International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.4, pp. 275-280
Section 3.4.4. Software packages for indexing and examples of their use^{a}Institute of Crystallography – CNR, Via Amendola 122/o, Bari, I-70126, Italy |
The different strategies and methods described in Section 3.4.3 have been implemented in a variety of automatic indexing programs (Bergman et al., 2004). Almost all use one of the two different approaches working in parameter space (i.e., unit-cell parameters) or index space (i.e., reflection indices). Only the EFLECH/INDEX program (Bergman, 2007), applying the scan/covariance strategy, works in both spaces: in parameter space from cubic down to monoclinic, switching to index space for triclinic. The different indexing methods are classified according to Shirley (2003) in Table 3.4.3. Alternative classifications can be made by considering whether a program works in direct or reciprocal space, or uses Bragg diffraction line positions or the whole experimental diffraction profile. Using the whole experimental diffraction profile requires a lot of computing time, but has become possible as a consequence of recent increases in the speed of computers. In the following, descriptions of the principal (default) steps of the most widely used indexing programs are given. Several non-default options are available for each program. The chances of success of the indexing step increase if more than one program is used.
This program is based on the zone-indexing strategy and uses the Runge–Ito–de Wolff–Visser method of decomposition of the reciprocal space into zones, as described in Section 3.4.3.1.1.
The following steps are executed by the program:
ITO is very efficient at indexing patterns with low symmetry and is only weakly sensitive to impurity peaks, if they occur at high angles. The most frequent causes of failure are inaccuracy or incompleteness of the input data.
Classified by Shirley (1980, 2003) as semi-exhaustive, TREOR90 is based on the index-heuristics strategy (see Section 3.4.3.1.3) and uses a trial-and-error approach. It performs the following steps:
The success of the program is related to the use of some suitable standard sets of parameter values (maximum unit-cell volume, maximum cell axis, tolerance of values etc.) arising from the accumulated experience of the authors; they can be easily changed by the user via suitable keywords in the input file.
This program works in direct space (down to the monoclinic system) by using the successive-dichotomy search method (see Section 3.4.3.1.5), which was introduced for the automatic indexing of powder diffraction patterns by Louër & Louër (1972). DICVOL91 has been defined as exhaustive by Shirley (1980, 2003). Its main steps are:
The program is fast at performing exhaustive searches in parameter space (except for the triclinic case); on the other hand, its efficiency is strongly related to the quality of the data and to the presence of impurities (in fact, impurities are not permitted).
Implemented in the EXPO program (Altomare et al., 2013) to perform the powder pattern indexing step, N-TREOR09 is an update of N-TREOR (Altomare et al., 2000), which in turn is an evolution of TREOR90, and preserves the main strategies with some changes introduced to make the program more exhaustive and powerful. In particular:
This program is also able to index powder patterns from small proteins: see Example 4 in Section 3.4.4.6.2.
The most recent of a series of versions, DICVOL14 is the successor of DICVOL04 (Boultif & Louër, 2004) and DICVOL06. DICVOL06 includes DICVOL04 with its optimized search procedure and an extended search in shells of volumes. DICVOL04 represented an improvement of DICVOL91. Among the features of DICVOL06 are:
No formal limits on the number of input Bragg peaks have been established but, for reliable indexing, it is recommended that 20 or more peaks (in the low-2θ region) are used.
Compared to DICVOL04/DICVOL06, DICVOL14 includes: an optimization of filters in the final stages of the convergence of the successive dichotomy process; an optimization and extension of scanning limits for the triclinic case; a new approach for zero-point offset evaluation; a detailed review of the input data from the resulting unit cells; and cell centring tests. DICVOL14 has been improved particularly for triclinic cases, which are generally the most difficult to solve with the dichotomy algorithm.
