Wings rescue a three-nanometre table only to its deepest column

A lamp’s spectrum is published as a table, one number every few nanometres, and a table cannot hold a narrow line’s peak. What it can hold is a bound: for each window between its entries, a ceiling on how high the true spectrum can rise, computed from the entries and a stated width for the narrowest line the lamp contains. For a fluorescent tube whose mercury lines are reversed — notched at their centres by cooler vapour absorbing its own light — a ceiling that knows the notch holds to the deepest column found that a one-nanometre table needs the notch built into its ceiling once the column is deep, and a wide table needs no notch to borrow found tables three to five nanometres wide needing none for 1.2-nanometre lines.

A broad line brings the notch back to a wide table broadened the lines’ Gaussian cores and found the rule that governs it: a table needs no notch while its slit is more than about two and a half times its lines’ core, and at a core of 2.4 nm the three-nanometre table, at a ratio of 1.25, failed from a column depth of two. Wings make a broad line easier to bound from its core then gave the 2.4-nanometre lines Lorentzian wings, as collisions in a discharge do, and found the rule’s threshold moving in rather than out. Wings move a line’s power from its peak into a skirt that a wide entry averages, so the entry carries a larger share of the peak; every four-, five- and six-nanometre table’s ceiling built for the core gained eight to twelve points of margin a step, and with wings a quarter of the core’s width all three held at every depth.

The three-nanometre table was left out of that, and it is the hard case. The prediction was that it still fails with a quarter’s wings, from a depth of about three, and holds at every depth only with wings half the core’s width — 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 had already held it, the wings’ lift would be set by the line rather than the slit, and any table whose lines carry collision wings would need no notch at all.

Half a core of wings falls short at the deepest column #

With bare lines, the three-nanometre table’s ceiling built for the core fails from a column depth of two, its worst window 29.9 per cent below the true peak at a depth of five. Wings a tenth of the core’s width fail it from two as well, at worst 23.3 per cent short; a quarter’s from 2.5, at worst 14.3 short; a half’s only at five, by 1.1 per cent. With half a core of wings the table holds at every depth to 4.5, its last margin 0.4 per cent.

• The first half of the prediction holds, the failure starting at 2.5 rather than three.
• The second half fails by a hair. Half a core of wings carries the three-nanometre table to 4.5 and not to five.
• The prediction’s mechanism holds. Each step of wing lifts the narrow table less than it lifts any wider one.

The four curves have one shape. Each starts above the true peak at a shallow column — 8.4 per cent above bare, 30.6 with half a core of wings — holds roughly level to a depth of 1.5, and then falls steadily as the notch splits each line into two shoulders the table’s entries catch less well. The wings lift the whole curve and leave its slope: from a depth of two to five every curve falls by about thirty points, bare or winged. A deep enough column beats any lift the wings give, and for the three-nanometre table the lift runs out just before the deepest column the census runs.

Every table climbs; this one climbs slowest #

With wings half the core’s width, the four-nanometre table’s least margin is +28.5 per cent, the five’s +24.8 and the six’s +48.7; the three-nanometre table’s −1.1. Bare, the four tables start at −29.9, −5.7, −11.6 and +7.3. Every table’s margin rises with the wings at every step and none ever falls; the three-nanometre table starts lowest and ends lowest, and the gap between it and the next table narrows only a little as the wings grow.

From none to a tenth of the core’s width, the wings add 6.6 points to the three-nanometre table’s least margin, 7.9 to the four’s, 8.3 to the five’s and 8.2 to the six’s; from a tenth to a quarter, 9.0, 10.7, 11.3 and 12.4; from a quarter to a half, 13.2, 15.5, 16.8 and 20.8. At every step the three-nanometre table’s lift is the smallest. Each lift grows with the step, because the steps double the wings’ width each time, and a broader skirt moves more of the line’s power out of its peak.

The growth of the lifts is itself worth reading. Each step of wing here doubles or more the Lorentzian’s width — from nothing to a tenth of the core, to a quarter, to a half — and each lift is a third to two thirds larger than the one before for every table. If the lift grew in proportion to the wing’s width it would grow by two and a half times from the second step to the third; it grows by a half. The wings’ benefit is closer to logarithmic in their width than linear, which is what a skirt whose share of the line’s power grows as the Lorentzian widens, but whose height falls, would give: a broader skirt moves more power out of the peak, and spreads it further from the windows that must catch it.

Two nanometres hide the yellow’s notch from any fit found that a plain fit on a two-nanometre table admits the reversed yellow line at every threshold and that its ceiling fails from a depth of two and a half — the narrowest tables are where reversed lines are hardest to bound, wings or no wings, and the three-nanometre table here sits on the same side of that line.

The three-nanometre table needed thirty points and got twenty-nine. A wing three quarters of the core’s width — two nanometres of Lorentzian full width on a 2.4-nanometre core — would add, at the same rate of growth, something like seventeen more points and carry it clear; but a mercury line in a low-pressure discharge has wings far narrower than its Doppler core, and the census’s half is already generous.

The wings push the failure deeper #

The three-nanometre table fails from a column depth of two with bare lines and with a tenth’s wings, from 2.5 with a quarter’s and from five with a half’s. The four-nanometre table fails from four bare and never once it has any wings; the five-nanometre table from 3.5 bare, from 4.5 with a tenth’s, and never with a quarter’s. Each table’s failure moves deeper as the wings widen, and each moves past the deepest column the census runs at some wing width — a tenth for the four-nanometre table, a quarter for the five, and for the three-nanometre table more than a half.

