At one half-width a flank can lose nothing or nearly everything

A clear mercury lamp lights a surface with four visible lines and nothing between them — few colours lost, but a mercury lamp’s few losses are deep — and the colours it lets a pigment show are set by how the pigment reflects at those four wavelengths. The hardest are the ones that split the green line at 546.5 nanometres from the yellow at 578 — a pair of lines asks for a cliff, not a gap — and a pigment built from absorption bands meets them with a band’s flank: absorb one line fully, let the flank fall away before it reaches the other.

A band must be nearly two to one before its steep side pays skewed each band, one side steeper than the other, and a band is only as wide as its steep side found what the skew had been doing. Pigments of three widths and many skews lost the same splitting directions whenever their steep sides had the same half-width — fourteen nanometres losing the same four at any full width — because the band’s wide side falls where the lamp has no line at all. Placed by the steep side’s half-width, fourteen pigments lay on one staircase.

Every flank in that census was a Gaussian, and a Gaussian’s half-width fixes its whole profile, from its shoulder to its foot. The staircase could not say which part of the flank it was measuring. A dye’s absorption band is often fitted by a Gaussian, but a Lorentzian, or a mix of the two, is just as common, and at the same half-width a Lorentzian’s wing reaches far further. The prediction was that the foot decides: that the count follows a single band’s error, which is set by the absorbance the flank still carries at the passed line, so that a Lorentzian steep side of fourteen nanometres loses most of the twenty-five directions and a sharper-footed one none. The alternative was three profiles losing about the same at the same half-width, which would mean the census responds to the flank’s middle and the staircase is a law any band obeys.

Nought to twenty-two at one half-width #

With the wide side held at a 26-nanometre Gaussian and the steep side fourteen nanometres wide at half depth, the band pigment loses none of the 25 splitting directions when its steep side is a Gaussian of power four, four when it is a Gaussian, nineteen when it is a squared hyperbolic secant and twenty-two when it is a Lorentzian — the order of their single bands’ errors, which are 0.0006, 0.52, 5.9 and 36 per cent. Drawn instead at the half-widths that give a single band the Gaussian’s error, the secant at 7.73 nanometres loses two directions and the power-four Gaussian at 39.1 nanometres loses seven, five of them directions the Gaussian keeps.

• The prediction holds. The Lorentzian loses all but three directions and the sharp foot loses nothing, at one and the same half-width.
• The alternative fails outright. Half-width is not what the census measures.
• The single band carries most of the account and not all of it. At matched error the counts spread from two to seven, where at matched half-width they spread from nought to twenty-two.

Read as a staircase, the four fourteen-nanometre flanks span it from top to bottom. The power-four Gaussian’s count, nought, is the count the Gaussian staircase reached only at a steep side of twelve and a quarter nanometres; the Lorentzian’s, twenty-two, is more than the symmetric forty-nanometre band lost with its twenty-nanometre sides. The half-width that the earlier essay’s staircase was drawn in could be the same and the pigment anything from perfect to useless.

What changes between them is not the half-width but how fast each flank falls beyond it. A Gaussian falls as the exponential of the square of the distance, a power-four Gaussian as the exponential of its fourth power, a squared secant as a plain exponential, and a Lorentzian only as the inverse square. More than three half-widths from the band’s centre — where the passed line sits once the band is placed as well as it can be, its centre beyond the line it absorbs — those rates leave absorbances that differ enormously: at 3.3 half-widths a Gaussian keeps four ten-thousandths of its peak, a squared secant about a hundredth, a Lorentzian eight hundredths, and a power-four Gaussian nothing measurable.

Four feet below one half-width #

Drawn with their half depths at the same wavelength, the four bands are indistinguishable at 580 nanometres and nowhere else. At the green line, 33.5 nanometres below, the power-four flank reflects 100.0 per cent, the Gaussian 99.6, the secant 88.6 and the Lorentzian 38.3. The Lorentzian is still absorbing most of the green, at a distance where the Gaussian has let it go, and no placement of the band can fix that: moving the band towards the yellow to spare the green lets the yellow out first.

The figure is the reason the earlier essays’ parameter worked. Every band in this collection’s census before this one was a Gaussian, and among Gaussians the half-width and the foot move together — narrow the half-width and the foot retreats in proportion. The staircase was a staircase in the foot all along, labelled by the one number that, for that family, moves with it.