The indexing programs described above are based on using, for a limited number of lines, the measured positions of peak maxima as directly obtained from the experimental powder diffraction pattern. Conograph (Oishi-Tomiyasu, 2014b), which has been more recently proposed, also belongs to that group of programs. A brief description of Conograph follows
Conograph is based on the topographs method, and its main functions are the determination of the primitive unit cell and lattice symmetry, and refinement of lattice parameters. Among the main features we note:
Programs that use only the measured positions of peak maxima are particularly vulnerable to experimental errors in the measured peak positions and to the presence of impurity peaks. For these reasons, at the end of the 1990s new indexing strategies were developed that do not require the peak locations in the experimental pattern. These approaches are completely different from the methods described above because they use the whole diffraction profile. They try to explore the parameter space (direct space) exhaustively by applying different optimization techniques in order to find the cells in best agreement with the experimental powder diffraction pattern. Some of the most widely used indexing programs in direct space are described here.
The use of genetic algorithms (GAs) for indexing powder diffraction data by exploiting the diffraction geometry (as in the traditional indexing methods) was firstly proposed by Tam & Compton (1995) and Paszkowicz (1996). Subsequently, Kariuki et al. (1999) applied GA techniques by using whole profile fitting with the aim of exploring the parameter space {a, b, c, α, β, γ} and finding the global minimum of the R-factor {a, b, c, α, β, γ} hypersurface, yielding the parameter set able to generate the best agreement between the observed and calculated powder diffraction patterns.
This new strategy has been implemented in the program GAIN (Harris et al., 2000), whose main features are:
This approach is robust at handling the problems that may affect the experimental powder pattern: peak overlap, (hkl)-dependent effects and zero-point errors. It is time consuming (particularly in the case of low symmetry) but not very sensitive to the presence of minority impurity phases.
The information in the whole powder diffraction profile is exploited by the program McMaille (Le Bail, 2004), which is based on the random generation of cell parameters and uses the Monte Carlo optimization technique. Once the trial cell parameters have been generated and the Miller indices and the peak positions have been calculated, the quality of the cell is assessed by using, as figure of merit, the conventional Rietveld profile reliability factor R_{p} (Young, 1993) or McM_{20} (see Section 3.4.2.1). The program uses some tricks that can increase the success of the Monte Carlo algorithm:
This procedure is not sensitive to impurity lines, provided that the sum of their intensities is less than 10–15% of the total intensity. A zero-point error up to 0.05° is tolerated. To reduce the long computing time required to successfully complete the procedure, a significant increase in speed has been obtained by using idealized profiles generated by applying simplified line profiles to extracted line positions. A parallelized version of McMaille has also been developed. The indexing problem can usually be solved in few minutes if: (a) no triclinic symmetry is handled (because this requires more computing time); (b) the cell volume is less than 2000 Å^{3}; (c) no cell length is longer than 20 Å.
The Crysfire suite (Shirley, 2002) is a multi-program indexing facility. It can perform a self-calibration, which is aimed at detecting and correcting 2θ zero errors, and is able to strip out weak lines. Its single unified user interface and data-file format make a wide set of indexing packages accessible with minimal effort (especially to non-specialists). Crysfire provides a list of the possible cells suggested by each indexing program, suitably ranked. The Crysfire 2003 suite supports a total of 11 programs (Bergmann et al., 2004), among which are ITO, TREOR90, DICVOL91 and McMaille. The possibility of using different indexing programs, working in parameter space or index space and adopting different indexing approaches increases the probability of finding the correct cell.
This commercial indexing program (Coelho, 2003a), which uses the Monte Carlo method, is part of the TOPAS (Coelho, 2003b) suite from Bruker AXS. The reciprocal-cell parameters in equation (3.4.2) are found by using, in an iterative way, the singular value decomposition (SVD) approach (Nash, 1990) to solve linear equations relating (hkl) values to d spacings. The method is particularly useful in cases for which there are more equations than variables. All the observed lines in the powder pattern are involved in the indexing procedure. It is claimed that the program is relatively insensitive to impurity peaks and missing high d spacings; it performs well on data with large diffractometer zero errors.
