A slow pigment splits mercury's lines only if its swing fits between them

A clear mercury lamp is four visible lines and nothing else: violet at 405 nanometres, blue at 437, green at 546 and yellow at 578. Under it the object-colour solid — every colour a surface can have — is shaped by those four wavelengths alone, and since the limits assume a pigment that switches instantly a run of essays has found it the strangest of the census’s nine lamps for a slow pigment. A mercury lamp’s few losses are deep found that it costs a slow pigment few directions of the solid but costs them heavily, and that every deep loss asks the pigment to reflect one of the green and yellow lines and absorb the other. A pair of lines asks for a cliff, not a gap found that what such a direction needs is a single steep flank between the two lines, and that fifteen directions stayed short however steeply a band pigment’s flank was cut.

Its closing section suspected the census itself. The spectrum is sampled every five nanometres, and the mercury lines in the lamp model are six nanometres wide, so each line is spread across one band and a little of its neighbours. A cliff that should fall in the gap between the green and yellow lines can only fall between samples, and the samples lie close to the lines. The slope limit was exact all along then showed that the slope-limited pigment’s count is exact on the five-nanometre grid — exact on its sampling of reflectance — and pointed at this as the one grid it did not touch.

The prediction was that on a one-nanometre grid the slope limit’s mercury count falls by about a third at the census’s line width, because a cliff can then sit wholly between the lines, and that it falls further as the lines are narrowed — so that the lamp’s residual cost lies in the width of its lines, which a high-pressure lamp broadens and a low-pressure one keeps narrow.

The grid moves nothing; the gap decides #

Sampled every nanometre — the CIE 1931 functions interpolated by Sprague’s fifth-order formula, the lamp drawn line by line, the ideal recomputed and the slope limit solved exactly at the census’s 40 nm swing — the mercury lamp loses the same 21 directions by more than one per cent as on the five-nanometre grid, and 12 by more than five against 11. Narrowed to three nanometres, its lines cost 18 directions; to one and a half, 17. With the narrowest lines, a pigment whose swing from nought to one takes 30 nm or less loses no direction at all, one taking 31 nm loses three and one taking 34 nm loses fourteen; the green and yellow lines are 31.5 nm apart. Every direction lost, at every width and swing, splits the green line from the yellow. Tungsten and three narrow emitters, sampled every nanometre, lose within two directions of their census counts.

• The prediction fails on its first claim. Nothing about the five-nanometre sampling was holding the cliff back at the census’s width.
• Narrow lines help only a pigment that can use them. A 40 nm swing cannot fit between lines 31.5 nm apart however thin they are.
• The losses are a property of the lamp’s spacing against the pigment’s swing, and the census’s sampling of either was never the issue.

The three curves are the same census with the lines drawn at three widths, and they part in the middle. With the narrowest lines the count is nought for every swing up to 30 nm, three at 31, eight at 32, twelve at 33 and fourteen at 34 — a step a few nanometres wide, sitting on the green-to-yellow gap marked on the axis. With lines three nanometres wide the step begins earlier and climbs more slowly, five lost at 30 nm. With the census’s six-nanometre lines there is no step at all: two lost at a 24 nm swing, sixteen at 30, twenty-one at 38 and 40.

Above the gap the curves come back together. At 40 nm, the census’s swing, they read 21, 18 and 17; at 48 nm, 23, 21 and 19. A pigment that needs longer than the gap to swing loses the splitting directions whatever the lines’ width, because it cannot go from fully reflecting one line to fully absorbing the next in the space between them.

Sampling every nanometre moves almost nothing #

The first result is a non-result, and it rests on getting the finer grid right. The colour-matching functions are published at five nanometres in the census’s tables; at one nanometre they are Sprague’s interpolation, the fifth-order formula the CIE recommends for exactly this, which passes through every tabulated value and reproduces a fifth-degree polynomial exactly away from the table’s two ends. The mercury lamp drawn on the five-nanometre grid as four six-nanometre lines is the census’s lamp to fifteen decimal places. On the one-nanometre grid it loses 21 directions, exactly the census’s count; tungsten loses 78 against 77, and the three emitters 33 against 31. Those emitters are the lamp three lines spare a slow pigment found gentle on a slow pigment until its transitions grew as wide as the lines’ spacing — the arithmetic this essay finds under mercury, met first on a lamp whose lines are further apart.

