A table can refuse a pair it cannot resolve

A narrow table declares its own lines took a fluorescent tube’s spectral table, measured through a triangular slit as narrow as its grid step, and fitted the seven entries round each peak with one Gaussian line of free width, centre and height on a sloping bed. Through a one-nanometre slit the entries admitted only widths from 1.17 to 1.21 nm against the true 1.2, at a precision of half a per cent; through two, about 1.1 to 1.3. The narrowest width any peak admitted became the table’s own declaration of its lamp’s narrowest feature, and a bound on the table’s bandpass error built on it cost two per cent over the bound built on the maker’s stated width — the width that a declared width buys a factor of two had first asked the lamp’s maker for.

The essay named the assumption that made this work: every table peak was one line standing on a smooth bed, and the fit’s success was the only test of that. A peak can be two lines closer together than the slit — a real mercury lamp’s yellow is a pair, at 576.96 and 579.07 nm, which the earlier model of the tube had written as one line at 578 — or it can be a narrow phosphor band. Either might fit as a single wider line, and a line read wider than it is sits in the unsafe direction for a bound that needs the lamp’s narrowest feature. The prediction was that through one nanometre the pair makes a peak no single line fits within half a per cent, so the table refuses to declare from it, and that through two it fits as one line about 2.5 nm wide — admitted, and wrong.

The table refuses a pair split by its step, and admits one split by less #

Through one and two nanometre slits the table refuses the mercury pair: the best single line misses the entries by 10.6 and 1.7 per cent of the peak. Through three it admits the pair as one line 3.24 nm wide, within a tenth of a per cent. Swept apart, a pair of 1.2 nm lines is refused at every slit once it is split by the grid step and admitted when split by less than three quarters of it — and every admitted pair is read as a single line wider than its lines, by up to 3.3 times. A 3 nm europium band is read at its own width through every slit, and through three nanometres it and the pair look the same. The table’s declaration survives all of it, because the tube’s other lines are single.

• The two-nanometre half of the prediction fails the safe way: the pair is refused there, not admitted.
• Refusal has a simple rule. A pair is caught when the table’s grid could put a point between its two lines, and not before.
• What the table admits, it reads too wide — every time.
• A table cannot tell a pair from a band once both are narrower than its slit, and does not need to for a declaration made from its narrowest peak.

The figure is the census in one picture. Each slit’s curve climbs steeply as the two lines separate, and crosses the half-per-cent line between three quarters of its step and one full step. To the left of each crossing the table admits a single line; to the right it refuses one. The curves fall again at the far right only where the table has resolved the pair into two separate peaks, each of which the fit then tries to read on its own, with its neighbour still inside the seven entries.

Three tubes #

The census uses the model tube and two changes to it. The tube is the one every earlier essay on table widths used: mercury lines at 404.7, 435.8, 546.1 and 578.0 nm, each 1.2 nm wide, standing on three broad phosphor bands. The split tube replaces the 578 line with the real mercury pair at 576.96 and 579.07 nm, 2.11 nm apart, each 1.2 nm wide and each carrying half the power. The europium tube adds to the split tube a band at 611 nm, 3 nm wide — the narrow red emission of the europium-doped phosphor that gives a triphosphor lamp its red, as a correction travels with its lines described the class of lamps these tables belong to.

Each is tabulated as before, the lamp averaged under a triangular slit of half-width one, two or three nanometres at every point of a grid of the same step — the integration the slit is what makes it legal showed a table needs, rather than point samples — and every peak the table holds is fitted by exactly the earlier essay’s method. For the one-line tube the census reproduces that essay’s fits to the digit.

Through a two-nanometre slit the pair makes a single table peak, as the prediction expected. The slit has smeared the two lines into one hump, and the seven entries round its top look, at a glance, like a broadened line. They are not quite one. The best single line through them is 3.64 nm wide and misses them by 1.67 per cent of the peak entry, more than three times the half per cent the earlier essay used as the test of a fit. The reason is visible in the figure: the pair’s hump is flatter on top and steeper at the sides than any Gaussian of its width, and seven entries two nanometres apart reach far enough down both sides to see it.

Through one nanometre the pair is resolved into two peaks, and each is refused worse, by 10.6 per cent: the fit tries to read one line at 577 while the other, at 579, sits inside the same seven entries. Through three it is one hump again, and now a single line 3.24 nm wide fits it within a tenth of a per cent. The prediction’s one-nanometre half holds, its two-nanometre half does not, and the admission it expected at two happens at three.

