# I What causes magnetism at the atomic level?

#### LouisL

Lets assume we have a 1 foot square bar of Iron.

I realize that unpaired valence shell electrons in an atom of a substance like Iron, all with the same electron spin---either +1/2 or -1/2---are consistent within the same atom, as indicated from the aufbau principle and experiments. So for an Iron atoms 4 unpaired valence electrons, let's arbitrarily say the spin on each of these 4 electron in the one atom is +1/2. I know these 4 electrons, all spinning in the same direction in this one atom, add up and give this particular Iron atom a magnetic field.

Question: Do all Iron atoms that are adjacent to this original atom mentioned above have 4 unpaired valence electrons with the same spin quantum number as this original iron atom OR does each iron atom form it's spin quantum # of these 4 unpaired valence electrons independently of one another?

I assume in a magnetic domain of a small section of this Iron bar there would be Iron atoms all having their unpaired electrons with the same spin quantum number? This would give this particular domain the same magnetic field? Adjacent domains would have different spin quantum numbers, by chance, and for the entire material, and these would cancel out unless an outside magnet is applied.

Correlary:

Could a domain be considered to be made from only one atom OR is it considered to be made from 1000s of atoms?

Thanks for any insight.

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#### Pallavi

I am not a professional but according to me it is not just the spin quantum numbers in action.

#### SpinFlop

There is a lot to unpack here given that magnetism in condensed matter is a complicated subject. However, I will step through your post and try to provide an outline for properly working through this.

I realize that unpaired valence shell electrons in an atom of a substance like Iron, all with the same electron spin---either +1/2 or -1/2---are consistent within the same atom, as indicated from the aufbau principle and experiments. So for an Iron atoms 4 unpaired valence electrons, let's arbitrarily say the spin on each of these 4 electron in the one atom is +1/2. I know these 4 electrons, all spinning in the same direction in this one atom, add up and give this particular Iron atom a magnetic field.
The important thing to note here is that you are talking about a SINGLE atom and effectively applying Hund's rule to work out that the 4 valence electrons will all align parallel to one another. This is a perfectly fine starting point, however the end picture for most materials (made from an immense number of atoms) is quite different. This can be easily deduced by the fact that 4 spin polarized electrons per iron atom should give us a moment of $4\mu B$, however the saturated magnetic moment of bulk iron is only $2.2\mu B$.

Question: Do all Iron atoms that are adjacent to this original atom mentioned above have 4 unpaired valence electrons with the same spin quantum number as this original iron atom OR does each iron atom form it's spin quantum # of these 4 unpaired valence electrons independently of one another?
Given that iron spontaneously orders ferromagnetically, the individual atoms absolutely do not select a spin arrangement (or spin quantum numbers) independent of one another. In order for the spins of these atoms to align, they must have some way to 'talk' to one another. Thus we again come to the conclusion that we cannot just think of magnetism in a material as a collection of independent atoms. The question now is how do these atoms talk to one another. There are two different extremes on how this communication takes place:

1) Local moment picture/Heisenberg model, (Insulators): Electrons are almost entirely bound to a particular atom, however there exists enough orbital overlap such that some hopping between neighbors can occur. Recall that for a 1D quantum box of length $L$, the energy is proportional $L^{-2}$, thus hopping allows electrons to effectively wander through a larger box, thereby reducing the kinetic energy. This means that the single-atom atomic orbitals you started with need to be replaced with multi-atom molecular orbitals. You often find that spin arrangements that break Hund's rule generate better hopping.
2) Itinerant moment picture/Stoner model, (metals): Electrons travel unbound through the material. In the far limit of an electron gas, there are no atoms at all and Hund's rules do not apply at all. Instead you would find that in a valence band half the electrons are spin up and half are spin down. However, if you applied a magnetic field then the energy of one spin direction would go up, while the other went down. This would result in a so called spin split band. It turns out that it is possible for an internal molecular field to spin split the band resulting in a favored spin state, this increases the molecular field further which in turn further spin splits the band and bootstraps a spontaneous ferromagnetic order.

Given that iron is a metal, the Stoner model is a more appropriate description. Indeed, the saturated moment is only $2.2\mu B$ because only electrons near the Fermi surface play a role in the magnetic ground state. However, it should be noted that many bulk properties of metals do not follow the Stoner model and this suggests that local moment correlations play a role. Indeed, the above two pictures are simple clean asymptotic solutions to magnetism in materials, in between exists a much more complicated world and somewhere inside this spectrum is where you will find most materials reside.

I assume in a magnetic domain of a small section of this Iron bar there would be Iron atoms all having their unpaired electrons with the same spin quantum number? This would give this particular domain the same magnetic field? Adjacent domains would have different spin quantum numbers, by chance, and for the entire material, and these would cancel out unless an outside magnet is applied.
Yes, a ferromagnet will break up into small domains. In a single domain the moments all alignment in the same direction which generates an external magnetic field. Other domains point in other directions and generate other magnetic fields. The sum of these fields all conspire to extinguish the total external magnetic field. However, note that you used the words 'by chance' whereas I used the word 'conspire'. This is because the material is actively trying to reduce the external field to zero, as opposed to just some chance random alignment of domains. This is because any external magnetic field will store a significant amount of energy. Thus, energy is saved by generating domains.

