A five-nanometre flank carries a fiftieth of Lorentzian

A clear mercury lamp’s light is mostly in a few narrow lines, the green at 546 nm and the yellow at 578 the brightest, and the most saturated surface colours under it reflect one of those lines and absorb the other. Of the directions in colour space that ask for that split, twenty-five can be reached by an ideal reflectance; the question this run of essays has followed is how nearly a pigment built from smooth absorption bands at least forty nanometres wide can reach them, and it has turned out to depend almost entirely on the steep side of one band — the edge that falls between the two lines.

A hundredth of Lorentzian undoes a Gaussian flank found that edge, fourteen nanometres wide at half depth, losing four directions by more than one per cent as a Gaussian and eleven with a single per cent of Lorentzian wing in it. A narrower flank moves its wing in with it narrowed the edge and found the Gaussian losing nothing at ten, seven and five nanometres while a twentieth of Lorentzian still lost thirteen, twelve and seven, because the narrower band slides fourteen nanometres in towards the line it must pass to keep the other line absorbed, and its wing there falls fourfold rather than eightfold. One band’s error falls roughly in proportion to the edge’s width, and at five nanometres its centre barely moves as the Lorentzian share changes.

If the band’s placement is fixed at five nanometres, the wing it carries to the passed line is simply the Lorentzian share times a geometric factor, and the single band’s error should grow in proportion to the share. The prediction was that the pigment’s count of lost directions follows that error: about two directions lost at a fiftieth of Lorentzian, none at a hundredth. If a hundredth still lost three or more, the count would not be following the single band, and the twenty-five directions would each meet the wing at a distance of their own.

None lost at a hundredth, and none at a fiftieth #

With a hundredth of Lorentzian in a five-nanometre steep side the pigment loses none of the twenty-five directions; its worst falls 0.31 per cent short of the ideal. With a fiftieth, the census’s search loses one direction, by 1.4 per cent — but searched again from the best pigments found for that direction at a hundredth and a twentieth, it reaches 99.84 per cent of the ideal, and the pigment loses none; its worst then falls 0.56 per cent short. With a twentieth the census’s search loses seven, and the second search recovers two of them: five are lost.

• The hundredth’s half of the prediction holds.
• The fiftieth’s half fails, the welcome way. A fiftieth loses nothing; the one loss the first search reported was a search that stopped short.
• The twentieth’s count, reported as seven, is five. The essay that measured it carries a correction.

Beside the fourteen-nanometre flank the five-nanometre one is a different pigment. At a hundredth, a fiftieth and a twentieth of Lorentzian the fourteen-nanometre steep side loses eleven, fourteen and eighteen directions; the five-nanometre side loses none, none and five. Narrowing the edge from fourteen nanometres to five buys a tolerance of a fiftieth of Lorentzian — a pigment whose bands carry a couple of per cent of collision-broadened wing can still split the lines cleanly if its edge is sharp enough — and at a twentieth it still loses well under a third of what the broad edge loses.

The search that stopped short #

The first search’s one loss at a fiftieth looked wrong before it was checked. Direction 35 reached 99.91 per cent of the ideal at a hundredth of Lorentzian and 99.67 at a twentieth, by the first search, and only 98.59 at a fiftieth. A pigment with less Lorentzian in its edge can always do at least nearly as well as one with more, because its edge is closer to the Gaussian that loses nothing; a reach that dips between its neighbours is the search’s, not the pigment’s. Started from the pigments it found at the neighbouring shares, the second search finds direction 35 at 99.84 per cent. Every other direction moves by under four hundredths of a point, so the second search changed one answer by a point and a quarter and left the other twenty-four as they were.

The census’s search for each direction starts from fourteen structured guesses — bands placed over the gaps the direction asks for, bands with their steep sides facing the passed line, and the best pigment of a milder skew — and runs Nelder–Mead from each. That is a thorough search for a problem with thirteen numbers, and it still misses a better valley now and then. A harder search pulls the band pigments away from the slope limit found the same once before: band pigments searched from more starts lost a sixth of their losses. The second search here is cheaper and more pointed — it starts from answers already known to be good for a nearly identical problem — and it finds the misses where the first search’s answers jump between neighbouring shares.

