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

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The phase shifts given in Table A1.4.4.1[link] 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[link]. 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)[link] (see also IT A, 1983[link]).

Let the direct-space transformation be given by[{\bf r}_{\rm new} = {\bi T}{\bf r}_{\rm old} + {\bf v}, \eqno(1.4.4.3)]where T is a nonsingular [3 \times 3] 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 [{\bf r}_{\rm new}] [({\bf r}_{\rm old})] 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 [({\bi P}_{\rm new}, {\bf t}_{\rm new})] and [({\bi P}_{\rm old}, {\bf t}_{\rm old})], respectively, we have[\eqalignno{({\bi P}_{\rm new}, {\bf t}_{\rm new}) &= ({\bi T}, {\bf v}) ({\bi P}_{\rm old}, {\bf t}_{\rm old}) ({\bi T}, {\bf v})^{-1} &(1.4.4.4)\cr &= ({\bi TP}_{\rm old} {\bi T}^{-1}, {\bf v} - {\bi TP}_{\rm old} {\bi T}^{-1} {\bf v} + {\bi T}{\bf t}_{\rm old}). &(1.4.4.5)}%1.4.4.5]It follows from (1.4.4.2)[link] and (1.4.4.5)[link] that if the old entry of Table A1.4.4.1[link] is given by[(n)\, {\bf h}^{T}{\bi P} : - {\bf h}^{T}{\bf t},]the transformed entry becomes[(n)\, {\bf h}^{T} {\bi TPT}^{-1} : {\bf h}^{T} {\bi TPT}^{-1}{\bf v} - {\bf h}^{T}{\bf v} - {\bf h}^{T} {\bi T}{\bf t}, \eqno(1.4.4.6)]and in the important special cases of a pure change of setting [({\bf v} = 0)] or a pure shift of the space-group origin (T is the unit matrix I), (1.4.4.6)[link] reduces to[(n)\, {\bf h}^{T} {\bi TPT}^{-1} : - {\bf h}^{T} {\bi T}{\bf t} \eqno(1.4.4.7)]or[(n)\, {\bf h}^{T} {\bi P} : {\bf h}^{T} {\bi P}{\bf v} - {\bf h}^{T}{\bf v} - {\bf h}^{T}{\bf t}, \eqno(1.4.4.8)]respectively. The rotation matrices P are readily obtained by visual or programmed inspection of the old entries: if, for example, [{\bf h}^{T}{\bi P}] is khl, we must have [P_{21} = 1], [P_{12} = 1] and [P_{33} = 1], the remaining [P_{ij}]'s being equal to zero. Similarly, if [{\bf h}^{T}{\bi P}] is kil, where [i = -h -k], we have[(kil) = (k, -h -k, l) = (hkl) \pmatrix{0 &\bar{1} &0\cr 1 &\bar{1} &0\cr 0 &0 &1\cr}.]The rotation matrices can also be obtained by reference to Part 7[link] and Tables 11.2.2.1[link] and 11.2.2.2[link] in Volume A (IT A, 2005[link]).

As an example, consider the phase shifts corresponding to the operation No. (16) of the space group [P4/nmm] (No. 129) in its two origins given in Volume A (IT A, 1983[link]). For an Origin 2-to-Origin 1 transformation we find there [{\bf v} = (\,{1 \over 4}, -{1 \over 4}, 0)] and the old Origin 2 entry in Table A1.4.4.1[link] is (16) khl (t is zero). The appropriate entry for the Origin 1 description of this operation should therefore be [{\bf h}^{T}{\bi P}{\bf v} - {\bf h}^{T}{\bf v} = k/4 - h/4 - h/4 + k/4 =] [ - h/2 + k/2], as given by (1.4.4.8)[link], or [-(h + k)/2] if a trivial shift of 2π is subtracted. The (new) Origin 1 entry thus becomes: (16) khl: [-110/2], as listed in Table A1.4.4.1[link].

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.








































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