Permissible angles for coincidence-site-lattice rotations

In addition to symmetry rotations (Σ = 1), lattices may admit coincidence-site-lattice (c.s.l.) rotations of various degrees of coincidence, Σ. The conditions for the occurrence of c.s.l, rotations are formulated in general terms by introducing the metric matrix of a lattice. When all lattices are considered, it is found that there are 4Σ possible values for the angle of rotation which give rise to a degree of coincidence Σ. These angles have cosines which are integral multiples of 1/2Σ. A particular lattice admits only some of these c.s.l, rotations. Cubic lattices are discussed in detail and it is shown that the number of permissible rotation angles for each odd value of Σ is approximately 3.414 × √Σ. Conversely, a particular rotation angle originates a c.s.l, of degree of coincidence Σ in those lattices which satisfy particular metric conditions. Finally, the effect of a uniform strain due to temperature or pressure changes is analysed, and it is shown that while symmetry rotations are invariant to this form of strain, only very few c.s.l, rotations are unaffected.