The hooks that weigh an absorption line

Every refractive index is made of absorption lines. The constant that depends on how fast it is asked built the index of a material from bound electrons that resonate at particular frequencies: far from a resonance each one adds a little to the index, near it the index swings wildly, and at it the material absorbs. Glass is transparent in the visible because its resonances lie in the ultraviolet, and its gentle rise of index towards the blue — the dispersion that the colour at which glass disappears used to make a filter — is the distant shoulder of those ultraviolet lines.

A vapour of atoms brings a resonance into the visible, and with it the region where the index runs wild. It is the clearest place in physics to see how absorption and refraction are two faces of one response. It is also a place where the obvious measurement fails. How strongly a strong line absorbs says very little about how many atoms are absorbing, because its core is already black. The refraction beside the line does not saturate. The trick is reading it.

An index that runs backwards #

Near a single resonance at wavelength $\lambda_0$, the vapour’s refractive index follows the classical oscillator. Far from the line on either side, in the region where the vapour is transparent,

$$n - 1 \approx \frac{r_e N f \lambda_0^3}{4\pi(\lambda - \lambda_0)},$$

where $N$ is the number of atoms per unit volume, $f$ the line’s oscillator strength — the fraction of a classical electron’s response this transition carries — and $r_e$ the classical electron radius. On the red side of the line, at longer wavelengths, the index is above one and rises towards the line; on the blue side it is below one and falls towards it. Inside the line, within its width, the index runs the other way, from its maximum on the red side down through one to its minimum on the blue: the region of anomalous dispersion.

August Kundt saw anomalous dispersion in the 1870s by passing light through prisms of dye solutions and finding the colours on either side of an absorption band in the wrong order, and Robert Wood saw it in sodium vapour with a prism of vapour itself. It was a shock at the time. Normal dispersion — index rising steadily towards the blue — was so universal that the order of colours in a prism, red bent least and violet most, was taken as a law, and the achromatic lens that two glasses that cancel a derivative described depends on two glasses both behaving that way. The angle the rainbow has to be put red on the outside of the bow for the same reason. A substance whose colours came out in the wrong order showed that the order was not a law of light but a consequence of where the material’s resonances lay: on the far side of an absorption line, the order reverses.

The swing inside the line is the dramatic part, and it is the useless part for measurement: inside the line the light is absorbed, and in a dense vapour nothing gets through to measure. The useful part is the region outside, where the index still feels the line, falls off only as one over the distance from it, and the vapour is transparent.

The factor in front of the $1/(\lambda - \lambda_0)$ is the quantity worth having. It contains the number of atoms times the oscillator strength, $Nf$, and nothing about the line’s width or shape. Measure the index in the transparent wings and the product $Nf$ follows directly — the atomic physicist’s oscillator strength if the density is known, or the density if the oscillator strength is.

Fringes that turn back #

Measuring an index a few parts per million from one, as a function of wavelength across a fraction of a nanometre, is a job for an interferometer. Rozhdestvensky’s arrangement passes white light through an interferometer — a Jamin or Mach–Zehnder, two separated arms recombined — with a long tube of the vapour in one arm and a plate of glass in the other, and then through a spectrograph. In the spectrograph’s image, interference fringes run across the spectrum: at each wavelength the bright fringes sit where the two arms’ path difference is a whole number of wavelengths, and moving along the slit changes the path difference steadily, so the fringes are lines across the image.

The glass plate tilts the fringes. Its own dispersion, and the extra path it adds, changes the number of fringes by a steady amount per nanometre of wavelength, so far from any line every fringe is a straight, sloping line. The vapour adds its own contribution in fringe orders, $(n-1)l/\lambda$ for a column of length $l$, which near the line is

$$m_{\text{vapour}} = \frac{A}{\lambda - \lambda_0}, \qquad A = \frac{r_e N f \lambda_0^2 l}{4\pi}.$$

The total order is the vapour’s $A/(\lambda - \lambda_0)$ plus the plate’s $K(\lambda - \lambda_0)$, and the fringe is where that total equals a whole number. On each side of the line the two contributions pull in opposite directions — the vapour’s rising steeply towards the line, the plate’s steadily away from it — and each fringe reaches a turning point and doubles back on itself.

