A ceiling built for the notch holds the three-nanometre table

A lamp’s spectrum is published as a table — its power in windows a few nanometres wide — and a table hides what happens inside each window. For a discharge lamp that matters: a mercury line is far narrower than any published window, so a reader who wants the lamp’s true peak, for an efficacy calculation or a colorimeter’s linearity, has to bound it from the table. The run of essays this one continues built that bound as a ceiling: the window’s own entry, or it and a neighbour, divided by the least share of a line’s peak that an entry of that slit can carry. A ceiling that knows the notch holds to the deepest column found that a reversed lamp — one whose lines have a notch cut into their cores by cooler vapour near the tube’s wall — needs a ceiling built for the notched line, because a notch moves a line’s peak onto its shoulders.

The essays since have widened the lines and the slits. A broad line brings the notch back to a wide table found tables up to five nanometres needing no notch while the slit is more than about two and a half times the line’s Gaussian core; wings make a broad line easier to bound from its core gave the lines Lorentzian wings and found them lifting every four-, five- and six-nanometre table’s ceiling. Wings rescue a three-nanometre table only to its deepest column tried the narrowest of those slits. With a 2.4 nm core and a ceiling built for that core, the three-nanometre table fails from a column depth of two with bare lines, two and a half with wings a quarter of the core’s width, and only at five with wings half of it, by 1.1 per cent.

A three-nanometre window is only a quarter wider than the core, so it can sit across a notched line with one shoulder inside it and one out, and a ceiling built for an unnotched core does not allow for that. A coarse table can borrow a notch, but not a phantom gave a two-nanometre table the notch a one-nanometre fit had read, scaled to every line by its strength, and found it safe. The prediction was that the same move — each window’s ceiling built for its nearest line’s notched profile, at the column’s depth — holds the three-nanometre table at every depth with any wing, its least margin above two per cent, because the notch-aware factor covers the shoulders the core-built factor missed. If it still failed at five with bare lines, the three-nanometre windows would be too wide to resolve a reversed line’s shoulders at all.

Built for the notch, the table holds at every depth #

The ceilings here are built for the true notch — the column’s optical depth at the green line, scaled to every other line by its strength — which is the best any borrowed reading could supply.

Built for the notch, the three-nanometre table’s ceiling lies above the true peak in every window at every column depth to five, bare and with wings. With bare lines its least margin is +0.3 per cent, at a depth of 1.5, and at five +8.1, where the core-built ceiling fell 29.9 per cent short. With wings a quarter of the core’s width the least margin is +12.8 per cent; with a half, +24.8.

The first half of the prediction holds and the second does not, for bare lines. The table is held at every depth, including the deepest column that defeated every core-built ceiling, so the alternative — that the windows are too wide to resolve a reversed line’s shoulders — is ruled out. But the margin was predicted above two per cent and with bare lines falls to three tenths. The notch-aware ceiling is safe and only just.

Every step of wing adds a dozen points #

The least margins are +0.3 per cent with bare lines, +12.8 with wings a quarter of the core’s width and +24.8 with a half; at every wing the margin is least at a column depth of one or one and a half and grows toward the deepest columns. The wings add about twelve points a step, as they did to the core-built ceiling, because the factor is built for the core alone and the wings put power in the window that the factor does not count on.

So with wings the notch-aware ceiling is conservative by a wide margin, at every depth the census runs, and only the bare lines test it. That is a useful division. A real discharge line has some wing, and the bare case is the limit the ceiling must survive as the wing goes to nothing; a ceiling that survives it by three tenths of a point survives every real line by more.

A deeper notch lowers both factors, so the ceiling rises #

Why the margin is least at a moderate column is visible in the factors themselves.

The single entry’s factor is 0.322 for the unnotched core and falls to 0.216 at a column depth of five; the flanking pair’s is 0.719 and falls to 0.434. But at a depth of a half the pair’s factor rises to 0.746, above the core’s. A ceiling divides the table by these factors, so a lower factor makes a higher, safer ceiling.

