Crystallographic analysis of apatite and tourmaline structure types: Rules of maximum symmetrization and coherent assembly

Following the maximum symmetrization principle (maximum decrease in the degrees of freedom of atoms in the structure), the results of the crystallographic analysis of structures of the very abundant apatite type (P6₃/m), rare krasheninnikovite mineral (P6₃/mcm), and also abundant tourmaline (R3m) are discussed. The configuration of cations defining the structure in the first two minerals is created by a set of “skeletal” planes of the cubic and hexagonal sublattices, whereas tourmaline contains only a cubic sublattice, with a great similarity to the former in the cation framework geometry.