The last mile: from a symbol to actual atoms
In the previous guide you learned to read a space-group symbol like Fm-3m or Pnma — a centering letter followed by the symmetry along each key direction. That tells you which symmetries the crystal obeys. But a symbol is a grammar, not a sentence: it does not yet say where the atoms are. To pin down a real crystal you must place atoms, and the whole gift of the space group is that you only place a few — its operations, including the screw axes and glide planes met at the start of this rung, clone each atom you supply into all of its symmetry partners.
Hold on to the running picture. A crystal is an infinite 3D wallpaper, and the unit cell is the single stamp that tiles all of space by translation. Now zoom inside one stamp. Even the stamp has internal symmetry — mirrors, rotation axes, inversion — so the picture printed on it is itself built from repeated pieces. The smallest piece you must draw by hand, from which the cell's own symmetry regenerates everything else, is the asymmetric unit. Draw it once; the space group inks in the rest of the stamp, and translation then prints that stamp across all of space.
The asymmetric unit: the crystal's smallest stamp
The asymmetric unit is the smallest fraction of the unit cell that, acted on by every symmetry operation of the space group, rebuilds the complete cell — and hence, by lattice translation, the whole crystal. If the space group makes M copies of a point that sits in open space, then the asymmetric unit is exactly 1/M of the cell volume. Everything outside it is a symmetry-generated echo; you never type those coordinates in yourself.
This is the same divide-and-copy idea as lattice-plus-motif, pushed one level deeper. Earlier, translation alone tiled the motif across space. Now, inside a single cell, the point-symmetry operations plus the rung's built-in translations first tile the asymmetric unit up to the full cell. The stronger the symmetry, the tinier the piece you must supply. In Pnma the asymmetric unit is 1/8 of the cell; in the far more symmetric face-centered cubic group Fm-3m it shrinks to just 1/192 of the cell — you name a couple of atoms and symmetry does the other 99.5 percent of the work.
Wyckoff positions: multiplicity and site symmetry
When you drop an atom into the cell, where it lands decides how many copies appear. A Wyckoff position is a family of points that the space group makes equivalent, all sharing the same local symmetry, and it carries two numbers you read straight off the International Tables. The multiplicity is how many equivalent points there are per conventional cell. The site symmetry is the little point group of operations that leave that exact point unmoved — the symmetry the atom itself is sitting on.
These two numbers trade off against each other by an exact law: the multiplicity times the order of the site symmetry is a constant for the whole space group — and that constant is the multiplicity of the general position. The intuition is a headcount. The space group has a fixed number of operations acting inside the cell; each one either carries a point to a fresh partner or, if the point lies on that element, pins it in place. The more operations pin a point, the fewer fresh copies survive — so higher site symmetry buys lower multiplicity, in exact proportion.
- Read the multiplicity — the number of equivalent atoms this row places in the conventional cell.
- Read the Wyckoff letter — just a label, assigned a, b, c... from the bottom up, so 'a' is the most symmetric site and the last letter is the general one.
- Read the site symmetry — the point group that fixes the point (a bare 1 means nothing but the identity holds it).
- Read the coordinate triplet — fully free (x, y, z) for a general point, or partly locked (like x, 1/4, z) for a special one.
- Sanity-check: multiplicity times the order of the site symmetry should equal the general multiplicity, every single time.
General versus special positions
A general position is an atom sitting on no symmetry element at all: its site symmetry is 1, its coordinates are completely free (x, y and z all adjustable), and its multiplicity is the largest in the group. It is always the last Wyckoff letter. In Pnma — point group mmm of order 8, with one lattice point per cell — the general position is 8d: eight equivalent atoms, with fractional coordinates (x, y, z) fully free.
A special position is an atom that has drifted onto a symmetry element — a mirror, an axis, or an inversion centre — so that some operations now map it to itself. Its multiplicity drops and its coordinates lock. In Pnma, an atom lying on the mirror plane occupies 4c at (x, 1/4, z): the mirror pins y to 1/4, halving the count from 8 to 4 — exactly the factor of 2 by which the site symmetry rose. An atom sitting on an inversion centre instead takes 4a or 4b, also multiplicity 4. Only an atom off every element needs the roomy 8d.
Counting atoms: the formula units Z per cell
Add up the multiplicities of the occupied Wyckoff positions and you have counted every atom in the cell — which hands you Z, the formula units per cell. Take rock salt, NaCl, in Fm-3m: put Na+ at 4a (0,0,0) and Cl- at 4b (1/2,1/2,1/2). That is 4 sodium plus 4 chlorine — 8 atoms, or 4 NaCl formula units, so Z = 4. Each Na+ sits in an octahedral hole of the FCC Cl- array with coordination 6, exactly as the structures rung promised. Both 4a and 4b carry the full site symmetry m-3m (order 48), and the bookkeeping checks: 4 times 48 = 192, the general multiplicity.
Fm-3m (No. 225) m-3m point group, F-centred = 4 lattice pts/cell
Mult Wyck Site symm Representative coordinate(s)
---- ---- --------- ----------------------------
192 l 1 (x, y, z ) <- general position
: : : ...
8 c -43m (1/4,1/4,1/4) tetrahedral hole (F- in CaF2)
4 b m-3m (1/2,1/2,1/2) <- Cl- in NaCl
4 a m-3m (0, 0, 0 ) <- Na+ in NaCl
---- ---- ---------
check: 4 x 48 (order of m-3m) = 192 = general multiplicity
cell : Na 4 + Cl 4 = 8 atoms -> Z = 4 formula unitsZ is the bridge from structure to theoretical density: the mass of Z formula units divided by the cell volume. For NaCl with a = 5.64 angstrom, Z = 4 gives 4 times 58.44 g/mol over (6.022 times 10^23 times (5.64 times 10^-8 cm)^3), which is about 2.16 g/cm^3 — spot on the measured 2.17. And when you meet the hexagonal metal group P6_3/mmc from the previous guide, the two atoms of an HCP metal sit at Wyckoff 2c, so Z = 2 — the symbol and the site count agree.
The International Tables, and what to watch for
Every one of the 230 groups gets its own page in the International Tables for Crystallography, Volume A — a single dense sheet holding the symmetry diagram, the general and special positions with their coordinates, the chosen asymmetric unit, the generators, and a column of reflection conditions. Learn to read one page and you can read all 230. Those reflection conditions are the systematic absences — the diffraction spots forced to vanish by centering, screw axes and glide planes — which the next and final guide of this rung turns into a fingerprint for identifying an unknown space group.
A few honest cautions. Multiplicities are quoted for the conventional, possibly centred cell: because an F cell holds 4 lattice points, its multiplicities are 4 times the primitive count — the same structure written in a primitive setting would list smaller numbers. Wyckoff letters are only labels in alphabetical order, with no physical meaning beyond 'a is most symmetric.' A site symmetry is a genuine point group, but it need not be the whole crystal's point group. And knowing the space group and the occupied Wyckoff letters still does not tell you which element sits where, or the values of the free x, y, z — those come from diffraction intensities and the structure factor, because diffraction measures |F|^2 and loses the phase. Symmetry hands you the labelled slots; the experiment must fill them.