InternationalSpace-group symmetryTables for Crystallography Volume A Edited by M. I. Aroyo © International Union of Crystallography 2016 |
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. Flack
^{b} |

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) and David & Shankland (2008) 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 to see how they apply to powder diffraction.

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) p.d.f.'s for space groups *P*1 and 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; Altomare, Caliandro, Camalli, Cuocci, Giacovazzo *et al.*, 2004; Altomare *et al.*, 2005, 2007, 2009) 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, 2012) 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) made software generally available for their approach. Vallcorba *et al.* (2012) 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.*A

**64**, 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.*A

**57**, 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.*A

**68**, 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.