What a position becomes on the way down

The points a group treats differently sorts a cell by how much of the group leaves each point alone: almost every point has an orbit as long as the group, and the exceptions — the points some operation fixes — are where atoms sit and where a careless motif destroys the group it was meant to illustrate.

The descent of symmetry is about a crystal losing operations, and how many domains a transition makes counts what the loss produces in the crystal as a whole. Neither asks what happens to the positions, and the answer is the thing a structural crystallographer actually needs at a phase transition.

Nothing moves, and everything changes #

The picture above is the whole idea and it is worth stating in words, because the arithmetic afterwards is only bookkeeping.

A Wyckoff position is not a place. It is a class of places — an orbit, the set of points a group carries onto one another — and to say that two atoms occupy the same position is to say that the symmetry requires them to be identical: same element, same environment, same displacement parameters, no independent existence at all.

Take half the operations away and the requirement weakens. The points are where they were; what has gone is the guarantee that some of them are copies of the others. One orbit of eight becomes two orbits of four, and the crystal now has two independent sites where it had one.

That is what an ordering transition needs. A parent structure with one site cannot hold two kinds of atom in an ordered arrangement, because the symmetry insists they are the same. Lose enough symmetry that the site splits, and there are two sites to put two elements on — which is exactly the arrangement an ordered superlattice has, and the reason its extra reflections exist.

The rule that never varies, and the one that does #

Running the computation over every subgroup relation among the seventeen gives two statements of very different character.

The general position always splits into exactly the index. A point with no symmetry has an orbit as long as the group, so a group of half the size gives orbits of half the length and there must be two of them. That is orbit–stabiliser and nothing else, it holds in every one of the forty-five relations computed, and it needs no case analysis.

What happens to the special positions is not predictable from the index. A position whose site symmetry survives the descent whole splits into the full index; a position where the descent removes an operation from the site itself keeps its multiplicity, does not split, and becomes less special instead. Which of the two happens depends on which operations were removed and where the site sits, and there is no shortcut: it has to be computed.

The two things a descent can do to a position #

It is worth naming the two outcomes, because a structure report distinguishes them and the words are easy to run together.

Splitting. The orbit breaks into shorter orbits. Multiplicity falls, the number of independent sites rises, and the atoms that were required to be identical are now free to differ. This is what makes ordering, and the sites it produces are the ones a refinement will treat separately.

Loss of site symmetry. The orbit stays the same length and the stabiliser shrinks. No new site appears, but the atom acquires freedom it did not have: a point on a mirror line can only move along the line, and if the mirror goes it can move anywhere. This is what makes a displacive transition, where nothing orders and everything shifts a little.

The two are exactly the two ways a transition is classified in practice, and they fall out of the same computation. In p4m to p4 the general position splits and every special position merely loses symmetry, so that descent is a splitting one at the general position and a displacive one everywhere else — which is a statement about what an atom is allowed to do rather than about what it does.

What the computation is #

The arithmetic is exact and short, and worth setting out because it makes the claims checkable.

Every point of a grid of twelfths is a pair of rationals; every operation is an integer matrix with a rational translation; applying one to the other is exact, so an orbit is a set rather than a cluster of nearby points. The grid is closed under every plane group’s operations, which is not automatic — a grid offset by half a step is closed under the square lattice’s operations and not under a three-fold rotation, and that is the trap the fundamental-domain machinery hit on this site two phases in.

Three things are then asserted rather than assumed.

Orbit–stabiliser, at every orbit: its length times the order of its stabiliser is the order of the group. That is the theorem the whole subject rests on and it is cheap to check everywhere rather than once.

Conservation: the pieces of a split orbit have lengths summing to the parent’s. A scheme in which the pieces do not add up is a scheme with a point in two orbits at once.

And containment: the child’s operations must actually be a subset of the parent’s. That check earns its place because the arithmetic works perfectly well without it — hand the machinery two groups where neither contains the other and it produces a table that looks entirely reasonable and means nothing. It is refused instead.

Where this is used #

Three uses, and they are the reason splitting schemes are tabulated in the International Tables rather than left as an exercise.

