Re: Upended quantum physics in the news

anon1@xxxxxxx wrote:

[snip]

Ferromagnets turn out to have the property that the electron
spins in them prefer to all be aligned parrallel to each other
over very long distances ... it turns out that this configuration
gives the minimum Gibbs free energy at low enough temperatures.

I'll have to just take your word for that, since I didn't get that far
in college physics to be able to make even the most primitive Gibbs
calculation. But at least it's not really counter-intuitive.

Oh, what's the key factor that makes one chemical ferromagnetic and
another chemical not? Like why are iron and nickel (and cobalt??), and
some iron oxides but not others, and chromium dioxide but not atomic
chromium, ferromagnetic?

This is an extremely subtle and very complex question which
requires giving you some real details. I will do my best,
but you should understand that it involves complex theory to
give a fuller answer. First you have to consider the atomic
structures of Fe, Co, Ni.

Iron has 26 electrons. Its basic electronic configuration
can be written for short as:

[Ar].3d(6).4s(2)

This notation means that it consists of an argon-like core
plus some outer electrons. The core electron shells, first
the K-shell consisting of the 1s, and the L-shell consisting
of the 2s and 2p orbitals, are completely filled up. In the
M shell the 3s and 3p orbitals are also completely filled
up.

That's the electronic structure of argon [Ar], which has 18
electrons (2 in the 1s, 2 in the 2s, 6 in the 2p, 2 in the
3s, and 6 in the 3p, taking account of the two orientations
possible for the electron spins, and the angular momentum
degeneracy of each orbital.

Here s stands for l=0, p for l=1, d for l=2 and so on, and
the degeneracy of an orbital is 2l + 1, so that, e.g., a p
orbital has three possible places each of which can hold two
electrons, for a total of six possible quantum states. 1s
indicates the principal quantum number is 1 and refers to a
scheme in which the basic orbitals are numbered like those
in hydrogen.

For atoms that only fill up to the 3d orbital, the energy
levels of the orbitals go up in the order that I wrote them
out.

But for atoms that have electrons at the level of the 3d
orbital a strange thing happens.

The 4s orbital actually comes down a bit lower in energy
than the 3d orbital due to the interactions among all of the
electrons and the nucleus, and so it gets filled up first.

Then, at a very slightly higher energy than the 4s is the
comes the open 3d orbital, which could hold up to 10
electrons total.

This makes the order in which these two orbitals fill up
with electrons a bit complicated as there is interaction
among these outer electrons and their spins in the outer
orbitals.

However: the 4s can hold 2 electrons, and it fills up
first. This brings the total up to 20. For atoms with from
20 to 23 electrons the electrons all go into the 3d orbital,
but at 24 you suddenly get five in the 3d and one in the 4s.
25 electrons has again two in the 4s and five in the 3d.

Finally iron, at 26 has six in the 3d and two in the 4d.
(a total of 8 electrons outside the argon core).

Cobalt then has seven in the 3d and Nickel has eight in the 3d.

A further thing to know is that iron likes to lose its
electrons when it forms oxides, and that this happens such
that it likes to lose either 2 or 3 electrons. It has thus
two possible valences, and the electron configurations are

[Ar].3d(5): the configuration for Iron III which when
combined with oxygen makes haematite (Fe2O3). In this case
you combine 3 oxygens with 2 irons, and there is a transfer
of 6 total electrons to make the covalent bond.

and the other is:

[Ar].3d(6): which is the configuration for Iron II, which
combines with a single oxygen to make iron oxide (FeO).

There is also a third oxide of iron, called triiron
tetroxide (Fe3O4), which is more generally called magnetite
or lodestone. This is the famous magnetic oxide of iron
from which the ancients first learned about magnetism.

Note that lodestone has some of the irons in the II and some
of them in the III valence state.

But also notice that what happens in all of the oxides is
that the two 4s electrons tend to leave the iron atom and go
onto the oxygen atom, while the 3d orbital on the iron is
still left at least partly filled.

As to the crystal structures of these oxides: they are all
different for the different oxides. Ordinary iron oxide
(FeO) has a simple cubic structure, while the others are all
more complicated.

