Crystal Lattices and Unit Cells

To describe the internal, geometric arrangement of particles in a crystalline solid precisely, chemists use the

idea of a crystal lattice. A crystal lattice is a regular, three-dimensional arrangement of points in space,

where each point (called a lattice point) represents the position of one constituent particle — an atom, ion,

or the centre of a molecule. The defining feature of a lattice is that every lattice point has an environment

identical to every other lattice point; the whole crystal can be generated by translating (repeating) a small,

representative block over and over in three dimensions.

That smallest repeating block is the unit cell — the fundamental building unit which, when stacked together

edge-to-edge in three dimensions with no gaps or overlaps, reproduces the entire crystal lattice. A unit cell is

described completely by six parameters: the lengths of its three edges, conventionally labelled $a$, $b$ and $c$,

and the three angles between those edges, labelled $\alpha$ (between $b$ and $c$), $\beta$ (between $a$ and $c$)

and $\gamma$ (between $a$ and $b$). Depending on the relative values of $a$, $b$, $c$ and $\alpha$, $\beta$,

$\gamma$, crystals fall into seven crystal systems (cubic, tetragonal, orthorhombic, and so on); this chapter

focuses mainly on the cubic system, where $a = b = c$ and $\alpha = \beta = \gamma = 90^\circ$, since it

covers most of the metals and simple ionic solids met at this level.

It helps to build the idea up from two dimensions before jumping to three. In two dimensions, a square lattice is

generated simply by repeating a single square unit cell — with one lattice point at each corner — along two

perpendicular directions; the resulting grid of points, extended indefinitely, is the 2-D lattice. In three

dimensions, the same idea is extended along a third direction: stacking cubic unit cells along all three mutually

perpendicular axes generates a 3-D simple cubic lattice, with lattice points sitting at the corners of every

cube in the stack.

Within the cubic system, unit cells are further classified by where additional lattice points sit besides the

corners. A primitive (simple) unit cell has lattice points only at its eight corners. A body-centred cubic (bcc) unit cell has lattice points at the eight corners and one additional point at the centre of the cube's …