More recently, two indexing methods have been introduced in TOPAS: LSI (least-squares iteration), an iterative least-squares process which operates on the d-spacing values extracted from reasonable-quality powder diffraction data, and LP-Search (lattice parameter search), a Monte Carlo based whole-powder-pattern decomposition approach independent of the knowledge of the d-spacings (Coelho & Kern, 2005).
This commercial program is part of the Materials Studio suite from Accelrys (Neumann, 2003). To perform an exhaustive search, like DICVOL, the program uses the successive-dichotomy approach. Its principal features are:
The program is described as `virtually exhaustive'; it is expected to work well when faced with missing lines, impurities and errors.
The program DICVOL06, as implemented in the WinPLOTR/FULLPROF suite (Roisnel & Rodríquez-Carvajal, 2001) and recently introduced into EXPO, was applied to two experimental diffraction patterns.
Example 1
Norbornene (Brunelli et al., 2001). Published information: C_{7}H_{10}, monoclinic, a = 7.6063 (9), b = 8.6220 (1), c = 8.749 (1) Å, β = 97.24 (1)°, P2_{1}/c, experimental range 5–60° 2θ, λ = 0.85041 Å, RES = 1.0 Å (where RES is the data resolution), synchrotron data, indexed by Fzon (Visser, 1969).
The 2θ values of the first 25 peaks, in the range 5–25°, were determined by WinPLOTR and supplied to DICVOL06. The first 20 peaks were used for searching for the solution. No plausible cell was found when assuming that no impurity was present and exploring all the systems (from cubic to triclinic). DICVOL06 was also unsuccessful when the non-default strategies of extended search and data correction for zero-point error were considered (by setting some flags to 1 in the input file). If it was supposed that two impurity lines might be present among the peaks (by setting the flag corresponding to the maximum number of accepted impurity/spurious lines to 2), DICVOL06 was able to find the following monoclinic cell: a = 8.7480 (36), b = 8.6313 (32), c = 7.6077 (26) Å, β = 97.201 (33)°, with two unindexed lines, M_{18} = 41.5, F_{18} = 125(0.0041, 35). The refinement of the cell by considering all the 25 lines gave a = 7.6087 (26), b = 8.6295 (30), c = 8.7459 (34) Å, β = 97.201 (34)°, which is very similar to the published one; 23 indexed lines, M_{20} = 30.1, F_{20} = 102.6(0.0048, 41). The presence of the two impurity lines has been ascribed by the authors to a small amount of hexagonal plastic phase.
Example 2
Cu(II)–Schiff base complex (Banerjee et al., 2002). Published information: Cu(C_{15}H_{12}NO_{2})_{2}, triclinic, a = 11.928 (4), b = 12.210 (5), c = 9.330 (5) Å, α = 102.54 (4), β = 111.16 (5), γ = 86.16 (4)°, , experimental range 6–100° 2θ, λ = 1.54056 Å, RES = 1.22 Å, high-quality X-ray laboratory data, indexed by DICVOL91. The 2θ values of the first 30 peaks, in the range 6–25°, were determined by WinPLOTR and supplied to DICVOL06. The first 20 peaks were used for searching for the solution. If it was assumed that no impurity was present, no plausible cell was found down to the monoclinic system. When the triclinic system was explored, DICVOL06 suggested only one plausible solution: a = 12.2157 (73), b = 12.2031 (77), c = 9.3071 (41) Å, α = 65.798 (46), β = 102.572 (59), γ = 95.711 (61)°, with no unindexed lines, M_{20} = 27.0, F_{20} = 77.0(0.010, 26). The refinement of the cell considering all the 30 lines gave a = 12.2125 (65), b = 12.1989 (61), c = 9.3016 (32) Å, α = 65.826 (33), β = 102.569 (40), γ = 97.755 (44)°, no unindexed lines, M_{20} = 27.9, F_{20} = 72.8(0.0106, 26). For this,the corresponding conventional cell is a = 11.93313 (61), b = 12.2125 (65), c = 9.3016 (32) Å, α = 102.569 (40), β = 111.152 (33), γ = 86.151 (44)°, similar to the published one.