The two behaviours are two different things a notch does to a line. A shallow notch flattens the line’s top: the peak falls a little and the power stays spread across the core, so a window carries a larger share of the peak than it would of an unnotched core’s, the pair’s factor rises, and a ceiling built for the shallow notch is lower — and nearer the truth — than one built for the core. A deep notch splits the line into two shoulders: the peak sits off the line’s centre, a window can hold one shoulder’s peak with little of the line’s power, the factors fall, and a ceiling built for the deep notch is higher. The core-built ceiling assumes neither, and is too generous at shallow depths and too stingy at deep ones; the notch-built ceiling is exactly right about the notch and has nothing to spare when the notch is moderate and the flattening and splitting nearly cancel.

At its tightest the ceiling grazes one window #

At a column depth of 1.5 with bare lines, the tightest window is the one centred at 401 nm, beside mercury’s violet lines, holding by +0.27 per cent; the next tightest, at 440 nm, by +10.5. The windows at the green and yellow lines hold by +76 and +66 per cent. The table’s whole margin rests on one window, and every other window has ten points or more to spare.

That window is an edge case in the plain sense: it ends under a nanometre short of the 404.7 nm line, so its entry carries only that line’s near skirt while its true peak, inside the window’s reach, is the line’s shoulder. The ceiling’s single-entry factor is built for the least share of a line’s peak a window can carry, and this window carries nearly the least. Every other window either holds a line’s core or lies far enough from one that the bed, not the line, sets its peak.

Knowing the notch costs the shallow columns and pays the deep ones #

With bare lines the notch-built ceiling holds by 3.9 points less than the core-built one at a column depth of a half and 5.5 less at one, by 7.9 points more at two, and by 38.1 more at five. At every wing the notch costs margin at depths of a half and one and repays it from two.

The notch-built ceiling becomes the safer between column depths of 1.5 and 2 with bare lines, and between 1 and 1.5 with wings. Below that the core-built ceiling is the more generous, and since it holds there anyway, nothing is lost by using it; above, only the notch-built ceiling holds. A practical table could carry both and use the larger. Taking the larger of the two ceilings in every window holds the bare three-nanometre table by at least 1.4 per cent at every depth — still under the prediction’s two per cent, and at least five times the notch-built ceiling’s own least margin.

Where the run of tables now stands #

The census has now bounded the reversed tube through every slit from one nanometre to six, and the three-nanometre table was the last one without a ceiling that held at every column depth. It is worth setting the results side by side, because together they say when a table needs to know about a notch at all.

At the fine end, two nanometres hide the yellow’s notch from any fit found that only a table about a nanometre fine can read a notch from its own entries, and the notch fit declares the true width, and the true width runs out found what that reading costs. The one-nanometre table, given a ceiling built for the notch it reads, holds to the deepest column; the two-nanometre table, built for its core, failed from a depth of three and a half and held once it borrowed the fine table’s notch. At the coarse end, a wide table needs no notch to borrow: a slit two and a half times the line’s core averages a notch away, and the core-built ceiling holds.

The three-nanometre table sits between the two regimes with a 2.4 nm core: its slit is only 1.25 times the core, too narrow to average the notch away and too wide to read it. Built for the core, it fails from a depth of two; built for the notch, it holds at every depth, by three tenths of a per cent with bare lines. That is the regime boundary the census had placed by ratio, now crossed with the right factor in hand, and it is crossed with almost nothing to spare. Read the other way, it is a statement about which tables a laboratory can publish safely for a reversed lamp: a table fine enough to read its own notch, a table coarse enough to average it away, or — between them — a table that states the notch it was built for, since without that statement no reader can rebuild the ceiling that holds.

