International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 3.4, p. 170

General setting considerations for three and fourcircle diffractometers have been discussed by Busing & Levy (1967). In principle, crystals can be placed on a fourcircle diffractometer in any general orientation, although it is often useful to have a setting such that the reciprocallattice axis lies parallel to the rotation axis. This setting is a prerequisite for effective use of the empirical absorption correction method of North, Phillips & Mathews (1968).
In the case where the crystal orientation is not precisely determined, setting is normally achieved using automatic procedures that involve finding a set of general reflections and generating a UB matrix from their angular positions. Generation of the UB matrix can be achieved by finding the shortest noncoplanar reciprocallattice vectors and assigning these as the reciprocalcell axes (Hornstra & Vossers, 1974). The resulting unit cell is always primitive, and additional manipulations are required to determine the conventional cell and type of Bravais lattice. This reciprocalspace method is adopted by the Nonius CAD4 diffractometer software (CAD4 Manual, 1989). Alternatively, the `autoindexing' method originated by Sparks (1976, 1982) and Jacobson (1976) can be used whereby directlattice vectors are generated, again through an initial cell. Clegg (1984) has described an enhancement of the directlattice vector method so that the initial cell is used to produce directlattice vectors systematically. In order to confirm that a generated vector is a true direct vector, the condition is applied that the scalar multiplication of a true direct vector and any true reciprocal vector (i.e. the observed reflection vectors) results in an integer. If a great majority of the products of a putative direct vector and each of the measured observed reflection vectors are integers, the direct vector is accepted. The final cell can be obtained from the set of accepted direct vectors. Subsequently, Duisenberg (1992) developed a method of autoindexing that is particularly applicable to difficult cases such as twin lattices, incommensurate structures, fragmented crystals, long axes, and even unreliable data. Finding the reciprocal lattice from a distribution of reciprocallattice points (i.e. observed reflections) is reduced to finding elementary periods in onedimensional rows, obtained by projecting all observed points onto the normal to the plane formed by any three of these points. Row periodicity and offending reflections are readily recognized. Each row, by its direction and reciprocal spacing, defines one directaxis vector, based upon all cooperating observations. A primitive cell can be obtained from the direct vectors and refined against the fitting reflections, resulting in one main lattice, or a main lattice and a set of alien reflections (see also Subsection 3.4.2.6).
References
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