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. 270281
https://doi.org/10.1107/97809553602060000949 Chapter 3.4. Indexing a powder diffraction pattern^{a}Institute of Crystallography – CNR, Via Amendola 122/o, Bari, I70126, Italy This chapter provides a general overview of the process of identifying cell parameters from a powder diffraction pattern. The main theoretical, computational and applicative approaches are described. In particular, the chapter deals with the basic concepts of a crystalline lattice, indexing equations, figures of merit, and traditional and nontraditional methods for solving the indexation problem. The most widely used indexing programs are discussed and examples of their applications are given. 
The crystal structure solution process presupposes that the crystal cell and the space group are known. In other words, the first step in the solution pathway is the identification of the unitcell parameters. Knowledge of the crystal structure strongly depends on the determination of the cell: a cell incorrectly defined does not lead to the solution. The celldetermination process (which operates in a 6dimensional continuous parameter space) is also called `indexing' because it consists of assigning the appropriate triple (hkl) of Miller indices of the lattice plane to each of the N_{l} experimental diffraction lines (in a 3N_{l}dimensional integervalued index space) (Shirley, 2003). (In this chapter, `line' and `peak' are used synonymously.) In the case of powder diffraction, the determination of the cell parameters is not a trivial task, and it is much more difficult than in the singlecrystal case. This is because the information about the threedimensional reciprocal space is compressed into the onedimensional experimental powder pattern. Whatever the method used, working in the parameter or index space, powder pattern indexing aims to recover the threedimensional information from the positions of the diffraction peaks in the observed profile. In particular, the experimental information used for carrying out the indexing process is the d_{hkl} interplanar spacings, which are related to the diffraction angles by the well known Bragg law:
In theory, if we had available an experimental pattern at infinite resolution with well resolved peaks with no overlapping, the determination of the six cell parameters corresponding to a problem with six degrees of freedom would be easy (Shirley, 2003). In practice, only the first 20–30 observed lines are useful for two main reasons: (1) they are less sensitive to small changes in the cell parameters than the higherangle lines; and (2) higherangle lines (even if they seem to be single peaks) actually often consist of more than one overlapping peak and their positions cannot be accurately evaluated. Using higherangle lines is therefore unwise. The successful outcome of powder pattern indexing is correlated to which and how many d_{hkl} values derived from the peaks in the diffraction pattern are selected and how reliable they are. Precision and accuracy in detecting peak positions are essential conditions for successful indexing (Altomare et al., 2008). Unfortunately they can be degraded by different sources of errors: peak overlap, poor peak resolution, 2θ zero shift, errors in measurement, and a low peaktobackground ratio. Moreover, impurity lines (i.e. peaks from a different chemical phase in the sample to the compound being studied) can hinder the attainment of the correct result. The history of indexing, having its origin in the early 20th century (Runge, 1917), has produced several methods and software packages (Shirley, 2003; Werner, 2002) with surprising progress in strengthening and automating the celldetermination process. Innovative approaches that aim to reduce the dependence on the d_{hkl} values (by avoiding the peaksearch step and considering the full information contained in the diffraction pattern) have also been developed. Despite great advances, powder pattern indexing is still a challenge in many cases. Factors that affect the success or failure of the process include: the presence of diffraction peaks from unexpected phases, the precision in the peakposition value, the size of the unit cell to be identified (indexing is easier if the unit cell is not too big) and the symmetry (indexing a pattern from a compound with high symmetry is generally more reliable than for a compound with lower symmetry). Before the zeroth step of the indexing process (the searching for peaks in the experimental pattern) it is always necessary to obtain goodquality diffraction data. Of course, the use of synchrotron radiation is preferable, but conventional laboratory Xray data are usually suitable. Whether automated or manual, the peak search and each successive step of the indexing process must be carefully checked. For example, in a first attempt the positions that correspond to overlapping peaks could be set aside. If one attempt fails, the most useful tactic is to try another software package, since the programs available at present are based on different approaches.
