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, p. 272
Section 3.4.2.2. Geometrical ambiguities^{a}Institute of Crystallography – CNR, Via Amendola 122/o, Bari, I70126, Italy 
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.)
References
Altomare, A., Giacovazzo, C. & Moliterni, A. (2008). Indexing and space group determination. In Powder Diffraction Theory and Practice, edited by R. E. Dinnebier & S. J. L. Billinge, pp. 206–226. Cambridge: RSC Publishing.Google ScholarBuerger, M. J. (1957). Reduced cells. Z. Kristallogr. 109, 42–60.Google Scholar
Buerger, M. J. (1960). Note on reduced cells. Z. Kristallogr. 113, 52–56.Google Scholar
Karen, V. L. & Mighell, A. D. (1991). Conversetransformation analysis. J. Appl. Cryst. 24, 1076–1078.Google Scholar
Kroll, H., Stöckelmann, D. & Heinemann, R. (2011). Analysis of multiple solutions in powder pattern indexing: the common reciprocal metric tensor approach. J. Appl. Cryst. 44, 812–819.Google Scholar
Mighell, A. D. (1976). The reduced cell: its use in the identification of crystalline materials. J. Appl. Cryst. 9, 491–498.Google Scholar
Mighell, A. D. (2001). Lattice symmetry and identification – the fundamental role of reduced cells in materials characterization. J. Res. Natl Inst. Stand. Technol. 106, 983–995.Google Scholar
Mighell, A. D. & Santoro, A. (1975). Geometrical ambiguities in the indexing of powder patterns. J. Appl. Cryst. 8, 372–374.Google Scholar
Niggli, P. (1928). Handbuch der Experimentalphysik, Vol. 7, Part 1. Leipzig: Akademische Verlagsgesellschaft.Google Scholar
OishiTomiyasu, R. (2014a). Method to generate all the geometrical ambiguities of powder indexing solutions. J. Appl. Cryst. 47, 2055–2059.Google Scholar
OishiTomiyasu, R. (2016). A table of geometrical ambiguities in powder indexing obtained by exhaustive search. Acta Cryst. A72, 73–80.Google Scholar
Santoro, A. & Mighell, A. D. (1970). Determination of reduced cells. Acta Cryst. A26, 124–127.Google Scholar
Santoro, A. & Mighell, A. D. (1972). Properties of crystal lattices: the derivative lattices and their determination. Acta Cryst. A28, 284–287.Google Scholar
Santoro, A., Mighell, A. D. & Rodgers, J. R. (1980). The determination of the relationship between derivative lattices. Acta Cryst. A36, 796–800.Google Scholar