A tube spares a pigment through its phosphors

The lamp outweighs the pigment model counted, for three models of what a real colorant can do, how many directions of the object-colour solid each cannot reach within one per cent of the ideal pigment’s reach. A slope limit caps how fast reflectance may change with wavelength; absorption bands no narrower than forty nanometres build reflectance from a few smooth dips; bands whose minimum width is fixed in energy do the same with widths that grow towards the red. Under three narrow emitters the models lost 31, 23 and 20 of about 150 directions. Under daylight they lost 89, 73 and 70, and under tungsten and a white LED about as many. The lamp mattered more than the model.

Three lines spare a slow pigment had given the reason the emitters are cheap: a pigment that cannot switch instantly between absorbing and reflecting pays for its slowness only where there is light to see the transition, and three narrow lines leave most of the spectrum dark. The broadband essay closed on the lamp that is both kinds at once. A fluorescent tube is narrow mercury lines standing on phosphor bands, and the question was which of its two parts decides. The prediction was that the tube sits much nearer the broadband lamps — about two thirds of daylight’s count for each model — because its phosphor bed lights the gaps between its lines, and that doubling the lines’ share of its power moves it only a quarter of the way towards the emitters. If so, the spareness the emitters grant is a property of darkness between lines rather than of lines, and a lamp designer who wants it has to take the bed away.

The phosphors are the tube’s lines, and its mercury is the cost #

A triphosphor tube sits with the emitters, not the broadband lamps: forty-nanometre bands lose 26 of its 154 directions against the emitters’ 23 and daylight’s 73, and the other two models are the same — a third of daylight’s share, not two thirds. Its phosphor coating alone, the mercury lines taken out, loses fewer still, 14, 22 and 18; the mercury lines at their normal strength add 19, 4 and 7 directions, and at twice their strength more, 33, 32 and 27. A halophosphate tube, which carries the same mercury lines on two broad phosphor bands, loses 82, 71 and 73 — as much as daylight — at a rendering fidelity within two points of the triphosphor tube’s.

• The prediction’s last clause is right and its numbers are not. Spareness is darkness between features, and a triphosphor tube’s bed is already dark between its phosphors’ bands.
• The mercury lines cost rather than spare. They put light in the phosphors’ gaps, where the ideal pigment’s edges sit.
• A tube is as spare as its coating. The same lines on a broad coating cost what daylight costs.
• A rendering index does not price it. Two tubes of fidelity 81 and 82 differ by two and a half to three times in what a slow pigment loses.

A later search moves the band models’ numbers. The census searched each direction’s band pigments from a handful of structured starts. Searched from 594 starts, a harder search pulls the band pigments away from the slope limit found the triphosphor tube costing bands in nanometres 19 directions rather than 26 — fewer than the emitters’ 23, against daylight’s 63 rather than 73 — and bands in energy 22 rather than 25. The coating alone falls from 22 to 13 for the band model, which removes the one place where the band model came out dearer than the slope limit; the doubled lines cost 23 rather than 32, still more than the tube’s ordinary lines; and the halophosphate tube costs 61 against daylight’s 63. The essay’s conclusions hold, and the tube’s sparing is stronger than stated. The numbers below are the census’s own.

The bars are the census. The top five lamps — three emitters, mercury’s lines alone, the triphosphor coating, the tube and the tube with doubled lines — cost every model between an eighth and a fifth of the solid. The bottom four — the white LED, tungsten, daylight and the halophosphate tube — cost between a third and a half. The line between the two groups does not run between line lamps and band lamps, and it does not run between tubes and everything else.

Five lamps from one tube #

The census is the broadband essay’s, unchanged: the same directions through the object-colour solid, the same ideal step pigments, the same three searches from the same starts, and a direction lost when a model’s best reach falls more than one per cent short of the ideal. The lamps are new, built from the same fluorescent model.

The triphosphor tube is a blue phosphor band near 450 nm, fifty nanometres wide; terbium’s green, a narrow band at 543 nm eighteen nanometres wide; europium’s red, narrow lines at 611, 626 and 651 nm; and mercury’s four visible lines at 405, 437, 547 and 578 nm. The tube with its lines doubled gives the mercury lines twice their strength. Its coating alone takes them out. Its mercury lines alone takes the coating away and keeps the lines at the tube’s scale — a clear mercury lamp’s visible output. The halophosphate tube replaces the three rare-earth phosphors with the two broad bands of the older coating, near 480 and 578 nm and a hundred nanometres wide, under the same mercury lines.

