A mercury lamp's few losses are deep

The ideal pigment — reflectance that jumps from nothing to everything at a wavelength — reaches further in every direction of colour space than any real material can, and the limits assume a pigment that switches instantly began the census of how much further. A real pigment is slow: its reflectance takes tens of nanometres to change. The census asks, direction by direction through the object-colour solid, how far a slow pigment reaches against the ideal one, and counts a direction as lost when the slow pigment falls more than one per cent short.

That count has carried a series of findings about lamps. Three lines spare a slow pigment: under a lamp whose light sits in three narrow emitters, a slow pigment loses a third as many directions as under daylight, because between the lines nothing is measured and the pigment can do its changing there. The lamp outweighs the pigment model: the choice of lamp moves the count more than the choice among three models of what a real pigment is. And a tube spares a pigment through its phosphors: a triphosphor tube is as spare as the emitters, a clear mercury lamp — its four visible lines alone — sparer still, and the share of the spectrum a lamp leaves dark ranks the nine lamps’ counts at 0.87.

Each of those essays said, in its limitations, that a count at one per cent is not a depth. A lamp could lose many directions by a little or a few by a great deal, and the count cannot tell them apart. This essay reads the same nine censuses for depth.

The darkest lamp loses least and deepest #

Under the clear mercury lamp a slope-limited pigment loses 14 per cent of the directions at one per cent — fewer than under the emitters or the triphosphor tube — and still 7 per cent at five, where the emitters cost 3 and the tube 2. Half its losses are deeper than five per cent, and a third to a half for the two band models; under every other lamp fewer than three in ten are. Its worst direction falls 28 per cent short. Its mean shortfall over every direction is 1.30 per cent, against 0.81 under the emitters and 0.68 under the tube — though for the two band models, whose losses under it are shallower, its mean shortfall is no larger than theirs. Every direction any model loses under it is one whose ideal pigment reflects one of mercury’s green and yellow lines and absorbs the other. The phosphor coating alone, at the other extreme, loses nothing by more than five per cent under any model. And the share of the spectrum a lamp leaves dark, which ranked the counts at one per cent at 0.82 to 0.87, ranks the losses at five per cent at 0.62 to 0.75.

• The count and the depth disagree about one lamp. By count at one per cent the mercury lamp is the sparest lamp but one; counted at five per cent it is dearer than every lamp but the four whose light fills the spectrum, for all three models, and for the slope limit its mean shortfall says the same.
• Its losses are one colour question. They are the directions that tell 546 nm from 578, two lines 32 nm apart.
• A lamp’s darkness predicts how many directions it costs better than how much.
• The triphosphor tube’s sparing survives the reading. Its losses are shallow as well as few; its coating’s are all under five per cent.

Later searches move the band models’ numbers. The census searched each direction’s band pigments from a handful of structured starts. Searched from 680, a pair of lines asks for a cliff, not a gap found the band model reaching further in seventeen of the twenty-five directions that split the green line from the yellow, and its worst loss 7.1 per cent rather than 10.3. Searched from 594 starts under every lamp, a harder search pulls the band pigments away from the slope limit found the band model’s deep losses under the mercury lamp four in eighteen and the energy model’s four in seventeen, from six and eight — under a quarter each, not the third or more stated here. The slope limit, solved exactly, keeps eleven of its twenty-one losses deep, and the mercury lamp’s losses remain one colour question. The numbers below are the census’s own.

Each lamp has three bars, the shares lost at three thresholds. For eight of the nine the bars step down steeply: daylight from 58 to 25 to 13 per cent, the emitters from 20 to 5 to 3. The mercury lamp’s step down least, 14 to 9 to 7. At one per cent it has the second-shortest top bar in the figure; at five, it is longer than the emitters’ and the triphosphor tube’s in all three of its forms.

The same censuses, read for depth #

Nothing is recomputed. The nine lamps and three pigment models are those of the tube essay: daylight D65, tungsten A, a single-phosphor white LED, three narrow emitters, the triphosphor tube, the same tube with its mercury lines doubled, its phosphor coating alone, its four mercury lines alone, and a halophosphate tube. The models are a slope limit, absorption bands no narrower than forty nanometres, and bands whose minimum width is fixed in energy. Each census records, for about 150 directions through the object-colour solid, how far each model reaches against the ideal pigment in that direction; the shortfall is one less that ratio. Counted at one per cent, the tables here give the published counts to the direction, which is checked.

The readings are four: the share of directions lost at one, three and five per cent; the mean shortfall over every direction, lost or not, which weighs each loss by its depth; and the share of each lamp’s losses that are deep — missed by more than five per cent. The last is the one the count cannot see. A lamp whose losses are mostly shallow is one where a slightly better pigment would recover them; a lamp whose losses are mostly deep is one where the slow pigment is not nearly good enough.

