short communications Acta Cryst. (2006). A62, 411–412 doi:10.1107/S0108767306025293 411 Acta Crystallographica Section A Foundations of Crystallography ISSN 0108-7673 Received 16 March 2006 Accepted 30 June 2006 # 2006 International Union of Crystallography Printed in Great Britain – all rights reserved Recurrence properties of O-lattices and the classification of grain boundaries David Romeu* and Alfredo Go ´ mez-Rodrı ´guez Instituto de Fı ´sica, UNAM, Me ´xico. Correspondence e-mail: romeu@fisica.unam.mx.uk A recurrence relation is shown to exist between O-lattices of rotation-related grain boundaries (GBs) when a suitable parametrization of the rotation angle is introduced. This relation allows the basis vectors of any O-lattice to be calculated by a simple vector addition if the basis vectors of any two orientations are known. Its main usefulness, however, lies in the fact that it induces a partition of the angular space into disjoint sets, which groups grain boundaries into a finite number of equivalence classes, each represented by a special singular boundary (normal form). This shows that the O-lattice theory contains within it a much sought after general classification scheme for interfaces independent of the crystal system and therefore completely general. 1. Introduction Perhaps the most important goal in the grain-boundary (GB) field is to find a general theory able to relate the physical properties to the atomic structure. One problem has been the difficulty in finding a general (system-independent) theory capable of providing structural information as a function of measurable external parameters such as the crystallographic structure and relative orientation of the parent grains. One theory that deserves special mention is Bollmann’s O-lattice theory (Bollmann, 1970), which provides the dislocation content of arbitrary interfaces. Bollmann’s theory is completely derived from first principles, sustained by a solid mathematical foundation and completely general. Its main drawback, however, is that it does not provide a detailed (atomistic) picture of the interface. In the hope of determining the detailed structure of GBs, a common course of action has been to classify GBs into property-related classes such as symmetry (Pond & Bollmann, 1979; Pond & Vachlavas, 1983) and structural units configuration (Sutton & Vitek, 1980). However, in spite of these efforts, in practice GBs are still crudely classified into three main groups: low angle, special or singular and general. The purpose of this paper is to show that if the rotation angle is properly parameterized then a recurrence relation between O-lattices of rotation-related GBs emerges. The parametrization induces a partition of the angular space into an effectively finite number of disjoint angular intervals. The recurrence relation links the O-lattices of GB lying in adjacent intervals. The partition of the angular range into disjoint intervals actually groups GBs into equivalence classes; all GBs contained within a given interval belong to a class that shares structural features. This means that the O-lattice theory alone produces a classification of GBs that is independent of the crystal system and is therefore completely general. In this scenario, each equivalence class is associated with a special GB (or normal form), which is a special (singular) boundary in the sense that it contains only primary dislocations and has a particularly simple structure. 2. Angular parametrization Consider two lattices L 1 and L 2 such that L 2 ¼ RL 1 , with R denoting a rotation through an angle  around a given crystallographic axis hhkli. For any rotation angle  between L 1 and L 2 , we can always write (Romeu, 2003) tanð=2Þ¼ N 1=2  1 ; ð1Þ where N ¼ h 2 þ k 2 þ l 2 and  is a real number. As we shall see, the above parametrization has important consequences, but for the time being notice that it is useful in that it allows the separation of the axis and angle contributions into the variables N and . If we define x as the closest integer to  and  ¼   x as the fractional part of  contained in the interval ð 1 2 ; 1 2 Þ, then  ¼ x þ  ð2Þ and equation (1) becomes tanð=2Þ¼ N 1=2 1  ¼ N 1=2 1 x þ  : ð3Þ If we restrict  to take on only rational values then the previous equation reduces to the well known Ranganathan equation tanð=2Þ¼ N 1=2 p=q, where N ¼ jhhklij 2 and p, q are integers (Ranganathan, 1966) giving the possible angles between rotation- related coincidence boundaries in the cubic system. It must be noted that, owing to symmetry considerations, different integers p, q in Ranganathan’s equation may yield the same structure. For example, for rotations around h001i, N ¼ 1 and  ¼ 2 tan 1 ðp=qÞ but, since any rotation by 90  leaves the structure unchanged, the same structure is obtained for ðq  pÞ=ðq þ pÞ, ðq þ pÞ=ðq  pÞ and q=p, corresponding to the angles 90   , 90  þ  and 180   . In general, if the rotation angle is restricted to the interval ½0; =2 with  = symmetry angle of the rotation axis, then the quotient p=q is uniquely determined. Note that, while Ranganathan’s equation is valid only for the cubic case, the above parametrization is valid always. If additionally  is an integer ð ¼ 0Þ, then the resulting interface is a singular coincidence boundary that contains only primary dislocations. Conversely, if  6¼ 0, then the interface contains secondary dislocations whose spacing is a function of  (Romeu, 2003). If  is small, x þ  is a large number and its fractional part  can be neglected. Hence, although small-angle boundaries are indeed singular boundaries (recall they are composed of primary disloca- tions), they are so close together (  x) that it is no longer justifiable