A reversed line is read as one line

A spectral table records a lamp at a grid of wavelengths, each entry the lamp seen through a slit as wide as the grid step — the slit that makes a coarse table legal — and everything a colour calculation does with a line lamp depends on how much the table has smoothed away. A narrow table declares its own lines found that a table fine enough can say how wide its lines are: fit each peak with one Gaussian line, and the narrowest width any peak admits within half a per cent is a width the table can declare. A narrow table can vouch for its own peaks built a ceiling on that width — how high the lamp can rise between entries, given lines no narrower than the declaration — and the ceiling is safe as long as the declared width is not wider than the lines really are. A declared width is worth having: a declared width buys a factor of two found one number of it cutting the bound on a colour engine’s error by more than half.

Two essays since have tested what could push the declaration past the truth. A table can refuse a pair it cannot resolve put two lines under one peak and found the table either refusing the peak — no single Gaussian fits — or reading it as one wider line; either way the declaration survived, because it is a minimum over peaks and the tube’s other lines are single. A line’s wings widen the width a table declares gave every line Lorentzian wings and found the declaration rising to the lines’ own half-maximum width — above the Gaussian core — with the ceiling still safe, because a winged line is lower at its centre than a Gaussian of the same half-maximum width. That essay’s closing section named the remaining shape a physical line can take.

A strong line in a dense discharge absorbs its own light on the way out. The atoms between the glowing region and the tube wall are the same atoms, and they absorb most strongly at the line’s centre. The emerging line is flatter on top than it was emitted; if the absorbing column is dense enough, its centre dips below its shoulders and the line becomes two humps. The prediction was that through a one-nanometre slit a centre optical depth above about one makes the top flat enough that no Gaussian fits within half a per cent, so the table refuses the line, while through two nanometres it is read as a Gaussian about the width of its flattened top — and that the ceiling built on that width stays safe, because a flat-topped line is lower at its centre than any Gaussian of its width. If so, the only single-line shape that can make a declaration unsafe is one narrower at its peak than its entries suggest, and a physical line becomes that only by being two lines.

The table admits a split line, and reads it too wide #

Through one nanometre the table refuses the 546 nm line from a column depth of one, as predicted: its best Gaussian misses by 0.74 per cent at a depth of one and by 3.5 at two. Through two nanometres it admits the line at every depth up to four, where the line has split into two humps with its centre eighty per cent below them, and refuses it only beyond. While the line’s top is flattening, the two-nanometre table reads it narrower than its own half-maximum width, and the ceiling built on that width holds — at a depth of three by nine per cent. At three and a half and four the width the table admits is too wide for the humps, and the ceiling falls below the line’s true peak, by two per cent and eleven. The table’s declaration, the narrowest width any of its peaks admits, never moves, because the other lines set it.

• The prediction is right about refusal. A flat top is a shape no Gaussian fits through a narrow slit.
• It is right about the two-nanometre reading only while the line is still one peak. Once it dips deeply the table reads the pair of humps as one wider line.
• The unsafe width arrives with the split. A reversed line is two humps, each narrower than the whole; this is the shape the wings essay said only two lines could make.
• The declaration survives for the same reason it survived the pair: it is a minimum, and the tube’s other lines are not reversed.

The solid lines are the ceiling built on the width the table admits for the reversed line; they stop where the table refuses it. Through two nanometres the solid line rises while the top flattens — a flat top is easier to bound than a sharp one — and turns down as the dip deepens, crossing one between depths of three and three and a half. The dashed lines are the ceiling built on the emitted line’s own 1.2 nm, which is what a table would use if it had declared its width from the other lines. Through two nanometres it holds everywhere; through one it falls below the peak from three and a half — at depths where the one-nanometre table has already refused the line.

A line seen through its own atoms #

The tube is the one every essay on the tables has used: four mercury lines, each a 1.2 nm Gaussian, on three broad phosphor bands. Only its green line at 546.1 nm is given the column. The emitted profile V passes through atoms with the same profile, so the fraction that survives at each wavelength is exp(−τ·V/V(0)), where τ is the column’s optical depth at the line’s centre; the line is then rescaled to carry the power it carried before, so that nothing changes but its shape.