How the losses fall away #

Each curve starts at the share lost at any depth and falls as the threshold rises. Daylight’s starts high and falls fast: most of its losses are between one and five per cent, as the lamp outweighs the pigment model found when it reported the models’ shortfalls averaged a couple of per cent. The emitters’ and the tube’s start low and fall to almost nothing by five per cent: their few losses are shallow too. The mercury lamp’s starts lowest and falls slowest, and it is the last curve still off the axis at twenty per cent.

That shape is the whole of the finding. A threshold at one per cent sees where each curve starts; a threshold at five sees how far along it still stands. For the broadband lamps and the triphosphor tubes the two readings rank the lamps alike, because their losses are shallow in similar proportion. For the mercury lamp they rank it at opposite ends.

Under the mercury lamp, 52 per cent of a slope-limited pigment’s losses are deep, 33 per cent of a forty-nanometre band model’s and 47 per cent of an energy band model’s. Under the other eight lamps the highest share for any model is 27 per cent, under tungsten and the halophosphate tube for the slope limit, and most sit between eight and eighteen. The coating alone has none: no direction under it is missed by more than five per cent, under any model — its worst misses are 4.4 to 4.7 per cent.

What the mercury lamp’s losses are #

Under a lamp of four lines the ideal pigment in any direction is decided line by line: the direction weighs each line’s light by how much it moves the colour that way, and the ideal pigment reflects every line whose weight is positive and absorbs the others. The lamp’s four visible lines come in two close pairs, violet-blue at 405 and 436 nm and green-yellow at 546 and 578. In 25 of the census’s 148 directions the ideal pigment splits the green-yellow pair — reflects one line and absorbs the other. In none does it split the violet-blue pair.

Every direction any model loses under the mercury lamp is one of those 25: all 21 the slope limit loses, all 18 the band model loses and all 17 the energy model loses. Every direction that keeps the pair together, the slow pigments reach. And of the 25, the slope limit loses 21, the band model 18, the energy model 17.

The reason is a width. To split the pair a pigment has to reflect at 546 and absorb at 578, or the reverse, so its reflectance has to make most of a full transition inside 32 nm. The ideal pigment does it instantly. A slope-limited pigment that takes forty nanometres to switch cannot; the best it can do is a compromise that reflects part of one line and absorbs part of the other, and because under this lamp almost the whole of the direction’s reach comes from those two lines, a compromise on them is a large fraction of the reach. Under daylight the same compromise would cost a sliver of the spectrum among many; under two lines it costs most of what there is.

That is the same mechanism the gap has to be dark, not the line narrow found for emitters, read from the other side. There, a pigment’s slowness was free in a dark gap between lines; here, a gap too narrow for the pigment to cross is where all of its cost sits. Three lines spare a slow pigment said the spareness lasts until the pigment’s transitions are as wide as the gaps; the mercury lamp’s green-yellow gap is narrower than the forty nanometres the models take.

The emitters and the tube, by contrast #

The three emitters sit at wavelengths chosen for a display’s white, far apart: no pair is as close as mercury’s. Their losses are 31, 23 and 20 directions, and only 9 to 16 per cent of those are deep. The triphosphor tube carries the same two mercury pairs on its phosphor bands, and its losses are 33, 26 and 25 directions, 8 or 9 per cent deep: the phosphors’ light round the lines means that a direction splitting the green and yellow lines also has the green phosphor’s band and the red phosphor’s to reach with, and a compromise on the lines costs a smaller share. The tube essay found the mercury lines adding losses to the coating — directions whose edges sit in the troughs between the phosphors — and those added losses are shallow.

So the tube essay’s recommendation, that a lamp designer who wants a slow pigment spared should choose narrow phosphors and keep the discharge’s lines out of their gaps, survives the depth reading. Its remark that a clear mercury lamp is kind to pigment makers does not: it is kind by count and unkind by depth, and a pigment maker who cannot make a colour that tells mercury’s green from its yellow will find it very visibly short.

Darkness orders the count, not the depth #

The tube essay’s summary was that how much of the spectrum a lamp leaves dark orders what it costs a slow pigment — ranked against the counts at one per cent at 0.82 for the slope limit and 0.87 for the two band models. Against the share lost at five per cent the same darkness ranks at 0.62, 0.73 and 0.75; against the mean shortfall at 0.63, 0.78 and 0.73. The mercury lamp is most of the difference: it leaves 89 per cent of 420 to 680 nm below a twentieth of its peak, the darkest lamp in the census by far, and it is not the cheapest by depth.

Darkness is a statement about how much of the spectrum a pigment’s slowness can hide in. It says nothing about where the lamp’s light is, and the mercury lamp’s light is concentrated in a place — two lines close together — where slowness cannot hide at all. A dark lamp spares a pigment only if its light is spread across features the pigment can separate. The emitters and the phosphors are; mercury’s green and yellow are not.

