Molecular Geometry & Bonding
what the shape decides
Concept

Exchange coupling — where it appears

The interaction that lines two spins up or opposes them, written as a splitting between a singlet and a triplet. It is far too large to be magnetic: it comes from hopping and the Pauli principle, and goes as the square of the hopping over the repulsion.

Named by 15 essays across 2 fields — each of them below, with the objects they name alongside it.

A coupling that is second order in the hopping. The singlet–triplet splitting of a two-site Hubbard model, and the same quantity multiplied by U. The product settles on −4t² — -4 at U = 64 — which is what makes the coupling a second-order effect rather than a term somebody put in.

What couples two spins

Two magnetic ions a few ångströms apart interact far too strongly to be doing it magnetically — the dipole–dipole energy is about 0.06 wavenumbers and the measured couplings run to hundreds. What couples them is hopping, which the Pauli principle allows for antiparallel spins and forbids for parallel ones, and the exact answer is −4t²/U.

applied · Magnetism The same sample, fitted over four temperature ranges. A pair coupled at -50 cm⁻¹, its susceptibility computed exactly, fitted to a Curie–Weiss law over four ranges. The moment and the Weiss temperature the fit reports both depend on which range was used, and the quality of the fit does not warn about it.

The moment a fit invents

One coupled pair of spins, its susceptibility computed exactly, fitted to a Curie–Weiss law over four temperature ranges. The moments reported are 2.471, 2.535, 2.566 and 3.590 Bohr magnetons, and the Weiss temperatures −51, −78, −78 and −292 K — from one sample, measured perfectly, with three of the four fits agreeing with their own data to better than a part in three hundred.

applied · Magnetism The coupling a fit reports, and the coupling the sample has. Exact susceptibilities of Heisenberg chains of two, four, six and eight spins, every one of them coupled at -50 cm⁻¹, each fitted with the two-spin expression over 80–600 K. The two-spin sample returns its own coupling exactly; every longer chain returns one too large, by more the longer it is, up to 20.9 per cent. Every one of those fits has an R² above 0.99, so nothing in the fit reports that anything is wrong.

The model is what is fitted

Fit a pair of coupled spins with the two-spin expression and it hands back the coupling exactly, from any temperature range. Fit a chain of eight with the same expression and it hands back −66.7 where the sample has −50, with a residual of 0.998 and a g factor of 1.973 — three numbers of which only the last says anything is wrong, and it is the one nobody looks at.

applied · Magnetism How much of a curve each extra parameter has left to work with. The singular values of the design matrix for a susceptibility curve, for two, three and four parameters fitted to the same data, on a logarithmic scale. With four they run 12.411, 2.026, 0.149, 0.025 — a span of 500 — so one per cent data fix the first two to under 3 per cent and the last to 37. Each value is what is left of the measurement after the directions above it have taken their share, so a short bar is not a hard parameter but an absent one.

How many parameters a curve is worth

A susceptibility curve routinely carries four fitted parameters and the question of whether it can support them is never asked. It has an arithmetic answer: the four directions the fit sees span a factor of five hundred, so one per cent data fix the first two to under three per cent and the last to thirty-seven — and forty points reaching two kelvin are worth more than sixteen thousand starting at twenty.

applied · Magnetism The four things a susceptibility curve measures, in order. The four directions in the parameter space, best fixed first, each written as the product of powers it is. The Jacobian is logarithmic, so a direction is a set of exponents and a combination is a product — which is why the answer can be printed. The best-determined is g · J^-0.33, fixed to 0.08 per cent by a curve measured to one per cent; the worst is tip · rho^-0.40, fixed to 40. Neither is one parameter's own axis.

The product a curve measures

A susceptibility curve's fourth parameter is undetermined, and the question is which combination the free direction actually is. It is a product of powers, because the Jacobian is logarithmic — and at the usual window it is the temperature-independent term divided by the 0.40 power of the monomer fraction, fixed to forty per cent, while the product one place up is fixed to 6.7. A paper could print that instead of four numbers.

applied · Magnetism The exponent's sign is the ring's parity. The power the temperature-independent term carries in the free combination, for an open chain and for rings of five to ten spins. Every odd ring is negative and every even ring and the chain is positive — so the free product is ρ·χ_TIP raised to a power whose SIGN changes, which is a different combination rather than a shifted one. Nothing here is a near miss: the closest pair on either side of zero are +0.23 and −0.29.

The sign a frustrated ring changes

There is a sharper question than whether a low-temperature feature buys back a fourth parameter: does it change which combination is free? It does, and by a sign. Every odd ring of spins leaves free the monomer fraction times a negative power of the temperature-independent term, and every even ring a positive power, with no case in between.

applied · Magnetism The sign follows the count, on a ring and on a chain alike. The free combination's exponent against the number of spins, for open chains and for rings. Every even count is positive and every odd count is negative, whichever topology it is — and an open chain has no frustration at all. The usual comparison sets frustrated odd rings against an unfrustrated chain of eight, which varies the frustration and the parity together.

It was the count, not the frustration

The free combination's exponent comes out negative on every odd ring and positive on every even ring and on an open chain, and the sign was put down to frustration. The control was a chain of eight. A chain of five is not frustrated in any sense — a chain is bipartite and every bond can be satisfied — and its exponent is −0.792.

applied · Magnetism Nine clusters, two candidate rules, and one of them survives. Every cluster's exponent, with whether it is frustrated and whether its count is even. Frustration is decided by whether the coupling graph is bipartite, since an antiferromagnet can satisfy every bond exactly when a two-colouring exists. All four decisive cases come out positive — 0.2136, 0.2084, 0.1924, 0.2048 — so the parity rule survives and the frustration account does not. The gap between the lowest positive exponent and the highest negative one is 1.0497.

The frustrated cluster with an even count

A parity account of the exponent's sign replaced a frustration account and left the two still confounded: every case tested had frustration and odd parity aligned. A tetrahedron of four spins is frustrated and even. Its exponent is +0.2084, and so are those of three more clusters the two accounts disagree about — but the mechanism proposed with the parity rule is refuted along with the account it replaced.

applied · Magnetism Sixteen clusters, and the couplings at which each one's sign changes. The sign of the exponent for every cluster across couplings from 2 to 1000 cm⁻¹, read at a 20–300 K window, with each change of sign marked. No cluster keeps one sign across the range. The clusters with a ground spin change sign twice below 150 cm⁻¹, at couplings that fall as the ground spin rises; the singlets first change sign above 300 cm⁻¹. The spin rule — positive for a singlet ground state, negative otherwise — is right on all sixteen at once only between 20.1 and 47.0 cm⁻¹, and the working point every earlier reading used, 50 cm⁻¹, is just outside it.

The sign rule holds between two poles

The sign of a susceptibility fit's exponent was put down to the parity of the spin count, then to the spin of the ground state, and every cluster tried had the two aligned. A star of four spins is even with a ground spin of one, and its exponent is −6.84. But swept across the coupling, every cluster's sign changes, through poles the ground spin places, and the spin rule is right on all sixteen clusters only between 20.1 and 47.0 cm⁻¹.

applied · Magnetism Where the nuisance parameters separate hardly depends on purity. For each of the seven clusters with a ground spin, the two couplings below 150 cm⁻¹ at which the monomer fraction and the temperature-independent term are uncorrelated, against the monomer fraction from half a per cent to sixteen, on logarithmic axes. Every cluster keeps both at every fraction. For ground spins of one and above they move by under three per cent; for the three doublets by about a tenth. A circle marks a pole of the third direction's exponent there and a square a zero.

Purity renames the poles

A susceptibility fit's sign poles were located at one monomer impurity, two per cent, and a real sample's impurity is rarely known. Swept from half a per cent to sixteen, the couplings where the fit's nuisance parameters separate barely move for ground spins of one and above and move a tenth for doublets. But the same separation stops being a pole and becomes a zero of the exponent at a few per cent, and above six to eight per cent every singlet acquires separations of its own.

applied · Magnetism Stars drift above the band and K₂,ₙ graphs fall below it. The upper separation coupling times (S + ½) against ground spin for every cluster computed, with the seven that defined the band shaded between 114.7 and 129.3 cm⁻¹. The stars, from four centres to eight, run 129.3, 114.7, 117.3, 127.2, 140.3 — down and then steadily up. The K₂,ₙ graphs, from K₂,₃ to K₂,₆, run 114.8, 124.0, 100.9, 90.0 — up and then steadily down. K₃,₅ sits at 113.9. Seven clusters of up to six centres happened to lie where the two families cross.

Five more clusters break the band

Seven spin clusters put their upper separation coupling times the ground spin plus a half inside a band from 115 to 129 cm⁻¹, which looked like a law of where a susceptibility fit's nuisance parameters decouple. Five clusters built to test it — two larger stars and three larger bipartite graphs — land inside it once. Stars drift above and bipartite graphs below, and neither the shape of the curve nor the first excitation places the separation instead.

applied · Magnetism The upper separation sits where the first spin-raising multiplet does. Every five-centre doublet, every six-centre triplet and the three stars, by the energy of the lowest multiplet with a larger spin than the ground state, in coupling units, and the coupling at which the fit's two nuisance parameters separate for the second time. On logarithmic axes a constant product is a line of slope minus one; the line drawn puts that multiplet 359 cm⁻¹ above the ground, and the forty-four clusters lie within a factor of one and a half of it.