At a depth of one the line’s top is flat — the centre loses just enough to match its shoulders — and its full width at half maximum has grown from 1.19 to 1.74 nm. At a depth of two the centre is a quarter below the shoulders, at three nearly three fifths, and at four eighty per cent: the line is two humps 1.7 nm apart with a trough between them, and its half-maximum width is 2.42. The humps are not two lines. They are the two flanks of one line, left standing where the column absorbs least.

Through one nanometre the table refuses a flat top #

Through a one-nanometre slit the best single Gaussian misses the seven entries round the peak by 0.05 per cent with no column, 0.43 at a depth of three quarters, and 0.74 at one. At one the table refuses the peak at half a per cent, as the prediction said it would; by two the miss is 3.5 per cent and by five eighteen. A one-nanometre slit sees enough of the top to know it is flat.

The picture at a depth of one shows why, and it is the difference a grid is not a resolution insisted on between a step and what it resolves. The entries at the peak and one nanometre either side are nearly equal — the flat top, sampled three times — and a Gaussian that matches all three is too wide to fall as fast as the entries two and three nanometres out do. It cannot fit both. The table could have been fooled by a line whose flatness lay between its entries; at one nanometre, with the flat stretch wider than a grid step, it is not.

Through two nanometres it reads the line as one #

Through a two-nanometre slit the story is different, because the slit averages over more of the line than its flat top spans. A narrow slit shrinks the error, not the bound found the slit’s width deciding what a table can say about the lamp; here it decides whether the table can see the line’s shape at all. The best Gaussian misses the entries by under a fifth of a per cent to a depth of three, by 0.38 per cent at four — where the line has an eighty-per-cent dip — and passes half a per cent only at four and a half. A two-nanometre table admits the reversed line as one Gaussian at every depth to four.

While the line is flattening the table reads it narrower than its half-maximum width: 1.47 nm against 1.74 at a depth of one, and 2.17 against 2.30 at three. That is the prediction’s reading — about the width of the flattened top — and it is the safe side of the line’s width for the same reason the winged line’s was: most of a flat-topped line’s power sits within its half-maximum width, and a Gaussian that covers the top has no weight left for the shoulders. At three and a half the reading reaches the half-maximum width, 2.35 against 2.36, and at four it overtakes it, 2.54 against 2.42.

The split line is read too wide #

The picture at a depth of four is the failure the ceiling measures. The line itself is two sharp humps with a deep trough; the slit, two nanometres wide, averages the humps and the trough into a single rounded peak; and a Gaussian 2.54 nm wide fits that rounded peak’s seven entries to within 0.38 per cent. The table has admitted a line whose peaks are far sharper than the Gaussian it read. The ceiling built on that width assumes nothing in the window can rise more steeply than a 2.49 nm line — the narrowest width the table admits — and each hump does: in the windows round 546 nm the ceiling falls 11.5 per cent below the reversed line’s true peak.

At three and a half the shortfall is two per cent, and at three the ceiling still clears the peak by nine. The unsafe range is narrow — depths from about three and a third to four and a half, where the line is split deeply enough that its humps are sharper than the whole but not so deeply that the two-nanometre fit notices — but inside it the declaration a single peak would give is wrong in the direction the bound cannot tolerate.

Why the declaration survives anyway #

The table does not declare the reversed line’s width. It declares the narrowest width any of its peaks admits, and the tube’s other three mercury lines are unabsorbed 1.2 nm Gaussians that admit widths down to 1.16 nm through one nanometre and 1.07 through two. At every depth the table’s declaration is exactly the unabsorbed tube’s, and the ceiling built on it holds in the reversed line’s windows too — the dashed line in the first figure, at 1.2 nm, stays at least 39 per cent clear of the peak through two nanometres at every depth.

That is the same rescue the pair of lines had, and it is worth being exact about what it rests on. A table whose strongest line is reversed and whose other lines are too weak to fit — a lamp with one dominant line, or a table read only in a window round that line — would declare the reversed line’s width, and between depths of three and a half and four and a half that width is unsafe. A table in which every strong line is reversed alike would declare an unsafe width however many lines it had.

