From wallpaper to a single tile
In the first guide of this rung we drew the line that trips everyone: a lattice is an infinite array of identical points — pure geometry, the where — while the motif is the group of atoms hung on each point, the what, so that crystal = lattice + motif. That picture is true but unwieldy: the lattice runs on forever. The whole job of this guide is to tame that infinity — to describe an endless pattern with a small handful of numbers you could scribble on a napkin.
The trick is repetition. Pick any lattice point as home and choose three non-coplanar translation vectors a, b and c that step from that point to its neighbours. Then every lattice point in the whole crystal is reached by R = u a + v b + w c, where u, v and w are whole numbers — positive, negative or zero. The small box spanned by a, b and c is the unit cell: one tile that, stacked edge to edge in all three directions, rebuilds the infinite pattern. It is the single rubber stamp that prints the entire wallpaper.
The six lattice parameters
How big is that box, and what shape? Six numbers pin it down completely, and together they are the cell's lattice parameters. Three are edge lengths — a, b, c — the lengths of the three vectors. Three are the interaxial angles: alpha between b and c, beta between a and c, and gamma between a and b. Fix those six and the cell's volume, its plane spacings, and its density all follow; nothing else about the box is free.
UNIT CELL - one tile that stacks to fill all space
.-----------------. SIX LATTICE PARAMETERS
/ /| a, b, c = the three edge lengths
/ / | alpha = angle between b and c
.-----------------. | beta = angle between a and c
| | | gamma = angle between a and b
| | |
c | | / b cubic: a = b = c and
| |/ alpha = beta = gamma = 90 deg
.-----------------.
a corner point shared by 8 cells:
8 x (1/8) = 1 -> primitive cell
A worked feel for the numbers: copper is cubic, so a = b = c = 3.615 angstrom and alpha = beta = gamma = 90 degrees — one length says it all, because the high symmetry forces the rest. A low-symmetry (triclinic) crystal is the opposite extreme: all six parameters differ and every one must be measured. And how are they measured? Here is the honest hand-off to the diffraction rungs later — the positions of the peaks in a diffraction pattern give you the cell parameters, while the peaks' intensities encode the motif inside. Positions size the box; intensities furnish it.
Placing the atoms: fractional coordinates
Now hang the motif inside the box. We give each atom's position not in angstrom but as fractional coordinates (x, y, z), each running from 0 to 1, measured as a fraction of the way along a, b and c. A corner sits at (0, 0, 0), the body centre at (1/2, 1/2, 1/2), the centre of a face at (1/2, 1/2, 0). The beauty is that these numbers are dimensionless: rescale the cell — say, heat it so a grows — and the fractions do not move. And because stepping by one whole cell just adds 1 to a coordinate, a single (x, y, z) automatically repeats through all of space.
A tiny example: body-centred iron lists just two atom positions, an Fe at (0, 0, 0) and an Fe at (1/2, 1/2, 1/2); every other atom in the metal is a translated copy of those two. Rock salt is only a little longer — a handful of Na and Cl coordinates and you have specified an infinite crystal exactly.
When the crystal also has internal symmetry — mirrors and rotations, the subject of the very next rung — you need list even less. Give only the asymmetric unit: the smallest group of atoms from which the symmetry operations regenerate all the rest, like one wedge of a paper snowflake that the folds copy around. So the complete recipe for any crystal is startlingly compact — six cell parameters, a short list of fractional coordinates for the asymmetric unit, and the symmetry — which is, almost word for word, what a real crystal-structure file (a CIF) stores.
Counting lattice points: primitive versus centered
Here is a question that sounds trivial and is not: how many lattice points does one cell actually own? A point sitting on a corner is shared by the eight cells that meet there, so it counts as only 1/8 of a point. The cell has eight corners, giving 8 times 1/8 = 1. A cell that contains exactly one lattice point is a primitive cell (labelled P) — the leanest tile there is.
- Corner point — shared by 8 cells, so it counts 1/8. Eight corners give 8 times 1/8 = 1 point.
- Face-centre point — shared by 2 cells, so it counts 1/2. An edge point is shared by 4 cells, counting 1/4.
- Body-centre point — lies wholly inside one cell, so it counts a full 1. Add the pieces for each cell type: primitive P = 1, body-centred I = 2, base-centred C = 2, face-centred F = 4.
Why would anyone pick a cell with more than one point when a primitive one exists? Because you can deliberately add extra lattice points — one at the body centre, or one on every face, or one on a single pair of faces — to build a conventional cell whose right angles put the lattice's symmetry on open display. This deliberate adding of interior points is lattice centering, and it is exactly why the humble cube reappears as body-centred and face-centred versions. The true primitive cell of those lattices does exist, but it is a squashed, skew rhombohedron that hides the cubic symmetry the eye wants to see. Guide 3 unpacks the trade in full.
The Wigner-Seitz cell, and what comes next
There is one more primitive cell worth meeting, and it needs no choice of axes at all: the Wigner-Seitz cell. To build it, stand on a lattice point, draw a straight line to each neighbouring point, and slice each line in half with a perpendicular plane. The smallest region those planes fence off is the cell. Read plainly, it is simply the patch of space closer to your point than to any other — each lattice point's own territory, like the tiles a map draws around competing radio towers. It always holds exactly one lattice point, so it is primitive, and it always carries the full symmetry of the lattice.
Unlike a box you had to choose, the Wigner-Seitz cell is unique and wears the lattice's symmetry on its face: for a body-centred cubic lattice it is a truncated octahedron, for face-centred cubic a rhombic dodecahedron. This is not a curiosity for its own sake. Build the very same construction in the crystal's 'shadow world' of reciprocal space and you get the Brillouin zone, the natural arena in which a solid's electrons and their energy bands are described — a direct pointer to the reciprocal-lattice rung waiting further up this ladder.
Step back and the shape of the rest of the rung appears. A cell is pinned by six parameters and a centering label — and those choices are not endless. The axial relations among a, b, c and their angles sort every conceivable cell into just seven crystal systems, and layering on the allowed centerings yields exactly fourteen Bravais lattices — the subjects of the final two guides here. 'Exactly fourteen' is a proved, complete enumeration, not a 'so far': it is the same rigidity that forbids a periodic lattice from ever having 5-fold symmetry (the crystallographic restriction theorem) — which is why quasicrystals, flaunting five-fold patterns that never quite repeat, were so shocking when they turned up.