InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 1.4, p. 120
## Section 1.4.4.3. Effect of direct-space transformations U. Shmueli
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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 bywhere ** T** is a nonsingular matrix describing the change of the coordinate system and

**v**is an origin-shift vector. The components of

**and**

*T***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 haveIt follows from (1.4.4.2) and (1.4.4.5) that if the old entry of Table A1.4.4.1 is given bythe transformed entry becomesand in the important special cases of a pure change of setting or a pure shift of the space-group origin (

**is the unit matrix**

*T***), (1.4.4.6) reduces toorrespectively. The rotation matrices**

*I***are readily obtained by visual or programmed inspection of the old entries: if, for example, is**

*P**khl*, we must have , and , the remaining 's being equal to zero. Similarly, if is

*kil*, where , we haveThe 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.