ANALYSIS OF EXTRA O-POINTS FOUND IN LOW ANGLE 110 TWIST BOUNDARIES: A GENERAL SOLUTION FOR O-LATTICES IN THREE DIMENSIONS D. Romeu and A. Go ´mez Instituto de Fisica, Universidad Nacional Autonoma de Me ´xico, Apartado Postal 20-364, Mexico, D.F. 01000, Mexico (Received December 7, 2000) (Accepted in revised form February 5, 2001) Keywords: Grain boundaries; Dislocations 1. Introduction Since its discovery by Bollmann [1], the O-lattice has played a central role in the field of interfaces given its ability to predict the dislocation structure of interfaces. Although its definition is very simple (see Eq. 1 below), its interpretation is not always straightforward and its casual use may lead to confusion. For example, Goodhew et al. [4] have observed dislocation networks in twist grain boundaries that show “extra” O-points in addition to those predicted by Eq. 1. In order to explain these, they introduce an ad-hoc inhomogeneous term (lattice translation) that explains the set of additional O-points but not the original (homogeneous) set, so that in order to account for all observed O-points, Eq. 1 has to be solved simultaneously for two different interfaces (with and without translation). The confusion arises because the approach normally taken to deal with the singular nature of the transformation T in Eq. 1 may lead to an under estimation of the number of O-points in some interfaces. In this paper we show that this problem can be avoided by using a general solution of Eq. 1 in three dimensions (3D) developed here for the O-lattice of rotation related lattices. The procedure, based on a previous numerical pseudo inverse approach [2], accounts for all possible O-points without the need of ad-hoc transformations, and provides a deeper insight into the physics and geometry of the problem. The results of Goodhew et al. are revisited to show the applicability of the equations, and the physical significance of the additional O-points is analyzed with the aid of a recently introduced structural model [2,6]. Also, as a corollary of the method, an expression for the inverse of the Frank’s formula is presented. 2. The General Solution for O-lattices of Rotations 2.1. The Displacement Field in 2 Dimensions For completeness, we will briefly review here the equations and assumptions made in deriving the O-lattice. Given two crystal lattices, L (1) , L (2) and a transformation A (restricted here to be a rotation A = R but the same ideas can be straightforwardly extended to include other transformations), such that L (2) = RL (1) , in the absence of translations, the O-lattice is defined as the set of points x (0) that satisfy Scripta mater. 44 (2001) 2617–2622 www.elsevier.com/locate/scriptamat 1359-6462/01/$–see front matter. © 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S1359-6462(01)00970-8