The excitation that raises the spin

A susceptibility fit's upper separation — the coupling at which its monomer fraction and temperature-independent term decouple for the second time — was placed by a band that broke, and the lead it left was that at fixed ground spin the first excitation ordered it. Every connected cluster of four, five and six spin-½ centres answers that. Among twenty five-centre doublets the first excitation ranks the separations at 0.36 and fewer bonds does better; the first excitation that raises the total spin ranks them at 0.86, and its product with the separation stays within a factor of one and a half where the separation itself spans three and a half. The separation sits where the first higher-spin multiplet is about 355 cm⁻¹ above the ground.

applied · Magnetism The window's top moves every separation with it. The median of the upper separation times the first spin-raising gap, over the forty-one clusters, for windows starting at 20 K and ending between 150 and 500 K. The dashed line is the reading that it sits 1.7 kT above the ground at the window's top; the dotted line is strict proportion through the 300 K value. The medians are 189, 243, 354, 463, 569 cm⁻¹: the earlier prediction of about 240 at 200 K and 475 at 400 K comes out 243 and 463. The rise is slightly slower than proportion — as the top temperature to the power 0.92.

The window sets the constant

A susceptibility fit's upper separation, multiplied by the gap to the first spin-raising multiplet, came to about 355 cm⁻¹ across forty-one clusters, and that was read as a position: the raising multiplet sitting 1.7 kT above the ground at the top of a window ending at 300 K. The reading predicted about 240 at a 200 K top and 475 at 400 K. Re-read under nine windows, the forty-one clusters give 243 and 463, and the raising gap collapses them equally well under every one. But the constant is not quite a multiple of kT at the top — it drifts from 1.81 to 1.64 as the top rises — and the drift is the window's width: windows that are scaled copies of one another give the same constant to a per cent, because the model has no energy scale of its own.

applied · Magnetism How the temperatures are spaced moves the constant more than how wide the window is. The median of the upper separation times the spin-raising gap, in units of kT at the window's top, for windows of span 7.5, 15 and 25 with forty temperatures spaced three ways: evenly in the temperature, in its logarithm and in its inverse. At a span of fifteen they give 1.70, 1.49, 1.36. From span 7.5 to 25 even spacing falls by 9.5 per cent and logarithmic by 14.0: the logarithmic design leans more, not less.

The spacing sets the constant

A susceptibility fit's upper separation, in units of kT at the window's top, was found to depend on the window's span — 1.81 at a span of 7.5, 1.64 at 25 — and every window read had forty temperatures evenly spaced, so a wider window was also a sparser one at its cold end. Holding the span and changing the design separates the two. The number of temperatures barely matters: twenty, forty and eighty converge on one value from above, and a design with the census's own step reproduces the forty-point constant at both ends of the range. How the temperatures are spaced matters a great deal. Spaced evenly in their logarithm, as many measurements are, the constant is twelve per cent lower and leans more, not less; spaced evenly in their inverse, lower again.

applied · Magnetism The chain of four's strongest line is a hole's standing wave read against the singlet's ends. For the half-filled chain of four at infinite repulsion, each removal line's weight is a sum over the four sites of two factors. The first is the hole's standing wave squared on each site: the inner levels at ±0.62 put more of the hole on the end sites, the outer levels at ±1.62 more on the middle ones. The second belongs to the spin chain alone: removing an up electron at an end site leaves the stronger spin doublet with weight (2 + √3)/8 = 0.4665, at a middle site 1/4. Their sums are 0.4067 and 0.3098, which are 1/4 + √3/16 ± √15/80, and the limit of the contrast is their ratio, 1.312498.

The limit was two chains at once

At infinite repulsion every removal line of a half-filled chain is a hole's standing wave read against the spin chain the hole moves through, and every hole level carries exactly half an electron. On the chain of four that makes the contrast's limit (20 + 5√3 + √15)/(20 + 5√3 − √15) = 1.3124978 — not 21/16, which it misses in the sixth figure. On chains of eight and twelve, which no exact spectrum had reached, the limit is 1.0458 and 1.0195, and an energy that vanishes in the limit decides it.

wrong · Photoelectron

Named alongside it

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

Model limitUnderdeterminationMagnetic susceptibilitySpin stateExact diagonalisationConditioningTemperatureConventionLeast-squaresMagnetic momentApproximationDegeneracy

All concepts