International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shumeli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B, ch. 3.1, p. 349
Section 3.1.6. Permutation tensors^{a}Department of Chemistry, University of Kentucky, Chemistry–Physics Building, Lexington, Kentucky 40506-0055, USA |
Many relationships involving vector products may be expressed compactly and conveniently in terms of the permutation tensors, defined as Since represents the volume of the parallelepiped defined by vectors , it follows that vanishes if any two indices are equal to each other. The same argument applies, of course, to . That is, If the indices are all different, for even permutations of ijk (123, 231, or 312), and for odd permutations (132, 213, or 321). Here, for right-handed axes, for left-handed axes, V is the unit-cell volume, and is the volume of the reciprocal cell defined by the reciprocal basis vectors .
A discussion of the properties of the permutation tensors may be found in Sands (1982a). In right-handed Cartesian systems, where , and , the permutation tensors are equivalent to the permutation symbols denoted by .
References
Sands, D. E. (1982a). Vectors and tensors in crystallography. Reading: Addison Wesley. Reprinted (1995) Dover Publications.