Maximum-entropy-method analysis of neutron diffraction data

The maximum-entropy method (MEM) is a very powerful method for deriving accurate electron-density distributions from X-ray diffraction data. The success of the method depends on the fact that the electron density is always positive. In order to analyse neutron diffraction data by the MEM, it is necessary to overcome the difficulty of negative scattering lengths for some atoms, such as Ti and Mn. In this work, three approaches to the MEM analysis of neutron powder diffraction data are examined. The data, from rutile (TiO₂), have been collected previously and analysed by the Rietveld method [Howard, Sabine & Dickson (1991). Acta Cryst. B47, 462-468]. The first approach is to add an artificial large constant to the scattering-length density to maintain that density positive, then to subtract the same constant at the completion of the MEM analysis. This approach, however, proves unsuccessful since unrealistic density distributions result. In the second approach, the observed structure factors are amended so that the sign of the contribution from the Ti atoms is reversed. This method produces plausible maps of scattering-length density but suffers the disadvantage that the observations must be corrected by a model-dependent calculated factor before the MEM analysis can proceed. The third approach is based not on scattering-length densities but on nuclear densities, which are always positive. Two equations are obtained, one for atoms with nuclei of positive scattering length and the other for atoms with negative scattering length. From these two equations, the nuclear densities of Ti and O atoms can be calculated separately. This procedure, like its X-ray counterpart, requires no structural model. MEM analysis of the rutile data by this approach has been successfully completed. As expected, and in contrast to the electron-density distribution obtained by the MEM [Sakata, Uno, Takata & Mori (1992). Acta Cryst. B48, 591-598], the map shows both Ti and O nuclear densities localized in very small regions around the atomic centres. It is concluded that the MEM applied to neutron powder diffraction data is superior to conventional Fourier transformation and, in spite of the longer computation time, is very well worthwhile.