Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.4, pp. 270-281

Chapter 3.4. Indexing a powder diffraction pattern

A. Altomare,a* C. Cuocci,a A. Moliternia and R. Rizzia

aInstitute of Crystallography – CNR, Via Amendola 122/o, Bari, I-70126, Italy
Correspondence e-mail:

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 non-traditional methods for solving the indexation problem. The most widely used indexing programs are discussed and examples of their applications are given.

3.4.1. Introduction

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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 unit-cell 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 cell-determination process (which operates in a 6-dimensional 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 Nl experimental diffraction lines (in a 3Nl-dimensional integer-valued index space) (Shirley, 2003[link]). (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 single-crystal case. This is because the information about the three-dimensional reciprocal space is compressed into the one-dimensional experimental powder pattern. Whatever the method used, working in the parameter or index space, powder pattern indexing aims to recover the three-dimensional 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 dhkl interplanar spacings, which are related to the diffraction angles by the well known Bragg law:[2d_{hkl}\sin\theta_{hkl}=\lambda.]

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[link]). 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 higher-angle lines; and (2) higher-angle 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 higher-angle lines is therefore unwise. The successful outcome of powder pattern indexing is correlated to which and how many dhkl 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[link]). Unfortunately they can be degraded by different sources of errors: peak overlap, poor peak resolution, 2θ zero shift, errors in measurement, and a low peak-to-background 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[link]), has produced several methods and software packages (Shirley, 2003[link]; Werner, 2002[link]) with surprising progress in strengthening and automating the cell-determination process. Innovative approaches that aim to reduce the dependence on the dhkl values (by avoiding the peak-search 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 peak-position 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 good-quality diffraction data. Of course, the use of synchrotron radiation is preferable, but conventional laboratory X-ray 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[link] is mainly devoted to the basic concepts of a crystalline lattice, the main indexing equations and figures of merits; Section 3.4.3[link] discusses the traditional and non-traditional methods developed for indexing a powder pattern, and Section 3.4.4[link] discusses some applications, referring to the most widely used indexing programs.

3.4.2. The basic concepts of indexing

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We now describe some concepts that are fundamental in crystallography and useful for understanding the indexing process. The measured diffraction intensities correspond to the reciprocal-lattice points[{\bf r}_{hkl}^* = {h}{\bf a}^* + {k}{\bf b}^* + {l}{\bf c}^*. ]The Miller indices (hkl) identify the plane of the direct lattice and [{\bf a}^*], [{\bf b}^*] and [{\bf c}^*] are the three vectors of the reciprocal lattice, which are related to the direct lattice by[{\bf a} = {{\bf b}^* \times {\bf c}^*\over V^*},\quad {\bf b} = {{\bf c}^*\times{\bf a}^* \over V^*},\quad {\bf c} ={ {\bf a}^*\times {\bf b}^*\over V^*}, ]where[V^* = {\bf a}^* \cdot {\bf b}^* \times {\bf c}^*]is the reciprocal-cell volume ([V^*] is the inverse of the direct-unit-cell volume V).

In case of single-crystal data, the three-dimensional nature of the experimental diffraction data makes it easy to identify [{\bf a}^*,{\bf b}^*,{\bf c}^*], from which the direct-space unit-cell vectors are derived (Giacovazzo, 2011[link]).

In case of powder diffraction, the three-dimensional nature of the diffraction data is compressed into one dimension in the experimental pattern, and the accessible experimental information is the dhkl values involved in the Bragg law and related to the diffraction angles by[d_{hkl}=\lambda/(2\sin\theta_{hkl}).]dhkl, the spacing of the planes (hkl) in the direct lattice, is obtained by the dot products of the reciprocal-lattice vectors with themselves:[\eqalignno{({\bf r}_{hkl}^*)^2 &= {1 \over d_{hkl}^2} = h^2a^{*2} + k^2b^{*2}+ l^2c^{*2} + 2hka^*b^*\cos \gamma^* &\cr&\quad+ 2hla^*c^*\cos \beta ^* + 2klb^*c^*\cos \alpha ^*,&(3.4.1)}]where [\alpha^*] is the angle between [{\bf b}^*] and [{\bf c}^*], [\beta^*] is the angle between [{\bf c}^*] and [{\bf a}^*], and [\gamma^*] is the angle between [{\bf a}^*] and [{\bf b}^*]. If we introduce[Q(hkl) = {10^4 \over d_{hkl}^2} ][where Q(hkl) differs from [\sin ^2\theta _{hkl}] by a scale factor of (200/λ)2], (3.4.1)[link] becomes[Q(hkl) =h^2 A_{11} +k^2 A_{22} +l^2 A_{33} +hk A_{12} +hl A_{13} +kl A_{23},\eqno(3.4.2)]where[\displaylines{A_{11} = {10^4}{a^{*2}}, \ A_{22} = {10^4}{b^{*2}},\ A_{33} = {10^4}{c^{*2}},\cr A_{12} = 2 \times {10^4}{a^*}{b^*}\cos \gamma ^*,\ A_{13} = 2 \times {10^4}{a^*}{c^*}\cos \beta ^*, \cr A_{23} = 2 \times{10^4}{b^*}{c^*}\cos \alpha^*.}]The number of parameters [{A_{ij}}] in (3.4.2)[link] depends on the type of symmetry: from 1 in the case of cubic symmetry to 6 for triclinic symmetry (see Table 3.4.1[link]).

