A band pigment must be narrower than the gap it splits

A clear mercury lamp is four visible lines and nothing else, and under it the colours a surface can have 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 it against the ideal pigment that switches instantly between reflecting and absorbing. A mercury lamp’s few losses are deep found the losses concentrated in the directions that ask a pigment to reflect the green line and absorb the yellow, 31.5 nm away, or the reverse. A pair of lines asks for a cliff, not a gap found that what those directions need is one steep flank between the two lines, and that a pigment built from absorption bands no narrower than forty nanometres still fell short by more than a per cent in fifteen directions however deep its cliff.

A slow pigment splits mercury’s lines only if its swing fits between them answered the same question for the other model, a pigment whose reflectance may change only so fast. On a one-nanometre grid with the lines narrowed, it lost no direction while its full swing from nought to one took 30 nm or less, and fourteen once it took 34 — a step at the lines’ 31.5 nm spacing. The pigment’s losses were arithmetic: a swing longer than the gap cannot reflect one line fully and absorb the other — the arithmetic three lines spare a slow pigment first met under three emitters, where the lamp stopped helping once the pigment’s transitions were as wide as the lines were far apart.

A band pigment does not swing at a fixed rate. Its reflectance is the exponential of minus its absorbance, and its absorbance a sum of Gaussian bands, so its flanks are steep where a band is deep and shallow in the band’s tails. The prediction was that the band pigment’s stubborn directions are the same arithmetic: its count of losses falls to nought once its band is narrow enough that its flank fits in the gap — near a band thirty nanometres wide. If the count fell well before that, the band’s steep middle would be doing the splitting and the gap arithmetic would be the slope limit’s alone.

Narrower bands lose fewer, and none at twenty-five #

Searched on the one-nanometre grid, the mercury lines one and a half nanometres wide and the bands no deeper than the census’s twelve, the band pigment loses fifteen of the 25 splitting directions by more than one per cent with bands no narrower than 40 nm, eleven at 35, nine at 30, and none at 25; at 20 nm one direction is lost by just over a per cent, which a narrower band could always match and the search did not find. In every lost direction the pigment already splits the pair, reflecting most of one line and a few per cent of the other, and falls short on both. One Gaussian band twelve deep, placed as well as it can be to absorb one line and pass the other 31.5 nm away, errs on one of them by 3.2 per cent at forty nanometres wide, 1.8 at thirty-five, 0.8 at thirty and 0.2 at twenty-five. Allowed twice the depth, the bands lose fourteen directions at forty nanometres, nine at thirty-five and two at thirty: depth recovers losses only where a band is already narrow enough nearly to split the pair, and at every width narrowing the bands by five nanometres lowers the mean shortfall more than doubling their depth.

• The prediction holds in kind and misses in size. The count does fall to nought as the band narrows, but at twenty-five nanometres, not thirty.
• The stubborn fifteen are the census’s fifteen. Forty-nanometre bands lose exactly as many directions on the fine grid as the coarse search left short.
• The limit is the band’s tail, not its flank. A Gaussian band deep enough to absorb one line fully still absorbs a little of the line beside it, and only a narrower band gets its tail out of the way.

The solid curve is the census’s band pigment, its depth capped at twelve. It falls from fifteen at forty nanometres through eleven and nine to nought at twenty-five, a slope rather than a step: the slope-limited pigment’s count jumped within a few nanometres of the gap, and the band pigment’s runs down across fifteen. The dashed curve lets the bands go twice as deep. It starts almost where the solid one does — fourteen at forty nanometres against fifteen — and parts from it as the bands narrow: nine against eleven at thirty-five, two against nine at thirty, and none at twenty-five or twenty, where the search at depth twelve missed one. The gap, marked at 31.5 nm, sits partway down the curve, not at its foot.

Every loss splits the lines and falls short at both #

The slope-limited pigment lost a direction by failing to complete its swing: it reflected one line and could only part-absorb the other. The band pigment’s losses look different. In every one of the fifteen it reflects 93 to 99.5 per cent of the line it should reflect and under 1 to 26 per cent of the line it should absorb, and the shortfall — between 1.2 and 9.9 per cent of the ideal’s reach — is split between the two. Eight of the fifteen ask for the green reflected and the yellow absorbed, seven the reverse, and the three deepest losses are all of the second kind. It is not that the pigment cannot make a cliff between the lines; it makes one, and the cliff is not clean on either side.