Could a domain be considered to be made from only one atom OR is it considered to be made from 1000s of atoms?
Domains will consist of many atoms. This is also due to energetics. Namely, in general it takes different amounts of energy to point spins along different directions in a material. The direction that takes the least energy is called the easy axis and the ferromagnetic alignment will be along this direction. Thus, you would expect domains to consist of spins that are 180deg rotated from one another (ie: they are always along this axis). It is possible to have more than one easy axis direction, but we need not consider that for the following argument. Within a domain wall you have spins that gradually twist from one domain direction to the other. This costs energy since these spins do not point along the easy axis. Thus, every domain wall will cost you energy, but this is counterbalanced by the fact that every additional domain reduces the external field energy. The total domain number is whatever generates the lowest energy between these two competing energy terms. Although calculating domain structure is extremely difficult, suffice it to say you can safely expect domains to be more than just one atom in size. Indeed, if this were the case I would be reticent to even call this a ferromagnet. Technically the site-to-site exchange coupling is always the same, so it would be a ferromagnetic. However, the end result would be more akin to something like an Ising spin glass where all spins are frozen into arbitrary directions, but with the a random site-to-site exchange governed by a magnetocyrstalline anisotropy.

#### Pallavi

There is a lot to unpack here given that magnetism in condensed matter is a complicated subject. However, I will step through your post and try to provide an outline for properly working through this.

The important thing to note here is that you are talking about a SINGLE atom and effectively applying Hund's rule to work out that the 4 valence electrons will all align parallel to one another. This is a perfectly fine starting point, however the end picture for most materials (made from an immense number of atoms) is quite different. This can be easily deduced by the fact that 4 spin polarized electrons per iron atom should give us a moment of $4\mu B$, however the saturated magnetic moment of bulk iron is only $2.2\mu B$.

Given that iron spontaneously orders ferromagnetically, the individual atoms absolutely do not select a spin arrangement (or spin quantum numbers) independent of one another. In order for the spins of these atoms to align, they must have some way to 'talk' to one another. Thus we again come to the conclusion that we cannot just think of magnetism in a material as a collection of independent atoms. The question now is how do these atoms talk to one another. There are two different extremes on how this communication takes place:

1) Local moment picture/Heisenberg model, (Insulators): Electrons are almost entirely bound to a particular atom, however there exists enough orbital overlap such that some hopping between neighbors can occur. Recall that for a 1D quantum box of length $L$, the energy is proportional $L^{-2}$, thus hopping allows electrons to effectively wander through a larger box, thereby reducing the kinetic energy. This means that the single-atom atomic orbitals you started with need to be replaced with multi-atom molecular orbitals. You often find that spin arrangements that break Hund's rule generate better hopping.
2) Itinerant moment picture/Stoner model, (metals): Electrons travel unbound through the material. In the far limit of an electron gas, there are no atoms at all and Hund's rules do not apply at all. Instead you would find that in a valence band half the electrons are spin up and half are spin down. However, if you applied a magnetic field then the energy of one spin direction would go up, while the other went down. This would result in a so called spin split band. It turns out that it is possible for an internal molecular field to spin split the band resulting in a favored spin state, this increases the molecular field further which in turn further spin splits the band and bootstraps a spontaneous ferromagnetic order.

Given that iron is a metal, the Stoner model is a more appropriate description. Indeed, the saturated moment is only $2.2\mu B$ because only electrons near the Fermi surface play a role in the magnetic ground state. However, it should be noted that many bulk properties of metals do not follow the Stoner model and this suggests that local moment correlations play a role. Indeed, the above two pictures are simple clean asymptotic solutions to magnetism in materials, in between exists a much more complicated world and somewhere inside this spectrum is where you will find most materials reside.

Yes, a ferromagnet will break up into small domains. In a single domain the moments all alignment in the same direction which generates an external magnetic field. Other domains point in other directions and generate other magnetic fields. The sum of these fields all conspire to extinguish the total external magnetic field. However, note that you used the words 'by chance' whereas I used the word 'conspire'. This is because the material is actively trying to reduce the external field to zero, as opposed to just some chance random alignment of domains. This is because any external magnetic field will store a significant amount of energy. Thus, energy is saved by generating domains.

Domains will consist of many atoms. This is also due to energetics. Namely, in general it takes different amounts of energy to point spins along different directions in a material. The direction that takes the least energy is called the easy axis and the ferromagnetic alignment will be along this direction. Thus, you would expect domains to consist of spins that are 180deg rotated from one another (ie: they are always along this axis). It is possible to have more than one easy axis direction, but we need not consider that for the following argument. Within a domain wall you have spins that gradually twist from one domain direction to the other. This costs energy since these spins do not point along the easy axis. Thus, every domain wall will cost you energy, but this is counterbalanced by the fact that every additional domain reduces the external field energy. The total domain number is whatever generates the lowest energy between these two competing energy terms. Although calculating domain structure is extremely difficult, suffice it to say you can safely expect domains to be more than just one atom in size. Indeed, if this were the case I would be reticent to even call this a ferromagnet. Technically the site-to-site exchange coupling is always the same, so it would be a ferromagnetic. However, the end result would be more akin to something like an Ising spin glass where all spins are frozen into arbitrary directions, but with the a random site-to-site exchange governed by a magnetocyrstalline anisotropy.
Amazingly detailed reply.Hats off to you

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