energy of paramagnets

["John S. Denker" <jsd@MONMOUTH.COM>]

Bernard Cleyet wrote:

> Using a mag. field to align formerly random, atoms reduces the
> entropy?

Yes.  That's a critical step in the operation of
ye olde adiabatic demagnization refrigerator.

> They are also now at a reduced energy. (the para case)

Yes, assuming you haven't done anything too
unconventional with your choice of zero-energy
reference.

> I'd appreciate a detailed analysis of this starting with the energy
> required to "create" the initial mag. field.

Wow, what an interesting question.
The details are a bit tricky.
Here goes.....

I start by putting on my electrical
engineer disguise, and looking at it that
way.

The coil, when it contains a paramagnetic
material, has a higher inductance.  The
power I must supply during ramp-up is
       dE/dt = V I
so the energy I must supply is
       E = integral dE
         = integral V I dt
         = integral L (d/dt I) I dt
         = integral L .5 (d/dt I^2) dt
         = integral .5 L d(I^2)
         = .5 L I^2            (not a surprise)

To keep things simple, let's imagine that
there's enough paramagnetic material to double
the inductance.  So there's twice as much energy.

The origin of the extra inductance is quite
undertandable in terms of physics.  The applied
field induces some spins to flip over and align
with the applied field.  When they flip, their
magnetic field lines flip past the coil windings,
inducing a voltage therein.  So there's more
induced voltage for any given (d/dt I).  It's
exactly like having more turns -- the same I
makes more flux.

Now let's look at the field energy.  At first
glance, everywhere we look there's twice as
much field.  Half comes from the coil, and
half from the induced moment in the material.
So the field energy is higher.

Now, at first glance, you would think (by the
principle of virtual work) that the system
would try to lower its energy by pushing the
material out of the field.  But that's the
exact opposite of what happens.  So we have
a riddle to solve.

There's another riddle as well:  We put in twice
as much electrical energy, but the field energy
is not doubled, it's quadrupled.  Field energy
goes like B squared.

The answer to both riddles is the same.  Previous
remarks labelled "at first glance" didn't look
closely enough.  If we look inside the material
we see lots of little dipoles.  Think about
what the dipole field looks like.  If the field
everywhere outside the dipole (where it's easy
to see) is pointing generally northward, there
will be a bit of very intense field threading
the core of the dipole, pointing generally
southward.  The energy in this field is totally
non-negligible.  The dipole lowers its energy
by aligning its _core_ field with the applied
field.  The increase in energy in the far-field
is overwhelmed by the decrease in energy in the
core field.

So here's the energy budget summary:

Without material:
       Energy from battery:  1X (one unit)
       Energy in easily-observed field:  1X
       energy is all accounted for

With material:
       Energy from battery:  2X
       Energy in easily-observed field:  4X
       Energy in dipole core fields:  -2X
       energy is all accounted for


> How does this relate to the absorption from an EM field in EPR?

Well, dipole energy is dipole energy.

If that answer isn't helpful, please ask a
more-detailed question.

> Is the absorption when the field "randomizes" the formerly aligned
> atoms?

Absorption, especially in a magnetic resonance
system, doesn't necessarily randomize the spins.
I can take a bunch of spins that are aligned in
the -Z direction and flip them so that they're
aligned in the +Z direction.  Huge increase in
energy, but negligible change in entropy.