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The Pauli exclusion principle is related to the electrons spin. The quantum numbers are complete only with the spin quantum number. I am curious whether anyone has attempted to relate the intrinsic property of the magnetic moment of the electron to the above-mentioned properties of spin.

If there aren’t sources, I have an additional question. How one imagine the orientation of the magnetic dipoles of the electrons in each shell? If one does not deny these dipole moments (and there is no reason for that?) then we would also have to deal with their effects.

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    $\begingroup$ It's only been nearly a hundred years. Look up explanations of the fine structure and hyper-fine structure. $\endgroup$ – dmckee Jul 29 at 19:57
  • $\begingroup$ @dmckee That is how a magnetic field influence the atomic structure. But how the electrons magnetic dipoles are oriented inside the atom? $\endgroup$ – HolgerFiedler Jul 29 at 20:13
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    $\begingroup$ The two questions are inseparable. Where do you think the energy differences come from? $\endgroup$ – dmckee Jul 29 at 20:15
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    $\begingroup$ Most textbooks on atomic physics cover this. e.g. there are good ones by G. K. Woodgate and by C.J. Foot. $\endgroup$ – Andrew Steane Jul 29 at 21:35
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    $\begingroup$ Holger if you think the standard texts don't cover this then you haven't understood them. Calculation of the fine and hyper-fine structure are all about the interaction of magnetic moments of various parts of atomic systems with fields from other parts of the system. $\endgroup$ – dmckee Jul 30 at 15:39
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To be crystal clear:

Has anyone tried to incorporate the electrons magnetic dipole moment into the atomic orbital theory?

YES. They've tried and they've succeeded. The electron's spin magnetic dipole is a standard part of atomic physics and quantum chemistry. Anybody attempting to claim that this isn't the case is simply describing their own ignorance about atomic physics rather than the subject as it is known.

Also, to be crystal clear:

  • The effects are perfectly well-known and have been described for the past 80+ years, but still
  • The effects are weak, and they are secondary to all sorts of other interactions that happen in atoms, including:

    • The electrostatic interaction between the electrons and the nucleus
    • The kinetic energy of the electrons
    • The electrostatic interaction between the electrons
    • The Pauli exclusion principle
    • The coupling between the (spin-induced) magnetic dipole moment of the electron and the magnetic dipole moment associated with the orbital motion of the electron (a.k.a. spin-orbit coupling)
    • The relativistic effects associated with the kinetic energy of the electrons, particularly in inner shells of large atoms
    • The effects caused by coupling to the QED vacuum (a.k.a. the Lamb shift)
    • The nuclear recoil from one electron's motion affecting the other electrons
    • The interaction between the electrons' (spin-induced) magnetic dipole moment and the magnetic dipole moment of the nucleus
    • The strict constraints imposed by the quantum mechanics of angular momentum on how two different quantized angular momenta can relate to one another, as applied to the relationship between spins and orbital angular momenta as well as to the spins among themselves

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    Essentially everything in that list takes precedence to spin-spin interaction via the electrons' magnetic dipole moments, particularly if you're looking for effects on the shape of the spectrum instead of just tiny shifts in the energies of the different levels. "The effect exists" and "the effect is well-understood" do not lead to "the effect matters".

And also, to be crystal clear: the existing theory of atomic physics agrees perfectly with experiment, often predicting the energies of atomic levels to eighteen significant figures (and climbing, as theoretical and experimental precision increase). If you have an alternative theory or you think that the calculations have been done incorrectly, then you need to be sure that your calculations reproduce the existing experimental data (say, this, matching to these calculations) ─ all of it, to its full eighteen significant figures where that is the experimental precision.


A quick note before moving on: full-blown atomic physics is a highly technical subject, and conciseness in technical communications here (as in every technical field) is at a very steep premium. As regards the magnetic interactions of electrons, in particular, this means that nobody in the technical literature is going to say "intrinsic magnetic dipole moment" where saying "spin" will suffice. That's the case for electrons' intrinsic magnetic dipole moment, which is always proportional to their spin.

This means that, if you want to play in the big leagues, you need to get behind the usage of the term "spin" as synonymous with "intrinsic magnetic dipole moment", as in e.g. "spin-spin coupling" and "spin-spin interaction", the technical terms for the interactions that you repeatedly (and incorrectly) claim are not considered by the literature. If that usage of terminology bothers you, then tough luck.


Now, the final item on the list above is particularly relevant to your second question:

How one imagine the orientation of the magnetic dipoles of the electrons in each shell?

Here the answer from QM is simply: in complicated ways.

Angular momentum in QM is complicated, particularly because its components are incompatible (i.e. do not commute) with each other, which means that the direction in which the spin is pointing is not something that QM allows its frameworks to talk about. (Yes, there are questions for which QM explicitly dictates that there are no answers available. Deal with it.)