Table 3.4.1| top | pdf |
Expressions for Q(hkl) for different types of symmetry

Cubic (h2 + k2 + l2)A11
Tetragonal (h2 + k2)A11 + l2A33
Hexagonal (h2 + hk + k2)A11 + l2A33
Orthorhombic h2A11 + k2A22 + l2A33
Monoclinic h2A11 + k2A22 + l2A33 + hlA13
Triclinic h2A11 + k2A22 + l2A33 + hkA12 + hlA13 + klA23

The quadratic form (3.4.2)[link] relates the observed Q(hkl) values to the reciprocal cell parameters and, consequently, to the direct cell. It is the basic equation used in powder-indexing procedures. Therefore the indexing problem (Werner, 2002[link]) is to find Aij and, for each observed Q(hkl) value, three crystallographic indices (hkl) satisfying (3.4.2)[link] within a suitable tolerance parameter Δ:[\eqalignno{Q{\rm{(}}hkl{\rm{)}} -\Delta &\,\lt\,h^2 A_{11} +k^2 A_{22} +l^2 A_{33} +hk A_{12} +hl A_{13} +kl A_{23} &\cr&\,\lt\, Q(hkl)+\Delta. &(3.4.3)}]The importance of using accurate Q(hkl) values in (3.4.3)[link] is obvious. Moreover, it is worth noticing that (3.4.3)[link] must lead to physically reasonable indexing – low-angle 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. Figures of merit

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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[link]) made an important contribution in this direction. He developed the M20 figure of merit defined by[M_{20} = {Q_{20} \over 2\langle\varepsilon\rangle N_{20}},\eqno(3.4.4)]where Q20 is the Q value corresponding to the 20th observed and indexed peak, N20 is the number of different calculated Q values up to Q20, and [\langle\varepsilon \rangle] 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 M20 is as follows: the better the agreement between the calculated and the observed peak positions (the smaller the [\langle\varepsilon \rangle] value) and the smaller the volume of the unit cell (the smaller the N20 value), the larger the M20 value and consequently the confidence in the proposed unit cell. A rule of thumb for M20 is that if the number of unindexed peaks whose Q values are less than Q20 is not larger than 2 and if M20 > 10, then the indexing process is physically reasonable (de Wolff, 1968[link]; Werner, 2002[link]). 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 high-angle peak positions, which are more affected by errors. M20 is statistically expected to be 1 in case of completely arbitrary indexing. It has no upper limit (it can be very large when [\langle\varepsilon \rangle] is very small).

Smith & Snyder (1979[link]) proposed the FN criterion in order to overcome the limits of M20 with respect to its dependence on the 20 lines and on crystal class and space group. The FN figure of merit is given by[F_N = {1 \over\langle |\Delta 2\theta | \rangle} {N \over N_{\rm poss}}, ]where [\langle |\Delta 2\theta | \rangle] is the average absolute discrepancy between the observed and calculated 2θ peak position values and Nposs is the number of possible diffraction lines up to the Nth observed line. The values of [\langle |\Delta 2\theta | \rangle] and Nposs, ([\langle |\Delta 2\theta | \rangle], Nposs), are usually given with FN. With respect to M20, FN is more suitable for ranking the trial solutions and less for indicating their physical plausibility (Werner, 2002[link]).

Both M20 and FN, 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[link], 1971[link]). Among the recently developed FOMs, we mention:

  • (1) Qpartial (Bergmann, 2007[link]):[Q_{\rm partial} = \sum\limits_{i} \min \left[w_i,\left({x_i - \hat{x_i} \over \delta _i} \right)^2 \right], ]where the summation is over the number of observed lines, wi is the observed weight of line i, x and [ \hat{x_i}] are the observed and simulated line positions, respectively, and δi is the observed random error of line i. Qpartial is multiplied by a factor that depends on the symmetry of the simulated lattice (triclinic, …, cubic), on the unit cell volume and on the number of ignored peaks.

  • (2) McM20 (Le Bail, 2008[link]):[McM_{20}=[100/(R_pN_{20})]B_rS_y,]where N20 is the number of possible lines that might exist up to the 20th observed line (for a primitive P lattice). Rp is the profile R factor (Young, 1993[link]). Br is a factor arbitrarily set to 6 for F and R Bravais lattices, 4 for I, 2 for A, B and C, and 1 for P. Sy is a factor equal to 6 for a cubic or a rhombohedral cell, 4 for a trigonal, hexagonal or tetragonal cell, 2 for an orthorhombic cell, and 1 for a monoclinic or triclinic cell.

