Maximum-entropy-method analysis of neutron diffraction data

Abstract

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][463][464][465][466][467][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 modeldependent calculated factor before the MEM analysis can proceed. The third approach is based not on scatteringlength 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.

A General Maximum-Entropy Method for Model-Free Reconstructions of Magnetization Densities from Polarized Neutron Diffraction Data

“…Gull & Daniel, 1978;Wilkins, 1983;Sakata & Sato, 1990;Sakata, Uno, Takata & Howard, 1993;Kumazawa, Takata & Sakata, 1995). These can be quite slowly converging and unstable, so that second-order variants exist (e.g.…”

“…pj is not positive definite) so that the entropy functional (3) must be modified. Various suggestions have been made (Sakata, Uno, Takata & Howard, 1993;Gull & Skilling, 1989;Steenstrup & Hansen, 1994) and our spin-density reconstructions are based on the MEMSYSIII subroutine library (Gull & Skilling, 1989), which we have modified to handle the complex non-linear case. It treats the case of nonpositive-definite distributions by assuming that a map may be expressed as the difference between two subsidiary positive-definite distributions, leading to a modified form for S (Gull & Skilling, 1989).…”

“…Many are restricted to linear reconstruction problems, where the Fourier coefficients are directly measurable (e.g. Zhuang, Ostevold & Haralick, 1987;Sakata & Sato, 1990;Kumazawa, Kubota, Takata & Sakata, 1993;Sakata, Uno, Takata & Howard, 1993;De Vries, Briels & Feil, 1994;Steenstrup & Hansen, 1994;Takata, Sakata, Kumazawa, Larsen & Iversen, 1994;Kumazawa, Takata & Sakata, 1995) or are specialized to the phase problem in crystallography (e.g. Wilkins, Varghese & Lehmann, 1983;Wilkins, 1983;Navaza, 1985;Gilmore, Bricogne & Bannister, 1990).…”

Application of Maximum Entropy Method to Semiconductor Engineering

“…On the other hand, MEM is tolerant to noise and provides only positive electron density, which enables us to determine the precise position of light elements such as hydrogen. In other MEM application examples, it has been used to extract the decay constant distribution from fluorimetry [8] and to analyze X-ray scattering [9,10], neutron diffraction [11,12] and scattering [13,14]. In such applications it provides very detailed structural information.…”

The Two-Channel Maximum-Entropy Method Applied to the Charge Density of a Molecular Crystal: α-Glycine

“…The maximum-entropy method has been proposed as the method of choice for enhancement of the electron density beyond the experimental resolution (Sakata & Sato, 1990;Sakata, Uno, Takata & Howard, 1993) and would thus appear the method of choice to optimize the information that can be extracted from an experimental data set and to produce the least possible biased density. Diffraction techniques make wide use of Fourier syntheses to obtain model-free information about various scattering densities from sparse and noisy sets of structure factors, provided that the latter are suitably phased.…”