Two examples of powder diffraction pattern indexing by using N-TREOR09, as implemented in the EXPO program, will be described. To activate the procedure some specific instructions must be given to EXPO via the input file or the graphical interface. As a first step, the peak-search procedure is automatically performed on the experimental powder pattern and the list of corresponding d values are supplied to N-TREOR09. During the indexing process a correction for zero-point error is automatically carried out (positive and negative shifts are taken into account). Both the examples below were successfully indexed by a default run of EXPO.
Example 3
Decafluoroquarterphenyl (Smrčok et al., 2001). Published information: C_{24}H_{8}F_{10}, monoclinic, a = 24.0519 (9), b = 6.1529 (3), c = 12.4207 (5) Å, β = 102.755 (2)°, I2/a, experimental range 7–80° 2θ, λ = 1.79 Å, RES = 1.39 Å, medium-quality X-ray laboratory data. The first 43 peaks (in the range 7–67°) with intensities greater than a default threshold were selected (an intensity-based criterion is automatically adopted). The first 25 lines were used to find a possible cell that was then refined by considering all the 43 peaks. At the end of the automatic indexing procedure, N-TREOR09 suggested two possible cells ranked according to WRIP20 [equation (3.4.5)], as shown in Fig. 3.4.1 (WRIP20 is denoted as FOMnew in N-TREOR09). The first one in the list is the correct cell. It is worth mentioning that the classical M_{20} figure of merit was not able to pick up the solution. The best cell parameters, found according to FOMnew, were a = 24.0951 (50), b = 6.1697 (21), c = 12.4578 (37) Å, β = 102.724 (18)°, similar to those reported in the literature, with FOMnew = 0.61, M_{20} = 12; all the lines in the pattern were indexed. The program provided the solution thanks to its automatic check for a zero-point correction (2θ zero shift = 0.04°) and was able to correctly identify the extinction group (I_a_). For the second suggested cell (the wrong solution) FOMnew = 0.41, M_{20} = 15, and two lines were unindexed.
Example 4
Hexagonal turkey egg-white lysozyme (Margiolaki et al., 2005). Published information: hexagonal, a = 71.0862 (3), c = 85.0276 (5) Å, P6_{1}22, experimental range 0.4–12° 2θ, RES = 3.35 Å, synchrotron data. The first 94 peaks (in the range 0.4–6°, λ = 0.700667 Å) with intensities greater than a default threshold were selected. An intensity-based criterion was automatically adopted. The first 25 lines were used to find possible cells that were then refined by considering all 94 peaks. Five possible unit cells were automatically suggested by the program in the following systems: hexagonal (1), orthorhombic (1) and monoclinic (3). The highest value for WRIP20 was 0.99, and was for the correct hexagonal cell parameters: a = 71.0922 (4), c = 85.0269 (7) Å, which are similar to those reported in the literature; all the 94 selected lines in the pattern were indexed. For this cell, the program detected a geometrical ambiguity (see Section 3.4.2.2) between hexagonal and orthorhombic lattices and automatically selected the higher-symmetry one.
