A band must be nearly two to one before its steep side pays

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 four wavelengths. The hardest colours to reach are those that ask a pigment to reflect one of the green and yellow lines and absorb the other, 31.5 nanometres away — a pair of lines asks for a cliff, not a gap — and a band pigment must be narrower than the gap it splits found a pigment of Gaussian absorption bands no narrower than forty nanometres losing fifteen of the twenty-five such directions by more than one per cent, because a band’s tail reaches the line it should spare.

A steep side splits mercury’s lines at forty nanometres then made each band lopsided, one side a third of its width and the other two thirds, and let the search turn the steep side either way. The pigment lost one direction instead of fifteen, and in every fit the absorbing band faced its steep side towards the line it had to pass. A single such band, placed as well as it can be, split the two lines within 0.37 per cent, where a symmetric band of the same width erred by 3.2.

Two to one is a strong skew. Real dye bands are often lopsided, but by less, and the practical question is how much of the gain survives a milder asymmetry. The prediction was that the count of lost directions falls roughly in step with a single band’s error — to about eight at sides of 0.55 and 0.45 of the width, three at 0.6 and 0.4, one or two at 0.65 and 0.35 — so that a skew of three to two already halves the symmetric band’s losses. The alternative was that the count stays near fifteen until the skew passes 0.6, in which case a pigment needs a markedly steep side before its band shape matters, and a dye’s modest natural asymmetry buys little under a line lamp.

Slowly, then at once #

Searched on the one-nanometre grid as before, with bands forty nanometres wide whose wide side is 0.55, 0.6 and 0.65 of the full width and the steep side’s direction chosen in the search, the band pigment loses thirteen, ten and five of the 25 splitting directions by more than one per cent — against sixteen with symmetric bands searched the same way and one at two to one. The directions recovered per twentieth of skew rise at every step: three, three, five and twelve. Wherever a single band of the same skew splits the two lines with an error above one per cent, the pigment loses ten or more directions; wherever below, five or fewer, and the single band crosses one per cent near a skew of 0.62. The worst direction’s shortfall falls from 9.9 per cent to 3.6 at 0.6 and 1.2 at two to one. At every mild skew the absorbing band’s steep side faces the passed line in at least twenty-four of the 25 fits.

• The prediction is too optimistic by about half at every step. Three to two removes six losses of sixteen, not thirteen.
• The alternative is nearer the truth. The count falls steadily but slowly until the skew passes 0.6, and most of the gain comes after.
• The mechanism the prediction named holds: the count does follow one band’s error — measured against the census’s own one-per-cent line rather than in proportion.

The found curve and the predicted one start together and part at once. The found count falls three directions for each of the first two twentieths of skew, five for the third and twelve for the last, where the prediction expected the largest drop first. The curve bends downward, and the prediction’s bends the other way. A dye chemist reading the prediction would have expected most of the benefit from a mild asymmetry; the census says most of it comes only near two to one.

What one band says #

The single band’s error falls smoothly as the band skews: 3.2 per cent symmetric, 2.1 at 0.55, 1.16 at 0.6, 0.52 at 0.65, 0.37 at two to one. On a logarithmic scale it is nearly a straight line, steepening a little at the end. Nothing in the single band’s behaviour is sudden. What is sudden is the census’s count, and the reason is the line it counts against.

A direction is lost when the pigment falls short of the ideal by more than one per cent. The splitting directions differ in how much they ask — some need a nearly perfect cut between the lines, others only a rough one — and a direction whose demands are close to what one band can do will be lost while that band’s error is above about one per cent and reached once it is below. The census’s count is the number of directions whose demands lie above the single band’s error, and that number changes fastest where the error passes through the band of demands the directions actually make.

The count against the error #

Plotted against the single band’s error, the five counts fall in order along one curve, and the one-per-cent line divides them cleanly: ten or more lost while the single band errs by more than one per cent, five or fewer once it errs by less. The prediction was right that the count follows the single band’s error and wrong about the scale: it assumed the count would fall in proportion to the error, while the census’s count falls in proportion to how many directions’ demands the error has passed.

That gives a rule a pigment designer can use without a census. Compute the error a single band of the proposed shape makes splitting the lamp’s closest pair of lines; if it is well above the tolerance the colours will be judged by, the pigment will miss most of the colours that pair defines, whatever else it does. The skew that matters is the one that brings that error under the tolerance — for mercury’s green and yellow at one per cent, a wide side about 0.62 of the width.

