The Blog is taking a detour into “fine structure” – a term that refers to the fine structure of emission spectra, but that has come to mean the progressive approximation of more and more of the relativistic and perturbative elaborations in the derivation of spectral emission features due to atomic electron and even muon interactions. This area also maybe eventually touches a little on the recent hot topic of the gyromagnetic ratio in the perturbative expansion of muon interactions and the possible implications of the need to add something to the Standard Model to account for observed small deviations from the expected ratio. So we will see how far this goes. The plan is to show how the early Bohr-Sommerfeld model for “fine structure” is the basis of this set of progressive approximations and then to emerge to Einstein’s 1917 description of photon emission and absorption and from there to the BKS model of emission and absorption and Compton’s role in supporting Einstein’s photon model using cloud chamber techniques in 1925. This should position us to reach the realm of the mesons fairly soon.

These days, in observations that use the
analysis of emissions to learn about distant objects or even the aurora
borealis, all aspects of fine structure (ie the splitting or intensity of
emission lines or wavelengths) can be summarized by four “quantum numbers” – a
term that goes back to the original Bohr-Sommerfeld atomic model and the
associated traditional descriptions of spectra.
For example, if we look at *An Introduction to Cosmochemistry*, a
textbook from 1995, we find on page 231 that “a single electron may be
described by four quantum numbers” (given variously as n, l, m with subscripts and j depending on
other factors). “n and l stand for the
principle quantum number and the angular momentum quantum number of orbital
motion,” which, again, is a picture or description based on Sommerfeld’s
revision of Bohr’s model in 1916.

So far so good. The structures that are “fine” are the fine
splitting of the spectral emission lines (first noted by Michelson and Morely
in 1887). So for example, (as is noted
in the Haken and Wolf textbook on page 112 and in *The Origin of Spectra* on page 25), the Hydrogen emission line at about 6564 Angstroms (a visible, if
reddish, “Balmer line”) is actually, at a fine enough resolution, six lines
(though 3 are very faint if present at all because they are unlikely or
“Forbidden” transitions according to the “selection rules”) very close
together. The other transitions are
between other electron states and should be reflected with single emission
lines according to the original Bohr mode because the energies involved are the
same. So, same energy and same
transition but a set of emission lines rather than just one. Well, why?
As Sommerfeld asked himself in 1916.

The answer (at least in terms of current introductory texts, the world of 1916 and a set of approximations good enough for astrophysical spectral analyses) is that electrons with the same energies may be traveling at different speeds at different points in their orbits and, due to relativistic effects, this results in different emissions. In the original Bohr-Sommerfeld model, this was imagined as the precession of electrons in elliptical orbits and the associated “quantum number” was thought to reflect the minor semiaxis of the elliptical orbit. Of course, one isn’t supposed to seriously imagine that atomic electrons travel in orbits, but the image is so useful for basic approximations that it has remained as part of the imagery of electrons for over a century. Anyway, here is the description of this first level of approximation in two pages from *The Origin of Spectra* in 1922: