Every essay — page 28
Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search
When the molecule does not stop
A chain of two hundred atoms is a molecule and behaves like a solid. Bands, gaps, metals, defects and surfaces, every one of them out of a finite matrix — and a clear account of what that route cannot reach.
The share that was read as a line
The share of defect runs sitting in a resonant pair was read as a straight line in the logarithm of the chain length, rising by a density times a reach per decade and never saturating. The exact rise is that amount times the share of runs not yet resonant. At the concentration first studied the difference is small over nine decades and large beyond them; swept to the concentration with the most runs, the rise falls to a third, and half of all runs are resonant ten orders of magnitude later than the straight line says.
The commonest runs reach it in a grain
Half the runs of six low sites in a disordered chain sit in a resonant pair only at 10⁴¹ sites, which no piece of matter reaches. Runs of four are four times as common and bind less deeply, and both of those shorten the answer, so they reach the same state at 10¹⁵ sites — a grain thirty micrometres across. What the essay before it found was not a fact about the resonant share but a fact about half filling and runs of six: reachability is a diagonal line across run length and concentration, and at seven tenths of the sites low it passes above runs of seven.
A plane spends the same reach on a disc
Everything this argument has computed about defect bands is one-dimensional, and the one-dimensional part is not the coupling law or the level spacing — it is the count of how many candidate twins lie within a given distance. On a plane that count grows as the area, so the same reach that finds one twin in a chain finds a disc full of them, and half the runs of six are in a resonant pair at 10³·⁹ sites rather than 10⁴⁰·⁸. That is thirty-seven orders of magnitude, it is a patch ninety-two sites square, and it means no two- or three-dimensional material has a partly resonant defect band at any size.
The rate that turned back
Opening the Peierls fit's window to half the chain added forty to eighty bonds to every stiff profile, and those bonds were read as tail. Most of them are not. The local decay rate on an opened stiff profile falls to a minimum and climbs again, because the bulk it is measured against is read from the chain's middle, which still carries both ends' healing — and a model of two ends and that reference, built from the quarter window alone, puts the minimum within two bonds of where it is. Read only to the turn, the fall of a seventh in the stiff exponents disappears, and one of the two cases the wider window was credited with resolving is not resolved.
No moment sees the winner change
On a frustrated net the arrangement that binds best at half filling changes at a contrast of 3.790, and the gap at the Fermi level does not follow until 4.849. The natural replacement for the gap is a moment of the whole band. It fails more completely than the gap did. The second and third moments are identical for every arrangement at the composition; the fourth is exactly the count of unlike bonds; and every moment from the fourth to the twenty-fourth keeps one sign from a contrast of 1 to 8. An account of the band that matches its first sixteen moments puts the change at 3.980, and only the one that resolves every level gets 3.790.
A ring is not the bulk
The local decay rate on a stiff Peierls chain turned back because its bulk was read from the chain's own middle. The remedy proposed was a ring of the same length, with no ends to heal from. A ring gets the soft chains wrong by thirty thousand times the floor — because the chain's relaxation holds its length fixed, and the ends' extra alternation is paid for by a shift in every bond that goes as one over the length. With that shift in the bulk, the soft middles agree to 10⁻¹¹, no stiff profile turns back, two tails describe every profile to its middle, and the stiff exponents keep rising where the published ones had turned down.
The residual was the stiffness
A matched Peierls collapse leaves 1.6 to 2.1 per cent between five rings that hold the same number of coherence lengths, and the question was how much of it the even-site rounding explains. Twenty-five rings answer it: a ring's curve moves by a few hundredths of a per cent per coherence length, so the rounding explains a hundredth of the residual. The rest is the stiffness itself — a term linear in the bulk amplitude, which no choice of sizes can match away — and its intercept at zero amplitude is BCS's gap curve to a fifth of a per cent.
The disc was a band with holes in it
The plane's defect band was built from a chain's coupling law spent on a disc of twins. Computed instead from the plane's own Green's function, two runs of six couple with a direction and a parity: a twin in the same row couples thirty times as strongly as one stacked above at the same distance between centres, and offsets across rows alternate between strong and weak. The region of twins strong enough to matter is a ragged band eleven sites long and six high, and half the runs are resonant at 10³·⁰⁹ sites rather than 10³·⁹³. The thirty-seven decades between a chain and a plane survive and grow.
The chain that pays for nothing
Held to a fixed length, a Peierls chain pays for its ends' extra alternation with a compression of every bond. Let the length go and nothing is paid: the middle is the free infinite chain to twelve figures, the ends adjust locally — and every tail exponent comes out where the fixed chain put it, to under a thousandth. The constraint was never in the exponents. Nor is the drift in them a missing 1/b term: fitting one makes the exponent wander three times further.
What a spectrum settles
A spectrum is a list of positions, and how many there can be is decided by the shape before any of them is measured. Counting them, and reading a structure back out.
Normal modes are not bond stretches
Water has two stretching frequencies and two O–H bonds, and it is almost irresistible to pair them off. Computed, each mode is exactly half in one bond and half in the other, and neither frequency belongs to a bond at all.
The isotope shift is arithmetic
Replace hydrogen with deuterium and every frequency drops. The usual rule says by a factor of the square root of two — and of water's three modes, not one of them does that. What is exact is a different identity, and the force constants cancel out of it.
Group frequencies, and where they stop
A carbonyl band sits near 1,700 wavenumbers in every ketone anybody looks at, and that regularity is real. Computed for a set of small molecules, four of twenty-one distinct frequencies belong to a single internal coordinate — and the four are exactly the coordinates symmetry leaves alone.