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

International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 9-10

## Section 1.1.3.7. Outer product

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France

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#### 1.1.3.7.1. Definition

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The tensor defined by is called the outer product of vectors x and y. (Note: The symbol is different from the symbol for the vector product.) The analytical expression of this tensor of rank 2 is

The components of this tensor satisfy the properties It is an antisymmetric tensor of rank 2.

#### 1.1.3.7.2. Vector product

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Consider the so-called permutation tensor of rank 3 (it is actually an axial tensor – see Section 1.1.4.5.3) defined by and let us form the contracted product It is easy to check that

One recognizes the coordinates of the vector product.

#### 1.1.3.7.3. Properties of the vector product

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Expression (1.1.3.4) 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 of Volume B of International Tables for Crystallography (2008)].

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 . In the space of n dimensions, one would write

The number of independent components of is equal to or 3 in the space of three dimensions and 6 in the space of four dimensions, and the independent components of 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

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 (2008). Vol. B, Reciprocal Space, edited by U. Shmueli. Heidelberg: Springer.