A steep side splits mercury's lines at forty nanometres

A clear mercury lamp is four visible lines and nothing else, and under it the colours a surface can show are set by how it reflects at 405, 437, 546 and 578 nanometres. A run of essays has compared what real pigments can reach under that lamp against the ideal pigment that switches instantly between reflecting and absorbing. A mercury lamp’s few losses are deep found the census’s pigment models falling short in a handful of directions, and a pair of lines asks for a cliff, not a gap traced them to the green and yellow lines: the hard directions ask a pigment to reflect one and absorb the other, 31.5 nanometres away.

Two essays then asked what limits each model there. A slow pigment splits mercury’s lines only if its swing fits between them found a pigment whose reflectance may change only so fast losing exactly the directions whose cliff it cannot fit between the lines. A band pigment must be narrower than the gap it splits found the same for a pigment built from up to three Gaussian absorption bands: fifteen of the twenty-five splitting directions lost with bands no narrower than forty nanometres, nine at thirty, none at twenty-five, because a Gaussian band placed to absorb one line reaches the other with its tail.

A Gaussian band is symmetric, and real dye bands often are not. Many absorb with a long gentle tail on their short-wavelength side and a steep fall on their long side, or the reverse, because the molecule’s vibrational structure piles absorption up on one side of its strongest transition. The proposal was a band whose one side is narrowed to half the other’s width, its steep side free to face either way, and the prediction was that a skewed band forty nanometres wide loses about as many directions as a symmetric thirty, because what matters is the half-width on the side facing the passed line — a skewed forty’s steep side is about thirteen nanometres, a symmetric thirty’s fifteen. The prediction added that the skew helps only the directions that pass the green and absorb the yellow, where the steep side can face the green. The alternative named was that the skew helps both kinds of direction equally, in which case the pigment is using its other bands to shape the cliff.

One lost at forty, none narrower #

Searched on the one-nanometre grid with three bands each split at its centre — the steep side a third of the full width, the other two thirds, the steep side’s direction chosen in the search — the band pigment loses one of the 25 splitting directions by more than one per cent with bands forty nanometres wide, and none at thirty-five or thirty. The same four-number search with symmetric bands loses sixteen, ten and eight, within one or two of the band census’s fifteen, eleven and nine. In every one of the 25 skewed fits at forty nanometres, the band that absorbs one line turns its steep side towards the other. Of the fourteen directions that pass the green line the skewed pigment loses none, where the symmetric one lost nine; of the eleven that pass the yellow it loses one, where the symmetric lost seven. A single skewed band forty nanometres wide and twelve deep, placed as well as it can be, splits the two lines within 0.37 per cent; a symmetric band forty wide errs by 3.23 per cent, thirty wide by 0.80, and must narrow to about twenty-seven to match the skewed forty.

• The prediction understates the skew. A skewed forty does not match a symmetric thirty; it beats it, nine directions to one.
• Its second half fails. Both kinds of direction gain, each by turning the steep side the right way.
• The alternative is half right. Both kinds gain equally — but because the steep side can face either line, not because other bands shape the cliff.

The three curves answer three questions. The census’s symmetric bands and the four-number control agree within two directions at every width, so the control is a fair one: adding a number for the steep side’s direction does not, of itself, help a symmetric band. The skewed curve sits on the floor, one direction at forty nanometres and none narrower — where the symmetric curve needs twenty-five nanometres to reach nought. A skewed band forty wide splits mercury’s lines as a symmetric band twenty-five wide does, give or take one direction.

What one band can do #

The single band is the census in miniature. A band must sit with its centre near the line it absorbs and let its flank fall before it reaches the line 31.5 nanometres away, and whichever line it misses by more sets its error. A symmetric forty errs by 3.2 per cent; its skewed twin by 0.37, under the one-per-cent line the census counts losses against. To err as little, a symmetric band must narrow to about twenty-seven nanometres, thirteen narrower than the skewed band. At twenty-five nanometres and below the skewed band’s error falls under a hundred-thousandth, off the bottom of the chart.

