Crystallography & Symmetry
patterns derived from their groups
Concept

Accidental overlap — where it appears

Two reflections at the same spacing by arithmetic rather than by symmetry, which no experiment at any resolution separates. It is the reason a powder pattern loses more than the collapse of symmetry-equivalent reflections accounts for.

Named by 2 essays across 2 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 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

Named alongside it

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

IndexingPowder diffractionCentred latticeDecidabilityFigure of meritInterplanar spacingMeasurementMultiplicityRietveld refinementSystematic absenceUnit cell

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