A pair of lines asks for a cliff, not a gap

The object-colour solid is every colour a surface can have under a light, and its boundary is reached only by ideal pigments — surfaces that reflect everything at some wavelengths and nothing at the rest, switching instantly between them. The limits assume a pigment that switches instantly asked what a real colorant loses by not switching instantly, and three models of a real one have since been set against the ideal: a slope limit, on how fast reflectance can change per nanometre; absorption bands no narrower than forty nanometres; and bands whose floor is set in energy rather than in wavelength. Each is searched, direction by direction, for the furthest it can reach against the ideal pigment.

A mercury lamp’s few losses are deep read those censuses for depth under nine lamps. By the usual count — directions reached less than ninety-nine per cent as far as the ideal — a clear mercury lamp, four visible lines and nothing else, spares a pigment as well as three narrow emitters do. By depth it was the dearest of the nine: a third or more of every model’s losses fell more than five per cent short, where a tube spares a pigment through its phosphors had found the same lines, with a coating under them, costing almost nothing deep. Every one of them was a direction whose ideal pigment reflects one of the lamp’s green and yellow lines, at 546 and 578 nanometres, and absorbs the other. The two lines are 32 nanometres apart, and each model describes a pigment by how fast it can switch.

That essay’s closing section observed that a real colorant need not be one transition. Two absorption bands close together leave a reflectance peak between them that can be far narrower than either band, which is how a dye can look a clean colour with broad absorptions. The prediction was that the band model, allowed two bands with a narrow gap between them, would recover most of the deep losses, because a narrow peak is exactly what splitting two close lines seems to need; the directions it could not recover would be those needing the split and a second edge elsewhere. If it recovered them, the mercury lamp’s depth would be a property of single-transition models.

The lamp asks for an edge, and depth buys it #

No direction the lamp’s census loses asks for a narrow peak. Of mercury’s four visible lines, the blue pair is never split by a losing direction, so every splitting direction asks for one transition between green and yellow and none within 110 nanometres of it: a cliff, not a gap. The lamp leaves twenty-five nanometres for the cliff, between the last grid sample carrying a quarter of the green line’s power and the first carrying a quarter of the yellow’s. A forty-nanometre band at the census’s cap of twelve takes 27.8 nanometres to fall from reflectance 0.05 to 0.95; two stacked at one centre take 24.9. Searched from 680 starts, the census’s own band model does that in every splitting direction — puts its bands beyond the line it absorbs, stacked two or three deep at one centre in twenty-four of twenty-five — and reaches further than the census found in seventeen, the worst loss falling from 10.3 per cent to 7.1. The five losses still deeper than five per cent all absorb the blue lines too and spend a band there. A fourth band, or a doubled cap, leaves at most one direction deep.

• The prediction’s mechanism is not the one the lamp asks for. A gap would help a pigment that must reflect one line and absorb lines on both sides of it within a few tens of nanometres; mercury’s lines never ask that.
• Two bands do recover the losses — stacked, not spaced. Stacking adds depth, and depth steepens a flank.
• The census’s search had stopped short. The model could already do what the prediction proposed adding to it; the census had not found it.
• Depth is bought slowly. The fall shortens as the square root of the logarithm of the depth, so doubling it buys a few nanometres.

The pale curve is the lamp: four lines, and nothing between them. The dashed steps are the ideal pigment for one direction of the solid — here one that asks for the green line and none of the others. The solid curve is the best three-band pigment the search found for it: a band near 427 nanometres takes out both blue lines, and two bands stacked at 607 nanometres make a single steep flank falling between 550 and 575, from the green line’s reflection into the yellow line’s absorption. From about 470 nanometres up to the green line the pigment reflects freely, and beyond 650 it reflects again, because the lamp has no light in either stretch and nothing a pigment does there costs it anything — the darkness the gap has to be dark, not the line narrow found was what spares a slow pigment under line lamps. There is no peak between two bands anywhere near the pair. With a fourth band the search puts a third band into the stack at 607, and the flank steepens further.

What four lines ask of a pigment #

Under a lamp of four narrow lines, a pigment’s colour depends on its reflectance at four places and nowhere else: 405, 436, 546 and 578 nanometres, each smeared over two or three grid samples by the line’s width. A direction of the solid is a weighting of those four, and its ideal pigment reflects the lines whose weight is positive and absorbs the rest. Twenty-five of the census’s 148 directions ask for the green line and the yellow differently, and they come in four patterns: reflect the green alone, reflect everything but the yellow, reflect everything but the green, reflect the yellow alone. In every one the blue pair is treated alike.