At a twentieth it recovers directions 30 and 37, from 98.9 and 97.7 per cent to 99.3 and 99.35, and leaves directions 49, 54, 61, 120 and 131 lost. The seven-nanometre and ten-nanometre counts a narrower flank moves its wing in with it reported have not been searched again, and they may be overstated by a direction or two in the same way.

One band’s error grows in proportion to the share #

At five nanometres the single band’s error is 0.22 per cent at a hundredth of Lorentzian, 0.43 at a fiftieth and 1.05 at a twentieth — about a fifth of a per cent for each per cent of Lorentzian, within a quarter of proportion across the range. At fourteen it is 1.2, 2.0 and 4.2 per cent. The prediction’s premise holds: at five nanometres the error grows as the share does, because the band sits where it sits whatever the share.

With no Lorentzian the five-nanometre band’s centre sits 31.5 nm from the passed line — exactly the lines’ spacing, the absorbed line at the band’s centre; with a hundredth it moves to 36.4 nm, and with a twentieth to 37.4. The first hundredth moves the band by five nanometres, as it trades a little of its absorption of the green line for distance from the yellow; every further share moves it by a nanometre. The fourteen-nanometre band moves from 46.8 to 48.3 to 51.0 nm over the same shares, never settling. That is why the five-nanometre pigment follows proportion and the fourteen-nanometre one does not: one band holds its place and lets the wing grow, the other keeps moving to escape it.

The pigment’s shortfall follows the band #

Searched twice, the five-nanometre pigment’s mean shortfall over the twenty-five directions is 0.01 per cent with no Lorentzian, 0.15 at a hundredth, 0.26 at a fiftieth and 0.55 at a twentieth; its worst direction falls short by 0.15, 0.31, 0.56 and 1.2 per cent. The fourteen-nanometre pigment’s mean is 0.52, 1.2, 1.7 and 3.0. From a hundredth to a twentieth the mean grows by 3.7 times while the single band’s error grows by 4.7: the pigment’s shortfall follows the band within a third, as the prediction’s mechanism required.

So the count of lost directions is a threshold applied to a quantity that grows smoothly. The worst direction’s shortfall crosses one per cent between a fiftieth and a twentieth of Lorentzian, at about four per cent by a straight line between them; below that share no direction is lost, and the count’s prediction of two at a fiftieth placed the threshold too early.

The whole spread of directions rises together. With no Lorentzian every direction lies within 0.15 per cent of the ideal; at a hundredth within 0.31; at a fiftieth within 0.56; at a twentieth, five directions lie past one per cent and the rest are spread up to it. The directions do not meet the wing at distances of their own, which was the alternative the prediction named; they meet it in proportion, and the ones nearest the line are the ones the twentieth pushes over.

Where the band pigment now stands #

This run began with a pigment that could not split the lines at all. A mercury lamp’s few losses are deep found every direction a slow pigment loses badly under this lamp asking it to reflect one of the two lines and absorb the other; a band pigment must be narrower than the gap it splits found symmetric bands forty nanometres wide losing fifteen of the twenty-five splitting directions, and narrower bands fewer. The steep side was the way past the band-width floor: a steep side splits mercury’s lines at forty nanometres gave the band one edge sharper than the rest, and a band must be nearly two to one before its steep side pays found the losses falling slowly with the skew at first and fastest in the last step to two to one.

Every step since has been about that one edge, and the pigment that results is easily described. A band forty nanometres wide, one side broad and the other a few nanometres wide at half depth, with its steep side’s profile close to Gaussian — under a fiftieth of Lorentzian — reaches every one of the twenty-five directions to within one per cent. That is a statement about a model pigment, but it has a physical reading: a dye whose absorption band has a steep, inhomogeneously broadened edge, little affected by collisions, placed so that edge falls between 546 and 578 nm. Whether any real colourant has such a band is a question for spectroscopy, not for this census; the census can say only what such a band would buy, and that the buying stops at about four per cent of Lorentzian whatever else is done to the edge. A pigment designer who can narrow an edge but not change its line shape gets a fiftieth’s tolerance at five nanometres and none at fourteen.

