Turning and climbing at once

Put a point somewhere in a crystal. Turn it a quarter turn about a vertical line, and at the same time lift it a quarter of the way up the cell. Do it again, and again, and again. After four moves the point has turned all the way round and climbed exactly one cell — so it is sitting directly above where it started, which in a periodic crystal is the same as being where it started.

That is a 4₁ screw axis, and the whole of this field is downstream of it.

The thing a rotation has that this does not #

A rotation about an axis leaves the axis alone. Every point on it stays exactly where it is, which is what it means for a line to be the axis.

A screw leaves nothing alone. Take a point on the line and apply the operation: it turns by nothing, since it is on the axis, and then it climbs. It has moved. Apply the operation to any point in the crystal, in fact, and it moves — there is no fixed point anywhere, and this is the property that puts the operation outside every point group.

A point group is a group of operations that all fix a common point. That is the definition, and a screw fails it immediately. So the point group of a crystal containing a screw axis does not contain that screw; what it contains is the rotation part of it, obtained by throwing the climb away. And throwing the climb away is exactly what the quotient by the translations does.

Why the climb has to be a simple fraction #

The rise cannot be anything. It is forced to a rational fraction of the cell with a small denominator, and the argument is two lines.

Take an axis of order n: the rotation part turns by 2π/n, and applying it n times gives a full turn, which is the identity. So applying the screw n times gives a pure translation along the axis — the linear parts have cancelled and only the climbs are left, n of them.

That translation has to be a lattice vector, because it is a symmetry of the crystal and every symmetry with no rotation in it is a lattice translation. So n times the rise is a whole number of cells. Call that number m, and the rise is m/n.

Two constraints, and everything else follows. The order is constrained by the crystallographic restriction to 1, 2, 3, 4 or 6, because the rotation part is an integer matrix and its trace has to be an integer. The rise is constrained by the argument above to a whole number of n-ths. Between them they leave a small, countable set of screws, and counting it is the next rung.

The order of a screw is not the order of its rotation #

There is a subtlety in the last figure worth pulling out, because it is where the word “order” becomes ambiguous and the ambiguity matters.

A 6₃ screw turns by a sixth and climbs by a half. Apply it twice: the point has turned by a third and climbed a whole cell, which is a lattice translation — so the square of a 6₃ is a pure threefold rotation, modulo the lattice. Apply it three times: turned by a half, climbed one and a half cells, which is a 2₁ screw.

So a single 6₃ axis carries a sixfold screw, a threefold rotation and a two-fold screw, all on the same line. Its order as an operation of the quotient group is six; the order of the rotation left after two applications is three; and neither number is wrong.

The same happens whenever the rise and the order share a factor. A 4₂ squares to a pure translation by one cell, so its square is the identity of the quotient and it behaves, in that one respect, like a two-fold. A 6₂ cubes to a translation by two cells. The screws whose m and n are coprime — 2₁, 3₁, 3₂, 4₁, 4₃, 6₁, 6₅ — are the ones where nothing shorter than n applications gets back to the axis.

This is why the enumeration reports the reduced order alongside the symbol. It is a property of the pair (m, n) rather than of either alone, and it is the arithmetic reason 6₃ is not a relabelled 3₁ even though both climb a third of a cell in two turns.

Why an origin cannot help #

The property that makes a screw a screw is not “the axis is in an awkward place”. It is worth insisting on this, because the intuition that a screw is a rotation seen from the wrong angle is very natural and completely wrong.

Move the origin anywhere. The axis moves with it — that is what moving an origin does. But the rise does not change, because the rise is what happens to the separation between a point and its image, and a translation of the whole coordinate system does not change separations.

The arithmetic version of this is the split: an operation’s translation is the sum of an intrinsic part, which is one n-th of the sum of the operation applied to itself n times, and a location part, which is everything a change of origin can produce. The intrinsic part of a rotation is zero. The intrinsic part of a screw is its rise. Nothing moves a number from one column to the other.

The helices that are already famous #

The word “screw axis” sounds like a technicality of crystallographic notation. It is not — it is the symmetry of every helix anybody has heard of.

