Adding Translation: Screw Axes and Glide Planes

The last ingredient: translation

Two earlier rungs handed you two separate boxes of parts. The point-symmetry rung gave the 32 point groups — every allowed way to combine rotations, mirrors and inversions about a single fixed point. The lattice rung gave the 14 Bravais lattices — every allowed way to fill space with repeating points. A point group tidies up what sits at one spot; a lattice says where to copy it. The question that closes this whole rung is the obvious one: what happens when you let those two boxes talk to each other?

Point-group operations all share one feature: they leave at least one point exactly where it was. Rotate a snowflake and its centre never budges; reflect it and the mirror line stays put. But a crystal is infinite and periodic, so it also owns a whole family of pure translations that shift every atom bodily into the next cell. The rich part appears when you weld a point operation onto a fraction of a translation — a rotation stapled to a partial slide, or a mirror stapled to a partial slide. These hybrids are genuine symmetries of the infinite pattern even though they leave no point fixed at all, and there are exactly two of them: the screw axis and the glide plane.

The screw axis: rotate, then slide

Picture the spiral ramp of a multi-storey car park. To climb from one level to the next you rotate part of the way around and rise a fixed step at the same time, and every level looks identical. A screw axis does exactly this to atoms. Its symbol is n with a subscript m, written n_m: rotate by 360/n degrees, then translate along the axis by the fraction m/n of the lattice repeat in that direction. Neither the rotation alone nor the slide alone is a symmetry — only the two performed together, as one indivisible move.

The simplest is the two-fold screw 2_1: turn 180 degrees and slide half a cell (m/n = 1/2). Do it twice and you have turned a full 360 degrees and slid one whole cell — landing exactly on the equivalent atom in the next cell up, which is the check that it really is a symmetry. A richer case you already met by name: hexagonal close packing is space group P6_3/mmc, and the 6_3 in that symbol is a six-fold screw — rotate 60 degrees, slide 3/6 = half a cell along c. Apply it to the atom at fractional height (1/3, 2/3, 1/4) and it lands on its partner at (2/3, 1/3, 3/4); that single operation is what stitches the ...ABAB... stacking of oranges into one seamless pattern.

1. Start with an atom at some height along the axis — call it level 0.
2. Rotate it by 360/n degrees about the axis: for a 2_1 that is 180 degrees, for a 6_3 that is 60 degrees.
3. Immediately slide it along the axis by m/n of the cell repeat: for a 2_1 that is half a cell, for a 6_3 that is 3/6 = half a cell.
4. The atom now sits where the crystal insists another atom must be. Repeat the move n times: you have turned 360 degrees and slid a whole number of cells, returning to an equivalent atom — proof the screw is a real symmetry.

The glide plane: reflect, then slide

Now staple a partial slide onto a mirror instead of a rotation and you get a glide plane. The everyday picture is a line of footprints in wet sand: each print is the mirror image of the one before — left, right, left, right — but shifted forward by half a stride. Neither the reflection alone (which would drop one foot straight onto the other) nor the half-step alone is a symmetry; only reflect-and-then-step is. A glide plane does this to atoms: reflect across the plane, then translate by a fixed fraction of a lattice vector that lies in the plane. A plain mirror is just the special case where the slide is zero.

The size and direction of the slide give the glide its name. An axial glide (labelled a, b or c) slides by half a cell edge — half of a, half of b, or half of c. A diagonal glide n slides by half of a face or body diagonal, such as (a+b)/2. The diamond glide d slides by a quarter of a diagonal, like (a+b)/4, and turns up in diamond and in spinel — the very structure that gave it its name. Every one of these erases a characteristic set of reflections in a diffraction pattern, a point we return to at the end.

OPERATION     SYMBOL   MOVE  =  point part    +  translation part
-----------   ------   ----------------------    ------------------
screw axis    2_1      rotate 180 deg            + c/2   (half repeat)
              3_1      rotate 120 deg            + c/3
              6_3      rotate  60 deg            + c/2
glide plane   a,b,c    reflect                   + a/2, b/2 or c/2
              n        reflect                   + (a+b)/2  (diagonal)
              d        reflect                   + (a+b)/4  (diamond)
mirror        m        reflect                   +  0   -> plain point op

Putting it together: the 230 space groups

Now assemble the full toolkit. Take the 14 Bravais lattices, drop one of the 32 point groups onto every lattice point, and let rotations and mirrors be promoted into screw axes and glide planes wherever periodicity permits it. Enumerate every distinct result and you get exactly 230 space groups — the complete list of ways atoms can be symmetrically arranged in a periodic three-dimensional crystal. Like the 14 lattices and 32 point groups before them, 230 is a proven, closed count (settled independently by Fedorov, Schoenflies and Barlow around 1890), not a running tally that might one day grow.

The 230 fall into two kinds. In a symmorphic space group — 73 of them — you can find one single point where all the symmetry elements pass through: no screw, no glide, just the point group parked cleanly on the lattice, and rock salt's Fm-3m is one. In a nonsymmorphic space group — the other 157 — at least one operation is an unavoidable screw or glide, so no single point can carry the full symmetry; HCP's P6_3/mmc and the common mineral packing Pnma are both nonsymmorphic. That clean 73 + 157 = 230 split is the sharpest way to see exactly what translation bought us: 157 groups that could not exist without it.

The names themselves are a compact code called Hermann-Mauguin notation: a leading centring letter (P, I, F or C) followed by the symmetry along each characteristic direction. So Fm-3m reads 'face-centred, with a mirror, a three-bar and a mirror along the cubic directions', and P6_3/mmc reads 'primitive, a 6_3 screw with a mirror across it, then mirrors and a c-glide.' Every space group's full anatomy — its symmetry diagram, its list of Wyckoff positions with their multiplicities and site symmetries, and how many formula units Z fit in the cell — is laid out in the International Tables for Crystallography. The next four guides open that book one door at a time: guide 2 tours the 230, guide 3 teaches you to read a symbol, guide 4 unpacks Wyckoff positions and the asymmetric unit, and guide 5 turns to the fingerprint.

The fingerprint: systematic absences

So how do you discover which of the 230 a real crystal belongs to? You cannot see the screw axes and glide planes directly, but they leave an unmistakable signature in a diffraction pattern (the full machinery of which arrives in a later rung — here only the idea matters). Because a screw or glide relates two atoms by a half-step, the waves they scatter arrive exactly out of phase for certain reflections and cancel completely, so those spots are simply missing. These blanks are the systematic absences, and each translational element writes its own rule: body-centring wipes out every (hkl) with h+k+l odd; a 2_1 screw along b erases the (0k0) reflections with k odd; a c-glide perpendicular to b erases the (h0l) reflections with l odd. Read the pattern of what is absent and you read the crystal's translational symmetry straight off.

Be honest about the reach of this fingerprint, though. Diffraction records intensities — the squared amplitudes |F|^2 — and throws away the phase of each wave, the notorious phase problem, so the absences pin down the translation parts but cannot settle everything on their own. A diffraction pattern also always looks centrosymmetric (Friedel's law), so on its own it usually cannot tell a centrosymmetric space group from a non-centrosymmetric one, nor a left-handed screw from its right-handed twin. In practice the absences narrow the 230 down to a short list of candidates — often just one or two — and other evidence closes the gap. Guide 5 works this backwards, reasoning from missing spots to the space group itself.