A line's wings widen the width a table declares

A table can refuse a pair it cannot resolve tested the assumption under a narrow spectral table’s declaration of its own line width: that each peak is one line. A pair of lines closer than the grid step passes as a single wider line, and is refused once the step could separate them; a narrow phosphor band is read at its own width. The declaration survived both, because it is the narrowest width any peak admits, and a mercury tube has single lines to give it.

That essay ended on the case a minimum cannot escape: a departure every peak shares. A mercury line in a discharge is broadened by motion, which gives a Gaussian core, and by collisions, which give a Lorentzian — and the line’s profile is the two convolved, a Voigt, whose wings fall away as the inverse square of the distance rather than exponentially. Every line in the tube carries them. A Gaussian of free width fitted to seven table entries can only accommodate weight two and three entries out by growing wider, so every peak’s admitted width should rise, and the minimum with them. The prediction was that wings a quarter of the core’s width would raise every line’s admitted width past the core’s 1.2 nm through a one-nanometre slit while still fitting within half a per cent, so that the declaration rises above the truth — the unsafe direction, reached by every peak at once.

The declaration rises, and the bound built on it holds #

The declaration rises as predicted: with wings a quarter of the core’s width, to 1.36 nm through a one-nanometre slit and 1.41 through two, against a core of 1.2. But it follows the winged line’s own full width at half maximum — 1.37 at a quarter, and within three and a half per cent of the one-nanometre declaration at every width of wing at which that table declares at all — and the bound the earlier essays built on a declared width stays safe on it. The ceiling on every window’s peak stays at least 3.2 per cent above the winged lamp’s true peak, and the variance a window is allowed at least 18 per cent above the lamp’s own wherever the lines carry wings. Through one nanometre the fit also notices the wings: the misses grow fifteenfold, and wings half the core’s width are refused at every line.

• The prediction is right about the width and wrong about the danger. A declaration above the core’s width is not a declaration the bound cannot use.
• What the bound needs is the line’s width as the slit sees it, and that is what the table measures.
• The core’s own width becomes the loose choice: safe, and looser the wider the wings.
• A one-nanometre table can see the wings; a two-nanometre table cannot.

The figure is the finding. The two declared widths climb steadily with the wings and stay close to the dashed curve, the winged line’s own width at half its peak. The flat line at 1.2 is the core, the width the lamp’s maker would state and the width the earlier bound assumed; from wings a tenth of the core’s width onwards, both declarations sit above it.

A mercury line with wings #

The census gives the model tube — mercury lines at 404.7, 435.8, 546.1 and 578.0 nm on three broad phosphor bands, the lamp a lamp is not a blackbody used to show how little a colour temperature says about a spectrum — a Voigt profile on every line: the 1.2 nm Gaussian core convolved with a Lorentzian of full width 5, 10, 15, 25, 37.5 or 50 per cent of the core’s, each line keeping its power. It is tabulated through triangular slits of half-width one and two nanometres on grids of the same step — the integration the slit is what makes it legal found every table needs — and every peak is fitted exactly as a narrow table declares its own lines fitted it: one Gaussian of free width, centre and height on a sloping bed, against the seven entries round the peak.

The logarithmic scale shows what a Lorentzian adds. At its centre the winged line is a little lower than the Gaussian, since the same power is spread wider; three nanometres out it is still almost a hundredth of its peak, where the Gaussian has fallen to three parts in a hundred million. Its full width at half maximum is 1.37 nm with wings a quarter of the core’s, 1.55 with wings half of it. On a linear scale the difference is a slightly broader line with slightly raised skirts; on a table whose entries sit one nanometre apart, the skirts are where three of the seven fitted entries lie.

Every line widens, and the minimum with them #

The table’s declaration is the narrowest width any peak admits at half a per cent. With no wings it is 1.157 nm through a one-nanometre slit and 1.069 through two, both at or below the true 1.2, as before. It rises with every step of wing width. Through one nanometre: 1.19 at wings five per cent of the core, 1.24 at ten, 1.28 at fifteen, 1.36 at a quarter, 1.47 at three eighths. Through two: 1.15, 1.20, 1.28, 1.41, 1.56, and 1.71 at half.

