Undergraduate · 15 min

Building crystals: unit cells and packing

Rotate six crystal structures, count atoms per cell and compare how tightly they pack.

Goal

Find Z (atoms per unit cell), the coordination number and the packing fraction for each structure.

Apparatus and reagents

A simulation only. Atom sizes follow the nearest-neighbor distance; in the ionic structures the two ions have different sizes.

Procedure

Select simple cubic, then body-centered, then face-centered cubic. Read Z, CN and the packing percentage.
Set the atom size to 1 to make neighboring atoms touch, then use 2 or 3 cells per edge to see the pattern repeat.
Look at NaCl and CsCl (two kinds of ions) and at diamond.

What to observe

Packing fractions: simple cubic 52 %, body-centered 68 %, face-centered 74 %, diamond only 34 %.
Z = 1, 2 and 4 for the three cubic lattices; NaCl has 4 Na⁺ and 4 Cl⁻ per cell with 6 : 6 coordination, CsCl has 1 + 1 with 8 : 8.

Explanation

Atoms on a corner count $1/8$ , on an edge $1/4$ , on a face $1/2$ and inside the cell $1$ ; adding them gives $Z$ . The packing fraction is $Z\cdot\frac{4}{3}\pi r^3/a^3$ with the atoms touching along the shortest contact: along the edge for simple cubic ( $a = 2r$ ), the body diagonal for BCC ( $a = 4r/\sqrt3$ ) and the face diagonal for FCC ( $a = 2\sqrt2\,r$ ). Diamond is open (34 %) because each atom has only four tetrahedral neighbors, the price of directional covalent bonding. Crystal structures like these are what X-ray diffraction measures.

History of the experiment

In 1912 Max von Laue showed that X-rays diffract from crystals, proving both that they are waves and that solids are ordered arrays. Within a year William Henry and William Lawrence Bragg reinterpreted the spots with the law

2d\sin\theta = n\lambda

and solved the first simple structures, starting with NaCl. The compact packing models of this simulation — BCC, FCC, diamond — are the same atom-by-atom maps that their diffraction data first revealed.

Chemists behind it

Dorothy Crowfoot Hodgkin Rosalind Franklin