The aim of this chapter is first to illustrate the background of the topic and the main theoretical approaches used to carry out the powder pattern indexing, and then to give some examples of applications. Section 3.4.2 is mainly devoted to the basic concepts of a crystalline lattice, the main indexing equations and figures of merits; Section 3.4.3 discusses the traditional and nontraditional methods developed for indexing a powder pattern, and Section 3.4.4 discusses some applications, referring to the most widely used indexing programs.
We now describe some concepts that are fundamental in crystallography and useful for understanding the indexing process. The measured diffraction intensities correspond to the reciprocallattice pointsThe Miller indices (hkl) identify the plane of the direct lattice and , and are the three vectors of the reciprocal lattice, which are related to the direct lattice bywhereis the reciprocalcell volume ( is the inverse of the directunitcell volume V).
In case of singlecrystal data, the threedimensional nature of the experimental diffraction data makes it easy to identify , from which the directspace unitcell vectors are derived (Giacovazzo, 2011).
In case of powder diffraction, the threedimensional nature of the diffraction data is compressed into one dimension in the experimental pattern, and the accessible experimental information is the d_{hkl} values involved in the Bragg law and related to the diffraction angles byd_{hkl}, the spacing of the planes (hkl) in the direct lattice, is obtained by the dot products of the reciprocallattice vectors with themselves:where is the angle between and , is the angle between and , and is the angle between and . If we introduce[where Q(hkl) differs from by a scale factor of (200/λ)^{2}], (3.4.1) becomeswhereThe number of parameters in (3.4.2) depends on the type of symmetry: from 1 in the case of cubic symmetry to 6 for triclinic symmetry (see Table 3.4.1).

The quadratic form (3.4.2) relates the observed Q(hkl) values to the reciprocal cell parameters and, consequently, to the direct cell. It is the basic equation used in powderindexing procedures. Therefore the indexing problem (Werner, 2002) is to find A_{ij} and, for each observed Q(hkl) value, three crystallographic indices (hkl) satisfying (3.4.2) within a suitable tolerance parameter Δ:The importance of using accurate Q(hkl) values in (3.4.3) is obvious. Moreover, it is worth noticing that (3.4.3) must lead to physically reasonable indexing – lowangle peaks should correspond to small integer values for h, k and l and the values of the cell parameters and cell volume should be reasonable.
An important task is the introduction of a figure of merit (FOM) that is able to (a) describe the physical plausibility of a trial cell and its agreement with the observed pattern, and (b) select the best cell among different possible ones. de Wolff (1968) made an important contribution in this direction. He developed the M_{20} figure of merit defined bywhere Q_{20} is the Q value corresponding to the 20th observed and indexed peak, N_{20} is the number of different calculated Q values up to Q_{20}, and is the average absolute discrepancy between the observed and the calculated Q values for the 20 indexed peaks; the factor 2 is a result of statistics, explained by the larger chance for an observed line to sit in a large interval as compared with sitting in a small interval. The rationale behind M_{20} is as follows: the better the agreement between the calculated and the observed peak positions (the smaller the value) and the smaller the volume of the unit cell (the smaller the N_{20} value), the larger the M_{20} value and consequently the confidence in the proposed unit cell. A rule of thumb for M_{20} is that if the number of unindexed peaks whose Q values are less than Q_{20} is not larger than 2 and if M_{20} > 10, then the indexing process is physically reasonable (de Wolff, 1968; Werner, 2002). This rule is often valid, but exceptions occur. The use of the first 20 peaks is a compromise (coming from experience) between introducing a quite large number of observed peaks (depending on the number of parameters of the unit cell) and avoiding the use of highangle peak positions, which are more affected by errors. M_{20} is statistically expected to be 1 in case of completely arbitrary indexing. It has no upper limit (it can be very large when is very small).
Smith & Snyder (1979) proposed the F_{N} criterion in order to overcome the limits of M_{20} with respect to its dependence on the 20 lines and on crystal class and space group. The F_{N} figure of merit is given bywhere is the average absolute discrepancy between the observed and calculated 2θ peak position values and N_{poss} is the number of possible diffraction lines up to the Nth observed line. The values of and N_{poss}, (, N_{poss}), are usually given with F_{N}. With respect to M_{20}, F_{N} is more suitable for ranking the trial solutions and less for indicating their physical plausibility (Werner, 2002).
