Atoms in a magnetic field

Question

If we consider electrons circulating around the nucleus in circular orbits, they constitute current loops and hence have magnetic moments (due to their orbital motion). In a paramagnetic material, for example, those magnetic moments point everywhere so the net moment is zero. Turning on a magnetic field should produce a non-zero magnetic moment because it tends to align the magnetic moments with the field direction by producing torques on the orbits (or the current loops).

First of all, how accurate is the above as a description of paramagnetism?
Second, generally, the torque on a current loop would depend on the direction of the current. So the magnetic moments end up being either parallel or anti-parallel with the field. If this is correct, then shouldn't the application of the field have no effect on the net magnetic moment? If, before the field is on, the moments are randomly oriented in all directions, isn't it reasonable to expect that some of them would end up parallel to the field and others anti-parallel in such a way that they cancel out?
Third, why does the fact that "the electrons in diamagnetic materials are all paired up and their spins cancel out" matter? If we're considering orbital motion, why are the spins relevant? In other words, are the magnetic properties of such materials manifested due to orbital motion or spin? (at least for the large part, which one are they produced by?)
Fourth, following up on the last question, how does the "unpaired" electron in a paramagnetic atom contribute to its characterization as paramagnetic?

"If we consider electrons circulating around the nucleus in circular orbits" Except of course: they don't. See https://en.wikipedia.org/wiki/Atomic_orbital — Gert, Oct 13 '21 at 23:32
@Gert well yeah but that doesn't mean there's nothing we can get out of thinking about it in this way. Plus, how, then, do we explain the nature of various kinds of magnetic materials, whether the electrons are "actually" circulating or not? — EM_1, Oct 13 '21 at 23:46
@user626542 by the algebra of angular momentum and spin in quantum mechanical terms. https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Book%3A_Introduction_to_Inorganic_Chemistry/06%3A_Metals_and_Alloys-_Structure_Bonding_Electronic_and_Magnetic_Properties/6.07%3A_Atomic_Orbitals_and_Magnetism — anna v, Oct 14 '21 at 03:54
"@Gert well yeah but that doesn't mean there's nothing we can get out of thinking about it in this way." Trouble is that it is very, very outdated thinking. 'Circular orbits' is basically the long defunct Bohr model, replaced by solutions to the Schrodinger equation. — Gert, Oct 14 '21 at 07:54

score 1 · Answer 1 · answered Oct 13 '21 at 23:50

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Orbital magnetic moments do exist, but that is not the main contributor to macroscopic magnetism. The truth is, each and every electron by itself is a dipole magnet. A lot of people don't realize this. So these "intrinsic magnetic moments," which are aligned with spin direction, have to line up. This video does a better explanation that I can do here https://youtu.be/hFAOXdXZ5TM

answered Oct 13 '21 at 23:50

RC_23

9,096

Atomic orbital magnetic responses are important in non-magnetic materials. Most of the macroscopic diamagnetism comes from it. – Mauricio Oct 18 '21 at 12:35
What prevents us from considering the magnetic dipole of subatomic particles to be as fundamental as the charge: RC_23, could you take a look at this? – HolgerFiedler Oct 19 '21 at 03:32
And there is another question Is the magnetic dipole moment in charged particles the more survivable property than the electric charge? – HolgerFiedler Oct 19 '21 at 03:35

Roger V. · Answer 2 · 2021-10-18T11:52:37.833

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Electrons in solid state possess orbital and spin magnetic moments - the former could be roughly described by the first paragraph in the OP (although any non-quantum description applied to microscale is incorrect, strictly speaking), whereas spin has no clasical analog.

The orbital momentum, behaving as a current loop, tends to counteract the external magnetic field, thus resulting in diamagnetic response. On the other hand, spins align with the magnetic field, producing paramagnetic response. In some materials the amount of electrons with spins of different orientation may be different even without external magnetic field applied, in which case we talk about ferromagnetism.

Whether paramagnetic or diamagnetic response dominates, depends on the specifics of material. Usually, the diamagnetic materials have all the states in their shells pair, so that re-orientation of spins along the magnetic field is prohibited. If this is not the case, the paramagnetism may win over.

Update
Here is a good answer to a similar question.

edited Oct 18 '21 at 11:52

answered Oct 14 '21 at 11:58

Roger V.

58,522

This is technically not completely accurate. The contribution from orbital angular momentum can also be paramagnetic. The Zeeman like term $\vec \mu\cdot \vec B$ takes into account both orbital and spin angular momentum and leads to a paramagnetic response. – Mauricio Oct 18 '21 at 12:33
@Mauricio thanks for this clarification. I have very much liked your answer to another question - which is why I linked it here. – Roger V. Oct 18 '21 at 12:47
Forces due to magnetic fields tend to align magnetic moments with the field, right? But the magnetic moment points from south to north. So by aligning with the field, it's counteracting it. Is that what you mean? If so, then why wouldn't spin magnetic moments aligning with the field cause negative magnetization too? – EM_1 Oct 20 '21 at 18:57
Why does it matter what kind of magnetic moment we have? It seems like orbital magnetic moments and spin magnetic moments respond to external fields in different ways and I still don't really understand why that is. – EM_1 Oct 20 '21 at 18:59
@user626542 spins are essentially magnetic dipoles with constant mzgnetic moment, whereas orbital magnetic moment is like current loops - it changes in response to magnetic field. – Roger V. Oct 20 '21 at 19:07

Mauricio · Answer 3 · 2021-10-18T13:09:28.083

Electrons orbiting atoms have two types of angular momentum. One is orbital angular momentum (the classical analog of a particle orbiting the nucleus) and one intrinsic spin angular momentum, associated also to an intrinsic magnetic moment.

Both types of angular momentum sum up to produce a total magnetic moment $\boldsymbol \mu = \frac{\mu_{\rm B}}{\hbar}(\mathbf L+g\mathbf{S})$, where $\mu_{\rm B}$ is the intrinsic magnetic moment of the electron (Bohr's magneton) and $g$ is the $g$-factor. This total magnetic moment couples to the magnetic field $\mathbf B$ through the Zeeman interaction as

$$H_{\rm p}=- \boldsymbol \mu\cdot B$$

and leads (in general) to a paramagnetic response (coming from both orbital and intrinsic angular momentum).

Yet this effect can be opposed by the Larmor precession: as you turn on a magnetic field, the current associated to the motion of the electron about an atom will try to induce a counter magnetic field. This is called Larmor diamagnetism and it is of the form $$H_{\rm d}\propto B^2\langle r^2\rangle $$ where $\langle r^2\rangle$ is the mean radius of the orbit squared. You can think of it as the action of Faraday-Lenz's law.

In each material these two terms compete. Some materials will be diamagnetic and some will be paramagnetic. Your description above is missing the diamagnetic term.

If the magnetic moments are randomly oriented, by turning the magnetic field you will have them orient in the direction of the field. Some will indeed point against the field, but energetically many will prefer to point in the direction of the field (while the orbital part will contribute diamagnetically always).

If you have an atom with an unpaired electron, this electron has an extra spin that can orient in the direction of the field and provide a paramagnetic contribution, but this does not assure you that the total response will be paramagnetic (see for example the case of bulk noble metals, which are diamagnetic and have unpaired electrons).

Atoms in a magnetic field

3 Answers3