The turning points are the hooks. They occur where the slope of the vapour’s contribution cancels the plate’s slope, $A/(\lambda-\lambda_0)^2 = K$, at

$$\lambda - \lambda_0 = \pm\sqrt{A/K},$$

so the hooks are separated by $\Delta = 2\sqrt{A/K}$, and every fringe has them at the same places. Turned round,

$$N f l = \frac{\pi K \Delta^2}{r_e \lambda_0^2}.$$

The plate’s constant $K$ is read off the photograph itself, from the spacing of the fringes far from the line. Everything else is a constant of nature. A photograph of the hooks, measured with a ruler, gives the number of atoms in the light’s path times the oscillator strength, with no calibration.

The plate is what makes the hooks. Without it the fringes run straight across the spectrum, and the vapour merely bends them up on one side of the line and down on the other, smoothly, with nothing to mark a particular wavelength. The plate’s steady slope gives the vapour’s steepening curve something to cancel against, and the cancellation happens at a definite point. The eye finds such a point well: a fringe turning back on itself is vertical at its turning point, and the position of a vertical tangent can be read on a photographic plate to a small fraction of the fringe width.

A worked case shows the scale. For the $D_2$ line, with $10^{14}$ atoms per cubic centimetre in a ten-centimetre column, $A$ is half a nanometre, so the vapour adds ten fringes fifty picometres from the line and a hundred at five. A plate giving fifty fringes per nanometre puts the hooks at $\sqrt{0.5/50}$ nm, a hundred picometres either side. Measuring the separation of two hooks two hundred picometres apart to two per cent gives $Nfl$ to four per cent, because the separation enters squared — a precision that absorption spectroscopy of a saturated line could not approach.

The square root of the number of atoms #

The hooks move apart as the vapour grows denser, but slowly: the separation goes as the square root of $Nfl$.

The square root is useful. It gives the method a very wide range: a hundredfold change in density moves the hooks by a factor of ten, and the plate can be chosen thicker or thinner to keep them at a convenient separation — a thicker plate, giving more fringes per nanometre, pulls them in towards the line, and a thinner one lets them out. In practice the hooks were placed where they were well clear of the line’s core and well resolved by the spectrograph, typically a few tenths of a nanometre apart, and the method was used from about $10^{12}$ to $10^{17}$ atoms per cubic centimetre.

A measurement that ignores the line’s shape #

The deepest virtue of the hooks is where they sit. A spectral line has a shape that depends on everything happening to the atoms — the natural width set by the excited state’s lifetime, which the width that is a lifetime followed; the Doppler broadening from the atoms’ thermal motion; the pressure broadening from collisions with other atoms; the line’s hyperfine structure. Each of these changes the line’s core. None of them changes its far wings, where the dispersion of every line, whatever its shape, falls off as the same $A/(\lambda-\lambda_0)$.

That is a consequence of a sum rule. The answer that cannot come first found that causality ties the refractive index at every frequency to the absorption at all frequencies; one of its consequences is that the total absorption of a line, integrated over the whole line, is fixed by $Nf$ alone, however the absorption is spread out, and that the dispersion far from the line is fixed by the same integral. A line can be narrow and deep or broad and shallow; its far-wing dispersion is the same. The hooks, being in the far wings, measure the integral and nothing else.

Where absorption gives up #

The contrast with measuring the absorption itself is sharpest for strong lines. The light a line removes from a continuous spectrum — its equivalent width — grows in proportion to the number of atoms only while the line is weak. Once the centre absorbs almost everything, adding atoms deepens a line that is already black, and the equivalent width grows only through the wings, slowly and in a way that depends on the line’s shape. Astronomers call the relation the curve of growth, and reading abundances from saturated lines requires a model of the line profile that the line itself cannot supply.

The shape of the absorption curve in the figure is the classical curve of growth that Marcel Minnaert and others used to read stellar abundances. At the lowest densities the equivalent width grows in proportion to the atoms. Then the Doppler core saturates and the width grows only logarithmically, because adding atoms merely pushes the edge of the black core a little further into the Gaussian wings. At the highest densities the Lorentzian wings of the natural line shape, which fall off far more slowly than a Gaussian, begin to absorb, and the width grows as the square root of the number of atoms. Each regime needs a different model of the line to invert, and the flat middle regime, where most strong lines in stellar spectra sit, is the one in which the abundance is least well determined by the absorption.

Sodium’s $D_2$ line in a ten-centimetre column is already saturated at its centre at ten thousand million atoms per cubic centimetre. By $10^{12}$ an absorption measurement calibrated in the weak-line regime underestimates the density by a factor of twenty-six, and by $10^{14}$ by a factor of nearly a thousand. The hooks, at those densities, are a hundred picometres from a line a few picometres wide, in vapour that is perfectly transparent, and they report the density to a per cent with nothing assumed about the line. That is why the hook method became, for most of the twentieth century, the standard way of measuring oscillator strengths of strong atomic lines, and of measuring vapour densities in furnaces and discharges where no gauge could be put.

