A hundredth of Lorentzian undoes a Gaussian flank

A clear mercury lamp gives a surface four visible lines and nothing between them, and the colours it lets a pigment show are decided at four wavelengths. The hardest to reach split the green line at 546.5 nanometres from the yellow at 578 — reflect one, absorb the other — and a pigment made of absorption bands does it with a band’s flank: its centre absorbs one line, and its side must fall away before it reaches the other, 31.5 nanometres off.

A band is only as wide as its steep side found Gaussian band pigments losing the same splitting directions whenever their steep sides were equally wide at half depth, and at one half-width a flank can lose nothing or nearly everything found why. At one half-width of fourteen nanometres, a power-four Gaussian lost none of the twenty-five directions, a Gaussian four, a squared secant nineteen and a Lorentzian twenty-two — in the order of how much absorbance each flank still carries at the passed line. The half-width had stood for the flank’s foot only because, among Gaussians, the two move together.

The Gaussian and the Lorentzian are the two ends of the profile spectroscopists fit to absorption bands. A Voigt profile is their convolution: a Gaussian core, from the spread of the absorbers’ environments, and Lorentzian wings, from the finite lifetime of the excited state. A dye in a solid binder is usually close to Gaussian at its centre; how much Lorentzian its wings carry decides, on the earlier essay’s evidence, nearly everything under a line lamp. The prediction was that a twentieth of Lorentzian already doubles the Gaussian’s four losses and a fifth reaches the secant’s nineteen, because the Lorentzian’s wing, scaled down by its share, still carries more absorbance at the passed line than the Gaussian’s. The alternative was that a twentieth costs nothing, which would mean a dye’s wing can be mostly Gaussian and still split the lines.

The wing decides at a hundredth #

With its steep side a pseudo-Voigt fourteen nanometres wide at half depth — the Gaussian and the Lorentzian of that half-width summed in proportion to the share — the band pigment loses eleven of the twenty-five splitting directions at a Lorentzian share of one per cent, fourteen at two, eighteen at five, nineteen at ten and twenty at twenty, against four with no Lorentzian and twenty-two with all of it. A single band’s error rises from 0.52 per cent to 1.23 at a hundredth and 4.2 at a twentieth. The worst direction’s shortfall rises at every step, from 1.6 per cent to 2.8, 3.8, 7.1, 11.3, 18.6 and 51.4.

• The prediction’s direction holds and its size does not. A twentieth does not double the losses but more than quadruples them; a fifth passes the secant’s nineteen.
• The alternative fails at the first step. A hundredth of Lorentzian costs seven directions.
• The count saturates; the shortfall does not. From a twentieth to a full Lorentzian the count rises by four, while the worst direction falls seven times further short.

Read on a logarithmic scale of share, the count climbs steeply and then levels. The first hundredth of Lorentzian costs seven directions, the next hundredth three, the next three hundredths four, and the last eight tenths of the way to a full Lorentzian cost two. The prediction’s two points, eight at a twentieth and nineteen at a fifth, sit well below the found curve at the first and just below it at the second: it expected a slow start and the census found the loss arriving almost at once.

That shape is the arithmetic of a thin tail. The best-placed Gaussian band sits with its centre beyond the line it absorbs — that line falls fifteen nanometres into the steep side, still several optical depths deep — so the passed line lies forty-seven nanometres from the centre, 3.3 half-widths out. There a fourteen-nanometre Gaussian has fallen to four ten-thousandths of its peak; a Lorentzian of the same half-width still carries eight hundredths, about two hundred times as much. Mixed in at one part in a hundred, the Lorentzian contributes twice what the whole Gaussian does at the passed line, and the flank’s foot there becomes the Lorentzian’s, diluted, rather than the Gaussian’s. Every further share adds absorbance where the line is, but the line was already being lost.

The single band, share by share #

The single band, swept through every placement in fiftieths of a nanometre, sees the share from its very first fraction. Its error is 0.52 per cent with no Lorentzian, 0.58 at a thousandth, 1.23 at a hundredth, 4.2 at a twentieth and 36 with a full Lorentzian — a curve that rises nearly in proportion to the share once the share passes a few thousandths, the signature of an error dominated by the Lorentzian’s diluted wing rather than by the Gaussian’s foot.

A thousandth of Lorentzian, already a tenth larger in single-band error, would on the Gaussian staircase cost under one direction. A hundredth, at 1.23 per cent, is where a forty-nanometre Gaussian with a sixteen-nanometre steep side stood, and that pigment lost nine. The single band places the Voigts on the staircase before the census is run; the census says how closely they sit there.

