Wings make a broad line easier to bound from its core

A lamp’s spectrum is published as a table: one number every few nanometres, each the lamp’s light seen through a slit of that width. A table that sampled a narrow line’s sharp peak would need to be impossibly fine, so the essays this one continues have asked a weaker question of it — not the peak, but a bound on the peak that the table itself can certify. A window’s ceiling is the largest the true spectrum can rise anywhere inside that window, given the table’s entries and a stated width for the narrowest line the lamp can contain. A reversed tube declares from its least reversed line and the essays after it built that ceiling for a fluorescent tube whose mercury lines are reversed — notched at their centres by cooler vapour absorbing its own light.

A wide table needs no notch to borrow found tables read through slits of three to five nanometres bounding the reversed tube’s peaks with a ceiling built for an unnotched line of the lines’ core width, by 4.7 per cent at least, because a wide entry carries so small a share of a 1.2-nanometre line’s peak that a ceiling built on it covers whatever a notch does. A broad line brings the notch back to a wide table broadened the lines’ cores and found the five-nanometre table failing from a column depth of 3.5 when the core is 2.4 nm, a slit-to-core ratio of 2.08, while every ratio of 2.5 or more held — so that a table needs the notch wherever its slit is less than about two and a half times its lines’ core.

Those lines were Gaussian. A mercury line in a discharge is broadened by collisions as well as by the atoms’ motion, and its profile is a Voigt: the Gaussian core convolved with a Lorentzian, whose wings fall as the inverse square of the distance rather than exponentially. A line’s wings widen the width a table declares found a table declaring the line’s half-maximum width, which Lorentzian wings raise. The prediction was that the threshold moves out in proportion to the declared width — to a ratio of about 2.5 against the core with wings a quarter of its width — and that a ceiling built on the declared width keeps the threshold where it is. If the wings moved the threshold further than the declared width does, no single width could serve, and a wide table would have to carry the notch whatever its ratio.

The threshold moves the other way #

With every line’s 2.4 nm core convolved with a Lorentzian a tenth of its width, the ceiling built for the core holds through the four- and six-nanometre tables at every depth to five and through the five-nanometre table to a depth of four, failing from 4.5; with a Lorentzian a quarter of the core’s width it holds through all three tables at every depth, the five-nanometre table by 8.0 per cent at least. Without wings it failed the four-nanometre table from a depth of four and the five from 3.5.

• The prediction fails in direction. Wings do not push the threshold out; they pull it in, and a quarter’s wings remove it altogether for these tables.
• The declared width is the wrong width to build on. A ceiling built for the Voigt line’s own half-maximum width fails three of the six winged tables, including both five-nanometre ones.
• The alternative also fails. A single width serves every winged table: the core’s.

The three tables’ lines rise together. The four-nanometre table’s least margin goes from −5.7 per cent bare to +2.2 with a tenth’s wings and +13.0 with a quarter’s; the five-nanometre table’s from −11.6 to −3.3 to +8.0; the six-nanometre table’s from +7.3 to +15.4 to +27.8. Each step of wing adds about the same margin to every table, eight to twelve points, and the order of the tables never changes: the six-nanometre table is the safest throughout, the five the least safe.

That order is itself a small departure from the ratio rule. The five-nanometre table, at a slit-to-core ratio of 2.08, is less safe than the four-nanometre table at 1.67, both bare and with wings. A broad line brings the notch back to a wide table found the ratio nearly deciding across twelve tables, with the threshold a little above two; within that band, which window of the grid happens to catch the worst-placed line decides between neighbours, and the five-nanometre grid’s phase against the tube’s strongest lines is the worse of the two.

Five nanometres at every depth #

At a column depth of nought the five-nanometre table’s worst ceiling stands 3.4 per cent above the true peak with bare lines, 10.4 with a tenth’s wings and 21.0 with a quarter’s. Bare, it peaks at 13.4 per cent at a depth of 1.5 and falls to nought at 3.0 and below it from 3.5; with a tenth’s wings it peaks at 21.5 and falls through nought between 4.0 and 4.5; with a quarter’s it peaks at 34.3 and is still 8.0 above at five. The wings lift the whole curve, and the deeper the notch, the more of the lift survives.

