International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shumeli

International Tables for Crystallography (2006). Vol. B, ch. 3.1, p. 349   | 1 | 2 |

Section 3.1.7. Components of vector product

D. E. Sandsa*

aDepartment of Chemistry, University of Kentucky, Chemistry–Physics Building, Lexington, Kentucky 40506-0055, USA
Correspondence e-mail: sands@pop.uky.edu

3.1.7. Components of vector product

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As is shown in Sands (1982a[link]), the components of the vector product [{\bf u} \wedge {\bf v}] are given by [{\bf u} \wedge {\bf v} = \varepsilon_{ijk} u^{i} v\hskip 2pt^{j} {\bf a}^{k}, \eqno(3.1.7.1)] where again [{\bf a}^{k}] is a reciprocal basis vector (some writers use [{\bf a}^{*}, {\bf b}^{*}, {\bf c}^{*}] to represent the reciprocal axes). A special case of (3.1.7.1)[link] is [{\bf a}_{i} \wedge {\bf a}_{j} = \varepsilon_{ijk} {\bf a}^{k}, \eqno(3.1.7.2)] which may be taken as a defining equation for the reciprocal basis vectors. Similarly, [{\bf a}^{i} \wedge {\bf a}\hskip 2pt^{j} = \varepsilon^{ijk} {\bf a}_{k}, \eqno(3.1.7.3)] which completes the characterization of the dual vector system with basis vectors [{\bf a}_{i}] and [{\bf a}\hskip 2pt^{j}] obeying [{\bf a}_{i} \cdot {\bf a}\hskip 2pt^{j} = \delta_{i}\hskip-1pt^{j}. \eqno(3.1.7.4)] In (3.1.7.4)[link], [\delta_{i}\hskip -1pt^{j}] is the Kronecker delta, which equals 1 if [i = j], 0 if [i \neq j]. The relationships between these quantities are explored at some length in Sands (1982a[link]).

References

Sands, D. E. (1982a). Vectors and tensors in crystallography. Reading: Addison Wesley. Reprinted (1995) Dover Publications.








































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