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

• Published in: Acta crystallographica. Section A, Foundations of crystallography Vol. 65; no. 6; pp. 443 - 455
• Main Authors: Zimmermann, Helmuth, Fischer, Karl F.
• Format: Journal Article
• Language: English
• Published: 5 Abbey Square, Chester, Cheshire CH1 2HU, England International Union of Crystallography 01.11.2009; Wiley Subscription Services, Inc
• Subjects: Experimental data; Fourier analysis; homometric solutions; Parameter estimation; parameter spaces; phase problem eliminated; structure determination; unique solution
• ISSN: 0108-7673; 1600-5724; 1600-5724; 2053-2733
• DOI: 10.1107/S0108767309030293

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.