A screw keeps a line and a glide keeps a plane

Where a gradient term survives at the edge of the zone computed a condition that the cell a zone-boundary mode doubles had only stated. A transition can freeze a commensurate structure continuously only if the free energy has no term linear in the gradient of the order parameter. Such a term, a Lifshitz invariant, makes a slow modulation cost less than none at all, so the crystal takes up a long wave instead of the structure the wavevector names. Over the seventeen plane groups and every special point of their zones, that condition failed in exactly ten places: where one glide acts and no second glide crosses it, and at the zone corner of three hexagonal groups with no glide at all.

The essay ended with two predictions for space, both stated before anything was computed. An order parameter where exactly one screw or glide is active should carry a term along that operation’s direction, and a second such operation crossing it should remove the term. And the three-fold axis at the corner of a hexagonal zone should admit a gradient term in some symmorphic groups and not in others.

The 219 types of space group are available here with their operations, so the census can simply be run, and it took less than two minutes. Of the two predictions, the second is right. The first is right in the monoclinic groups, wrong about direction everywhere, and wrong about the second operation from the orthorhombic groups upward. And the largest thing the census found was something neither prediction mentioned, in the part of the zone the plane had settled as empty.

What was counted #

An order parameter is a representation: at wavevector $k$ it is a physically irreducible representation of the space group, carrying the star of $k$ and of its negative. A Lifshitz invariant is antisymmetric in the components of the order parameter and linear in one gradient, and the number of them is given by the same character formula as in the plane, an average over the group of the antisymmetric part of the character times the trace of the operation on a vector.

The one thing that had to change is the size of the group, and the change is a simplification. Translations must act trivially on a term, and a product of two arms of a star survives that only when the two wavevectors add to zero. So terms live on one arm and its negative, and the group that matters is the set of operations carrying $k$ to $k$ or to $-k$, taken modulo the translations on which $k$ has a whole phase. That group has at most 192 elements for any type, and usually far fewer, which is why a census of this size runs at all.

A special point here means a wavevector no line of the zone passes through with the same symmetry, once the sign is taken into account. These are found rather than listed: every wavevector on a grid of twenty-fourths of the reciprocal axes is tried, and it is kept if the operations carrying it to plus or minus itself leave no direction fixed. That gives 1,146 special points over the 219 types and 3,933 order parameters, of which 767 carry a term.

The zone centre was not empty #

In the plane, no order parameter at the centre of the zone has a Lifshitz term, a fact which way the order parameter points computed for every plane group. The natural reading, and the one the plane essay’s prediction took for granted, was that the question belongs to the zone boundary. A Lifshitz invariant was thought of as a thing that happens where a screw or a glide is active.

In space it is not true. Eight of the thirty-two classes have a term at the zone centre, the picture at the head of this essay. They are 3, 4, 6, 32, 422, 622, 23 and 432, with the same answer for every space group in each class, since a zone-centre representation depends on the point group alone. A cubic group in 23 or 432 has a three-dimensional order parameter with one term for each copy of that representation; the others have a two-dimensional one.

The reason the plane could not have this is dimension. In a plane group, a two-dimensional order parameter has an antisymmetric square that is one-dimensional and behaves as a rotation about the axis normal to the plane, and the vector representation contains nothing that behaves that way. Space has a third direction, and in a rotation group the normal component of a vector and the rotation about that normal behave identically. A term linear in a gradient is the antisymmetric product of an order parameter with itself, and only in space can such a product match a vector. Put concretely, the term is a curl: the order parameter twists as it advances along an axis.

A curl distinguishes a right-handed twist from a left-handed one, so any operation that reverses hand forbids it. That is why all eight classes lie among the eleven with no improper operation, the ones a structure of one hand may sit in. $4mm$, which has the same four-fold axis as 422 and the same two-dimensional representation, has no term, because its mirrors reverse the twist. The three chiral classes without a term, 1, 2 and 222, have no two-dimensional order parameter to twist. This is the algebra behind the long helices of chiral magnets, where the group $P2_13$, in class 23, is the standard example. The census counts representations of the space group and says nothing about magnetic moments, and the term it finds there is the shape of the known one.

What the plane predicted, tested #

Move out to the zone boundary, and take a precise meaning for the word “active”. An operation of order two that fixes $k$ squares to a translation. If $k$ has phase $-1$ on that translation, the operation squares to $-1$ on any order parameter at $k$, which forces the representation to be at least two-dimensional, the same arithmetic by which a glide sticks two levels together. Call that operation active. An operation with no intrinsic translation, or with one on which the phase is whole, is not.

