Sphere hexagonal-closest

Tc or c cubic closest-packing of spheres Th or h hexagonal closest-packing of spheres Ts stacking sequence AA... of hexagonal layers Qs stacking sequence AA... of square layers... 

There are four spheres, four octahedral interstices and eight tetrahedral interstices per unit cell. Therefore, their numerical relations are the same as for hexagonal closest-packing, as well as for any other stacking variant of closest-packings one octahedral and two tetrahedral interstices per sphere. Moreover, the sizes of these interstices are the same in all closest-packings of spheres. 

As an example of the prevalence of high-symmetry structures we can take the closest packings of spheres only in the cubic and the hexagonal closest-packing of spheres are all atoms symmetry equivalent in other stacking variants of closest-packings several nonequivalent atomic positions are present, and these packings only seldom occur. 

According to the discussion in Section 17.3, many structures can be derived from the hexagonal closest-packing of spheres by occupying a fraction of the octahedral interstices with other atoms. If the X atoms of a compound MX form the packing of spheres, then the fraction 1 fn of the octahedral interstices must be occupied. The unit cell of the... 

P63/m2/m2/c, which is the space group of the hexagonal closest-packing of spheres, has only one maximal subgroup in which the position 2a is split into two independent positions, namely P 3 2/m 1. If one of these positions is occupied and the other one remains vacant, this corresponds to the Cdl2 type. 

Group-subgroup relations from hexagonal closest-packing of spheres to some MX3 and M2X3 structures. The boxes represent octahedral voids, with the coordinates as given at the top left. The positions of the octahedron centers are labeled by their Wyckoff letters. Gray boxes refer to occupied voids. The dots indicate how the atoms Ru, P and N are shifted from the octahedron centers parallel to c... 

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