Permutation tensors

Sands, D. E.

doi:10.1107/97809553602060000559

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

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

Section 3.1.6. Permutation tensors

D. E. Sands^a ^*

^a Department 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 $[\eqalignno{ \varepsilon_{ijk} &= {\bf a}_{i} \cdot {\bf a}_{j} \wedge {\bf a}_{k} &(3.1.6.1)\cr \varepsilon^{ijk} &= {\bf a}^{i} \cdot {\bf a}\hskip 2pt^{j} \wedge {\bf a}^{k}. &(3.1.6.2)}%(3.1.6.2)]$ 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 $[{\bf a}^{i}, {\bf a}\hskip 2pt^{j}, {\bf a}^{k}]$ .

A discussion of the properties of the permutation tensors may be found in Sands (1982a). 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.Google Scholar

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