Crystallography & Symmetry
patterns derived from their groups
Series

Fundamental domain — the series

11 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A fundamental domain for p4m. One representative from every orbit of p4m, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.

    The fundamental domain

    The smallest piece of a pattern from which the group rebuilds the rest. Drawing one is easy and drawing one correctly is not, because a region with a gap or an overlap looks exactly like a region without.

    part 1 · operations
  2. Special positions in p4m. Every point of a 12×12 grid inside the cell of p4m, drawn at a size set by how many operations fix it. 80 of the 144 are general — nothing but the identity leaves them alone, so their orbit is the full 8 points. The other 64 are special, and fall into 3 kinds: 60 points fixed by 2 operations, with orbits of 4; 2 points fixed by 4 operations, with orbits of 2; 2 points fixed by 8 operations, with orbits of 1.

    The points a group treats differently

    Almost every point of a cell has an orbit as long as the group. The exceptions are the points some operation leaves alone, and they are where atoms sit, where a structure's formula comes from, and where a careless motif destroys the group it was meant to illustrate.

    part 2 · operations
  3. p4 inside p4m, by area. A fundamental domain for p4m beside one for p4, drawn by the same construction on the same grid. p4 sits inside p4m with index 2: it has 8 ÷ 4 = 2 times as few operations to rebuild the pattern with, so it needs 2 times as much of the cell to rebuild it from — measured here at 1.51 times as many samples. The two domains are one region and 2 copies of it, and the ratio between them is the index rather than a coincidence of shape.

    Domains of a subgroup

    A group with half the operations needs twice as much of the cell to rebuild the pattern from. That single sentence is the index arithmetic of the whole classification, and it turns the containments among the seventeen into a statement about area.

    part 3 · operations
  4. The general positions of P2₁/c. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. 12 general positions, the orbit of a three-point asymmetric motif, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

    One part in however many, and why it is never quite that

    A crystal's contents are the asymmetric unit repeated by the group. The unit's volume is the cell's divided by the order of the group — except that it is always a little more, and the excess is exactly the special positions counted whole.

    part 4 · operations
  5. p4: a domain of 38 cells with 7 walls. The fundamental domain of p4 on a grid of 12ths, with the walls it shares with its neighbouring copies marked. Each wall names the element that carries this copy onto the copy across it, and there are 7 distinct such elements. Those elements generate the whole group — checked by closing them up and requiring every coset and the whole translation lattice to be reached, not assumed — which is Poincaré's theorem, and it means the generators of a wallpaper group can be read off a picture. The domain is pixelated rather than a polygon, so the wall count is a property of this domain and not of the group.

    Every wall names a generator

    The copies of a fundamental domain tile the plane and stand in one-to-one correspondence with the elements of the group. So the elements that carry the home copy across a wall generate everything — and the generators of a wallpaper group can be read off a picture rather than looked up.

    part 5 · operations
  6. p4m: 10 of 36 wavevectors have to be visited. The Brillouin zone of the square lattice with a grid of 36 wavevectors on it, of which 10 are drawn solid: one per star, which is everything a calculation over a p4m-symmetric operator has to visit. The share is 27.8 per cent against the 12.5 per cent that the order of the point group would give, and it is larger for a reason worth naming — the wavevectors on the boundary of the wedge have short stars, so they are over-counted by any argument that only divides by the group order. The identity that is checked is that the star sizes add to the whole grid.

    The domain in reciprocal space

    A fundamental domain is the piece of a pattern the group repeats, and several have been drawn in the plane. The same idea in reciprocal space is what makes a calculation over a crystal affordable — and its share of the zone is larger than one part in the group's order, for a reason worth measuring.

    part 6 · operations
  7. p4m: which distortions the atoms of each position can make. Every Wyckoff position of p4m, with the number of independent displacement patterns of each symmetry its atoms supply. A zero is a distortion those atoms cannot make however their amplitudes are chosen — an atom pinned at a rotation centre cannot move in a way that keeps less symmetry than the centre has. The row sums, weighted by the dimensions along the top, come to twice the number of atoms in the cell, which is the check that nothing has been lost. The last column is the number of free coordinates the position has, and it equals the multiplicity of the identity representation in that row: a displacement that keeps every symmetry is exactly a move of the position within its own Wyckoff set.

