Lattice Centering and Coordination Number

The lattice system of a crystalline solid is defined by the shape of its unit cell. Each lattice system represents one or more lattice types, which are distinguished by the positions of the lattice points.

A primitive, or simple, lattice has lattice points only on the corners of its unit cell. Each lattice system has a primitive lattice.

Some lattice systems also have centered lattices, which have additional lattice points on the faces of or entirely enclosed within their unit cells. There are three types of centering: body-centered, base-centered, and face-centered.

Body-centered, base-centered, and face-centered lattices have additional lattice points at the center of the unit cell, on the top and bottom faces, and on each of the six faces, respectively.

The coordination number of an atom in a unit cell is the number of neighboring atoms of interest with which it is in direct contact, or its ‘nearest neighbors’. The nearest neighbors may be within the same unit cell or in other unit cells.

Assuming identical atoms positioned at all lattice points, centered lattices have higher packing efficiencies and coordination numbers than primitive lattices do.

Consider the cubic lattice system, which includes primitive, body-centered, and face-centered lattices.

Assuming all lattice points are occupied by identical atoms, a primitive cubic unit cell contains one atom. This atom contacts four atoms from the same layer, one in the layer above, and one in the layer below for a coordination number of six.

A body-centered cubic unit cell has two atoms owing to the additional atom at the center of the unit cell. Each atom contacts four atoms in the layer above and four in the layer below for a coordination number of eight.

A face-centered cubic unit cell has four atoms owing to the atoms at all six faces, which corresponds to three additional atoms being assigned to the unit cell. Each atom contacts four atoms in the vertical plane in front of it, four in its own plane, and four in the plane behind it for a coordination number of twelve