Crystal Lattices and Unit Cells

Think of tiles laid to cover a floor: they generate a repeating pattern. If you mark the same point on every tile (say its centre) and look only at the marked points, you get a regular set of points — the "scaffolding" on which the tile pattern was built.

• This scaffolding of points is a space lattice (also called a crystal lattice).
• The structural unit placed on each point — a tile in the analogy, and an atom, ion or molecule in a real crystal — is called the basis or motif.
• Placing identical motifs on the lattice points, in the same orientation everywhere, generates the crystal structure.

Fig. 1.5 shows a motif, a two-dimensional lattice, and the crystal structure that results when the motif is placed on every lattice point; Fig. 1.6 shows that different spatial arrangements of points give rise to different lattices. In a real (three-dimensional) crystalline solid the space lattice is a three-dimensional array of points, and every motif has the same structure, orientation and surroundings as every other (except at the surface).

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Characteristics of a crystal lattice

• Each point in the lattice is a lattice point (or lattice site).
• Each lattice point represents one constituent particle — an atom, an ion, or a molecule (a group of atoms).
• Lattice points are joined by straight lines to bring out the geometry of the lattice.

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The unit cell

We do not need the whole lattice to describe a crystal — a small repeating portion is enough. This is the unit cell. Normally we choose the cell with the shortest perpendicular sides from which the entire crystal can be reconstructed by simply translating (shifting) the cell in three dimensions, filling all space with no gaps. Fig. 1.7 illustrates this tiling in two dimensions.

In two dimensions the unit cell is a parallelogram with sides of length $a$ and $b$ and an angle $\gamma$ between them (Fig. 1.8). A portion of a three-dimensional lattice and its unit cell are shown in Fig. 1.9.

A three-dimensional unit cell is characterised by: …