A broad line brings the notch back to a wide table

A spectral table can promise a ceiling: from the width of the narrowest line it holds, an upper bound on how high any line’s true peak rises between its entries. A declared width buys a factor of two made that promise, and the essays since have tested it on a tube whose every line is reversed — notched at its centre by a cooler column of the same atoms — where the promise needs the notch as well as the width.

A coarse table can borrow a notch, but not a phantom found the two-nanometre table’s ceiling failing without the notch from a column of three and a half, and holding on a notch borrowed from a finer table. A wide table needs no notch to borrow then widened the slit and found the opposite of what the trend promised. Through slits of three, four and five nanometres the ceiling built for an unnotched line held at every column depth to five, by 4.7 per cent at least, because a wide entry carries so small a share of a 1.2-nanometre line’s peak — an eighth, at five nanometres — that the ceiling built on it sits far above anything a notch does to the line’s top.

That account rests on the line being narrow against the slit. The census’s lines are 1.2 nanometres wide at their cores, and a lamp run at higher pressure, or a source of broader emitters, has lines several times that. The prediction was that broader lines bring the notch back: that at a core of 3.6 nanometres the five-nanometre table’s unnotched ceiling fails from a column of about three, as the two-nanometre table’s did for the narrow core, because what matters is the ratio of slit to core, and a table needs the notch wherever that ratio falls under about two. The alternative was that the five-nanometre ceiling still holds at every core, which would mean its safety comes from averaging over the empty spectrum between lines and the ratio is not the rule.

The ratio, almost exactly #

With every line’s core 3.6 nanometres wide, the five-nanometre table’s unnotched ceiling holds below a column of three and fails from three on, reaching 0.840 of the true peak at five. Across all twelve tables — cores of 1.2, 2.4 and 3.6 nanometres, slits of two to five — the first failing column rises with the slit-to-core ratio: a half at a ratio of 0.56, one and one and a half at 0.83, two and two and a half near 1.2, three at 1.39, three and a half and four at 1.67, three and a half at 2.08; every table at 2.5 or more never fails. With the true notch every table holds at every depth, the least by 0.3 per cent.

• The prediction holds for the five-nanometre table, to the column.
• Its rule holds in shape and not in its number. The ratio orders the failures, and the threshold lies between 2.08 and 2.5, a little above two.
• The alternative fails. Averaging over empty spectrum does not save a wide slit once the line fills it.

Plotted against the ratio, twelve tables built from three cores and four slits fall nearly on one rising track. The narrowest ratio, a two-nanometre slit on a 3.6-nanometre core, fails at the first column the census runs; the widest ratios, three to five nanometres on the narrow core, never fail. Between them the first failing column climbs with the ratio, and the three cores’ points interleave rather than forming three separate curves, which is what the prediction’s account requires: a table and a lamp scaled together behave alike.

The interleaving is not perfect. A four-nanometre slit on a 3.6-nanometre core, at a ratio of 1.11, first fails at two and a half, later than a three-nanometre slit on a 2.4-nanometre core at 1.25, which fails at two; and a five-nanometre slit on a 2.4-nanometre core, at 2.08, fails at three and a half, sooner than a four-nanometre slit on the same core at 1.67. Where the census’s lines fall against a coarse table’s entries depends on the table’s step, and a step that lands an entry on a line’s notch fails sooner than its ratio alone would say.

Five nanometres, core by core #

The five-nanometre table is where the prediction placed its test. With 1.2-nanometre lines its unnotched ceiling never comes within 4.7 per cent of the true peak; with 2.4-nanometre lines it stands at 1.0002 at a column of three and fails from three and a half; with 3.6-nanometre lines it fails from three, falling to 0.840 at five. The broad-line curves rise with the first column, as the narrow-line one does, and then fall through the peak.

The rise and fall are the notch’s two effects. A shallow column flattens a line’s top, and a flattened top is easier for two wide entries to cover, so the ceiling’s margin grows; a deep column splits the top into two shoulders with a trough between them, and once the shoulders are far enough apart for an entry to fall between them, the entry carries less of the peak than an unnotched line would give it. For a narrow line the shoulders stay well inside a five-nanometre entry at every depth; for a broad one they spread apart with the core and come to lie in different entries.