The shape of each curve is the reversed line’s: as the column deepens, the line’s centre is absorbed and its light moves to two shoulders either side, so a table entry that caught the centre now catches less of the new peaks, and the ceiling falls towards them. A ceiling that knows the notch holds to the deepest column found that on a one-nanometre table the unnotched rule fails from a depth of two and a half and that a ceiling built for the notch itself holds to the deepest column; the wide tables here never needed that repair while the margin lasted, and the wings decide how long it lasts. The rise at shallow depths, before the fall, is the notch’s first effect: a shallow absorption flattens the line’s top before it splits it, and a flatter top is easier for an entry to cover. A notch is the failure’s cause in all three; the wings decide how much margin the ceiling had before the notch began to eat it.

Where the wings’ margin comes from #

A line’s power is fixed by the discharge; its shape decides how that power is shared between its peak and the rest. Convolved with a Lorentzian a tenth of the core’s width, a line of the same power peaks at 0.914 of the bare core’s height, and with a quarter at 0.806. The wings carry the difference outwards: at ten nanometres from the centre, where the bare core has nothing, the winged lines still emit a thousandth of their peak or more.

A table entry is an average of the spectrum across its slit. Moving a line’s power from its peak into a skirt a few nanometres wide changes the average little, because the skirt still falls inside the slit; it lowers the peak a good deal. The ceiling is the entry divided by a factor the stated width sets, and the true peak is the peak, so the ratio the census reads — ceiling over peak — rises as the peak falls and the entry holds.

For one line notched at a column depth of three and placed where the five-nanometre table sees it worst, the best entry carries 0.262 of the line’s true peak with a bare core, 0.285 with a tenth’s wings and 0.316 with a quarter’s. A ceiling built for an unnotched line of the core’s width assumes 0.232; built for the declared width, 0.241 with a tenth’s wings and 0.256 with a quarter’s. The four- and six-nanometre slits tell the same story at their own scale: the entry’s share rises with the wings by a fifth, and the core-built factor does not move.

That is the whole mechanism in two numbers. The core-built ceiling divides each entry by a factor computed for the narrowest line that could be there, and wings make the real lines wider in their skirts while leaving their cores as narrow as before, so every real line is easier to cover than the narrowest possible one. A ceiling built for the declared width divides by a larger factor, because a wider stated line puts more of its peak into an entry, and so builds a lower ceiling — but the real line’s core is as narrow as ever, and where a notch has sharpened its shoulders the lower ceiling falls under them.

The declared width is the unsafe one #

The Voigt line’s half-maximum width is 2.53 nm with a tenth’s wings and 2.74 nm with a quarter’s, five and fourteen per cent above the core. Built for those widths, the five-nanometre table’s ceiling fails from a column depth of 3.5 and 4.5; built for the core, it fails from 4.5 with a tenth’s wings and never with a quarter’s. At every depth the declared-built ceiling lies five to fifteen points below the core-built one.

A line’s wings widen the width a table declares found the table’s own declaration rising with the wings, and called the direction unsafe because a Voigt line is a sum of shifted copies of its Gaussian core: the bound that covers the core covers the sum, and a declaration wider than the core can be beaten by the core’s sharpness. This calculation measures that unsafety in the ceilings. A table that declares its lines’ half-maximum width and builds its ceilings on what it declares fails the five-nanometre table at depths where the core’s ceiling would have held, and with a quarter’s wings fails it where the core’s never does.

The notch fit declares the true width, and the true width runs out found a fitted declaration converging on the true line width only to have that width stop being enough once the notch deepened. The wings add a second sense in which the fitted width is the wrong one to build on: it is wider than the narrowest feature in the line, and a ceiling must be built for the narrowest.

Eighteen tables and builds #

Of the nine tables built on the core, three fail — the four- and five-nanometre tables with bare lines and the five-nanometre table with a tenth’s wings — and every table with a quarter’s wings holds. Of the six winged tables built on the declared width, three fail: the four- and five-nanometre tables with a tenth’s wings, by 2.2 and 7.9 per cent at worst, and the five-nanometre table with a quarter’s, by 3.6. The four-nanometre table with a tenth’s wings is the clearest single case: built on the core it holds by 2.2 per cent, and built on the declared width it fails by the same 2.2.

