Structural Relationships by Group-Subgroup Relations

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. 

The given conditions do not always allow for equivalent positions for all atoms. Take as an example the following conditions composition MX5, covalent M-X bonds, all X atoms bonded to M atoms. In this case all X atoms can only be equivalent if each set of five of them form a regular pentagon around an M atom (as for example in the XeFj ion). If this is not possible for some reason, then there must be at least two non-equivalent positions for the X atoms. According to the symmetry principle the number of these non-equivalent positions will be as small as possible. 

As was shown in various previous chapters, many structures of solids can be regarded as derivatives of simple, high-symmetry structure types. Let us recall some examples  

Body-centered cubic sphere packing = CsCl type = superstructures of the CsCl type 

Closest-packings = closest-packings with occupied octahedral interstices (e.g. Cdl2 

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