A narrower flank moves its wing in with it

A clear mercury lamp puts most of its visible light into a handful of narrow lines, and the two brightest, green at 546 nm and yellow at 578, sit thirty-one and a half nanometres apart. A surface colour under that lamp is a set of numbers about how much of each line the surface returns, so the brightest colours a pigment can make there are the ones that reflect one line almost completely and absorb the other almost completely. Of the directions in colour space that ask for that split, twenty-five can be reached by the ideal reflectance, which switches from nought to one between the lines; the question this run of essays has followed is how close a pigment built from smooth absorption bands can come to them.

A steep side splits mercury’s lines at forty nanometres found the answer in a band with one steep side, and a band is only as wide as its steep side found that what matters is the steep side’s own half-width. At one half-width a flank can lose nothing or nearly everything held that half-width at fourteen nanometres and changed the side’s shape: a Gaussian lost four of the twenty-five directions by more than one per cent, a Lorentzian twenty-two. A hundredth of Lorentzian undoes a Gaussian flank mixed the two, as real absorption bands are mixed in a Voigt profile, and found a single per cent of Lorentzian taking the losses from four to eleven and a twentieth taking them to eighteen, because the best-placed band leaves the passed line more than three half-widths out, where even a small Lorentzian wing outweighs the whole Gaussian.

The wing’s absorbance falls as the square of the distance over the half-width. Narrow the whole steep side and the passed line is further out in half-widths, and the wing there falls with the square of the narrowing. The prediction was that a twentieth-Lorentzian steep side seven nanometres wide loses four directions again — its wing at the passed line a quarter of the fourteen-nanometre wing’s, matching the Gaussian’s single-band error — and that at five nanometres it loses none. If the seven-nanometre Voigt still lost ten or more, the named alternative was that the shoulder a narrow flank leaves behind matters more than its wing, and that a dye with any Lorentzian in its band cannot split close lines however sharp its core.

The losses barely fall #

Searched in all twenty-five splitting directions, the forty-nanometre pigment with a twentieth-Lorentzian steep side loses thirteen directions by more than one per cent at ten nanometres, twelve at seven and seven at five, against eighteen at fourteen. With no Lorentzian in the side it loses none at ten, seven or five nanometres, where at fourteen it lost four.

• The prediction fails at both widths. Seven nanometres loses three times the four predicted, and five loses seven where none was.
• The alternative’s outcome arrives. A seven-nanometre Voigt loses more than ten directions, as the alternative said it would if the narrowing did not pay.
• Its explanation does not. Every lost direction is still lost to the wing, as the next sections show; there is no shoulder in the account.

The two lines on the figure part company completely. The Gaussian pigment’s losses vanish as soon as its steep side narrows past fourteen nanometres: at ten its worst direction falls short of the ideal by 0.16 per cent, and at seven and five by 0.15 per cent, the residue of a search on a one-nanometre grid rather than of the shape. The Voigt’s losses fall by a third from fourteen nanometres to ten, by one more direction from ten to seven, and by five from seven to five. Halving the width removes about two thirds of them. Nothing like the square of the width is at work.

One correction belongs here. A five-nanometre flank carries a fiftieth of Lorentzian searched every direction at five nanometres a second time, from the best pigments found at neighbouring Lorentzian shares, and found two of the seven directions lost at a twentieth recoverable: at a twentieth the five-nanometre pigment loses five, not seven. The counts at seven and ten nanometres have not been searched again and may be overstated in the same way. None of that changes the argument — five is still nothing like the none predicted — but the figure’s last point sits two directions high.

The counts are counts of directions past a one-per-cent threshold, and a count can move slowly while the shortfall beneath it moves fast. It does not here. The twentieth-Lorentzian pigment’s worst direction falls short of the ideal by 7.1 per cent at fourteen nanometres, 3.6 at ten, 2.1 at seven and 2.3 at five, and its mean shortfall over the twenty-five directions by 3.0, 1.6, 0.95 and 0.64 per cent. The mean falls about as fast as the width, and the worst direction stops falling at seven nanometres.

One band, falling in step with its width #

The simplest test of the wing’s square is a single band, twelve deep, its wide side a twenty-six-nanometre Gaussian and its steep side the Voigt, placed along the wavelength axis wherever it best absorbs the green line and passes the yellow. Its larger error on the two lines is a clean measure of how well one band of that shape can split them, without a search in many directions.

