A notch in a line is never refused

A spectral table that records a lamp at one-nanometre or two-nanometre steps cannot see inside a mercury line, which is 1.2 nm wide, and the collection’s tables have been built to say so honestly. A narrow table declares its own lines found that a table can nevertheless declare how wide its lines are: fit each peak’s seven entries with one Gaussian line blurred by the slit, and the narrowest width any peak admits within half a per cent is a width the table can stand behind. A narrow table can vouch for its own peaks turned that width into a ceiling — how high the lamp can rise between entries, given lines no narrower than the declaration — and the ceiling is safe as long as no line is sharper than the width it was built on.

A reversed line is read as one line tested the one thing a real discharge does to a strong line that could break that promise. The atoms between the glowing core and the tube wall are the same atoms, and they absorb the line hardest at its centre; with a dense enough column the centre sinks below the shoulders and the line becomes two humps. Through a one-nanometre slit the table refused such a line from a column depth of one, because its top was visibly flat. Through two nanometres it admitted the line as one Gaussian up to a depth of four, and between three and a half and four the width it admitted was wider than the humps could bear: the ceiling built on it fell eleven per cent below the true peak.

That census gave the column the emitted line’s own profile, which is what a column of the same atoms at the same temperature has. A real absorbing layer is cooler than the core it covers. Doppler width grows with the square root of temperature, so the atoms near the wall absorb over a narrower band than the core emits over, and their trough is a notch cut inside the line rather than a broad dip across it. The prediction the earlier essay closed on was that a notch makes the humps sharper at any given depth, so that the ceiling fails from a shallower depth — perhaps two — while the one-nanometre refusal stays where it was, because the flat top it detects is set by the emission’s width, not the absorption’s.

The refusal stays, the ceiling fails a unit sooner, and nothing stops it #

Behind a column half or a third as wide as the line, a one-nanometre table refuses the 546 nm line from a depth of one, exactly where it refused the uniform column’s. A two-nanometre table does something the prediction did not say: it never refuses the notched line at all. To a column depth of five — where ninety-eight and ninety-nine per cent of the centre are gone — its best single Gaussian misses the peak’s seven entries by at most 0.14 per cent. And the ceiling built on the width it admits falls below the line’s true peak from a depth of 2.34 behind the half column and 2.35 behind the third, a whole unit shallower than the uniform column’s 3.42, and stays below it at every depth to five, reaching three quarters of the true peak.

• The prediction is right about the refusal. A one-nanometre slit sees the line’s top, and at a depth of one every column has made the top unfit for a Gaussian.
• It is right about the direction of the ceiling’s failure, and about its onset, within half a unit. The notch makes the humps sharper, and the unsafe reading arrives at about two and a third.
• It missed the end of the unsafe range. Behind the uniform column the two-nanometre table eventually noticed the split and refused the line at four and a half. Behind a notch it never does, so the unsafe range has no far end inside the census.
• The table’s own declaration still survives, because it is set by the unreversed lines. The rescue is the same one the uniform column had, and it rests on the same thing.

The picture carries the whole finding. The pale line is the uniform column’s, and its history is the earlier essay’s: it rises while the line’s top flattens — a flat top is easy to bound — turns down as the dip deepens, crosses one between three and three and a half, and stops at four, where the table refuses the line. The two darker lines are the notches. They turn down sooner, cross one within a hundredth of each other at about two and a third, and carry on falling to the end of the census, the half column’s to ×0.73 and the third’s to ×0.80. Nothing in the two-nanometre table ever says that anything is wrong.

What a cooler layer cuts #

The tube is the one every essay on these tables has used: mercury’s four visible lines at 404.7, 435.8, 546.1 and 578.0 nm, each a 1.2 nm Gaussian, on three broad phosphor bands. Only the green line is given a column. The fraction of its light that survives at each wavelength is exp(−τ·A), where τ is the column’s optical depth at the line’s centre and A is the column’s absorption profile, one at the centre: the emitted line’s own profile for the uniform column, and a Gaussian 0.6 nm or 0.4 nm wide for the cooler ones. The line is then rescaled to carry the power it carried before, so that nothing changes but its shape. The census runs every half unit of depth from nought to five.