  • (3) WRIP20 (Altomare et al., 2009[link]):[{\rm WRIP}20 = {\rm RAT}_{Rp}^2\times {\rm RAT}_{\rm Ind}\times {\rm RAT}_{\rm Pres}\times w_u\times {\rm RAT}_{M20}^{1/2}.\eqno(3.4.5)]Based on M20 (M20 and FN remain the most widely used FOMs), WRIP20 has been developed for exploiting the full information contained in the diffraction profile. The factors that appear in (3.4.5)[link] are[\displaylines{{\rm RAT}_{Rp} = {1 - R_p \over 1 - (R_p)_{\rm min}}, \quad {\rm RAT}_{\rm Ind} = {{\rm PERC}_{\rm Ind} \over ({\rm PERC}_{\rm Ind})_{\rm max}}, \cr {\rm RAT}_{\rm Pres} = {({\rm PERC}_{\rm Pres})_{\rm min} \over {\rm PERC}_{\rm Pres}},\quad w_u= (N_{\rm obs } - N_u)/N_{\rm obs},\cr {\rm RAT}_{M20} = {M_{20} \over (M_{20})_{\rm max}},\quad {\rm PERC}_{\rm Pres} = \textstyle\sum\limits_{\rm Pres} {\rm mult} /\textstyle\sum\limits_{\rm all} {\rm mult}.}]Rp is the profile-fitting agreement calculated after the Le Bail (Chapter 3.5[link] ) decomposition of the full pattern using the space group with the highest Laue symmetry compatible with the geometry of the current unit cell and no extinction conditions. PERCInd, the percentage of independent observations in the experimental profile, is estimated according to Altomare et al. (1995[link]). For each extinction symbol compatible with the lattice geometry of the current unit cell, normalized intensities are calculated and subjected to statistical analysis in order to obtain a probability value associated with each extinction symbol in accordance with Altomare et al. (2004[link], 2005[link]). For the extinction symbol with the highest probability value, the value of PERCPres is calculated: [\textstyle\sum_{\rm all}{\rm mult}] is the total number of reflections (symmetry-equivalent included) for the space group having the highest Laue symmetry and no extinction conditions. (It varies with the volume of the unit cell and the data resolution.) [\textstyle\sum_{\rm Pres} {\rm mult}], which varies according to the extinction rules of the current extinction symbol, coincides with the number of non-systematically absent reflections (with the symmetry equivalents included). The subscripts min and max mark the minimum and the maximum values of each factor respectively, calculated for the possible unit cells that are to be ranked. Nobs and Nu are the number of observed and unindexed lines, respectively. All the terms in (3.4.5)[link] are between 0 and 1, so ensuring that WRIP20 also lies between 0 and 1. In addition, WRIP20 has the following properties: (a) it is continuous, that is, definable in any interval of the experimental pattern; (b) it takes into account the peak intensities, the number of generated peaks and their overlap, and the systematically absent reflections (through the extinction-symbol test); and (c) it is not very sensitive to the presence of impurity lines (these usually have low intensities). WRIP20 is effective in finding the correct cell among a number of possible ones and selecting the corresponding most probable extinction symbol (see Example 3[link] in Section[link]).

  • (4) Two new figures of merit based on de Wolff's method, the reversed figure of merit ([M_n^{\rm Rev}]) and the symmetric figure of merit ([M_n^{\rm Sym}]), have recently been proposed (Oishi-Tomiyasu, 2013[link]). As observed by Oishi-Tomiyasu, the de Wolff figure of merit Mn does not use the observed and calculated lines in a symmetrical way, consequently it is (a) insensitive to computed but unobserved lines (i.e., extinct peaks) and (b) sensitive to unindexed observed lines (e.g., impurity peaks). [M_n^{\rm Rev}] and [M_n^{\rm Sym}] aim to compensate for the disadvantages of Mn. In particular, [M_n^{\rm Rev}] has characteristics opposite to those of Mn with regard to sensitivity to extinct reflections and impurity peaks, and [M_n^{\rm Sym}] has intermediate properties between Mn and [M_n^{\rm Rev}]. They prove useful in selecting the correct solution, particularly in case of presence of impurity peaks. (See also Section[link].) Geometrical ambiguities

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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[link]) 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[link], 1960[link]; Santoro & Mighell, 1970[link]; Mighell, 1976[link], 2001[link]). 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[link]). An algorithm based on the converse-transformation theory has been developed and implemented in the Fortran program NIST*LATTICE for checking relationships between any two cells (Karen & Mighell, 1991[link]).

It is very important to recognize that two lattices are derivative of each other, because many crystallographic problems (twinning, indexing of powder patterns, single-crystal 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[link]).

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[link]). 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 reciprocal-lattice vector and not the three-dimensional vector itself, is geometrical. It mainly occurs for high-symmetry cells (from orthorhombic up). The lattices having this property are related to each other by rotational transformation matrices. In Table 3.4.2[link] some examples of lattices giving geometrical ambiguities and the corresponding transformation matrices are given (Altomare et al., 2008[link]). Where there are geometrical ambiguities, additional prior information (e.g., a single-crystal study) may be useful in order to choose one of the two possible lattices.

