The commonest runs reach it in a grain

The resonant share does not rise for ever: each decade of chain length adds a fixed amount times the share of runs not yet resonant, so the curve bends over, and half the runs of six low sites are in a resonant pair at a chain of 10⁴¹ sites rather than the 10³¹ a straight line predicted.

Ten orders of magnitude is a satisfying correction and it left a reading that is now worth challenging. 10⁴¹ sites is not a piece of matter, so the conclusion drawn was that the saturated regime is unreachable and the straight line’s error is an academic matter about a limit nobody visits.

The quantity itself is worth restating, because everything below turns on which of its three ingredients moves. A run of L consecutive low sites holds a level in the gap at a depth the run’s length fixes; two runs of the same length split that level into a pair by an amount falling exponentially with their separation; and the pair counts as resonant when that splitting exceeds the mean spacing of all the gap levels around it, which is the criterion that decides when many defects are a band. The spacing falls as one over the chain length, so a longer chain buys reach — and the share of runs with a twin inside that reach is the quantity.

That conclusion was about runs of six, at half filling, and it does not survive either change.

Runs of six are not the runs a material has most of #

The density of runs of exactly L low sites is $x^{L}(1-x)^{2}$ per site, which falls with every extra site in the run. At half filling runs of three occur once in thirty-two sites, runs of six once in two hundred and fifty-six, and runs of eight once in a thousand. Six was chosen in the essay that first put a defect band together because it is comfortably long enough to hold a level well inside the gap and short enough to be common — a reasonable choice, and a choice.

The runs a real material’s disorder actually presents are the short ones, in overwhelming numbers. So the question the earlier essay answered for runs of six is worth asking for the runs that are there.

There is a floor on how short, and it is the reason nothing below three is computed. A run has to be long enough for its box level to sit inside the gap the other runs’ levels define, and a run of one is an impurity rather than a box — its level depends on the site energy and not on a length at all. Three is where the sequence of box levels starts, and it is also where the level spacing’s own definition starts, since the spacing here counts every run between three and nine.

At half filling the answers are 10^9.5 sites for runs of three, 10^15.0 for runs of four, 10^24.4 for runs of five, 10^40.7 for runs of six, 10^70.0 for runs of seven and 10^123.3 for runs of eight.

Twenty-five orders of magnitude separate runs of four from runs of six, and the sequence of exponents itself grows geometrically — each extra site multiplies the number of decades required by about 1.7. That is a steeper dependence than anything in the underlying physics, and it comes from a division: the separation needed is inversely proportional to the density, and the decades needed are that separation divided by the reach bought per decade.

What the numbers are in sizes #

A number of sites is not an answer until it is a size. A solid holds about 5 × 10²² atoms in a cubic centimetre, which fixes the rest.

10^9.5 sites is a cube about a micrometre across — a small crystallite, or a grain boundary’s worth of material. 10^15.0 is a grain thirty micrometres across, which is an ordinary crystallite in an ordinary polycrystalline sample. 10^24.4 is a litre of solid. 10^40.7 is a rocky body a dozen kilometres across. And 10^123.3 is forty orders of magnitude past every atom in the observable universe.

So runs of three and runs of four reach the saturated regime inside a real sample and runs of five upward do not. The essay below’s finding survives for the run length it was measured at and reverses two sites down, which is the whole of this essay’s first half: the statement a real material never reaches it is true of the object it was measured on and false of the objects a material mostly contains.

Two causes, and they compound #

Why a shorter run gets there so much sooner has two halves, and the second is easy to miss because it points the same way as the first.

The obvious half is the density. Runs of four are sixteen times as common as runs of eight at half filling, so a given run has sixteen times as many candidate twins within any separation, so the separation at which half of them have one is sixteen times smaller — 44 sites against 709.

The second half is the binding. A run of four holds its state less deeply in the gap than a run of six does, and a less deeply bound state has a shorter decay length: ξ is 1.438 for runs of four against 2.556 for runs of eight. A shorter decay length means the coupling falls off faster with separation, which means the reach the chain buys per decade of length is smaller — 3.31 sites a decade against 5.89.

That second effect works against the first, and loses. The decades required are the separation divided by the reach per decade: 44/3.31 is thirteen and 709/5.89 is a hundred and twenty. So the density’s factor of sixteen becomes a factor of nine in the decades, because the shallower binding gives back part of it.