Both M_{20} and F_{N}, being based on the discrepancies between observed and calculated lines, are less reliable if there are impurity peaks; if the information about the unindexed lines is not taken into account, the risk of obtaining false solutions increases. Alternative FOMs based on joint probability have also been proposed (Ishida & Watanabe, 1967, 1971). Among the recently developed FOMs, we mention:
Before discussing the concept of geometrical ambiguity in indexing, it is useful to introduce the definition of a reduced cell. While a unit cell defines the lattice, a lattice can be described by an unlimited number of cells. The Niggli reduced cell (Niggli, 1928) is a special cell able to uniquely define a lattice. Methods and algorithms have been derived for identifying the reduced cell starting from an arbitrary one (Buerger, 1957, 1960; Santoro & Mighell, 1970; Mighell, 1976, 2001). The reduced cell has the advantage of introducing a definitive classification, making a rigorous comparison of two lattices possible in order to establish whether they are identical or related (Santoro et al., 1980). An algorithm based on the conversetransformation theory has been developed and implemented in the Fortran program NIST*LATTICE for checking relationships between any two cells (Karen & Mighell, 1991).
It is very important to recognize that two lattices are derivative of each other, because many crystallographic problems (twinning, indexing of powder patterns, singlecrystal diffractometry) stem from the derivative properties of the lattices. Derivative lattices are classified as super, sub or composite according to the transformation matrices that relate them to the lattice from which they are derived (Santoro & Mighell, 1972).
A further obstacle to the correct indexing of a powder pattern is the problem of geometrical ambiguities. It may occur when `two or more different lattices, characterized by different reduced forms, may give calculated powder patterns with the identical number of distinct lines in identical 2θ positions' (Mighell & Santoro, 1975). The number of planes (hkl) contributing to each reflection may differ, however. Such ambiguity, due to the fact that the powder diffraction pattern only contains information about the length of the reciprocallattice vector and not the threedimensional vector itself, is geometrical. It mainly occurs for highsymmetry cells (from orthorhombic up). The lattices having this property are related to each other by rotational transformation matrices. In Table 3.4.2 some examples of lattices giving geometrical ambiguities and the corresponding transformation matrices are given (Altomare et al., 2008). Where there are geometrical ambiguities, additional prior information (e.g., a singlecrystal study) may be useful in order to choose one of the two possible lattices.

A recent procedure developed by Kroll et al. (2011) aims to reveal numerical and geometrical relationships between different reciprocal lattices and unit cells. The procedure is based on the assumption that distinct unit cells with lines in the same 2θ positions are derivatives of each other. However, two nonderivative lattices can have identical peak positions. Very recently, OishiTomiyasu (2014a, 2016) has developed a new algorithm able to obtain all lattices with computed lines in the same positions as a given lattice. (See also Section 3.4.4.3.)
Indexing methods aim to reconstruct the threedimensional direct lattice from the onedimensional distribution of d_{hkl} values. Systematic or accidental peak overlap, inaccuracy of peak positions, zero shift in the 2θ_{hkl} Bragg angles and/or the presence of impurity peaks make the reconstruction difficult. Data accuracy is fundamental for increasing the probability of success; as emphasized by de Wolff: `The `indexing problem' is essentially a puzzle: it cannot be stated in rigorous terms (…). It would be quite an easy puzzle if errors of measurements did not exist' (de Wolff, 1957).
Different approaches have been proposed for solving the indexing puzzle since the pioneering work of Runge (1917). As suggested by Shirley (2003), indexing procedures work in parameter space, or in index space, or in both spaces. As a general consideration, the parameter space allows the inclusion of the cell information and constraints, while the index space is more suitable in cases where there are accidental or systematic absences (Shirley, 1980). In this section an outline of the strategies and search methods adopted by the main traditional and nontraditional indexing approaches is given. For more details see the papers by Shirley (2003) and Bergmann et al. (2004).
