A reversed tube declares from its least reversed line

A spectral table records a lamp at a grid of wavelengths through a slit as wide as the grid step — the slit that makes a coarse table legal — and a line lamp’s lines are narrower than either. A narrow table declares its own lines found a way for the table to say how narrow: fit each peak’s entries with one Gaussian line blurred by the slit, and take the narrowest width any peak admits within half a per cent. A narrow table can vouch for its own peaks built a ceiling on that declaration — how high the lamp can rise between the table’s entries, if no line is narrower than declared — and the ceiling is safe as long as no line is.

Two essays have tested what a dense discharge does to that promise. The atoms between the glowing core and the tube wall absorb a strong line’s centre on the way out, and a reversed line is read as one line found the green 546 nm line, reversed by a column of its own atoms, admitted by a two-nanometre table as a single Gaussian too wide for its humps. A notch in a line is never refused made the column cooler and so narrower than the line, and found the two-nanometre table never refusing the notched line and the ceiling on its width failing from a depth of about two and a third. In both, the table’s own declaration survived — it is a minimum over peaks, and the tube’s blue and yellow lines, unreversed, admitted narrow widths and set it.

That rescue rested on the other lines being left alone, and a real tube does not leave them alone. The violet, blue and green lines at 404.7, 435.8 and 546.1 nanometres are one triplet from a single upper level into the three levels of mercury’s lowest triplet term, two of which are metastable and heavily populated in a running discharge; a lamp dense enough to reverse one reverses the others, each in proportion to how strongly it absorbs. The notch essay closed on that case. The prediction was that a one-nanometre table would refuse every strong line from the depth at which the green’s refusal begins and declare from the weakest line, 404.7 nanometres, whose column is shallowest; and that a two-nanometre table would admit every reversed line and declare a width wider than any of them from a green-line depth of about two and a half, with every one of its fits passing.

The declaration comes from the yellow, and fails where no line was checked #

Behind columns in proportion to each line’s strength, the violet line is never read as a peak by either table, so it never declares anything. The yellow line, whose column is under half the green’s, sets the declaration at every depth, and its own reversal widens it: through two nanometres from 1.07 nanometres to 1.81 behind a column as wide as the lines and to 1.69 behind one half as wide. Through one nanometre the table refuses the blue and green lines from a green-line depth of one, as predicted, but goes on declaring from the yellow for a little longer — and at every depth it does so, the ceiling on its declaration falls below the true peak, worst in the violet line’s window. Then it refuses everything and declares nothing. Through two nanometres behind the cooler column the table refuses nothing at any depth, and its declaration is unsafe from two and a half to the end of the census, falling to 0.69 of the true peak in the blue line’s window.

• The prediction is right about the two-nanometre table: behind a cooler layer it declares an unsafe width from a depth of two and a half, and every fit passes.
• It is wrong about which line declares. The weakest line is too weak to be a peak; the declaration comes from the least reversed line that is one.
• The one-nanometre table does not escape. Between refusing the lines it can see are reversed and refusing them all, it declares a width set by the one still admitted — too wide for the violet line, which the table never examined.
• Nothing is left to rescue the declaration. A minimum over peaks is safe only if some peak is an honest line, and in a reversed tube none is.

The four lines are four tables. The two dashed ones, through one nanometre, are short: they start above one, dip below it within a unit of depth, and stop where the table has refused every peak and has nothing left to declare. The two solid ones, through two nanometres, run the whole census. Behind a column as wide as the lines the two-nanometre declaration holds to a depth of four and fails at four and a half. Behind the cooler column it crosses one at about two and a fifth and falls steadily to 0.69. No line reaches the census’s end above one except by refusing to declare.

Four lines, four depths #

The tube is the one every essay on these tables has used: mercury’s four visible lines as 1.2 nm Gaussians on three phosphor bands, the green strongest, then the blue, the yellow and the violet, at relative strengths of 2.40, 1.85, 1.10 and 0.55. Each line is given a column of its own atoms whose depth at its centre is the green line’s depth times its strength over the green’s: at a green-line depth of three, the blue line’s column is at 2.31, the yellow’s at 1.38 and the violet’s at 0.69. The columns are as wide as the lines, or half as wide for a cooler absorbing layer, and each line is rescaled to hold its power.