Ordering, as above. A transition that puts two elements onto what was one site needs the site to split, and the scheme says whether the intended subgroup can do it. A proposed low-symmetry structure whose ordering requires a split the scheme does not provide is a proposal that can be rejected on symmetry alone.

Counting parameters. The number of free coordinates in a structure is a sum over its occupied positions of the freedom each one has, and both change on descent: more sites, and more freedom at each. That count is what decides whether a refinement is overdetermined, and it is the same arithmetic the unknowns against the observations does at the level of the whole structure.

Reading a spectroscopy. A technique that sees the local environment of an atom — a nuclear resonance, a Mössbauer spectrum — counts sites directly. One line above a transition and two below it is a splitting, seen without any diffraction at all, and the scheme says which subgroups are consistent with the number of lines observed.

A worked descent, position by position #

The p4m to p4 relation is the smallest one with both outcomes in it, and going through it is quicker than any general statement.

The four-fold points. The corner of the cell and its centre are fixed by eight operations in p4m — the four rotations and the four mirrors — so each is a one-point orbit. In p4 only the four rotations survive, so the stabiliser drops from eight to four; the orbit is still one point, because a point that was alone is still alone. No split, and the site symmetry has halved.

The two-fold points. The edge midpoints form an orbit of two, fixed by four operations. Losing the mirrors takes the stabiliser to two and leaves the orbit at two. Again no split.

The mirror-line points. A general point on a mirror line has an orbit of four and a stabiliser of two — itself and the mirror. In p4 the mirror is gone, the stabiliser is one, and the orbit is still four. That point has stopped being special altogether: it is now an ordinary point of p4 that happens to lie where a mirror used to be, and nothing prevents it from moving off.

The general point. Eight images in p4m, stabiliser one. In p4 the stabiliser is still one and the orbit is four, so the eight points must fall into two orbits — and they do, alternating around the cell in the figure at the top of this page.

So one descent, four positions, and three different things happening: a stabiliser halved with no split, a speciality lost entirely, and a genuine split. A rule that predicted any one of them from the index alone would be wrong about the other two.

The same arithmetic in the other direction #

A splitting scheme can be read upwards as well, and the upward reading is what a crystallographer does when a structure looks more symmetric than it was solved in.

Two independent sites in a low-symmetry structure that turn out to have the same environment, the same occupancy and coordinates related by an operation the group does not have are two orbits that want to be one. Merging them is asserting a symmetry the refinement did not impose, and the test is exactly the splitting scheme run backwards: is there a supergroup in which these two orbits are one orbit, and are the coordinates consistent with it to within their uncertainties?

That is the standard cure for a structure refined in too low a symmetry — a mistake with a characteristic signature, since a structure with a symmetry it has not been given refines to a set of correlated parameters and unreasonable displacement ellipsoids. The same pattern, described twice is about a milder version of the same confusion, where one structure has many descriptions rather than one description having too little symmetry.

The upward reading is harder than the downward one for a reason worth naming: a group has finitely many subgroups of each index and infinitely many supergroups is not quite the problem — the problem is that a near coincidence of coordinates is evidence rather than proof, and deciding it needs a tolerance. Near-symmetry, and the tolerance that is not here is the essay about what that costs. Going down is arithmetic; coming back up is a judgement.

Where the exactness stops #

Computed here: for every containment among the seventeen plane groups, the orbits of both groups on a grid of twelfths, the map between them, the number of pieces each parent orbit breaks into, and the site symmetry each piece has. Orbit–stabiliser and conservation are asserted at every orbit rather than at an example.

A coarser statement than the Tables make. Positions are grouped here by the order of their stabiliser, which is the coarse invariant a grid can settle. Whether two orbits with the same stabiliser order are the same Wyckoff position in the Tables’ sense is a question about conjugacy in the normaliser — the same question one crystal and sixteen coordinate lists is about — and this does not answer it.

And the subgroups here keep the lattice. A descent can also give up translations rather than point operations, which multiplies the cell and splits positions in a different way; the two ways down is the essay about the distinction, and everything above is the first kind only.

What the count of sites is worth #

There is a number in every splitting scheme that a reader can check against a real structure without any crystallography at all, and it is worth pointing at because it is the scheme’s most testable consequence.

