International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 3.4, p. 170

## Section 3.4.2.7. Diffractometer-setting considerations

P. F. Lindleya

aESRF, Avenue des Martyrs, BP 220, F-38043 Grenoble CEDEX, France

#### 3.4.2.7. Diffractometer-setting considerations

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General setting considerations for three- and four-circle diffractometers have been discussed by Busing & Levy (1967). In principle, crystals can be placed on a four-circle diffractometer in any general orientation, although it is often useful to have a setting such that the reciprocal-lattice 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 non-coplanar reciprocal-lattice vectors and assigning these as the reciprocal-cell 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 reciprocal-space method is adopted by the Nonius CAD4 diffractometer software (CAD4 Manual, 1989). Alternatively, the auto-indexing' method originated by Sparks (1976, 1982) and Jacobson (1976) can be used whereby direct-lattice vectors are generated, again through an initial cell. Clegg (1984) has described an enhancement of the direct-lattice vector method so that the initial cell is used to produce direct-lattice 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 auto-indexing 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 reciprocal-lattice points (i.e. observed reflections) is reduced to finding elementary periods in one-dimensional 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 direct-axis vector, based upon all co-operating 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

Busing, W. R. & Levy, H. A. (1967). Angle calculations for 3- and 4-circle X-ray and neutron diffractometers. Acta Cryst. 22, 457–464.Google Scholar
CAD4 Manual (1989). Enraf–Nonius, Scientific Instruments Division, PB 483, NL-2600 AL, Delft, The Netherlands. Google Scholar
Clegg, W. (1984). Enhancements of the auto-indexing' method for cell determination in four-circle diffractometry. J. Appl. Cryst. 17, 334–336.Google Scholar
Duisenberg, A. J. M. (1992). Indexing in single-crystal diffractometry with an obstinate list of reflections. J. Appl. Cryst. 25, 92–96.Google Scholar
Hornstra, J. & Vossers, H. (1974). Philips Tech. Rundsch. 33, 65–78.Google Scholar
Jacobson, R. A. (1976). A single-crystal automatic indexing procedure. J. Appl. Cryst. 9, 115–118.Google Scholar
North, A. C. T., Phillips, D. C. & Mathews, F. S. (1968). A semi-empirical method of absorption correction. Acta Cryst. A24, 351–359.Google Scholar
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