The run’s arithmetic also says what the next table would need. A slit between 1.25 and 2.5 times the core is the band where neither the plain nor the notched ceiling is comfortable — the plain one because the notch is not averaged away, the notched one because the flattening and splitting of a moderate notch nearly cancel. A four-nanometre table on this core, at 1.67 times, held with a core-built ceiling once the lines had wings; with bare lines it is the case the census has not yet tried with a notch-built one.

What a spectroradiometrist should take from this #

A notched line needs a notched factor, and a three-nanometre slit can carry one. The windows are only a quarter wider than the line’s core, yet a ceiling built for the notched profile bounds the reversed lamp at every column depth the census holds; no slit width here is too coarse for the notch, given the notch.

Expect the tightest case at a moderate notch, not the deepest. Flattening and splitting move the factors in opposite directions, and between them, at a column depth of about one and a half, the notch-built ceiling has least to spare.

Wings are a margin, not a requirement. Every step of wing adds about twelve points to the notch-built ceiling; a table built for bare notched lines is safe for every real line, which has some wing.

How the table and ceilings were computed #

The census’s reversed tube has mercury’s lines on a broad bed, each line a Gaussian core 2.4 nm wide at half maximum, convolved with a Lorentzian whose full width is none, a quarter or a half of the core’s, and notched by a column of optical depth scaled from the green line’s by the line’s strength, the column a Gaussian half the core’s width, each line’s power held. The table is the lamp through a triangular slit of half-width three nanometres, on windows every three nanometres from 380. A window’s ceiling is the larger of its own entry over the single-entry factor and the larger flanking pair over the pair factor, the factors being the least ratios of a slit-averaged entry, and of two neighbouring entries, to the line’s peak, over every placement of the line; the notch-built factors use the nearest line’s notched core at its column depth. Each ceiling is set against the lamp’s true peak within the window’s reach.

What this does not reach #

The notch is the true one. A notch borrowed from a fine fit carries the fit’s error; a coarse table can borrow a notch, but not a phantom found one phantom notch failing a two-nanometre table, and at three tenths of a point the bare three-nanometre table has no room for one.

The wings are not in the factor. The notch-built factors are for the notched core; building them for the notched Voigt line would spend the twelve points a step of wing adds, and is not done here.

One lamp, one column width. The tightest window’s position beside the violet lines is this lamp’s; another lamp’s tightest window would fall wherever a line ends a window’s reach.

Still open: whether a noisy borrowed notch keeps the bare table #

The bare table holds by 0.27 per cent at a column depth of 1.5, with the true notch. A notch read by a one-nanometre fit carries noise, and a notch read too shallow at that depth moves the factors toward the core’s and lowers the ceiling.

The calculation is the bare three-nanometre table’s notch-built ceiling at every column depth with the notch depth taken from each of the one-nanometre census’s noisy readings of a 2.4 nm line, scaled by strength as here, counting the readings that leave a window under its true peak. The prediction is that at depths from one to two a reading a tenth too shallow fails the 401 nm window, so that a fifth or more of the noisy readings fail there, while at depths of three and more every reading holds, because the deep columns’ margin is eight points or more. If no noisy reading fails at 1.5, the 401 nm window’s margin is set by a factor insensitive to the notch’s depth there, and the ceiling can borrow any reading.

A bound that had nothing to spare #

The habit here is about where a bound is tight.

The prediction asked for two per cent of margin everywhere and expected the deepest column to be the test, because that is where every earlier ceiling had failed. The deepest column held by eight points. The test was a moderate column, where the notch flattens and splits a line about equally and the notch-built factor, right about both, has nothing left over — and within it a single window that ends a nanometre short of a line. A bound built exactly for the feature it bounds is tight wherever the feature is exactly as hard as its model says, which is somewhere in the middle.

The failure mode is to test a bound only at the extreme its predecessor failed at. The test is to sweep the parameter the bound depends on and look for the least margin, which for a bound built to the feature can lie where neither extreme would point.