The same site under two names

The normaliser is not a function of the group counts how many coordinate lists describe one arrangement, and finds that the count depends on the cell as well as the group. The points a group treats differently sorts every point of a cell by how much of the group leaves it alone, and stops short of the tables’ letters on the grounds that a letter is an index into a table rather than a property of the group.

Those two computations meet at a question neither of them asks, and it is the question a structure report raises every time it is read. A report does not give an atom’s coordinates alone. It says which Wyckoff position the atom occupies, and a database indexes structures by those positions. If one arrangement has several correct coordinate lists, it may have several correct position assignments as well — and then the position is not something the crystal has, but something one description of it says.

It can, and the positions that can trade places are exactly computable. A motion in the normaliser rewrites a description without changing what is described. It carries the group’s operations onto the group’s operations, so it carries each point’s stabiliser onto another point’s stabiliser, and a whole Wyckoff position onto a whole Wyckoff position with the same site symmetry. Two positions it exchanges are one position written from two origins, or along two sets of axes. The orbits of the normaliser on the positions are called Wyckoff sets, and a set, not a position, is what an arrangement actually determines.

The table has two drops in it and they are different in kind. The first, from seventy-two to fifty-three, is the one every structure report is exposed to whatever its cell: nineteen positions across the seventeen groups are another position seen from a different origin. The second, from fifty-three to fifty-one, happens in exactly two groups and only on a cell more symmetric than those groups require. Most of what follows is about why those two drops happen where they do, and why the second one is not the first one made larger.

A position is its stabiliser, not its stabiliser’s order #

The positions have to be found before anything can be done to them, and the obvious shortcut gets the count wrong.

A point’s site symmetry is the set of operations that leave it where it is. Points whose site symmetries have the same number of operations look alike, and it is tempting to call them one position. In pmm the four points (0, 0), (½, 0), (0, ½) and (½, ½) each sit where two mirrors cross, and each is fixed by four operations. Sorted by that number they are one kind of point. They are four positions, because no operation of pmm carries any of them onto any other: the group’s translations are whole cells, and its mirrors and half-turns hold each of the four exactly where it is.

So a point is keyed by its stabiliser as a set of motions, exactly, and not by how many motions the set contains. Two points are in one position when an operation of the group carries one to the other, or when the identical set of motions fixes both after one of them is shifted by a whole cell — which is how a mirror line that crosses the cell in two separate pieces is recognised as one line. Done that way over a grid of twelfths, the seventeen groups have seventy-two positions, which is the number the International Tables print for them group by group. Sorting by stabiliser order alone gives fewer than the true count on twelve of the seventeen.

pmm has nine positions, then: four crossing points, four positions on mirror lines, and the general position that every point off the mirrors belongs to. Moving the origin by half a cell along either edge changes none of the group’s operations. A mirror through x = 0 and a mirror through x = ½ are both already in the group, and a shift by a half exchanges them, so the operation set comes back identical and the shift is in the normaliser.

The first such shift carries (0, 0) to (½, 0); the second carries it to (0, ½); together they reach (½, ½). The four crossing points are one set. The same two shifts carry the horizontal mirror at height 0 onto the one at height ½, and the vertical mirror at 0 onto the one at ½. What no shift can do is turn a horizontal line into a vertical one. That gives four sets — the crossings, the horizontal mirrors, the vertical mirrors, and everything else — out of nine positions.

That count has a concrete reading. A pmm structure with one kind of atom on the crossing points can be reported with the atom at (0, 0) or with it at (½, ½), and both reports are correct. So can a structure with its atoms on a horizontal mirror, reported at height 0 or at height ½. A comparison that matches structures by the position a report names will call each of those pairs two structures, and it will be wrong in exactly the way a comparison of raw coordinates is wrong: it is the same mistake, made one level up.

What an equivalent origin does to the positions #

p4m shows the mechanism with a single move. Its normaliser on the square cell holds one shift that is not already a translation of the group: moving the origin to the centre of the cell, by (½, ½). The four-fold centres at the corners and the one at the centre are both centres of the group, the mirrors through each are both mirrors of the group, and so conjugating by the shift returns every operation unchanged. It is a change of description and nothing else.

What the shift does not move is as instructive as what it does. The two-fold points at the edge midpoints, (½, 0) and (0, ½), are carried onto each other — but those two points are already one position, because the group’s own quarter-turn exchanges them, so the shift maps that position onto itself. The diagonal mirrors are carried onto themselves as a whole. Of the seven positions, four trade places in two pairs and three stay put, and seven positions become five sets.

