(ii) Its edges are along symmetry directions of the lattice. (i) Its basis vectors a, b, c define a right-handed axial setting. This is a unit cell that satisfies the three conditions below (ITA1.3.2): The translational symmetry is captured by the conventional cell. The defining operation corresponds to p = q = 0 and the symbol g (½, 0, 0) x, y,0 is replaced by the special symbol a x, y,0. For example, consider the geometric element x, y, 0 (a plane) with an element set composed of an infinite number of glide reflections g with glide vectors ( p + ½, q, 0), where p and q are integers. It specifies the name (mirror plane, glide plane, rotation axis, screw axis) and the symbol (alphanumeric and graphic) of the symmetry element. Among the symmetry operations sharing the same geometric element, the simplest one is called the defining operation of the symmetry element. A symmetry element is defined as the combination of a geometric element with the set of symmetry operations having this geometric element in common (the so-called element set) (de Wolff et al., 1989, 1992 Flack et al., 2000 ). Finally, in the case of an inversion, the inversion centre serves as geometric element. For a rotoinversion, the geometric element is the line of the corresponding rotation axis together with the unique inversion point on the axis fixed by the rotoinversion. Thus, for (glide) reflections and (screw) rotations the geometric elements are planes and lines, respectively. Except for the identity and for translations, a geometric element is attached to each symmetry operation, which is closely related to the set of fixed points of the operation. In the former case, the operation is a rotation, reflection or rotoinversion and has at least one fixed point, while in the latter case it is a screw rotation or glide reflection and does not leave any point fixed.Ī symmetry operation of an object is an isometry (congruence) which maps the object onto itself. Depending on whether their intrinsic part is zero or not, symmetry operations are of finite or infinite order. This is found to be a 180° screw rotation about a line ¼, y, ¼, parallel to the b axis but passing through x = ¼, z = ¼, with an intrinsic (screw) component ½ parallel to b. Represents an operation mapping a point with coordinates x, y, z to a point with coordinates + ½, y + ½, + ½. The column w can be decomposed into two components: an intrinsic part, which represents the screw or glide component of the operation (screw rotation or glide reflection), and a location part, which is non-zero when the rotation axis or reflection plane does not pass through the origin. A rotation, reflection or rotoinversion about the origin is represented by ( W, 0), where 0 is the zero column. A translation is represented as ( I, w), where I is the identity matrix. Our reference is Volume A of International Tables for Crystallography (Aroyo, 2016 ), whose chapters are indicated henceforth as ITA X where X is the number of the chapter.įor the following discussion we need to remind the reader that, with respect to a coordinate system, a symmetry operation is represented by a matrix–column pair ( W, w), where the (3 × 3) matrix W is called the linear or matrix part and the (3 × 1) column w is the translation or column part. In this article we present a brief panoramic overview of well known and less well known crystallographic terms in a practical and concrete approach our aim is also to help the less theoretically inclined crystallographer to understand and apply the concepts expressed by these terms. Unfortunately, incorrect definitions and sloppy terminology are not rare in textbooks and scientific manuscripts, and frequently lead to misunderstandings. Crystallography is no exception: crystal structures and their symmetry groups are arranged in a hierarchical way into classes, systems and families that emphasize common features used as classification criteria. (`La hiérarchie c'est comme une étagère, plus c'est haut, plus c'est inutile.') While this may well apply to social sciences, in exact sciences hierarchy is behind fundamental concepts like taxonomy and phylogenetics. A French proverb states that hierarchy is like shelves: the higher they are, the less useful they are.
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