(The last piece to the puzzle. Previous sections can by found here: Part 1: Magnetic Field, Part 2: Magnetic Dipoles, Part 3: Magnetic Force, Part 4: Quantum Mechanical Spin)

This post has been sitting on the back burner for a very long time now. I established the physical theory for what a magnetic field is in the very first post. I then talked about how magnetic dipole moments arise from basic magnetic fields. Next, I spoke about the physics of why magnetic dipoles induce forces upon one another and how those forces act. I then slipped the surly bonds of reality and spoke about the existence of quantum mechanical spin, relating where exactly the tiny dipole moments that underpin magnetism come from. This gets us to a place where a very subtle collection of “other stuff” still resides that no one pretty much ever actually talks about. Turns out that this other stuff which fills in the remaining gaps is pretty difficult to understand on its own and is required for a subtle reason that is somewhat hard to explain. I’ve percolated over it for a long time trying to decide how best to tackle it. Realistically, I would say that I don’t fully understand the math because it is very complicated, so I will mainly be writing about the parts of this set of ideas that I do understand.

I’ll start by trying to motivate where this weird, freaking hard gap resides in the description I’ve pushed thus far. In part 4, I’ve given you a magnetic dipole moment that just IS. Spin is a purely quantum mechanical effect that can be considered the means by which you give feature and shape to effectively point-like particles that inhabit the puffy underbelly of reality. If you drop two electrons on top of one another so that they are positionally and energetically identical, they break symmetry from one another by inhabiting two separate fundamental ‘orientations.’ These orientations can be distinguished by immersing these electrons in a magnetic field: one gives a dipole moment to the electron that effectively points with the external field, while the other points against it. A spatial wavefunction that is otherwise indistinguishable is made distinguishable at a finer structure level with the electron energies offset by some splitting quantity related to spin and strength of the magnetic field (Otherwise called the Zeeman effect). If you were to trap a single electron all by itself and look really closely at it, you would discover that it has a tiny magnetic field on its own that looks exactly like the magnetic dipole I constructed in part 2; this is The electron dipole moment –(cue eerie music). You might also know it as the Bohr Magneton. If you were to gather a bunch of electrons together and orient these so that they all face in the same direction, you would get a construct like a bar magnet, which is to say a measurable magnetic field that can exert forces on nearby dipolar magnets, like a compass needle.

The problem with trying to do this, gather electrons together, is that electrons all have negative charge. They all have negative charge, without exception. Pushing two electrons together, their charges cause them to repel against one another. So, you take a bunch of electrons, gather them into the same location and let go. Boom, they explode apart. All the negative charge repels and they fly away from each other, no forces present to prevent it. So, for electrons alone, there’s not a clear way why a bunch of electron spins might be proximal enough to each other to reinforce into a powerful dipole field.

If you want to gather two electrons together stably in the same location, one way around the repulsion effect is to exploit electrostatics and neutralize the charges of electrons so that they can be near one another. For example, you get a naked helium nucleus. The nucleus has a charge of +2 and therefore exerts enough force to pull two separate electrons into close proximity with one another. They drop in around the nucleus until they become close enough to each other that their repulsion stops them from moving closer together, as balanced by the attractive force exerted by the nucleus. As helium is spherically symmetric and the electrons are essentially shapeless, both end up seeing exactly the same environment as the other and they tend to occupy physically identical states… up to a point. Because these electrons obey Fermi counting statistics (that is, they are fermions) they are required to exist in antisymmetric state superpositions. The molecular orbitals they inhabit are indistinguishable in shape, except for spin. So, one electron ends up in the alpha spin state, while the other drops into beta… rather, they both end up in superpositions of this where an electron is in “alpha” and another is in “beta,” but you can’t tell which is which. I call them “alpha” and “beta” here because you can’t tell that one is “up” and one is “down” until these electrons are subjected to an external magnetic field.

This presents two problems. First, electrons gathered by these means innately turn their spins opposite to one another, canceling out any magnetic field seen far away. Second, without some external means of orienting the system, the system is spherical and cannot be seen to point in any direction in particular without some external means of setting a direction, say by adding an external magnetic field. Reality is that this isn’t even a bad thing since the phenomenon of diamagnetism is almost exactly this: a piece of non-magnetic matter becomes magnetically poled by induction from an external field. Problem with that is that diamagnetism is very very weak and you can’t witness it as easily as the ferromagnetism of a bar magnet.