One error, three half-widths #

The single band, swept through every placement in fiftieths of a nanometre, gives each profile’s error as a function of its half-width. The Gaussian’s error at fourteen nanometres, 0.52 per cent, is reached by the secant at 7.73 nanometres and by the power-four Gaussian at 39.1 — a steep side wider at half depth than the 26-nanometre wide side it is paired with. The Lorentzian never reaches it at any half-width drawn: 3.9 per cent at two nanometres, 36 at fourteen.

That makes a sharper test than the prediction asked for. If the single band’s error were all the census measured, these three profiles — at half-widths more than five times apart — would lose the same directions as the Gaussian. The earlier essay’s twins, two Gaussians of different full widths with the same steep side, lost the same four directions by the same shortfalls to a hundredth of a point. These are twins in single-band error, and much less alike in every other respect.

Twins in error, strangers in shape #

Drawn together, the three matched flanks agree at the green line — every one reflects 99.6 per cent there, which is what the single band’s error measures — and part between the green and the yellow. At 565 nanometres the secant reflects 77 per cent, the Gaussian 54 and the power-four flank 39. The power-four flank, broad at half depth and sharp at its foot, starts absorbing sooner and keeps a wide shoulder; the secant, narrow at half depth with an exponential foot, holds its reflectance until close to the yellow and then falls.

Those shoulders lie where the lamp has no line, between 546.5 and 578, and a single band’s error, read only at the lines, cannot see them. The census sees them through the directions’ other demands: a pigment searched for the whole of the object-colour solid does not only split the two lines, it places several bands and a floor, and a shoulder that reaches across the gap constrains where its neighbours can go.

Which directions each one loses #

The secant loses directions 61 and 131, both among the Gaussian’s four; the power-four flank loses seven — 54, 61, 118, 122, 126, 128 and 134 — sharing only 54 and 61 with the Gaussian. Its new losses are by 1.2 to 2.4 per cent, short and shallow, and none of them is a direction the Gaussian staircase lost anywhere below a steep side of eighteen nanometres.

Five of the power-four flank’s seven losses are directions in the 118 to 134 range, which pass the yellow line and absorb the green. A pigment for those turns the band around — its steep side towards the yellow, its centre on the green — and the band’s shoulder then reaches up towards 560 nanometres, where the yellow-passing directions do not want absorption either. For the Gaussian the shoulder is mild; for the power-four flank, broad at half depth, it is not, and those directions pay for it.

The staircase, with strangers on it #

Placed by single-band error, the new flanks sit near the Gaussian staircase rather than on it. The fourteen-nanometre secant, at 5.9 per cent and nineteen lost, and the Lorentzian, at 36 per cent and twenty-two, lie past the staircase’s top, where the Gaussians at twenty nanometres lost sixteen at 3.2 per cent; the power-four flank at fourteen sits at the bottom with nought. The two matched flanks lie either side of the Gaussian at the same error, two directions below it and three above.

So the single band sorts the pigments correctly — no flank with a larger error loses fewer directions than one with a much smaller error — and does not rank them exactly. Across the range the census spans, from a power-four flank that loses nothing to a Lorentzian that loses nearly everything, the single band’s error accounts for the ordering of twenty-two directions’ worth of difference; at a fixed error the shoulder adds or removes a handful.

Three ways of saying a pigment is slow #

These essays have written a slow pigment down three ways, and the flank result says how they relate. The limits assume a pigment that switches instantly introduced slowness as a slope limit: a reflectance may not swing from nought to one in less than a stated distance. It constrains the edge directly, and under this lamp a slow pigment splits mercury’s lines only if its swing fits between them: forty nanometres of swing do not fit into a gap of thirty-one and a half, and the slope limit was exact all along found that account unmoved by any finer grid.

The band model wrote slowness as a width instead, and a band pigment must be narrower than the gap it splits found its symmetric bands failing for the same reason the slope limit did: a band’s flank is too wide for the gap. But a sharp edge is bought with depth had already found the band model reaching further than the slope limit elsewhere, because a band deep enough draws a reflectance edge steeper than its own width — the reflectance turns over far down the flank, where a deep band’s absorbance is still falling fast.

The flank census adds the third way, and it is the one the other two were approximating. What a band presents to a line lamp is the reflectance its flank leaves at the passed line, and that is set by the profile far out on the flank — by the foot, which a width does not specify and a slope limit specifies only for edges that are straight. A slope limit of forty nanometres is roughly a Gaussian flank’s statement of its foot; a Lorentzian of the same width is a much slower pigment in every sense that matters here, and a power-four Gaussian a much faster one.