Stated that way it is a competition, and the reason it reads as a compounding is the third quantity. The half-way separation is not simply 1/ρ; it is ln 2 divided by −ln(1 − ρ), and the decades needed are measured from where the coupling reaches anything at all, which itself moves with ξ. The three together give the factor of 2.7 in exponents between runs of four and runs of six, and none of the three is negligible.

Where the boundary actually is #

Neither the run length nor the concentration decides reachability on its own.

At three tenths of the sites low, runs of three reach half at 10^21.3 — just inside a cubic centimetre — and runs of four need 10^54.9, which is more than the Earth has atoms. At half filling the boundary falls between runs of four and runs of five. At seven tenths, runs of seven reach half at 10^20.1 and runs of eight at 10^25.2, so the boundary has moved up by three sites of run length.

Three decades of run density move the boundary by four sites of run length, which is a strong dependence and a useful one: it says which materials this question is live in. A lightly disordered solid, with a few per cent of substituted sites, has no resonant band at any size for any run long enough to hold a level. A heavily disordered one — an alloy near equal composition, or a heavily doped semiconductor — has resonant bands of its short runs in every crystallite.

That is the reading the essay before it could not reach, because one concentration and one run length is one point and a boundary needs two axes.

What a measurement would actually see is the share, not the boundary, and the shares are worth putting beside the boundary because they are less dramatic. In a grain thirty micrometres across at half filling, the resonant share is 69 per cent for runs of three, 50 for runs of four, 33 for five, 21 for six, 12 for seven and 7 for eight. Not one of those is near zero and not one is near one. A gap spectrum of that material contains, simultaneously, a saturated band of short runs, a half-formed one of runs of four, and a set of essentially isolated levels from the long runs — which is a more interesting object than either limit, and is what the boundary language obscures by asking a yes-or-no question of a quantity that slides.

The boundary is still the right thing to compute, because it says which run lengths a change of sample size can move. Below it the share is near saturation and a larger sample changes nothing; above it the share is small and a larger sample changes it proportionally. What sits on it is the run length whose band is being made, and at half filling that is runs of four and five.

How long the transition takes #

There is a second quantity in the sweep that is worth reporting on its own, because it says what kind of thing this is.

The coupling reaches nothing until the level spacing has fallen far enough, which happens at a definite length — 200 sites for runs of six at half filling. From there to the half-way point is thirty-eight decades. For runs of eight it is a hundred and twenty; for runs of three, eight.

A transition that takes thirty-eight decades is not a transition. What it means in practice is that a sample’s resonant share is set by the sample’s own size and not by the material at all: double the crystallite and the share moves by the third of a point a decade gives. The quantity is a property of the specimen, which is exactly the kind of thing a measurement cannot be compared against a theory of the material.

That is a different complaint from the one the earlier essay made and it is the more serious of the two. The straight line was wrong about where the half-way point is; the window says the half-way point is not a feature. A quantity with a sharp threshold has a value that belongs to the material either side of it, and this one has no threshold at any run length: the share slides from nothing to everything over between eight and a hundred and twenty decades, and every real sample sits somewhere in the middle of that slide.

For runs of three and four the window is eight and thirteen decades, which is still enormous, and this is the honest limit on the first half of this essay. Runs of four reach the half-way point in a grain thirty micrometres across, at 50.3 per cent, and in a cubic centimetre of the same material they are at 66.5 per cent — sixteen points apart for a sample eight decades larger. The share is reachable; it is still not a number the material owns.

The straight line was worse elsewhere #

One last reading, which is the earlier essay’s own finding generalised.

The straight-line reading is early at every run length, and by more the longer the run: 0.8 decades at runs of four, 3.5 at runs of five, 8.1 at runs of six, 16.3 at runs of seven and 31.2 at runs of eight. Runs of three are the exception in a way that is worth stating, because their share has already passed a half at 10¹² and the reading is then an interpolation rather than an extrapolation — and there the straight line is late by two thirds of a decade.

So the instance that was corrected was among the mildest available. The error in a linear extrapolation grows with how far it has to run, and the runs that need the most decades are the ones where the extrapolation is asked to cover the most ground. Had the essay before it picked runs of eight it would have found thirty-one decades rather than eight, and had it picked runs of three it would have found the straight line very nearly right and drawn the opposite conclusion.

