General Features of Twinning
In general we can distinguish two types of twins, twins by merohedry and general
twins.
1. Twins by Merohedry
The reciprocal lattices of the different twin domains superimpose exactly. The
twinning is not detectable from the reflection pattern (no splitting of
reflections).
The twin element (symmetry element of twinning)
belongs to the holohedry of the lattice, but not to the point
group of the crystal. The holohedry is always the point group of highest symmetry
in a given crystal system:
Holohedry in the triclinic crystal system: -1
Holohedry in the monoclinic crystal system: 2/m
Holohedry in the orthorhombic crystal system: 2/m 2/m 2/m
Holohedry in the tetragonal crystal system: 4/m 2/m 2/m
Holohedry in the trigonal crystal system: -3 2/m
Holohedry in the hexagonal crystal system: 6/m 2/m 2/m
Holohedry in the cubic crystal system: 4/m -3 2/m
Example: The point group of a crystal is 2. The symmetry element
"inversion" does not belong to this point group, but to the holohedry 2/m of the
monoclinic system. The crystal of point group 2 can therefore form merohedrical
twins with the inversion as twin element.
1.1 Merohedrical Twins, Type 1
The twin element belongs to the Laue class of the crystal, but not to the
point group of the crystal. All merohedrical twins of type 1 can be described
as inversion twins. Therefore, merohedrical twinning of type 1 is only
possible in the 21 non-centrosymmetric point groups, but not in the 11
Laue groups.
Note: All merohedrical twins in the triclinic, monoclinic and
orthorhombic crystal system belong to type 1.
1.2 Merohedrical Twins, Type 2
The twin element belongs to the holohedry of the lattice, but not to the Laue
class of the crystal. (The Laue class of the crystal ist lower than the Laue
class of the lattice.)
Note: Merohedrical twinning of type 2 is only possible in the following
point groups:
tetragonal: 4, -4, 4/m
cubic: 23, m-3
trigonal P: 3, -3, 32, 3m, -3m
trigonal R: 3, -3
hexagonal: 6, -6, 6/m
2. General Twins
The reciprocal lattices of the different twin domains do not superimpose
exactly. The twinning can be detected from the reflection pattern (broad
reflections, split reflections etc.).
2.1 Pseudomerohedrical Twins
The metric specialization generates a higher symmetry. The twin element belongs
to the higher (pseudo-)symmetry of the lattice. The reflections of the different
twin components lie so close together, that they often cannot be seperated by
the resolution of the diffractometer. In my opinion, pseudo-merohedrical twins
are the most difficult cases to detect and to handle.
Some
examples of pseudo-merohedrical twins can be found here.
2.2 General Twins
The twin element does not belong to the holohedry of the crystal. Some
reflections overlap, others not.
A main problem here is the indexing of the reflections, because they belong to
two or more lattices. A valuable tool in this situation is the program
DIRAX.
If the reflections are indexed, the twin law(s) can be derived from the
orientation matrices.
Further information
Information about twins in the WWW
Literature about twinning
Lecture about twinning (Bruker User Meeting, Madison,
2003) - Powerpoint
Page maintained by Martin Lutz. Last change 19-JAN-06.
e-mail to Martin Lutz