Simple lattices and their unit cells
Simple Cubic (SC) - There is one host atom ("lattice point") at each corner of a cubic unit cell. The unit cell is described by three edge lengths a = b = c = 2r (r is the host atom radius), and the angles between the edges, alpha = beta = gamma = 90 degrees. There is one atom wholly inside the cube (Z = 1). Unit cells in which there are host atoms (or lattice points) only at the eight corners are called primitive.
Body Centered Cubic (BCC) - There is one host atom at each corner of the cubic unit cell and one atom in the cell center. Each atom touches eight other host atoms along the body diagonal of the cube (a = 2.3094r, Z = 2).
Face Centered Cubic (FCC) - There is one host atom at each corner, one host atom in each face, and the host atoms touch along the face diagonal (a = 2.8284r, Z = 4). This lattice is "closest packed", because spheres of equal size occupy the maximum amount of space in this arrangment (74.05%); since this closest packing is based on a cubic array, it is called "cubic closest packing": CCP = FCC.
FCC Primitive - It is also possible to choose a primitive unit cell to describe the FCC lattice. The cell is a rhombohedron, with a = b = c = 2r, and alpha = beta = gamma = 60 degrees. [A cube is a rhombohedron with alpha = beta = gamma = 90 degrees!]
Simple Hexagonal (SH) - Spheres of equal size are most densely packed (with the least amount of empty space) in a plane when each sphere touches six other spheres arranged in the form of a regular hexagon. When these hexagonally closest packed planes (the plane through the centers of all spheres) are stacked directly on top of one another, a simple hexagonal array results; this is not, however, a three-dimensional closest packed arrangement. The unit cell, outlined in black, is composed of one atom at each corner of a primitive unit cell (Z = 1), the edges of which are: a = b = c = 2r, where cell edges a and b lie in the hexagonal plane with angle a-b = gamma = 120 degrees, and edge c is the vertical stacking distance.
Closest Packing
Hexagonal Closest Packing (HCP) - To form a three-dimensional closest packed structure, the hexagonal closest packed planes must be stacked such that atoms in successive planes nestle in the triangular "grooves" of the preceeding plane. Note that there are six of these "grooves" surrounding each atom in the hexagonal plane, but only three of them can be covered by atoms in the adjacent plane. The first plane is labeled "A" and the second plane is labeled "B", and the perpendicular interplanar spacing between plane A and plane B is 1.633r (compared to 2.000r for simple hexagonal). If the third plane is again in the "A" orientation and succeeding planes are stacked in the repeating pattern ABABA... = (AB), the resulting closest packed structure is HCP.
HCP Coordination - Each host atom in an HCP lattice is surrounded by and touches 12 nearest neighbors, each at a distance of 2r: six are in the planar hexagonal array (B layer), and six (three in the A layer above and three in the A layer below) form a hexagonal prism around the central atom.
Cubic Closest Packing (CCP) - If the atoms in the third layer lie over the three grooves in the A layer which were not covered by the atoms in the B layer, then the third layer is different from either A or B and is labeled "C". If a fourth layer then repeats the A layer orientation, and succeeding layers repeat the pattern ABCABCA... = (ABC), the resulting closest packed structure is CCP = FCC. Again, the perpendicular spacing between any two successive layers is 1.633r.
CCP Coordination - Each host atom in a CCP lattice is surrounded by and touches 12 nearest neighbors, each at a distance of 2r: six are in the planar hexagonal (B) plane, and six (three in the C layer above and three in the A layer below) form a hexagonal anti-prism (also known as a distorted octahedron) around the central atom.
Rhombohedral (R) lattice - If, in the (ABC) layered lattice, the interplanar spacing is not the closest packed value (1.633r), then the primitive (Z = 1) unit cell is a rhombohedron with a = b = c <> 2r and alpha = beta = gamma <> 60 degrees. The non-primitive hexagonal unit cell (Z = 3).may also be chosen.
2- & 3-layer repeats - There is only one way to produce a repeat pattern (crystal lattice) in two layers of hexagonally closest packed planes: (AB) = HCP. Likewise, there is only one way to produce a repeat pattern in three layers of hexagonally closest packed planes: (ABC) = CCP.
4-layer repeats - However, there are two ways to produce a closest packed lattice in four layers: (ABAC) and (ABCB). By extension, there are increasing numbers of ways to produce closest packed lattices in five layers, six layers, etc., up to and including non-repeating random stacking. Thus, there are many closest (and pseudo-closest) packings in natural and artificial materials.
