InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B, ch. 1.1, pp. 5-7
## Section 1.1.4. Tensor-algebraic formulation |

The present section summarizes the tensor-algebraic properties of mutually reciprocal sets of basis vectors, which are of importance in the various aspects of crystallography. This is not intended to be a systematic treatment of tensor algebra; for more thorough expositions of the subject the reader is referred to relevant crystallographic texts (*e.g.* Patterson, 1967; Sands, 1982), and other texts in the physical and mathematical literature that deal with tensor algebra and analysis.

Let us first recall that symbolic vector and matrix notations, in which basis vectors and coordinates do not appear explicitly, are often helpful in qualitative considerations. If, however, an expression has to be evaluated, the various quantities appearing in it must be presented in component form. One of the best ways to achieve a concise presentation of geometrical expressions in component form, while retaining much of their `transparent' symbolic character, is their tensor-algebraic formulation.

We shall adhere to the following conventions:

A familiar concept but a fundamental one in tensor algebra is the transformation of coordinates. For example, suppose that an atomic position vector is referred to two unit-cell settings as follows: and Let us multiply both sides of (1.1.4.1) and (1.1.4.2), on the right, by the vectors , *m* = 1, 2, or 3, *i.e.* by the reciprocal vectors to the basis . We obtain from (1.1.4.1) where is the Kronecker symbol which equals 1 when and equals zero if , and by comparison with (1.1.4.2) we have where is an element of the required transformation matrix. Of course, the same transformation could have been written as where .

A *tensor* is a quantity that transforms as the product of coordinates, and the *rank* of a tensor is the number of transformations involved (Patterson, 1967; Sands, 1982). *E.g.* the product of two coordinates, as in the above example, transforms from the **a**′ basis to the **a** basis as the same transformation law applies to the components of a contravariant tensor of rank two, the components of which are referred to the primed basis and are to be transformed to the unprimed one:

The expression for the scalar product of two vectors, say **u** and **v**, depends on the bases to which the vectors are referred. If we admit only the covariant and contravariant bases defined above, we have four possible types of expression:

There are numerous applications of tensor notation in crystallographic calculations, and many of them appear in the various chapters of this volume. We shall therefore present only a few examples.

### References

*International Tables for Crystallography*(2004). Vol. C.

*Mathematical, physical and chemical tables,*edited by E. Prince. Dordrecht: Kluwer Academic Publishers.

Patterson, A. L. (1967). In

*International tables for X-ray crystallography*, Vol. II.

*Mathematical tables,*edited by J. S. Kasper & K. Lonsdale, pp. 5–83. Birmingham: Kynoch Press.

Sands, D. E. (1982).

*Vectors and tensors in crystallography.*New York: Addison-Wesley.

Schomaker, V. & Trueblood, K. N. (1968).

*On the rigid-body motion of molecules in crystals. Acta Cryst.*B

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