What the method needed #

The method asks for two things the early twentieth century could just supply. The interferometer’s two arms must stay stable to a fraction of a wavelength over the exposure, with a furnace full of hot metal vapour in one of them; Rozhdestvensky’s arms were separated by a few centimetres and the whole apparatus rested on stone. And the spectrograph must resolve a few tenths of a nanometre near the line, which a good prism or grating instrument of the time did. Against that, the method needed no absolute intensity measurement, no knowledge of the detector’s response and no model of the line — a ruler on a photographic plate was the whole of the reading.

The plate’s constant $K$ is measured on the same photograph, from the fringe spacing far from the line, so the glass’s own dispersion need not be known. The column length $l$ is the length of the furnace, and the density is the only real unknown if the oscillator strength is known, or the oscillator strength if the density is. The hooks of two lines on the same photograph compare their strengths directly, since the density and the path cancel: sodium’s $D_2$ and $D_1$ lines, whose upper levels have four and two magnetic substates, should have oscillator strengths in the ratio two to one, and the hook separations, squared, give that ratio without anything else being known. Later work combined the hook method with independent measurements of vapour pressure to obtain oscillator strengths of iron, chromium, titanium and other metals of importance to astronomy, where the abundances of elements in the Sun and stars are read from lines whose oscillator strengths must be known first. The fringe and the spectrum are one measurement found a similar economy in Fourier spectroscopy: an interferometer’s fringes, read correctly, are themselves a spectrum, and here they are a spectrum and an index measurement at once.

Steep dispersion as a tool #

The hooks are a way of measuring dispersion near a line. A century later, dispersion near a line became a way of controlling light. The group velocity of a pulse depends not on the index but on how fast the index changes with frequency, and between the hooks, where the index is changing fastest, the group index can be far from one. Inside an absorption line light is absorbed, so the steepest part of the curve is unusable; but if a second laser opens a narrow window of transparency in the middle of a line — electromagnetically induced transparency — the index across that window changes very steeply in the normal direction while the vapour stays clear, and a pulse inside it travels slowly. In 1999 Lene Hau’s group slowed light in an ultracold sodium vapour to seventeen metres per second by exactly this route. The wavelength a fibre does not smear followed the opposite use of the same physics, finding the wavelength where a fibre’s dispersion vanishes so that pulses keep their shape; here the dispersion is made as large as possible so that pulses crawl.

What the drawings leave out #

The figures use a single resonance, the $D_2$ line of sodium, and ignore its neighbour $D_1$ six tenths of a nanometre away, whose own dispersion adds a slowly varying background at the hooks; a real analysis subtracts it, or places the hooks so that the neighbour’s contribution is small. The vapour is taken as a uniform column, while real furnaces have colder ends where the density falls. The line’s dispersion is the classical oscillator’s, which is exact for the far wings of any line but not for the core; since the hooks are in the far wings, nothing they measure depends on the core’s details. The domain of the drawings is sodium at $10^{11}$ to $10^{16}$ atoms per cubic centimetre in a 10 cm column, a plate giving 50 fringes per nanometre, and line widths up to a few tens of picometres.

Still open: how well oscillator strengths are now known #

The hook method has largely been replaced by measurements of excited-state lifetimes with lasers, which give oscillator strengths through the same sum rules, and by branching ratios measured with Fourier spectrometers. The precision of the best laboratory oscillator strengths is now a few per cent for many lines, but the abundances astronomers derive for elements in stars still depend on lines whose oscillator strengths are uncertain by tens of per cent, especially for heavy elements with complicated spectra. Which laboratory methods can be extended to those spectra, and how much of the remaining disagreement between stellar abundances derived from different lines is laboratory error rather than stellar physics, is unsettled.

The geometry underneath is simple. Beside an absorption line a vapour adds $A/(\lambda - \lambda_0)$ fringe orders, with A proportional to the number of atoms times the oscillator strength; a glass plate adding $K(\lambda - \lambda_0)$ makes every fringe turn back at $\lambda - \lambda_0 = \pm\sqrt{A/K}$ — ±100 pm for 10¹⁴ sodium atoms per cubic centimetre in 10 cm — so $N f l = \pi K \Delta^2 / r_e \lambda_0^2$, read with a ruler, unchanged for any line much narrower than the hooks and unaffected by the saturation that makes the line’s own absorption report a thousandth of the truth. The atoms that a black line hides are counted by the light that passes beside it.