You could perhaps get away from this pesky fermion situation by switching to a boson. Two bosons can drop into the same fundamental spatial quantum state. You could get them into the same location somehow, say by the force of gravity, but you lose out on the directional spin in most cases or become unable to manipulate the particle in question. Photons and W and Z particles all have integer spin, meaning they spin in some up or down fashion, but there’s no known way to gather these and hold them in some location. Composites of a proton and electron are effectively a boson, but the system has a huge amount of other structure, meaning that they don’t just occupy up or down spin and the directions of their spin are pretty much not controllable under most conditions. It is thought that metallic hydrogen at the cores of Jupiter and Saturn are responsible for the magnetic fields of those planets, so I can’t say it doesn’t happen with bosons, but that the physics are not the same as what is seen in bar magnets –with regard to metallic hydrogen, I also can’t say that the magnetic field here is strictly a result of spin since it might be electric current too. One paper I’ve found says that metallic hydrogen may well be ferromagnetic.

Neglecting the possibility of bosons, you can also get around the fermion conundrum by going to situations where you have an odd number of electrons. Now, the unpaired electron would have a free spin. Unfortunately, you also end up tripping over the second problem above where the system is not necessarily oriented. So you’ve got two atoms or molecules each with an odd number of electrons, there is nothing to say that they must point their free electron spins in the same direction…

In fact, since magnetic fields contain energy, a thermodynamically governed system at its minimum free energy, in absence of some effect contributing an off-set in entropy, will prefer to be at its minimum energy. And that is to say possessing no magnetic field or canceling its magnetic fields out internally. The Up and Down directions of spin only matter in the sense that you have a field or some means of establishing a polarity for the spin directions. Without some means of “poling” the spins, all you have is two indistinguishable flavors of spin not known to point in any particular direction –except, of course that they are in opposition when in the same spatial orbital, canceling out each other’s magnetic dipole moment.

Now, there’s nothing actually wrong with that. You can have systems with unpaired electrons that have magnetic moments. Molecular oxygen, for instance, has unpaired electron spins. And, when you put oxygen into a magnetic field, it orients with the field and you get a detectable magnetic moment that is driven by the unpaired spins. This phenomenon is called “paramagnetism.” But, again, you can’t just build a bar magnet of oxygen which always has a magnetic field.

And so, we hit the crux of the puzzle. To have a bar magnet with an detectable magnetic field, the electron spins associated with atoms inside the material must be spontaneously oriented by some means. If you stop and actually try to figure out how that happens, it is freaking mind bending! So mind bending, in fact, that I’ve stalled over writing this post for several years.

How does it work?

I’ve debated for a very long time how best to describe this situation in a manner that is both accessible and lucid. The math really does turn out to be a nightmare and without some very specialized skill, it is not going to be that accessible to the casual reader. I therefore have chosen to try to be mostly non-mathematical in this post and to wield contextual examples wherever possible.

The problem here splits into two components that I will try to address separately. The first is why electron spins end up pointing in the same direction in magnets at all. The second is why groups of spins pointing in the same direction can be lined up along some preferred direction relative to the outside of the bar magnet.

The first part depends on the Pauli Exclusion Principle. Named for Wolfgang Pauli, who earned a Nobel Prize in 1945, Pauli Exclusion Principle is an extension of the fermion nature of electrons. One of the consequences of fermion counting is that such particles enter into a wave function in anti-symmetric superpositions of states. The most famous result is that no two electrons end up able to present the same set of quantum numbers, or that no two electrons can occupy the same quantum state. When you try to represent a collection of fermions in a wave function, the wave function must be constructed so that it is a -1 eigenstate of the exchange operator. This is one way to address anti-symmetry; that if two electrons in the wave function are exchanged with one another, that the wave function inverts on itself, or produces a -1 eigenvalue when hit with the exchange operator. A wave function that obeys exchange anti-symmetry can be constructed by creating a sum of terms where the eigenstates of the electrons are permuted among the electrons, creating what is called a single-determinant wave function, or a Slater Determinant. In this, the operation of forming a determinant, as expressed in linear algebra, is used to fabricate the wave function. This operation tends to create wave functions that are utterly inexpressible, where permutations for a relatively small number of particles in the wave function churn out unfathomably enormous numbers of terms… as many as 10^50 for a molecule as simple as benzene.