In the monoclinic groups the prediction holds with no exceptions at all. No active operation means no term, on 47 special points. One active operation means a term, on 22 of 22. Two active operations mean none, on 2 of 2. The rule stated about a plane in terms of glides carries over to space with screws added, and it predicts every monoclinic point.

The orthorhombic row shows where the rule stops. The first line still holds: no active operation, no term, on all 107 points. But with one active operation there are three points with no term out of sixty, and with two active operations, where the prediction says the term should be gone, 87 of 104 points keep one. In a monoclinic group there is little room for a second active operation to cross the first; they are nearly parallel by construction. In an orthorhombic group they may act along different axes, and the two can each leave a direction the other does not disturb. The second half of the plane’s rule was a fact about having no third dimension to escape into. One line in the plane and a second crossing it share everything. Two screws along different axes in space do not.

So the usable statement is the one half. An active operation is necessary in those two systems, not sufficient, and the count of them is only a guide above the monoclinic.

The direction was the wrong thing to predict #

The prediction was that a term would run along the operation’s direction. For a glide in the plane that is the same as saying along the line, because the line has only one direction. In space the two statements part.

In the monoclinic groups the result is again exact. A screw gives one term at each of the ten special points where one is active, always along its axis. A glide gives two terms at each of its twelve, both in the glide plane. So $Pc$, with its glide translating along the axis $c$, has a term along $c$ as predicted and a term along $a$, where no translation points at all.

The reason takes one line. The two-dimensional order parameter forced by an active operation $g$ has an antisymmetric square that $g$ acts on with determinant one, which is trivially. A gradient along a direction $d$ can pair with it only if $g$ leaves $d$ alone. A screw leaves one direction alone and reverses the other two. A glide, being a reflection, leaves two directions alone and reverses one. So what decides the term’s direction is what the operation keeps, and the translation that made it active plays no further part. The translation decides whether there is a term. What the linear part keeps decides where.

Restricted to the plane containing the glide’s direction and its normal, a glide in space has one term, the plane’s answer, and that is a check on both computations. The plane essay’s glide rule was a correct statement of the only case available to it.

In the orthorhombic groups, which have several operations acting at once, the direction rule becomes a tendency. Of 331 direction-and-term pairs over the two systems, 286 run along a direction every active operation keeps, and 45 do not. It would be a mistake to promote that to a law, and it is reported here as the count it is.

Two causes, and no third #

The monoclinic rule requires an active operation, but the second prediction was about groups with no screws or glides at all. The two are consistent once the terms away from the zone centre are sorted by what is available at their special point.

Of the 711 order parameters away from the centre that have a term, 587 sit at a point with an active operation. The other 124 have none, and at every one of those points the group carrying $k$ to $\pm k$ contains an operation of order above two. No term anywhere in the census lacks both. The second cause is the same one that fills the zone centre: an operation of order three, four or six has complex characters, so a real order parameter is two-dimensional or more with no help from any translation, and there is something for the antisymmetric square to be. Triclinic groups have neither cause and carry no term out of their 24 order parameters. That is a count rather than a theorem, and it is the census’s main summary: a Lifshitz term in space comes from an operation that squares to a phase, or from a rotation whose characters are not real.

The corner, and the groups it divides #

The second prediction said the corner of a hexagonal zone would admit a term in some symmorphic groups and not others, by the arithmetic of cube roots of unity. That is correct.

In eleven of the sixteen the corner is a special point. Eight have terms: $P6$, $P6/m$, $P622$, $P321$, $P6/mmm$, $P\bar62m$, $P\bar3m1$ and $P6mm$. Three do not: $P\bar6$, $P\bar6m2$ and $P3m1$. All have no screw and no glide. All have the three-fold axis at the corner. The difference is what connects $K$ to $-K$, since that pairing is how a complex representation at $K$ becomes a real order parameter, and the available direction must be one that the operations keeping $K$ preserve and the operations reversing it turn over. A mirror in the plane of the zone, as in $P\bar6$ and $P\bar6m2$, reverses the normal while keeping $K$ fixed, which removes the one direction a one-dimensional representation at $K$ could use. $P\bar62m$ has the same mirror and keeps a term regardless, and the figure shows how: its corner carries four-dimensional order parameters, from a two-dimensional representation at $K$ that $P\bar6m2$ does not have. The two groups differ only in how their two-fold lines lie against the lattice.