What a table should do with a line it cannot see inside #

Refuse, or declare from the narrowest line. Through one nanometre the table refuses the reversed line from the depth at which its top flattens, and a refusal is safe: it says the table cannot vouch for that peak and leaves the bound to a stated width. Through two nanometres the table cannot see the reversal and has to rely on the minimum over peaks.

Distrust a declared width that is set by one line. When one peak alone decides the declaration, its shape is the whole of the evidence, and a reversed line’s shape through a coarse slit is indistinguishable from a wide one. The tables cannot bound what they discarded was about what a table loses between its entries; a reversed line loses its trough there, and the entries cannot say it was ever there.

Know the discharge. A mercury tube is the lamp a lamp is not a blackbody begins with, a third of its light in four lines, and those lines are exactly the ones a discharge can reverse. Whether the tube’s green line reaches a centre optical depth of three is a question about the lamp — the mercury vapour’s density, the tube’s diameter, the population of the level the green line ends on — and not about the table. A table built for a lamp known to self-reverse should be read at the finer slit or not trusted to declare its own width.

How the census was taken #

The tube, the tables and the fits are those of a line’s wings widen the width a table declares with no wings: 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 as wide as the grid step, one or two nanometres, integrated at a tenth of a nanometre. The column acts on the 546.1 nm line alone, as exp(−τ·V/V(0)) with the depth τ from nought to five, and the line is rescaled so its power is unchanged. Each peak is fitted exactly as before: one Gaussian of free width blurred by the slit, plus a straight background, fitted to the seven entries round the peak, the width scanned a per cent at a time from 0.6 to 3 nm; the peak is admitted at a width if the fit misses by no more than half a per cent of the peak entry. The ceiling is the peak essay’s: the nearest three entries over the width’s $\kappa$ and the flanking pairs over $\kappa_2$, the smaller of the two, in every window within four nanometres of the line, against the reversed lamp’s true maximum in that window. With no column the tables are the wings census’s unwinged tube to the last digit, which is checked.

What this leaves out #

The column is uniform and the same atoms. A real discharge’s absorbing layer is cooler than its emitting core, and its absorption profile is narrower than the emission’s; that would put the trough in a narrower notch and make the humps sharper still for the same depth. The unsafe range could start shallower.

Only one line is reversed. In a real tube the other visible lines self-absorb too, each by as much as its lower level is populated; if they were reversed as deeply, the declaration’s minimum would not rescue anything.

The lines have no wings. The wings essay’s Lorentzian wings would add to the reversed line’s shoulders; whether they widen the reading or refuse the peak sooner has not been computed.

Still open: whether a narrower absorbing layer moves the unsafe range #

The reversal here comes from a column of the same atoms at the same temperature as the emitters, so the trough has the emitted line’s own width. A cooler absorbing layer has a narrower Doppler width, and its trough is a notch inside the emitted line rather than a broad dip across it.

The calculation is this census with the column’s profile narrower than the emission’s — a half and a third of its width — at the same range of depths, with the table’s reading and the ceiling built on it. The prediction is that a narrow notch makes the humps sharper at a given depth, so that the ceiling falls below the peak from a shallower depth, perhaps two, while the one-nanometre table refuses the line from about the same depth as here, because the flat top it detects is set by the emission’s width. If the unsafe range moves shallower while the refusal does not, a two-nanometre table’s reading of a self-reversed line is unsafe for the lamps most likely to reverse.

A shape is safe until it becomes another shape #

The habit is about how far a reason for safety reaches.

The wings essay found a winged line safe because it is lower at its centre than a Gaussian of the same width, and the reasoning carried over to a flattened line, which is lower at its centre still. It held — until the centre dipped so far that the line stopped being one peak. At that point “lower at its centre” was true of the wrong thing: the centre was low, but the line’s peaks were no longer at its centre.

The failure mode is to carry a reason for safety past the shape it was a reason about. The test is to follow the parameter until the shape changes kind, and ask the question again there.