The figure shows what the prediction had in mind and why it misjudged it. The tube does have a bed under its lines, but the bed is not smooth: the coating is itself three narrow emitters and a moderate band, with troughs almost to nothing between 490 and 525 nm, between 565 and 595, and past 665. Mercury’s green line at 547 nm sits on terbium’s green; its yellow line at 578 nm sits in the trough between the green and the red. The phosphor bed does not light the gaps between the lines; the lines light gaps in the bed.

A triphosphor tube is as spare as three emitters #

Under the triphosphor tube the slope limit loses 33 directions of 154, bands in nanometres 26 and bands in energy 25. Under the three emitters the same models lose 31, 23 and 20 of 157. As shares, the tube is within two to four points of the emitters for every model, and at 0.21, 0.17 and 0.16 of the solid it costs a third of what daylight costs — 0.58, 0.47 and 0.45. The prediction’s two thirds of daylight would have been about 60, 49 and 47 directions.

The ordering of the models survives too, as it did under every lamp the broadband essay tried: the slope limit dearest, the band models close together and cheaper. Under this tube the two band models differ by one direction — the closeness a sharp edge is bought with depth found when a band model turned out no stronger than the slope limit it was meant to tighten, and that the pigment decides the trade, not the band traced to the pigment rather than to the lamp.

The mercury lines cost directions, and doubled they cost more #

The prediction’s test of the lines was to double them. Doubled, they move the tube away from the emitters, not towards them: 33, 32 and 27 directions lost, against 33, 26 and 25 at their ordinary strength. And taking the lines out altogether moves it the other way: the coating alone loses 14, 22 and 18 — for the slope limit, less than half what the emitters cost.

The directions the lines cost say why. Twenty directions that a slope-limited pigment reaches under the coating alone it loses under the full tube, and none is gained in exchange. The ideal pigment for each of them — the one that reaches furthest in that direction — switches between absorbing and reflecting at wavelengths that gather in two places: 550 to 600 nm, round the 578 nm line in the trough between the green and red phosphors, and 470 to 510, on the long-wave flank of the blue band. Those are the edges a real pigment cannot make instantly, and under the coating alone they sat in the dark, where a slow transition costs nothing. The yellow line puts light exactly there.

So the lines’ contribution to a tube’s spareness is negative: they are narrow, but they are placed in the coating’s gaps rather than on its peaks, and a gap lit by a narrow line is still a gap lit. The mercury lines alone, without the coating, lose only 21, 18 and 17 — as few as the emitters — because alone they leave everything else dark.

A clear mercury lamp spares a pigment and renders badly #

The mercury lines taken on their own are a lamp too — a clear mercury discharge’s visible output, four lines and nothing else — and they make the distinction between the two questions sharp. Under the lines alone the three models lose 21, 18 and 17 directions of 148, as few as under the three emitters, and the lamp leaves nine tenths of the range from 420 to 680 nm below a twentieth of its peak. Its rendering fidelity is 59, far below the tubes’ 81 and 82 and the emitters’ 79.

The two numbers answer different questions and move in opposite directions here. A rendering index asks whether surfaces look under a lamp as they do under a reference, and four lines of light make most surfaces look wrong. The census asks how much of the colour a real, slow pigment could have made is still distinguishable from what an ideal one could make, and a lamp that is dark almost everywhere hides almost every slow edge. A lamp can be kind to pigment makers and unkind to the people looking at the result, and the clear mercury lamp is the extreme case of both. A triphosphor tube is a compromise that keeps most of the spareness at a decent rendering, and it keeps it because its phosphors are nearly as dark between their bands as the mercury lines are between theirs.

The models’ order survives here as everywhere, slope limit dearest; under the mercury lines the two band models differ by a single direction. Written in energy, the bands spare the green and a limit written in energy charges the reds found the energy-width model’s advantage depending on where a lamp’s light sits; under a lamp of four widely spaced lines there is little light at the reds or greens for the difference to act on.

A halophosphate tube costs what daylight costs #

The halophosphate tube is the prediction’s picture of a tube: narrow lines on a broad bed. It loses 82, 71 and 73 directions of 148, shares of 0.55, 0.48 and 0.49 — within eight points of daylight’s share for every model, and more than two and a half times the triphosphor tube’s. Its bed lights the whole spectrum from 430 to 650 nm, and there the pigments’ slow edges are charged wherever they fall, lines or no lines.

The two tubes’ rendering fidelities are 82.4 and 81.2 — the same, as a lamp’s datasheet would report them. A lamp is not a blackbody set a halophosphate tube beside a white LED of the same white and found them rendering surfaces differently; here two tubes with the same rendering index differ by nearly three times in a question the index was never built to ask: how much of a surface colour a real pigment can reach under them at all.

