International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 1.6, pp. 127-128

Section 1.6.5.3. Space-group determination from powder diffraction

H. D. Flackb

1.6.5.3. Space-group determination from powder diffraction

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In powder diffraction, the reciprocal lattice is projected onto a single dimension. This projection gives rise to the major difficulty in interpreting powder-diffraction patterns. Reflections overlap each other either exactly, owing to the symmetry of the lattice metric, or approximately. This makes the extraction of the integrated intensities of individual Bragg reflections liable to error. Experimentally, the use of synchrotron radiation with its exceedingly fine and highly monochromatic beam has enabled considerable progress to be made over recent years. Other obstacles to the interpretation of powder-diffraction patterns, which occur at all stages of the analysis, are background interpretation, preferred orientation, pseudo-translational symmetry and impurity phases. These are general powder-diffraction problems and will not be treated at all in the current chapter. The reader should consult David et al. (2002[link]) and David & Shankland (2008[link]) or the forthcoming new volume of International Tables for Crystallography (Volume H, Powder Diffraction) for further information.

It goes without saying that the main use of the powder method is in structural studies of compounds for which single crystals cannot be grown.

Let us start by running through the three stages of extraction of symmetry information from the diffraction pattern described in Section 1.6.2.1[link] to see how they apply to powder diffraction.

  • (1) Stage 1 concerns the determination of the Bravais lattice from the experimentally determined cell dimensions. As such, this process is identical to that described in Section 1.6.2.1[link]. The obstacle, arising from peak overlap, is the initial indexing of the powder pattern and the determination of a unit cell, see David et al. (2002[link]) and David & Shankland (2008[link]).

  • (2) Stage 2 concerns the determination of the point-group symmetry of the intensities of the Bragg reflections. As a preparation to stages 2 and 3, the integrated Bragg intensities have to be extracted from the powder-diffraction pattern by one of the commonly used profile analysis techniques [see David et al. (2002[link]) and David & Shankland (2008[link])]. The intensities of severely overlapped reflections are subject to error. Moreover, the exact overlap of reflections owing to the symmetry of the lattice metric makes it impossible to distinguish between high- and low-symmetry Laue groups in the same family e.g. between 4/m and 4/mmm in the tetragonal family and [m\overline{3}] and [m\overline{3}m] in the cubic family. Likewise, differences in intensity between Friedel opposites, hkl and [\overline{h}\overline{k}\overline{l}], are hidden in a powder-diffraction pattern and the techniques of Section 1.6.5.1[link] are inapplicable. It is also known that experimental results on structure-factor statistics described in Section 1.6.2.2[link] are sensitive to the algorithm used to extract the integrated Bragg intensities from the powder-diffraction pattern. One procedure tends to produce intensity statistics typical of the noncentrosymmetric space group P1 and another those of the centrosymmetric space group [P\overline{1}]. In all, nothing much can be learnt from stage 2 for a powder-diffraction pattern. As a consequence, space-group determination from powder diffraction relies entirely on the Bravais lattice derived from the indexing of the diffraction pattern in stage 1 and the detection of systematic absences in stage 3.

  • (3) Stage 3 concerns the identification of the conditions for possible systematic absences. However, Bragg-peak overlap causes difficulties with determining systematic absences. For powder-diffraction peaks at small values of [\sin\theta/\lambda], the problem is rarely severe, even for low-resolution laboratory powder-diffraction data. Potentially absent reflections at higher values of [\sin\theta/\lambda] often overlap with other reflections of observable intensity. Accordingly, conclusions about the presence of space-group symmetry operations are generally drawn on the basis of a very small number of clear intensity observations. Observing lattice-centring absences is usually relatively easy. In the case of molecular organic materials, considerable help in space-group selection comes from the well known frequency distribution of space groups, where some 80% of compounds crystallize in one of the following: [P2_{1}/c], [P\overline{1}], [P2_{1}2_{1}2_{1}], [P2_{1}] and [C2/c]. Practical methods of proceeding are described by David & Sivia (2002[link]). It should also be pointed out that Table 1.6.4.1[link] in this chapter may often be found to be helpful. For example, if it is known that the Bravais lattice is of type cP, Table 1.6.4.1[link] tells us that the possible Laue classes are [m\overline{3}] and [m\overline{3}m] and the possible space groups can be found in Tables 1.6.4.25[link] and 1.6.4.26[link], respectively. The appropriate reflection conditions are of course given in these tables. All relevant tables can thus be located with the aid of Table 1.6.4.1[link] if the Bravais lattice is known.