DNA in its B form has ten base pairs per turn and a rise of 3.4 ångströms each, so the double helix is invariant under a rotation of 36° combined with a climb. That is a screw operation with n = 10, which no crystal can have, because ten-fold rotation is forbidden by the lattice. A single DNA molecule is not a crystal and does not have to obey the restriction; a crystal of DNA, which is how the structure was determined, has to arrange the molecules so that the crystal’s own symmetry is one of the permitted ones, and the molecule’s tenfold screw is not among them.

An α-helix in a protein has 3.6 residues per turn — a non-integer, and worse, an irrational-looking one. Again a symmetry of the molecule that cannot be a symmetry of a crystal containing it.

Quartz is where the crystal itself is the helix: chains of silicon–oxygen tetrahedra spiralling about threefold screw axes, in one of two mirror-image arrangements. Here the screw is crystallographic, n = 3, and the two directions of the spiral are the enantiomorphic pair P3₁21 and P3₂21.

The pattern across the three is worth noticing. The restriction constrains what a crystal can be, not what a molecule can be, and a molecule with a forbidden screw simply gets packed into a crystal that does not share it. That is the same distinction this site makes about quasicrystals and about the fivefold molecules whose symmetry no crystal reproduces.

Composition, and the screws nobody chose #

A group with a screw in it can be built by putting one there. It can also acquire one without anybody asking, and in three dimensions that is the commoner case.

Two mirrors at right angles compose to a two-fold rotation about their line of intersection. Give one of them a slide along that line and the composition becomes a two-fold screw — the rotation is unchanged and the slide survives the composition, so the product has an intrinsic translation neither generator had along that direction.

Composition producing operations nobody put in is the subject of its own essay, and it is the reason the machinery here always runs the closure rather than working from a list of generators. A list of generators is not a group.

What it does to a diagram #

The International Tables mark an axis perpendicular to the page as a polygon with as many sides as its order — a lens for two, a triangle for three, a square for four, a hexagon for six. A screw is the same polygon left open, with tails, and the number of tails is m.

That last observation deserves a moment. A fourfold screw is not one operation, it is a coset of them: the turn, its inverse and its square all live on the same axis, and their rises are a quarter, three quarters and a half. So a single 4₁ axis carries a 4₁, a 4₃ and a 2₁ all at once, and the Tables draw one mark for the axis rather than three marks on top of each other. The mark names the axis by its generating operation, which is the highest order with the smallest rise.

The figures on this site collapse them the same way, and the collapsing is a real decision rather than a tidy-up. Drawing all three produces a shape that is none of them.

Counting what a cell holds #

The last practical consequence is the one that shows up first in a real structure determination, and it follows from the screw having no fixed point.

A rotation axis has points on it, and an atom sitting on one is fixed by the rotation — so it contributes one atom to the cell rather than a pair. A screw fixes nothing, so no atom can sit on a screw axis in the sense of being fixed by it. Every atom in a crystal whose only symmetry is a screw belongs to an orbit as long as the group.

That is a hard constraint on how much can be in a cell. In P2₁ the group has two operations and there are no special positions at all, so the number of molecules per cell is even, always. Measure a density, work out the cell volume, divide, and if the answer is an odd number of molecules the space group is wrong or the density is wrong. Crystallographers have used that arithmetic as a first sanity check since long before structures could be solved.

The plane has exactly two groups that act freely — p1 and pg — and the Wyckoff essay found them by sampling every point of a cell and asking what fixes it. In space there are more, and the useful ones for chemistry are P2₁ and P2₁2₁2₁, which between them hold the majority of published structures of chiral molecules.

Where the picture is a lie #

Three things about the drawings above that a reader should not take literally.

The helix is not part of the crystal. The curve in every screw figure here is the path a point would follow if the operation were applied continuously, and the operation is not continuous. What exists are the discrete images — the dots — and the curve is an aid to seeing which dot came from which. A crystal has no spiral in it unless the atoms happen to lie on one, and in quartz they roughly do and in most screw-containing structures they do not at all.

The scale between panels is not comparable. A 6₅ screw climbs five whole cells in six turns and a 2₁ climbs one in two, so drawing them at a common scale would waste four fifths of most panels or run the tall ones off the top. Each panel is scaled to its own advance, and the figure that puts the rises on a common axis is the enumeration, where the comparison is the point.

