Analysis of the structural continuity in twinned crystals in terms of pseudo-eigensymmetry of crystallographic orbits

The reticular theory of twinning gives the necessary conditions on the lattice level for the formation of twins. The latter are based on the continuation, more or less approximate, of a substructure through the composition surface. The analysis of this structural continuity can be performed in terms of the eigensymmetry of the crystallographic orbits corresponding to occupied Wyckoff positions in the structure. If [Formula: see text] is the space group of the individual and [Formula: see text] a space group which fixes the twin lattice obtained as an intersection of the space groups of the individuals in their respective orientations, then a structural continuity is obtained if (1) the eigensymmetry of an orbit under [Formula: see text] contains the twin operation; (2) the eigensymmetry of a union of orbits under [Formula: see text] contains the twin operation; (3) the eigensymmetry of a split orbit under [Formula: see text] contains the twin operation; or (4) the eigensymmetry of a union of split orbits under [Formula: see text] contains the twin operation. The case of the twins in melilite is analysed: the (approximate) restoration of some of the orbits explains the formation of these twins.