When operating the Schrodinger equation onto a Slater Determinant, the structure of the permutation can be worked into the equation such that you can actually skip ever forming the determinant directly. This gives rise to energetic interaction terms between pairs of electrons occupying pairs of states. The first electron-electron interaction is the coulombic repulsion term, which is simply the electrostatic repulsion pressure one electron has on another when they are in proximity to each other. The second interaction term is called the “Exchange Interaction” and it is an energy describing the situation where one indistinguishable electron switches identities with another and they exchange states. Yes, this is the epitome of quantum weirdness! This term specializes specifically to electrons that are not distinguishable, which is electrons that are of the same spin that are near to one another.

1 Exchange energy

This is the exchange energy term, just so that you know what it looks like. This term looks very similar to the potential energy between two repulsive electron charges. The difference is in the Phis; electrons ‘1’ and ‘2’ exchange between spatial states ‘a’ and ‘b’ where the strength of the interaction falls off as the reciprocal of the distance between the electrons. The integral emerges because both electrons end up being distributed across space in some non-trivial way, requiring you to weight every combination of positions those electrons occupy based on the wave functions they are in.

Electron exchange ends up being the basis of ferromagnetism!

The crux is simply this. Remember where I said above that minimizing the strength of the magnetic field also minimizes the energy? This is one way to consider why fermions, like electrons, almost always drop into the same spatial orbital in opposite spin states. Most of the time, if they fall in with opposite spins, they minimize their magnetic energy by canceling out their magnetic dipole moments as a pair. In a ferromagnet, for metallic atoms of Iron, Nickel and Cobalt, it turns out that this is actually not the case. If nearby spins drop into orbitals where they have the same spin value, they achieve a lower overall energy than they would if they dropped into the same orbital in opposite spins. The reason is specifically to facilitate exchange energy minimization. They would rather have greater magnetic energy because in doing so they have lower exchange energy and the combination is enough that their overall energy is lower than it would be otherwise!

Some of the papers I’ve looked at talk in terms that are peppered with references to condensed matter physics: strongly bound bands and anti-symmetry in Wannier orbitals and such. I tried to wade into that and decided that it would probably not help my discussion here because it becomes fairly opaque. I’m not a condensed matter physicist and, while I have a basic understanding of “density of States” and “Bloch quantization,” this is not something that I understand exceptionally well. One advantage that I do have is that I’m really not in the same place I was when I first started writing this series on how magnets work several years ago.

I’ve gotten access to some tools that are absolutely killer for trying to model these sorts of situations on a molecular level!

The first thing I did, which failed kind of spectacularly, was to try to model and crystal cell from magnetite using Gaussian 16. Magnetite is among the most famous magnetic crystals and is probably the material present in whatever bar magnet you happen to own. The problem is that the unit cell has 56 atoms and 880 electrons. Including periodic boundaries ended up taxing my computer resources so spectacularly that a little more memory was never quite enough.

2 magnetite cell
Magnetite. X-ray structure taken from this paper. Haavik et al Am Min 85 (3-4): 514–523 (2002)

Here is the unit cell for magnetite. The red atoms are doubly charged (-2) oxygens while the purple atoms are a combination of Fe(III) and Fe(II) iron atoms (octahedral coordination for III, tetrahedral for II). Even going for a small basis set, this is beyond my computational resources in large part because there are many atoms and because the iron needs d-shells to be done even remotely correctly.

As I worked my way into this, I learned some things about metal liganding that made representing the system somewhat easier. I dialed back my demands and decided to focus on the bare minimum system. The next trial involved two iron atoms liganded with hydroxide… this ended up being a bit of a failure too, but for the reason that I didn’t fully understand the chemistry I was dealing with. With the system involving hydroxide, one iron atom preferred an octahedral geometry, while the other involved trigonal bipyrimidal.

3 hydroxide and iron
Fe(V) and Fe(VI) with hydroxide and oxygen ions, Gaussian 16 uB3LYP/3-21G.