Where the share shows #

Drawn as reflectance, the profiles are one curve over the band’s core and part only in the foot. At the green line the Gaussian band reflects 99.6 per cent, the one-per-cent Voigt 98.6, the five-per-cent Voigt 94.9 and the full Lorentzian 38.3. A spectrophotometer measuring the band’s half-width and its shape near the peak would not tell the Gaussian from the one-per-cent Voigt; the difference is a point of reflectance at the green line, forty-seven nanometres from the band’s centre, at the level of an instrument’s stray light.

This is the practical difficulty the result creates. The share that decides a pigment’s reach under a line lamp is a property of the band’s wing, a decade or two below its peak absorbance, and a band fitted from its core and shoulder can carry any share the wing measurement cannot rule out. A chemist who wants to know whether a dye will split mercury’s lines needs its absorbance measured, not fitted, three or more half-widths from the band’s centre, where a thousandth of the peak is the quantity that matters.

The staircase, a step or two high #

Placed by single-band error, the Voigts climb the Gaussian staircase and continue it past its top. Where their errors overlap the Gaussians’, at a hundredth and a fiftieth of Lorentzian, the Voigts lose 1.9 and 2.0 directions more than the staircase at the same error; past the Gaussians’ largest error, 3.2 per cent, they carry the staircase on, flattening towards the twenty-two a pure Lorentzian loses.

The earlier essay found matched-error flanks of other shapes scattering two below and three above the Gaussian — the shoulder adding or removing a handful once the foot was fixed. The Voigts’ shoulders are wider than the Gaussian’s at the same half-width, because the Lorentzian’s share broadens the flank above half depth as well as below it, and they sit on the high side of the scatter. Single-band error orders every flank this census has drawn; at a given error, a broader shoulder costs a direction or two.

The count stops, the shortfall does not #

A count of lost directions saturates by construction: there are twenty-five, and some ask so little of the split that no flank loses them. The worst direction does not saturate: it is 2.8 per cent short at a hundredth of Lorentzian, 7.1 at a twentieth, 18.6 at a fifth and 51.4 with a full Lorentzian, and the mean over all twenty-five rises from 1.2 to 11.0 per cent over the same range. Past a twentieth of Lorentzian the pigment is losing the same directions by more, not more directions.

A mercury lamp’s few losses are deep found the same character in this lamp’s account from the start: it loses few colours, and those it loses it loses badly. A Lorentzian wing makes that sharper. A dye designer comparing two dyes by how many mercury colours they miss would see little difference between a tenth and a fifth of Lorentzian; one comparing them by how far the worst colour falls short would see the fifth twice as bad.

Green first #

At a hundredth of Lorentzian the losses are nearly even between the two kinds, six green-passing and five yellow-passing; by a twentieth the green-passing directions have lost eleven of fourteen and the yellow-passing seven of eleven, and only at a fifth, ten and ten, do the yellow-passing directions catch up. A green-passing direction asks for a band on the yellow with its steep side turned towards the green; a yellow-passing one asks for a band on the green with its steep side turned towards the yellow.

The asymmetry is in the lamp and the eye rather than in the band. The green line carries more of the lamp’s luminance than the yellow, so a green-passing direction loses more of its reach for the same fraction of the green absorbed, and a wing that steals a little of the passed line costs green-passing directions first. The census reports the order; the luminance weighting is the likeliest account of it.

Depth, skew and wing are one quantity #

The run of essays this one closes has varied three things about a band, and each turned out to act through the same number. A sharp edge is bought with depth found a deeper band drawing a reflectance edge steeper than its own width, because the edge sits where the band’s absorbance falls through about one, and a deeper band reaches that level further down its flank. A band must be nearly two to one before its steep side pays skewed the band, and the steep-side census found the skew acting only through the steep side’s half-width. The flank census and this one changed the profile, and found the profile acting through its wing at the passed line.

All three are statements about one quantity: the absorbance the band leaves at the line it should spare, at the placement that best absorbs the other. Depth moves the placement, skew narrows the flank on the side that faces the line, and the profile decides how fast the flank falls once it is narrow. The single band’s error is that quantity read off directly, which is why it orders every pigment any of these censuses has searched.