It rises because every line rises together. At a quarter of the core, the three peaks through one nanometre admit widths from 1.36 to 1.40 at their narrowest; none of them is left at the core’s width to hold the minimum down. The rule that made the declaration safe against a pair — take the narrowest over peaks — has nothing to work with when the departure is in all of them.

The figure shows the fit’s compromise at one line. The table’s entries lie on the winged line as the slit reads it, and the best single Gaussian, 1.40 nm wide, passes within 0.42 per cent of the peak entry: close to the three central entries, a little under the fourth and fifth on each side, where the wings hold weight a Gaussian cannot. The width it chooses is the width that best spreads the core’s peak and the wings’ skirts together, and that is near the width at which the line is half its peak.

That relation holds across the census. Through one nanometre the declared width lies within three and a half per cent of the Voigt line’s half-maximum width at every width of wing at which it declares at all: 1.19 against 1.23, 1.24 against 1.27, 1.28 against 1.30, 1.36 against 1.37, 1.47 against 1.46. Through two nanometres it runs above it once the wings pass a quarter, by three per cent at a quarter and a tenth at a half.

The bound built on the declared width holds #

The prediction’s worry was concrete. A narrow table can vouch for its own peaks built a ceiling on each window’s peak from the table’s entries and a declared width: a feature no narrower than the width cannot be flattened by the slit below a computable fraction of its height, so the entries round a window cap how tall the lamp can be inside it. A declared width wider than the lamp’s features makes that fraction too generous and the ceiling too low — which is exactly what a table that reads 1.36 for a 1.2 nm core ought to do.

It does not happen. Built on the width the table declares, the ceiling is never below the winged tube’s true peak in any window, at any width of wing or either slit; its closest approach is 1.032 times the true peak, through one nanometre with wings a tenth of the core’s. The other place the width enters the bound — the most variance a lamp of features that wide could hide inside a window, the quantity the tables cannot bound what they discarded first needed and a narrow slit shrinks the error, not the bound found so loose — never falls below the lamp’s own either: the allowance’s closest approach, 1.08 times it, comes with no wings at all; with wings it never comes nearer than 1.18.

The reason is the one the half-maximum relation already gave. A Voigt line is a Gaussian core with its peak lowered and its power pushed into the skirts, and through a slit that is what a wider Gaussian looks like too: a lower peak for its entries, and less variance hidden between them than the narrow core alone would hide. The ceiling needs the ratio of a feature’s true peak to what the slit makes of it, and the table’s fit measures exactly that ratio’s shape. What the prediction called the truth — the core’s 1.2 nm — is the right width only for a lamp with no wings. For this lamp, the width the table declares is the one that describes it.

The core’s width grows loose #

The dashed curves in the figure are the same ceiling built on the core’s 1.2 nm, the width a maker would state. It is safe on the winged tube too, and grows looser as the wings widen: through two nanometres its closest approach to a window’s peak goes from 1.08 with no wings to 1.40 with wings half the core’s. A width narrower than the lamp’s effective width is always safe — narrower features are the harder case — and the price of safety is slack.

So the two widths change places. With no wings the maker’s width is the truth and the table’s declaration a slightly narrower, slightly looser estimate of it. With wings, the table’s declaration is the tighter of the two and still safe, and the maker’s core width, however accurate as a statement about the line’s core, is the conservative one. A declared width buys a factor of two asked the maker for one number; for a lamp with wings, the number to give is not the core’s.

A one-nanometre table can see the wings #

The fit’s miss says something the width does not. Through one nanometre the worst line’s best miss grows from 0.05 per cent with no wings to 0.80 per cent with wings half the core’s width, fifteen times, crossing the half-per-cent test between wings a quarter and half of the core. At three eighths two of the three lines are refused; at a half, all three, and the table declares nothing. Through two nanometres the misses stay under half a per cent at every width the census runs: the wings are real, and the wider slit has smoothed away the evidence of their shape while keeping their weight.

That is the same division a table can refuse a pair it cannot resolve found for pairs, from the other side. A narrow table notices structure in a peak that a single Gaussian cannot imitate; a wider one reads the peak as a Gaussian of whatever width fits — the distinction a grid is not a resolution drew between what a table’s step can see and what its slit smooths away. For a pair, the width it then reads is dangerous. For a winged line, it is not.