The worst direction #

The worst direction tells the same story in another unit. Its shortfall falls from 9.9 per cent with symmetric bands to 6.4 at 0.55, 3.6 at 0.6, 1.6 at 0.65 and 1.2 at two to one; the mean over all 25 from 2.9 per cent to 0.38. Unlike the count, these fall steadily from the first step, because they measure how short the pigment falls rather than whether it falls short by more than a threshold. A colour scientist judging the pigment by its worst colour would see a mild skew helping at once; one judging it by how many colours it misses would see little until the skew is strong. Both are right about the same pigment, and the count is the stricter reading.

The band’s flank at each skew #

Drawn at a fixed centre, the three bands show where the gain comes from. At the green line the symmetric band reflects 79 per cent, the 0.6 band 97 and the two-to-one band nearly all of it. The flank retreats from the green as the steep side steepens, and the first step of skew moves it further than the last. The search does not hold the centre fixed: each pigment places its band to balance both lines, and the less skewed bands must sit closer to the green to absorb the yellow fully, giving back some of what the retreat bought. That balance is what keeps the mild skews’ counts high.

Both kinds of split #

The two kinds of direction recover together. With symmetric bands nine of the fourteen green-passing directions and seven of the eleven yellow-passing ones are lost; at 0.6, six and four; at 0.65, three and two; at two to one, none and one. In at least twenty-four of the twenty-five fits at every mild skew the absorbing band turns its steep side towards the passed line, as it did at two to one. The rule the stronger skew showed — steep towards what must be spared — holds from the first step, and both kinds of split benefit equally because the search can turn the band either way.

Whether the search found the shapes it was given #

A census of searched pigments counts what its search found, and a count that falls slowly could mean the mild skews help little or that the search finds their best placements less often. A harder search pulls the band pigments away from the slope limit is the warning: searched from hundreds of starts rather than a few structured ones, the symmetric band pigments reached further in most of the directions they had lost, and under this same mercury lamp their deep losses halved.

Two readings say the mild counts are not search failures. The first is the steep side’s direction. A search that failed to settle would leave the steep side facing either way at random, and in twenty-four or twenty-five of every twenty-five fits it faces the passed line, which is the geometry the single band says is best. The second is the agreement with the single band: the one-band calculation is not a search at all, but a sweep of every offset in fiftieths of a nanometre, and the census’s counts fall on the side of the one-per-cent line the sweep predicts at all five skews. A search losing directions it should have reached would put mild-skew counts above that line’s prediction without regard to it; they sit where the sweep says they should.

What the harder search could still do is lower all five counts together, as it did for the symmetric bands. That would move the curve down without straightening it, because the knee comes from where a single band’s error crosses the tolerance, and no search can make one band split two lines better than its best placement does.

The slope limit’s reading of the same pair #

The band pigment is one of two models of a slow pigment this collection compares, and the other is a slope limit: a reflectance that may not change by more than its full swing over forty nanometres. A sharp edge is bought with depth found the band model reaching further than the slope limit in most directions, because a band deep enough draws a reflectance edge steeper than its own width, and the slope limit was exact all along found the slope limit’s counts unmoved on every finer grid. Under the clear mercury lamp a slow pigment splits mercury’s lines only if its swing fits between them: forty nanometres of swing do not fit into thirty-one and a half, and on the census’s five-nanometre sampling every direction the slope-limited pigment loses is a split of the green and yellow lines.

A skewed band sits between the two. Its steep side draws an edge narrower than forty nanometres where the lines need one, and its wide side keeps the band’s total width. At two to one the steep half-width is about thirteen nanometres, and the edge it draws at twelve deep fits comfortably between the lines; at 0.55 the steep half-width is eighteen, and the edge is nearly as wide as the gap. The census’s knee is where that edge first fits, which is the slope limit’s own condition, stated for a band’s steep side rather than for the whole pigment.

Other lamps, other gaps #

The arithmetic carries to other lamps. Three lines spare a slow pigment found three narrow emitters helping a slow pigment until its transitions grew as wide as the lines were far apart, and the lamp outweighs the pigment model found the lamp mattering more than which pigment model is used. Mercury’s green and yellow are the closest pair of strong lines the clear lamp presents, which is why they define its hardest colours, and why a symmetric forty-nanometre band fails them where it passes emitters spaced further apart.