Table 3.4.2| top | pdf |
Examples of lattices leading to geometrical ambiguities

P = {Pij} is the transformation matrix from lattice I to lattice II, described by the vectors {ai} and {bi}, respectively, with [{\bf b}_i = \textstyle\sum_{j} P_{ij}{\bf a}_{j}].

Lattice ILattice IIP
Cubic P Tetragonal P [\pmatrix{ 0 &-1/2 &1/2\cr 0 &1/2&1/2 \cr -1& 0 & 0}]
Cubic I Tetragonal P [\pmatrix{ 0& -1/2 &1/2\cr 0 &-1/2 & -1/2\cr 1/2 & 0 & 0}]
Orthorhombic F [\pmatrix{-1/3 & -1/3 & 0\cr 0 & 0 & -1 \cr 1 & -1& 0}]
Orthorhombic P [\pmatrix{ 1/4 & -1/4 & 0 \cr 0 & 0 & 1/2 \cr -1/2 & -1/2 & 0}]
Cubic F Orthorhombic C [\pmatrix{ -1/2 & 0 & 1/2 \cr 0 & 1& 0 \cr -1/4 & 0 &-1/4}]
Orthorhombic I [\pmatrix{ -1/6 & 0 & -1/6\cr 1/2 & 0 &-1/2 \cr 0 &-1 &0}]
Hexagonal Orthorhombic P [\pmatrix{1/2 & 1/2 & 0 \cr 1/2 & -1/2& 0 \cr 0 & 0 &-1} ]
Rhombohedral Monoclinic P [\pmatrix{ -1/2 & 0 & -1/2 \cr 1/2 & 0 &-1/2 \cr 0 & -1 &0 }]

A recent procedure developed by Kroll et al. (2011[link]) 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 non-derivative lattices can have identical peak positions. Very recently, Oishi-Tomiyasu (2014a[link], 2016[link]) has developed a new algorithm able to obtain all lattices with computed lines in the same positions as a given lattice. (See also Section[link].)

3.4.3. Indexing methods

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Indexing methods aim to reconstruct the three-dimensional direct lattice from the one-dimensional distribution of dhkl values. Systematic or accidental peak overlap, inaccuracy of peak positions, zero shift in the 2θhkl 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[link]). As suggested by Shirley (2003[link]), 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[link]). In this section an outline of the strategies and search methods adopted by the main traditional and non-traditional indexing approaches is given. For more details see the papers by Shirley (2003[link]) and Bergmann et al. (2004[link]).

Among the main indexing procedures, zone indexing (Section[link]), SIW heuristic (Section[link]), successive dichotomy (Section[link]), the topographs method (Section[link]) and global-optimization methods (Section[link]) operate in the parameter space; index heuristics (Section[link]) and index permutation (Section[link]) work in the index space; and scan/covariance (Bergmann, 2007[link]) 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 non-exhaustive 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 semi-exhaustive (Table 3.4.3[link]).

Table 3.4.3| top | pdf |
Classification of indexing methods

Method (strategy/search)SpaceExhaustive
Zone indexing Parameter No
SIW heuristic Parameter Semi
Scan/covariance Parameter (for cubic to monoclinic); index (for triclinic) To monoclinic
Index heuristics Index Semi
Index permutation Index Yes
Successive dichotomy Parameter Yes
Grid search Parameter Yes
Genetic algorithms Parameter No
Monte Carlo Parameter No

Indexing procedures can also be classified as traditional and non-traditional. 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[link]). Traditional indexing methods

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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. Zone-indexing strategy

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The zone-indexing strategy was originally developed by Runge (1917[link]), successively proposed by Ito (1949[link], 1950[link]), generalized by de Wolff (1957[link], 1958[link]) and enhanced by Visser (1969[link]). 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 [{\bf r}_{hkl}^*] and [{\bf r}_{h'k'l'}^*] 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 [m {\bf r}_{hkl}^* \pm n {\bf r}_{h'k'l'}^*,] where m and n are positive integers. If ω is the angle between [{\bf r}_{hkl}^*] and [{\bf r}_{h'k'l'}^*], the squared distance of B from O (i.e., [{Q_{m,n}}]) can be expressed by (de Wolff, 1958[link]; Visser, 1969[link])[Q_{m,n} = m^2 Q_{\rm A} + n^2 Q_{\rm A'} \pm m n R,\eqno(3.4.6)]where [Q_{\rm A} = Q(hkl)] and [Q_{\rm A'}=Q(h'k'l')] are the squared distances of A and A′ from O, respectively, and [R = ] [2(Q_{\rm A} Q_{\rm A'})^{1/2} \cos \omega ]. R can be derived as[R = |{Q_{m,n}} - {m^2} {Q_{\rm A}} - {n^2} {Q_{\rm A'}}|/m n.\eqno(3.4.7)]

