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. 7-8
## Section 1.1.5. Transformations |

It happens rather frequently that a vector referred to a given basis has to be re-expressed in terms of another basis, and it is then required to find the relationship between the components (coordinates) of the vector in the two bases. Such situations have already been indicated in the previous section. The purpose of the present section is to give a general method of finding such relationships (transformations), and discuss some simplifications brought about by the use of mutually reciprocal and Cartesian bases. We do not assume anything about the bases, in the general treatment, and hence the tensor formulation of Section 1.1.4 is not appropriate at this stage.

Let and be the *given* and *required* representations of the vector **r**, respectively. Upon the formation of scalar products of equations (1.1.5.1) and (1.1.5.2) with the vectors of the second basis, and employing again the summation convention, we obtain or where and . Similarly, if we choose the basis vectors , *l* = 1, 2, 3, as the multipliers of (1.1.5.1) and (1.1.5.2), we obtain where and . Rewriting (1.1.5.4) and (1.1.5.5) in symbolic matrix notation, we have leading to and and leading to and

Equations (1.1.5.7) and (1.1.5.9) are symbolic general expressions for the transformation of the coordinates of **r** from one representation to the other.

In the general case, therefore, we require the matrices of scalar products of the basis vectors, ** G**(12) and

**(22)**

*G**or*

**(11) and**

*G***(21) – depending on whether the basis or ,**

*G**k*= 1, 2, 3, was chosen to multiply scalarly equations (1.1.5.1) and (1.1.5.2). Note, however, the following simplifications.

It should be noted that the above transformations do not involve any shift of the origin. Transformations involving such shifts, notably the symmetry transformations of the space group, are treated rather extensively in Volume A
of *International Tables for Crystallography* (2005) [see *e.g.* Part 5
there (Arnold, 2005)].

This example deals with the construction of a Cartesian system in a crystal with given basis vectors of its direct lattice. We shall also require that the Cartesian system bears a clear relationship to at least one direction in each of the direct and reciprocal lattices of the crystal; this may be useful in interpreting a physical property which has been measured along a given lattice vector or which is associated with a given lattice plane. For a better consistency of notation, the Cartesian components will be denoted as contravariant.

The appropriate version of equations (1.1.5.1) and (1.1.5.2) is now and where the Cartesian basis vectors are: , and , and the vectors and are given by where and , *i*, *k* = 1, 2, 3, are arbitrary integers. The vectors and must be mutually perpendicular, . The axis of the Cartesian system thus coincides with a direct-lattice vector, and the axis is parallel to a vector in the reciprocal lattice.

Since the basis in (1.1.5.12) is a Cartesian one, the required transformations are given by equations (1.1.5.10) as where , *k*, *i* = 1, 2, 3, form the matrix of the scalar products. If we make use of the relationships between covariant and contravariant basis vectors, and the tensor formulation of the vector product, given in Section 1.1.4 above (see also Chapter 3.1
), we obtain

Note that the other convenient choice, and , interchanges the first two columns of the matrix ** T** in (1.1.5.14) and leads to a change of the signs of the elements in the third column. This can be done by writing instead of , while leaving the rest of unchanged.

### References

*International Tables for Crystallography*(2005). Vol. A.

*Space-group symmetry,*edited by Th. Hahn. Heidelberg: Springer.Google Scholar

Arnold, H. (2005).

*Transformations in crystallography*. In

*International tables for crystallography,*Vol. A.

*Space-group symmetry,*edited by Th. Hahn, Part 5. Heidelberg: Springer.Google Scholar