planar atomic density
Now instead of a line, lay a plane flat inside the crystal like a floor and ask how tightly it is tiled with atoms. The planar atomic density is the number of atom centres lying in the plane per unit area, and the related planar packing fraction is the share of the floor actually covered by atomic discs. Some planes are paved edge to edge; others are sparse.
You count the atoms whose centres lie within a chosen patch of the plane and divide by that patch's area: PD = atoms per area. Compare two planes of face-centred cubic. On the (100) face, an area a^2 holds four corner atoms shared four ways plus one face-centre atom, giving two atoms, so PD = 2 / a^2 and the discs cover about 79 percent. On the (111) plane the atoms sit in a tighter hexagonal net, giving PD = 4 / (sqrt(3) times a^2), roughly 2.31 / a^2 — a higher density, with the discs covering about 91 percent. The (111) plane is the more crowded floor.
The densest plane is where the action is. The plane of highest planar density is the close-packed plane, and it is both the plane on which slip occurs and, in many crystals, the plane along which the material cleaves. So counting atoms per area, like counting atoms per length, quietly predicts how a crystal will deform and split.
In face-centred cubic, the (111) plane has planar density 4/(sqrt(3) times a^2) — higher than the (100) plane's 2/a^2. That is why FCC metals slip on the close-packed {111} planes: the densest floor is the easiest one for atoms to shear across.
Atoms per area picks out the close-packed plane, which becomes the slip and cleavage plane.
As with linear density, count only atoms whose centres lie in the plane. The densest plane is where slip and cleavage prefer to happen — geometry, not chemistry, points the way.