The prediction’s arithmetic was that the side facing the passed line is what matters: a skewed forty’s steep half-width is 13.3 nanometres, close to a symmetric thirty’s 15. That arithmetic undersold the skew, because a skewed band gains twice. Its steep side reaches less far towards the passed line, and its wide side, two thirds of the width on the far side of the absorbed line, lets the band sit further from the passed line while still absorbing the other fully. A symmetric band must centre itself between the two demands; a skewed one leans away from the line it must spare.

The steep side faces the passed line #

Drawn as reflectance, the two bands make the mechanism plain. Both absorb the yellow line completely. The symmetric band’s short-wavelength flank begins its fall before the green line and takes a few per cent of it; the skewed band’s flank, as steep as a band thirteen nanometres wide, holds the green fully reflected and falls in the gap. On the band’s far side the skewed band’s wide flank runs further into the red than the symmetric band’s, which costs nothing here because no mercury line lies there.

The search found this arrangement by itself in every one of the 25 directions. Wherever a direction asked for the yellow absorbed and the green reflected, the absorbing band centred itself beyond 578 nanometres with its steep side short, facing the green; wherever it asked the reverse, the band centred below 546 nanometres with its steep side long, facing the yellow. The direction of the skew was free and the optimiser always chose the same rule: steep towards what must be spared.

Both kinds of split gain #

The prediction expected the skew to help only the directions that pass the green, imagining the steep side facing the green. With the steep side free, the census finds the opposite kind gaining as well: none of the fourteen green-passing directions is lost with skewed bands, against nine with symmetric ones; one of the eleven yellow-passing directions is lost, against seven. The one survivor passes the yellow and absorbs the green, missing by just over a per cent; why that direction keeps a shortfall the census does not separate, beyond its absorbing band sitting below 546 nanometres with its wide side reaching into the blue-green, where the direction’s other weights lie.

At thirty nanometres the symmetric bands still lose four of each kind and the skewed bands none. The symmetric bands’ losses had been spread evenly between the two kinds all along, which is what the prediction’s account would not have expected; the account was right about the mechanism and wrong to give it a preferred direction.

Every direction gains #

Ordered by how badly the symmetric pigment does, the directions show the skew’s gain is not confined to the hard cases. The symmetric forty’s worst direction reaches 90.1 per cent of the ideal and its mean shortfall is 2.9 per cent; the skewed forty’s worst reaches 98.8 and its mean shortfall is 0.38. Every direction gains, the worst by most: the directions that needed the cleanest cut between the lines were the ones a symmetric tail spoiled most.

That worst direction asks for the green absorbed and the yellow reflected. The symmetric pigment’s best attempt absorbs the green almost completely and reflects only 93 per cent of the yellow, its band’s long flank still falling at 578 nanometres; the skewed pigment’s long flank is steep, reaches the yellow at 99 per cent, and keeps the green under a fifth of a per cent. The two reflectance curves differ only in the twenty nanometres between the lines, and that is where all of the difference in reach comes from.

Shape against width and depth #

The band census had already compared two levers. Narrowing the bands by five nanometres lowered the mean shortfall more than doubling their depth did, at every width from forty to thirty, and doubling the depth helped only near the threshold where one band could nearly split the pair already. Depth deepens the trough but does not shorten the tail, and the tail is what reaches the passed line. Shape is a third lever, and the strongest of the three here: at a fixed width of forty nanometres, skewing the band two to one does more than narrowing it to thirty and far more than doubling its depth.

That ranking matters because the three levers cost different things in a real colorant. A narrower band asks for a molecule with less vibrational broadening, which chemistry does not always supply; a deeper band asks for more colorant, which a film can hold only so much of. A steeper side asks for the band’s shape — and dye chemists do shape bands, by the substituents that shift one transition relative to its neighbours. The census cannot say how hard a two-to-one skew is to make, but it can say it is worth more under a line lamp than either of the obvious alternatives.

Where this leaves the pigment models #

The limits essays have carried three models of what a slow pigment can do: a slope limit on how fast reflectance may change, bands of a fixed minimum width in wavelength, and bands whose width is fixed in energy. Written in energy, the bands spare the green found the energy model cheaper under a four-emitter lamp because its bands narrow towards the blue, and the lamp outweighs the pigment model found the lamp mattering more than the choice among the three. All three share one assumption the skewed band breaks: that a pigment’s absorption is symmetric about each of its features. A slope limit treats rising and falling edges alike; a Gaussian band is symmetric by construction; an energy band is symmetric in energy and only mildly asymmetric in wavelength.

