Tables for
Volume C
Mathematical, physical and chemical tables
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International Tables for Crystallography (2006). Vol. C, ch. 1.3, pp. 12-14

Section 1.3.4. Twinning by merohedry

E. Kocha

aInstitut für Mineralogie, Petrologie und Kristallographie, Universität Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

1.3.4. Twinning by merohedry

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A twin is called a twin by merohedry if its twin operation belongs to the point group of its vector lattice, i.e. to the corresponding holohedry. As each lattice is centrosymmetric, an inversion twin is necessarily a twin by merohedry. Only crystals from merohedral (i.e. non-holohedral) point groups may form twins by merohedry; 159 out of the 230 types of space groups belong to merohedral point groups.

For a twin by merohedry, the vector lattices of all twin components coincide in direct and in reciprocal space. The twin index is 1. The maximal number of differently oriented twin components equals the subgroup index m of the point group of the crystal with respect to its holohedry.

Table[link] displays all possibilities for twinning by merohedry. For each holohedral point group (column 1), the types of Bravais lattices (column 2) and the corresponding merohedral point groups (column 3) are listed. Column 4 gives the subgroup index m of a merohedral point group in its holohedry. Column 5 shows m − 1 possible twin operations referring to the different twin components. These twin operations are not uniquely defined (except for point group 1), but may be chosen arbitrarily from the corresponding right coset of the crystal point group in its holohedry. It is always possible, however, to choose an inversion, a reflection, or a twofold rotation as twin operation.

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Possible twin operations for twins by merohedry

m is the index of the point group in the corresponding holohedry; point groups allowing twins of type 2 are marked by an asterisk.

HolohedryBravais latticePoint groupmPossible twin operations
1 aP 1 2 [\bar 1]
2/m mP, mS 2 2 [\bar 1]
m 2 [\bar 1]
mmm oP, oS, oI, oF 222 2 [\bar1]
mm2 2 [\bar 1]
4/mmm tP, tI [^*4] 4 [\bar1, .m., .2.]
[^*\bar4] 4 [\bar1, .m.,.2.]
[^*4/m] 2 [.m.]
422 2 [\bar 1]
[4mm] 2 [\bar 1]
[\bar42m/\bar4m2] 2 [\bar 1]
[\bar3m] hR [^*3] 4 [\bar1, .m, .2]
[^*\bar3] 2 .m
32 2 [\bar1]
3m 2 [\bar1]
6/mmm hP [^*3] 8 [\bar1], .m., .2., m.., ..m, 2.., ..2
[^*\bar3] 4 .m., m.., ..m
[^*321/312] 4 [\bar1,m..,..2/.2.]
[^*3m1/31m] 4 [\bar1,m..,..m/.m.]
[^*\bar3m1/\bar31m] 2 m..
[^*6] 4 [\bar1,.m., .2.]
[^*\bar6] 4 [\bar1, .m., ..m]
[^*6/m] 2 .m.
622 2 [\bar1]
6mm 2 [\bar1]
[\bar62m/\bar6m2] 2 [\bar1]
[m\bar3m] cP, cI, cF [^*23] 4 [\bar1, ..m,..2]
[^*m\bar3] 2 ..m
432 2 [\bar1]
[\bar43m] 2 [\bar1]

A twin that is not a twin by merohedry as defined above but, because of metrical specialization, has a twin lattice with twin index 1 is called a twin by pseudo-merohedry.

Two kinds of twins by merohedry may be distinguished.

Type 1: The twin can be described as an inversion twin. Then, only two twin components exist and the twin operation belongs to the Laue class of the crystal. As a consequence, the reciprocal lattices of the twin components are superimposed so that coinciding lattice points refer to Bragg reflections with the same [|F|^2] values as long as Friedel's law is valid. In that case, no differences with respect to symmetry, or to reflection conditions, or to relative intensities occur between two sets of Bragg intensities measured from a single crystal on the one hand and from a twin on the other hand (whether or not the twin components differ in their volumes). If anomalous scattering is observed and the twin components differ in size, the intensities of Bragg reflections are changed in comparison with the untwinned crystal but the symmetry of the diffraction pattern is unchanged. For equal volumes of the twin components, however, the diffraction pattern is centrosymmetric again. The occurrence of anomalous scattering does not produce additional difficulties for space-group determination. The change of the Bragg intensities in comparison with the untwinned crystals, however, makes a structure determination more difficult.

