🚀 go-pugleaf

columbus@strident.think.com (Michael Weiss) writes:
>>Alison Chaiken writes:
>>(This is really an indirect
>>manifestation of the Coulomb repulsion between like charges.)
>I don't understand the parenthetical remark.  The exclusion
>principle applies even to uncharged fermions, no?

I guess you were looking for a more detailed answer ;-).  The simplest
details that I know come from the Hartree-Fock theory, which is the
zeroth-order many-body theory of electrons in solids (or atoms).
Here's how one derives the Hartree-Fock equations for the electron
energies: First, form a wavefunction called a Slater determinant which
guarantees that the Exclusion Principle will be satisfied (by taking
linear combinations of single-particle states which change sign when
particles are exchanged).  Next, take the <psi/H/psi> product to
evaluate the energies, where H, the Hamiltonian, involves kinetic and
Coulomb potential energies.  Because of the form of the Slater
determinant, there are TWO types of terms with coefficient e^^2, where
e is the electronic charge.  One is the simple repulsion of an
individual electron to all the others.  The other term is called the
exchange energy because it involves an integral over products like
(a_up b_down -- b_up a_down), where a and b are single-particle
wavefunctions and up, down refer to spin direction.  This exchange
energy is the fundamental origin of the itinerant electron
ferromagnetism of transition metals like Fe, Ni, Co.  It is not a
concocted toy model like the Ising model; it comes from fundamental
QM.

>I wrote:
>    The converse of the preceding statement is that if you have a low-density
>    electron gas, there is an effective net force aligning the spins.

>>Is there a rough-and-ready intuitive explanation for this effect?  Naively,
>>it would seem that the exclusion principle would tend to *anti-align*
>>spins.

That IS the rough-and-ready explanation; the other explanation I know
requires equations.  Exclusion does anti-align spins if they can be
put in the same energy level.  However, you get Hund's rules (of
atomic energy level filling) wrong if you ignore exchange energy; it's
an experimental fact.  I think that exchange energy is one of those QM
concepts that is not particularly intuitive; I view it as an
expression of the delocalization of electrons.

>Obviously this line of thought is too naive.  Is there a qualitative
>explanation that shows how Pauli exclusion --> aligned spins?

Sorry if you think this is too naive, but further discussion may be
beyond the scope of this newsgroup.  I would suggest that you look at
a QM textbook like _Intermediate QM_ by Bethe and Jackiw under
"Hartree-Fock" for more details about exchange energy.  If you want a
really detailed reference specifically about FM in Fe, I can dredge
one up.  Neville Mott's papers about FM are the best.
-- 
Alison Chaiken 				alison@wsrcc.com
(510) 422-7129 [daytime] 		or chaiken@cmsgee.llnl.gov
Look if you like, but you will have to leap.