The line that is really two

A sodium street lamp emits one colour, and the colour has a name and a wavelength: 589 nanometres, the D line. Put its light through a good enough grating and there are two lines, at 588.995 and 589.592 nanometres.

The gap is 0.597 nanometres, which is one part in a thousand. It is not a defect of the lamp, not a Doppler effect and not an artefact of the grating. It is the atom.

What the gross structure leaves out #

The ladder that produces a spectrum is built on one number per level.

That calculation assumes three things that are each slightly false, and each falsity is of the same size.

The first two of them are about the electron itself. It is not a point sitting at a radius: what replaced the Bohr orbit is a probability distribution, and every quantity below is an average over one.

The electron is not slow. In the ground state of hydrogen its speed is $\alpha c$, about one part in 137 of the speed of light. The kinetic energy is therefore not exactly $p^2/2m$; the next term in the expansion is smaller by $(v/c)^2 = \alpha^2$.

At $\beta = 0.0073$ — the innermost electron of hydrogen — the relativistic and non-relativistic energies are indistinguishable on any axis that shows both. That is why fine structure is fine: the corrections enter at $\alpha^2$, about one part in twenty thousand, and everything about the gross structure survives them intact.

The electron has a magnetic moment. It carries an angular momentum that is not the angular momentum of anything going round, and a magnetic moment with it.

There is no continuum of orientations; there are two, and the atom’s energy in a magnetic field is different for each. That two-valuedness is what supplies the moment doing the splitting — and it is the point at which the classical account of fine structure stops working entirely, because a classical angular momentum would give a continuum of energies and a smeared line rather than a doublet.

And the electron is moving through an electric field. In its own instantaneous rest frame the nucleus is going round it, and a moving charge is a current.

A circulating charge is a magnetic dipole producing a field along its axis, and from the electron’s point of view the proton makes exactly such a loop. The field at the electron’s position is therefore of order a tesla — enormous, and produced by the atom on itself — and the doublet is the electron’s own moment sitting in it at two orientations.

The three terms, and why one of them is not enough #

Each of the three gives a shift of order $\alpha^2$ times the level’s energy, and they are usually named separately: the relativistic kinetic correction, the spin–orbit coupling, and the Darwin term, which affects only states that reach the nucleus.

Calling fine structure “the spin–orbit interaction” is the standard shorthand and it is wrong in a way that matters, because spin–orbit alone does not reproduce the observed pattern. The three together do something none of them does alone: the answer depends on the total angular momentum $j$ and not on how it was assembled. A 2s state with $j = \tfrac12$ and a 2p state with $j = \tfrac12$ come out at exactly the same energy, though the first has no orbital angular momentum at all and the second has one unit of it.

$$\Delta E_{n,j} = -\frac{Z^2\,\mathrm{Ry}}{n^2}\,\frac{(Z\alpha)^2}{n^2}\left(\frac{n}{j+\tfrac12} - \frac{3}{4}\right).$$

That is Dirac’s result, from a theory in which spin is not added by hand but forced by requiring the wave equation to be first order in time and to respect relativity. The three terms of the older treatment are what its expansion looks like, and their conspiring to depend on $j$ alone is a sign that the split into three was an artefact of the expansion rather than a feature of the atom.

For $n = 2$ the gap between $j = \tfrac12$ and $j = \tfrac32$ comes out as $\mathrm{Ry}\,\alpha^2/16$, which is 45.3 microelectronvolts, or 10.9 gigahertz. Measured: 10,969 megahertz.

Why sodium’s is four orders of magnitude bigger #

Hydrogen’s fine structure needs a radio-frequency measurement. Sodium’s is visible with a school spectrometer, and the difference is a factor of forty thousand in the relative splitting.

An s state has a finite probability density at the nucleus and a p state has none, so the two sample the nuclear charge quite differently. In sodium the outer electron spends part of its time inside the closed shells, where it sees a charge far larger than the screened one — and since the splitting goes as the fourth power of the effective charge, that penetration is what makes sodium’s doublet four orders of magnitude wider than hydrogen’s.