The Gaussian band’s error falls from 0.52 per cent at fourteen nanometres to 0.031 at ten and 0.0006 at seven — almost a thousandfold over half the width. The twentieth-Lorentzian band’s falls from 4.2 per cent to 2.8, 1.7 and 1.0 per cent at five: a factor of four over a factor of 2.8 in width. A Gaussian’s foot falls as the exponential of the square of the distance, so narrowing it empties the passed line completely; a Voigt’s falls as the inverse square of the distance, and the prediction expected that inverse square to be read at a fixed distance. The single band’s error falls roughly in proportion to the width instead, and that rate, not the square, is what the pigments’ counts follow.

The band moves in #

The reason is where the band sits. A band placed to split two lines has two errors to balance: the absorbed line must be absorbed deeply, and the passed line must not be absorbed at all. Narrowing the steep side changes the balance, and the band does not stay where the wider one sat.

At fourteen nanometres the best band’s centre sits 51.0 nm from the passed line, 3.64 half-widths; at ten, 44.8 nm and 4.48 half-widths; at seven, 40.3 nm and 5.75; at five, 37.4 nm and 7.48. As the steep side narrows the band moves fourteen nanometres in towards the line it lets through, and the passed line ends up seven and a half half-widths out instead of the ten it would have reached had the band stayed put.

The move is forced by the other line. The absorbed line lies on the same steep side, thirty-one and a half nanometres nearer the band’s centre, and a narrower side absorbs it less at the same distance. To keep that line deeply absorbed the band has to bring its centre closer — from 19.5 nm short of it at fourteen nanometres to 5.9 nm at five — and in doing so it brings the passed line closer too. The gap between the lines is fixed, and the band can narrow its flank only by sliding its whole profile towards both.

Left where the fourteen-nanometre band sits, the wing’s absorbance at the passed line would fall from 0.042 to 0.0057 by five nanometres, an eighth. Placed as well as it can be, it falls to 0.0105, a quarter. At every width the band’s whole absorbance at the passed line is the wing’s to four decimal places; the Gaussian part contributes nothing there from ten nanometres down. The wing is the entire error, as it was at fourteen nanometres, and the prediction’s arithmetic was right about the wing and wrong about the distance at which the wing would be read.

So the alternative named in advance, that a narrow flank’s shoulder takes over from its wing, is not what happens. There is no shoulder in the account at all: the Gaussian core of a five-nanometre side is gone by the passed line, and the remaining absorbance there is exactly the Lorentzian share over one plus the square of seven and a half. What defeats the narrowing is geometry — two lines a fixed distance apart, both on one steep side — and it would defeat any profile whose foot falls as a power of the distance rather than exponentially.

Four bands over the two lines #

Drawn over the two lines, the four best bands show the slide directly. The fourteen-nanometre band’s edge falls gently across the whole gap, reaching ninety-odd per cent reflectance only near the yellow line; the five-nanometre band’s falls sharply, its edge almost a step, but the step sits well inside the gap rather than in its middle, because the band has moved in to hold the green line. The reflectance at the yellow line, 95.8 per cent for the fourteen-nanometre band and 99.0 for the five, is what the error figures describe. Each narrowing buys a steeper edge and spends part of it on placement.

The same slide happens for the Gaussian, and costs it nothing. A Gaussian edge as steep as five nanometres falls from full absorption to none within a few nanometres, so wherever the band slides within the gap the passed line is untouched. The Lorentzian share changes that, because its contribution never reaches nought: a band that must sit close to the passed line to hold the absorbed one carries its wing there whatever its width.

Which directions come back #

Of the fourteen directions that pass the green line and absorb the yellow, the twentieth-Lorentzian pigment loses eleven at fourteen nanometres, eight at ten, seven at seven and five at five; of the eleven that pass the yellow and absorb the green, it loses seven, five, five and two. The yellow-passing directions come back proportionally faster: by five nanometres two of their seven losses remain, against five of the green-passing eleven.

The asymmetry is the census’s, and the census does not trace its cause. A pair of lines asks for a cliff, not a gap found every deep loss under this lamp asking a pigment to reflect one of the two lines and absorb the other, with a single transition between them, so that each direction leans on one band’s one edge; which edge, and how far from it the released line falls, differs between the two kinds of direction, and nothing here measures it. What the census does establish is that both kinds come back slowly — neither at the square of the width — and the yellow-passing ones less slowly.

The gap has decided every width before this one #

The fixed distance between the lines has been the deciding number at every step of this run, and it is worth seeing the present result as the latest instance rather than a surprise. A mercury lamp’s few losses are deep found that every colour a slow pigment loses badly under this lamp is one that reflects one of the two lines and absorbs the other, thirty-two nanometres apart, so that the whole difficulty of the lamp is concentrated in one spacing. A band pigment must be narrower than the gap it splits found the band pigment’s losses falling with the band’s width — fifteen at forty nanometres, eleven at thirty-five, nine at thirty — as a band narrow enough to fit inside the gap stopped absorbing the line it should pass.