Among the main indexing procedures, zone indexing (Section 3.4.3.1.1), SIW heuristic (Section 3.4.3.1.2), successive dichotomy (Section 3.4.3.1.5), the topographs method (Section 3.4.3.2.1) and globaloptimization methods (Section 3.4.3.2.2) operate in the parameter space; index heuristics (Section 3.4.3.1.3) and index permutation (Section 3.4.3.1.4) work in the index space; and scan/covariance (Bergmann, 2007) operates both in index and parameter space. Each method can be classified as exhaustive or not. An exhaustive method systematically and rigorously searches in the solution space; a nonexhaustive method exploits coincidences and relations between the observed lines with the aim of finding the solution quickly. The classification is not rigorous: approaches that try to combine rigour and speed can be defined as semiexhaustive (Table 3.4.3).

Indexing procedures can also be classified as traditional and nontraditional. Each indexing method generates a list of possible cells. Their reliability is assessed by FOMs with the aim of selecting the correct one (see Section 3.4.2.1).
The traditional indexing approaches adopted over the last century are based on the following strategies and search methods: (1) zone indexing, (2) SIW heuristic, (3) index heuristics, (4) index permutation and (5) successive dichotomy. All of them exploit information about a limited number of observed peak positions.
The zoneindexing strategy was originally developed by Runge (1917), successively proposed by Ito (1949, 1950), generalized by de Wolff (1957, 1958) and enhanced by Visser (1969). This approach is based on the search for zones, i.e., crystallographic planes, in the reciprocal lattice, defined by the origin O and two lattice points. If and are two vectors in reciprocal space, i.e. the positional vectors of the lattice points A and A′, they describe a zone containing any lattice point B whose positional vector is of type where m and n are positive integers. If ω is the angle between and , the squared distance of B from O (i.e., ) can be expressed by (de Wolff, 1958; Visser, 1969)where and are the squared distances of A and A′ from O, respectively, and . R can be derived as
The method is applied as follows: and are chosen among the first experimental Q_{i} values; the {Q_{i}}, up to a reasonable resolution, are introduced in (3.4.7) in place of ; and a few positive integer values are assigned to m and n. Equation (3.4.7) provides a large number of R values; equal R values (within error limits) define a zone, for which the ω angle can easily be calculated. The search for zones is performed using different () pairs. The R values that are obtained many times identify the most important crystallographic zones. The zones are sorted according to a quality figure, enabling selection of the best ones. In order to find the lattice, all possible combinations of the best zones are tried. For every pair of zones the intersection line is found, then the angle between them is determined and the lattice is obtained.
The method has the advantage of being very efficient for indexing lowsymmetry patterns. The main disadvantage is its sensitivity to errors in the peak positions, particularly in the low 2θ region.
This needs only one single wellestablished zone, then it arbitrarily chooses the 001 line from the firstlevel lines. The indexing problem is thus lowered to two dimensions and an exhaustive search is carried out.
The indexheuristics strategy searches for the correct cell via a trialanderror approach, assigning tentative Miller indices to a few experimental peak positions (basis lines), usually belonging to the low 2θ region of the experimental pattern. It was first proposed by Werner (1964), then successively refined (Werner et al., 1985) and made more robust and effective (Altomare et al., 2000, 2008, 2009). This approach, which works in the index space, was defined by Shirley as semiexhaustive (Shirley, 1980). The search starts from the highestsymmetry crystal system (cubic) and, if no plausible solution is found, it is extended to lower symmetry down to triclinic. The number of selected basis lines increases as the crystal symmetry lowers. A dominant zone occurs when one cell axis is significantly shorter than the other two; in this case most of the first observed lines (in terms of increasing 2θ_{hkl} values) can be indexed with a common zero Miller index. Special shortaxis tests, aimed at finding twodimensional lattices, have been proposed for monoclinic symmetry in order to detect the presence of dominant zones (Werner et al., 1985). The indexheuristics method is based on the main indexing equation [see equation (3.4.2)] that can be rewritten (Werner et al., 1985) aswhere {x_{i}} = X is the vector of unknown parameters, which are derived by solving a system of linear equationswhere M is a matrix of Miller indices and Q is the vector of the selected Q(hkl) values corresponding to the basis lines. The dimensions of M, X and Q change according to the assumed symmetry. From the inverse matrix M^{−1} the corresponding X is obtained via X = M^{−1}Q. In the case of monoclinic and higher symmetry, {x_{i}} are calculated by Cramer's rule. Different X vectors are derived by using a different selection of basis lines. The possible solutions are checked by using the full list of peak positions (up to the first 25 experimental lines). The method is sensitive to errors on peak positions and to the presence of impurities (the presence of only one impurity peak is not critical). The correctness of the {x_{i}} strongly depends on the accuracy of the observed Q values, especially for low2θ region lines, which are the most dominant ones for this indexing procedure. The possibility of testing different combinations of basisline sets enables the correct cell to be found by bypassing the cases for which errors in the basis lines occur.