There is a second factor of the same kind. The spin–orbit energy depends on the field the electron sees, and that field is the nuclear electric field seen from a moving frame — so it grows as the charge and falls steeply with distance. An electron that spends its time far out sees almost nothing; an electron that dips in sees a great deal, briefly, and the average is dominated by the dips. Two states of the same energy can therefore have quite different splittings if one penetrates and the other does not, which is why sodium’s p states are split and its d states are barely split at all.

The splitting goes as the fourth power of the effective charge the electron sees, and the outer electron of sodium — nominally seeing $Z_{\text{eff}} = 1$ far out — dips into the core, where the eleven protons are barely screened at all. A modest rise in effective charge is a large rise in its fourth power.

One more ingredient is doing silent work. Which states are occupied at all is settled by the rule that forbids two electrons the same state, and sodium’s single outer electron — the whole reason its spectrum is simple enough to be a school demonstration — is a consequence of that rule applied to eleven of them.

Sommerfeld’s right answer for the wrong reason #

The formula above was first obtained in 1916, twelve years before Dirac, by Arnold Sommerfeld — from a model in which the electron runs round an ellipse rather than a circle and its speed therefore varies round the orbit, so that the relativistic mass correction differs from orbit to orbit and lifts the degeneracy.

It is worth dwelling on, because the result is exactly right and the model is entirely wrong. Sommerfeld’s atom has no spin in it; the electron follows a trajectory, which nothing does; and the quantum number that labels his orbits is not the one that labels the states. Two errors cancel — the spin–orbit term he had no way to include, and the relativistic term he double-counted by treating the orbital motion classically — and they cancel exactly, for every level, in a way that has no reason within his framework.

The episode is the standard cautionary tale about agreement with experiment. Sommerfeld’s formula fitted the measured splittings of hydrogen and of ionised helium to the accuracy then available, was taken as strong evidence for the elliptical-orbit picture, and led to a decade of work on quantisation rules for orbits that turned out to be a dead end. The number that survived him is $\alpha$ — he introduced it, and named it the fine-structure constant, for exactly the role it plays in the hero figure.

Seeing it #

Splitting a line by one part in a thousand requires an instrument that can distinguish wavelengths that close, and the requirement has a number.

The resolving power of a grating is the order times the number of illuminated rulings, and separating the sodium D lines needs about a thousand — which a hand-held spectroscope has. That is why this particular doublet is the standard demonstration: it is wide enough to see with modest equipment and narrow enough that seeing it feels like an achievement.

What is being counted at the far end is photons: light arrives in lumps, and a spectrum is a histogram of their energies. A line’s width in that histogram is what an instrument competes against, and the fine structure is a gap between two of them.

That is the general shape of the relationship between fine structure and instruments, and the two have driven each other for a century. Michelson’s interferometer was built to measure line shapes; the sharpness of the sodium doublet was one of the first things it settled; and the modern definition of the second is a hyperfine splitting of caesium, measured to a part in $10^{16}$.

What decides which splittings appear #

Not every pair of levels produces a line, and the missing lines are as informative as the present ones.

A current loop’s magnetic moment is its current times its area, and it is the quantity that couples to a field. The atomic version of the same statement decides which splittings appear in a spectrum: a transition shows only where the moment can couple, so the selection rules are statements about which pairs of states the operator connects, and lines that would be there on energy grounds alone are simply absent.

Sodium’s yellow light comes from the 3p level dropping to the 3s. The 3p is split into $j = \tfrac12$ and $j = \tfrac32$; the 3s has only $j = \tfrac12$. Two upper levels and one lower gives two lines, and that is the doublet. Had the lower level been split too, there would have been four candidate transitions and the selection rules would have forbidden one of them.

What the same physics does in a field #

Once a level is split by an internal field, the obvious next question is what an applied one does, and the answer has two regimes rather than one.

If the applied field is much weaker than the internal field the electron already sits in — a tesla or so — then the spin and the orbit stay coupled to each other, the total $j$ remains a good label, and each fine-structure level splits into $2j+1$ evenly spaced sublevels. Sodium’s $j = \tfrac32$ level becomes four, its two $j = \tfrac12$ levels become two each, and the yellow doublet becomes ten components. That is the anomalous Zeeman effect, and it is called anomalous only because it was discovered before anybody knew about spin and so had no explanation.