The lamp’s own shape enters through the same door. A pair of lines asks for a cliff, not a gap found the clear mercury lamp’s hardest colours set by its closest pair of strong lines, and the gap has to be dark, not the line narrow found a slow pigment spared by darkness between lines rather than by narrow lines. A band’s wing reaching across the gap is harmless where the lamp is dark and costly exactly at the line, which is why a line lamp is the instrument that reads a dye’s wing most sharply. The slope limit was exact all along is the reminder of what a simpler model of the same pigment says: a slope limit has no wing at all, and its account of this lamp was settled by whether its whole swing fits between the lines.

What a dye chemist should take from this #

Under a line lamp, a band’s Lorentzian share matters at the one per cent level. A steep side that is a Gaussian fourteen nanometres wide splits mercury’s lines in twenty-one of twenty-five directions; the same side with one per cent of Lorentzian in it splits them in fourteen.

Measure the wing. The share is invisible in a band’s core and shoulder, and a fitted profile will report whatever share the fit’s model allows. What matters is the absorbance three half-widths and more from the band’s centre, a few ten-thousandths of the peak, and it has to be measured there.

Judge the worst colour, not the count. Past a few per cent of Lorentzian the number of colours missed hardly changes, and how badly they are missed keeps getting worse.

How the flanks were searched #

The pigment is the band census’s — three bands and a floor, no band deeper than twelve, searched by Nelder–Mead for two thousand steps from the census’s structured starts and from the best fit the forty-nanometre Gaussian pigment at 0.65 found in the same direction — with each band’s wide side a Gaussian 26 nanometres wide at half depth and its steep side the pseudo-Voigt $(1-\eta)\,e^{-u^2\ln 2} + \eta/(1+u^2)$, $u$ the distance from the centre over fourteen nanometres, at Lorentzian shares $\eta$ of 0.01, 0.02, 0.05, 0.1 and 0.2. The Gaussian and the Lorentzian at fourteen nanometres are the earlier essay’s. The lamp is clear mercury, its lines one and a half nanometres wide on a one-nanometre grid, and a direction is lost when the pigment’s best reach falls more than one per cent short of the ideal’s. The single band is twelve deep, swept through every offset in fiftieths of a nanometre, and its larger error kept.

What this does not reach #

A pseudo-Voigt, not a Voigt. A true Voigt is the convolution of the two profiles and its wing at a given half-width differs from the weighted sum’s; the share that matters here is the wing’s, and a convolution with the same wing would lose about as many directions.

Structured starts. Each share was searched once from the census’s starts and the Gaussian’s best fits; a search from the neighbouring share’s answers, as the steep-side census did, might restore a direction or two at the shares where the count moves fastest.

One half-width. Every flank here is fourteen nanometres wide. A narrower Gaussian core carries a Lorentzian wing that is narrower too, and the share it can bear would be larger.

Still open: how narrow a Voigt must be #

A Lorentzian wing of a given share falls as the square of the distance over its half-width, so narrowing the whole flank pushes the wing’s absorbance at the passed line down in proportion to the half-width squared. A Voigt with a Lorentzian share of a twentieth might split the lines as a pure Gaussian does if its half-width is small enough.

The calculation is the forty-nanometre pigment with its steep side a pseudo-Voigt of a twentieth Lorentzian at half-widths of ten, seven and five nanometres, searched as here, beside each flank’s single band’s error. The prediction is that at seven nanometres the twentieth-Lorentzian flank loses four directions again — its wing at the passed line, a quarter of the fourteen-nanometre wing’s, matching the Gaussian’s error — and that at five it loses none. If the seven-nanometre Voigt still loses ten or more, the shoulder the narrower flank leaves behind matters more than its wing, and a dye with any Lorentzian in its band cannot split close lines however sharp its core.

A small share of the wrong shape #

The habit here is about the size of a contamination that matters.

A Voigt with one per cent of Lorentzian is, by every measure of its core, a Gaussian: the same half-width, the same peak, a profile indistinguishable above a tenth of its height. The prediction treated the share as a small perturbation and expected a small effect, scaled up only slowly. What the lamp measured was not the band’s core but its far wing, and in the far wing the small share of the wide-tailed shape is the whole of the profile.

The failure mode is to judge a mixture’s effect by the share of the mixture when the measurement reads the mixture where one component dominates. One per cent of a heavy-tailed shape is one per cent of the profile near its centre and almost all of it far out. The test is to ask where the measurement reads — here, more than three half-widths from the band’s centre — and to compute each component’s contribution there before deciding which share is small.