That settles what shape a pigment needs. Reflecting the green and not the yellow needs reflectance to fall somewhere between them; it does not matter what it does between the blue and the green, where the lamp is dark. The nearest other transition any pattern asks for is between 436 and 546 nanometres, a hundred and ten apart. A narrow peak — reflectance rising and falling within a few tens of nanometres — is what a pigment would need to reflect one line and absorb lines close on both sides of it, and mercury’s four lines have no such arrangement. The prediction’s gap is a solution to a problem this lamp never poses.

How steeply a band can fall #

A band model’s pigment absorbs as the sum of Gaussian bands, and its reflectance is the exponential of minus that sum. Where one band’s flank crosses the pair, the fall from absorbing to reflecting is set by two things: the band’s width, floored at forty nanometres, and its depth, capped at an absorbance of twelve. A deeper band falls further in the same distance, so the stretch over which it passes from 0.05 to 0.95 reflectance shortens as it deepens — which is what a sharp edge is bought with depth found for the band model against the slope limit.

The lamp leaves twenty-five nanometres. The green line carries its power at 540, 545 and 550 nanometres on the collection’s five-nanometre grid, the yellow at 575, 580 and 585; the last strong green sample and the first strong yellow one are 550 and 575. One band at the cap takes 27.8 nanometres to fall from 0.05 to 0.95 — too long for the gap, so it must either spill absorption onto the green or leave reflection on the yellow. Two bands stacked at one centre act as one band of twice the depth and take 24.9; three take 23.6. The model allows three bands, each capped at twelve, and nothing stops it putting them in the same place.

The curve also shows why depth is expensive. A Gaussian flank’s position at a given reflectance goes as the square root of the logarithm of the depth, so each doubling buys less: twelve bands’ worth of depth stacked together would still take 20.4 nanometres. The mercury pair sits just inside what two or three bands can reach, and that is why its losses are deep rather than absent or total.

The census’s own model, searched harder #

The census searched each direction’s band pigment with a Nelder–Mead descent from a few structured starts, built from where the direction’s ideal pigment absorbs. Here the same model — the same three bands, the same forty-nanometre floor and cap of twelve, the same objective — is searched from 680 starts: two bands placed on a grid of centres across the visible, at half and full depth, the third empty. The harder search reaches at least as far as the census in every splitting direction and further in seventeen. The worst direction’s shortfall falls from 10.3 per cent to 7.1 and another from 9.9 to 3.2; the mean over the twenty-five falls from 3.1 per cent to 2.4. The count short by more than one per cent does not move, eighteen either way, but the count deeper than five per cent falls from six to five.

That is a correction to the earlier census, and the published essay now carries it. The prediction proposed giving the band model a freedom it already had. The census had not used it, because a descent started from the places a direction’s ideal pigment absorbs looks for bands where the absorption is, and the best answer here puts all its bands on one side of the absorption, stacked. A search that finds a local answer can be read as a limit of the model, and the mercury lamp’s depth was partly that.

The bands fall into three clusters. Directions that absorb the yellow put their bands at 607 to 623 nanometres, beyond it; directions that absorb the green put theirs at 510 to 523, below it — each stack’s near flank falling across the gap. The two or three bands of a stack sit within six nanometres of one another, at or within two nanometres of the width floor, and nearly all at the cap. The third cluster, round 420 to 440 nanometres, is the one band spent by directions that must also absorb the blue lines. Across all twenty-five no pigment leaves a reflectance peak narrower than 32 nanometres anywhere near the pair.

A later search moves one count. Searched from a grid of 594 starts under all nine lamps, a harder search pulls the band pigments away from the slope limit found the band model under the mercury lamp losing eighteen directions by more than one per cent, as here, and four by more than five — one fewer than the five below, a direction that grid brings to 4.3 per cent. The mechanism described here is unchanged.

Why the deep ones are deep #

The five directions still more than five per cent short all absorb both blue lines, and every one of them spends one of its three bands below 460 nanometres to do it. That leaves two bands for the cliff, which is just enough in the arithmetic of the flank and not enough once the flank must also clear the green line’s own smear to 540 or the yellow’s to 585. Directions that reflect the blue lines spend nothing there and stack all three bands at the cliff; none of them is more than 2.5 per cent short.