The figure shows why. The direction reached least well wants the yellow reflected and the green absorbed, and falls 9.9 per cent short. To absorb the green line the forty-nanometre pigment parks a band at its full depth at 523 nm, 23 nanometres short of the green, so that the green sits on the band’s long-wavelength flank, where the absorbance is still nearly five and lets through one per cent; the yellow line, 31.5 nm further on, sits on the band’s tail, where the absorbance has fallen to 0.07 — and 0.07 of absorbance is seven per cent of the yellow’s light. Moving the band nearer the green absorbs it better and reaches further into the yellow; moving it away does the reverse, and 523 nm is the balance. Moved by the handle, the narrower bands sit closer to the green — at 528, 532, 536 and 540 nm — absorb it as fully and leave more of the yellow alone, and at twenty-five nanometres the reach is 99.3 per cent.

One band’s arithmetic #

The mechanism can be priced without searching anything. One Gaussian band of full width w and peak absorbance twelve, placed to minimise the larger of its two errors — light let through on the absorbed line, light lost on the passed one — errs by 3.2 per cent at w = 40 nm, 1.8 at 35, 0.8 at 30, 0.5 at 28 and 0.2 at 25. The error falls steeply as the band narrows, because a Gaussian’s tail falls as the exponential of the square of distance over width: halving the band’s width does not halve its reach into the neighbouring line, it squares away most of it.

That puts the threshold for a one-per-cent split near thirty-one nanometres — at the gap itself, and close to the prediction’s thirty. The census’s directions ask more than each line split to one per cent, because the two lines do not weigh the same. A direction’s reach is measured against what the ideal pigment gathers from the lines it reflects, and the line it absorbs can carry several times that weight, so every per cent the pigment leaks there counts several times over. That is why at thirty nanometres the single-band arithmetic clears one per cent and nine directions still do not; at twenty-five both have room to spare.

The direction the census reaches least well shows the weighting plainly. It wants the yellow reflected and the green absorbed, and its weight on the green line is nearly three times its weight on the yellow. With bands thirty nanometres wide the pigment parks one at 532 nm, absorbs all but 0.3 per cent of the green and passes 98.3 per cent of the yellow — cleaner on the absorbed line and less clean on the passed one than the single-band arithmetic’s even 0.8 per cent each, a trade the green’s heavier weight rewards. Yet the direction is still 2.5 per cent short: 0.84 of it from the green’s small leak, multiplied by the green’s weight, and 1.66 from the yellow’s loss. None of the shortfall comes from the violet or blue lines, where the pigment’s second band, 67 nanometres wide near 430 nm, does what the direction asks. Every bit of what the band pigment loses under mercury is lost on the two lines 31.5 nanometres apart, and a direction that weighs the absorbed line heavily needs a cleaner split than one per cent a line.

Depth buys a logarithm, width buys a square #

Doubling the depth a band may reach is the other way to sharpen its edge, and a sharp edge is bought with depth found that the edge a band draws in reflectance depends on how much colorant is in the film, not only on the band’s width. The census’s bands may go to an absorbance of twelve. Allowed twenty-four, one band errs by 2.0 per cent at forty nanometres, 1.0 at thirty-five, 0.39 at thirty and 0.09 at twenty-five — against 3.2, 1.8, 0.8 and 0.2 at twelve. Doubling the depth cuts the error by between a third and three fifths; narrowing the band by five nanometres cuts it by more at every width.

The reason is in the Gaussian’s tail. A band of peak absorbance $K$ and width $w$ has an absorbance at distance $x$ from its centre of $K\exp(-4\ln 2\, x^2/w^2)$. To absorb one line strongly and pass the other, the band must put the absorbed line where that absorbance is still large and the passed line where it has fallen to almost nothing, 31.5 nm further on. Doubling $K$ moves the place where the absorbance reaches any given value outward only by the square root of a logarithm; narrowing $w$ pulls the whole tail in proportionally. A deeper band can afford to sit a little further from the line it absorbs, and its tail then reaches a little less far into the line it passes, but the gain shrinks as the depth grows. A narrower band shortens everything at once.

The search shows the same thing on the 25 directions, with a twist. Allowed twenty-four deep, the band pigment loses fourteen directions at forty nanometres, nine at thirty-five, two at thirty and none at twenty-five or twenty. At forty nanometres the extra depth recovers one direction; at thirty it recovers seven of nine. Depth helps most where the band is already narrow enough nearly to split the pair, and least where it is not; and by the mean shortfall over the 25 directions, narrowing the bands by five nanometres beats doubling their depth at every width from forty to thirty.