This system ended up showing signs of ferromagnetism, but not because I knew what I was dealing with. The charge state was set to neutral charge, which, given the charges of -1 on the hydroxide and -2 on the oxygen ions, requires the Iron to be of +5 and +6 charge states, or Fe(V) and Fe(VI). This is different from the magnetite model I had originally tried to base my work on, which contains only Fe(II) and Fe(III). To get a convergent wave function, I also needed to set the spin for a dectet, which I didn’t fully understand at the time when I did it. Shoot first and look later, right? Truth is that I’ve never really dealt with iron in any formal capacity and I needed to learn a lot about d-shells before I got anything that made real sense. The basic design here, however, seemed sound because it allowed me to place two iron atoms in close enough proximity in a similar charge state to mimic what can happen in a crystal of magnetite. I understand the purpose of the spin state setting now, but not at the time.

5 periodic table 2
Periodic Table, taken from here.

In this version of the periodic table, I’ve highlighted iron so that you can see where it is. Iron is in the transition metals there in column 8. The valence structure of iron is in the d-shell: counting from the left, neutral iron has 6 electrons and iron begins filling its period 4 s-shell before it starts filling its period 3 d-shell.

The way that the periodic table labels the s, p, d periods, each shell orbital is filled in closure with one version of each spin type in each orbital. s-shell has 1 spatial orbital, and 2 spins, giving column 1 and 2 on the right side of the table (except helium, which has only a filled s-shell and goes into the Noble gas column). p-shell has 3 spatial orbitals and 2 spins, giving 6 columns and only starting on the second row after Be with columns 13 to 18. d-shell has 5 spatial orbitals and 2 spins, giving 10 columns, but not starting until the 4th row because the 3rd p-shell and 4th s-shell both fill with electrons before the 3rd d-shell. d-shell are columns 3 to 12. This idea of “closure” is basically the magnetic cancellation that I mentioned earlier, but is better considered as angular momentum closure, which is to say that configurations with minimal angular momentum tend to be more stable than otherwise… which is why the Noble gases on the right are particularly stable; they have the most completely canceled angular momentum of any atoms since all their occupied shells are closed.

One thing that makes this a little more complicated is that in a system containing degenerate states at some energy level will tend to fill that level according to what’s called Hund’s rule. This follows from the first Hund’s Rule in the wikipedia link. For a level, the electrons tend to singly occupy each orbital before they start doubling up… which is important for how the d-shell fills.

For d-shells, there is also one other crazy feature which seems to add to the fog of war, which is that atoms preferring octahedral liganding, like Iron, tend to create a split in the d-shell where two orbital levels are offset to a slightly higher energy than the first three. If the energy splitting between the two sets of states is sufficiently large, Hund’s rule allows electrons to spread across only the first three states before ever filling the last two states. The system that only fills the first three states is called Low spin, while a system that can fill all five is called High spin.

5 spin states fe0

This diagram shows how the states could be filled in uncharged iron. The low spin configuration has no angular momentum and would produce a spin singlet. The high spin version, on the other hand, has four orbitals occupied singly and would produce a spin quintet. The quintet splitting comes from the fact that the singletons could be either up or down spin and that there are five possible fine structure energies that could emerge ranging from all up to all down with combinations of up and down in between.

I came to focus on combinations of Fe(II) with Fe(III). The oxidation state sign here reflects the number of electrons that oxygen has stolen from iron and directly correlates in this case with the charge on the iron: Fe(II) = +2 charge, Fe(III) = +3 charge. If you ignore Hund’s rule to start with and simply invoke spin closure, this would be the d-shells of each.

6 spin states feii and feiii

This would be a ground state with a spin doublet. As it turns out, I was unable to converge geometry for any structure in this state. Solving wave functions depends quite strongly on having a good guess at the initial geometry and at the initial wave function… if you’re lacking on either, no solution will be possible. The doublet may well just be too much a violation of the reality to craft any wave functions. I was able to get a singlet for an Fe(III) – Fe(III) complex, but no multiplets ended up possible.

Instead of looking further at the spin doublet, I went to the next multiplicity of states in the Fe(II) – Fe(III) complex, the Low spin combination. I adopted a similar geometry to what was seen above with the hydroxide, but supplemented water for hydroxide at a large enough frequency to make an uncharged complex with Fe(II) and Fe(III). In this case, the liganding geometry starts to fall apart and trigonal bipyrimidal geometry is no longer visible. Moreover, the waters appear to be undergoing some acid-base chemistry in the process of the simulation by exchanging around protons which actually never come back to a fully bonded length (this is omitted from my images here because I filled in bonding to make it sensical).