At the same depth, a narrower column cuts a deeper, narrower notch. At a depth of three the uniform column’s line has dipped by 59 per cent and stands as two humps 1.51 nm apart; behind the half column it has dipped by 88 per cent with its humps 1.14 nm apart, and behind the third by 92 per cent with its humps 0.87 nm apart. The reason is plain once the profile is drawn: the uniform column absorbs across the whole line and spreads its loss over the flanks as well as the centre, while a narrow column spends the same central depth on a narrow band and leaves the flanks almost untouched. The humps a notch leaves are the line’s own flanks, standing close together, with a gap between them narrower than a nanometre.

The half-maximum width tells the same story from the other side. The uniform column widens the line from 1.19 nm to 2.30 at a depth of three and 2.52 at five, because its broad dip lowers the peak and pushes the half-maximum points outward. A notch barely widens it: 1.79 nm behind the half column and 1.55 behind the third at a depth of three, and still only 1.87 and 1.59 at five. The notched line stays the emitted line’s size on the outside and becomes two sharp spikes on the inside. That combination — an ordinary outline and a sharp interior — is exactly the shape a table cannot see, and it is worth holding on to while the tables read it.

Through one nanometre the refusal does not move #

Through a one-nanometre slit every column’s line is admitted at a depth of a half and refused at a depth of one. The misses at a half are 0.22, 0.39 and 0.32 per cent for the whole, half and third columns, all under the half-per-cent threshold; at one they are 0.74, 1.10 and 0.82, all over it. The notch does not make the one-nanometre table refuse any sooner, and the reason is the one the prediction gave. A one-nanometre table’s peak has three entries within a nanometre of the centre and four more out to three nanometres, and what it detects is that the three central entries are nearer to one another than a Gaussian that also fits the outer four allows. That is a statement about the line’s top, and the top’s width is set by the emitted line. A notch changes what lies between the central entries without changing how flat they read.

The dashed lines in the picture rise steeply after refusal — to eleven and six per cent at a depth of five behind the half and third columns — but they rise less steeply than the uniform column’s eighteen, because a notch leaves the outer entries where the emitted line put them. Past refusal none of this matters to the ceiling. A table that refuses a peak says it cannot vouch for it, and the bound falls back on a stated width.

The picture at a depth of one behind the half column shows the mechanism at the moment of refusal. The notch has already cut a third of the centre away; through a one-nanometre slit the peak becomes a shallow plateau across the three central entries, and the Gaussian that best matches them, 1.61 nm wide, is too broad to fall away as fast as the entries two and three nanometres out. It misses by 1.10 per cent. A grid is not a resolution drew the line between the step a table is sampled at and the features it can resolve; here the one-nanometre table resolves enough of the top to know it is not a Gaussian’s, which is all a refusal needs.

Through two nanometres the notch disappears #

Through a two-nanometre slit the story is entirely different, and the difference is in the slit’s reach. A triangular slit two nanometres wide averages the lamp over four nanometres of spectrum. A narrow slit shrinks the error, not the bound found the slit’s width deciding what a table can say about a lamp between its entries; here it decides whether the table can see the line’s interior at all. Behind the half column the best Gaussian misses the entries by at most 0.14 per cent at any depth, and behind the third by at most 0.04. The uniform column’s miss crossed half a per cent at a depth of four and a half. A notch’s never comes near it.

The uniform column was eventually refused because its dip was broad enough to change the blurred peak’s shape: at four and a half its humps stood 1.77 nm apart, nearly the slit’s own two nanometres, and the blurred peak had the flattened top a Gaussian cannot fit. A notch’s humps never stand that far apart. Behind the half column they are 1.25 nm apart at a depth of five and behind the third 0.94, and the slit’s four-nanometre reach averages the pair and the gap between them into a single rounded peak that is, to within a seventh of a per cent, a Gaussian. A two-nanometre table’s reading of a notched line is not a reading that happens to be close to a Gaussian. It is a Gaussian, blurred, and the notch is in none of its entries.

At a depth of four behind the half column the line itself is two spikes 1.20 nm apart with ninety-five per cent of the centre gone between them. The slit sees a smooth peak, and a Gaussian 2.08 nm wide passes through all seven entries within 0.11 per cent. A reader shown only the entries — which is all a table’s user is ever shown — would describe a slightly broadened line with nothing unusual about it.