The rule every motion here has to obey is that it carries each position onto exactly one position with the same site symmetry, and it is checked for every motion of every normaliser used rather than assumed. It has to hold, for a reason worth stating once: if a motion carries the group onto itself, it carries the stabiliser of a point onto the stabiliser of that point’s image, and a conjugate of a four-fold axis with four mirrors is again a four-fold axis with four mirrors. Conjugation is the test of sameness inside a group, and this is the same test applied to the stabilisers. A shift that breaks the rule is not in the normaliser. Offered to pmm, a shift by a quarter of a cell carries the crossing points onto bare mirror lines, and it is refused.

A normaliser can be larger than its group and fuse nothing at all. p4g, p31m and p6 each have two descriptions of every arrangement and exactly as many sets as positions. In each of them every position is the only one with its site symmetry — p6 has one six-fold position, one three-fold, one two-fold and the general one — and a normaliser can only exchange positions with the same site symmetry, so the second description moves each position onto itself. A table of indices would show those three groups beside p4m as groups with two descriptions; a table of sets separates them.

The cell decides which positions fuse #

In pmm on a rectangular cell the horizontal and vertical mirror positions stay apart, and the reason given above — no shift turns a horizontal line into a vertical one — is incomplete. A quarter-turn would do it, and so would a mirror along the diagonal. Neither is a symmetry of a rectangle, so neither is available. A pmm pattern drawn on a square cell has eight descriptions rather than four, because the square’s holohedry supplies exactly those motions and half of the new ones normalise the group.

So the number of genuinely different places a single kind of atom can be put in a pmm crystal depends on a measurement. On a rectangular cell, a structure with atoms on the horizontal mirrors and a structure with atoms on the vertical mirrors are different structures: in one the atoms repeat along the short edge, in the other along the long edge, and the distances between them differ. On a square cell they are the same structure turned through a right angle, and the two reports differ only by a relabelling of the axes — the freedom one group, three symbols is about.

cmm is the other group this happens to, and the geometry is different enough to be worth seeing.

The two cases are one case. In both, the group has two positions with the same site symmetry lying along two different directions; the general cell makes those directions inequivalent by giving them different lengths; and the special cell takes the difference away. That is also the whole list. Of the seventeen groups only pmm and cmm have a more symmetric cell on which any positions fuse, and each has exactly one.

A cell that has been reduced by its metric alone flags both cases before any positions are compared, because a reduction that returns a square type for a group that asked only for a rectangle or a rhombus is saying precisely that the extra motions exist. The procedure that gets the descriptions right gets the sets right as well, with no step added.

Two answers to one change of cell #

The specialised-metric count found four groups whose descriptions a special cell raises — p2 by up to six, and pmm, pgg and cmm by two. It would be natural to expect the positions to follow the count, so that a group gaining descriptions also gains fusions. They do not, and sorting every group on every cell it may sit on shows the two answers to be independent.

p2 on a hexagonal cell is the sharpest case: six times as many descriptions, and not one more position fused. p2 has five positions — its four half-turn points and the general position — and the shifts by half a cell already carry the four points onto one another on the most general cell there is. Every one of the twenty descriptions a hexagonal cell adds is a rotation or reflection of the axes, and by the time they arrive the half-turn points are one set. A larger normaliser can only act on positions, and there is nothing left for it to act on. pgg on a square cell is the same story at a factor of two: its two positions of half-turn points were already one set.

The empty cell is a theorem rather than an accident of the seventeen. The normaliser on a special cell contains the normaliser on the general one, because the special lattice’s symmetries include the general lattice’s. If the description count does not rise, the two normalisers have the same number of motions and one contains the other, so they are the same motions — and the same motions exchange the same positions. A cell can raise the descriptions without fusing any positions, and cannot fuse positions without raising the descriptions.

So the two numbers answer different questions. The count of descriptions is what a comparison of coordinate lists has to divide out, and the sets are what a comparison of position assignments has to divide out. Knowing that one of them changed says nothing definite about the other, and a procedure that updates one when the cell turns out to be special has to update the other separately.

The affine answer, and why it is not the answer for one crystal #

The International Tables group positions into Wyckoff sets with a larger normaliser still: the affine normaliser, every change of basis that carries the group onto itself whether or not it preserves lengths. For pmm that includes exchanging the two axes of a rectangular cell. That is not an isometry — it turns the long edge into the short one — but it carries the group’s operations onto themselves, which is all the affine definition asks. The affine sets of pmm fuse its horizontal and vertical mirror positions whatever the cell.

Searched over integer matrices with entries up to two and translations on twelfths, the affine sets come to fifty-one over the seventeen groups. For every group they are not merely as many as the sets on its most symmetric cell but the same partition of the same positions. Every identification the affine normaliser makes between two plane-group positions is made by an isometry of some cell that group can sit on. That is a fact about the plane, checked group by group rather than argued, and not one to carry into three dimensions by analogy.