A band must be nearly two to one before its steep side pays found the same spacing behind a threshold in skew: a band forty nanometres wide recovered its directions slowly as one side steepened — three per twentieth of skew at first — and twelve in the last step, to two to one, where the steep side is about fourteen nanometres wide. In every case the band’s useful edge has to fit into thirty-one and a half nanometres, and in every case the profile’s foot is read at whatever distance the band’s placement leaves between that edge and the released line.

What is new here is that the placement moves. With a Gaussian edge the placement hardly matters, because the foot is gone within a few half-widths and the band can sit anywhere inside the gap; every earlier result was read off Gaussian or nearly Gaussian edges, and none of them needed to say where the band sat. With a Lorentzian share the foot never goes, and the band’s position inside the gap is no longer free. It is set by the absorbed line, which pulls the band towards it as the edge sharpens, and the released line goes with it.

What a pigment chemist should take from this #

Narrowing a Voigt band is worth about its width, not its width squared. Halving a steep side’s half-width halves the error a twentieth of Lorentzian causes across close lines; it does not quarter it.

The Lorentzian share matters far more than the width. At every width from fourteen nanometres to five, a twentieth of Lorentzian loses seven to eighteen directions that a pure Gaussian of the same width does not lose at all. A dye whose band is collision-broadened enough to carry a few per cent of Lorentzian will not split lines thirty nanometres apart cleanly at any practicable sharpness.

The geometry is the general part. Any band that must hold one line and release another a fixed distance away moves in as its edge sharpens, and any profile with a power-law foot carries its foot to the released line. The result is a property of the lines’ spacing and the profile’s foot, not of mercury.

How the pigments were searched #

Each pigment is up to three absorption bands at least forty nanometres wide on a constant background, every band a twenty-six-nanometre Gaussian on its wide side and on its steep side a pseudo-Voigt — the Gaussian and the Lorentzian of the same half-width at half depth, summed with weights nineteen twentieths and one twentieth, or with no Lorentzian at all — at half-widths of ten, seven and five nanometres, scaled with the band’s width over forty. The reflectance is the exponential of minus the summed absorbance on a one-nanometre grid. In each splitting direction the pigment’s reach is its colour’s projection onto that direction under the clear mercury lamp, through the CIE 1931 observer, over the ideal reflectance’s; the search maximises it by Nelder–Mead from fourteen starts — the earlier essays’ structured starts and the milder skews’ best fit — 2,000 steps each. A direction is lost when the best reach falls more than one per cent short. The single band is twelve deep and placed on a fifty-thousandth of a nanometre grid to minimise its larger error on the two lines.

What this does not reach #

Other Lorentzian shares at the narrow widths. A twentieth is the share the earlier essay found most damaging for its size; a hundredth or a fiftieth at five nanometres has not been searched.

Real profiles. A pseudo-Voigt is a sum, not the convolution of a Gaussian with a Lorentzian, and the two differ in the wing by a few per cent; the conclusions depend on the wing’s power law, which both share.

Other lamps. Lines closer together than thirty-one and a half nanometres would force the band further in, and lines further apart less.

Still open: how little Lorentzian a sharp flank can carry #

A five-nanometre steep side loses nothing as a Gaussian and seven directions with a twentieth of Lorentzian, and its error falls roughly in proportion to the share: the wing at the passed line is the share times a geometric factor the band’s placement fixes.

The calculation is the forty-nanometre pigment with its steep side five nanometres wide and a hundredth and a fiftieth of Lorentzian, searched as here. The prediction is that the count falls in proportion to the share’s effect on the single band’s error — about two directions lost at a fiftieth and none at a hundredth — because at five nanometres the band’s placement changes little with the share. If a hundredth still loses three or more, the count is not following the single band, and the twenty-five directions each meet the wing at a distance of their own.

A prediction right about the law and wrong about the setting #

The habit here is about where a scaling law is read.

The wing’s inverse square was the right law, and the arithmetic built on it was correct for a band that stays put. The band did not stay put: the quantity being optimised had another term, the absorbed line, and narrowing the flank changed the best placement, so the law was read at a distance that moved with the parameter being varied. Read at the new distance the law fits to four decimal places.

The failure mode is to apply a scaling law at the operating point of the old optimum, when the optimum itself moves with the parameter being scaled. The test is to re-optimise at each setting and read where the optimum went before reading what the law predicts there.