The smooth lamps move by a direction or two in both directions of the reach: the ideal pigment can place its own edges more precisely on a finer grid, and so can the slow one, and the two refinements nearly cancel. The mercury lamp does not move at all. Its splitting directions ask for a swing between two lines that are thirty-one nanometres apart on either grid, and the slow pigment’s forty-nanometre swing does not fit on either grid.

The cliff a splitting direction asks for #

The direction the census reaches least well among the splitting ones wants the yellow line reflected and the green absorbed: its weights are strongly negative at 546 nm and positive at 578. The ideal pigment does exactly that, absorbing everything to just past the green line and reflecting everything from there on. The slow pigment with a 40 nm swing must also absorb the green fully — its weight there is the largest — and then can climb only thirty-one fortieths of the way by the yellow line, reflecting 78 per cent there and reaching 77.3 per cent as far as the ideal. With a 30 nm swing, moved by the handle, the climb completes between the lines and the pigment reaches 99.5 per cent; what it still loses is the lines’ own width, a little light on the green line’s long flank that a climbing pigment cannot avoid reflecting.

The picture is the whole of the mechanism. The pigment’s swing is a length on the wavelength axis, and the lamp’s gap is another length on the same axis. When the swing is shorter than the gap the cliff fits, and the only cost is the tail of each line spilling onto the slope. When the swing is longer, the cliff falls short by the difference, and the shortfall is set by the ratio of the gap to the swing, not by how finely the axis is drawn.

One ratio predicts the depth #

The cliff figure gives the shortfall a size that can be worked out on the back of an envelope. A pigment that must absorb the green line fully and may then climb by one swing’s worth over the swing’s length reaches, at the yellow line, the fraction of a full climb that the gap allows: 31.5 nanometres of a 40 nm swing is 0.79, and the computed reflectance on the yellow line is 0.78, the difference being the green line’s own half-width, which the climb must also clear before it starts. The direction the census reaches least well carries nearly all of its positive weight on the yellow line, so its reach is close to that same fraction, and the computed 77.3 per cent sits a point below it for the same reason.

The ratio also says how the losses should grow with the swing, and why the step in the knee figure is sharp only for thin lines. Every splitting direction asks for the same cliff, between the same two lines, but they weight the lines’ flanks differently: a direction with weight on the green line’s long-wavelength tail wants the climb to start a little later, one with weight on the yellow line’s short-wavelength tail wants it to finish a little earlier. With lines one and a half nanometres wide those tails are short, every splitting direction asks for nearly the same gap, and the count jumps from nought to fourteen over four nanometres of swing. With six-nanometre lines the tails overlap the gap itself, the effective room between the lines differs by several nanometres from direction to direction, and the same jump is spread over fifteen. The census’s line width turned a threshold into a slope, which is why the five-nanometre counts never showed a gap to find.

Narrow lines help the fast pigment, not the slow one #

The prediction’s second claim was that narrower lines would keep lowering the count, and that the mercury lamp’s residual cost was in the width of its lines. That is true of one kind of pigment. At a 30 nm swing, narrowing the lines from six nanometres to one and a half takes the count from sixteen to nought and the deep losses from one to none. At the census’s 40 nm swing it takes the count from 21 to 17 and the deep losses from 12 to 10. For a pigment whose swing fits in the gap, line width is the whole of the problem; for one whose swing does not, it is a small part. The gap has to be dark, not the line narrow found the same under three emitters: broadening them cost a slow pigment through the light they spilled into the gap, not through their width as such.

So a low-pressure lamp’s narrow lines help a sharp pigment a great deal and a slow one hardly at all, and a high-pressure lamp’s broadened lines cost a sharp pigment a great deal and a slow one a little more. The census’s slope limit of forty nanometres is on the slow side of mercury’s gap, which is why every census reading of the mercury lamp has found its losses stubborn. A colorant chemist designing for a mercury-line source has a target: a full swing shorter than thirty-one nanometres, and the narrower the lamp’s lines the closer to that the swing may be.

Every loss splits the green from the yellow #

The earlier mercury essays found every deep loss splitting the green-yellow pair. On the one-nanometre grid every loss of any depth does, at every line width and every swing from 24 to 48 nm: of the census’s 148 directions, 25 ask the pigment to reflect one of those lines and absorb the other, and the slow pigment loses only from among those 25. The other close pair, violet at 405 and blue at 437, is 31 nm apart as well, and no direction that splits it but not the green-yellow pair is lost — presumably because the violet line is weak and the colour-matching functions small there, so a direction splitting that pair has little of its reach riding on the split.