The width it reads is wider than the line #

Behind a notch, the width a two-nanometre table admits rises with depth faster than the line’s half-maximum width, and from a depth of three it is the wider of the two. Behind the half column the table admits widths from 1.64 nm at a depth of two, when the line’s half-maximum width is 1.73; by three they are 1.85 against 1.79, and by five 2.10 against 1.87. Behind the third the admitted width passes the half-maximum width a little sooner, between two and two and a half, and ends at 1.77 against 1.59. While the uniform column’s line was flattening, the table read it narrower than its half-maximum width — the safe side, and the reason the earlier census’s ceiling held to a depth of three. A notch puts the reading on the unsafe side almost at once.

It is worth being exact about what the admitted width is being compared with. A half-maximum width is a property of the line’s outline, and for a notched line the outline is misleading in the dangerous direction: each hump is far narrower than the line it belongs to. The ceiling needs a width no wider than the sharpest feature in the window, and the sharpest feature here is a hump whose own flanks fall over a few tenths of a nanometre. Reading the notched line as a Gaussian wider than its own outline puts the declaration further from the humps than even the outline would. That is why the ceiling fails.

Why the half column is worse than the third #

The two notched columns cross into the unsafe range at almost the same depth — 2.34 and 2.35 — and then separate. At a depth of three the ceiling built on the half column’s admitted width reaches 89 per cent of the true peak and the third’s 93; at five, 73 and 80. The half column fails further at every depth from three.

The third column cuts a narrower notch, and its humps stand closer together — 0.94 nm apart at a depth of five against the half column’s 1.25. Two spikes less than a nanometre apart sit inside a single entry’s view even of a one-nanometre slit, and through two nanometres their pair behaves more like one line than the half column’s wider pair does: the admitted width stays nearer the pair’s combined extent, and the ceiling is less wrong about how high the window’s true peak can rise. That is a reading of the figures rather than a separate calculation, and the census does not compute a column narrower than a third. What it does establish is that the danger is not monotone in how narrow the absorbing layer is. A column as wide as the line is eventually noticed; a column a third as wide is never noticed but is less wrong; a column half as wide is never noticed and is the most wrong of the three.

Why the declaration still survives #

As with the uniform column, the table does not declare the notched line’s width. It declares the narrowest width any of its peaks admits, and the tube’s three other mercury lines are unabsorbed Gaussians that admit widths down to 1.157 nm through one nanometre and 1.069 through two. Behind every column and at every depth the whole table’s declaration is exactly the unabsorbed tube’s, and a ceiling built on the emitted 1.2 nm clears the notched line’s true peak in every window near it through two nanometres, by at least ten per cent. Through one nanometre that ceiling does fall below the notched line’s peak — from a depth of three behind the third column and four behind the half — but only at depths where the one-nanometre table has already refused the line.

The rescue is the same one a table can refuse a pair it cannot resolve found, and it rests on the same assumption: that at least one strong line in the table is unreversed and single. The census reverses only the green line. A real tube’s other lines self-absorb too, each by as much as its lower level is populated. The violet, blue and green lines at 404.7, 435.8 and 546.1 nm are one triplet, falling from a single upper level into the three levels of mercury’s lowest triplet term, and two of those three are metastable and heavily populated in a running discharge. A table whose every strong line is notched alike has no unreversed line to set its declaration, and on a two-nanometre grid it would declare a width the ceiling cannot stand on — and give no sign of it.

What a table owes a lamp it cannot see inside #

Read at the finer slit, or do not let the table declare its own width. The one-nanometre table refuses a notched line at the depth where the notch first matters, and refuses it however narrow the absorbing layer is. The two-nanometre table, on the evidence here, cannot refuse a notched line at any depth, and a declaration it makes from such a line is unsafe from a depth of about two and a third.

Distrust an outline. The tables cannot bound what they discarded argued that a table’s entries cannot say what lay between them; a notch is the extreme case, since the entries preserve the line’s outline exactly and discard only its inside. The half-maximum width of the notched line is ordinary, the fit is excellent, and both are uninformative about the spikes.

