# Hemitropy

### From Online Dictionary of Crystallography

Hémitropie (Fr). Hemitropie (Ge). Emitropia (It). Hemitropía (Sp).

The term hemitropy (or hemitropism), from the Greek ἡμι- ‎(hēmi-, half) and -τροπία ‎(-tropía) [from τρόπος ‎(trópos, a turn)], was introduced by René-Just Haüy and adopted by Auguste Bravais to indicate the geometrical relation between two individuals (domains) in a twin. These authors assumed that all twins could be explained by a binary (180° rotation) about a given direction (the twin axis) and classified twins in three types:

• normal hemitropy: the twin axis is normal to a face of the crystal, which acts both as twin plane and as composition plane;
• parallel hemitropy: the twin axis is parallel to a zone axis of the crystal and located within the composition plane; the latter is not necessarily a face of the crystal;
• complex hemitropy: the twin axis is normal to an edge of the crystal and located within the composition plane.

This classification was criticized and rejected by Mallard and later by Friedel on the basis of the following arguments:

• rotation twins occur not only by a binary rotation but also, although more rarely, by a threefold, fourfold or sixfold rotation, and they cannot be accounted for by the concept of hemitropy;
• a reflection twin can be equivalently described, from the reticular viewpoint (not necessarily from the structural viewpoint) as a rotation twin about a direction perpendicular to the twin plane only when that direction is rational. This occurs only in TLS twinning, whereas for TLQS twinning there is no rational direction perpendicular to the twin plane. Because a twin element is always a direct lattice element (Mallard's law), the concept of hemitropy simply does not apply in this case.

The classification of twinning in terms of hemitropy has been replaced by the reticular classification and today is only of historical interest. Nevertheless, it sometimes appears even in recent textbooks and articles, mainly in the mineralogical community.