The axis is drawn where the arithmetic put it, which is not always where the eye expects. An operation’s element is located by solving for the set it fixes, and the translates of an operation by lattice vectors have elements at fractional positions — a fourfold axis composed with a cell translation has its axis at the cell centre, not one cell along. Every mark in every plan diagram in this field is placed that way rather than by copying a mark and sliding it, which is the difference between P4/mmm having axes at its corners and centre and having them only at its corners.

What the machinery decides, and what it does not #

The operation is exact and two things about it are worth stating as limits rather than as caveats.

The rise is a fraction of this cell. A rise of a quarter means a quarter of whatever the c axis happens to be, and the c axis is a property of the crystal being measured. So “4₁” is a statement about a ratio and not about a distance, and two crystals with the same space group and different cell dimensions have screws that climb by different amounts in ångströms. That is obvious once said and it is the reason the whole subject works in fractional coordinates rather than in lengths.

Nothing here says a screw is physically realised by anything. A screw axis is a symmetry of an arrangement, and the arrangement is a set of positions. Whether atoms actually spiral, whether bonds follow the helix, whether the structure is best described as helical — all of that is chemistry and this machinery cannot see it. Quartz has threefold screws and is genuinely helical; a great many structures have screw axes and nothing about them looks like a spiral, because the operation relates parts of the structure that are not bonded to each other at all.

The screws are counted next, the glides after that, and then the question of what a closure produces that nobody asked for.

A screw has infinite order #

The word order was pulled apart above for one reason and it deserves pulling apart for a second, because the honest answer changes the relationship between a space group and its point group.

A rotation of order $n$ satisfies $R^n = 1$. Apply it $n$ times and every point is exactly where it started. That is what makes a point group finite: it has finitely many operations and each of them, repeated, comes back.

A screw never comes back. Apply an $n_m$ screw $n$ times and every point has been carried $m$ cells up the axis. Apply it $n$ more times and it is $2m$ cells up. There is no power of a screw that is the identity, so a screw is an element of infinite order — the same kind of element a translation is, dressed in a rotation.

So a space group is infinite in more ways than one. Its translations are infinite in number and each has infinite order, and that much is expected. What is less expected is that a non-symmorphic group contains infinitely many non-translational elements of infinite order as well, one for every screw and every glide it holds.

And the point group is a quotient rather than a subgroup. This is the sentence the whole essay has been circling. Take a space group and forget the translation attached to each operation; what is left is the point group, and forgetting is a quotient map. For a symmorphic group there is also a copy of the point group sitting inside the space group — pick the right origin and every point operation appears with translation zero. For a non-symmorphic group there is no such copy: the fourfold screw has no fourth power equal to the identity, so nothing in the group behaves like the fourfold rotation the point group contains.

That is why P4₁ and P4 have “the same point group” and are not the same anything else. The point group 4 is what forgetting leaves, in both cases. Only in P4 can it be found again by looking.

Why the plane has no screws #

The plane’s classification has four non-symmorphic groups and every one of them owes its status to a glide. There is no plane screw, and the reason is worth a paragraph because it explains what a screw actually needs.

A screw is a rotation composed with a translation along its own axis. The axis is the line the rotation fixes, and the translation runs in the same direction. Both ingredients are required to point the same way.

In the plane a rotation fixes a point, not a line. The rotations of a plane pattern turn about centres, and the direction “along the axis” is out of the page — a direction the pattern does not occupy. Translating along it moves the pattern off the plane entirely, which is not a symmetry of a plane pattern.

So the only translation a plane rotation can be composed with is one in the plane, perpendicular to the axis rather than along it, and the previous section shows what that produces: a rotation of the same angle about a shifted centre. It has a fixed point, so it is a rotation, and nothing new has been made.

The mirror is the one operation with a fixed line in the plane, and a line supports a translation along itself. That is the glide, and it is the whole of the plane’s non-symmorphic supply.

Adding a dimension adds fixed lines. A rotation in space fixes an axis and a reflection fixes a plane, so both acquire directions along which they can slide — hence screws as well as glides, hence eleven screws and five glides, and hence two hundred and thirty where the same argument gave seventeen.