What is computed, and what it rests on #

Every number here comes from three closed forms and one fit. The density $x^{L}(1-x)^{2}$ is exact and is checked against counts in generated chains. The geometric separation law is exact for an independent-site chain and is checked the same way. The level spacing of an N-site chain falls as 1/N, with the constant fixed by the energy range the gap levels of runs between three and nine occupy. The fit is the coupling’s decay length ξ and prefactor, obtained from pairs of runs at measured separations.

The reach is where a fit enters and where the run-length dependence lives. ξ is fitted separately for each run length, from the splitting of two runs of that length at a range of separations, and the six values run from 1.164 to 2.556. They are not a guess: a run of L holds a level at a definite depth and the decay length of a state at that depth is determined, so ξ(L) is a computed quantity with a fitted constant rather than a parameter. The distribution those fits came out of spans a hundred and seventy decades in coupling, which is why the quantity being computed is a count and not an average of anything.

The nine-decade window for the straight line is the earlier essay’s choice and is kept. A linear reading taken over more decades would be less wrong, and the point of keeping the same window is that it is the window a reader would have.

Where this stops #

Independent sites. Every site is low with probability x regardless of its neighbours, which is what makes the density and the separation law exact. A real alloy has short-range order — like sites clustering or avoiding each other — and that changes the run-length distribution without changing anything else here. Which direction it moves the answer depends on the sign of the ordering, and nothing in this model has a knob for it.

One run length at a time. The resonant share of runs of four is computed as though runs of four only coupled to other runs of four, which is what makes them a set of levels at one energy. Runs of different lengths hold levels at different energies and couple too, more weakly for being off resonance, and a real gap has all lengths in it at once. The level spacing used here already counts every run between three and nine, so the spacing is the mixed one and the coupling is not.

And the box level is a box level. A run of four low sites is treated as a particle in a box of four sites, which is what the level depth and therefore ξ come from, and the approximation is better the deeper the level. At a run of three it is at its worst, and the two conclusions that depend on runs of three — the micrometre cube and the straight line’s reversal of sign — are the two least secure in this essay.

A chain. Everything above is one-dimensional, which is where the geometric separation law comes from.

A statement about a case, read as a statement about a quantity #

The habit: when a model gives an extreme answer, check whether the extremity belongs to the quantity or to the case it was evaluated at.

10⁴¹ sites is an extreme answer and it invites a general conclusion — the saturated regime is unreachable, the correction is academic. The extremity was in the choice of run length, and it was a choice made three essays earlier for a different reason entirely, which is why nothing flagged it. Two sites shorter and the same calculation lands inside a crystallite.

The corollary is about which way a sweep should be run. The essay below swept the concentration, which moves the density across four decades, and found that the linear reading’s error grows with density. Sweeping the run length moves the density across the same range and moves ξ as well — so it is the sweep that shows the two causes, and it was available for the cost of a loop.

Who counted the runs, and when #

The geometric distribution of gaps between runs in a Bernoulli chain is elementary and old. The exponential decay of a defect state’s coupling with separation is standard tight-binding; the criterion that a set of levels is a band when the coupling exceeds the spacing is the one every account of a band uses. The atoms-per-cubic-centimetre figure that converts a site count to a size is the ordinary one for a solid.

What is computed here is the half-way length for six run lengths at three concentrations, the decomposition of its run-length dependence into density and reach, the reachability boundary in that plane, and the straight-line error across the same grid.

The numbers worth carrying are 10¹⁵ and 10⁴¹ — the same quantity, two sites of run length apart.

Still open: short-range order, and the mixed gap #

The obvious open question is the one the independent-site assumption forbids. Every density here is $x^{L}(1-x)^{2}$, which is exact for sites that do not know about each other, and a real alloy’s sites do: a tendency for like sites to cluster lengthens the runs and one for them to alternate shortens them, and either moves the whole argument above by a shift in effective run length. A single ordering parameter — the probability that a site’s neighbour is like it, against x — would let the run-length distribution be computed rather than assumed, and the interesting quantity is not the shift but whether the reachability boundary moves parallel to itself or tilts.

The nearer question is the mixed gap. The level spacing used here counts the runs of every length between three and nine, so it is already the mixed quantity, and the coupling is not: runs of four are treated as coupling only to runs of four. Two runs of different lengths hold levels a definite distance apart in energy, and a coupling between levels that far apart mixes them by the coupling over the detuning — which is computable from the same splitting law and the same box levels, and would say whether a run of four is better off with a distant twin or a near neighbour of the wrong length. On the numbers above the second is far commoner, so the answer decides whether the single-length calculation is a good approximation or a different problem.