So the first question is, why is pure iron ferromagnetic?

Here I'm going to cheat a very great deal in the explanation
and just give you the rough idea.

Pure iron forms a body centered cubic structure, in which
the iron atoms are brought pretty close together, thus
bringing a lot of those outer electrons that we noticed that
iron likes to lose in its oxides very close to other iron
atoms. This happens in both the 4s electrons and in the 3d
electrons.

Since the inter-iron distances are not too great in the
crystal, what then happens is that the Pauli principle comes
into effect on the outer electrons that are in adjacent
atoms in the crystal. The wavefunction must be totally
anti-symmetric, when you switch an outer electron in the 3d
orbital on one atom with an electron on the next atom in the
crystal.

Now it turns out that the lowest energy configuration is
obtained by making the two electron spins line up in the
same direction, and the antisymmetry is enforced by putting
a node into the relative spatial wavefunction of the
electrons, so that the wavefunction is a spatially odd
function.

The inter-iron distances are just right at low enough
temperatures that this `exchange interaction' always favours
lining up the spins. At high temperatures the distances
increase, and the effect goes away. The temperature at which
it goes away is called the Curie temperature.

This could have gone exactly the other way, and it does so
in some materials, which like to line up the spins
anti-parallel. These materials are called anti-ferromagnetic.

This lining up of spins due to the exchange interaction
actually happens for all of the 3d electrons in iron. They
like to line up not only on neighboring atoms, but also on
the same iron atom.

Now cobalt and nickel work basically the same way, and there
are two other elements which have the right electronic
structure to do this: these are gadolinium and dysprosium,
and they fall into the same group of 3d transition metals.

So that in a nutshell, is why iron is ferromagnetic.

In the various oxides, the inter-iron distances are
different due to the different crystal structures, and there
may or may not be a sufficiently large exchange interaction
for the 3d electrons to produce ferromagnetism.

Thankfully for the development of world history, Fe3O4
works out right!

If you want more detailed theory, let me say that you can
basically spend a lifetime working on these details.

Why did the distances work out as they did and so on
.... there are at least thousand questions to be analyzed.










Is it some subtle balance of two computed
values, just like when you compute which isotopes are stable and which
not based on total binding energy as-is vs. decayed? Or is it something
more absolute/discrete like fermion/boson based on odd/even half-spins?


It is subtle, but I hope you see the main point. There are a lot
of unpaired electrons in the 3d orbital in materials which may
be ferromagnetic and the distances have to work out
right.

everything is entirely consistent with the Pauli principle when you
break everything down into the description in terms of single
electron states.

Except, as you pointed out, for long-range interactions *only* you also
need to consider bound states whereby several bound particles act
globally as if they were a single particle but *only* while they really
are bound. That helped immensely in resolving my puzzlement.
(It also seems to explain why there's a temperature and magnetic limit
to superconductivity: High temperature tends to un-bind systems. Strong
magnetic field tends to perturb Keplerian orbits to cause them to swing
out to escape on one side of the orbit. Is that OK at layman-level?)


Yes, that's a very good basic description. Glad what I said helped.

to get a real understanding you really need to look into the
equations.

Unfortunately directly applying Schroedinger's wave equation to model
systems is beyond my ability. Spherical harmonics is just about the
borderline of what I can just barely sort-of understand and deal with
at a conceptual level if I can cheat when it comes to the actual math
by looking up the already-worked-out answer. I like potential-energy
expressions, which make it easy to decide via "landscape" metaphor how
much thermal or quantum energy is needed to spill from one local
minimum over a "mountain pass" to a neighboring local minimum.
(Actually I even have to look up the table of spherical harmonics. I
can't produce them from first principles. They're much more difficult
than one-dimension Fourier terms which are simple harmonics.)
.

If you want and we had a week or so I could definitely teach
you to construct the spherical harmonics by hand ... there
are at least two very elegant ways to go about it. They
are the eigenfunctions for the angular momentum operator,
and you can easily construct shift operators which let
you generate them all ...

Cheers!

David

.

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