At a green-line depth of three behind the cooler column the green line has lost 88 per cent of its centre and the blue 78 per cent; both are two sharp humps. The yellow has lost half of its centre and is still recognisably one peak with a dent. The violet has lost a sixth. The ordering is the whole story that follows. The table’s declaration is a minimum over the peaks it admits, and the peak that admits the narrowest width is the one least changed — so the declaration tracks whichever line is least reversed among those the table can see.

The violet line cannot vouch for anything #

A table reads as peaks only the entries that stand above a floor on the lamp’s scale; lower bumps are treated as background, because a fit to a bump barely above the phosphor bed says nothing about a line’s width. The violet line, the weakest of the four and blurred by the slit, never rises above that floor through either slit, at any depth — through one nanometre its highest entry is 0.39 against a floor of 0.4, so it misses being read by a hair, and a table on a slightly different scale would have read it. The prediction had it declaring from the depth at which the stronger lines were refused, because its column is shallowest. It cannot declare at all.

That matters twice. It means the declaration falls to the yellow line, which is reversed. And it means the violet line’s own window is a window the table has never checked. A ceiling built on a declared width is safe in a window only if the lamp there is no narrower than the width; the violet line is only lightly reversed, still close to its emitted 1.2 nanometres, and a declaration widened by the yellow’s reversal is too wide for it.

Through one nanometre: refused, then silent #

Through one nanometre the blue and green lines are refused from a green-line depth of one, behind either column, as the green alone was refused in the earlier census: a one-nanometre slit sees their tops flatten or split. The yellow line’s column is under half as deep, and the table admits it a little longer — to a depth of two behind a column as wide as the lines, to one and a half behind a half-width one. While it does, it is the only peak admitted, and it declares.

That declaration is unsafe at every depth it is made. Behind the full column the table declares 1.30 nanometres at a depth of one, 1.40 at one and a half and 1.51 at two, and the ceiling built on those widths reaches 0.988, 0.965 and 0.881 of the true peak — the first two in the violet line’s window at 406 nanometres, the third in the blue’s at 434. Behind the half column it is unsafe even at a depth of a half, before anything is refused: 0.994 of the true peak at 406. The yellow’s reversal has widened the declaration past what the violet line can bear, and the violet line cannot object, because the table never reads it. From a depth of two and a half the table refuses the yellow as well, and a table that refuses every peak declares nothing — which is safe, and useless.

Through two nanometres: nothing refused #

Through two nanometres the slit averages every line’s notch into a rounded top, as the notch essay found for the green alone, and behind the cooler column the table admits all three peaks at every depth to five. Each admits a width that grows as its line reverses: the green’s from 1.09 nanometres to 2.10, the blue’s to 1.96, the yellow’s to 1.69. The yellow, least reversed, stays narrowest and sets the declaration throughout — 1.34 at a depth of two, 1.47 at three, 1.69 at five.

Behind a column as wide as the lines the pattern is the same with the failure later. The table declares from the yellow throughout, 1.07 to 1.81 nanometres; it refuses the green only at four and a half and the blue at five; and its declaration holds to four, failing at four and a half and five. A broad absorbing layer flattens the lines’ tops enough for the two-nanometre fits to notice late; a narrow one never lets them notice.

Where the declaration fails #

At every depth where the two-nanometre declaration fails behind the cooler column, its worst window is the blue line’s, at 438 nanometres. The picture at a depth of four shows why: across the phosphor bands the ceiling clears the lamp by a wide margin, because a smooth band never rises steeply between entries; round each line it closes in, and round the blue it goes under. The blue line is reversed nearly as deeply as the green — its column is at three quarters of the green’s depth — so its humps are almost as sharp, and the declaration, set by the yellow at 1.59 nanometres, is far too wide for them. Which of the two strongly reversed lines fails first is a matter of where their humps fall against the grid; in this tube it is the blue at every depth.

It is worth asking what an honest declaration would have been. The lines were emitted at 1.2 nanometres, and a table that somehow knew that and declared it would be building its ceiling on the truth about the emission. That too fails at depth: through two nanometres behind the cooler column a ceiling on 1.2 nanometres falls below the true peak from a depth of four, to 0.91 at five, and through one nanometre from two and a half, to 0.78. The reversed humps are sharper than the emitted line. There is no single width a reversed tube can declare that is safe everywhere at depth — the problem is not only that the table misreads the width but that a reversed line has no width a Gaussian ceiling can stand on.

