International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 10

Section 1.1.3.7.3. Properties of the vector product

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.3.7.3. Properties of the vector product

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Expression (1.1.3.4)[link] of the vector product shows that it is of a covariant nature. This is indeed correct, and it is well known that the vector product of two vectors of the direct lattice is a vector of the reciprocal lattice [see Section 1.1.4[link] of Volume B of International Tables for Crystallography (2001)[link]].

The vector product is a very particular vector which it is better not to call a vector: sometimes it is called a pseudovector or an axial vector in contrast to normal vectors or polar vectors. The components of the vector product are the independent components of the antisymmetric tensor [p_{ij}]. In the space of n dimensions, one would write [v_{i_{3}i_{4}\ldots i_{n}} = \textstyle{1 \over 2} \varepsilon_{i_{1}i_{2}\ldots i_{n}} p^{i_{1}i_{2}}.]

The number of independent components of [p^{ij}] is equal to [(n^{2} - n)/2] or 3 in the space of three dimensions and 6 in the space of four dimensions, and the independent components of [p^{ij}] are not the components of a vector in the space of four dimensions.

Let us also consider the behaviour of the vector product under the change of axes represented by the matrix [\pmatrix{\bar{1} & 0 & 0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}.]

This is a symmetry with respect to a point that transforms a right-handed set of axes into a left-handed set and reciprocally. In such a change, the components of a normal vector change sign. Those of the vector product, on the contrary, remain unchanged, indicating – as one well knows – that the orientation of the vector product has changed and that it is not, therefore, a vector in the normal sense, i.e. independent of the system of axes.

References

International Tables for Crystallography (2001). Vol. B. Reciprocal space, edited by U. Shmueli. Dordrecht: Kluwer Academic Publishers.








































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