Theory

Relations between structure types specified by their space group and occupied Wyckoff positions. Definitions of the aristotype and hettotypes for a given crystal family. Relations between the space group, structure type and structure. Construction of a descending Bärnighausen tree beginning with an aristotype and an ascending tree beginning with a hettotype. Determination of archetypes as the dead ends of the ascending Bärnighausen tree. Hettotypes with non-characteristic orbits. Equivalent structure types. Bärnighausen trees for two cases: symmetry relationships between different phases of a given compound, symmetry relationships between different structures with varied composition derived from a given parent crystal. Computer tools on the Bilbao Crystallographic server: MAXSUB, MINSUP, CELLSUB, CELLSUPER, WYCKPOS, WYCKSPLIT, WYCKSETS, SUBGROUPGRAPH

Exercices

Bärnighausen tree construction for two structure-types taken as parent structures:

1. Anatase structure type G141 (I4₁/amd) (4a, 8e). Find the minimal supergroups of the anatase space group and show that it is the archetype of the tree. Find maximal subgroups of the anatase space group, occupied Wyckoff position splittings, structure types with non-characteristic orbits. Find possible paths to the cottunite structure type G62 (Pnma) (4c, 4c+4c).
2. Wurtzite structure type G186 (P6₃mc) (2b, 2b). For the wurtzite parent structure, find possible (GaN)_m(AlN)_n superlattice families specified by one of the maximal subgroups in the Bärnighausen tree. Determine the superlattice growth direction, i.e. the direction of the unit cell multiplication. Find the possible combinations of occupations of the splitted Wyckoff positions.

Supplementary Material