Computation of coincident and near-coincident cells for any two lattices – related DSC-1 and DSC-2 lattices

Abstract

A computation method is presented for determining: (i) pairs of non-primitive cells M1 and M2, constructed on three translation vectors of a lattice 1 and three vectors of a lattice 2 respectively, such that the sizes of M1 and M2 are (almost) identical; (ii) Z~ (E2), defined by the number of primitive cells of lattice 1 (lattice 2) contained in M1 (M2); (iii) a characteristic relative orientation of the two lattices for which M1 and M2 coincide exactly or approximately, for which the transformation relating M1 to M2 (denoted A in general) is a pure deformation, whose principal strains are calculated; (iv) base vectors for the DSC-1 and DSC-2 lattices, so that the Burgers vectors of intrinsic phase (or grain) boundary dislocations are determined. The DSC-1 lattice is constructed by summing the vectors of lattice 1 and lattice 2', deduced from lattice 2 by A-1. The DSC-2 lattice is derived from the DSC-1 lattice by A. Tables of results are presented for a lattice 1/lattice 2 of Zn/Zn, up to ZI = Z2 = 25, and for Ni3A1 (cubic)/Ni3Nb (orthorhombic), up to E1 =21 and Z 2 = 10.

Grain boundaries in materials with a hexagonal, rhombohedral or tetragonal lattice: the connection between different treatments of approximate coincidence

“…For boundaries between grains of the same phase with a hexagonal, rhombohedral or tetragonal lattice we obtained that the rotational part of A is trivial (Ro = I), that e2 = 0, i.e. e3 =-el = e,* *The fact that 82=0 and e3 =-e, = e was stated by Bonnet, Cousineau & Warrington (1981) for grain boundaries in hexagonal materials and by Lartigue (1988) for grain boundaries in rhombohedral materials. and that e = A sin~.…”

“…Let us compare this result with the general result of Bonnet & Durand (1975) and Bonnet & Cousineau (1977), valid also for boundaries between different phases of arbitrary symmetry. They write A as…”

“…[See Warrington (1975), Bonnet, Cousineau & Warrington (1981), Bleris, Nouet, Hag~ge & Delavignette (1982), Grimmer & Warrington (1985) and Grimmer (1989b) for hexagonal lattices, Doni, Fanides & Bleris (1986) and Grimmer (1989a, d) for rhombohedral lattices, Erochine & Nouet (1983) for tetragonal lattices.] In these cases, a coincidence rotation with axis n and angle O has a multiplicity that is independent of the axial ratio r = c~ a of the lattice if either n is parallel to the principal (i. e. 6-, 3-or 4-fold) symmetry axis of the lattice or O = 180 ° and n is perpendicular 0108-7673/90/060510-05503.00 to the principal axis.…”

N-Dimensional coincidence-site-lattice theory

“…Attempts at finding the general solution of this equation were made by Santoro & Mighell (1973) and by Bonnet & Cousineau (1977); and special methods were developed for two identical cubic lattices (Grimmer, 1974b;Bleris & Delavignette, 1981) and hexagonal lattices (Bonnet et al, 1981). In the following discussion, it will be assumed that a particular rational solution X 0 of this equation has been determined.…”

“…Most of the work has been on CSL's of two identical three-dimensional lattices, especially cubic lattices (Ranganathan, 1966;Fortes, 1972;Grimmer, 1973;Grimmer, Bollmann & Warrington, 1974;Bleris & Delavignette, 1981) and hexagonal lattices (Fortes, 1973;Warrington, 1975;Bonnet, Cousineau & Warrington, 1981), although attention has also been given to the general case of two different three-dimensional lattices (Bucksch, 1972;Santoro & Mighell, 1973;Grimmer, 1976;Iwasaki, 1976;Bonnet & Cousineau, 1977;Fortes, 1977;Bacmann, 1979). These lattices are, of course, of special importance in solid-state physics and metallurgy, but recently attention has been given to lattices of higher dimension (e.g.…”