jMol series: Unit cell tutor

Crystallographic unit cells are three-dimensional "tiles" inside a crystal that can fill space completely, at the same time marking atom positions by repeating themselves translationally in all directions. Unit cells generally have the form of parallelepipeds (six sides in parallel pairs). Two such unit cells are illustrated in the model of a table salt crystal shown above, where the sodium atoms are purple and the chlorine atoms are green. Such unit cells are used to describe the detailed way in which ordered solids are put together on the atomic scale, which in turn makes possible the wide variety of properties of solid materials (e.g. strength, flexibility, hardness, porosity) as well as of their surfaces (e.g. roughness, reflectivity, reactivity and adhesion).

Lattice parameters: If you draw scale bars from a starting point along three non-parallel edges of this tile, they will tell you the lengths of a set of unit cell axes. These axis lengths are commonly called a, b and c. In the model below, double-clicking on the far end of the axis you decide to call a, single-clicking on the axes' origin, and then double-clicking on the far end of axis b will mark the angle between a and b (commonly called gamma). Similarly measuring alpha (the angle between b and c) and beta (the angle between c and a) will then give you the full set of lattice parameters {a, b, c, alpha, beta, gamma} for your unit cell.

Primitivity: A unit cell is primitive if a unit cell containing fewer atoms is not possible. The Figure above shows a marked-up unit-cell that is not primitive, but has the advantage of a cubic symmetry, along with a smaller non-cubic primitive cell. These concepts are illustrated in the interactive model below, which allows you to re-orient the lattice with your mouse for a better look.

Click below to mark one possible cubic but non-primitive unit cell. The second button illustrates some tiling across the specimen. Given that face atoms count 1/2, edge atoms 1/4, and corner atoms 1/8, this unit cell contains 1+12/4=4 sodium atoms and 6/2+8/8=4 chlorine atoms. A second click removes the marking.
Click below to mark one of the possible primitive non-cubic unit cells. The second button illustrates its tiling across the specimen. This cell contains 2/2=1 sodium atom and 8/8=1 chlorine atom, suggesting that the asymmetric unit for this lattice is a single NaCl molecule.
Finally, click below to mark another of the possible primitive non-cubic unit cells. The second button again illustrates tiling. This cell also contains 4/4=1 sodium atom and 8/8=1 chlorine atom.

Hit reload to clear all markings, and try finding and marking a unit cell of your own in the model NaCl lattice above. A mark will be added whenever you first double-click on a pair of atoms in sequence.

What are your unit-cell's lattice parameters? The good news is that the lattice is unambiguously described by any space filling unit cell that you manage to find! The bad news, however, is that the unit cell you find by this method is not unique, and hence won't necessarily be the conventional one.

Is your unit cell primitive? One way to check this is to ask: How many Na and Cl atoms does it contain?