What is Lattice and basis? — Physics

Definition 23.1 University Physics — Year 3 · Chapter 23 — Crystalline Solids

A crystal is a repeating arrangement: a lattice (the grid of mathematical points $\vect r = n_1\vect a_1 + n_2\vect a_2 + n_3\vect a_3$ , integers $n_i$ ) decorated by a basis (the atom or group hung identically on every point). The parallelepiped spanned by $\vect a_1, \vect a_2, \vect a_3$ is a unit cell; its edge $a$ is the lattice constant, a few $\text{Å}$ . Three cubic stackings carry most of this course: simple cubic (points at cube corners — rare: one atom per cell); body-centred cubic (bcc: corners $+$ centre, 2 atoms per cell — iron, chromium); face-centred cubic (fcc: corners $+$ face centres, 4 atoms per cell — copper, aluminium, silver, gold: the densest cubic packing, filling 74 % of space). Diamond is fcc with a two-atom basis; rock salt is fcc with an Na–Cl pair.

The three cubic lattices. Corner atoms are shared among eight cells, face atoms between two, the body centre by one: count them and the cells hold 1, 2 and 4 atoms respectively — the bookkeeping behind every density calculation in this chapter.

Examples

Example 23.2 (Weighing a unit cell)

Copper is fcc with $a = 3.61\,\text{Å}$ . Its cell holds 4 atoms of molar mass $63.5\,\mathrm{g}/\mathrm{mol}$ , so

\rho = \frac{4M/N_{\text{A}}}{a^3} = \frac{4\times1.055 \times 10^{-25}\,\mathrm{kg}}{4.71 \times 10^{-29}\,\mathrm{m}^{3}} \approx 8960\,\mathrm{kg}/\mathrm{m}^{3} ,

the handbook value. Run backwards, the same arithmetic turns a measured $a$ and a bench-top density into $N_{\text{A}}$ — the route by which X-rays first counted atoms (Exercise 23.6).

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