The rule is the grid step #

Sweeping the separation rather than using the one pair nature supplies turns the three results into a rule. At half a per cent, every pair of 1.2 nm lines split by at least the grid step is refused, through every slit; every pair split by less than three quarters of the step is admitted. A one-nanometre table refuses a pair split by 0.9 nm and admits one split by 0.6; a two-nanometre table refuses at 1.8 and admits at 1.5; a three-nanometre table refuses at 3 and admits at 2.5.

The rule has a plain reading. A table sees structure between its grid points only through the slit’s averaging, and a pair split by less than a step can sit wholly between two grid points: every entry then sees the two lines together, from nearly the same distance, and their sum through the slit is a smooth hump that one wider line imitates well. Split by a full step, the pair puts a grid point between its lines, and the entry there sees a dip that no single line has. A table refuses exactly the pairs its grid could have resolved. A grid is not a resolution made the general point that a table’s step and its slit are two different limits; here the step decides what the table can catch and the slit only how cleanly.

The half-per-cent precision matters to where the edge falls, as it did for the width declaration. A table whose entries are good to a fifth of a per cent catches a pair a little closer; one good to one per cent, a little further apart. The shape of the rule does not change.

What the table admits it reads too wide #

The admitted pairs are the dangerous ones, and every one of them is read wider than its lines. A pair split by only 0.6 nm reads as a line 1.42 nm wide through every slit, eighteen per cent too wide. A pair split by 1.5 nm reads as 2.37 through a two- or three-nanometre slit. The widest admitted reading is a pair split by 2.5 nm through a three-nanometre slit, read as one line 3.94 nm wide — 3.3 times the width of either line in it.

That is the direction the earlier essay was careful about. Its bound — the Cauchy–Schwarz bound the tables cannot bound what they discarded set up, tightened by a narrow slit shrinks the error, not the bound — needs a width no wider than the lamp’s narrowest feature, because a narrower feature is taller for the same area, and a table that assumes wider features than it has will cap each window’s peak too low. A pair read as a 3.9 nm line is a peak whose true features are 1.2 nm wide, and a bound that took the width from this peak alone would be unsafe by a factor the table itself gives no hint of: the fit is within two tenths of a per cent.

A band is a line of its own width #

The europium band is the other thing a peak can be, and it causes no trouble at all. Through every slit it is admitted and read at its own width: 2.99 to 3.11 nm at half a per cent, against a true 3. A band of Gaussian shape is, to the fit, a line that happens to be wider, and its width is exactly the quantity the fit reports. Its narrowest feature is its own width, and the table declares it correctly. A phosphor band is also the part of a tube that a lamp is not a blackbody treated as smooth, and at 3 nm this one is closer to a line than to the bed.

That leaves the census’s picture of a real triphosphor tube simple. Through a one-nanometre slit the blue and green mercury lines are read at 1.17 to 1.21 nm, the europium band at 2.99 to 3.11, and the yellow pair is refused in both its peaks. Through two the same, with the pair refused as one peak. Through three the single lines admit anything from 0.2 nm up, as the earlier essay found, and the pair and the band are both admitted as features about three nanometres wide.

Through three nanometres a pair and a band are the same peak #

The prediction asked whether a table can say which of its peaks are lines. Through three nanometres the answer is no, and the figure shows why. The pair reads as a line 3.24 nm wide and the band as one 2.99 nm wide, both within two tenths of a per cent; scaled to the same height and with their backgrounds removed, their seven entries lie on nearly the same curve. One is two narrow lines, the other one broad band, and at this slit nothing in the table distinguishes them.

Through one and two nanometres the table can say: the band is admitted and the pair refused. So the answer to the question is the grid step again. A table can tell a pair from a band, and both from a single line, when the structure inside the peak is at least a step wide; below that, every peak is a line of some width, and the width is the widest the peak could be.

The declaration survives because the lamp has single lines #

None of this reaches the earlier essay’s result, and it is worth being exact about why. The declaration is the narrowest width any peak admits, and splitting the yellow removes a peak from that minimum without adding one above it. Through one nanometre the split tube declares 1.17 nm, the same as the one-line tube, from its blue and green lines. Through two it declares 1.12 where the one-line tube declared 1.08 — the one-line tube’s narrowest admission had come from its yellow line, which the split tube refuses. Both stay at or below the true 1.2. The europium band, at 3 nm, never comes near the minimum.

So a table’s own declaration is safe in the direction the bound needs for as long as at least one of the lamp’s narrowest features is a single, isolated line. That is true of a mercury tube, whose green line at 546.1 nm stands alone. It would not be true of a lamp every one of whose narrow features was an unresolved pair: such a table would declare the width of its pairs, too wide, and the fit would give no warning, because the pairs would be admitted within tolerance. The rule that caught the mercury pair — refusal from a separation of one grid step — is also the rule that says which lamps are dangerous: those whose lines come in pairs closer than the step.

