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. 270272
Section 3.4.2. The basic concepts of indexing^{a}Institute of Crystallography – CNR, Via Amendola 122/o, Bari, I70126, Italy 
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.)
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