Molecular Geometry & Bonding
what the shape decides
Concept

Coherence length — where it appears

The distance over which a disturbance in an ordered state heals back to its bulk value. It is set by the gap that stabilises the order — a smaller gap means a longer length — and it is what a decay profile has to be measured in before two systems can be compared.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The exponent against how far the tail can be seen. The fitted power against the reduced reach — how many coherence lengths of the decay survive above the floor before the excess is numerical noise. The six cases with a reach past six give a power between 0.41 and 0.57 and are drawn solid; the rest are hollow and their fitted powers are off this scale in the negative direction. The reach is not a choice — it falls as the gap closes, because the excess the tail starts from falls with it.

The exponent was the floor

Fitting the local decay rate against the reciprocal distance reads a power off the slope. It runs from 0.41 to 0.66 across ten stiffnesses and appears to settle near two thirds. It is not settling. The tail is dropping below the arithmetic's own floor sooner at every step, so each case's power is taken over a shorter piece of the curve than the last.

solids · Peierls distortion The exponent against where the fit is allowed to start. For each stiffness, the fitted tail exponent as the near end of the fitting window is moved outward from two bonds to thirty. The standard start is six, by a rule of thumb — three coherence lengths — and the question was whether that choice is doing any work. Below six the exponent rises steeply; from six outward it is nearly flat. The rule of thumb sits on a plateau.

The rule of thumb was on the flat part

Fitting the Peierls tail discards the first few bonds of every profile, on a rule of thumb — three coherence lengths. Does that unexamined choice hide a second exponent? It does not. From six bonds outward the fitted power moves by half a per cent to nine; below six it moves seven times as much, and starting at two would have halved the very trend the fit reports.

solids · Peierls distortion The far end stops mattering, abruptly. The fitted tail exponent as the far end of the window is opened from twenty bonds to a hundred and twenty. Each curve is flat past a stiffness-dependent point and exactly flat past it — because beyond a profile's own reach there are no more local rates to add, so a larger window is the same fit. The standard sixty bonds is inside the flat part for every case.

The other window was a plateau too

Sweeping where the fit begins finds a plateau. The far end is the other window and nobody had swept it: inside each profile's own reach the exponent moves by at most 5.3 per cent, and past that reach every larger window returns exactly the same fit — because there are no more points to add. What the reach is depends on the stiffness, and for half the series it is an arbitrary rule rather than the physics.

solids · Peierls distortion

Named alongside it

The objects these essays reach for when they reach for this one.

Bond alternationExtrapolationNumerical precisionPeierls distortionTight-binding modelsApproximationModel limitBand gap

All concepts