What a table owes a high-pressure lamp #

Refusing is the only safe answer, and only the fine slit gives it. Through one nanometre the table refuses every reversed line from a moderate depth and declares nothing; between the first refusal and the last it declares from the least reversed line, and that brief declaration is unsafe. A procedure that declares only when every strong peak is admitted — rather than whenever any is — would have stayed silent through that stretch.

Check the lines the table cannot see. The violet line never entered the declaration and was where the declaration first failed; the tables cannot bound what they discarded, and a line below the table’s floor is something it discarded. A declared width is a claim about every line in the lamp, and a table that reads only its strong peaks has checked it against those alone.

A minimum over peaks assumes one honest peak. The declaration survived every earlier test because one line was left unreversed. In a lamp whose every line is changed alike by the same physics, the minimum picks the least changed and has no way to know it was changed. A declared width buys a factor of two found the declaration worth having; it is worth having for lamps whose lines keep their shape, and a high-pressure mercury lamp is not one.

How the census was taken #

The tube, slits and fits are the notch census’s: mercury’s lines at 404.7, 435.8, 546.1 and 578.0 nm as 1.2 nm Gaussians on three phosphor bands, tabulated through triangular slits one and two nanometres wide on grids of the same step, integrated at a tenth of a nanometre. Each line’s surviving fraction is the exponential of minus its column’s depth times the column’s absorption profile, of unit height at the line’s centre — the line’s own profile for a column as wide as the lines, a Gaussian 0.6 nm wide for the half-width one — and each line’s depth is the green line’s times the line’s strength over the green’s; each line is rescaled to hold its power. A table’s peaks are its local maxima above 0.4 on the lamp’s scale; each is fitted with one Gaussian of free width and centre blurred by the slit, plus a straight background, over its seven nearest entries, and admitted at a width if the fit misses by no more than half a per cent of the peak entry. The declaration is the narrowest width any admitted peak admits. The ceiling is the peak essay’s, built on the declaration rounded to a thousandth of a nanometre, in every window of the table against the reversed lamp’s true maximum there. With every column at depth nought the tables and declarations are the tube’s exactly, which is checked.

What the census leaves out #

Depths in proportion to strength. A line’s self-absorption depends on how populated its lower level is and on its oscillator strength, and the violet, blue and green lines’ lower levels are populated differently; a real lamp’s depths are not in proportion to the lines’ emitted strengths. The proportion is the simplest rule that reverses the strong lines most, and the finding — that the least reversed peak declares — does not depend on the exact ratios.

The yellow line is one line. Mercury’s yellow is a pair, 577.0 and 579.1 nanometres, which a table can refuse a pair it cannot resolve treated separately; here it is one line as the collection’s tube has it. As a pair it would be refused sooner through one nanometre and read wider through two.

One layer, one width per case. A real discharge’s absorbing column is a gradient, and the two cases here bracket it.

Still open: whether a stricter fit refuses what a looser one admits #

Through two nanometres the reversed lines are admitted because the slit averages their notches into tops a Gaussian fits within half a per cent — the notch census found misses of a seventh of a per cent at worst. Half a per cent is the collection’s threshold, chosen before any line was reversed.

The calculation is this census with the admission threshold lowered to a fifth and a tenth of a per cent, with the declaration and its ceiling recomputed. The prediction is that at a tenth of a per cent the two-nanometre table refuses the green and blue lines behind the cooler column from a depth of about three, while the yellow is admitted to four; that between them it again declares unsafely from the yellow, as the one-nanometre table did; and that a stricter fit also refuses unreversed lines on a noisy table, so that the threshold low enough to catch a reversal is too low for a table read from a real instrument. If so, a two-nanometre table cannot be made safe for a reversed lamp by fitting it harder.

A safeguard is a claim about the case it guards #

The habit here is about what a rescue depends on.

The declaration’s safety came from being a minimum: if any line was honest, the minimum would be at least as narrow as the truth. That is a good safeguard for a lamp with one misbehaving line, and it was tested only on lamps with one. A discharge that misbehaves in every line removes the premise without leaving a trace in the table — every fit still passes, and the minimum is still a minimum.

The failure mode is to test a safeguard against one failure at a time when the failures share a cause. The test is to ask what makes the failure happen and whether the same cause reaches the thing meant to catch it.