Two changes to the model test the reading. Given a fourth band at the same cap, every direction that spends one on the blue puts the new one into its stack, and no direction is left deeper than five per cent; the worst falls to 5.0. Raising every band’s cap to twenty-four leaves one deep direction, and to forty-eight none, with the worst at 4.2. Neither change brings the count short by one per cent below fifteen: the cliff is steeper, not vertical, and a direction whose ideal is a perfect step between 550 and 575 is still missed by a little. The pair’s cost is a depth budget — bands times cap — and not a limit of how a single transition is shaped.

What a colorant chemist should take from this #

Under a line lamp, look at where the lines are before designing a shape. The mercury lamp asks for one steep edge between two lines and nothing narrower anywhere, and a pigment whose absorption rises very steeply between 550 and 575 nanometres — a deep band, or overlapping bands, beyond the yellow — serves every splitting direction it can. A lamp with lines crowded on both sides of a third would ask for a gap, and the prediction’s design would then be the right one.

Depth is what an edge costs. A colorant’s absorption can be made steeper only by making it deeper, and the returns fall slowly — the same currency in which the pigment decides the trade, not the band found band-shape pigments beating a slope limit under four-emitter lamps; the lamp outweighs the pigment model found that which lamp is chosen moves a pigment’s reach more than which model describes it, and here the lamp decides even where the pigment’s depth has to go.

Search before concluding. A shortfall found by a search is an upper bound on the model’s shortfall, and a census that reports it as the model’s is making a claim about its search. Three lines spare a slow pigment and the censuses after it all searched the same way; this essay has re-searched one lamp’s splitting directions and no others.

How the census was taken #

The lamp, directions and band model are the depth census’s: the clear mercury lamp as the collection’s triphosphor tube less its coating, on the five-nanometre grid from 380 to 780 nanometres; every twelfth direction of a Fibonacci sweep of the solid under the CIE 1931 observer, 148 of them with a reachable boundary; and a pigment’s reflectance as the exponential of minus its absorbance, a constant plus up to three Gaussian bands, each with its own centre, at least forty nanometres wide, and of peak absorbance at most twelve. The splitting directions are those whose ideal pigment’s weights at 545 and 580 nanometres differ in sign. Each is searched with the census’s own Nelder–Mead from 680 starts — two bands at centres every ten nanometres from 470 to 660 and every twenty from 380 to 700, each at half or full depth, the third empty — and the best kept. The four-band search starts from the three-band optimum with the fourth band at every twenty nanometres, at depths of six and twelve. A pigment’s reach is its support in the direction over the ideal’s; its shortfall is one less that. The flank’s fall is the distance between the wavelengths at which a single Gaussian band of forty nanometres and a stated depth gives reflectance 0.05 and 0.95.

What the census leaves out #

The search is still a search. Six hundred and eighty starts found more than the census’s few; they are not a proof of the optimum, and one direction’s best answer uses only one band at its cliff where two would seem better.

Gaussian bands. Real absorption bands are not Gaussians. A band with a steeper flank than a Gaussian’s of the same width — a skewed band, or one broadened unevenly — would fall faster for its depth, and the essay’s arithmetic of depth would change with it.

One lamp, one model. The slope and energy models have not been searched again, and nor have the other eight lamps. The correction here is to the band model under the mercury lamp.

Still open: whether the yellow line’s smear sets the floor #

The fifteen directions still short by more than one per cent at every depth tried are short because the cliff must fall across twenty-five nanometres at most, and the lamp’s lines are not points: the green line’s power reaches 540 nanometres and the yellow’s 585, a smear of the grid and of the lines’ own 6 nm width in the collection’s lamp model.

The calculation is this census with the lamp’s lines narrowed — 3 nm and 1.5 nm wide, on a grid fine enough to carry them — and the band model searched as here. The prediction is that the room for the cliff grows towards the lines’ 32-nanometre separation as they narrow, that two stacked bands then fit it with room to spare, and that the count short by one per cent falls from fifteen towards the handful whose ideal also needs the blue; so that the mercury lamp’s residual cost is in the width of its lines, which a high-pressure lamp broadens and a low-pressure one keeps narrow.

A shape answers the question it was built for #

The habit here is about fitting a remedy to a question.

A narrow gap between two bands is the right shape for a colour that must pass one narrow band of light between two others. Splitting two lines looked like that question and was not: with nothing to absorb on the green line’s far side, what the pigment needed was one steep side, and the answer was depth.

The failure mode is to design for the difficulty a problem seems to have rather than the one it has. The test is to write down exactly what the problem asks — here, the four signs a direction puts on four lines — before choosing a shape to answer it.