If the applied field is much stronger than the internal one, the spin and the orbit stop noticing each other and respond to the applied field separately. The pattern collapses back to the simple three-line form that a spinless theory predicts. The crossover between the two happens where the applied field matches the atom’s own, which is how a spectrum becomes an instrument for measuring a field: the pattern of a line tells how strong a field the atoms are sitting in, whether they are in a laboratory magnet or in a sunspot four hundred million metres wide.

A ratio that can be asked about the past #

The splitting is $\alpha^2/4$ of the level it splits. Both quantities are energies of the same atom, so their ratio is a pure number, and a pure number is the only kind of constant it is meaningful to ask whether the universe has changed. Whether the metre or the second has changed is not a question; whether $\alpha$ has is.

That question is answerable, and the instrument is the doublet on this page. Light from a quasar passes through gas clouds on its way here, and each cloud stamps its own absorption lines on the spectrum, redshifted by however far away it is. Every wavelength is stretched by the same factor, so the redshift cancels out of a ratio — and the ratio of a doublet’s separation to the transition’s own wavelength is $\alpha^2$ times a number the atomic physics fixes.

So measuring a doublet in a cloud whose light set out ten thousand million years ago measures $\alpha$ ten thousand million years ago, with the awkward astronomy divided away. The refined version compares transitions in several different ions, whose splittings respond to $\alpha$ with different coefficients and even with different signs, which turns a small shift into a pattern that instrumental error cannot easily imitate.

The results are worth stating carefully, because this is a field with a contested claim in it. One group’s analysis of one telescope’s spectra reported a change of about a part in a hundred thousand; a second telescope’s data did not reproduce it, and the discrepancy has never been fully resolved. What is agreed is that any variation is smaller than that, and laboratory comparisons of optical clocks — which watch two transitions with different $\alpha$ dependence drift against each other over a year — now bound the rate of change to below a part in $10^{17}$ per year.

Either way, the measurement exists because the fine structure is a fixed fraction of the gross structure and the fraction is $\alpha^2$. A constant that only appears with units attached could not have been interrogated at all.

The star that is made out of this line #

The sodium doublet has one industrial use, and it is the reason large telescopes work.

Ninety kilometres up there is a layer of sodium atoms a few kilometres thick, left by meteors ablating in the upper atmosphere. Shine a laser tuned to the D2 line — 589.159 nanometres, the shorter of the two — up through the telescope, and those atoms absorb and re-emit it. The result is an artificial star, at a known place in the sky, bright enough to measure the atmosphere’s distortion against and therefore to correct it. Every large observatory now has one or several.

Two details of the design come straight off this page. The laser has to be tuned to the line rather than to the doublet, because the two components are 0.6 nanometres apart and only one of them is being used — and the tuning has to hold to a fraction of the atomic linewidth, which is why the light is generated by sum-frequency mixing of two solid-state lasers rather than taken from anything simpler.

And the hyperfine structure — the splitting underneath the splitting, mentioned above as an afterthought — turns out to decide the brightness. Sodium’s ground state is split into two hyperfine levels 1.77 gigahertz apart, and an atom that scatters a few photons is liable to end up in the one the laser is not addressing, at which point it stops scattering and goes dark. The fix is to add a second frequency about 1.7 gigahertz away carrying a tenth of the power, which pumps those atoms back and roughly doubles the return. A splitting that is invisible in a school spectrometer is a line item in the specification of a twenty-watt laser.

What the guide star cannot do is also instructive. The beam goes up through the same air it comes back down through, so any overall tilt of the atmosphere displaces the outgoing beam and the returning image equally and cancels. A laser star therefore measures every distortion except the one that moves the image sideways, and a real star has to be found nearby to supply that.

What it costs #

Perturbation theory is being used on something that is not always small. For hydrogen $(Z\alpha)^2$ is $5 \times 10^{-5}$ and the expansion is superb. For hydrogen-like uranium it is 0.45, the series barely converges, and the Dirac equation has to be solved exactly rather than expanded — which is a large part of why heavy highly-charged ions are studied at all.

