More Fine Structure

Still taking a bloggy 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.  Just as Foote and Mohler did in 1922 when writing The Origin of Spectra, we are going to avoid discussing the Stark and Zeeman effects which are due to the impact of external fields.  With fine structure we are staying pretty much inside the energies of the atom itself.

We have noted that one way of visualizing and approximating what is happening with fine structure is to see the energies of the electron as being distributed in an elliptical orbit.  This orbital imagery seems to work fine for uses in current astrophysics more than a century after Sommerfeld’s introduction of the notion of a relativistic interpretation of precessing elliptical orbits.  Moreover, as Kragh points out in his book on Bohr, in working out the relativistic correction that produced the series of spectral lines, Sommerfeld introduced the fine structure constant as the measure of this relativistic correction – the ratio of the velocity of the electron in the first circular orbit of the Bohr model to the speed of light.  Of course, since, these days, the Bohr orbits have no fundamental explanatory role, the fine structure constant is demonstrated or derived in other ways.

In introductory textbooks, it is generally accepted that imagining electrons as spinning spheres following various orbits is inevitable and not a problem as long as the one doing the imagining bears in mind that their picture is not the whole story of what is happening with an electron in an atom.  No one seems to think of the visual construct of electrons in concentric “shells” around the nucleus is a problem at all, at least in terms of the imagery involved.  On the other hand, for one-electron atoms and ions at least, after 1925, the Schroedinger equation and Born’s probability interpretation offered an image of the electron as a pulsating sphere of probability centered on the nucleus.  In Photons and Electrons  (1950), K. H. Spring, notes the significance of the electron shells as shown in X-ray spectrography and describes the shells (or modernized versions of Bohr’s orbits) as spherical modes of vibration that give the probable relation of the electron and the nucleus.  

But the paradise of the circular orbits, precessing ellipses and pulsating spheres was not to last very long before better approximations incorporating electron spin arrived in the late 1920s.  First, Charles Darwin’s grandson, Charles Galton Darwin worked out two small situations that had some impact on electron energies.  Rather confusingly these small shifts are referred to as non-relativistic expansions of the Dirac equation which was derived at about the same time, but all these expansions, approximations and corrections have tended to become pretty confusing so that by the late 1940s, when Foldy and Wouthuysen were resolving some issues with the use of the Dirac equation, they felt a need to explain they were not talking about the non-relativistic impacts or the relativistic aspects of Zitterbewegung  (zig-zagging electrons) or the Lamb Shift (the first experimental work suggesting a focus for the need for some explanation via perturbative quantum electrodynamics), but about the Dirac equation in a general way – though the Foldy-Wouthuysen transformation of the Dirac equation is sometimes seen as a “heuristic” explanation of Zitterbewegung  and the Darwin Term.  Anyway, we are still pretty much in 1922 in this blog so we won’t be getting to the Dirac equation until after some more fine structure, more puzzling over the photon, the Klein-Nishina equation and the Schroedinger equation, at the very least.  So, more fine structure:

Griffiths, in his Introduction to Quantum Mechanics (many editions after 2004), organizes these extended approximations by powers of a, the small structure constant.  Since the constant is one over 137 or so, increasing the power makes it much smaller, so a squared is about 1/137 times 1/137, a relatively small number.  So the fine structure corrections (Sommerfeld’s and Darwin’s) are at about a to the fourth power and the Lamb Shift is at about a to the fifth power.  These days, Sommerfeld’s correction is referred to as the relativistic correction and Darwin’s corrections are split into spin-orbit coupling and the Darwin Term.

Napolitano, in his most recent revision (2021) of J. J. Sakurai’s Modern Quantum Mechanics, acknowledges that the best way to actually resolve all these approximations and corrections is to use computing “to obtain the desired solution numerically to the requisite degree of accuracy” – which might well be more accuracy than anyone needs for say, astrophysics.  He runs through fine structure on his way through approximation and on to the Dirac equation, leaving the “Darwin Term” to the side, which is sensible since there are a lot of ways of describing what is happening with that term.  One is Zitterbewegung which seems to have become slightly controversial recently, not sure why exactly – maybe the Foldy-Wouthuysen transformation has become a bit too “historical” (to quote Wikipedia).   Anyway, next post we will look at least a little into that possibly excessive rendering off and away unto old heuristic history of the Foldy-Wouthuysen transformation and the Darwin Term.