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In studying introductory atomic physics I have come across fine structure splittings in energy levels due to spin-orbit coupling. Which has a sub-structure called hyperfine structure which comes from a coupling of the nuclear spin and the total angular momentum.

My question is, is there any theoretical reason (as I assume there's no experimental evidence) that there's a sub-structure to the hyperfine levels in some atoms, in certain conditions?

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As a matter of fact the list of the corrections to the hydrogen atom goes on and on. This is a list of corrections to the hydrogen atom and their order of magnitude for comparison.

  1. Bohr energy, which is very rough version of the hydrogen atom $\sim\alpha^2m_ec^2$
  2. Spin orbit coupling (AKA Fine structure of hydrogen) $\sim \alpha^4m_ec^2$
  3. Hyperfine splitting $\sim \alpha^4m_ec^2 \left( \frac{m_e}{m_p}\right)$
  4. Lamb shift because of the vacuum fluctuations of the electromagnetic field, $\sim \alpha^5m_ec^2$

where $\alpha$ is the fine structure constant, $m_e$ and $m_p$ mass of electron and proton respectively and $c$ the speed of light. Notice that $\alpha \approx 1/137$ and $\frac{m_e}{m_p}\approx1/2000$. This should give an idea how the corrections get smaller and smaller. However with current technology we are able to experiment with much better precession so they are also experimentally confirmed. Furthermore I also believe that there are also further corrections but I personally don't know any of it.

At the very least you can do relativistic corrections „although the mean speed of the electron in hydrogen is only 1/137th of the speed of light” (Wikipedia) and with the development of quantum gravity I believe there will be other corrections.

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