The method is applied as follows: [Q_{\rm A}] and [Q_{\rm A'}] are chosen among the first experimental Qi values; the {Qi}, up to a reasonable resolution, are introduced in (3.4.7)[link] in place of [{Q_{m,n}}]; and a few positive integer values are assigned to m and n. Equation (3.4.7)[link] 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 ([Q_{\rm A},Q_{\rm A'}]) 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 low-symmetry patterns. The main disadvantage is its sensitivity to errors in the peak positions, particularly in the low 2θ region. Shirley–Ishida–Watanabe (SIW) heuristic strategy

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This needs only one single well-established zone, then it arbitrarily chooses the 001 line from the first-level lines. The indexing problem is thus lowered to two dimensions and an exhaustive search is carried out. Index-heuristics strategy

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The index-heuristics strategy searches for the correct cell via a trial-and-error 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[link]), then successively refined (Werner et al., 1985[link]) and made more robust and effective (Altomare et al., 2000[link], 2008[link], 2009[link]). This approach, which works in the index space, was defined by Shirley as semi-exhaustive (Shirley, 1980[link]). The search starts from the highest-symmetry 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 2θhkl values) can be indexed with a common zero Miller index. Special short-axis tests, aimed at finding two-dimensional lattices, have been proposed for monoclinic symmetry in order to detect the presence of dominant zones (Werner et al., 1985[link]). The index-heuristics method is based on the main indexing equation [see equation (3.4.2)[link]] that can be rewritten (Werner et al., 1985[link]) as[Q(hkl) = {h^2}x{}_1 + {k^2}x{}_2 + {l^2}x{}_3 + hk{x_4} + hl{x_5} + kl{x_6}, ]where {xi} = X is the vector of unknown parameters, which are derived by solving a system of linear equations[{\bf MX}={\bf Q},\eqno(3.4.8)]where 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−1Q. In the case of monoclinic and higher symmetry, {xi} 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 {xi} strongly depends on the accuracy of the observed Q values, especially for low-2θ region lines, which are the most dominant ones for this indexing procedure. The possibility of testing different combinations of basis-line 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[link], 2009[link]) by introducing new procedures that are able to increase the probability of successful indexing (see Section[link]); 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 M20 in identifying the correct solution among a set of possible ones (see Section[link]); (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 F-centred cells) (such information is used in the successive steps). Index-permutation strategy

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This strategy was proposed by Taupin (1973[link]), and is based on a systematic permutation of indices associated to observed lines for obtaining candidate cells. Because this trial-and-error strategy is similar to the index-heuristics approach, we do not describe it here. Successive-dichotomy search method

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The successive-dichotomy method, first developed by Louër & Louër (1972[link]), 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 n-dimensional 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 2n subdomains via a successive-dichotomy procedure. Each subdomain is analyzed and discarded if it does not contain a solution. The method was originally applied to orthorhombic and higher-symmetry systems (Louër & Louër, 1972[link]), but it has been successively extended to monoclinic (Louër & Vargas, 1982[link]) and to triclinic systems (Boultif & Louër, 1991[link]). 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 A3. For triclinic symmetry ΔV is related to the volume Vest suggested by the method proposed by Smith (1977[link]), which is able to estimate the unit-cell volume from only one line in the pattern:[V_{\rm est} \simeq {0.60 d^3_N \over {1\over N} - 0.0052},]where dN is the value for the Nth observed line; in the case N = 20 the triclinic cell volume is [V_{\rm est} \simeq 13.39 d_{20}^3].

Let us consider, as an example, the monoclinic case; in terms of direct cell parameters, Q(hkl) is given by (Boultif & Louër, 1991[link])[Q(hkl) = f(A,C,\beta) + g(B), ]where [f(A,C,\beta) = h^2/A^2 + l^2/C^2 - 2hl\cos \beta /(AC)], [A = a\sin \beta ], [C = c\sin \beta ], [g(B) = {k^2}/{B^2}] and B = b. The search using the successive-dichotomy method is performed in a four-dimensional space that is covered by increasing the integer values i, l, m and n in the intervals [A, A+] = [A = A0 + ip, A+ = A + p], [B, B+] = [B = B0 + lp, B+ = B + p], [C, C+] = [C = C0 + 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 A0, B0 and C0 are the lowest values of A, B and C (based on the positions of the lowest-angle 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 [\left [{Q{}_ - (hkl),{Q_ + }(hkl)} \right]]. D is retained only if the observed Qi values belong to the range [Q(hkl) − ΔQi, Q+(hkl) + ΔQi], where ΔQi is the absolute error of the observed lines (i.e., impurity lines are not tolerated). If D has been accepted, it is divided into 24 subdomains by halving the original intervals [A, A+], [B, B+], [C, C+] and [β, β+] and new limits [\left [{Q{}_ - (hkl),{Q_ + }(hkl)} \right]] 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 successive-dichotomy method; therefore the basic indexing equation (3.4.2)[link] is used. In this case, the [Q(hkl), Q+(hkl)] intervals are set in reciprocal space according to the Aij parameters of (3.4.2)[link]. 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 successive-dichotomy 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[link]; Louër & Boultif, 2006[link], 2007[link]), see Section[link]). Non-traditional indexing methods