Two of these groups form a pair worth pausing on. $P321$ has terms at the corner and $P312$ does not even have the corner as a special point, although the two share a class and differ only in how the two-fold axes lie relative to the lattice. The same thing happens with $P3m1$ and $P31m$. The pair that swaps its rows in the plane is a well-known difference with no visible consequence for a dot pattern. Here it decides whether a transition can freeze a structure at the corner of the zone.

Nine groups the wave cannot read #

If active operations are one cause of a term, the groups with screws or glides but no active operation should have no term off the zone centre, and a list of those is a sharp prediction. There are nine such groups among the 146 non-symmorphic types.

All nine are on body-centred or face-centred lattices, and the reason is the same in each. On such a lattice, the real wavevectors are restricted, and the translation that a screw or glide squares to has a whole phase at all of them. The two groups in the first rows, $I2_12_12_1$ and $I2_13$, are exactly the two that no single operation reveals, the groups that extinguish no reflection their symmorphic partners do not. That essay identified them by looking for operations whose translations leave no trace on a diffraction pattern. This census identifies them, with seven others, by looking for operations whose translations leave no trace on a gradient. Those are two different measurements of one thing, which is how visible a translation part is to a wave.

The other seven are not hidden in the earlier sense, because they do cause absences. $Fd\bar3m$, the group of diamond, is among them, and it is an arresting member, with d glides and four-one screws. The wave at these points has a phase that sees the operation’s translation only in multiples that cancel. So an order parameter at a special point of diamond’s zone meets no gradient term of this kind. This statement covers the special points only, and the census says nothing about how diamond behaves at any general point.

What this does not show #

A term is a permission. A Lifshitz invariant permits a long modulation by making it lower the energy at first order, and the census counts the invariants. It does not say how large a coefficient is, how long the resulting wave would be or whether a crystal gets close to that instability. A small coefficient gives a modulation too long for an experiment to distinguish from a commensurate structure.

The picture of a twist is not a computation of one. The diagrams of a screw and a glide show what is kept, which is the reason given, and they show nothing about the character formula that actually decided each case. They are true to the monoclinic results and cannot be drawn for an orthorhombic point where several operations act together.

Off the special points this says nothing. The argument that a general wavevector always fails is the plane essay’s, it applies here, and it has not been rerun: the census is over special points only. And “special” has been taken in one definite sense, isolated once the sign is allowed. A wavevector on a line with its own symmetry is a separate and larger question.

Physical irreducibility is assumed, with time reversal as the pairing. The order parameters here are real representations of the space group, which fits a structural transition. A magnetic order parameter is odd under time reversal, and the groups that apply are the magnetic ones, of which there are 1,651 and which this census does not touch. The agreement with $P2_13$’s helices is agreement in form.

How it was checked, and what had to be refused #

The representations are not read from a table. A random real symmetric element of the group’s commutant splits the regular representation into eigenspaces, each carrying one real irreducible representation, and the eigenspaces whose translations act with the right phases are the order parameters at $k$. The count of invariants is required to be a whole number for each, which is a real test of a floating-point computation.

Results known from elsewhere have to come out. The three-dimensional representation of $P2_13$ must have one term. The class 432 must have terms and $m\bar3m$ none. The zone-centre verdict must agree for every space group of a class, which it does for all 219, since the representation depends on the point group; the two routes to that fact are the group’s own quotient and the class. And a glide in space has to reduce to the plane’s answer when restricted to the plane’s two directions.

Types were checked for their total. The groups come from the arithmetic classes through the normaliser’s orbits on the cohomology, and the positions of those same 219 types reproduce the 1,731 Wyckoff positions the Tables print. A census over a wrong list of groups would have been a meaningless one, and that number was the check on the list.

And the natural summary had to be turned away. “Along the operation’s direction” is what a reader of the plane’s result would carry to space. $Pc$ has a term across its glide’s direction, and that is computed rather than argued.

Still open: what the second operation really does #

The monoclinic groups need one number, and the orthorhombic groups do not obey it. There are 104 orthorhombic points with two active operations, and 87 keep a term; there should be a rule that says which, and it probably depends on the angle between the two operations’ kept directions rather than on their count. A screw along $a$ keeps only $a$, and a screw along $b$ keeps only $b$, and together they keep nothing, so they should remove any term; a glide and a screw whose axis lies in its plane keep a shared direction. Whether a shared kept direction is both necessary and sufficient, across all 331 pairs of direction and term that the count above already contains, would turn the count into a theorem, and the 45 exceptions to the simple form are where to start. Past that lie the tetragonal and cubic groups, where both causes can act at once on one order parameter, and a count of the two together has not been separated here.