Two and Three Layer Repeat

There is only one way to produce a repeat pattern (crystal lattice) with layers of hexagonally closest packed planes in just two orientations: (AB). If the interlayer distance is such that c:a = CPIS, then (AB) = HCP. Otherwise, the unit cell is still hexagonal with Z = 2, but the lattice is quasi-closest packed.

Likewise, there is only one way to produce a repeat pattern in three layers: (ABC). Again, if
c:a = 1.5.CPIS, then (ABC) = CCP = FCC. Otherwise, the lattice is quasi-closest packed and can be described with either the hexagonal unit cell (Z = 3) or the rhombohedral unit cell (Z = 1).

[Note: these are not stereo pairs]

It is important to understand that apparently different permutations of the letter codes for hexagonal closest packed layers do not necessarily represent unique lattice patterns. For example, the six permutations (AB), (AC), (BA), (BC), (CA), and (CB) all represent the same HCP lattice in different orientations and choices of unit cell axes. Likewise, the six permutations of (ABC) all represent the same CCP lattice.

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