First, here is the low spin orbital configuration:

7 spin states feii and feiii 2

I was able to find a converged geometry, starting from this electron configuration.

8 low spin complex
Low spin complex Fe(II), Fe(III) with water and hydroxide. Gaussian 16 uB3LYP/3-21G.

This is the low spin complex. It might have been possible to solve a structure for low spin with both atoms in octahedral form, but I really didn’t want to spend the time. This structure has a spin sextet. There’s a quirk to these solutions that I will mention in better detail after I introduce the high spin complex.

And, yes, I did find a high spin complex with the same atoms and electron count. Here is the electron configuration:

10 spin states feii and feiii 3

As you can see, the spins are now completely dispersed in low and high energy orbitals. This is now a spin octet. The structure itself doesn’t look very different from the low spin version of the same complex.

9 high spin complex
Low spin complex Fe(II), Fe(III) with water and hydroxide. Gaussian 16 uB3LYP/3-21G.

Now, before I go on, there’s something else that you need to understand about the orbital models that I started with.

They are completely wrong!

If you want to know more about the organic chemistry lie, that is here. The orbital ideas introduced above all hinge on the idea of closure. This is the notion that every spatial orbital you might discover is incomplete unless it is filled… or even designed to be filled… by two anti-parallel spins. This turns out to be a lie. Even down to the tool of the Periodic Table, as detailed above, it’s a lie! In reality, there is no reason at all that individual electrons can’t simply occupy individual orbitals!

The idea introduced above for Hund’s rule relies on a notion called “restricted open-shell” orbitals… or that only the orbitals which are occupied by one electron are singly occupied. This turns out to be a simplification of the reality that completely removes a big, important subtlety. The solutions that I found for these complexes were created by Unrestricted DFT (denoted by the “u” in the uB3LYP/3-21G signification). Unrestricted solutions take all alpha electrons and all beta electrons in the complex and find orbitals for them at whatever energy a given orbital happens to fall at. In this form of solution, no orbitals are doubly filled. You may glance at the high spin configuration and shrug your shoulders: well, all of those are singly occupied anyway! Well, yes, but that says nothing about every other electron in the complex, including those for the waters and the hydroxides! In the unrestricted solutions, all orbitals are singly occupied wherever they be, and they may not any of them have the same energies.

The reason this is important is because the solution removes the ambiguity in the spins introduced with Hund’s rule. For the energy diagrams, all the singly occupied orbitals are left ambiguous in that they could be “up” or “down.” For the unrestricted solutions, a given orbital is assigned “alpha” or “beta” with no ambiguity! If alpha is taken to be “up,” all spins of alpha flavor are strictly and unambiguously up!

What I gain by looking for unrestricted orbitals is a way to measure ferromagnetism on the level of atoms.

11 low spin spin occupancy
Low spin. Mulliken spin population, green=alpha, red=beta. Energy = -3196.738623 Ht.
12 high spin spin occupancy
High spin. Mulliken spin density, green = alpha, red = beta. Energy = -3196.755640 Ht.

These images are colored by Mulliken population spin density. This is a spin density representation which goes on a color gradient scale: extreme alpha spin is green, neutral is black and beta is red.

13 color scale

First, there is no particular spin polarity for most of the complex, except on orbitals localized at the iron atoms. For the low spin complex, one iron is strongly spin polarized to alpha, while the second is only weakly so. No part of the complex is polarized explicitly to beta. In the high spin complex, both iron atoms are strongly spin polarized to alpha. This would hardly matter but for the energies associated with these states… the high spin state has a lower energy! If you work the numbers, this difference is 0.54 eV, about half an electron volt. To understand how significant a difference that is, consider that the energy in a covalent bond is of the order of 1 or 2 eV. The low spin state could be reachable thermally, but you will mostly see the high spin state. Moreover, I was unable to find more weakly polarized states than these; I could not solve for a doublet. The trend implies strongly that the most prevalent state is with electrons around these nearby nuclei strongly spin coupled.

When you start looking at the molecular orbital populations, it turns out that both of these complexes have very lopsided orbital occupancy. The high spin version has 65 beta orbitals and 72 alpha, or seven more alpha spin-orbitals than beta.