Under smooth lamps the assumption costs little, since a band’s two flanks each cost a little everywhere. Under a line lamp it decides the count. Fifteen of twenty-five directions lost at forty nanometres was a statement about symmetric bands, and a harder search pulls the band pigments away from the slope limit had already shown how much those counts depend on what the model may do. Allowed a steep side, the band pigment loses one.

What this means for a pigment chemist #

A dye’s band shape matters as much as its width, and on the side that faces the colour it must spare. A band forty nanometres wide is broad for a dye, and under a line lamp a symmetric one cannot split two lines thirty nanometres apart; the same width with a steep side splits them as cleanly as a symmetric band twenty-five wide. Choosing or engineering a dye whose steep edge faces the light that must pass buys more than narrowing the whole band.

The skew does not need to be large. A two-to-one ratio between the sides moved the forty-nanometre pigment from fifteen losses to one. How much less skew would do is a question the census can now ask cheaply.

Under smooth lamps none of this matters. The splitting directions exist because mercury’s two lines are narrow and close; under a continuous spectrum a band’s tail costs a little everywhere and nowhere decisively, as the slope limit was exact all along found for the slope-limited pigment’s census under daylight.

How the pigments were searched #

The pigment is the band census’s: a reflectance e^(−a), where a is a floor of absorbance plus three absorption bands, each no deeper than twelve at its centre and no narrower than the stated width, searched by Nelder–Mead from structured starts — bands placed over each run of the direction’s negative weights, and a band placed with its flank in the gap at seven offsets — for two thousand steps, keeping the best. Each band here is split at its centre: on one side a Gaussian of half-width at half height two thirds of the stated full width, on the other one third, so that its full width at half height is the stated width; a fourth number per band chooses which side is steep. The control uses the same four numbers per band with both sides a half. The lamp is clear mercury with its lines one and a half nanometres wide, on a one-nanometre grid, and the 25 splitting directions are the census’s: those whose ideal pigment reflects one of the green and yellow lines and absorbs the other. A direction is lost when the pigment’s best reach is more than one per cent short of the ideal’s. The single band is placed at every offset in steps of a twentieth of a nanometre and its larger error kept.

What this does not reach #

One skew. Every skewed band has sides in the ratio two to one. A real dye’s band has its own ratio, and a milder skew would sit between the two curves.

Gaussian halves. Each side is a Gaussian. A real band’s steep side may be steeper still near its edge and its gentle side longer-tailed than a Gaussian, and the census does not model either.

One lamp. The splitting directions are clear mercury’s. Other line lamps put their pairs of lines at other spacings, and a skew that splits mercury’s 31.5 nanometres may not split a closer pair.

Still open: how little skew is enough #

A two-to-one skew took the forty-nanometre pigment from fifteen losses to one. The single-band arithmetic says the error falls steeply with the steep side’s half-width, so a much milder skew might already do most of the work.

The calculation is this census at forty nanometres with the bands’ sides in the ratios 0.55 to 0.45, 0.6 to 0.4 and 0.65 to 0.35 of the full width, each direction’s reach read as here. The prediction is that the count of lost directions falls roughly in step with the single band’s error — to about eight at 0.55, three at 0.6 and one or two at 0.65 — so that a skew of three to two already halves the symmetric band’s losses. If the count stays near fifteen until the ratio passes 0.6, the pigment needs a markedly steep side before its band shape matters, and a dye’s modest natural asymmetry buys little under a line lamp. A band must be nearly two to one before its steep side pays makes that calculation.

A preferred direction nobody needed #

The habit here is about a mechanism stated with a direction it did not have.

The account was right: what limits a band under a line lamp is how far its flank reaches into the line it must pass. The prediction then attached that mechanism to one geometry — the steep side facing the green — and expected one kind of direction to gain. With the steep side free, the search turned it whichever way each direction needed, and both kinds gained.

The failure mode is to carry a mechanism’s example into its consequences. The example was a band absorbing the yellow and sparing the green; the mechanism was a band sparing whatever line it faces. The test is to let the variable the mechanism names — here the side the steep flank faces — be chosen by the calculation, and to read which way it went.