The method has been recently enhanced (Altomare et al., 2000, 2009) by introducing new procedures that are able to increase the probability of successful indexing (see Section 3.4.4.2.1); among them are: (1) a correction for systematic errors in the experimental 2θ values (positive and negative trial 2θ zero shifts are taken into account); this correction should, in principle, describe a real diffractometer error; in practice, it also approximates the specimen displacement error well (perhaps coupled with transparency for organic samples); (2) a more intensive search in solution space for orthorhombic and monoclinic systems; (3) an improvement of the triclinic search; (4) a new figure of merit, WRIP20, which is more powerful than M_{20} in identifying the correct solution among a set of possible ones (see Section 3.4.2.1); (5) a check for geometrical ambiguities; (6) an automatic refinement of the possible cells; and (7) a statistical study of the parity of the Miller indices, performed at the end of the cell refinement, aimed at detecting doubled axes or additional lattice points (for A, B, C, I, R or Fcentred cells) (such information is used in the successive steps).
This strategy was proposed by Taupin (1973), and is based on a systematic permutation of indices associated to observed lines for obtaining candidate cells. Because this trialanderror strategy is similar to the indexheuristics approach, we do not describe it here.
The successivedichotomy method, first developed by Louër & Louër (1972), is based on an exhaustive strategy working in direct space (except for triclinic systems, where it operates in reciprocal space) by varying the lengths of the cell axes and the interaxial angles within finite intervals. The search for the correct cell is performed in an ndimensional domain D (where n is the number of cell parameters to be determined). If no solution belongs to D, the domain is discarded and the ranges for the allowed values of cell parameters are increased; on the contrary, if D contains a possible solution, it is explored further by dividing the domain into 2^{n} subdomains via a successivedichotomy procedure. Each subdomain is analyzed and discarded if it does not contain a solution. The method was originally applied to orthorhombic and highersymmetry systems (Louër & Louër, 1972), but it has been successively extended to monoclinic (Louër & Vargas, 1982) and to triclinic systems (Boultif & Louër, 1991). The search can be performed starting from cubic then moving down to lower symmetries (except for triclinic) by partitioning the space into shells of volume ΔV = 400 A^{3}. For triclinic symmetry ΔV is related to the volume V_{est} suggested by the method proposed by Smith (1977), which is able to estimate the unitcell volume from only one line in the pattern:where d_{N} is the value for the Nth observed line; in the case N = 20 the triclinic cell volume is .