Type 2: The twin operation does not belong to the Laue class of the crystal. Such twins can occur only in point groups marked by an asterisk in Table[link], i.e. in 55 out of the 159 types of space groups mentioned above. If the different twin components occur with equal volumes, the corresponding diffraction pattern shows enhanced symmetry. On the contrary, the reflection conditions are unchanged in comparison to those for a single crystal, except for [Pa\bar3]. As a consequence, for 51 out of the 55 space-group types, the derivation of `possible space groups', as described in IT A (2005[link], Part 3[link] ), gives incorrect results. For [P4_2/n], [I4_1/a] and [Ia\bar3], the combination of the simulated Laue class of the twin and the (unchanged) extinction symbol does not occur for single crystals. Therefore, the symmetry of these twins can be determined uniquely. In the case of [Pa\bar3], the reflection conditions differ for the two twin components. [This is because the holohedry of [Pa\bar3] is [m\bar3m] whereas the Laue class of the Euclidean normalizer [Ia\bar3] of [Pa\bar3] is [m\bar3]; cf. IT A (2005[link], Part 15[link] ).] As a consequence, the reflection conditions for such a twinned crystal differ from all conditions that may be observed for single crystals (hkl cyclically permutable: 0kl only with k = 2n or l = 2n; 00l only with l = 2n) and, therefore, the true symmetry can be identified without uncertainty.

In Table[link], all simulated Laue classes (column 1) are listed that may be observed for twins by merohedry of type 2. Column 2 shows the corresponding extinction symbols. The symbols of the simulated `possible space groups' that follow from IT A (2005[link], Part 3[link] ) are gathered in column 3. The last column displays the symbols of those space groups which may be the true symmetry groups for twins by merohedry showing such diffraction patterns.

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Simulated Laue classes, extinction symbols, simulated `possible space groups', and possible true space groups for crystals twinned by merohedry (type 2)

Twinned crystalSingle crystal
Simulated Laue classTwin extinction symbolSimulated `possible space groups'Possible true space groups
4/mmm P - - - [P422, P4mm,] [P\bar42m], [P\bar4m2,] [P4/mmm] [P4, P\bar4, P4/m]
[P4_2] - - [P4_222] [P4_2, P4_2/m]
[P4_1] - - [P4_122, P4_322] [P4_1, P4_3]
[Pn] - - [P4/nmm] [P4/n]
[P4_2/n] - - [P4_2/n]
I - - - [I422, I4mm, I\bar42m,] [I\bar4m2,] [I4/mmm] [I4, I\bar4, I4/m]
[I4_1] - - [I4_122] [I4_1]
[I4_1/a] - - [I4_1/a]
[\bar3m1] P - - - [P321, P3m1], [P\bar3m1] [P3, P\bar3]
[P3_1] - - [P3_121, P3_221] [P3_1, P3_2]
[\bar31m] P - - - [P312, P31m], [P\bar31m] [P3, P\bar3]
[P3_1] - - [P3_112, P3_212] [P3_1,P3_2]
[\bar3m] R - - [R32, R3m, R\bar3m] [R3, R\bar3]
[6/m] P - - - [P6, P\bar6, P6/m] [P3, P\bar3]
[P6_2] - - [P6_2, P6_4] [P3_1, P3_2]
[6/mmm] P - - - [P622, P6mm,] [P\bar6m2], [P\bar62m,] [P6/mmm] [P3, P\bar3, P321], [P312, P3m1], [P31m, P\bar3m1,] [P\bar31m, P6, P\bar6,] [P6/m]
[P6_3] - - [P6_322] [P6_3, P6_3/m]
[P6_2] - - [P6_222, P6_422] [P3_1, P3_2], [P3_121, P3_221,] [P3_112, P3_212], [P6_2, P6_4]
[P6_1] - - [P6_122, P6_522] [P6_1, P6_5]
P - - c [P6_3mc, P\bar62c,] [P6_3/mmc] [P31c, P{\bar 3}1c]
P - c - [P6_3cm, P\bar6c2,] [P6_3/mcm] [P3c1, P\bar3c1]
[m\bar3m] P - - - [P432,P\bar43m,] [Pm\bar3m] [P23, Pm\bar3]
[P4_2] - - [P4_{2}32] [P2_13]
[Pn] - - [Pn\bar 3 m] [Pn\bar 3]
I - - - [I432, I\bar43m,Im\bar3m] [I23, I2_13, Im\bar3]
Ia - - [Ia\bar3]
F - - - [F432, F\bar43m,] [Fm\bar3m] [F23, Fm\bar3]
Fd - - [Fd\bar3m] [Fd\bar3]
[P2_1/a,b] - - [Pa\bar3]


International Tables for Crystallography (2005). Vol. A, Space-group symmetry, edited by Th. Hahn. Heidelberg: Springer.

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