A single number for the effective charge is a fiction. Sodium’s outer electron does not see one charge; it sees a charge that varies from 11 at the nucleus to 1 outside the core, and any $Z_{\text{eff}}$ quoted for it is a fitted number chosen to reproduce one quantity. It will not reproduce a second.

The fourth-power law is a scaling and not a formula. The splitting of an alkali doublet goes as $Z_{\text{eff}}^4/n^3$ times a factor that depends on which orbital is involved, and the measured doublets of the alkali metals — sodium’s 0.6 nanometres, potassium’s 0.3, rubidium’s 1.5, caesium’s 4.2 — follow it well enough to be recognisable and not well enough to predict a fifth from four. Every departure is a fact about how a particular core screens.

And there is a splitting underneath the splitting. The nucleus has a magnetic moment too, three orders of magnitude smaller than the electron’s because it goes as one over a nuclear mass. Its interaction with the electron’s is hyperfine structure, and the D lines each have it.

Where the model stops #

Dirac’s degeneracy is not quite right. The theory says the 2s and 2p states with $j = \tfrac12$ have exactly the same energy. They do not: the 2s sits about a thousand megahertz above the 2p, which Lamb and Retherford measured in 1947. Nothing in a one-electron relativistic theory produces that. It comes from the electron’s interaction with the fluctuating electromagnetic field of the vacuum, and its measurement is the event that started quantum electrodynamics.

Occupation falls exponentially with energy, and in a discharge the population sits overwhelmingly in the lowest excited states — which is why a sodium lamp shows the D lines and almost nothing else. The spectrum of an element is not a list of its levels; it is a list of the transitions that are both allowed and populated, and the second condition does most of the excluding.

The electron’s magnetic moment is not exactly one Bohr magneton per half unit of spin. It is larger by about a part in a thousand, and the anomaly is the most precisely tested prediction in physics — twelve significant figures — and again it comes from the vacuum rather than from the atom.

The internal field has been quoted and not computed. The tesla-scale field the electron sits in is the nuclear electric field seen from a moving frame, and a magnetic field does no work — so it changes the energy of a state only through the orientation of a moment, which is why the effect is a splitting rather than a shift.

The nucleus has been a point charge throughout. It has a size, its charge is spread over it, and for heavy elements the correction from that is larger than the fine structure it is being compared with.

And every level here is infinitely sharp. A real excited state decays, so it has a width; the sodium 3p lives about 16 nanoseconds, giving a natural linewidth of 10 megahertz. That is far narrower than the fine structure and far wider than the hyperfine, which is exactly why the doublet is easy to see and the hyperfine is not.

What the pictures cannot show #

The hero figure draws the fine structure at a magnification of a hundred thousand beside the gross structure, and that magnification is a lie about proportion in the service of a truth about arrangement. Drawn to scale, the entire splitting would be a fifth of the thickness of the line representing the level.

Nor can a level diagram show what a level is. The horizontal lines are eigenvalues of an operator, and the vertical axis is an energy; the state has no place on the diagram at all. Everything this essay is about — which state has how much orbital angular momentum, how its spin is oriented relative to it, whether its wavefunction reaches the nucleus — is invisible in the one picture the subject is always drawn with.

Where this ladder goes next #

Two rungs stand on atomic-spectra. The first found that the lines are differences rather than values. This one finds that each of those lines is itself a difference of things that had been counted as one, and that the size of the split is set by how fast the electron is moving.

The habit worth carrying away is about small parameters. When a theory is corrected, the size of the correction is a measurement of something the original theory was assuming. Here $\alpha^2$ is not merely the size of an effect; it is the answer to “how non-relativistic was the atom?” — so the same number that gives the splitting says how good the calculation was that the splitting corrects, and there is no way to have one without the other.

What is left on this ladder is what happens when a field is applied from outside rather than generated within. A magnetic field splits each of these levels again by an amount proportional to the field, and the pattern it produces differs depending on whether the applied field is smaller or larger than the internal one — which turns a spectrum into an instrument for measuring fields in places nobody can reach.