Twin index

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Indice de macle (Fr). Zwillingsindex (Ge). Indice di geminazione (It). 双晶指数 (Ja). Índice de macla (Sp).


Contents

Definition

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 the 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 X, h, k, l, u, v and w, as in the following table.

Lattice typeCondition on hklCondition on uvwCondition on Xn
PnonenoneX oddn = X
X evenn = X/2
Ch+k oddnonenonen = X
h+k evenu+v and w not both evenX oddn = X
X evenn = X/2
u+v and w both evenX/2 oddn = X/2
X/2 evenn = X/4
Bh+l oddnonenonen = X
h+l evenu+w and v not both evenX oddn = X
X evenn = X/2
u+w and v both evenX/2 oddn = X/2
X/2 evenn = X/4
Ak+l oddnonenonen = X
k+l evenv+w and u not both evenX oddn = X
X evenn = X/2
v+w and u both evenX/2 oddn = X/2
X/2 evenn = X/4
Ih+k+l oddnonenonen = X
h+k+l evenu, v and w not all oddX oddn = X
X evenn = X/2
u, v and w all oddX/2 oddn = X/2
X/2 evenn = X/4
Fnoneu+v+w oddnonen = X
h, k, l not all oddu+v+w evenX oddn = X
X evenn = X/2
h, k, l all oddu+v+w evenX/2 oddn = X/2
X/2 evenn = 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 no longer coincides 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 of 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 et 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 for Crystallography, Volume C
  • Chapter 3.3 of International Tables for Crystallography, Volume D