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 7 essays across 5 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 All thirteen families of line groups, reached by rolling. Each of the seventeen plane groups, on its generic lattice, rolled into a tube along every lattice vector that gives a repeat, with a mark under each of the thirteen families of line groups some rolling lands in. The band above each column is the family's translational part: a general screw for families 1 and 5, a half-step screw for 4, 8 and 13, pure translations or glide planes for the rest. Every family is reached, six of them by a single plane group, and the two oblique groups reach nothing because an oblique lattice rolled in any direction never repeats.

The centring that turns into a screw

Rolled with no limit on the order, the seventeen plane patterns reach every one of the thirteen families of line groups — six of them through a single pattern each. And the families' translations are decided by one integer: a tube that keeps any reflection holds one or two cells of its sheet in a repeat, never more, and it holds two exactly when the sheet is centred along the rolling direction. The centring vector of the flat lattice is the half-step screw of the tube.

restriction · Finite groups Six matrices and eight: a monoclinic normaliser modulo two. For P2/m and C2/m, every 3 × 3 matrix over the field of two elements that preserves the reductions of the two eigen-sublattices of the class's two-fold or mirror — six on the primitive lattice and eight on the centred one — each drawn as a tile of nine squares filled where the entry is one. The whole infinite normaliser reduces into this set, and a dot under each tile marks that some normaliser element with entries −1, 0 and 1 reduces to it. Every tile is realised, and the same holds in all six monoclinic classes.

The normaliser only acts modulo two

The space groups were counted as orbits of a small, finite set of changes of basis — only those with entries −1, 0 and 1 — out of normalisers that are infinite in the monoclinic classes. That the counts came out right was evidence, not proof. It becomes proof in two steps. A missing change of basis can only leave the count too high, never too low, so the published totals already force every class. And in the monoclinic classes the infinite normaliser acts on the cohomology only through its reduction modulo two, where the small set already reaches everything.

restriction · Finiteness Six groups are made of glides, four of mirrors, eleven of both. For each of the twelve plane groups that contain a reflection, what its mirrors generate on their own, what its glides generate, and what both together generate, counting lattice translations as something the generators must produce. Four groups are generated by their mirrors alone, six by their glides alone — pg, pgg, p4g, p3m1, p31m and p6m — and eleven by the two kinds together; pm is the one they cannot make, because its mirrors are parallel and nothing produces the translation along them. Where a set falls short the cell says how: every coset reached but only part of the lattice, translations in one direction only, or a proper subgroup.

Six groups are made of glides

Four plane groups are generated by their mirrors. Six are generated by their glides, and they are mostly different ones, because a glide done twice is a translation where a mirror done twice is nothing. Closing the plane from located glide lines with their slides reaches six of the seventeen; mirrors and glides together reach eleven, every group with a reflection except pm. The classification does not fall out of the closure, and where it stops says what each kind of reflection can and cannot make.

operations · Composition Every group a symmetric motif can lift a plane group to. The seventeen plane groups arranged by lattice, with an arrow from each group to every group a motif of higher symmetry, placed at some position, turns it into, on the generic lattice of the first. Dashed arrows mark the lifts that also halve the cell, two of which return the same type on the smaller cell. p1 lifts only to p2, the rectangular groups to pmm, pmg and cmm, the square ones to p4g and p4m, and p3 to four different groups. Three groups, outlined, cannot be lifted at all without a special metric: p2, pmm and p6m.

Where a symmetric motif lifts the group

A motif with more symmetry than its site asks for can raise the group of the whole pattern — or not, and which one is decided by the site, not the motif. The criterion is one line: a symmetry of the motif survives into the pattern exactly when it normalises the group. Run over every Wyckoff position of the seventeen, it finds twenty-nine of seventy-two where a motif can lift the group, general positions only in the four groups with a direction to slide along, two sites where the same motif gives different groups depending on how it is turned, and seven lifts between groups that halve the cell.

classification · Accidental symmetry

Named alongside it

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

Glide reflectionPlane groupCentringLattice typeNormaliserPrimitive cellScrew axisSystematic absenceAccidental overlapAccidental symmetryArithmetic classBasis reduction

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