Crystallography & Symmetry
patterns derived from their groups
Concept

Hermann–Mauguin notation — where it appears

The crystallographic notation that names a group by reporting what lies along each of three families of directions, in a fixed order. It looks like a licence plate and is in fact a set of instructions, readable once the rules are in hand.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

The wallpaper group p4m. A pattern with the symmetry of p4m, generated by applying the group's 8 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.

Reading Hermann–Mauguin

p4g looks like a licence plate and is in fact a set of instructions. Half an hour with the rules turns the seventeen from a list to be memorised into a notation that can be read.

classification · Seventeen Reading 4/mmm off its own directions. Each position of 4/m2/m2/m reports one symmetry direction of the tetragonal system: the highest-order axis lying along it, and whether a mirror is perpendicular to it. Nothing is looked up — every row is computed from the group's own matrices.

Reading a class off its own axes

A Hermann–Mauguin symbol is not a name that was assigned. It is a report on three directions, read in order, and the whole of it can be derived from the group's matrices — with one genuine convention and one exception, and the exception is orthorhombic.

point-groups · Crystal classes P2₁/c under every cell choice. One group, described in each of the 6 bases that keep its cell the shape its system requires, with the symbol derived from the operations each time. The distinct symbols are P2₁/c, P2₁/a, P2₁/n — 3 names for one group. Nothing about the crystal has changed: the basis changes all have determinant ±1, so the lattice is untouched, and the census of rotations, screws, mirrors and glides is identical in every row, since conjugation cannot turn one kind into another. What changes is which lattice vector a glide's translation is half of, and the glide letter names exactly that.

One group, three symbols

P2₁/c, P2₁/a and P2₁/n are the same space group written on three choices of axes, and the literature contains all three as though they were different. Deriving a symbol from a group's own operations shows why — and found an entry on this site that had been carried under another setting's name for two phases.

space-groups · Settings Six classifications, and which are enumerated here. The families of symmetry groups by how many directions they repeat in and how many they live in. The thirty-two crystal classes, the seven friezes and the seventeen plane groups are each built from their own operations and counted. So are both subperiodic families: the seventy-five rod groups class by class, and the eighty layer groups as seventeen one-sided, seventeen mirrored and the forty-six two-colour classes of the plane group underneath. The two hundred and thirty space groups remain a number from the literature, marked as such wherever it appears: reaching it needs the translation extensions of seventy-three arithmetic classes in full.

A layer is not a wallpaper

A sheet repeats in two directions and lives in three, and its symmetry group is not one of the seventeen. There are eighty of them, the difference between one and another is a single sign per operation, and the arithmetic that supplies those signs is the arithmetic of a two-coloured pattern.

classification · Subperiodic The thirty-two, in both notations. Each class with the symbol crystallography uses and the symbol spectroscopy uses, both derived from the class's own matrices. The Hermann–Mauguin symbol is a report on three families of directions, read in an order the lattice system fixes. The Schoenflies symbol is a report on a construction: a principal axis of order n, whether there are n twofold axes across it, and which mirrors were added. Neither can be computed from the other without going back to the group, which is why the two lists are set beside each other rather than one derived from the other.

One class, two names

Hermann–Mauguin names directions and Schoenflies names a construction, and the two are derived here from the same integer matrices by computations that share no step. Neither can be obtained from the other without going back to the group — which is why a molecule has one kind of symbol and a crystal has both.

point-groups · Crystal classes Aem2: one plane, two glides. The plane of Aem2 that carries two operations, drawn edge-on with each slide beside it. The two differ by the A-centring translation, which lies inside this plane — that is the whole condition for a plane to carry two, and it is why no primitive group has one. Both slides are axial, b and c, so neither letter has a claim on the symbol, and before 1992 the Tables simply chose. The letter e is the choice being refused.

The plane that carries two glides

A plane can hold two glide operations at once, with slides that have no claim on each other. Every symbol printed before 1992 chose one of them, so the name recorded a convention rather than a group — and the International Tables invented a letter to stop it.

space-groups · Screws and glides Pnma has 6 names. The group Pnma with its three axes relabelled in each of the six possible ways. Every row is the same group, and each row is checked as it is drawn: the change of basis has determinant one, conjugating back gives the original operations exactly, and the census of screws, glides, mirrors and rotations is unchanged — and 6 different symbols come out. Each symbol is derived from the conjugated operations, not looked up, by the same routine that has to reproduce the symbol every group was entered under.

Six ways to name one group

Pnma is also Pmnb, Pbnm, Pcmn, Pmcn and Pnam. Nothing about the crystal changes between those six; what changes is which axis was called a. In an orthorhombic group the axes are inequivalent and unlabelled, and naming them is a choice made six ways.

space-groups · Settings Two dimensions to be ambiguous in, and one. Why a plane can carry two operations and a line cannot, side by side. Two reflections sharing a plane differ by a translation lying in that plane, and their slides are vectors in the plane — a two-dimensional space, in which a centring vector need not be a multiple of the slide. So the two slides can be genuinely different glides, b against c, and in 1992 the International Tables invented the letter e for the case where neither has a claim. Two rotations sharing an axis differ by a translation along that axis, because anything across it would move the line; their intrinsic parts are vectors along the line, a one-dimensional space in which every lattice vector is a whole multiple of the shortest. So the two differ by a whole number of repeats and are the same screw. The plane has one dimension of freedom left over and the line has none.

A line carries one screw

A plane can hold two glide operations at once, and in 1992 the International Tables invented a letter for the case where neither has a claim. The same question put to an axis has the opposite answer: across 3,388 axes, not one line carries two — and the reason is that a slide has two dimensions to be ambiguous in and an intrinsic translation has one.

space-groups · Screws and glides How many reflection operations a plane of each lattice can carry. Every mirror of each of the fourteen Bravais lattices' holohedries, sorted by how many classes of lattice vector lie in it modulo the conventional cell's own translations — which is how many reflection operations any plane of that orientation can carry in any space group on that lattice. Primitive lattices carry one on every plane. Centred ones carry two on the planes their centring vectors lie in: one plane of the C-centred orthorhombic lattice, the diagonal planes of the body-centred tetragonal and cubic ones, and every mirror plane of the face-centred lattices. The rhombohedral lattice, on the hexagonal axes the Tables use, carries three on each of its three mirror planes. None carries four.

Three on a plane, and never four

A reflection plane in a space group carries one operation for each class of lattice vector lying in it, and a face-centred lattice has planes holding two independent centring vectors — enough for four. None of those planes is ever a mirror. The fourteen lattices decide the question without the two hundred and thirty: one, two or three operations on a plane, and never four — with the three on every mirror plane of the rhombohedral lattice, where the Tables' own hexagonal axes put them and the letters have no name for them.

space-groups · Screws and glides

Named alongside it

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

CentringGlide planeSettingInternational tablesNotationSpace group symbolChange of basisCrystal classHolohedrySymmetry elementAxial classAxis permutation

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