There has been considerable progress since 2000 in the automated extraction by software of the set of conditions for reflections from a powder-diffraction pattern for undertaking stage 3 above. Once the conditions have been identified, Tables 1.6.4.2–1.6.4.30 are used to identify the corresponding space groups. The output of such software consists of a ranked list of complete sets of conditions for reflections (i.e. the horizontal rows of conditions given in Tables 1.6.4.2–1.6.4.30). Accordingly, the best-ranked set of conditions is at the top of the list followed by others in decreasing order of appropriateness. The list thus is answering the question: Which is the most probable set of reflection conditions for the data to hand? Such software uses integrated intensities of Bragg reflections extracted from the powder pattern and, as mentioned above, the results are sensitive to the particular profile integration procedure used. Moreover, only ideal Wilson (1949[link]) p.d.f.'s for space groups P1 and [P\overline{1}] are implemented. The art of such techniques is to find appropriate criteria such that the most likely set of reflection conditions is clearly discriminated from any others. Altomare et al. (Altomare, Caliandro, Camalli, Cuocci, da Silva et al., 2004[link]; Altomare, Caliandro, Camalli, Cuocci, Giacovazzo et al., 2004[link]; Altomare et al., 2005[link], 2007[link], 2009[link]) have used a probabilistic approach combining the probabilities of individual symmetry operations of candidate space groups. The approach is pragmatic and has evolved over several versions of the software. Experience has accumulated through use of the procedure and the discrimination of the software has consequently improved. Markvardsen et al. (2001[link], 2012[link]) commence with an in-depth probabilistic analysis using the concepts of Bayesian statistics which was demonstrated on a few test structures. Later, Markvardsen et al. (2008[link]) made software generally available for their approach. Vallcorba et al. (2012[link]) have also produced software for space-group determination, but give little information on their algorithm.

References

Altomare, A., Caliandro, R., Camalli, M., Cuocci, C., da Silva, I., Giacovazzo, C., Moliterni, A. G. G. & Spagna, R. (2004). Space-group determination from powder diffraction data: a probabilistic approach. J. Appl. Cryst. 37, 957–966.
Altomare, A., Caliandro, R., Camalli, M., Cuocci, C., Giacovazzo, C., Moliterni, A. G. G. & Rizzi, R. (2004). Automatic structure determination from powder data with EXPO2004. J. Appl. Cryst. 37, 1025–1028.
Altomare, A., Camalli, M., Cuocci, C., da Silva, I., Giacovazzo, C., Moliterni, A. G. G. & Rizzi, R. (2005). Space group determination: improvements in EXPO2004. J. Appl. Cryst. 38, 760–767.
Altomare, A., Camalli, M., Cuocci, C., Giacovazzo, C., Moliterni, A. G. G. & Rizzi, R. (2007). Advances in space-group determination from powder diffraction data. J. Appl. Cryst. 40, 743–748.
Altomare, A., Camalli, M., Cuocci, C., Giacovazzo, C., Moliterni, A. & Rizzi, R. (2009). EXPO2009: structure solution by powder data in direct and reciprocal space. J. Appl. Cryst. 42, 1197–1202.
David, W. I. F. & Shankland, K. (2008). Structure determination from powder diffraction data. Acta Cryst. A64, 52–64.
David, W. I. F., Shankland, K., McCusker, L. B. & Baerlocher, Ch. (2002). Editors. Structure Determination from Powder Diffraction Data. IUCr Monograph No. 13. Oxford University Press.
David, W. I. F. & Sivia, D. S. (2002). Extracting integrated intensities from powder diffraction patterns. In Structure Determination from Powder Diffraction Data, edited by W. I. F. David, K. Shankland, L. B. McCusker & Ch. Baerlocher. IUCr Monograph No. 13. Oxford University Press.
Markvardsen, A. J., David, W. I. F., Johnson, J. C. & Shankland, K. (2001). A probabilistic approach to space-group determination from powder data. Acta Cryst. A57, 47–54.
Markvardsen, A. J., David, W. I. F., Johnston, J. C. & Shankland, K. (2012). A probabilistic approach to space-group determination from powder data. Corrigendum. Acta Cryst. A68, 780.
Markvardsen, A. J., Shankland, K., David, W. I. F., Johnston, J. C., Ibberson, R. M., Tucker, M., Nowell, H. & Griffin, T. (2008). ExtSym: a program to aid space-group determination from powder diffraction data. J. Appl. Cryst. 41, 1177–1181.
Vallcorba, O., Rius, J., Frontera, C., Peral, I. & Miravitlles, C. (2012). DAJUST: a suite of computer programs for pattern matching, space-group determination and intensity extraction from powder diffraction data. J. Appl. Cryst. 45, 844–848.
Wilson, A. J. C. (1949). The probability distribution of X-ray intensities. Acta Cryst. 2, 318–321.








































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