14 highspin spectrum

In addition to this, because the fundamental model used to design the initial experiment is itself broken, I went on a fishing expedition. Noting that the sextet and octet geometries are very similar, I took the octet geometry and I did a spin scan. I scanned spins doublet (2), quadruplet (4), sextet (6), octet (8), dectet (10) and dodectet (12). The 2 and 4 spins failed to converge –I was expecting 2 to fail from my earlier work– but all the rest did not!

13 spin state occupancy

Rather than trying to wedge a 1920s model into a 2020 simulation, this is making use of the true power of quantum chemistry by creating every possibly wave function and checking their energies. It turns out that the octet isn’t quite the lowest energy state… the dectet is, but only by a bit.

15 super high spin iron
Super-high spin, alpha = green, neutral = black, beta = black. uB3LYP/3-21G

This configuration might be called the “super-high spin” geometry. I’ve left the waters in their partly broken state here, so protons are not always explicitly bonded to the oxygens and are sometimes at strained lengths. Again, this is because the simulation allows for bonds to break. There can be no doubt at all that the Iron spins are locked as alpha here and that this is the state that will be most frequently observed. In this plot, the sextet energy is attenuated high (1.8 eV versus 0.52 eV) because this is not the optimal geometry for the sextet… it’s the optimal geometry for the octet. That’s noteworthy because it’s also not the optimal geometry for the dectet, meaning that the energy for 10 will probably go down more if the geometry is optimized. On the other hand, I don’t know that I really care since perfecting the energies doesn’t actually change my point at all… it would be a good deal of computer time for very little pay-off (which is why I kept to B3LYP and 3-21G rather than going for a better functional or basis).

(Edit 5-24-20:

Because I had a moment where the computer was lying dormant, I came back and calculated geometries for the 10 and 12 multiplets and got energies for them in the process.

21 spin states accurate

This shows more accurately the relative energies between the spins because I have optimized geometries where all of these energies are taken. You can see that the 6-tet is now 0.5 eV higher energy than the 8-tet. The 10-tet and the 12-tet both decrease in energy relative to the 8-tet, as expected. However, the overall trend is not different from what I said originally. The dectet is still the most stable state.)

The transition from the restricted open-shell chemistry problem to the unrestricted model causes people’s chemistry intuition to break. The classical notion of a covalent bond involves this idea of angular momentum closure, where a covalent bond is two electrons with canceled anti-parallel spin. In the case of the iron complex above, this idea is completely broken in the valence band.

16 alpha HOMO
Alpha HOMO for octet configuration. uB3LYP/3-21G

This is the highest occupied molecular orbital for the alpha spins. That’s one electron literally ranging all over the complex, through the waters and between the irons! The orbital looks kind of like an anti-bonding orbital since it has nodes everywhere, but some of the lobes may well also be bonding. In some ways, it shouldn’t be too surprising because it does look kind of d-orbital-like around the iron on the right, but it goes everywhere! That the orbital connects between the two iron nuclei may well be confirmatory as to why the alpha spins appear to center on the irons. Of course the spins on the iron nuclei are locked; the orbitals that contain them are spread to both such nuclei. You can use rules like Hund’s rule to start trying to “understand” this, but be aware that the reality is actually somewhat messier.

The point overall is that the Fe(II) – Fe(III) combination can appear in a form where the spins of the electrons are coupled to one another and that this is an energetically favorable situation. I assert that Exchange energy is the reason for this, but not without support. Also from what I read, Magnetite contains a significant number of spin-coupled Iron atoms, though it should be noted that not all point the same direction. A majority point one direction, while the minority point the opposite direction, giving the crystal cell a significant net magnetization along this axis.

Now, I’ve tackled the first issue. Ferromagnetic coupling of electrons occurs because it is energetically favorable for them to spin orient in the same direction. Now, why is it that this can occur in a particular structural direction with respect to the material phase which harbors these spins?

It turns out that there are several interconnected features that need to be discussed here.