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New indexing procedures that provide alternatives to the traditional approaches outlined in Section[link] have recently been proposed. The topographs method

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This method (Oishi et al., 2009[link]) is based on the Ito equation (de Wolff, 1957[link]):[Q({\bf h}_1+{\bf h}_2)+Q({\bf h}_1-{\bf h}_2)=2[Q({\bf h}_1)+Q({\bf h}_2)], \eqno(3.4.9)]where Q(h) is the length of the reciprocal vector [{\bf r}_{hkl}^*] corresponding to the Miller index vector h = (hkl). It uses Conway's topograph (Conway & Fung, 1997[link]), a connected tree obtained by associating a graph to each equation of type (3.4.9)[link] and consisting of infinite directed edges. According to Ito's method, if quadrupoles (Q1, Q2, Q3, Q4) detected among the observed Qi values satisfy the condition 2(Q1 + Q2) = Q3 + Q4, two Miller-index vectors h1 and h2 are expected to exist such that Q1 = Q(h1), Q2 = Q(h2), Q3 = Q(h1h2) and Q4 = Q(h1 + h2). If an additional value Q5 satisfying the condition 2(Q1 + Q4) = Q2 + Q5 is found, the graph of the quadrupole (Q1, Q2, Q3, Q4) grows via the addition of the Q5 contribution; this procedure is iterated. If topographs share a Q value that corresponds to the same reciprocal-lattice vector, then a three-dimensional lattice is derived containing the two-dimensional lattices associated with the original topographs. Three-dimensional 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. Global-optimization methods

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Global-optimization 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 grid-search methods. Genetic-algorithm search method

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The use of genetic algorithms (GAs) for solving the indexing problem was proposed by Tam & Compton (1995[link]) and Paszkowicz (1996[link]). Since then, Kariuki and co-workers (Kariuki et al., 1999[link]) have combined GAs with a whole-profile-fitting 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 pattern-decomposition process is performed using the Le Bail algorithm (Chapter 3.5[link] ) 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 unit-cell parameters). The most plausible cell is therefore found by exploring a six-dimensional hypersurface [R_{wp}^{\prime}(a,b,c,\alpha, \beta, \gamma)] and searching for the global minimum of [R_{wp}^{\prime}] (see Section[link]). 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 [R_{wp}^{\prime}] 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. Monte Carlo search method

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The Monte Carlo approach has also been applied to indexing powder diffraction patterns (Le Bail, 2004[link]; Bergmann et al., 2004[link]; Le Bail, 2008[link]). 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. Rp and McM20, see Section[link]). The main drawback of this approach is the significant computing time required, in particular for triclinic systems. Grid-search method

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This performs an iterated `step-and-repeat search' in the parameter space. It has the advantage of being flexible, exhaustive and not particularly sensitive to impurities or errors, and the disadvantage of being slow (Shirley, 2003[link]).

3.4.4. Software packages for indexing and examples of their use

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The different strategies and methods described in Section 3.4.3[link] have been implemented in a variety of automatic indexing programs (Bergman et al., 2004[link]). 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[link]), 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[link]) in Table 3.4.3[link]. 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. Traditional indexing programs

| top | pdf | ITO (Visser, 1969[link])

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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[link].

The following steps are executed by the program:

  • (1) The potential zones are found always using the 20 lowest Bragg angle peaks. The program does not work with fewer peaks.

  • (2) All possible combinations of the six best zones (including the combination of each zone with itself) are found by searching for trial zones that share a row of common points. For every pair of such zones, the angle between them is found, thus giving a trial reciprocal lattice.

  • (3) The reduction of the resulting unit cells is carried out using the Delaunay–Ito method (Pecharsky & Zavalij, 2009[link]).

  • (4) The program tries to index the first 20 lines and repeats this check after least-squares refinement of the unit-cell parameters.

  • (5) The figures of merit are calculated to assess the quality of each trial unit cell and the four best lattices are provided.

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. TREOR90 (Werner et al., 1985[link])

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Classified by Shirley (1980[link], 2003[link]) as semi-exhaustive, TREOR90 is based on the index-heuristics strategy (see Section[link]) and uses a trial-and-error approach. It performs the following steps:

  • (1) Some basis lines are selected among the experimental d values, generally from the low-2θ region of the powder diffraction pattern. Five sets of basis lines are generally sufficient for orthorhombic tests, whereas more than seven sets may be necessary for the monoclinic system. At least 20–25 experimental d values are potentially required.

  • (2) The trial unit cells in the index space are searched by varying the Miller indices that are tentatively assigned to the basis lines.

  • (3) The analysis starts with cubic symmetry and, in a stepwise manner, tests for lower-symmetry crystal systems are performed. In the case of monoclinic symmetry, a special short-axis test is carried out.

  • (4) The solution of the linear system in equation (3.4.8)[link] gives the possible cell parameters. Different combinations of the basis lines are tested.

  • (5) Each possible solution is checked by using the full list of experimental lines.

  • (6) The quality of the trial cell parameters is mainly assessed by using the M20 figure of merit [see equation (3.4.4)[link]]. An effective rule for identifying a reliable solution is M20 > 10 and no more than one unindexed line.

