Re: Mechanism Underling Ferromagnetism

[Tim Sullivan x5830 <sullivan@KENYON.EDU>]

> 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