on average) in the bulk of the material, but the surface currentsadd. The surface currents
behave like the wires of a solenoid or sheet of current wrapped around the object toincreasethe
total field inside.

has the opposite sign altogether to anything you would expect classically.

Let us suppose that the permanent magnetic moments on two neighboring atoms can themselves
interact. This alone isn’t inconceivable – one creates a (weak) magnetic field at the location of the
other, although the actual direction of that field is determined by therelativeorientation of the
source dipole and the target location and hence not easy to imagine.We will further suppose that
the interaction is bilinear in the magnetic moments themselves, and since energy is a scalar, we’ll
make the bilinear product the scalar product for simplicity.

That is, let us suppose that the potential energy of interaction between two neighboring atoms
(labelled withiandjrespectively) has the general form:

Uij=−Jij~mi·~mj (632)

whereJijis theenergy couplingbetween the two moments. Note well that this form is by no means
unique or necessarily correct – it is more or less a hypothesis that we’d need to test against observed
materials.

IfJij>0, the two moments will have minimum energy when they arealigned(ferromagnetism).
IfJij <0, the two moments will have minimum energy when they point inopposite directions
(antiferromagnetism). As before, when the temperature goes down, the energy removed has to come
from somewhere, so low temperatures will favor a “paramagnetic”alignment or antialignment of the
moments. The interesting thing is that this alignment will occureven in the absence of an external
field!

The energetics of this are illustrated in figure 110. This is yet another cartoon representation
in two spatial dimensions, this time of “spins” in one dimension (each spin is associated with a
magnetic dipole moment more or less as usual by a relation such as:

~me= e

2 me

~s (633)

in a suitable system of quantized angular momentum units). In this kind of toy model, we only
let the spins point in one of two directions: up or down, to study only their tendency to align or
antialign at different temperatures. This is a “real” model of some importance in physics in the