Coincidence-site lattices and complete pattern-shift in cubic crystals

Two interpenetrating point lattices contain under certain conditions a common sublattice, i.e. a 'coincidence-site lattice' (CSL). The present study is restricted to cubic lattices [primitive cubic (p.c.), f.c.c. and b.c.c.]. It is shown that it is necessary for the existence of a CSL that the two lattices are related by a rotation R represented by a rational matrix in the cubic coordinate system of one of the lattices. It is further shown that the common denominator N in R is equal to the ratio Σ of the unit volume of the CSL referred to the unit volume of the crystal lattice. The 'complete pattern-shift lattice' (DSCL) is defined as the coarsest lattice which, in CSL orientation, contains both crystal lattices as sublattices. Further it is proved that the volume of the DSCL unit referred to the crystal unit is 1/Σ and that for the p.c. structure the CSL and the DSCL are reciprocal lattices. For both f.c.c. and b.c.c. the CSL and the DSCL have to be face-centred or body-centred respectively. Methods are described for all three cases to determine explicitly the CSL, the DSCL and the planar density of coincidence sites. A table is given for R, the CSL's and the DSCL's up to Σ = 49. This study is of importance for the investigation of grain boundaries in cubic crystals.