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 asSince 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), andfor 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.