Read as a design rule, the figure says how much collision broadening a table of each width needs before its core-built ceiling can be trusted to a column depth of five. The rule is steep. The four- and five-nanometre tables, at ratios of 1.67 and 2.08 against the core, need a tenth or a quarter of the core’s width in Lorentzian wings; the three-nanometre table, at 1.25, needs more than a half. The step from 1.67 to 1.25 in ratio costs a factor of five in wing width, which is a reminder that the ratio rule’s threshold near two is soft only within a narrow band; below it, the wings must do a great deal.

The column depth is also the lamp’s state, not the table’s. A tube run hot, its mercury vapour denser near the wall, reverses its lines more deeply; a reversed tube declares from its least reversed line found the table’s declaration set by the line reversed least, but the ceiling’s failure is set by the line reversed most. A three-nanometre table published for a tube at one operating state can be safe at that state and unsafe at a hotter one, and nothing in the table says which.

The entry carries more, and is asked for more #

For one notched line at a column depth of three, placed where each table sees it worst, the three-nanometre table’s best entry carries 0.416 of the line’s true peak bare, 0.439 with a tenth’s wings, 0.468 with a quarter’s and 0.508 with a half’s; its core-built ceiling assumes 0.322. Through five nanometres the entry carries 0.262 to 0.362 and the ceiling assumes 0.232. The narrow slit’s entry carries more of the peak than a wide slit’s does — a three-nanometre window is close to the line’s own width — and it is asked for more, because the ceiling’s factor is computed for the narrowest line the slit could see, and a narrow slit’s factor is high.

So the single line is not where the three-nanometre table fails. For one line alone its entry beats the factor by nine points bare and eighteen with half a core of wings, like the wider tables. The failure comes from the whole lamp rather than one line, and the census does not trace which window’s contents defeat it. A plausible reading is that at a slit of three nanometres the windows are close enough that a notched line’s two shoulders fall in different windows, so that a ceiling built for one unnotched line per window covers neither shoulder fully, and that a neighbouring line’s skirt and the phosphor bed add light the flanking pair must also cover.

Where the half-winged table runs out #

With wings half the core’s width — a Voigt line 3.11 nm wide at half maximum, equal to the slit — the three-nanometre table’s least margin runs +30.6 per cent at a column depth of nought, +28 to +31 to a depth of one, and then falls through +20, +14, +10, +7, +4, +2 and +0.4 to −1.1 at five. The fall is steady, about three points every half unit of depth from two onwards, and it crosses nought between 4.5 and five.

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 it stop being enough once the notch deepened; a coarse table can borrow a notch, but not a phantom found a two-nanometre table safe with a notch borrowed from a finer one. Both point to the same remedy for the three-nanometre table at its deepest column: build the ceiling for the notched line, not the bare core. The wings narrow how much a notch-aware ceiling has to carry; they do not remove the need for it at a ratio of 1.25.

What a spectral metrologist should take from this #

Below a slit-to-core ratio of about 1.7, collision wings do not save a core-built ceiling at deep reversal. A three-nanometre table of a 2.4-nanometre-core lamp fails from a column depth of two to five depending on the wings, and holds to five only with wings wider than half the core.

A narrow table’s ceiling likely needs the notch. Where the table’s windows are nearly as narrow as the lines, a ceiling built for an unnotched line runs out at deep reversal whatever the wings; the notch-aware ceiling the one-nanometre tables use is the natural remedy, and the next calculation to make.

The wings’ help is real and graded by the slit. Every step of wing lifts every table’s margin, and the lift grows with the slit’s width — a third more for a six-nanometre table than a three at the largest step.

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, a quarter or a half of the core’s, or left bare; each line’s power is held to the bare line’s. Each 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 held too, on the census’s broad phosphor bed. A table is the lamp read through a triangular slit of half-width three, 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 core’s width, the factors from the census’s 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 #

A notch-aware ceiling at three nanometres. The remedy the essays on one-nanometre tables found has not been applied to this table.

Narrower cores. At a core of 1.2 nm the three-nanometre table’s ratio is 2.5, inside the safe range; the calculation here is the hard case only.

Real wing widths. The wings here run to half the core’s width; a real low-pressure mercury line’s collision broadening is far smaller, and a high-pressure lamp’s far larger.

Still open: whether a borrowed notch saves the three-nanometre table #

At a three-nanometre slit a winged line’s ceiling runs out at the deepest column because the core-built ceiling covers neither of a notched line’s shoulders fully. 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 strength, and found it safe.

The calculation is the three-nanometre table, bare and with wings a quarter and a half of the core’s width, its ceiling built for the notched line at the column depth borrowed from a one-nanometre fit, at every depth to five. The prediction is that the borrowed notch holds the 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 the borrowed-notch ceiling still fails at five with bare lines, the three-nanometre windows are too wide to resolve a reversed line’s shoulders at all, and the table must be read through a narrower slit or declared unbounded at deep reversal.

A ceiling built for the notch holds the three-nanometre table made that calculation: the table holds at every depth with any wing, bare by three tenths of a per cent at a moderate column.

A remedy that ran out one step early #

The habit here is about how close a near-miss is.

The wings lifted the three-nanometre table at every step, the lift grew with each step, and the last step left it 1.1 per cent short at the last column. It would be easy to read that as a success — nearly every depth held — or as a failure that one more step of wing would fix. Either reading misses that the deficit at depth five is the end of a steady fall of thirty points from depth two, which the wings shift but do not flatten.

The failure mode is to read a near-miss as a remedy almost working, when the remedy only offsets a trend it does not change. The test is to ask whether the remedy changes the slope of what is failing or only its level, and to project the slope past the last point read.