So the rule that served Gaussian lines — build the ceiling for the core, and trust it while the slit is more than about two and a half times the core — serves Voigt lines better than it served the Gaussian ones. The core-built ceiling is never less safe with wings than without, and the declared-built ceiling is never safer than the core-built one.

What a spectral metrologist should take from this #

Build a table’s ceiling on the narrowest feature a line can have, its core, not on its half-maximum width. Lorentzian wings make a line wider at half maximum and no wider at its core; the core-built ceiling grows safer with the wings, and the declared-built one does not.

Collision broadening helps a coarse table bound a reversed line. Wings move power from a line’s peak into its skirts, which a wide slit averages, so the entry carries a larger share of the peak and the ceiling’s margin grows by eight to twelve points for each step of wing from a tenth to a quarter of the core.

The slit-to-core ratio is a guide, not a rule. Between ratios of about 1.7 and 2.1 the grid’s phase against the lamp’s strongest lines decides which table fails first; a five-nanometre table can be less safe than a four-nanometre one.

How the tables were built #

Every mercury line of the census’s tube has a 2.4 nm Gaussian core, convolved numerically on a hundredth-of-a-nanometre grid with a Lorentzian of full width a tenth or a quarter of the core’s, or left bare; each line’s power is held to the bare line’s. Each line is notched by a column of cooler vapour, a Gaussian absorption half the core’s width, at an optical depth set for the green line and scaled to every other line by its strength; the notched line’s power is held too. The lines sit on the census’s broad phosphor bed. A table is the lamp read through a triangular slit of half-width four, five or six nanometres, sampled every slit-width on the site’s grid. A window’s ceiling is the larger of two readings — the entry and its neighbours over the single-entry factor, and the flanking pair over the pair factor — for an unnotched Gaussian line of the stated width, the factors computed by the census’s own search over line placements; the true peak is the lamp’s maximum inside the window on a hundredth-of-a-nanometre grid, and a table’s figure is its worst window.

What this does not reach #

Other cores. The calculation holds the core at 2.4 nm, the width at which the bare table first failed; a 3.6 nm core, which failed every table more, has not been given wings.

Real line shapes. The column’s notch is a Gaussian absorption; a real self-reversed line’s absorbing layer has its own pressure broadening, and its notch’s wings would interact with the emitting line’s.

Narrower slits. Three-nanometre tables, at a slit-to-core ratio of 1.25, were left out; they are where a bare core fails earliest.

Still open: whether wings rescue a three-nanometre table #

A quarter’s wings lift the five-nanometre table’s least margin by nearly twenty points and every table here by eight to twelve points a step. The three-nanometre table, at a ratio of 1.25 against a 2.4 nm core, is the table a bare broad line failed earliest, and the wings’ lift might not reach it.

The calculation is the same tube read through a three-nanometre slit, bare and with wings a tenth, a quarter and a half of the core’s width, each window’s ceiling built for the core’s width, at every column depth to five. The prediction is that the three-nanometre table still fails with a quarter’s wings, from a depth of about three, and holds at every depth only with a half’s — because a narrower slit averages less of a line’s skirt, so each step of wing lifts its margin by less than it lifts a wider table’s. If a quarter’s wings already hold the three-nanometre table, the wings’ lift is set by the line rather than the slit, and any table whose lines carry collision wings needs no notch at all.

A width that was wider than the feature #

The habit here is about which width a bound should be built on.

A line’s half-maximum width is the natural number to quote, and the one a table’s fit declares. It was the wrong number for a ceiling, because a ceiling covers the sharpest thing a line contains, and a Voigt line’s sharpest thing is its core, whatever its wings do. Building on the core gave a bound that the wings only made safer; building on the half-maximum width gave one the core could beat.

The failure mode is to build a worst-case bound on a summary of a shape that is larger than the shape’s sharpest component. The test is to ask whether the shape can be written as a sum of narrower parts, and if it can, to build the bound on the narrowest part.