Indexing with the successive dichotomy method, DICVOL04

D. Louër¹ and A. Boultif²

¹Laboratoire de Chimie du Solide et Inorganique Moléculaire (UMR 6511 CNRS), Institut de Chimie,

Université de Rennes, Avenue du Général Leclerc, 35042 Rennes, France,

²Département de Cristallographie, Institut de Physique, Université Mentouri, 25000 Constantine, Algeria

Ab initio powder pattern indexing is required in many applications of the powder method, such as ab initio structure determination from powder X-ray diffraction data. The objective of pattern indexing is to reconstruct the 3-dimensional reciprocal lattice from the radial distribution of d-spacings in the diffraction pattern. In practice, the methodology consists in finding the crystal data of the material, i.e. the dimensions of the unit cell and lattice symmetry. Among approaches reported for pattern indexing, the efficiency of the successive dichotomy method introduced by Louër and Louër [1] has been proved by many years usage. The method is based on the variation, in parameter space, of the lengths of cell edges and inter-axial angles by finite ranges, which are progressively reduced according to a dichotomy algorithm. The absolute error on peak measurements is incorporated in the procedure, without any re-evaluation during the numerical calculations. Solutions are then searched exhaustively in an n-dimensional space, from n = 1 (cubic lattice) to n = = 6 (triclinic lattice). The method strategy is based on the search of solutions with smallest cell volumes. Data precision is a major factor of successful indexing and the de Wolff figure of merit acts as solution filter. The development of the computer program, using the dichotomy principle, has been carried out by stages according to the progresses of computing technology over forty years. The most recent program DICVOL91 [2] has been used for indexing hundreds of powder diffraction patterns, from which subsequent structure determinations were often carried out. Although the selection of the input parameters offers strategy choices to the user, indexing practices have revealed the need for new options.

The new facilities implemented in DICVOL04 [3] include (i) a tolerance for unindexed diffraction lines, (ii) the refinement of the ‘zero-point’ shift, (iii) an a priori analysis of input data to detect the presence of significant zero-point error, (iv) the use of the reduced cell concept to identify equivalent solutions in monoclinic and triclinic systems and (v) a reviewing of all available peak positions from the unit cell parameters found from, generally, the first twenty lines (if no dominant zone is present). Additionally, different strategies have also been applied, particularly to reduce the risk to miss a solution because of metric lattice singularity. Default values have been adapted to more convenient parameters according to the data precision available with high resolution powder diffractometers.

DICVOL04 has been tested with many powder data sets, most of them found in literature, e.g. in the NBS Monograph No. 25, the 71 data sets of Section 20 and all triclinic examples reported in the entire Monograph, powder data of pharmaceutical compounds collected with the capillary technique and monochromatic X-rays, and difficult cases reported in recent publications. The benefit of zero-shift refinement and a priori evaluation of zero-error for in situ powder data has been shown. The success rate of DICVOLO4 is high. Nevertheless, it should be reminded that data quality remains a major requirement. This is due to the nature of the mathematical problem which involves the restoration of a 3-dimensional object from 1-dimensional data.

[1] D. Louër, M. Louër, J. Appl. Cryst. 5 (1972) 271-275.

[2] A. Boultif, D. Louër, J. Appl. Cryst. 24 (1991) 987-993.

[3] A. Boultif, D. Louër, Submitted for publication.