The number of crystallographically distinct sites is the number of orbits, and a technique that counts environments counts orbits. A nuclear magnetic resonance spectrum of a solid has one line per distinct site, at first approximation; a Mössbauer spectrum has one doublet; an infrared spectrum has a number of bands fixed by the same arithmetic through the selection rules. So the prediction this position splits in two is falsifiable by a measurement that never sees a diffraction pattern.

That is worth having because the two kinds of evidence fail in different places. Diffraction sees the average over the whole crystal and can miss a splitting that is local or slightly disordered — the symmetry of an average is the essay about exactly that failure. A local probe sees each atom’s own environment and cannot see the long-range arrangement at all. A structure whose diffraction says one site and whose resonance says two is a structure with local order and no long-range order, which is a real and common state of affairs rather than a contradiction.

The splitting scheme is what makes the two counts comparable: it says how many sites each candidate subgroup would give, and the experiment says how many there are.

Who found it, and when #

The systematic tabulation is recent: the International Tables’ volume on subgroup relations, with splitting schemes for every maximal subgroup of every space group, appeared in 2004 — a century after the space groups themselves, and only because the computation is enormous by hand and trivial by machine.

The physics that made it worth tabulating is older. Landau’s theory of second-order transitions, from 1937, requires the low-symmetry phase to be a subgroup of the high-symmetry one, and everything about the resulting structure follows from which subgroup: the order parameter is a representation, the domains are cosets, and the sites are a splitting scheme. The three are one calculation done at three levels of detail.

Why the grid is fine enough #

One methodological point, because a computation on a grid invites the question of what it misses.

The grid is twelfths, which puts 144 points in a cell, and every special position of every plane group lies on it: the special positions are at points with coordinates in halves, thirds, quarters and sixths, and twelve is the least common multiple of all four. That is not luck — it is the crystallographic restriction again, since the orders available are 2, 3, 4 and 6 and a fixed point of an operation of order n has coordinates with denominator dividing n or 2n.

So the census of kinds of position is complete: nothing is missed by sampling, because there is nothing between the twelfths that a plane group treats specially. What a finer grid buys is more general points, which are all alike, and nothing else.

A grid of tenths would miss every three-fold and six-fold position, and the striking thing is what that failure would look like: not an error, but a census with fewer kinds of position in it, and a splitting scheme that was perfectly self-consistent and about the wrong group. That is the same shape of failure as the sampling grid that was not closed under the group, and the reason both are worth stating rather than fixing quietly.

Where the ladder goes next #

Back, to the positions themselves: the points a group treats differently, and domains of a subgroup, where the same containment is drawn as a statement about area rather than about orbits.

Sideways, to what the freedom at a position means for a physical property: which modes a site can carry counts the displacements a Wyckoff orbit permits, and the piece with multiplicity zero is the direction an atom at that site may not move in.

And forward, to the transition that does the splitting: an order parameter is a representation predicts which subgroup a crystal descends to, and this essay says what that descent does to every atom in the cell.

Schemes compose, which is why only the maximal ones are tabulated #

A reader opening the International Tables at a group and looking for the descent they care about often does not find it, and the reason is a small structural fact about these schemes rather than an omission.

A chain of descents composes. If $G > H > K$, then the orbits of $K$ refine the orbits of $H$, which refine the orbits of $G$, and refining twice is refining once. The scheme for $G \to K$ is therefore recoverable from the scheme for $G \to H$ followed by the scheme for $H \to K$: take each piece the first produces, and split it according to the second.

Every subgroup sits at the end of a chain of maximal ones, since a proper subgroup that is not maximal is contained in something proper and larger, and the index is finite so the descent cannot be refined forever. The Tables therefore list maximal subgroups only, and every other descent is a composition the reader performs.

The indices multiply along the chain, which is the check that a composition was done correctly: the general position must end up split into the product of the indices, whatever route was taken to get there. Two different chains from $G$ down to the same $K$ give the same final scheme — they must, because the scheme is a statement about $K$’s orbits and $K$ does not know which way it was reached.

That last point is the one worth carrying. The route is bookkeeping; the destination is the crystal.