A four-letter string that unpacks into a whole crystal
Guide 2 handed you the punchline: every periodic crystal in the universe belongs to one of exactly 230 space groups. That is a beautiful fact, but a list of 230 items is useless unless each one has a short, unambiguous name. The name is the space-group symbol, and it is written in Hermann-Mauguin notation — things like Fm-3m, P6_3/mmc, Pnma. At first glance they look like a licence plate. They are the opposite: not an arbitrary label but a compressed instruction manual. Read left to right, a symbol tells you the lattice you are stamping on and every symmetry element it carries, in the order a crystallographer expects.
The whole trick is that the symbol is positional. The first character is always the centering letter. Everything after it is a short queue of symmetry operators, and — this is the key idea — the slot an operator sits in tells you the direction it acts along, while the character itself tells you which operation it is. A crystallographer does not read 'Pnma' as four separate facts; they read a position, a lattice, and three directions each carrying a mirror or glide. Learn the slot rule once and the same reading works for all 230.
The leading letter: which lattice you stamp on
The first capital letter records the lattice centering — the same seven choices you met when the 14 Bravais lattices were built. P means primitive: one lattice point per cell, sitting at the corners only. I (from the German innenzentriert) is body-centered: an extra point at the cell's dead center, so 2 points per cell. F is face-centered: an extra point at the middle of every face, giving 4 points per cell. C is base-centered on the top and bottom faces (2 per cell); A and B are the same idea on the other face pairs. R is the rhombohedral centering used in the hexagonal setting, with 3 points per cell.
That single letter already carries physical weight. It fixes how many lattice points — and therefore how many copies of the motif — the cell repeats, and it is the first thing a diffraction pattern reveals, because centering wipes out whole families of reflections (F-centering, for instance, allows only reflections whose h, k, l are all even or all odd). So before you have read a single symmetry element, the leading letter has already narrowed the possibilities and told a diffractionist half of what they need. We meet those systematic absences head-on in guide 5; for now just notice that the very first character is already a fingerprint.
The remaining slots: symmetry read along named directions
After the centering letter come one, two, or three slots, and each slot is nailed to a specific crystallographic direction fixed by the crystal system. In the cubic system the three slots mean, in order, the cube edges <100>, the body diagonals <111>, and the face diagonals <110>. In the hexagonal system they mean the c-axis [001], the a-axes <100>, and the <1-10> directions. You do not guess the directions — the system dictates them, so 'the second slot' always means the same family of directions for that system.
Inside each slot the character names the operation. A bare number is a rotation axis (2, 3, 4, 6); a number with a subscript is a screw axis that rotates then translates (6_3 means a six-fold turn stapled to a slide of 3/6 = 1/2 of the axis repeat); a bar over a number (-3, written 3-bar) is a rotoinversion. A lowercase m is a mirror plane; the glide letters a, b, c, n, d are glide planes that reflect then translate, differing only in the direction and length of that translation. A slash, as in 6_3/m or 2/m, means 'axis with a plane perpendicular to it'. That is the entire alphabet.
- Read the first capital: that is the centering (P, I, F, C, A, B, or R) and fixes the lattice.
- Identify the crystal system from the pattern of symbols, which tells you what direction each remaining slot refers to.
- Walk the slots in order; for each, name the operation (rotation, screw, rotoinversion, mirror, or glide) and attach it to that slot's direction.
- Read any slash as 'a plane perpendicular to the axis before it', and note whether any screw axis or glide plane appears — that decides symmorphic vs nonsymmorphic.
Three real symbols read out loud
Start with Fm-3m (No. 225), the space group of copper and of rock salt. F says face-centered lattice. The cubic slots then read: m perpendicular to the cube edges <100>, a 3-bar rotoinversion along the body diagonals <111>, and m perpendicular to the face diagonals <110>. Every element here is a plain mirror or rotoinversion with no built-in translation, so Fm-3m is symmetric in the strict sense — a symmorphic space group, one where all the symmetry passes through a single common point.
Now P6_3/mmc (No. 194), the space group of magnesium in hexagonal close packing and of graphite. P is primitive. The hexagonal slots read: along the c-axis a 6_3 screw axis with a mirror perpendicular to it (that is the 6_3/m), a mirror m perpendicular to the a-axes, and a c-glide along the <1-10> directions. Because it contains a screw axis and a glide plane — operations that only exist once you allow translation — P6_3/mmc is nonsymmorphic. That distinction is not decorative: the 6_3 and the c-glide will each carve their own gaps into the diffraction pattern.
Finally Pnma (No. 62), an orthorhombic workhorse worn by olivine, cementite, and countless distorted perovskites. P is primitive. The three orthorhombic slots are the a, b, c axes in turn, and the plane characters read: an n-glide perpendicular to a, a mirror m perpendicular to b, an a-glide perpendicular to c. Two glides make it firmly nonsymmorphic. Be honest about one subtlety, though: Pnma is one setting of space group No. 62, and the very same group appears as Pbnm or Pnam if you relabel the axes. The abstract group is unique; its written symbol depends on which axis you call a, b, or c.
F m -3 m <- cubic, Fm-3m (Cu, NaCl) | | | | | | | +-- <110> face-diag : mirror m | | +------- <111> body-diag : 3-bar rotoinversion | +------------- <100> cube edge : mirror m +---------------- lattice : F (face-centered) P 6_3 /m m c <- hexagonal, P6_3/mmc (Mg, graphite) | | | | | | | | | +-- <1-10> dirs : c-glide | | | +------- <100> a-axes : mirror m | | +----------- perp to c : mirror /m | +--------------- [001] c-axis : 6_3 screw axis +------------------ lattice : P (primitive)
From symbol to atoms: Wyckoff positions, Z, and the International Tables
Reading the symbol tells you the symmetry; the payoff is that the symmetry tells you where atoms are allowed to sit. The reference that spells this out is the International Tables for Crystallography, one dense page per space group. Its heart is the list of Wyckoff positions: every distinct kind of site the symmetry permits, each labelled by a letter and a multiplicity — the number of equivalent points the symmetry generates from it. The most crowded site, where an atom lies on no symmetry element and is simply copied everywhere, is the general position; sites that sit on a mirror, axis, or inversion center are the special positions, and they have lower multiplicity because the symmetry maps them partly onto themselves.
Here is the accounting made concrete. In Fm-3m, rock salt puts Na+ at the Wyckoff site 4a (0,0,0) and Cl- at 4b (1/2,1/2,1/2); each site has multiplicity 4, so the conventional cell holds 4 Na and 4 Cl — that is 4 formula units, Z = 4, the formula units per cell. Copper in the same group puts one atom at 4a, giving Z = 4 atoms per cell (the FCC count you already knew). Magnesium in P6_3/mmc sits at the special position 2c, so Z = 2. Multiply Z by the formula mass and divide by the cell volume and you have the crystal's density from symmetry alone — a small miracle that the symbol set in motion.