International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shumeli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 3.1, pp. 404-405
Section 3.1.4. Angle between two vectors^{a}Department of Chemistry, University of Kentucky, Chemistry–Physics Building, Lexington, Kentucky 40506–0055, USA |
By (3.1.2.1) and (3.1.2.4), the angle between vectors u and v is given byAn even more concise expression of equations such as (3.1.4.1) is possible by making use of the ability of the metric tensor g to convert components from contravariant to covariant (Sands, 1982a). Thus,and (3.1.2.4) may be written succinctly asorWith this notation, the angle calculation of (3.1.4.1) becomesThe summations in (3.1.4.3), (3.1.4.4) and (3.1.4.5) include only three terms, and are thus equivalent in numerical effort to the computation in a Cartesian system, in which the metric tensor is represented by the unit matrix and there is no numerical distinction between covariant components and contravariant components.
Appreciation of the elegance of tensor formulations may be enhanced by noting that corresponding to the metric tensor g with components there is a contravariant metric tensor with componentsThe are contravariant basis vectors, known to crystallographers as reciprocal axes. Expressions parallel to (3.1.4.2) may be written, in which plays the role of converting covariant components to contravariant components. These tensors thus express mathematically the crystallographic notions of crystal space and reciprocal space [see Chapter 1.1 and Sands (1982a)].
References
Sands, D. E. (1982a). Vectors and Tensors in Crystallography. Reading: Addison Wesley. Reprinted (1995) Dover Publications.