What a declared width is a statement about #

The earlier essays called the declared width “the lamp’s narrowest feature”, and for a lamp of Gaussian lines that is what it is. This census changes the reading. The width a table declares is the width of the Gaussian that looks, through its slit, like the lamp’s narrowest peak — a statement about how the slit sees the lamp rather than about the lamp’s physics. For a Gaussian line the two coincide. For a winged line the declared width is the line’s half-maximum width, near enough, and it is the correct input to the bound; for a pair it is too wide, and the table refuses it only when the grid can see between the lines.

The practical rule is short. A colour engine may take a narrow table’s own declaration even for lamps whose lines are not Gaussian, provided the lamp’s narrowest peaks are single lines: wings broaden them, the declaration follows, and the bound stays safe. Where the lamp’s lines might be close pairs, the grid step decides whether the table can say so.

How the census was built #

Each mercury line is the tube’s Gaussian of full width 1.2 nm convolved numerically with a Lorentzian of full width a stated share of that, sampled at a hundredth of a nanometre to sixty nanometres either side and continued beyond by the Lorentzian’s own tail; each line keeps the power its Gaussian carried. The bed, the lines’ positions and powers, the triangular slits, the tabulation at a tenth of a nanometre and the peak finding are the earlier census’s. Each peak is fitted over widths a hundredth apart from 0.6 to 3 nm — finer than the earlier census’s four-per-cent steps, whose nearest scanned widths either side of 1.2 were 1.168 and 1.215. On the unwinged tube this finer scan puts every line’s best miss near 0.05 per cent; the earlier census’s quarter-per-cent misses were the distance from 1.2 to its nearest scanned width. The half-maximum width is read off each Voigt profile by bisection. The ceiling is the earlier essay’s — the three nearest entries over $\kappa$, the flanking pairs over $\kappa_2$, the smaller — checked against the lamp’s true peak within a slit’s width of every grid point at a hundredth of a nanometre; the variance allowance is the declared-width variance of the same essay, checked against the lamp’s own variance inside each window.

What this leaves out #

The whole bound was not recomputed. Its two inputs that depend on the declared width — the ceiling and the variance allowance — were checked window by window on the winged lamp; the full notch census of the earlier essays, which compares the finished bound with the actual error of each notch, was run only for the Gaussian tube.

Real lines are not these. The Lorentzian share here is swept, not measured: a low-pressure mercury discharge has narrow lines, and much of what a spectrometer records as a line’s shape is its own instrument function, which is often closer to a Voigt than to a Gaussian. The census’s result applies whichever of the two supplies the wings.

The wings are symmetric. A line with a shoulder on one side — a self-absorbed line, or an unresolved weak neighbour — shifts the fitted centre as well as widening it, and is closer to the pair of the earlier essay than to this.

Still open: whether a self-absorbed line reads as a pair #

A strong mercury line in a dense discharge absorbs its own centre on the way out of the tube: its profile flattens on top, and at high enough density dips at the centre, with two humps either side. That is a single line that looks, in shape, like a pair.

The calculation is this census with the 546 nm line given a self-absorbed profile — the emitted Voigt passed through a column of the same atoms, its optical depth at the centre swept from nothing to three — with the question whether the table refuses it as it refused the mercury pair, or reads it as a wider line as it read the winged ones. The prediction is that through a one-nanometre slit a centre optical depth above about one makes the line’s top flat enough that no Gaussian fits within half a per cent, so the table refuses it, while through two nanometres it reads as a Gaussian about the width of the flattened top — and that, because a flat-topped line is lower at its centre than any Gaussian of the same width, the ceiling built on that width stays safe here as it did for the wings. If it does, the only single-line shape that can make the declaration unsafe is one that is narrower at its peak than its entries suggest, and a physical line becomes that only by being two lines.

A worry about a quantity is a worry about its use #

The habit is about which comparison makes a departure dangerous.

The prediction measured the declaration against the line’s core and called any excess unsafe, because the earlier essays had called the declared width the lamp’s narrowest feature, and a core is a narrowest feature. By that comparison the prediction held exactly. The comparison that mattered was the one the declared width is used in: the ceiling and the variance allowance, set against the lamp they are meant to bound. By that comparison nothing failed, and the core, the width the prediction treated as the truth, turned out to be the loose one.

The failure mode is to judge an estimate against the physical quantity it was named after, when what it has to be right about is the quantity it is used to compute. A declared width is a parameter of a bound, and a bound is judged against the thing it bounds.