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Twin index

  • Page ID
    19616
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    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 X, h, k, l, u, v 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 u, v and w not all odd X odd n = X
    X even n = X/2
    u, v 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
    h, k, l not all odd u+v+w even X odd n = X
    X even n = X/2
    h, k, l 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


    Twin index is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Online Dictionary of Crystallography.

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