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Twin index

A twin operation overlaps both the direct and reciprocal lattices of the individuals that form a twin; consequently, the nodes of the individual lattices are overlapped (restored) to some extent (twinning). The reciprocal n of the fraction 1/n of (quasi)restored nodes is called twin index

Let (hkl) be the twin plane and [uvw] the lattice direction (quasi)-normal to it. alternatively, let [uvw] be the twin axis and (hkl) the lattice plane (quasi)-normal to it. For twofold operations(180º rotations or reflections) the twin index is:

n = X/f, X = |uh+vk+wl|

where f depends on the lattice type and on the parities of Xhkluv and w, as in the following table

Lattice type condition on hkl condition on uvw condition on X n
P none none X odd n = X
X even n = X/2
C h+k odd none none n = X
h+k even u+v and w not both even X odd n = X
X even n = X/2
u+v and w both even X/2 odd n = X/2
X/2 even n = X/4
B h+l odd none none n = X
h+l even u+w and v not both even X odd n = X
X even n = X/2
u+w and v both even X/2 odd n = X/2
X/2 even n = X/4
A k+l odd none none n = X
k+l even v+w and u not both even X odd n = X
X even n = X/2
v+w and u both even X/2 odd n = X/2
X/2 even n = X/4
I h+k+l odd none none n = X
h+k+l even uv and w not all odd X odd n = X
X even n = X/2
uv and w all odd X/2 odd n = X/2
X/2 even n = X/4
F none u+v+w odd none n = X
hkl not all odd u+v+w even X odd n = X
X even n = X/2
hkl all odd u+v+w even X/2 odd n = X/2
X/2 even n = X/4


When the twin operation is a rotation of higher degree about [uvw], in general the rotational symmetry of the two-dimensional mesh in the (hkl) plane does no longer coincide with that of the twin operation. The degree of restoration of lattice nodes must now take into account the two-dimensional coincidence index Ξ for a plane of the family (hkl), which defines a super mesh in the twin lattice. Moreover, such a super mesh may exist in ξ planes out of N, depending on where is located the intersection of the [uvw] twin axis with the plane. The twin index n is finally given by:

n = NΞ/ξ

References

  • Chapter 3.1.9 in International Tables for X-Ray Crystallography (1959)

 

History

  • Friedel, G. (1904). Étude sur les groupements cristallins. Extrait du Bullettin de la Société de l'Industrie minérale, Quatrième série, Tomes III e IV. Saint-Étienne, Société de l'imprimerie Thèolier J. Thomas et C., 485 pp.
  • Friedel, G. (1926). Leçons de Cristallographie. Berger-Levrault, Nancy, Paris, Strasbourg, XIX+602 pp.

See also

Chapter 1.3 of International Tables of Crystallography, Volume C
Chapter 3.3 of International Tables of Crystallography, Volume D