Re: Mechanism Underlying Ferromagnetism

[Allen Miller <amiller@SUHEP.PHY.SYR.EDU>]

Here is a response to the questions raised by Tim Sullivan, on my
response to Mark Sylvester's question as to the mechanism underlying
ferromagnetism.

(1) What is the physical model for the electrons? Electrons are partly
attached to particular sites. But they interact with electrons on nearby
sites.

(2) How can all the electron energy be kinetic? It cannot. The electrons
experience Coulomb repulsion with all other electrons, as well as
attraction from all the positively charged ions. However, in order to see
the effect of electron kinetic energy, there is pedagogic value in
imagining the Coulomb interactions to be turned off (free electron model).

The lowest energy state of the free electron gas has zero magnetic moment,
by the argument of the earlier post. Hence, in the actual metal, (with
interactions present), the effect of kinetic energy still is to thwart
ferromagnetism.

(3) You are right, in that the last sentence in my previous post could
have been worded better.  A better wording would be this:

"So, above the Curie temperature, electron spins prefer a random
arrangement, and the ferromagnetism is lost."

To see that entropy increases when we deviate from spin alignment,
consider N electrons, each on one site, capable of alignment (say, the 3d
electrons in nickel or iron). There is only one state of this system, for
which all the electrons are, say, "up". (The spatial state is fixed, for
each electron.) Hence, the entropy is zero.

But there are N states with one reversed spin, because one can choose any
one of the N sites to flip the electron spin. So, the entropy is raised to
k ln N.
 B

As Tim Sullivan notes, anti-alignment is also an ordered state. Here, we
alternate the spin direction up and down as we move from one site to its
neighbor. For a chain, there are only two ways to do this. So,
anti-alignment also has low entropy. But such a state describes
anti-ferromagnetism, not paramagnetism. Our interest is to show that
ferromagnetism yields to paramagnetism beyond a critical temperature.

(4) Does this apply to a real material, where the electrons participating
may be bound electrons, such as the 3d electrons in nickel, iron or
cobalt?

Actually, one cannot accurately describe these electrons as either bound
or free. They have aspects of both properties. In iron, nickel and cobalt,
the interactions of the 3d and 4s electrons with those on neighboring
sites plays a crucial role in both ferromagnetism and cohesion of the
metal. This is not in disagreement with the fact that the Coulomb
attraction of these electrons with the nucleus of its "parent" site is
also important. The latter contributes to the "bound-like" properties.

Allen Miller,
Physics
Syracuse University.


 On Tue, 8 Jun 1999, Tim Sullivan x5830 wrote:

> > I'll make an attempt on Mark Sylvester's question as to the mechanism
> > underlying ferromagnetism. We need to see if it is favorable for the
> > electron spins of a selected pair of electrons to be aligned or
> > anti-aligned.
>
> Thanks for volunteering your expertise. I hope you wouldn't mind answering a
> few clarifying questions.
>
> I don't understand exactly the physical model that you are analyzing. Electrons
> tied to lattice points, electrons in a periodic potential, free electron
> gas...?
>
> > If we ignore Coulomb interactions experienced by the electrons, then
> > anti-alignmnet is favored. This can be seen by noting that in this case,
> > all of the energy of the electrons is kinetic.
>
> This may go back to my previous question, but how can all electron energy be
> kinetic in a material?
>
> > No energy is shared, so
> > single-electron states can describe the situation. Now, for each state of
> > "up" spin, there is a state of the same kinetic energy of "down" spin, So,
> > if the these states are the lowest-energy state available, the latter
> > state is also available, if the former is. The electron pair can be
> > "placed" in these two states. The net magnetic moment is zero.
>
> > So, from this argument, we will never get ferromagnetism, unless we take
> > into account the Coulomb repulsion between the two electrons. This
> > repulsion favors alignment. The outcome of alignment vs. anti-alignmnet
> > is decided by this competition, and the winner varies, when we compare one
> > elemental metal against another.
>
> > Why does Coulomb repulsion favor alignment? The reason is due to an
> > interplay between the Pauli Exclusion Principle and the Coulomb
repulsion.
> > Due to the Exclusion Principle, parallel-spin electrons tend to stay
> > apart. This is the "Fermi hole". Otherwise, they would be occupying the
> > same state, violating the Pauli principle. By staying apart, their Coulomb
> > repulsion energy is diminished. This lowers the total energy of the
> > system. At low T, this produces the alignment.
>
> > At high T, one needs to maximize the entropy, as well as keep the energy
> > low. Electron spins that are aligned have a high degree of order, and
> > hence low entropy. So, beyond the Curie temperature, spin anti-alignment
> > is preferred and the ferromagnetism is lost.
>
> I suspect that the last sentence isn't exactly what you intend. Anti-alignment
> is order too and in fact one form of ordered state is antiferromagnetic
> materials. Above the Curie temperature spins are randomly aligned.
>
> Also, I'm not sure how this applies to real materials. The elemental
> ferromagnets have *bound* electrons in high *orbital* angular momentum states
> (unpaired d orbitals, if I recall correctly, in Fe). Yet I've read that the
> influence of the conduction electrons are crucial to the ordering.
>
> > Allen Miller,
> > Physics,
> > Syracuse University
>
>
> Thanks for tackling this. I've always suspected that the reason ferromagnetism
> is seldom treated on the undergrad level is that it is a fairly complex
> subject.
>
> Tim Sullivan
> sullivan@kenyon.edu