4.5.3.23: Twin index

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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
P	none	none	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
		u+v and w not both even	X even	n = X/2
		u+v and w both even	X/2 odd	n = X/2
		u+v and w both even	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
		u+w and v not both even	X even	n = X/2
		u+w and v both even	X/2 odd	n = X/2
		u+w and v both even	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
		v+w and u not both even	X even	n = X/2
		v+w and u both even	X/2 odd	n = X/2
		v+w and u both even	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
		u, v and w not all odd	X even	n = X/2
		u, v and w all odd	X/2 odd	n = X/2
		u, v and w all odd	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
h, k, l not all odd	u+v+w even	X even	n = X/2
h, k, l all odd	u+v+w even	X/2 odd	n = X/2
h, k, l all odd	u+v+w even	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

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