Coincidence orientations of crystals in tetragonal systems, with applications to YBa2Cu3O7

We have developed a method for the characterization of coincidence-site lattices (CSL's) in tetragonal or near-tetragonal orthorhombic structures, by suitable modifications to the method of Grimmer & Warrington [Acta Cryst. (1987), A43, 232-243]. We have applied our method to determine coincidence rotations and the associated information appropriate for forming constrained CSL's in the high-To superconductor YBazCu307-n. The unit cell is orthorhombic with lattice parameters a = 3.82, b = 3"89 0108-7681/90/020117-09503.00 © 1990 International Union of Crystallography 118 C O I N C I D E N C E O R I E N T A T I O N S IN T E T R A G O N A L SYSTEMS and c = 11.67 A for the nominal composition. We present tables of coincidence rotation angles, .Z, CSL, DSCL and associated step vectors up to ,~ = 50. We find our results to be reasonable, and representative of grain-boundary structure in YBa2Cu307-a in view of observations of grainboundary dislocation networks in this material. Introduction Coincidence-site lattices (CSL's) are geometrical models of grain-boundary structure and are formed by relative rotations of two congruent lattices, using a lattice site as the origin. The ratio of the unit-cell volume of the CSL to that of the original lattice is usually denoted by 2;. The displacement-shiftcomplete lattice (DSCL), which is a lattice of vectors representing the displacements of one crystal with respect to the other that leaves the boundary structure shifted but complete, is also of importance in defining grain-boundary geometry. If the relative orientation between two grains deviates by only a few degrees from a coincidence orientation with a low value of 2, then it has often been observed that the deviation from exact coincidence is accommodated by arrays of dislocations in the boundary. Small deviations from the axial ratios needed to form the CSL are accommodated in the same way, for non-cubic materials, as demonstrated by Chen & King (1988). The Burgers vectors of such grainboundary dislocations are often vectors of the DSCL. Another important geometrical feature besides the usual CSL and DSCL is the step vector associated with a DSC dislocation. The step vector is used in determining the height of the step in the grain boundary that is associated with the core of a grainboundary dislocation. The definition of the step vector and an extensive description of its properties was given by King & Smith (1980) and King (1982). Step vectors for grain-boundary dislocations in cubic materials were tabulated by King (1982) and for h.c.p, materials by Chen & King (1987). Interest in the structure and behaviour of grain boundaries in YBa2Cu307_ 8 is increasing due to the problems of achieving sufficiently high critical current densities in polycrystalline samples. Knowledge of the possible geometries of coincidence-related grain boundaries is therefore essential, and in this paper we indicate appropriate techniques for the determination of such information, and provide tabulations of such data. Although YBa2Cu307_ 8 is orthorhombic, it is very closely approximated by a tetragonal unit cell. We presume that coincidences of the tetragonal lattices can form the basis for 'constrained-coincidence' structures in Y B a 2 C u 3 0 7 8. General theory for tetragonal structures (a) The rotational matr ix We follow the development given by Grimmer & Warrington (1987) for hexagonal lattices. In the case of tetragonal crystals we find the following. The crystal basis e consisting of three mutually orthogonal vectors el, e2, e3 of lengths a, a and c = pa, is related to the cubic b a s i s , consisting of three mutually orthogonal vectors, each of length a, by e = ES, where 1 0 0 S = 0 1 0 (1) 0 0 p Consider a right-handed rotation by an angle 0 = 2~o, 0 ___ 0 ___ rr, about an axis with cubic components v satisfying v] + v~ + v~ = 1, with parameters [a, B, Y, 6] = [cos~o, v~sin~o, v2sin~o, v3sin~0]. (2) The general form of the rotation matrix in cubic coordinates for the rotation considered above may be found in Grimmer & Warrington (1987). The crystal components n of the rotation axis are given by: