Reflection conditions

Janssen, T.; Janner, A.; Looijenga-Vos, A.; Wolff, P. M. de

doi:10.1107/97809553602060000624

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 921-935

Section 9.8.3.3.2. Reflection conditions

T. Janssen,^a A. Janner,^a A. Looijenga-Vos^b and P. M. de Wolff^c ^‡

^a Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands,^bRoland Holstlaan 908, NL-2624 JK Delft, The Netherlands, and ^cMeermanstraat 126, 2614 AM, Delft, The Netherlands

9.8.3.3.2. Reflection conditions

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The indexing of diffraction vectors is a matter of choice of basis. When the basis chosen is not a primitive one, the indices have to satisfy certain conditions known as centring conditions. This holds for the main reflections (centring in ordinary space) as well as for satellites (centring in superspace). These centring conditions for reflections have been discussed in Subsection 9.8.2.1.

In addition to these general reflection conditions, there may be special reflection conditions related to the existence of non-primitive translations in the (3 + 1)-dimensional space group, just as is the case for glide planes and screw axes in three dimensions.

Special reflection conditions can be derived from transformation properties of the structure factor under symmetry operations. Transforming the geometric structure factor by an element $[g_s=(\{R|{\bf v}\}, \{R_I|\upsilon_I\})]$ , one obtains $[S_{\bf H}=S_{R^{-1}{\bf H}}\exp[-2\pi i ({\bf H}\cdot{\bf v}+H_I\cdot \upsilon_I)]. \eqno (9.8.3.10)]$ Therefore, if RH = H, the corresponding structure factor vanishes unless $[{\bf H}\cdot{\bf v}+H_I\cdot\upsilon_I]$ is an integer.

The form of such a reflection condition in terms of allowed or forbidden sets of indices depends on the basis chosen. When a lattice basis is chosen, one has $[H_s=({\bf H}, H_I)=\textstyle\sum\limits^4_{i=1}h_i{\bf a}^*_{si}, \eqno (9.8.3.11)]$ $[\upsilon_s=({\bf v}, \upsilon_I)+\textstyle\sum\limits^4_{i=1}\upsilon_i{\bf a}_{si}. \eqno (9.8.3.12)]$ Then the reflection condition becomes $[H_s\cdot\upsilon_s=\textstyle\sum\limits^4_{i=1}h_i\upsilon_i=\hbox{integer } \quad \hbox{ for }R{\bf H}={\bf H}. \eqno (9.8.3.13)]$

In terms of external and internal shift components, the reflection condition can be written as $[H_s\cdot\upsilon_s={\bf H}\cdot{\bf v}+H_I\cdot\upsilon_I={\bf H}\cdot{\bf v}+m\upsilon_I=\hbox{integer for }R{\bf H}={\bf H}. \eqno (9.8.3.14)]$ With $[{\bf H}={\bf K}+m{\bf q}]$ and $[\upsilon_I=\delta-\bf{q\cdot v}]$ , (9.8.3.14) gives $[{\bf K}\cdot{\bf v}+m\delta=\hbox{integer$\quad$ for }R{\bf H}={\bf H}. \eqno (9.8.3.15)]$ For $[{\bf v} =\upsilon_1{\bf a}+\upsilon_2{\bf b}+\upsilon_3{\bf c}]$ and $[{\bf K} =h{\bf a}^*+k{\bf b}^*+l{\bf c}^*]$ , (9.8.3.15) takes the form (9.8.3.13): $[h\upsilon _1+k \upsilon_2+l\upsilon_3+m\delta = \hbox{integer }\quad \hbox{ for }R{\bf H}={\bf H}. \eqno (9.8.3.16)]$ When the modulation wavevector has a rational part, one can choose another basis (Subsection 9.8.2.1) such that K′ = K + mq^r has integer coefficients: $[{\bf H}={\bf K}'+m{\bf q}^i=H{\bf a}^*_c + K{\bf b}^*_c + L{\bf c}^*_c+m{\bf q}^i.]$ Then, (9.8.3.15) with $[\tau=\delta-{\bf q}^r\cdot{\bf v}]$ becomes $[{\bf K}'\cdot{\bf v}+m\tau=\hbox{integer }\quad\hbox{ for }R{\bf H}={\bf H} \eqno (9.8.3.17)]$ and (9.8.3.16) transforms into $[H\upsilon'_1+K\upsilon'_2 +L\upsilon'_3 +m\tau=\hbox{integer }\quad\hbox{ for }R{\bf H}={\bf H}, \eqno (9.8.3.18)]$ in which $[\upsilon'_1,\upsilon'_2]$ , and $[\upsilon'_3]$ are the components of v with respect to the basis $[{\bf a}_c,{\bf b}_c]$ , and $[{\bf c}_c]$ .

As an example, consider a (3 + 1)-dimensional space-group transformation with R a mirror perpendicular to the x axis, $[\varepsilon=1,{\bf v}={1\over2}{\bf b}]$ , and $[\tau={1\over4}]$ with b orthogonal to a. The modulation wavevector is supposed to be $[(\,{1\over2}{1\over2}\gamma)]$ . Then $[\delta={1\over4}+{\bf q}^r\cdot{\bf v}={1\over2}]$ . The vectors H left invariant by R satisfy the relation 2h + m = 0. For such a vector, the reflection condition becomes $[{\bf K}\cdot{\bf v}+m\delta=\textstyle{1\over2}{\bf K}\cdot{\bf b}+{1\over2}m=\displaystyle{k+m\over2}={\rm integer, \quad or\ } k+m=2n.]$ For the basis $[{1\over2}({\bf a}^*+{\bf b}^*)]$ , $[{1\over2}({\bf a}^*-{\bf b}^*)]$ , c*, the rational part of the wavevector vanishes. The indices with respect to this basis are H = h + k + m, K = h − k, L = l and m. The condition now becomes $[{\bf K}'\cdot{\bf v}+m\tau={H-K+m\over4}= {\rm integer},]$ $[{\rm or}\quad H-K+m=4n, \quad {\rm for}\ K=-H.]$ Of course, both calculations give the same result: k + m = 2n for h, k, l, −2h and H − K + m = 4n for H, −H, L, m.

The special reflection conditions for the elements occurring in (3 + 1)-dimensional space groups are given in Table 9.8.3.5.

References

International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 921-935