    Which modes a site can carry

    An atom on a mirror cannot move in a way that breaks the mirror while its images move with it: the displacements of a Wyckoff orbit carry a representation, and some of its pieces have multiplicity zero. The count of those pieces is a character, and the one that breaks nothing is the position's own freedom.

    part 7 · operations
  8. One orbit of p4m, two of p4. The general position of p4m — 8 points in a cell, all equivalent under that group — with each point coloured by which orbit of p4 it belongs to. Losing half the operations does not move a single point; it changes which of them are related, and the one orbit becomes 2. An atom sitting on this position in the parent becomes 2 crystallographically distinct atoms in the child, which may then be different elements, or move independently, or order.

    What a position becomes on the way down

    Cool a crystal through a transition and it loses operations. Nothing moves — and one crystallographic site becomes two, which is how an ordering transition finds somewhere to put a second kind of atom.

    part 8 · operations
  9. The domain is a polygon, and its edges are elements. The Dirichlet domain of a point whose stabiliser is trivial: the set of points at least as close to it as to any other point of its orbit. It is a convex polygon, it is a fundamental domain, and each of its edges lies on the bisector of the base point and one image of it — so each edge already carries the element that produced it, with no search. Edges are drawn by kind: paired with another edge, fixed pointwise by a reflection, or folded in half by a half turn.

    The relations a polygon dictates

    Poincaré's theorem has two halves. The walls of a fundamental domain name the generators, which is the half this collection already computes; walking round its corners names the relations, which needs a domain with corners rather than a domain made of pixels. Building the Dirichlet polygon exactly gives a presentation of each of the seventeen — and coset enumeration says every one of them is right.

    part 9 · operations
  10. Which Laue groups have a sampling grid that touches no special point. Every Laue group on every Bravais lattice it occurs on, twenty-four pairs, with the smallest grid of wavevectors that keeps the whole group and has no point fixed by any operation but the identity. Such a grid exists for 1̅, 2/m, mmm and 4/m, at exactly as many points as the group has operations. It does not exist, as far as the search reaches, for 4/mmm or for any group with a three-fold axis: 3̅, 3̅m, 6/m, 6/mmm, m3̅ and m3̅m.

    A three-fold axis puts a grid point on itself

    A grid of wavevectors offset from the zone centre can keep a crystal's whole symmetry and touch no mirror, no axis and no point where time reversal folds the zone onto itself, and then its share of the zone is exactly one over the order of the group, with no boundary excess at all. Four Laue groups have such a grid, as small as the group itself. Every group with a three-fold axis has none, for a reason that fits in a line: the rotoinversion fixes one point of the plane across its axis, and above that point the three-fold rotation fixes the grid.

    part 10 · operations
  11. Every square-symmetric grid of wavevectors has a point on a mirror. One cell of the square reciprocal lattice, centred on the zone centre, with the mirror lines of 4mm through it and three grids the four-fold rotation keeps with an offset off the axis. The quarter turn allows that offset at one place only, the second four-fold centre of the grid's own lattice, ringed. On an aligned grid of odd side the rotation fixes a point; on an aligned grid of even side the offset lies on a diagonal mirror; on a grid turned by 45 degrees it lies on an axial mirror. No grid escapes all three.

    The offset a quarter turn allows lies on a mirror

    A census of sampling grids found free grids for the tetragonal group 4/m and none for 4/mmm, to a hundred and twenty-eight points, with no reason. The reason is two lines long. A quarter turn lets a symmetric grid be offset to exactly one place off its axis, the second four-fold centre of the grid's own lattice, and a square lattice that a mirror keeps can only be aligned with its mirrors or turned by half a right angle. Either way that centre lies on a mirror line.

    part 11 · operations

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