Know how the lamp is made. The depth of a mercury line’s self-absorption is a property of the discharge — the vapour pressure, the tube’s bore, the temperature gradient between core and wall — and it is not in the table. A lamp is not a blackbody opened the collection’s case against treating a lamp as a smooth curve; a high-pressure lamp whose lines are known to self-reverse is one that a two-nanometre table should not be trusted to describe between its entries, whatever its fits say. A declared width buys a factor of two found one number of declared width halving a colour engine’s error bound; the factor is only worth having from a table that can see the lines it declares.

How the census was taken #

The tube, the tables and the fits are those of the uniform-column census and of a line’s wings widen the width a table declares without wings: the four mercury lines 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 green line’s surviving fraction is exp(−τ·A(x)), with A the emitted profile over its own peak for the uniform column and a Gaussian of full width 0.6 or 0.4 nm for the cooler ones, the depth τ from nought to five in halves, and the line rescaled to hold its power.

Each peak is fitted as before: one Gaussian of free width and centre, 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 built on the narrowest width admitted, as 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 notched lamp’s true maximum in that window. The onsets are interpolated linearly between the census’s half-unit depths. With the column as wide as the line, every reading through both slits is the uniform-column census’s to the last digit, which is checked.

What the census leaves out #

The layer is one layer with one width, and the widths are bounds. Doppler width goes as the square root of temperature, so a column half as wide as the line is at a quarter of the core’s temperature and a third as wide at a ninth — colder than any real tube wall runs. The two cooler columns bracket the direction a real layer pushes the line, not the amount. A real discharge has a temperature gradient from core to wall, and its absorbing column is a stack of layers with widths from the core’s down to the wall’s. The trough such a stack cuts lies somewhere between the broad dip and the notch; the census brackets it rather than modelling it.

The line holds its power. The line is rescaled after absorption so that only its shape changes, which is why the notched lines’ humps rise above the emitted peak — by eight and fourteen per cent at a depth of five. A lamp whose output is regulated electrically behaves roughly like this; one whose line simply loses the absorbed light would have lower humps, and the ceiling’s ratios, which compare the ceiling with the true peak in the same lamp, would move less than the humps do.

Only one line is reversed, and no line has wings. Both were limits of the uniform-column census and both remain. The second matters less here than there: a notch leaves the flanks where the emission put them, and wings would add to the flanks.

Still open: whether a tube reversed in every line declares an unsafe width #

The declaration survives this census for one reason only — the tube’s other lines are single and unreversed and set the minimum — and a real mercury discharge reverses its strong lines together. The violet, blue and green lines are one triplet ending on the three levels of a single term, and a lamp dense enough to notch one is dense enough to notch the others, each in proportion to how strongly its own lower level absorbs.

The calculation is this census with every mercury line given a column, at depths in proportion to each line’s strength, through both slits, with the table’s own declaration — the narrowest width any peak admits — and the ceiling built on it across every window. The prediction is that the one-nanometre table refuses every strong line from the depth at which the green line’s refusal begins, and declares from the weakest line, 404.7 nm, whose column is the shallowest; and that the two-nanometre table admits every notched line and declares a width wider than any of them from a green-line depth of about two and a half — so that a two-nanometre table of a high-pressure lamp declares an unsafe width with every one of its fits passing. If the weak line keeps the declaration safe at two nanometres too, a table’s safety rests on its weakest line, which is the one a user would think least about.

A safe reading can be safe for the wrong reason #

The habit here is about what a passing fit certifies.

The uniform column’s census ended with a reason for safety carried past the shape it was a reason about: a flattened line is lower at its centre than a Gaussian of its width, until the centre has dipped so far that the line’s peaks are no longer at its centre. The notch carries that one step further. A two-nanometre table fits the notched line better than it fitted the flattened one — a seventh of a per cent at worst — and the fit is excellent precisely because the slit has averaged away the feature that makes the line dangerous.

The failure mode is to read a good fit as evidence about the thing fitted, when it is evidence only about the thing sampled. The test is to ask, of any check that passes, whether it could have failed on the case that matters — and for a two-nanometre table and a notched line, it could not.