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, p. 272

Section Geometrical ambiguities

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: 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].)


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