Novel treatment of the experimental data in the application of the maximum-entropy method to the determination of the electron-density distribution from X-ray experiments

Dysnomia, a computer program for maximum-entropy method (MEM) analysis and its performance in the MEM-based pattern fitting

“…Therefore, electron-or nuclear-density distributions in real space smooth down with increasing order, x (de Vries et al, 1994;Palatinus and van Smaalen, 2002). Weighting of d j 4 is reported to be the best in the case of single-crystal X-ray diffraction (de Vries et al, 1994).…”

“…., with n = 2 corresponding to the conventional F constraint, λ n is the relative weight for each of C Fn , N F is the number of reflections, m n is the even nth-order central moment of normal (Gaussian) distribution, w j is the weighting factor, and C wn is the criterion for convergence (Izumi and Momma, 2011). In addition, Dysnomia allows us to impose static weighting factors, w j , on σ(h j )'s on the basis of the lattice-plane spacing, d j (de Vries et al, 1994), or arbitrary weighting factors (Momma and Izumi, 2011a). Separation of w j from σ(h j ) makes it easier to separate the effect of weighting from differences in the overall scale of σ(h j ).…”

“…One of the errors is caused by a seriestermination effect arising from the availability of only a limited number of reflections (Jauch, 1994;de Vries et al, 1996;Palatinus and van Smaalen, 2002), although this effect is much less significant in MEM than in Fourier synthesis. On the analysis of X-ray diffraction data, MEM also tends to give large residual errors for some low-Q reflections that contribute most significantly to information entropy, S. Such a tendency of MEM often generates artifacts in calculated electron densities even if accurate standard uncertainties, σ(h j ), of F obs (h j )'s are obtainable (de Vries et al, 1994;Palatinus and van Smaalen, 2002). This problem can be partly solved either by changing a weighting scheme for the F constraint (de Vries et al, 1994;Palatinus and van Smaalen, 2002) or by imposing a constraint based on higher-order moments of normalized residuals of structure factors, ΔF j = |F obs (h j ) − F(h j )|/σ(h j ), rather than their second-order moment (Palatinus and van Smaalen, 2002).…”

The Influence of the Completeness of the Data Set on the Charge Density Obtained with the Maximum-Entropy Method. A Re-examination of the Electron-Density Distribution in Si

“…Recently, it has been claimed that non-physical fine features found in the MEM density are probably due to the non-uniform residual distributions of X 2 (Jauch & Palmer, 1993). From the simulation using the model charge density of a hypothetical crystal, De Vries, Briels & Feil (1994) have proposed a weighting scheme that leads to more uniform residual distributions and gives a smoother density. It is plausible that many factors may influence the MEM charge densities.…”

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

“…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).…”