The same single-band sweep answers for any pair. Place one band at the gap the lamp presents, read its error against the census’s tolerance, and the skew that brings it under is the skew the pigment needs; a wider gap than mercury’s would need a milder skew at forty nanometres’ width, and a narrower one a stronger skew than this census searched.

What a pigment chemist should take from this #

A modest asymmetry buys a few colours under a line lamp; a strong one buys nearly all of them. A band whose sides are 0.55 and 0.45 of its width — a skew many dyes have naturally — recovers three of sixteen directions; one at 0.65 and 0.35 recovers eleven. The return per unit of skew grows fourfold across the range, so a chemist tuning a dye’s band shape for line lamps should expect little until the steep side is close to half the other.

Judge the shape by one band against the tolerance. The census’s count follows a single band’s error against the one-per-cent line so closely that the single band — a calculation of a fraction of a second — predicts which side of the line a pigment’s colours fall.

This is mercury’s pair. Other lamps’ closest lines sit at other spacings; a pair further apart would let a milder skew cross the tolerance, a closer pair would ask for a stronger one.

How the pigments were searched #

The pigment is the band census’s — a reflectance e^(−a), a the sum of a floor and three absorption bands no deeper than twelve, searched by Nelder–Mead from structured starts for two thousand steps — with each band forty nanometres wide at half depth and split at its centre: on one side a Gaussian whose half-width at half height is the stated share of the full width, on the other the rest, a fourth number per band choosing which side is steep. The shares are 0.55, 0.6 and 0.65 here, beside the symmetric control and the two-to-one skew of the earlier essay, searched the same way. The lamp is clear mercury with its lines one and a half nanometres wide on a one-nanometre grid, the 25 splitting directions the census’s, and a direction 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 fiftieth of a nanometre and its larger error kept.

What this does not reach #

One width. Every band is forty nanometres wide. At thirty-five or thirty, where the symmetric bands already lose fewer directions, a milder skew may be enough.

Structured starts, and one direction they missed. Every count here is what the census’s structured starts found. A band is only as wide as its steep side searched every direction lost at 0.6 and 0.65 again from the best fits at the milder skews and found one of them — direction 30, lost here at both skews and reached at 0.55 — reached within half a per cent at 0.6 and a third of one at 0.65. The counts at those two skews are nine and four, not ten and five; the other losses stand, and the knee’s late arrival stands with them, one direction lower at each step.

A threshold of one per cent. The count depends on where the census draws the line; a stricter tolerance would move the knee towards stronger skews and a looser one towards milder.

Two-sided Gaussians. Each side is a Gaussian; a real band’s steep edge may be steeper near its foot, which the census does not model.

Still open: whether a narrower band needs less skew #

A band thirty-five nanometres wide loses eleven directions symmetric and none at two to one, and the single-band arithmetic suggests its error crosses one per cent at a milder skew than a forty-nanometre band’s, because its flank starts further from the line it spares. If a chemist can narrow a band a little and skew it a little, the two together may do what a strong skew does alone.

The calculation is the band census at thirty-five nanometres with sides of 0.55, 0.6 and 0.65, the counts read as here, and the skew at which a single thirty-five-nanometre band’s error crosses one per cent. The prediction is that the single band crosses one per cent near 0.57 and the census’s count falls to five or fewer by 0.6, so that a band five nanometres narrower needs a skew about a twentieth milder to do as well. If the thirty-five-nanometre count still needs 0.65, width and skew do not trade, and the steep side must be steep whatever the band’s width. A band is only as wide as its steep side makes that calculation.

A threshold turns a curve into a knee #

The habit here is about counting against a line.

The single band’s error falls smoothly with skew, and the prediction expected the count of lost colours to fall the same way. A count of cases beyond a tolerance does not inherit the smoothness of the quantity it thresholds: it moves when the quantity passes through the range of the cases’ own demands, and that range sat just below the tolerance here, near the strong end of the skews.

The failure mode is to read a thresholded count as if it were the measurement beneath it. A count can sit nearly still while the measurement improves and then fall all at once, and a design judged only by the count would be abandoned just before it worked. The test is to report the measurement beside the count — here, the single band’s error and the worst direction’s shortfall beside the number lost — so that the improvement is visible before the count shows it.