Tables for
Volume B
Reciprocal space
Edited by U. Shumeli

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

Section 3.1.6. Permutation tensors

D. E. Sandsa*

aDepartment of Chemistry, University of Kentucky, Chemistry–Physics Building, Lexington, Kentucky 40506-0055, USA
Correspondence e-mail:

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} &(\cr \varepsilon^{ijk} &= {\bf a}^{i} \cdot {\bf a}\hskip 2pt^{j} \wedge {\bf a}^{k}. &(}%(] 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(] If the indices are all different, [\varepsilon_{ijk} = PV,\quad \varepsilon^{ijk} = PV^{*} \eqno(] for even permutations of ijk (123, 231, or 312), and [\varepsilon_{ijk} = -PV,\quad \varepsilon^{ijk} = -PV^{*} \eqno(] 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}\hskip 2pt^{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}].


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

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