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

International Tables for Crystallography (2010). Vol. B, ch. 3.1, p. 405   | 1 | 2 |

Section 3.1.6. Permutation tensors

D. E. Sandsa*

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

3.1.6. Permutation tensors

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Many relationships involving vector products may be expressed compactly and conveniently in terms of the permutation tensors, defined as[\eqalignno{ \varepsilon_{ijk} &= {\bf a}_{i} \cdot {\bf a}_{j} \wedge {\bf a}_{k} &(3.1.6.1)\cr \varepsilon^{ijk} &= {\bf a}^{i} \cdot {\bf a}^{\,j} \wedge {\bf a}^{k}. &(3.1.6.2)}%(3.1.6.2)]Since [{\bf a}_{i} \cdot {\bf a}_{j} \wedge {\bf a}_{k}] represents the volume of the parallelepiped defined by vectors [{\bf a}_{i}, {\bf a}_{j}, {\bf a}_{k}], it follows that [\varepsilon_{ijk}] vanishes if any two indices are equal to each other. The same argument applies, of course, to [\varepsilon^{ijk}]. That is,[\varepsilon_{ijk} = 0,\quad \varepsilon^{ijk} = 0,\ \hbox{ if } j = i \hbox{ or } k = i \hbox{ or } k = j. \eqno(3.1.6.3)]If the indices are all different,[\varepsilon_{ijk} = PV,\quad \varepsilon^{ijk} = PV^{*} \eqno(3.1.6.4)]for even permutations of ijk (123, 231, or 312), and[\varepsilon_{ijk} = -PV,\quad \varepsilon^{ijk} = -PV^{*} \eqno(3.1.6.5)]for odd permutations (132, 213, or 321). Here, [P = +1] for right-handed axes, [P = -1] for left-handed axes, V is the unit-cell volume, and [V^{*} = 1/V] is the volume of the reciprocal cell defined by the reciprocal basis vectors [{\bf a}^{i}, {\bf a}^{\,j}, {\bf a}^{k}].

A discussion of the properties of the permutation tensors may be found in Sands (1982a[link]). In right-handed Cartesian systems, where [P = 1], and [V = V^{*} = 1], the permutation tensors are equivalent to the permutation symbols denoted by [e_{ijk}].

References

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








































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