That makes the mercury lamp’s census a single question. Twenty-five directions need a cliff between 546 and 578 nm, and a slow pigment reaches each of them in proportion to how much of its swing fits in thirty-one nanometres. At a 48 nm swing it loses 19 to 23 of them, at 40 nm 17 to 21, at 34 nm 14 to 19, at 30 nm none to sixteen depending on the lines; and none of the other 123 directions is ever lost.

What this means for the census #

The five-nanometre census was never the obstacle. The essays that counted the mercury lamp’s losses were counting a property of the lamp: two lines a little closer together than the pigment can turn. The census’s mercury numbers stand as they were published, and so does the conclusion that a clear mercury lamp costs a slow pigment few directions deeply. What changes is the explanation of the depth. It is not the five-nanometre grid smearing the lines, nor the lines’ width, though both add a little; it is the arithmetic of 31.5 against 40.

The prediction’s last clause asked whether the lamp’s residual cost lay in its lines’ width, which a high-pressure lamp broadens. For a pigment as slow as the census’s, it mostly does not. For a pigment fast enough to split the lines, it is the only thing left.

How the grid was refined #

The CIE 1931 2° functions, tabulated every five nanometres from 380 to 780, are interpolated to every nanometre by Sprague’s fifth-order formula, with the end intervals extrapolated by CIE 167’s coefficients; on a quintic the interpolation is exact to a part in a hundred million away from the last two intervals at each end, where the colour-matching functions are near nought. The mercury lamp is four Gaussian lines at 405.4, 436.6, 546.5 and 578.0 nm with the census’s relative strengths, six, three or one and a half nanometres wide at half height; tungsten is a 2,856 K radiator and the three emitters the census’s.

The directions are the census’s 148 for the mercury lamp, 148 for tungsten and 157 for the emitters. The ideal reach is the positive part of each direction’s weights summed over the grid; the slope-limited reach is the dynamic programme’s best on a grid of reflectance levels a fortieth apart, one level a nanometre for a 40 nm swing, exact by the corner argument of the previous essay because every corner’s values are whole fortieths. Other swings use one level in T per nanometre. A direction splits the green-yellow pair when its weights at 546 and 578 nm have opposite signs.

What this does not reach #

The band models. Only the slope limit is solved on the finer grid, because only it is exact and cheap there; the band models are searched, and a harder search pulls the band pigments away from the slope limit found their counts sensitive to how hard. Whether the band pigment’s fifteen stubborn directions are the same gap arithmetic is very likely and not computed.

Lines narrower than the grid. One and a half nanometres is the narrowest line a one-nanometre grid samples honestly; a low-pressure lamp’s lines are far narrower, and their limit is the 31.5 nm step itself.

A slope that varies. Real colorants turn faster in some regions of the spectrum than others; a pigment that is slow in the blue and fast in the green-yellow would split mercury’s pair at no cost elsewhere.

Still open: whether the band pigment’s stubborn directions are the same gap #

The band pigment’s flank is not a straight ramp; a forty-nanometre absorption band falls steepest at its half-height points. A pair of lines asks for a cliff, not a gap left fifteen directions short by more than a per cent however deep the band’s cliff was cut.

The calculation is the band model searched on the one-nanometre grid for the 25 splitting directions, its bands’ width fixed at forty nanometres and then narrowed to thirty, twenty-five and twenty, with the lines at one and a half nanometres. The prediction is that the stubborn count falls to nought once the band is narrow enough that its flank, from ninety per cent reflectance to ten, fits in the 31.5 nm gap — near a band width of thirty — and that the census’s fifteen are exactly the directions whose cliff a forty-nanometre band’s flank cannot fit. If the count falls well before that width, the band’s steep middle does the splitting and the gap arithmetic is the slope limit’s alone. A band pigment must be narrower than the gap it splits makes that search.

A grid is suspected until the answer’s scale is found #

The habit here is about deciding whether a discretisation is hiding something.

The census’s five-nanometre sampling was a reasonable suspect: it was coarser than the lines it sampled, and the losses it reported were concentrated exactly where two lines were close together. Refining the grid was the right test, and its answer was a null. What turned the null into an explanation was finding the length the answer depends on — the lines’ 31.5 nm spacing against the pigment’s 40 nm swing — and seeing that both are far larger than five nanometres.

The failure mode is to keep refining a grid because the result is concentrated where the grid is coarse. The test is to find the scale on which the answer changes and compare it with the grid’s step. When the scale is six times the step, the grid was not the cause, and the place the losses concentrate is the place the physics put them.