Let us consider, as an example, the monoclinic case; in terms of direct cell parameters, Q(hkl) is given by (Boultif & Louër, 1991)where , , , and B = b. The search using the successivedichotomy method is performed in a fourdimensional space that is covered by increasing the integer values i, l, m and n in the intervals [A_{−}, A_{+}] = [A_{−} = A_{0} + ip, A_{+} = A_{−} + p], [B_{−}, B_{+}] = [B_{−} = B_{0} + lp, B_{+} = B_{−} + p], [C_{−}, C_{+}] = [C_{−} = C_{0} + mp, C_{+} = C_{−} + p] and [β_{−}, β_{+}] = [β_{−} = 90 + nθ, β_{+} = β_{−} + θ], where the step values of p and θ are 0.4 Å and 5°, respectively, and A_{0}, B_{0} and C_{0} are the lowest values of A, B and C (based on the positions of the lowestangle peaks), respectively. Each quartet of intervals defines a domain D and, by taking into account the current limits for the parameters A, B, C and β, a calculated pattern is generated, not in terms of discrete Q(hkl) values but of allowed intervals . D is retained only if the observed Q_{i} values belong to the range [Q_{−}(hkl) − ΔQ_{i}, Q_{+}(hkl) + ΔQ_{i}], where ΔQ_{i} is the absolute error of the observed lines (i.e., impurity lines are not tolerated). If D has been accepted, it is divided into 2^{4} subdomains by halving the original intervals [A_{−}, A_{+}], [B_{−}, B_{+}], [C_{−}, C_{+}] and [β_{−}, β_{+}] and new limits are calculated; if a possible solution is found, the dichotomy method is applied iteratively. In case of triclinic symmetry the expression for Q(hkl) in terms of direct cell parameters is too complicated to be treated via the successivedichotomy method; therefore the basic indexing equation (3.4.2) is used. In this case, the [Q_{−}(hkl), Q_{+}(hkl)] intervals are set in reciprocal space according to the A_{ij} parameters of (3.4.2). To reduce computing time the following restrictions are put on the (hkl) Miller indices associated with the observed lines: (1) maximum h, k, l values equal to 2 in case of the first five lines; (2) h + k + l < 3 for the first two lines.
The outcome of the successivedichotomy method is not strongly influenced by the presence of a dominant zone. New approaches have been devoted to overcome the limitations of the method with a strict dependence on data accuracy and on impurities (Boultif & Louër, 2004; Louër & Boultif, 2006, 2007), see Section 3.4.4.2).
New indexing procedures that provide alternatives to the traditional approaches outlined in Section 3.4.3.1 have recently been proposed.
This method (Oishi et al., 2009) is based on the Ito equation (de Wolff, 1957):where Q(h) is the length of the reciprocal vector corresponding to the Miller index vector h = (hkl). It uses Conway's topograph (Conway & Fung, 1997), a connected tree obtained by associating a graph to each equation of type (3.4.9) and consisting of infinite directed edges. According to Ito's method, if quadrupoles (Q_{1}, Q_{2}, Q_{3}, Q_{4}) detected among the observed Q_{i} values satisfy the condition 2(Q_{1} + Q_{2}) = Q_{3} + Q_{4}, two Millerindex vectors h_{1} and h_{2} are expected to exist such that Q_{1} = Q(h_{1}), Q_{2} = Q(h_{2}), Q_{3} = Q(h_{1} − h_{2}) and Q_{4} = Q(h_{1} + h_{2}). If an additional value Q_{5} satisfying the condition 2(Q_{1} + Q_{4}) = Q_{2} + Q_{5} is found, the graph of the quadrupole (Q_{1}, Q_{2}, Q_{3}, Q_{4}) grows via the addition of the Q_{5} contribution; this procedure is iterated. If topographs share a Q value that corresponds to the same reciprocallattice vector, then a threedimensional lattice is derived containing the twodimensional lattices associated with the original topographs. Threedimensional lattices are also obtained by combining topographs. The probability that topographs correspond to the correct cell increases with the number of edges of the graph structure. The method is claimed by the authors to be insensitive to the presence of impurity peaks.
Globaloptimization methods, widely adopted for solving crystal structures from powder data, have also been successfully applied to indexing. Among them, we provide brief descriptions of genetic algorithms, and Monte Carlo and gridsearch methods.
The use of genetic algorithms (GAs) for solving the indexing problem was proposed by Tam & Compton (1995) and Paszkowicz (1996). Since then, Kariuki and coworkers (Kariuki et al., 1999) have combined GAs with a wholeprofilefitting procedure for indexing powder diffraction patterns. This approach exploits the information of the full powder diffraction pattern. It is inspired by the Darwinian evolutionary principle based on mating, mutation and natural selection of the member of a population that survives and evolves to improve future generations. The initial population consists of a set of trial cell parameters, chosen randomly within a given volume range; a full patterndecomposition process is performed using the Le Bail algorithm (Chapter 3.5 ) and the agreement between the calculated and observed profiles is derived and used for assessing the goodness of an individual member (i.e., a set of unitcell parameters). The most plausible cell is therefore found by exploring a sixdimensional hypersurface and searching for the global minimum of (see Section 3.4.4.3.2). In contrast to the main traditional methods, whose outcomes depend on the reliability of a set of peak positions, this procedure has the advantage of being insensitive to the presence of small impurity peaks that have a negligible influence on the agreement factor between the experimental and calculated profiles: the global minimum of is reached if the majority phase is correctly indexed. The main disadvantage of the method is the computing time required, in particular in the case of low symmetry.