What has been learned about the edge is that its sharpness and its shape trade against each other in a measurable way. At fourteen nanometres no Lorentzian at all was the only safe shape, and even the Gaussian lost four directions. At five, a fiftieth is safe and a twentieth loses five. Each narrowing pays for some Lorentzian, and the price is set by how far the band’s best placement leaves the passed line in half-widths: the further out the line sits, the more wing the edge can carry before the line feels it.

The slope limit was exact all along showed the census’s other model of a slow pigment — one whose reflectance may change only so fast per nanometre — needing no second search at all, because its best answer has a closed form. The band pigments have no closed form, and this essay’s correction is a reminder that every count of their losses is a count of what a search found. The counts here at five nanometres have now been searched twice and are the firmest the run has; the counts at other widths are as firm as one search makes them.

What a pigment chemist should take from this #

A sharp absorption edge tolerates a little collision broadening. At five nanometres half-width, a band whose profile carries up to a fiftieth of Lorentzian splits mercury’s two brightest lines in every direction within one per cent; at fourteen nanometres the same share loses fourteen directions of twenty-five.

Below about four per cent of Lorentzian, sharpness wins. The pigment’s shortfall grows in proportion to the share once the edge is sharp enough to hold its place, and the threshold of one per cent is reached between a fiftieth and a twentieth.

Check a census’s counts where they jump. A loss that appears at one share and vanishes on either side of it is more likely a search that stopped short than a property of the pigment, and a second search from the neighbours’ answers is cheap.

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 five nanometres wide at half depth — the Gaussian and the Lorentzian of the same half-width summed with weights one less the share and the share — 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 first search runs Nelder–Mead for 2,000 steps from fourteen structured starts; the second runs it for 4,000 steps from the first search’s best pigment for the same direction at each of the other shares, and the better of the two is kept. A direction is lost when its 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 #

The other widths’ counts. Only the five-nanometre pigment has been searched twice; the seven- and ten-nanometre counts may also be overstated.

Shares between a fiftieth and a twentieth. The threshold’s crossing near four per cent is a straight line between two searched shares.

Real dyes. A pseudo-Voigt is a sum, not a convolution, and a real dye’s band is neither; the conclusion rests on the wing’s power law, which both share.

Still open: whether the second search lowers the narrow flanks’ counts #

The second search recovered two of the five-nanometre pigment’s seven directions at a twentieth of Lorentzian, and the seven- and ten-nanometre pigments’ counts at the same share — twelve and thirteen — came from the same first search.

The calculation is the forty-nanometre pigment at steep half-widths of seven and ten nanometres with a twentieth of Lorentzian, every direction searched a second time from the best pigments found at the neighbouring widths, the counts read as here. The prediction is that each count falls by one or two, to about ten and eleven, and that the narrowing from fourteen nanometres to five is then worth about thirteen directions rather than eleven, with the fall in the count nearly linear in the half-width. If either count falls by four or more, the first search was weakest exactly where the edge is intermediate, and every count in this run between the Gaussian and the full Lorentzian needs searching again before its shape can be read.

Each direction falls short by one band’s ratio made that calculation: both counts stand, and every direction’s shortfall falls with the half-width by one band’s ratio, so the count is a ranking of constants read against a moving line.

A count that jumped between its neighbours #

The habit here is about reading a result that is out of line with its neighbours.

The first search reported a direction reachable at a hundredth and at a twentieth of Lorentzian and lost at a fiftieth between them. A pigment with less wing cannot be much worse than one with more, so the dip was a property of the search; searching that direction again from its neighbours’ answers removed it, and with it the only loss at a fiftieth. The same second search then lowered a count published one essay earlier.

The failure mode is to report an optimiser’s answer at face value where it breaks a monotonicity the problem guarantees. The test is to list the places where neighbouring settings’ answers jump against the expected order, search those again from the neighbours’ solutions, and report what moved.