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Lattice vs Crystal: Points and Motif

The single distinction that trips up every beginner: a lattice is an infinite array of identical points (pure geometry), while the crystal is what you get after you hang atoms — the motif — on each point. Crystal = lattice + motif, and everything in crystallography grows from that one idea.

Wallpaper, and the stamp that makes it

Earlier in this ladder we said a crystalline solid has long-range order: its atoms repeat in a regular, predictable pattern that marches on for millions of rows. Now we make that precise. Picture a roll of wallpaper — the same little drawing printed over and over at perfectly regular positions. A crystal is exactly that idea carried into three dimensions. To describe it, crystallographers split it into two things that beginners almost always fuse together: the grid of positions, and the picture printed at each position. Keeping those two apart is the single most important habit in the whole subject.

The rule is a slogan worth memorising: crystal = lattice + motif. The lattice is pure geometry — an infinite array of identical points with no atoms in it at all. The motif (also called the basis) is the actual group of atoms — one atom, or two, or two hundred — that you hang on each point. Stamp the same motif onto every lattice point and you rebuild the crystal. That is the lattice-plus-motif decomposition, and it is the frame we will hang the rest of this rung on.

The lattice: pure geometry of points

So what exactly is a lattice point? Not an atom, and not even a fixed place — it is an abstract point whose surroundings look exactly the same, in the same orientation, as every other lattice point. That 'identical environment' test is the real definition of a lattice point. Stand on any point and look around; move to any other point and look around; you must see precisely the same view. If two candidate points have even slightly different neighbours, only one of them can be a lattice point.

Because every point shares the same view, you can hop from any one to any other by adding whole numbers of just a few basic steps. Those steps are the translation vectors a, b and c. Every lattice point is reached from the origin by a vector R = u a + v b + w c, where u, v, w are whole integers — never fractions. That integer rule is the exact mathematical meaning of 'periodic': slide the whole infinite lattice by any lattice translation and it lands perfectly back on itself. The lattice is, quite literally, the complete set of all such R.

  LATTICE (points only)        MOTIF (2 atoms)        CRYSTAL = lattice + motif

   .    .    .    .              O-o                   O-o  O-o  O-o  O-o
   .    .    .    .                                    O-o  O-o  O-o  O-o
   .    .    .    .          (hang one motif           O-o  O-o  O-o  O-o
   .    .    .    .           on every point)          O-o  O-o  O-o  O-o
A 2D wallpaper in words: the bare lattice is only points (where), the motif is the atoms (what), and stamping the motif on every point makes the crystal.

The motif: what hangs on each point

The motif is whatever repeats. In a pure metal like copper the motif is a single atom, so the crystal looks almost like the bare lattice with one atom parked on each point. But the motif can be a molecule, an ion pair, or an entire folded protein of thousands of atoms — in a protein crystal the motif is the whole molecule, stamped identically at every lattice point. The number and identity of the atoms in the motif is what makes the chemistry; the lattice only ever says where, never what.

Take common table salt. Its rock-salt structure sits on a face-centred cubic lattice, and the motif is two ions: one Na+ and one Cl-. Hang that two-ion motif on every point of the FCC lattice and you get the familiar three-dimensional checkerboard of sodium and chlorine. Each Na+ ends up cradled by six Cl- neighbours (a coordination number of 6, sitting in an octahedral hole of the chlorine array), and each Cl- likewise. Here is the subtlety that pays off forever: the sodium and the chlorine are NOT two different lattice points — their surroundings differ, so they fail the identical-environment test — they are the two members of one motif, both hung on a single set of points.

  1. Pick any one atom as a reference, then find every other atom whose surroundings are truly identical to it — same neighbours, same distances, same orientation.
  2. Mark that set of identical points and forget the atoms for a moment: those points, on their own, are the lattice — pure geometry.
  3. Whatever sits between one lattice point and the next — a single atom or a whole cluster — is the motif, and it is identical at every point.
  4. Check yourself: lattice (where) + motif (what) must rebuild the crystal exactly. If two atoms have different surroundings, they are different members of the motif, not different lattice points.

The unit cell: one stamp that tiles all space

You never draw an infinite lattice. Instead you draw one small box that, stacked edge to edge with no gaps and no overlaps, fills all of space by pure repetition. That box is the unit cell — the single stamp of our wallpaper analogy. Its size and shape are fixed by six numbers, the lattice parameters: three edge lengths a, b, c and the three angles alpha, beta, gamma between those edges. Give those six numbers and the box is completely specified; the next guide is devoted entirely to them.

To say where each atom sits inside the box we use fractional coordinates — positions measured as fractions of the cell edges. A corner is (0,0,0), the centre of a face is (1/2,1/2,0), and the very middle of the box is (1/2,1/2,1/2). These fractions are wonderfully portable: they stay the same whether the cell edge is 3 angstrom or 30. In practice you list only the small handful of atoms you cannot generate from the others by the cell's symmetry — the asymmetric unit — and symmetry plus translation regrows all the rest.

Now a counting subtlety that surprises people. A corner of the box is shared by eight neighbouring cells, so each corner counts as only 1/8 of a lattice point; a point sitting on a face is shared by two cells, counting 1/2. A face-centred cubic cell therefore holds 8 corners times 1/8 + 6 faces times 1/2 = 1 + 3 = 4 lattice points, not one. That is what lattice points per cell means. Why choose this roomy four-point cube over a smaller box holding just one point? Because the cube displays the full cubic symmetry at a glance, and that clarity is worth more than being minimal — guide 3 unpacks exactly this trade-off between primitive and centred cells.

How many patterns are there? Seven systems, fourteen lattices

Two questions close this rung, and both have exact answers. First: how many distinctly-shaped boxes are there? Seven — the seven crystal systems (cubic, tetragonal, orthorhombic, hexagonal, and so on), sorted purely by the relations among a, b, c and the angles. Second: how many genuinely different point-lattices can fill three-dimensional space? Exactly fourteen — the fourteen Bravais lattices. Not 'fourteen so far' and not 'fourteen that we have found': it is a proven, complete enumeration, first worked out by Bravais in 1848. Guides 4 and 5 build the seven systems and the fourteen lattices one at a time.

Periodicity is a strict landlord. A lattice may have only 1-, 2-, 3-, 4- or 6-fold rotational symmetry, and never 5-fold — this is the crystallographic restriction. The intuitive reason is one you can feel with floor tiles: regular pentagons cannot tile a floor without leaving gaps, whereas triangles, squares and hexagons cover it perfectly. There is also a beautifully democratic way to draw a one-point cell: claim all the space that is closer to a given lattice point than to any other, and you carve out the Wigner-Seitz cell — the same construction that returns, as the Brillouin zone, when we reach reciprocal space later in this domain.