What a table measured through a narrow slit can say about itself #

Three things, and a boundary on each.

It can declare its lamp’s narrowest feature, as the earlier essay found, through a slit no wider than about twice that feature, provided the narrowest feature appears somewhere as an isolated line.

It can refuse a peak that is not a single line, through the same slits, provided the peak’s structure is at least one grid step wide. A table that refuses a peak is telling its user that the peak is two things; the refusal costs nothing, because the declaration is made from the peaks that remain.

It cannot tell a pair from a band, or either from a wider line, below its step. Every peak it admits is read as the widest single line consistent with it, which for a pair is always too wide.

For a colour engine taking a spectral table from a measurement, the practical version is short: a width read from a single peak is not a declaration. The narrowest over several peaks is, if the lamp is the kind whose narrowest lines stand alone, and a refused peak is information rather than a failure. A narrow table can vouch for its own peaks built its ceiling on one declared width for the whole table; this is the evidence that one width, taken as a minimum, is the right shape for that declaration.

How the census was built #

The tubes are the model tube and the two variants above, each written as Gaussian lines on three broad Gaussian phosphor bands, with the mercury pair at its published wavelengths and the europium band’s height chosen so its peak stands above the bed as the tube’s lines do. A table entry is the lamp averaged under a triangular slit of half-width one, two or three nanometres, integrated at a tenth of a nanometre, at every point of a grid from 380 to 780 nm with the same step. A peak is an entry higher than both neighbours and above 0.4 of the tube’s scale. At each peak the seven entries round it are fitted, for each of 105 widths from 0.2 to 12 nm spaced by four per cent, by a Gaussian line of that width blurred by the same slit, its centre scanned over a slit’s width either side of the peak in fortieths, its height and a sloping background solved exactly by least squares; the peak’s miss at that width is the best fit’s root-mean-square residual over the peak entry. A peak admits the widths whose miss is within the precision, and is refused when none is. The pair sweep centres two 1.2 nm lines of equal height on 578 nm at eleven separations from 0.3 to 4 nm, and reads the peak or peaks within four nanometres of 578.

What this leaves out #

Every feature is a Gaussian. A real mercury line has wings from pressure broadening that a Gaussian lacks, and a real europium emission is a cluster of lines rather than a smooth band. A Gaussian fit to a line with Lorentzian wings reads it wider than its core; whether that stays inside the half-per-cent test, and so reaches the declaration, is the next question.

The pair has equal halves. Mercury’s yellow lines are close to equal in a fluorescent tube; a pair with one strong and one weak line is harder to refuse, since the weak line shifts the hump rather than flattening it.

The precision is stated, not measured. A real table’s entries carry noise, and the refusal edge moves with it as the width declaration’s edge did.

Still open: whether a line’s wings reach the declaration #

The fit assumes a Gaussian line. A mercury line in a discharge is broadened by collisions as well as by motion, and its profile has Lorentzian wings that fall far more slowly than a Gaussian’s. Read through a narrow slit, the wings raise the entries two and three steps from the peak, and a Gaussian fit can only accommodate them by widening.

The calculation is this census with every mercury line given a Voigt profile — a 1.2 nm Gaussian core convolved with a Lorentzian whose width is swept from nothing to half the core’s — with the table’s declaration read at each. The prediction is that a Lorentzian a quarter of the core’s width raises every line’s admitted width past 1.2 nm through a one-nanometre slit while still fitting within half a per cent, so that the declaration, taken as a minimum over lines that all share the wings, rises above the core’s true width — the unsafe direction, reached not by one odd peak but by every peak at once. If so, the declaration needs a line shape before it needs a width, and a table measured through a narrow slit should be fitted with the lamp’s own profile rather than with a Gaussian.

A test that can only pass is a description #

The habit is about what a fit’s success is evidence of.

The width declaration was built on a fit that could fail, and the earlier essay used its failure — the half-per-cent test — as the table’s way of saying it could not read a width. That made the declaration safe where the fit failed. What the fit could not do was fail where the model was wrong in a way the model could absorb: a pair close enough to look like a wider line is not a failure of the fit but a success at the wrong answer. A test only detects the departures from its model that the model cannot imitate, and a Gaussian of free width can imitate a great deal.

The census found the boundary of what this test detects — structure a grid step wide — and the declaration survived because it never relied on a single peak. The failure mode is to trust a fit because it passed, without asking what else would have passed it.