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. DICVOL91 (Boultif & Louër, 1991[link])

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This program works in direct space (down to the monoclinic system) by using the successive-dichotomy search method (see Section[link]), which was introduced for the automatic indexing of powder diffraction patterns by Louër & Louër (1972[link]). DICVOL91 has been defined as exhaustive by Shirley (1980[link], 2003[link]). Its main steps are:

  • (1) The unit-cell volume is partitioned by moving from high to low symmetry. Shells of 400 Å3 of volume are scanned for all the symmetry systems except for triclinic; for the triclinic system the shells are based on the definition of volume proposed by Smith (1977[link]) (see Section[link]).

  • (2) A search using the successive-dichotomy method, based on suitable intervals (see Section[link]), is carried out. In the case of the triclinic system the general expression for [Q(hkl)] as function of the direct cell parameters is too complicated, and the reciprocal-space parameters in equation (3.4.2[link]) are used for setting the intervals.

  • (3) The derived cell parameters are refined using the least-squares method.

  • (4) The quality of each trial unit cell is evaluated by using the MN and FN figures of merit (see Section[link]).

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). Evolved indexing programs

| top | pdf | N-TREOR09 (Altomare et al., 2009[link])

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Implemented in the EXPO program (Altomare et al., 2013[link]) to perform the powder pattern indexing step, N-TREOR09 is an update of N-TREOR (Altomare et al., 2000[link]), 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:

  • (a) If the default indexing process fails, the unit-cell search is automatically repeated by changing some default choices, e.g., increasing the tolerance value on the observed d values. If still no solution is obtained, the maximum (hkl) Miller indices assigned to the orthorhombic or monoclinic base lines are increased and the tolerance limits of the default values are halved in order to avoid the generation of wrong large unit cells.

  • (b) At the end of the first run, whatever the obtained results, a possible 2θ zero-position shift is taken into account: the indexing process starts again by applying positive and negative 2θ zero-position shifts to the original peak search result.

  • (c) An exhaustive triclinic search is performed. The dominant-zone tests that are usually carried out for the monoclinic system have been extended to include the triclinic case.

  • (d) A new figure of merit, WRIP20, more powerful than the classical M20, is used. It is calculated when more than one possible cell is found and takes into account the M20 value, the full experimental pattern, the degree of reflection overlap, the systematically absent reflections and the number of unindexed lines (see Section[link]).

This program is also able to index powder patterns from small proteins: see Example 4[link] in Section[link]. DICVOL06 (Louër & Boultif, 2006[link], 2007[link]) and DICVOL14 (Louër & Boultif, 2014[link])

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The most recent of a series of versions, DICVOL14 is the successor of DICVOL04 (Boultif & Louër, 2004[link]) 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:

  • (a) A tolerance for unindexed lines that can result from the presence of unwanted additional phases or inaccurately measured peaks. The program can tolerate a user-defined number of unindexed lines. Care must be taken when using this option to avoid the possibility of generating erroneous cells. It is worth noting that the inclusion of the possibility of at least one unindexed peak has markedly increased the success rate of DICVOL06.

  • (b) A correction of the zero-point error in the measured data. Via an a priori zero-origin evaluation, two different approaches can be followed: (i) if there is a non-negligible zero shift (i.e., ~0.1°), the reflection-pair method is adopted (Dong et al., 1999[link]); (ii) if the shift is small (<0.03°), a refinement of the experimental data zero point together with the cell parameters is carried out as soon as a solution is found. In the monoclinic and triclinic systems, a reduced-cell analysis is performed to choose among equivalent solutions.

  • (c) When a solution is found in a 400 Å3 shell of volume, the exhaustive search is extended to the whole domain.

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. Non-traditional indexing programs

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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[link]), which has been more recently proposed, also belongs to that group of programs. A brief description of Conograph follows Conograph: indexing via the topographs method

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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:

  • (1) A new Bravais-lattice determination algorithm (Oishi-Tomiyasu, 2012[link]), which has been proved to be stable with respect to peak-position errors under very general conditions. The algorithm applies the Minkowski reduction to primitive cells and the Delaunay reduction (Delaunay, 1933[link]) to face-centred, body-centred, rhombohedral and base-centred cells in such a way that the computational efficiency of the process is better than the Andrews & Bernstein (1988[link]) method.

  • (2) The two figures of merit [M_n^{\rm Rev}] and [M_n^{\rm Sym}] proposed by Oishi-Tomiyasu (2013[link]) are used for for selecting the true unit cells. They are also used to estimate the zero-point shift.

  • (3) The use of many observed peaks in the default setting, which aims to make Conograph robust against dominant zones and missing or false peaks (Oishi-Tomiyasu, 2014b[link]).

  • (4) The method for exhaustively searching unit cells that involve geometrical ambiguity (Oishi-Tomiyasu, 2014a[link], 2016[link]). The geometrical ambiguities that are detected also include lattices with very similar calculated lines, because of the error tolerance in the d spacings.

