International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.4, p. 120   | 1 | 2 |

## Section 1.4.4.3. Effect of direct-space transformations

U. Shmuelia

#### 1.4.4.3. Effect of direct-space transformations

| top | pdf |

The phase shifts given in Table A1.4.4.1 depend on the translation parts of the space-group operations and these translations are determined, all or in part, by the choice of the space-group origin. It is a fairly easy matter to find the phase shifts that correspond to a given shift of the space-group origin in direct space, directly from Table A1.4.4.1 . Moreover, it is also possible to modify the table entries so that a more general transformation, including a change of crystal axes as well as a shift of the space-group origin, can be directly accounted for. We employ here the frequently used concise notation due to Seitz (1935) (see also IT A, 1983 ).

Let the direct-space transformation be given by where T is a nonsingular matrix describing the change of the coordinate system and v is an origin-shift vector. The components of T and v are referred to the old system, and  is the position vector of a point in the crystal, referred to the new (old) system, respectively. If we denote a space-group operation referred to the new and old systems by and , respectively, we have It follows from (1.4.4.2) and (1.4.4.5) that if the old entry of Table A1.4.4.1 is given by the transformed entry becomes and in the important special cases of a pure change of setting or a pure shift of the space-group origin (T is the unit matrix I), (1.4.4.6) reduces to or respectively. The rotation matrices P are readily obtained by visual or programmed inspection of the old entries: if, for example, is khl, we must have , and , the remaining 's being equal to zero. Similarly, if is kil, where , we have The rotation matrices can also be obtained by reference to Part 7 and Tables 11.2.2.1 and 11.2.2.2 in Volume A (IT A, 2005 ).

As an example, consider the phase shifts corresponding to the operation No. (16) of the space group (No. 129) in its two origins given in Volume A (IT A, 1983 ). For an Origin 2-to-Origin 1 transformation we find there and the old Origin 2 entry in Table A1.4.4.1 is (16) khl (t is zero). The appropriate entry for the Origin 1 description of this operation should therefore be  , as given by (1.4.4.8) , or if a trivial shift of 2π is subtracted. The (new) Origin 1 entry thus becomes: (16) khl: , as listed in Table A1.4.4.1 .

### References

International Tables for Crystallography (1983). Vol. A, Space-Group Symmetry, edited by Th. Hahn. Dordrecht: Reidel.
International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 5th ed., corrected reprint. Heidelberg: Springer.
Seitz, F. (1935). A matrix-algebraic development of crystallographic groups. III. Z. Kristallogr. 90, 289–313.