Two nanometres, core by core #

At the other end of the slits the broad lines fail almost at once. The two-nanometre table’s unnotched ceiling fails from three and a half with 1.2-nanometre lines, from one and a half with 2.4-nanometre lines and from a half with 3.6-nanometre lines. A slit narrower than its lines sees a line’s shape rather than its area, and a notch changes the shape from the first fraction of an optical depth.

The two-nanometre table on 1.2-nanometre lines and the five-nanometre table on 2.4-nanometre lines sit at nearly the same ratio, 1.67 and 2.08, and both first fail at three and a half — the closest pair in the census of two very different tables behaving alike because their lines and slits are in proportion. That is the ratio’s strongest evidence: a table of a narrow lamp at fine resolution and a table of a broad lamp at coarse resolution are, to the ceiling, the same table.

Why a broad line fills an entry #

The mechanism is the single-entry factor, the least share of a line’s true peak one entry carries however the line falls against it. A five-nanometre entry carries at least 0.125 of a 1.2-nanometre line’s peak, 0.232 of a 2.4-nanometre line’s and 0.304 of a 3.6-nanometre line’s — the last more than a two-nanometre entry carries of the narrow line, 0.273. A ceiling divides entries by this factor, so the broad line’s ceiling at five nanometres sits as close to its peak as the narrow line’s did at two, and is as easily fooled by a notch.

The factor is what the earlier essay’s account was about, made precise. A wide table was safe because its factor was small; its factor was small because the line was narrow against the slit; and the factor depends on the ratio of slit to core almost alone, because a Gaussian line and a triangular slit scaled together give the same share. The census’s threshold lies between a factor of 0.20, which a ratio of two and a half delivers and which holds, and 0.23, which a ratio of 2.08 delivers and which fails: below about a fifth, an entry’s share of the peak is small enough to cover every notch the census’s columns cut.

The true notch saves every table #

Built on the true notch, every table holds at every core and every column. The least margin is 0.3 per cent, at a three-nanometre slit on a 2.4-nanometre core; the broad cores’ tables keep at most 2.4 per cent and the narrow core’s between 4.6 and 8.6. The ceiling’s construction, given the notch, is sound at every ratio the census reaches — every failure in this essay is a failure to know the notch, never a failure of the ceiling that knows it.

The thinness of the broad cores’ margins has the same cause as their failures. A ceiling divides an entry by the share of the peak that entry must carry, and a broad line’s entries carry a large share; the ceiling built on them sits close to the true peak whether the notch is known or not. Knowing the notch removes the failure but cannot restore the generous margin a small share gives a narrow line. A table of broad lines, notch and all, is a tight bound — useful when a tight bound is wanted, and a quarter of a per cent from wrong when the notch it borrowed was read a little shallow.

So the earlier essays’ prescription survives with its domain stated. A table must carry the notch, borrowed from a finer reading as a coarse table can borrow a notch, but not a phantom did it, wherever its slit is less than about two and a half times its lines’ cores; above that it may leave the notch out and keep a wider margin than borrowing would give. A table maker who knows the lamp’s line widths knows which side of that line the table is on.

A line’s shape, scaled #

Drawn behind the same column, the three green lines are one shape at three scales. Each keeps its unreversed peak height, so the shoulders reach the same height and the notch falls to the same depth; the notch is six tenths of a nanometre across at a core of 1.2 and a nanometre and four fifths at 3.6. A five-nanometre entry spans the narrow line’s whole reversed top and averages it to something close to the line’s area; it spans only part of the broad line’s, and what it carries depends on whether it sits on a shoulder or in the trough.

Because each line keeps its peak, a broader line carries more power than a narrow one, and the lamp’s total grows with the core. That does not change the ceiling, which compares each window with that window’s own peak; it does mean the broad-line lamp is a brighter lamp, and a comparison of ceilings across cores is a comparison of shapes, not of lamps a buyer would choose between.

