Structure determination without Fourier inversion. V. A concept based on parameter space

Abstract

The parameter-space concept for solving crystal structures from reflection amplitudes (without employing or searching for their phases) is described on a theoretically oriented basis. Emphasis is placed on the principles of the method, on selecting one of three types of parameter spaces discussed in this paper, and in particular on the structure model employed (equal-atom point model, however usually reduced to one-dimensional projections) and on the system of 'isosurfaces' representing experimental 'geometrical structure amplitudes' in an orthonormal parameter space of as many dimensions as unknown atomic coordinates. The symmetry of the parameter space as well as of the imprinted isosurfaces and its effect on solution methods is discussed. For point atoms scattering with different phases or signs (as is possible in the case of X-ray resonant or of neutron scattering) it is demonstrated that the 'landscape' of these isosurfaces remains invariant save certain shifts of origin known beforehand (under the condition that all atomic scattering amplitudes have been reduced to 1 thus meeting the requirement of the structure model above). Partly referring to earlier publications on the subject, measures are briefly described which permit circumventing an analytical solution of the system of structure-amplitude equations and lead to either a unique (unequivocal) approximate structure solution (offering rather high spatial resolution) or to all possible solutions permitted by the experimental data used (thus including also all potential 'false minima'). A simple connection to Patterson vectors is given, also a first hint on data errors. References are given for practical details of various solution techniques already tested and for reconstruction of three-dimensional structures from their projections by 'point tomography'. We would feel foolish if we tried to aim at any kind of 'competition' to existing methods. Having mentioned 'pros and cons' of our concept, some ideas about potential applications are nevertheless offered which are mainly based on its inherent resolution power though demanding rather few reflection data (use of optimal intensity contrast included) and possibly providing a result proven to be unique.

Neutron radiography study of the dehydration kinetics of smectites in moulding sands

“…An AFM-study of the surface topography of anatase single crystals. Andreas Bürger a , Uta Magdans a , Hermann Gies a , a Department of Geology, Mineralogy and Geophysics, Ruhr-University of Bochum, Germany E-mail: andreas.buerger@rub.de Grazing Incidence X-ray diffraction (GIXRD) is a surface Xray diffraction method for the analysis of mineral surfaces and interfaces on atomic scale [1,2]. This method requires very flat and large single crystal surfaces with a rms-roughness well below 10 nm.…”

“…A quantitative evaluation of the simulation data will be presented. In the parameter space concept developed for the determination of 1-dimensional structures or structure projections without Fourier inversion, an arrangement of m equal scatterers is represented by a vector t(x 1 , x 2 , … x m ) defined in space P m [1,2]. Given a suitable number n ≥ m of observed intensities of reflections h, the principal objective of this approach is to find those test vectors t t for which the best overall agreements between the observed e(h) obs and the model e(h) calc are obtained.…”

Xtal-xplore-R: a graphical tool for exploring the residual function involved in crystal structure determination

“…Fischer and Kirfel have suggested another approach, using projection techniques and the symmetry of iso-surfaces to reduce the phase volume of the parameter space (Fischer et al, 2005;Zimmermann & Fischer, 2009). Yet, their approach seems most suitable for high-resolution single-crystal (neutron or synchrotron) diffraction data.…”

Surface and embedded 2-dim structures: a direct-space approach to modelling

“…A quantitative evaluation of the simulation data will be presented. [1,2]. Given a suitable number n ≥ m of observed intensities of reflections h, the principal objective of this approach is to find those test vectors t t for which the best overall agreements between the observed e(h) obs and the model e(h) calc are obtained.…”