The two answers are right for different jobs. The affine set is the right unit for classifying kinds of arrangement, because a kind of arrangement should not depend on the lengths of one crystal’s edges; the tables’ classification of point configurations is built on it. The Euclidean set on the cell as measured is the right unit for deciding whether two reports describe one crystal, because a pmm crystal with atoms on its horizontal mirrors and one with atoms on its vertical mirrors are, on a rectangular cell, two crystals with different interatomic distances. Using the affine set to compare them would identify structures that are genuinely different — a false positive, which is the worse of the two errors a comparison can make.

The case between the two is a familiar one. A pmm cell with one edge of 5.001 and the other of 4.999 is not square, and whether to treat its two mirror positions as one is exactly the tolerance decision near-symmetry and the tolerance declines to make. What the computation does settle is where the question can arise at all: in two groups, on one specialisation each, and nowhere else in the plane.

What a position in a report is evidence of #

Three things follow for anyone reading positions out of a structure report rather than coordinates.

A position name is a coordinate, not an invariant. It depends on the origin and axes chosen, and it can change under a change of description that changes nothing else. Within one set a position carries no information about the crystal; the set it belongs to does. The site symmetry does not change within a set, which is why what a molecule gives up can treat a site’s symmetry as a fact about the crystal, and why the modes a site can carry come out the same for every position of one set: the stabilisers are conjugate, and conjugate stabilisers decompose displacements identically.

Positions split and fuse in opposite directions, and the two must not be confused. What a position becomes on the way down follows one position of a group as it breaks into several positions of a subgroup when symmetry is lost. Sets run the other way: several positions of one group that were never different. A comparison of a structure above and below a transition that reads positions off two independent reports can see an atom move from one position to another when the two reports merely chose different origins. If the two positions lie in one set, nothing moved.

And the check is cheap. Given the group and the measured cell, the sets are found by applying each normalising motion to one point of each position, once. The motions are the same ones the same pattern, described twice already needs for coordinates, so a procedure that divides out equivalent descriptions has everything required to divide out equivalent positions, and no excuse not to.

How the positions were found, and what could have refused them #

For each of the seventeen plane groups, on a grid of twelfths in exact rational arithmetic, the stabiliser of every point was taken as a set of motions; the positions were the classes of points under the group and under identity of stabilisers after a whole-cell shift; and each position’s site symmetry, multiplicity and locus were read from its stabiliser. Then the Euclidean normaliser on every lattice the group may sit on, from the same conjugation search that counts descriptions, and the affine normaliser over integer matrices with small entries, were each applied to the positions and their orbits counted.

The positions have to agree with an independent list. The seventy-two found from stabilisers are compared group by group with the counts the tables print, and all seventeen agree. Sorting points by stabiliser order alone must also give fewer somewhere, or the work of comparing whole stabilisers would be decorating an answer the orders already gave; it gives fewer on twelve.

The grid must not decide the answer. Halving the spacing to twenty-fourths changes no count of positions and no count of sets on any group. The normalisers of p1, pm, pg and cm have free directions, so the search returns grid points for them rather than an index, but their sets are safe from that: sliding the origin along a mirror carries each mirror position onto itself, and the continuous part of those normalisers fuses nothing the discrete part had not.

The fusions have to grow with the normaliser. Each larger normaliser must fuse at least as much as a smaller one — positions, then sets on the group’s own cell, then on its best cell, then affine — and the totals run seventy-two, fifty-three, fifty-one, fifty-one. The affine search must not depend on its own bound either: matrix entries up to one and entries up to two give identical partitions for all seventeen groups.

And two motions must be turned away. A shift of pmm by a quarter cell is not in its normaliser, and it must not be accepted as a relabelling of positions; it is refused because it carries a crossing point onto a bare mirror line, which no conjugation can do. A quarter-turn is not an affine symmetry of pmg, whose two axes carry different kinds of operation — mirrors along one, glides along the other — and the affine search must not find one among the sixteen motions it does find.

Where the question goes next #

One dimension up is where the plane’s tidy answer is least safe. In space the affine normaliser of a monoclinic or triclinic group contains changes of basis that mix the cell’s axes, and it is not obvious that some metric realises each of its identifications as an isometry, the way some cell always does in the plane. Whether the Euclidean sets on the best cell reach the affine sets for every space-group type is a question the seventeen plane groups answer only for themselves.

Sideways, to two origins for one group, where the same arithmetic separates two things the tables treat very differently. A shift in the normaliser changes nothing, and a Wyckoff set is its positional form. A shift between two published origin choices changes every coordinate, and it has a positional form too: one position name standing for different points under the two choices. The first is bookkeeping, and the second is the one that does damage.