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Asking a question Has anyone tried to incorporate the electrons magnetic dipole moment into the atomic orbital theory?, I was curious whether anyone has attempted to relate the intrinsic property of the magnetic moment of the electron to the above-mentioned properties of spin.

In the extremely detailed answer (thanks to the author, who took the time despite the pointlessness of such a question) it is clarified that

The effects are weak, and they are secondary to all sorts of other interactions that happen in atoms,...

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.

Now there is a new question about Electron to electron interaction.

There is a critical distance

$$d_\text{crit}=\sqrt\frac{3\epsilon_0\mu_0\hbar^2}{2m^2}=\sqrt{\frac{3}{2}}\frac{\hbar c}{m}=\sqrt{\frac{3}{2}}\overline\lambda_C,$$

where $\overline\lambda_C$ is the reduced Compton wavelength of the electron, at which the two forces are equal in magnitude.

Since the Compton wavelength is a standard measure of where quantum effects start to be important, this classical analysis can't be taken too seriously. But it indicates that spin-spin interactions are important at short distances.

I wonder how these two points of view can be related.

HolgerFiedler
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1 Answers1

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They can be related by the fact that the Bohr radius of hydrogen is $1/\alpha\approx 137$ times larger than the reduced Compton wavelength of the electron. (Here $\alpha$ is the fine-structure constant. For helium, divide by 2 to get 68.5.) At this large a separation between the proton and electron, the magnetic interaction that I calculated is small compared to the electrostatic interaction.

G. Smith
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  • May you expand your answer to the electron-electron interactions in atoms? – HolgerFiedler Aug 19 '19 at 06:18
  • That's a messy subject that I don't want to get into. – G. Smith Aug 19 '19 at 06:21
  • Why it is a messy subject. Will I get an answer asking this on PSE? – HolgerFiedler Aug 19 '19 at 06:32
  • intuitively it is all due that there are no magnetic monopoles, and the magnetic dipoles are a higher order effect, whereas the Bohr orbits come from first order effects. – anna v Aug 19 '19 at 06:49
  • @annav So there is no chance to calculate the interactions of a model of electrons magnetic dipoles - say for Ne - numercally? – HolgerFiedler Aug 19 '19 at 07:28
  • look, this has been done for hydrogen lines , i.e the interaction of the spins proton/electron. http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydfin.html . For nuclei there is hyperfine structure https://en.wikipedia.org/wiki/Hyperfine_structure which has to take into account the nucleus + the electrons. It is not a simple problem – anna v Aug 19 '19 at 08:20
  • @Holger Those numerical calculations are referenced in the very passage you quote in this question. (Scaling up to neon just makes the problem numerically harder and it reduces the scientific gain that it might offer, but it's obviously the same calculation.) If that's the level of analysis that you're going to apply to this site's responses, why should anyone here spend their time to answer your questions? – Emilio Pisanty Sep 04 '19 at 22:23