References
Altomare, A., Campi, G., Cuocci, C., Eriksson, L., Giacovazzo, C., Moliterni, A., Rizzi, R. & Werner, P.-E. (2009). Advances in powder diffraction pattern indexing: N-TREOR09. J. Appl. Cryst. 42, 768–775.Google ScholarAltomare, A., Cuocci, C., Giacovazzo, C., Moliterni, A., Rizzi, R., Corriero, N. & Falcicchio, A. (2013). EXPO2013: a kit of tools for phasing crystal structures from powder data. J. Appl. Cryst. 46, 1231–1235.Google Scholar
Altomare, A., Giacovazzo, C., Guagliardi, A., Moliterni, A. G. G., Rizzi, R. & Werner, P.-E. (2000). New techniques for indexing: N-TREOR in EXPO. J. Appl. Cryst. 33, 1180–1186.Google Scholar
Andrews, L. C. & Bernstein, H. J. (1988). Lattices and reduced cells as points in 6-space and selection of Bravais lattice type by projections. Acta Cryst. A44, 1009–1018.Google Scholar
Banerjee, S., Mukherjee, A., Neumann, M. A. & Louër, D. (2002). Ab-initio structure determination of a Cu(II)-Schiff base complex from X-ray powder diffraction data. Acta Cryst. A58, c264.Google Scholar
Bergmann, J. (2007). EFLECH/INDEX – another try of whole pattern indexing. Z. Kristallogr. Suppl. 26, 197–202.Google Scholar
Bergmann, J., Le Bail, A., Shirley, R. & Zlokazov, V. (2004). Renewed interest in powder diffraction data indexing. Z. Kristallogr. 219, 783–790.Google Scholar
Boultif, A. & Louër, D. (1991). Indexing of powder diffraction patterns for low-symmetry lattices by the successive dichotomy method. J. Appl. Cryst. 24, 987–993.Google Scholar
Boultif, A. & Louër, D. (2004). Powder pattern indexing with the dichotomy method. J. Appl. Cryst. 37, 724–731.Google Scholar
Brunelli, M., Fitch, A. N., Jouanneaux, A. & Mora, A. J. (2001). Crystal and molecular structures of norbornene. Z. Kristallogr. 216, 51–55.Google Scholar
Coelho, A. A. (2003a). Indexing of powder diffraction patterns by iterative use of singular value decomposition. J. Appl. Cryst. 36, 86–95.Google Scholar
Coelho, A. A. (2003b). TOPAS. Version 3.1 User's Manual. Bruker AXS GmbH, Karlsruhe, Germany.Google Scholar
Coelho, A. A. & Kern, A. (2005). Discussion of the indexing algorithms within TOPAS. IUCr Commission on Powder Diffraction Newsletter, 32, 43–45.Google Scholar
Delaunay, B. (1933). Neue Darstellung der geometrischen Kristallographie. Z. Kristallogr. 84, 109–149.Google Scholar
Dong, C., Wu, F. & Chen, H. (1999). Correction of zero shift in powder diffraction patterns using the reflection-pair method. J. Appl. Cryst. 32, 850–853.Google Scholar
Harris, K. D. M., Johnston, R. L., Chao, M. H., Kariuki, B. M., Tedesco, E. & Turner, G. W. (2000). Genetic algorithm for indexing powder diffraction data. University of Birmingham, UK.Google Scholar
Kariuki, B. M., Belmonte, S. A., McMahon, M. I., Johnston, R. L., Harris, K. D. M. & Nelmes, R. J. (1999). A new approach for indexing powder diffraction data based on whole-profile fitting and global optimization using a genetic algorithm. J. Synchrotron Rad. 6, 87–92.Google Scholar
Le Bail, A. (2004). Monte Carlo indexing with McMaille. Powder Diffr. 19, 249–254.Google Scholar
Louër, D. & Boultif, A. (2006). Indexing with the successive dichotomy method, DICVOL04. Z. Kristallogr. Suppl. 23, 225–230.Google Scholar
Louër, D. & Boultif, A. (2007). Powder pattern indexing and the dichotomy algorithm. Z. Kristallogr. Suppl. 26, 191–196.Google Scholar
Louër, D. & Boultif, A. (2014). Some further considerations in powder diffraction pattern indexing with the dichotomy method. Powder Diffr. 29, S2, S7–S12.Google Scholar
