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  • 1
    Electronic Resource
    Electronic Resource
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 51 (1995), S. 450-455 
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: Diffraction experiments provide information on the Fourier components of microscopic density distributions in crystals. To obtain the spatial densities themselves, an inverse Fourier problem has to be solved. The procedure is complicated by the presence of noise and incompleteness of the data. The application of the maximum-entropy (MaxEnt) principle was a breakthrough in density reconstruction, allowing high-quality density maps to-be obtained without involving any a priori information concerning what the reconstructed density should look like. In this work, a procedure is proposed that incorporates a priori (e.g. theoretical) information into MaxEnt reconstructions of spin density distributions. It allows, on the one hand, the evaluation of the existing density models and, on the other, the precise investigation of what new information the experiment brings. Unlike traditional parameter-refinement techniques, the new method does not impose any strict constraints on the density to be reconstructed and is thus much more flexible. At the same time, it suppresses artifacts and yields high-quality density maps. The advantages of the new methods are illustrated by an example of spin density reconstruction based on real polarized neutron diffraction data.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 48 (1992), S. 244-246 
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: The inverse Fourier transform is a most prominent and widely used tool in crystallography. Used to retrieve Patterson densities and densities from measured intensities or phased structure factors, it is well known to suffer from major drawbacks among which is the neglect of known error bars. In this paper, it is shown how to incorporate measured experimental error bars in the computation of the Fourier syntheses. This simple novel procedure should be useful when a proper maximum-entropy analysis proves infeasible, due to either a high dynamical range of the data or a high dimension of the direct space in which the density is sought (at least five or six dimensions for quasicrystallography).
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 51 (1995), S. 295-300 
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: When density distributions in crystals are reconstructed from three-dimensional diffraction data, a problem sometimes occurs when the spatial resolution in one given direction is very small compared to that in perpendicular directions. In this case, a two-dimensional projected density is usually reconstructed. For this task, the conventional Fourier inversion method only makes use of those structure factors measured in the projection plane. All the other structure factors contribute zero to the reconstruction of a projected density. On the contrary, the maximum-entropy method uses all the three-dimensional data, to yield three-dimensional-enhanced two-dimensional projected density maps. It is even possible to reconstruct a projection in the extreme case when not one structure factor in the plane of projection is known. In the case of poor resolution along one given direction, a Fourier inversion reconstruction gives very low quality three-dimensional densities `smeared' in the third dimension. The application of the maximum-entropy procedure reduces the smearing significantly and reasonably well resolved projections along most directions can now be obtained from the MaxEnt three-dimensional density. To illustrate these two ideas, particular examples based on real polarized neutron diffraction data sets are presented.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 52 (1996), S. 397-407 
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: A two-channel maximum-entropy method (MEM), first used to enhance magnetization densities from phased polarized neutron data by Papoular & Gillon [(1990). Europhys. Lett. 13, 429–434], has been applied to the electron deformation density. The resulting entropic densities are compared with standard deformation densities and with dynamic and static deformation maps obtained from multipole refinements. The procedure is illustrated with simulated and real single-crystal X-ray data sets on the molecular crystal of α-glycine. Both a uniform prior and a prior equal to the MEM-enhanced dynamic model deformation density are used in the MEM procedure, the result of which does not depend on the starting density. The method is judged by the appearance of the resulting maps and the values of the molecular dipole moment before and after the MEM. Compared with the conventional deformation density, the MEM procedure sharpens the peaks in the bond but flattens the weaker features, especially when a uniform prior is used. The dipole-moment criterion shows the non-uniform prior to be preferable to the uniform prior in reproducing electrostatic properties. The usefulness of the MEM in charge-density analysis remains open to discussion.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 51 (1995), S. 739-746 
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: The maximum-entropy method (MEM) for structure determination is applied on the plastically crystalline phases of KOH and KOD characterized by delocalized H+ or D+ ions in a simple rock salt structure. The structure factors measured by neutron diffraction already give a consistent picture of the hydrogen distribution by conventional Fourier and modelling methods, which helps to understand the merit of the MEM. Owing to the negative sign of the scattering length of hydrogen, the MEM can, in a limited sense `model free', separate off the hydrogen density distribution. It is found, however, that the MEM cannot be applied naively for these compounds and reasons are given why the uniform density as a priori information has to be given up in the present case. The a priori information procedure to be used is discussed. In addition, and perhaps unexpectedly, the deuterium density is also obtained.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    [S.l.] : International Union of Crystallography (IUCr)
    Acta crystallographica 47 (1991), S. 293-295 
    ISSN: 1600-5724
    Source: Crystallography Journals Online : IUCR Backfile Archive 1948-2001
    Topics: Chemistry and Pharmacology , Geosciences , Physics
    Notes: The formulae for Fourier and inverse Fourier transforms have been generalized to take into account crystal symmetry using group theory. When the Fourier components (structure factors) are restricted to a given reciprocal plane, the two-dimensional inverse Fourier transform yields the projection of the charge/spin density in the unit cell in direct space, parallel to the axis normal to that plane. The formulae above are further generalized to this two-dimensional case. The latter case is central to a polarized neutron diffraction data analysis using maximum entropy.
    Type of Medium: Electronic Resource
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