The first thing to note is how a material substance, like your bar magnet, can be prepared to have a “net” magnetization. I specify net magnetization in the same sense as it is described for the magnetite crystal, that a majority of the spins happen to point in the same direction. This may not be a big majority, but just more than 50%. Any majority will give an observable external field. In most iron ore rock, the magnetic dipole moments are locally ordered, but globally disordered. The ordered spins become trapped in small sections of crystal called domains where the spins within the domain are very well ordered, but that the domains are not ordered with respect to one another. A plain old piece of iron scrap picked up off the side of the road may have no net magnetization and will not produce an observable magnetic field.

A ferromagnet produced in a factory is built using a special annealing process. The metal is heated until it is molten and then cooled into a solid in the presence of a powerful electromagnetic field. While molten, the electromagnetic field poles a majority of the spins within the metal. Given the quantum mechanical effects mentioned above, these spins like to be co-oriented relative to each other and given the physics of how magnetic dipoles respond to an external magnetic field, they will tend to orient relative to the field. When the metal is then cooled, the spins become “locked in.” Take the field away and the solid has a permanent magnetic dipole moment that is proportional to the numerical difference between the spins initially oriented with that external field and oriented against it.

This locking in turns out to be very dependent on what sort of crystal structure the substance has settled into. This should make some sense: as I showed with the molecular orbital in the image above, the magnetic spins are not necessarily localized exactly to the atoms, but are distributed among them. It turns out that the regular structures of crystal cells tend to have preferred directions where magnetization “likes” to be pointed. This is called magnetocrystalline anisotropy.

Magnetocrystalline anisotropy (from wikipedia)

The point of this is that if you magnetize the material while it’s molten, then cool it into a solid, as the solid crystallizes, the magnetization is most stable if the crystal is oriented relative to the external field. You could magnetize spins in any direction relative to the crystal that you want, but if you pick a particular direction, called the ‘easy’ direction, Entropy will tend to not disorder the spins in the crystal that quickly. It turns out that hexagonal crystals (middle panel) are highly preferred because they have a single axis along which magnetization turns out to be easy. For example, hexaferrites which contain ferrite on a hexagonal lattice, are well known for industrial applications. This material has strictly columnar easy magnetization.

Of note, magnetite probably has 8 directions (right panel) that are easy. If you look down any corner of the crystal unit cell, it has a hexagonal footprint:

17 magnetite hexagon
magnetite crystal cell regarded along a point of the cube (Gaussview).

This would suggest that magnetite is not necessarily the most industrially preferred magnetic material.

Naturally occurring lodestones come to exist in a process much like what is described above. Iron ore liquified in the interior of the Earth is poled by the Earth’s magnetic field and then allowed to cool. Allow it to sit undisturbed for the right amount of time and then pluck out a piece… boom, magnetic metal.

To make a good permanent magnet, you pick out a magnetic material that can be crystallized selectively on a hexagonal lattice. You would then want to learn conditions that allow for crystallization of large domains. To make your magnet, you would melt that material and crystallize it in the presence of a strong magnetic field under conditions that make big domains. If you’re really classy, you would then machine this material down to a monodomain (while cooling it to keep from heating it above its Curie temperature during the machining process). Rare-earth magnets are even classier because Lathanides have even better characteristics for orienting spins than iron does.

There are significant other details present in this discussion. Magnets are very subtle. But, I think I’ve left no part of this muddled topic untouched. Do you suppose Insane Clown Posse would change the lyrics in their song? Yeah, I know… the song specifically resists the notion that people without expertise can learn from people with expertise.

Never thought I’d write the following on this series of posts, but The End!



Because I think it’s cool, here’s some additional structure for the spin orbitals in the iron cluster above.

20 spin differences3
0.025 isodensity level on alpha minus beta spin-orbital difference density
19 spin differences2
0.004 isodensity level on alpha minus beta spin-orbital difference density
18 spin differences
0.0004 isodensity level on alpha minus beta spin-orbital difference density

This sequence of images depicts a density difference map. The density is all alpha orbitals minus all beta orbitals. Purplish zones are where alpha density is greater than beta density. Cyan zones are where beta density is greater than alpha density. At the highest isodensity level, it’s clear that the majority of the excess alpha spin is trapped on both of the iron atoms. However, atoms are less than rigid blocks and fringes of the density is spread across the entire complex. Interestingly, the excess alpha density is clearly clustered on the atoms rather than in between, suggesting that the excess alpha orbitals are predominantly anti-bonding. Kind of cool.

Published by foolish physicist

Low level academic enthralled with learning how things work.

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