🚀 go-pugleaf

columbus@strident.think.com (Michael Weiss) writes:
>1) The exchange energy is not simply another name for the Pauli
>    exclusion principle.  (Would it be correct to say it is due to
>    the need to treat an ensemble of electrons with Fermi-Dirac
>    statistics rather than Boltzman statistics?)

Yes, this is the essential point!  The exchange energy is an
electrostatic repulsion energy which comes about because of the
existence of electron spin.  This is hard to comprehend because spin
is a purely quantum phenomenon with no classical analog.  There is no
exchange energy or ferromagnetism in classical physics, which is why
it's hard to explain them simply.

>2)  The exchange energy affects the way electrons fill up energy levels in a
>    solid, again modifying classical expectations.  For iron, it turns out
>    the minimum energy configuration is with the electon "gas" having aligned
>    spins, and the bound electrons having spins aligned too, but aligned
>    opposite to the free electrons.

This is essentially right.  The simplest way to see the existence of
the different energies is to look at the sequence of level fillings as
you move through the periodic table.  Why for example is Cu 3d9 4s1,
not 3d10 4s0?  You have to go beyond the bare exclusion principle and
consider exchange to explain this.

>3)  There is no simple qualitative explanation of why iron should have this
>    sort of minimum energy state, while copper, for instance, doesn't.

Actually, there is a simple answer.  The highest occupied levels of Cu
consist of mixed s and d electron states with a low electron mass.
That's why Cu is highly conductive!  Simple math shows that low mass
=> low density-of-states (DOS).  Fe on the other hand has no s charge
carriers, but has several massive partially occupied d electron bands.
This is part of why Fe is a poorer conductor than Cu.  Because the d
bands are flat, Fe has a high DOS.

When a system undergoes an ordering, symmetry-breaking transition, an
energy gap is typically opened up at the top of the spectrum.    The
system undergoes the transition because it lowers its total energy by
an amount

	condensation energy = (DOS) * gap

This equation is just integrating under the DOS vs energy curve.
Basically it says that (energy gained in transition) = (number of
participating particles)*(energy/particle).   Since Cu has a low DOS,
its condensation energy for ferromagnetism (or superconductivity or
charge-density waves or . . .) would be very low.   Fe, on the other
hand, can significantly lower its energy by opening up an exchange gap
at the Fermi energy.    We see that poorly conducting metals are
likely to exhibit many interesting phase transitions, a hypothesis
which is borne out by experiments.

By the way, Dr. Baez, I'm a solid-state experimentalist, not a
theorist.  I have to be careful because real theorists are watching.
-- 
Alison Chaiken 				alison@wsrcc.com
(510) 422-7129 [daytime] 		or chaiken@cmsgee.llnl.gov
Look if you like, but you will have to leap.