Louër, D. & Louër, M. (1972). Méthode d'essais et erreurs pour l'indexation automatique des diagrammes de poudre. J. Appl. Cryst. 5, 271–275.Google Scholar
Margiolaki, I., Wright, J. P., Fitch, A. N., Fox, G. C. & Von Dreele, R. B. (2005). Synchrotron X-ray powder diffraction study of hexagonal turkey egg-white lysozyme. Acta Cryst. D61, 423–432.Google Scholar
Nash, J. C. (1990). Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol: Adam Hilger.Google Scholar
Neumann, M. A. (2003). X-cell: a novel indexing algorithm for routine tasks and difficult cases. J. Appl. Cryst. 36, 356–365.Google Scholar
Oishi-Tomiyasu, R. (2012). Rapid Bravais-lattice determination algorithm for lattice parameters containing large observation errors. Acta Cryst. A68, 525–535.Google Scholar
Oishi-Tomiyasu, R. (2013). Reversed de Wolff figure of merit and its application to powder indexing solutions. J. Appl. Cryst. 46, 1277–1282.Google Scholar
Oishi-Tomiyasu, R. (2014a). Method to generate all the geometrical ambiguities of powder indexing solutions. J. Appl. Cryst. 47, 2055–2059.Google Scholar
Oishi-Tomiyasu, R. (2014b). Robust powder auto-indexing using many peaks. J. Appl. Cryst. 47, 593–598.Google Scholar
Oishi-Tomiyasu, R. (2016). A table of geometrical ambiguities in powder indexing obtained by exhaustive search. Acta Cryst. A72, 73–80.Google Scholar
Paszkowicz, W. (1996). Application of the smooth genetic algorithm for indexing powder patterns – tests for the orthorhombic system. Mater. Sci. Forum, 228–231, 19–24.Google Scholar
Pecharsky, V. K. & Zavalij, P. Y. (2009). Determination and refinement of the unit cell. In Fundamentals of Powder Diffraction and Structural Characterization of Materials, 2nd ed., pp. 407–495. New York: Springer.Google Scholar
Roisnel, T. & Rodríquez-Carvajal, J. (2001). WinPLOTR: a windows tool for powder diffraction pattern analysis. Mater. Sci. Forum, 378–381, 118–123.Google Scholar
Shirley, R. (1980). Data accuracy for powder indexing. In Accuracy in Powder Diffraction, edited by S. Block & C. R. Hubbard, NBS Spec. Publ. 567, 361–382.Google Scholar
Shirley, R. (2002). User Manual, The Crysfire 2002 system for Automatic Powder Indexing. Guildford: Lattice Press.Google Scholar
Shirley, R. (2003). Overview of powder-indexing program algorithms (history and strengths and weaknesses). IUCr Comput. Comm. Newsl. 2, 48–54. http://www.iucr.org/resources/commissions/crystallographic-computing/newsletters/2 .Google Scholar
Smith, G. S. (1977). Estimating the unit-cell volume from one line in a powder diffraction pattern: the triclinic case. J. Appl. Cryst. 10, 252–255.Google Scholar
Smrčok, L., Koppelhuber-Bitschau, B., Shankland, K., David, W. I. F., Tunega, D. & Resel, R. (2001). Decafluoroquarterphenyl – crystal and molecular structure solved from X-ray powder data. Z. Kristallogr. 216, 63–66.Google Scholar
Tam, K. Y. & Compton, R. G. (1995). GAMATCH – a genetic algorithm-based program for indexing crystal faces. J. Appl. Cryst. 28, 640–645.Google Scholar
Visser, J. W. (1969). A fully automatic program for finding the unit cell from powder data. J. Appl. Cryst. 2, 89–95.Google Scholar
Werner, P.-E., Eriksson, L. & Westdahl, M. (1985). TREOR, a semi-exhaustive trial-and-error powder indexing program for all symmetries. J. Appl. Cryst. 18, 367–370.Google Scholar
Young, R. A. (1993). Introduction to the Rietveld method. In The Rietveld Method, edited by R. A. Young, pp. 1–38. Oxford: IUCr/Oxford University Press.Google Scholar