Darkness orders the cost #

The broadband essay’s explanation, and the emitters’ before it, was that a slow pigment pays only where there is light to see its transitions. The census makes that a measurement. The share of 420 to 680 nm a lamp leaves below a twentieth of its own peak orders the nine lamps’ cost to the band model at a rank correlation of 0.87. Daylight, tungsten, the white LED and the halophosphate tube leave almost nothing that dark, and lose nearly half the solid; the emitters, the mercury lines and the triphosphor lamps leave about a third or more dark, and lose under a quarter.

The ordering is not perfect, and its one clear misfit is informative. The tube with doubled lines leaves slightly more of the spectrum below a twentieth of its peak than the ordinary tube — its lines are taller, so its peak is higher and its bed falls further below it — and yet it costs more. A threshold on the lamp’s own peak counts a gap as dark when it is dim relative to the lines; what the pigment feels is whether there is enough light there to see its edge. The lines’ height does not darken the troughs they stand in.

What a lamp designer who wants the spareness should do #

Choose narrow phosphors, not fewer lines. The spareness a triphosphor tube grants a slow pigment is its phosphors’ doing, and a tube with the same mercury lines on broad phosphors grants none of it.

Keep the discharge’s lines out of the phosphors’ gaps, or accept that they cost. Mercury’s yellow line at 578 nm is fixed by physics, and a coating whose red and green phosphors bracketed it more tightly would light less of the trough round it; a discharge with fewer lines between the phosphors would cost less still.

Do not read the cost from a rendering index. Fidelity indices ask how a fixed set of surfaces looks under the lamp against a reference; the cost here is how much of the surface colours a real pigment can make is reachable at all. The limits assume a pigment that switches instantly is the essay about why the ideal solid is out of reach; this census says the lamp decides how far out of reach, and the index does not see it.

How the census was taken #

The directions, ideal pigments and models are those of the lamp outweighs the pigment model: 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 two band models’ best reach from Nelder–Mead searches over up to three absorption bands from the same starts, widths floored at forty nanometres or at the energy width of forty nanometres at 560. The lamps are the fluorescent model used before — phosphor bands and lines as Gaussians, mercury lines six nanometres wide at their relative strengths times a line strength of 0.55, doubled to 1.1, or zero for the coating alone — each normalised to its peak, which no count depends on; the mercury lines alone are the tube less its coating at the tube’s own scale. The dark share is the fraction of 5 nm samples from 420 to 680 nm at which the lamp is below 0.05 of its peak. Rendering fidelity is the twelve-sample index used throughout.

What this leaves out #

The tubes are models. The phosphor bands and the mercury lines are stated Gaussians, and a real triphosphor tube’s europium emission is a cluster of sharp lines rather than three; a real tube’s lines are much narrower than six nanometres. Narrower lines would light less of each trough, and the lines’ cost would fall.

The directions are a sample, as in every earlier census of pigment models: a stride of twelve through the solid, about 150 directions, each counted equally. A lamp could be cheap on this sample and dear on directions it misses.

The loss is a threshold at one per cent. The broadband essay found its models’ shortfalls shallow — a couple of per cent on average — and a threshold of three per cent separates the slope limit from the band models more than one per cent does. The tube’s position beside the emitters holds at one per cent; at three it has not been counted.

Still open: whether a narrower mercury line costs less #

The lines cost directions because they light the phosphors’ troughs, and a six-nanometre line lights more of a trough than a real mercury line does: in a low-pressure discharge the visible lines are far narrower than any phosphor band. The calculation is this census under the tube with its mercury lines drawn at 3, 1.5 and 0.5 nm, their power held, and the count of directions the lines cost against the coating alone. The prediction is that the cost falls with the lines’ width but does not vanish — about half the twenty slope-limit directions at half a nanometre — because an ideal pigment whose edge sits exactly on the 578 nm line still has to switch somewhere in the trough, and a line anywhere in the trough is light where its slow edge is. If the cost vanishes as the line narrows, a line’s cost is its width, and a lamp’s spareness can be read from its phosphors alone.

A bed is not a background #

The habit is about the word used to describe the part of a spectrum that is not the feature of interest.

The prediction called the phosphor coating a bed — the smooth thing the lines stand on — and reasoned about it as a background that fills the gaps. That is the halophosphate tube, and the census bore the reasoning out there exactly. The triphosphor coating is not a background: it is three narrow emitters and a moderate band, and its gaps are deeper than the lines are narrow. Calling it a bed assigned the tube’s structure to the wrong part of the spectrum.

The failure mode is to reason from the name of a component rather than its shape. The test is to take each component away and count what remains.