What is a twin? No space group With space group A twin is a heterogeneous edifice built by homogeneous crystals (individuals) of the same phase in different orientations, related by an operation (the twin operation) that does not belong to the point group of the individual.

Point space vs. Vector space Point space is the ordinary space (direct space, crystal space) where we describe the crystal structure. It has an origin and N basis vectors, N being the dimensions of the space. Vector space is an abstract space whose elements are vectors. It has no origin but in it the zero vector is defined. Face normals, translation vectors, reciprocal-lattice vectors are elements of vector space. Obviously, point space and vector space are dual.

Point space vs. Vector space Morphology and crystal physical properties Crystal structure Point groups Site-symmetry groups (Wyckoff positions) Face normals, translation vectors, reciprocal-lattice vectors Space groups Vector space Point space

Space groups, vector point groups and site-symmetry groups Space groups describe the global symmetry of the crystal structure and are defined in the point space. The “usual” point groups describe the morphological symmetry, as obtained through the face normals, as well as the symmetry of the physical properties of crystals. The are defined in the vector space and are more rigorously termed vector point groups. The symmetry of a Wyckoff position is a point group defined in the point space and for this reason it is a “point point group”, usually termed site-symmetry group.

Mapping of individuals in twins Reflection in (100) Rotation about [001] Reflection in {031} (cyclic twin) Reflection in {11-22}

Twin operation, twin element, twin law Twin operation: the movement mapping the orientation of one individual onto the orientation of another individual. Twin element: the geometrical element in direct lattice (plane, axis, center) about which the twin operation is performed. Correspondingly, twins are classified as reflection twins, rotation twins and inversion twins Twin law: the set of twin operations equivalent under the point group of the individual, obtained by coset decomposition.

Twinning by merohedry All nodes are restored by the twin operation: we say that the twin index is n = 1

Twinning by pseudo-merohedry All nodes are quasi-restored by the twin operation: we say that the twin index is n = 1

Definition of obliquity w

Twinning by reticular merohedry One node out of three is restored by the twin operation: we say that the twin index is n = 3

Twinning by reticular pseudo-merohedry One node out of three is quasi-restored by the twin operation: we say that the twin index is n = 3

Zero-obliquity (reticular) pseudo-merohedry b  c c 90º b d twin misfit

Classification criteria H = point group of the individual D = holohedral point group of the individual D(Lind) = point group of the lattice of the individual H* = intersection point group of the individuals in the respective orientations D(LT) = point group of the lattice of the twin K = (chromatic) point group of the twin Tind = translation group of the individual TT = translation group of the twin

Tind = TT (I) H < D D(Lind) = D(LT) = D H*=H K > H, K  D The twin operation belongs to D, not to H Twinning by syngonic merohedry ex. H =222, D(Lind) = D(LT) = D = mmm K=m'm'm'

Tind = TT (II) H < D D(Lind) = D(LT) > D (the lattice has accidentally specialized metric) The twin operation belongs to D H*=H K > H, K  D Twinning by syngonic merohedry ex. H = 4, D = 4/mmm, D(Lind) = D(LT) = m-3m K = 4m'm'

Tind = TT (III) H  D D(Lind) = D(LT) > D (the lattice has accidentally specialized metric) The twin operation belongs to D(Lind) = D(LT) H*=H K > D, K  D(Lind) Twinning by metric merohedry ex. H = 422, D = 4/mmm, D(Lind) = D(LT) = m-3m K = (4(2,1)3(3)2(2,1))(3)

Tind = TT (IV) Quasi-symmetry of D(Lind) D(LT)  D' > D(Lind) (the lattice is pseudo-symmetric) The twin operation belongs to D' H*=H K > D(Lind), K  D' Twinning by pseudo-merohedry ex. H = 2/m, D = 2/m, b = 91º, D(Lind) = D(LT)  mmm, K = 2'/m' 2/m 2'/m'

Computation of the obliquity w [hkl]* [uvw] L*(hkl) L(uvw) (hkl) L*(hkl)L(uvw)cosw = hkl|a*b*c*abc|uvw = |hu+kv+lw| w = cos-1|hu+kv+lw|/L*(hkl)L(uvw)

D(Lind)  D(LT), Tind > TT Tind > TT (the lattice of the twin is a sublattice of the lattice of the individual) D(Lind)  D(LT) (the individual and the twin have a different point group) The twin operation belongs to D(LT) Twinning by reticular merohedry Rotation about 111

Tind > TT, Quasi-symmetry of D(LT) Tind > TT (the lattice of the twin is a sublattice of the lattice of the individual) D(LT)  D'  D(Lind) (the twin lattice is quasi-symmetric) The twin operation belongs to D(LT) Twinning by reticular pseudo-merohedry Reflection in {11-22}

D(Lind) = D(LT), Tind > TT Tind > TT (the lattice of the twin is a sublattice of the lattice of the individual) D(Lind) = D(LT) (the individual and the twin have the same point group – they differ in the orientation) The twin operation belongs to D(LT) Twinning by reticular polyholohedry

Tind > TT, Quasi-symmetry of D(LT) (2) Tind > TT (the lattice of the twin is a sublattice of the lattice of the individual) D(LT)  D' = D(Lind) (the twin lattice is quasi-symmetric) The twin operation belongs to D(LT) Twinning by reticular pseudo-polyholohedry