Molecular Geometry & Bonding
what the shape decides
Concept

Frustration — where it appears

A coupling pattern in which no arrangement satisfies every bond at once, as in an odd ring of antiferromagnetically coupled spins. It leaves a ground state that is degenerate or nearly so, and it is a property of the graph of couplings rather than of any one bond.

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

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

Named alongside it

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

DegeneracyExact diagonalisationExchange couplingMagnetic susceptibilityModel limitUnderdeterminationApproximationConventionConvergenceGraphLeast-squaresReference state

All concepts