Crystallography & Symmetry
patterns derived from their groups
Concept

Centred lattice — where it appears

A lattice described on a cell with extra points at face or body centres, so that the conventional cell is a whole-number multiple of the primitive one. Doing this to each of the five plane lattices in turn is what shows the list is five rather than ten.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

One lattice, two cells, and the absences the choice creates. The same set of points described on a centred rectangular cell and on its primitive rhombic cell, with what the centred description scatters. Half the reflections vanish, and they vanish because of how the cell was drawn rather than because of anything the crystal does.

Centring, and why cm is not pm

A centred cell has a lattice point in the middle and twice the area it needs, and crystallography prefers it anyway. The preference has a price, and the price is paid in reflections that vanish for reasons that have nothing to do with the crystal.

lattices · Centring Centring the five lattices. Each of the five plane lattices with the midpoint of every cell added, and the type of lattice that results — read off the reduced basis of the new point set rather than looked up. Every centring halves the cell area, so the original lattice is a sublattice of index two in the centred one, and every centred lattice is again one of the five. Two of the five come back as themselves and are therefore no richer for being centred. The rectangular and rhombic lattices exchange, which is what makes them one family under two descriptions. And the hexagonal lattice centred is rectangular — its holohedry falls from 12 to 4, so centring destroys the symmetry it was meant to display.

Centring, counted as a sublattice

Adding the centre of every cell to a lattice produces another lattice, containing the first with index two. Doing it to each of the five in turn shows why the list is five rather than ten, and why only one of the five has a centred description worth keeping.

lattices · Centring 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.

CentringLattice typePrimitive cellSystematic absenceAccidental overlapBasis reductionDecidabilityFigure of meritHolohedryIndexingInterplanar spacingMeasurement

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