What a dye chemist should take from this #

Measure a band’s wing, not its width. Two dyes with the same half-width can differ by twenty directions under a line lamp, and the difference is the absorbance their bands still carry at the passed line, more than three half-widths from the band’s centre. A band fitted to a Gaussian may be described by its half-width; a real band should be described by its wing, measured.

A Lorentzian wing cannot split close lines. At any half-width the census drew, a Lorentzian steep side leaves at least four per cent of a line it should spare, more than any Gaussian steep side under twenty nanometres does. A dye whose bands are Lorentzian in their wings will miss most of a mercury lamp’s green–yellow colours however it is skewed.

Then look at the shoulder. Once the wing is clean, the shape between the lines decides a last handful of directions. A flank that starts absorbing early, even with a sharp foot, gives some of them up.

How the flanks were searched #

Each band is forty nanometres wide in the census’s sense and split at its centre: on the wide side a Gaussian 26 nanometres wide at half depth — the forty-nanometre band’s wide side at a skew of 0.65 — and on the steep side one of four profiles, each one half at its stated half-width: $e^{-u^2\ln 2}$ for the Gaussian, $e^{-u^4\ln 2}$ for the power-four Gaussian, $\operatorname{sech}^2(1.763\,u/2)$ for the secant and $1/(1+u^2)$ for the Lorentzian, $u$ being the distance from the centre over the half-width. The pigment is three such bands and a floor, no band deeper than twelve, searched by Nelder–Mead for two thousand steps from the census’s structured starts and, in addition, from the best fit the forty-nanometre Gaussian pigment at 0.65 found in the same direction. 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 reach falls more than one per cent short of the ideal’s. The Gaussian at fourteen nanometres is the earlier census’s forty-nanometre pigment at 0.65, as corrected there. The single band is twelve deep, swept through every offset in fiftieths of a nanometre, and its larger error on the two lines kept.

What this does not reach #

Searches from the census’s own starts. The Lorentzian and the secant at fourteen nanometres each lose direction 30 by exactly the 1.6 per cent the Gaussian census lost it by before a second search reached it. Their counts may each be one too high; a direction more or less does not move them off the top of the staircase.

One wide side. The wide side is held at 26 nanometres, Gaussian. Under this lamp it falls where there is no line, and the earlier essay found it doing nothing; a different wide side would matter under a lamp with light on that side.

Four profiles. A Voigt profile — the convolution of a Gaussian and a Lorentzian, and the usual shape of a real absorption line — falls between the Gaussian and the Lorentzian here, and where it falls depends on the mix.

Still open: how much Lorentzian a wing can carry #

Real absorption bands are commonly fitted by a Voigt profile, a Gaussian core with a Lorentzian share in its wings, and the census’s Lorentzian and Gaussian are its two ends. The Lorentzian at fourteen nanometres loses twenty-two directions and the Gaussian four; somewhere between them a small Lorentzian share already costs most of what the full Lorentzian does.

The calculation is the forty-nanometre band pigment with its steep side a pseudo-Voigt profile — a weighted sum of a Gaussian and a Lorentzian of the same fourteen-nanometre half-width — at Lorentzian shares of a twentieth, a tenth and a fifth, searched as here, beside each profile’s single band’s error. The prediction is that a twentieth already doubles the Gaussian’s four losses and a fifth reaches the secant’s nineteen, because the Lorentzian’s wing, scaled by its share, still carries more absorbance at the passed line than the Gaussian’s. If a twentieth costs nothing, a dye’s wing can be mostly Gaussian and still split the lines, and the census’s Gaussian staircase holds for the bands chemists actually make. A hundredth of Lorentzian undoes a Gaussian flank makes that calculation.

A parameter that described a family #

The habit here is about what a fitted parameter means outside the family it was fitted in.

The earlier essay found that one number, a Gaussian flank’s half-width, decided everything, and it named that number as the measure. Within Gaussians it was: every Gaussian flank of the same half-width has the same foot and the same shoulder. Outside the family the number stopped meaning anything, because what it had been standing for — the absorbance far down the flank — no longer moves with it.

The failure mode is to report a family’s parameter as the cause when it only indexes the cause within that family. The test is to leave the family and hold the parameter fixed, here by drawing four profiles at one half-width, and to see whether the result holds; when it does not, find the quantity that was moving with the parameter all along, here the absorbance at the passed line, and test that the same way — by matching it across shapes and seeing what is left.