A multisolution method of phase determination by combined maximization of entropy and likelihood. VI. Automatic likelihood analysis via the Student t test, with an application to the powder structure of magnesium boron nitride, Mg3BN3

Abstract

The multisolution method to solve powder structures ab initio from their X-ray diffraction data developed by Bricogne and Gilmore [Bricogne (1991). Acta Cryst. A47, 803–829; Gilmore, Henderson & Bricogne (1991). Acta Cryst. A47, 830–841] has been further tested by redetermination of the structure of the low-pressure phase of magnesium boron nitride, Mg3BN3, on which a previous attempt using the maximum-entropy (ME) procedure devised by Gull, Livesey & Sivia [Acta Cryst. (1987), A43, 112–117] had failed. In the successful application of the ME method presented here, the data were normalized using both overlapped and nonoverlapped reflections in the program MITHRIL [Gilmore (1984). J. Appl. Cryst. 17, 42–46; Gilmore & Brown (1988). J. Appl. Cryst. 21, 571–572]. After definition of the origin by the phase of a single reflection, seven reflections selected by a criterion of optimum second-neighbourhood enlargement were given permuted phases, thus generating 128 nodes of a phasing tree. Each node was subjected to constrained entropy maximization followed by the evaluation of a log-likelihood gain incorporating both overlapped and nonoverlapped reflections. These log-likelihood gains were analysed with the Student t test in which single-, double- and triple-phase indications were tested. Eight nodes survived the tests at the 2% significance level to give a subset of preferred nodes; the member of this subset with the highest log-likelihood gain gave a centroid map that revealed the positions of all the Mg, B and N atoms. Detailed examination of the phasing tree confirmed previous observations that the log-likelihood gain, not the entropy, is the most reliable criterion on which to base a multisolution phasing procedure.