The band models make the same point less sharply. Their losses under the mercury lamp are a third to a half deep and their share lost at five per cent is three to four times the emitters’ and the tube’s, but their mean shortfall — 0.52 and 0.61 per cent — is about the emitters’ and the tube’s, because a band pigment’s compromise on the green-yellow pair is gentler than a slope limit’s. A sharp edge is bought with depth found band pigments drawing sharper edges than their widths, when loaded heavily enough; that is what shortens their losses here, and the pigment decides the trade, not the band is why the ordering of the models survives the lamp.

The rank correlations are taken over nine lamps, and several lamps share a dark share: daylight and tungsten both leave nothing dark, the tube and its coating both leave 32 per cent. Ties are ranked in the census’s own order, as the tube essay’s were, so the numbers here and there are comparable; a different tie rule would move them by a few hundredths.

What a pigment maker should take from this #

Ask how deep, not only how many. A count at one per cent treats a direction missed by 1.1 per cent and one missed by 28 alike, and the two are not alike to anyone looking at the result. The mean shortfall is the single number that weighs them, and the depth curve is the whole answer.

Under a line lamp, look at the closest pair of lines. The directions a slow pigment loses under a lamp of lines are those that separate the lamp’s closest features, and how deep it loses them depends on how much of the lamp’s light those features carry. For a clear mercury lamp that is the green-yellow pair; for a lamp of emitters chosen for a display it is nothing close enough to matter.

A narrow phosphor bed rescues the pair. The same mercury lines on a triphosphor coating cost a slow pigment little, because the direction that splits the lines has other light to reach with. A fourth emitter spends the gap it fills found adding light to a gap costs a pigment directions; here, adding light beside a pair of lines too close to split is what makes their splitting cheap.

How the census was read #

The censuses are those of the lamp outweighs the pigment model and the tube essay, held unchanged: directions through the object-colour solid on a stride of twelve, kept where the ideal reach clears the boundary floor; the slope limit’s best reach from a dynamic programme over reflectance steps; the band models’ from Nelder–Mead searches over up to three absorption bands. Each direction’s shortfall is one less the model’s reach over the ideal’s, floored at nought. The ideal pigment’s choice at each wavelength under the mercury lamp is the sign of the direction’s weighting there — the lamp’s power times the direction’s combination of the colour-matching functions — read at 545 and 580 nm and at 405 and 435, the grid samples nearest mercury’s lines. The dark share is the tube essay’s: the fraction of 5 nm samples from 420 to 680 nm at which the lamp is below a twentieth of its peak.

What this leaves out #

The lines are six nanometres wide. A real mercury line is far narrower, and on the census’s 5 nm grid a narrower line would put its light in one sample rather than three. The pair would be as close and the split as hard; whether it would be harder — a slow pigment can compromise between two broad lines more gently than between two sharp ones — needs the census on a finer grid.

The directions are a sample. About 150 directions on a stride of twelve. The 25 that split the green-yellow pair are the sample’s; a finer stride would find more, and the share of them lost need not stay at two thirds to four fifths.

Depth is measured against the ideal reach. A direction missed by 28 per cent is 28 per cent short of what an impossible pigment reaches, not 28 per cent short of what any real one could; the census has no better pigment to compare against.

Still open: whether a two-band pigment splits the pair #

The three models all describe a pigment by how fast it can switch, and a pigment that must switch inside 32 nm fails all three. A real colorant is not always one transition: two absorption bands close together can make a narrow reflectance peak between them, and a peak narrower than 32 nm is exactly what splitting mercury’s green from its yellow needs.

The calculation is the mercury lamp’s 25 splitting directions with the band model allowed a narrow gap between two bands — the bands no narrower than forty nanometres, as before, but free to sit so close that the reflectance between them rises and falls within the pair’s 32 nm. The prediction is that it recovers most of the deep losses, because the gap between two broad bands can be narrow even when neither band is, and that the directions it cannot recover are those that need the split and a second edge elsewhere. If it recovers them, the mercury lamp’s depth is a property of single-transition models, and a colorant chemist with two bands to place has what the lamp demands.

A threshold is a depth chosen in advance #

The habit is about what a count at a threshold reports.

Every census here counted at one per cent because one per cent is small, and small seemed safe: anything that mattered would be counted. It was — every deep loss is also counted at one per cent. What a single threshold cannot do is say how the counted losses compare, and the lamp that looked kindest by count was the one whose losses were least like the others’.

The failure mode is to report a count at a threshold as if it were a cost. The test is to read the same census at a second threshold, or better as a distribution, and see whether the ranking survives. Here it survived for eight lamps and reversed for the ninth.