Where line widths come from #

The census’s 1.2-nanometre core is the narrowest feature the earlier essays declared for the tube, and the essays on how a table declares its lines have already met widths larger than that. A line’s wings widen the width a table declares found a line’s Lorentzian wings making a table declare a wider line than its core, and a stricter fit declares a wider line found the declared width growing as a table’s fit was made more careful. A declared width is what a ceiling is built on, and every widening of it moves a table’s ratio down towards the threshold this census finds.

A narrow table declares its own lines is the reminder that a table can measure its lines’ widths for itself when its slit is fine enough. Read together with this essay, the procedure a table maker would follow is short: declare the line width, divide the slit by it, and carry the notch — borrowed from a finer reading if the table cannot see it — whenever the ratio is under two and a half.

Applying the ratio to a table in hand #

The rule is meant to be used on a table someone else made, and it asks for two numbers. The first is the slit, which a published table usually states as its bandwidth or can be inferred from its step; the census’s tables use a triangular slit as wide at half height as the step, so a table published every five nanometres is a five-nanometre table here. The second is the narrowest line’s width, which the lamp’s maker can state, or which a narrow table declares its own lines shows a finer table can measure for itself.

Three cases show the range. A five-nanometre table of the census’s own tube, its lines 1.2 nanometres wide, has a ratio over four and needs no notch at any column the census runs — the earlier essay’s finding. The same table of a lamp whose lines are 2.4 nanometres wide sits at 2.08, below the threshold, and is safe only while the reversing column stays under three and a half optical depths; past that it needs the notch, and has no fit of its own that could read one. A two-nanometre table of that broader lamp sits at 0.83 and needs the notch from a column of one and a half, which is to say from the first visible reversal.

What none of the three needs is a census of its own. The ratio places a table, the first failing column says how deep a reversal it can tolerate without the notch, and the true-notch margins say that once the notch is carried, every table here is safe. A table maker who cannot tell whether the lamp’s lines are reversed at all can still read the ratio: above two and a half, it does not matter; below it, the question has to be answered, and the depth rule’s phantom-free borrowing is the answer the earlier essays found.

The census cannot say how many published tables fall on each side, because it does not hold their lamps’ line widths. It can say which question each table owes its reader: a table published at a step under two and a half times its narrowest line’s width owes a statement of whether that line is reversed.

What this does not reach #

Gaussian cores. Every line here is a Gaussian at three widths; a line with Lorentzian wings carries more of its power far from its peak, and a wide entry’s share of its peak falls more slowly with the slit.

One column shape. The column is half the line’s width at every core; a column relatively narrower cuts a sharper notch, which a wide entry would average away more easily.

Twelve tables. The threshold lies between 2.08 and 2.5 because the census holds no ratio between them; a slit of 5.5 nanometres on a 2.4-nanometre core would narrow it.

Still open: whether a line’s wings move the threshold #

The threshold sits between a slit-to-core ratio of 2.08 and 2.5 for Gaussian lines. A real discharge line has Lorentzian wings, and a line’s wings widen the width a table declares found a fifth of Lorentzian width raising the declared width by more than its share.

The calculation is the reversed tube with lines of a 2.4-nanometre Gaussian core and Lorentzian wings a tenth and a quarter of the core’s width, read through slits of four, five and six nanometres, the unnotched ceiling built on the declared width and on the core, set against the true peak at every column. The prediction is that the threshold moves out in proportion to the declared width — to a ratio of about 2.5 against the core with a quarter’s wings — and that a ceiling built on the declared width keeps the threshold where it is. If the wings move the threshold further than the declared width does, a ceiling cannot be built on any single width for a reversed line with wings, and a wide table must carry the notch whatever its ratio.

A safety that was a ratio #

The habit here is about where an instrument’s robustness comes from.

The wide tables were safe, and the earlier essay gave the reason as the width of the slit. The reason was the slit’s width compared with the line’s, and the census’s lines all had one width, so a property of the pair was read as a property of the instrument. Changing the line changed the instrument’s verdict without touching the instrument.

The failure mode is to credit an instrument with a robustness that belongs to the relation between the instrument and the thing it measured. The test is to vary the measured thing at a fixed instrument — here, the line’s core under one slit — and see whether the robustness travels; when it does not, state it as the ratio it was, and find where the ratio runs out.