In particular, that means that the relative orientation between any two electron spins in an atom (and therefore the relative orientation between their magnetic dipole moments) is a question that doesn't have an answer within QM. If that bothers you, go somewhere else.

To be more specific, there are two core (fatal) roadblocks in attempting to build an answer to the question of the relative orientation of the spins in an atom.

  • The first is that the only well-defined quantity for the entire shell is the total spin angular momentum. This is a consequence of how the addition of angular momenta works in QM, again as a result of the incompatibility of the different components of angular momentum, and it is a completely-well-established framework that is explained in detail in any QM textbook worth its salt. (With an additional spanner in the works coming from the requirement for wavefunction antisymmetry coming from the Pauli exclusion principle, I should add.)

    This means that, say, for the six electrons in the $2p$ shell in neon, the only quantities that have well-defined values are the magnitude of the total spin, $S^2$, and one component of the total spin, normally taken to be $S_z$. Everything else (including all of the components of the spins of all the individual electrons) generally doesn't have a value.

  • The second roadblock is the fact that the energy scales for spin-spin interactions are so much lower with the key drivers of atomic structure (namely, the kinetic energy of the electrons, and their electrostatic interactions with the nucleus and with each other), which means that the spin and orbital sectors of the quantum state decouple. This means that the total quantum state factors out to the form $$|\Psi\rangle = |\text{spatial dependence}\rangle \otimes |\text{total spin state}\rangle,$$ with a global spatial wavefunction governing the relative positions of the electrons in the atom that is completely decoupled from the spin state. Or, in other words, to the extent that each spin has an orientation (i.e. not very $-$ they are all essentially in a superposition of a wide range of orientations), this is independent of where that electron is in relation to the others.

    This has important implications to the question of how big of a player the electron spin-spin magnetic interaction is in atomic structure, because the magnetic dipole-dipole interaction, which has the form $$H = K \left[\frac{\mathbf s_1\cdot \mathbf s_2}{r^3}-3\frac{(\mathbf s_1\cdot \mathbf r)(\mathbf s_2\cdot \mathbf r)}{r^5}\right]$$ for $K$ a constant and $\mathbf r$ the relative position of the two spins, is sensitively dependent on the relative orientation of the spins with respect to the spins themselves. (Intuitively: two magnets whose orientations that are frozen in space might attract or repel, depending on how they're positioned relative to those orientations.) In atoms, however, the relative orientations are averaged over, as a consequence of the separation between the spatial and spin sectors of the wavefunction, which further weakens the interaction.

    Here it is important to emphasize that if the electron spin-spin interaction were strong enough to overwhelm the electrostatic components of the dynamics, then it would be conceivable that this separation would break down, and indeed it does break down in heavy atoms. This process is known as a change in angular momentum coupling scheme, from $LS$ coupling to $jj$ coupling, where each electron can have its own total angular momentum (with the individual electrons' angular momenta then further coupled into a single total atomic angular momentum), and it leaves clear traces in the spectrum. However, this is basically always driven by spin-orbit coupling, with spin-spin interaction being at best a minor contribution.


In any case, just to provide a specific reference that gives electron spin-spin interactions the full works of the attention of atomic physics, here you have one:

Mutual spin-orbit and spin-spin interactions in atomic structure calculations. M Jones. J. Phys. B: At. Mol. Phys. 4, 1422 (1971)

There's plenty more where that came from if you know how to search the literature (and if you don't, then to be frank the material is too technical and you should be reading textbooks until you can), though it does seem that these explicit non-relativistic hamiltonians seem to have been dropped in the more recent literature on the theory of precision spectroscopy, due to the requirement of full-blown quantum field theoretic calculations.

(Also, in case you're wondering just how weak: this paper calculates the energy shifts coming from electron spin-spin coupling for a range of two-electron systems. The largest is in helium, for which the coupling energy is of the order of $\sim 7 \:\mathrm{cm}^{-1}$, or about $0.86\:\rm meV$, as compared to typical characteristic energies of $\sim 20\:\rm eV$, some five orders of magnitude higher, for that system.)

In any case, Eq. (4) in Jones 1971 details the interaction hamiltonian for the process that you asked about, together with other similar contributions in Eqs. (2) and (3), and all of these are worked out in their full glorious detail in the ensuing pages. Does it all look like one big, unenlightening mess? well: boo hoo. Of course it does. The interaction is weak, as compared to the real movers-and-shakers of atomic structure, and it is only worth studying if you're working towards precision spectroscopy with the full strength of the formalism. Most of this will be illegible if you haven't spent significant lengths of time with textbooks and doing their exercises, but that's just the nature of technical communication.

And then again, we've been telling you that there is no going around the need of reading textbooks if you want to understand the physics for years on end, and you've roundly ignored us, so if you don't understand the primary literature, that's why.

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