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. GAIN: indexing via a genetic-algorithm search method

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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[link]) and Paszkowicz (1996[link]). Subsequently, Kariuki et al. (1999[link]) 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[link]), whose main features are:

  • (1) Starting from a population of Np sets of lattice parameters and using the evolutionary operations of mating, mutation and natural selection, the population is allowed to evolve through several generations, with the aim of generating sets of possible trial cell parameters.

  • (2) The search procedure, using a GA, is performed in restricted, sensible cell-volume ranges consistent with the knowledge of the system under study.

  • (3) For each set of trial parameters a calculated powder diffraction pattern is constructed. The peak positions and parameters describing the shape and width of each peak are used in the Le Bail profile-fitting procedure (Chapter 3.5[link] ).

  • (4) The pattern is split into different regions (defined by the user), and the weighted profile R factor is calculated for each region; all the values are summed to obtain the overall [R_{\rm wp}^{\prime}]:[R_{\rm wp}^{\prime} = \sum\limits_{\rm regions} \left[{\textstyle\sum_i w_i(y_i - y_{ci})^2 \over \textstyle\sum_i w_iy_i^2} \right] ^{1 / 2}, ]where the summation is over the regions, i runs over the experimental points belonging to each region and [{y_i}] and [{y_{ci}}] are the observed and calculated profile at the ith experimental step, respectively. Via the [R_{\rm wp}^{\prime}] formula the residual for each region is scaled according to the total intensity in the region, so a region with only low-intensity peaks can make an important contribution to [R_{\rm wp}^{\prime}].

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. McMaille: indexing via a Monte Carlo search method

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The information in the whole powder diffraction profile is exploited by the program McMaille (Le Bail, 2004[link]), 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 Rp (Young, 1993[link]) or McM20 (see Section[link]). The program uses some tricks that can increase the success of the Monte Carlo algorithm:

  • (1) Only the trial cells corresponding to a value of Rp that is smaller than a user-defined value (∼50%) are retained for successive refinement.

  • (2) If all the observed peaks, except for a user-defined number of tolerated impurity peaks, are `explained' whatever the Rp value, the cell is retained for successive examination.

  • (3) If either of the conditions (1) or (2) is fulfilled, the cell parameters are randomly changed in 200 to 5000 attempts (for cubic to triclinic cases, respectively) in which small random parameter variations via the Monte Carlo algorithm are carried out. The new parameters are preserved if an improvement of Rp is verified in 85% of the attempts.

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 Å3; (c) no cell length is longer than 20 Å. Crysfire: a suite of indexing programs

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The Crysfire suite (Shirley, 2002[link]) 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[link]), 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. Two commercial programs

| top | pdf | SVD-Index

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This commercial indexing program (Coelho, 2003a[link]), which uses the Monte Carlo method, is part of the TOPAS (Coelho, 2003[link]b) suite from Bruker AXS. The reciprocal-cell parameters in equation (3.4.2)[link] are found by using, in an iterative way, the singular value decomposition (SVD) approach (Nash, 1990[link]) 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[link]). X-CELL

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This commercial program is part of the Materials Studio suite from Accelrys (Neumann, 2003[link]). To perform an exhaustive search, like DICVOL, the program uses the successive-dichotomy approach. Its principal features are:

  • (1) the user can define how many impurity lines can be tolerated;

  • (2) a search for the zero-point shift of the diffraction pattern; and

  • (3) systematic absences are taken into account.

The program is described as `virtually exhaustive'; it is expected to work well when faced with missing lines, impurities and errors. Examples of applications of indexing programs

| top | pdf | Indexing using DICVOL06

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The program DICVOL06, as implemented in the WinPLOTR/FULLPROF suite (Roisnel & Rodríquez-Carvajal, 2001[link]) and recently introduced into EXPO, was applied to two experimental diffraction patterns.

Example 1

Norbornene (Brunelli et al., 2001[link]). Published information: C7H10, monoclinic, a = 7.6063 (9), b = 8.6220 (1), c = 8.749 (1) Å, β = 97.24 (1)°, P21/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[link]).

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, M18 = 41.5, F18 = 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, M20 = 30.1, F20 = 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[link]). Published information: Cu(C15H12NO2)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)°, [P \bar 1], 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, M20 = 27.0, F20 = 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, M20 = 27.9, F20 = 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. Indexing using N-TREOR09

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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[link]). Published information: C24H8F10, 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)[link]], as shown in Fig. 3.4.1[link] (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 M20 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, M20 = 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, M20 = 15, and two lines were unindexed.

[Figure 3.4.1]

Figure 3.4.1 | top | pdf |

The list of possible cells for the decafluoroquarterphenyl structure automatically found using N-TREOR09.

Example 4

Hexagonal turkey egg-white lysozyme (Margiolaki et al., 2005[link]). Published information: hexagonal, a = 71.0862 (3), c = 85.0276 (5) Å, P6122, 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[link]) between hexagonal and orthorhombic lattices and automatically selected the higher-symmetry one.

3.4.5. Conclusion

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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. Good-quality 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.

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