What this means for the two pigment models #

Both of the census’s slow pigments lose the mercury lamp’s splitting directions for one reason: their transition from reflecting to absorbing is longer than the lines’ spacing. For the slope limit that length is its swing, a fixed number. For the band pigment it is the distance over which a Gaussian band falls from absorbing a line nearly completely to absorbing almost none of it, a little longer than the band’s half-height width, and it depends on how deep the band may go. The pigment decides the trade, not the band found the two models parting widely under a four-emitter lamp, the band pigment losing a third fewer directions; under mercury’s close pair they fail for one reason. The census’s forty-nanometre bands and forty-nanometre swing are both on the wrong side of mercury’s 31.5 nm; a colorant designed for a mercury-line source wants a band under about thirty nanometres, or a swing under thirty.

The essay that found the slope limit exact left the band models’ counts as the census’s one uncertainty, searched rather than solved. Here the search is small — twenty-five directions — and its count at the census’s width agrees with a much larger earlier search’s, fifteen and fifteen. One count, at twenty nanometres, is a search miss: a narrower band can always do what a wider one does, so a loss there that twenty-five nanometres avoided is the search’s, and it is reported as found rather than corrected.

How the band pigment was searched #

The lamp, grid and directions are the one-nanometre census’s: the clear mercury lamp’s four lines one and a half nanometres wide, the CIE 1931 functions interpolated to one nanometre by Sprague’s formula, and the census’s 148 directions, of which 25 split the green line at 546.5 nm from the yellow at 578. The band pigment’s reflectance is e to the minus its absorbance, the absorbance a floor plus three Gaussian bands, each no narrower than the stated width and no deeper than twelve, or twenty-four. It is searched by Nelder–Mead from structured starts: one band over each run of negative weight, the same widened and deepened, and a band placed with its flank in the green–yellow gap at seven offsets on the side of the line to be absorbed — nine starts, fifteen hundred steps each. The reach is the best weighted sum over the ideal’s.

A single band’s arithmetic places a Gaussian of the stated width and depth at every offset from the absorbed line in twentieths of a nanometre and keeps the placement whose larger error — transmission at the absorbed line, absorption at the passed one — is least.

What this does not reach #

A search, not a solution. Every count is what nine starts found, and a harder search pulls the band pigments away from the slope limit showed how much a band model’s count can move with the effort spent finding it; the search at twenty nanometres missed one direction a narrower band must be able to reach, and others may be missed silently.

One lamp. The mercury lamp’s pair is the census’s sharpest test; a lamp with lines further apart, or a tube with mercury lines over a continuum, sets a different gap and a weaker penalty for missing it.

The bands’ shape. Real absorption bands are not Gaussian; a band with a steeper flank and a shorter tail — a Lorentzian is the opposite, a vibronic band often both — would split the pair at a greater width.

Still open: whether a band with a shorter tail splits the pair at forty #

The band pigment’s losses come from its Gaussian tail reaching the neighbouring line. Real dye absorption bands are often asymmetric, falling steeply on their long-wavelength side and trailing on their short; a band whose steep side faces the line it must pass would reach less far into it.

The calculation is this census with the bands given a skewed shape — a Gaussian whose long-wavelength side is narrowed to half the short side’s width — at forty, thirty-five and thirty nanometres of total width, its steep side free to face either way. The prediction is that a skewed forty-nanometre band loses about as many directions as a symmetric thirty, because what matters is the half-width on the side facing the passed line; and that the skew helps only the directions that pass the green and absorb the yellow, where the steep side can face the green. If the skewed band helps both kinds of direction equally, the pigment is using its other bands to shape the cliff, and the single-band arithmetic understates what three bands can do. A steep side splits mercury’s lines at forty nanometres makes that calculation.

A model’s limit is in its shape, not its label #

The habit here is about what a “width” means when two models are compared.

The census called both of its slow pigments forty nanometres — a slope limit with a forty-nanometre swing and a band pigment with forty-nanometre bands — and the earlier essay found the slope limit’s mercury losses set by that number against the lines’ spacing. The natural guess was that the band pigment’s forty meant the same thing. It nearly does: the band pigment needs a band of about twenty-five to thirty to split what the slope limit needs a swing under thirty to split.

The failure mode is to compare two models through a parameter that shares a name and a unit but describes different shapes. The test is to find the shape each model’s parameter controls at the point that matters — here how far a transition reaches into a line 31.5 nm away — and compare that. For the slope limit it is the whole swing; for the band pigment, a Gaussian tail, and the tail is what the lines see.