The Monte Carlo approach has also been applied to indexing powder diffraction patterns (Le Bail, 2004; Bergmann et al., 2004; Le Bail, 2008). It exploits all the information contained in the full pattern, randomly generates and selects trial cell parameters, and calculates peak positions to which it assigns the corresponding Miller indices. An idealized powder pattern consisting of peak positions d and extracted intensities I is considered to test the trial cell. The cell reliability is assessed by suitable figures of merit (e.g. R_{p} and McM_{20}, see Section 3.4.2.1). The main drawback of this approach is the significant computing time required, in particular for triclinic systems.
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., unitcell 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 nondefault 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 zoneindexing 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 semiexhaustive, TREOR90 is based on the indexheuristics strategy (see Section 3.4.3.1.3) and uses a trialanderror 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 unitcell 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 successivedichotomy 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, NTREOR09 is an update of NTREOR (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 low2θ 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 zeropoint 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 (OishiTomiyasu, 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 Rfactor {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 zeropoint 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 zeropoint 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 multiprogram indexing facility. It can perform a selfcalibration, which is aimed at detecting and correcting 2θ zero errors, and is able to strip out weak lines. Its single unified user interface and datafile format make a wide set of indexing packages accessible with minimal effort (especially to nonspecialists). 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 reciprocalcell 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 (leastsquares iteration), an iterative leastsquares process which operates on the dspacing values extracted from reasonablequality powder diffraction data, and LPSearch (lattice parameter search), a Monte Carlo based wholepowderpattern decomposition approach independent of the knowledge of the dspacings (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 successivedichotomy 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 & RodríquezCarvajal, 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 nondefault strategies of extended search and data correction for zeropoint 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, highquality Xray 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 NTREOR09, 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 peaksearch procedure is automatically performed on the experimental powder pattern and the list of corresponding d values are supplied to NTREOR09. During the indexing process a correction for zeropoint 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, mediumquality Xray laboratory data. The first 43 peaks (in the range 7–67°) with intensities greater than a default threshold were selected (an intensitybased 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, NTREOR09 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 NTREOR09). 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 zeropoint 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 eggwhite 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 intensitybased 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 highersymmetry one.
Indexing a powder diffraction pattern is sometimes described as a `gateway technology', because the determination of the cell parameters is so fundamental: if no cell has been identified the execution of the subsequent steps of the structure solution process is impossible, and if a wrong cell has been used the correct solution is unreachable. Therefore extremely close attention must be paid to the indexing step of the process. From the early 1970s, the increasing interest in powder pattern indexing and the progress seen, in terms of both methods and algorithms, have strongly contributed to opening the door to modern applications of powder diffraction techniques. The availability of a quite large number of software packages, based on different indexing strategies, enables the scientist interested in solving crystal structures to switch from one program to another when the first fails, so increasing the possibility of success. In some cases indexing is still a challenging process. Goodquality data are necessary and careful inspection of each indexing step, in particular in the selection of the experimental peak positions to be used, is advisable.
Supporting information
Powder diffraction data for norbornene in .pow format. DOI: 10.1107/97809553602060000949/Hach3o4sup1.txt
Powder diffraction data for a Cu(II)Schiff base complex in .pow format. DOI: 10.1107/97809553602060000949/Hach3o4sup2.txt
Powder diffraction data for decafluoroquarterphenyl in .pow format. DOI: 10.1107/97809553602060000949/Hach3o4sup3.txt
Powder diffraction data for hexagonal turkey eggwhite lysozyme in .pow format. DOI: 10.1107/97809553602060000949/Hach3o4sup4.txt
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