9.23: 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 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 = N Ξ/ξ