Crystallography & Symmetry
patterns derived from their groups
Concept

Powder diffraction — where it appears

Diffraction from many small crystals in every orientation, which collapses each set of equivalent reflections into one line. Reflections that had their own places arrive together, and some of the coincidences are exact and have nothing to do with symmetry.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

168 reflections, 26 rings, one axis. Every reflection of a square lattice out to indices of 6, drawn as a point of the reciprocal lattice with the rings of equal length it lies on. A single crystal gives each point its own place on a detector; a powder is every orientation at once, so what is measured is the ring rather than the point, and the 168 reflections arrive as 26 lines on the single axis at the right, with height the number sharing each. The rings are found by sorting the lengths and grouping equal ones, and every point is checked to sit on the ring it was given. 1 of the rings are marked: they hold reflections that no operation of the lattice's point group relates, so the collapse has added together two genuinely independent measurements and no experiment can take them apart again.

What a powder pattern loses

Grind a crystal up and every orientation is present at once, so a two-dimensional pattern of spots collapses onto a single axis. Reflections that had their own places arrive together, and some of the coincidences are exact and have nothing to do with symmetry.

diffraction · Diffraction The shells of the hexagonal lattice. Every point of the hexagonal lattice within a squared distance of 24, with a circle drawn at each length that occurs. The form is x² + xy + y², and the number of points on each circle is a coefficient of the lattice's theta series: 6 at 1, 0 at 2, 6 at 3, 6 at 4, 0 at 5, 0 at 6, 12 at 7, 0 at 8. The gaps matter as much as the counts — a circle with no points on it is a length the lattice does not have, and which lengths those are is a question in number theory rather than in geometry.

How many vectors of each length

Counting the lattice points at each distance from the origin turns out to be a question about divisors, and the answer explains something a crystallographer meets every day: why a cubic powder pattern has no line at seven.

lattices · Lengths 7 cells explain the lines; one of them is right. A line list from a face-centred cubic cell of 5.64 Å, with a realistic error added, handed to a sweep over every cubic cell between 2 and 12 Å in all three centrings. 7 distinct cells explain every line within the tolerance, and each is a genuine solution rather than a numerical accident. The true cell comes top by de Wolff's figure of merit — the last Q over twice the mean discrepancy times the number of lines the candidate says should have been visible — which punishes a candidate for predicting lines nobody saw. That is the whole of what makes indexing decidable in practice: not the arithmetic, which has many answers, but a criterion for preferring one.

Indexing a powder pattern

A powder pattern is a list of numbers and a cell is six. Getting the second from the first is the first step of every powder study and the one that fails — because the arithmetic has many answers, and choosing between them is a ranking rather than a deduction.

applied · Indexing The figure of merit a supercell always beats. One line list, indexed on the true cell and on five multiples of it. The mean discrepancy is not merely similar down the column, it is identical to every digit: the supercell's grid of allowed Q values contains the true cell's grid exactly, so each line lands on precisely the same place and misses by precisely the same amount. Any figure of merit built on agreement alone therefore returns one number for the whole family, and cannot prefer the true cell. What falls is the last column, and the only thing in it that the fit does not already contain is the count of lines the cell says should have been seen.

The figure of merit a supercell always beats

Indexing a powder pattern returns a ranked list rather than an answer, and the ranking needs a number. The obvious number — how well the cell accounts for the lines — is exactly the number a supercell cannot lose on, because the supercell's grid contains the true cell's grid and the discrepancies are identical to every digit. What has to be paid for is the lines nobody saw.

applied · Indexing Seven cubic antiferromagnets, and what a powder of each can say. Seven collinear antiferromagnets on the three cubic lattices, each with the size of its propagation vector's orbit, the number of the forty-eight cubic operations that keep the vector, the kind of every magnetic ring out to a fixed radius, summed from the actual magnetic reflections of one domain, and what a powder pattern can therefore read about the moments' direction. The two whose halving the cubic group keeps are blind in every ring. An orbit of three leaves every ring uniaxial about a cube axis, an orbit of four about a body diagonal, and the body-centred orbit of six leaves rings that read the direction. The cubic chemistry is the same in every row; the propagation vector decides.

Where a powder goes blind

A neutron sees only the part of a magnetic moment across its path, so each reflection knows which way the moments point. A powder adds up every reflection of a given length, and for a ring with cubic symmetry that sum comes out the same whichever way they point. But the ring's symmetry is not the chemistry's: it is the symmetry the propagation vector keeps, and most cubic antiferromagnets have lost enough of it to be read.

point-groups · Magnetic

Named alongside it

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

IndexingAccidental overlapFigure of meritMultiplicitySystematic absenceCentred latticeDecidabilityDivisor sumDomain stateGaussian integerHolohedryInterplanar spacing

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