Crystal field theory, what is spin?

[billiards]

Can anybody explain this thing that is "spin" to me?

I basically came across it whilst researching how we know about the deep earth. Mineralogists and crystallographers talk about this thing called spin, quantum physicists also talk about spin; yet as far as I am aware they are they are not talking about the same thing.

From what I understand it has something to do with the electron spin states. Electrons can orbit different spin states in an atom, which changes some properties of the atom? So at the base of the mantle if the perovskite (Mg,Fe)(Si,Al)O₃ has a "low" spin state, it changes the electrical and thermal conductivity as well as some chemical properties.

So is spin something to do with angular velocity? I mean, if the electrons were in a higher spin state and had more angular momentum, wouldn't they be more likely to fly off or something?

Please, I hope someone out there relates to this and can explain it to me, it would be really good if I could actually understand this as I'll be studying it in more detail next term and its assumed knowledge.

Cheers

 

[Claude Bile]

As far as I know, and I am by no means an expert - In a geological context, 'spin' is likely to refer to the alignment of ferromagntic filaments in minerals, geologists can use the alignment of these filaments to calculate the Earth's magnetic history. The term 'spin' comes from the fact that it is an intrinsic magnetic dipole that causes the response of these filaments to an external magnetic field, and classically, one obtains a magetic dipole by rotating a charged particle. The term 'spin' though is used to denote the presence of a magnetic dipole, rather than any physical rotation occurring.

Electron spin is slightly different, it refers to the intrinsic dipole of an electron, rather than whole atoms/groups of atoms as was referred to in the previous case. The spin of an electron can only be up or down with a fixed magnitude (h-bar/2 from memory), which corresponds to the intrinsic dipole aligning up or down in comparison to some external reference.

In reference to 'low' spin states and so on, this type of thing is starting to creep outside my range of knowledge, however I would venture that the 'low' spin is in reference to the intrinsic dipole of the atom or molecule, rather than the indivdual electrons that occupy the various orbitals.

So to sum up, there are two types of 'spin' in the picture, atomic spin and electron spin. Spin in this context likely refers to the presence of a magnetic dipole, rather than actual physical rotation of the particle.

Note my many disclaimers through my post! I would encourage you to verify this through your own reading on the topic.

Claude.

 

[billiards]

Yeah we had a discussion about it with an '"expert" in uni. He said the spin didn't change magnetic properties. The main implications of a change in spin were to do with thermal heat transfer and rheology. I can't remember but I think a low spin state allows less radiation and conduction which might cause convection to occur, this is important in understanding geodynamical models.

 

[Gokul43201]

Actually, the minerologists, crystallographers and quantum physicists are talking about the exact same thing.

The thing you need to learn here is a set of rules of thumb called the Hund's Rules, which help you determine how electrons occupy energy levels to make the ground state of a substance.

In transition metal perovskites (and more generally, in all transition metal salts) like the aluminosilicates you mention, the valence electrons of the transition metal elements come from the d-orbitals. Now in an isolated atom, these d-orbitals have a five-fold degeneracy (there are 5 d-orbitals of equal energy), but when you put these atoms in a crystal, the loss of symmetry from the environment (eg: there's O-atoms above and below an Mg atom, but Al-atoms lying along the same plane) results in a breaking of this degeneracy. What happens typically, in an orthorhombic crystal geometry is that the differences in the fields acting from different directions (hence the name "crystal field effect") due to nearby atoms causes the energy of the d-orbitals to split into two different levels (labeled $e_g$ and $t_{2g}$).

To find the ground state configuration, you take the valence electrons and fill them into these levels appropriately. If the crystal field is strong, the energy difference between the split levels is large. In that case, it is energetically more favorable to put as many electrons into the lower energy level before you start to fill the higher energy level (if you have to). Filling all the orbitals in the lower level means you have to pair them up (within each orbital) with opposite spins (Pauli Exclusion Principle). The effect of pairing electrons with opposite spin makes no addition to the total spin (the two opposite spins "cancel off"). This results in a low-spin state.

If the crystal field is weak, however, the energy difference between the levels is small. In that case, it becomes more favorable to fill all the orbitals (first the lower ones, then the higher ones) with just one electron (no pairing involved yet), and then, if you've got more electrons to put in, you start pairing up. Each unpaired electron contributes to the total spin, and you end up in a high-spin state.

So, to recap, there's two energy terms of interest:
(i) the crystal field energy ($\Delta$) is the energy difference between the doubly-degenerate and triply degenerate d-levels created by the environment;
(ii) the pairing energy - you can think of this as a spin-spin interaction energy or a simple Coulomb energy between electrons in the same orbital.
Since the aim of the game is to minimize energy, you chose pairing if the crystal field energy is large compared to the pairing energy (making a low spin state), and you chose unpairing, if the crystal field energy is sufficiently small (making a high spin state).

 

[billiards]

Thanks Gokul, I'll reread that tomorrow to try and digest it all then, but before I go to bed I'll just pick you up on one thing. You said that the crystallographers were talking about the same thing as the quantum guys, but when the quantum guys talk about spin isn't that an intrinsic property of a particle. For example I know that all electrons have spin 1/2, according to Hawking's book that means you have to spin it round twice before it'll look the same. How is this the same thing as the spin in crystal field theory?

 

[Gokul43201]

Thanks Gokul, I'll reread that tomorrow to try and digest it all then, but before I go to bed I'll just pick you up on one thing. You said that the crystallographers were talking about the same thing as the quantum guys, but when the quantum guys talk about spin isn't that an intrinsic property of a particle.

Yes, it is.

For example I know that all electrons have spin 1/2, according to Hawking's book that means you have to spin it round twice before it'll look the same.

Yes, both of these are true.

How is this the same thing as the spin in crystal field theory?

Any spin-1/2 particle can have two values of angular momentum (along some chosen direction), +1/2 or -1/2 (units of hbar). When you have a system of many particles (like the valence electrons in an atom), you can calculate a total spin-angular momentum from each of the individual spins. This then gives you the total spin-angular momentum of the atom (ignoring nuclear spins). The "high-spin" and "low-spin" labels refer to this total spin.

To really understand this a little better, you need to learn the basic ideas behind atomic structure. Any standard Modern Physics (eg: Tipler) or Physical Chemistry (eg: Atkins) text will cover this.

On spin: http://hyperphysics.phy-astr.gsu.edu/Hbase/spin.html

 

[pervect]

As far as spin goes, I would suggest reading about the Stern-Gerlach experiment. Spin is something that can be measured. It is highly non-classical though. When you measure the state of spin of an object (such as the silver ions in the Stern-Gerlach experiment) you find that the spin can have one of two values, usually called UP and DOWN.

This is quite different from the classical behavior of a spinning object - which can have any magnitude of spin, along any axis. One usually pictures spin by a three-dimensional vector, which represents the axis of the spin.

An unmeasured spin can, according to the Rules of Quantum Mechanics, be regarded as a superpositon of |up> and |down> "wavefunctions".

What happens from there is highly mathematical and not at all intuitive, unfortunately. But the mathematical quantum-mechanical description of spin, described by the complex amplitude of the |up> and |down> components (i.e by two complex numbers) can be put into a 1:1 correspondence with the usual non-classical idea of spin as being represented by a spin axis, which requires three real numbers. This allows quantum mechanics to approach classical mechanics in the "classical limit" of large objects.

 

[Petar Mali]

You don't measure spin. You measure magnetic moment. That is historical confusion. You don't needed spin you can always talk about magnetic moment.

 

[DrDu]

You also can measure spin, e.g. in the Einstein de Haas Experiment.

 

[alxm]

You're responding to a three year old thread.