Solving Crystal Structures without Fourier Mapping. I. Centrosymmetric Case

Pilz, Katrin; Fischer, Karl F.

doi:10.1107/s0108767397016176

Cited by 6 publications

(4 citation statements)

References 5 publications

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“…Leaving aside problems with obtaining K, the method is deductive and leads either to one unambiguous solution (plus its enantiomer) or to a finite set of all possible structures. Individual s(x j ) can be obtained from Monte Carlo calculations as described for the p1 case (Pilz & Fischer, 1998). For practical purposes, errors in the absolute scale factor K apparently play a significant role only if K is wrong by more than 10% (i.e.…”

Section: Discussion Conclusionmentioning

confidence: 99%

“…(All B a are assumed identical and known.) By analogy to Ott (1928) and also Pilz & Fischer (1998), a polynomial R(a) in a exp 2pix…”

Section: One-dimensional Casementioning

confidence: 98%

“…Solution of the algebraic phase problem is achieved on a basis similar to sign determination for the centrosymmetric case. Because j may have any value in the range Àp; :::; 0; :::; p (instead of just one of two for centrosymmetry), some differences occur in the determinants D 1 n corresponding to Equation (3a) compared with Equation (13) of Pilz & Fischer (1998)…”

Section: One-dimensional Casementioning

confidence: 98%

“…Equation (3) is a short-hand expression for Newton's relations (see Pilz & Fischer, 1998) which is suitable for easy handling with a computer.…”

Section: One-dimensional Casementioning

confidence: 99%

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Solving crystal structures without Fourier mapping. II. Non-centrosymmetric case

Pilz¹,

Fischer²

2000

Zeitschrift Für Kristallographie - Crystalline Materials

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An algebraic method for the determination of one-dimensional centrosymmetric structures (or projections) of equal atoms has been extended to crystals without a centre of symmetry. The technique is an (almost) ab initio one, provides higher resolution than a Fourier map using the same number of structure factors, and permits either a unique solution compatible with the data employed or provides all possible solutions. It appears suitable in particular for partial structures with not too many atoms per asymmetric unit (e.g. anomalous scatterers in a ªbiologicalº crystal structure). It only needs rather selected data: from central reciprocal lattice rows for one-dimensional projections plus from (at least) two rows parallel to the first ones for three-dimensional structures.

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Section: Discussion Conclusionmentioning

confidence: 99%

“…(All B a are assumed identical and known.) By analogy to Ott (1928) and also Pilz & Fischer (1998), a polynomial R(a) in a exp 2pix…”

Section: One-dimensional Casementioning

confidence: 98%

Section: One-dimensional Casementioning

confidence: 98%

“…Equation (3) is a short-hand expression for Newton's relations (see Pilz & Fischer, 1998) which is suitable for easy handling with a computer.…”

Section: One-dimensional Casementioning

confidence: 99%

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Solving crystal structures without Fourier mapping. II. Non-centrosymmetric case

Pilz¹,

Fischer²

2000

Zeitschrift Für Kristallographie - Crystalline Materials

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

Zimmermann

Fischer

2009

Acta Cryst Sect A

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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.

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Conditional ambiguity of one-dimensional crystal structures determined from a minimum of diffraction intensity data

Shkel¹,

Lee²,

Tsodikov³

2011

Acta Cryst Sect A

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When the number of intensities greatly exceeds the number of unknown atomic coordinates, the problem of obtaining a crystal structure from the intensities is overdetermined and, for a sufficiently small structure, a chemically meaningful solution can be found by direct methods. A difficulty in determining a structure has been historically attributed to the non-uniqueness of such a structure owing to multiple, or homometric, structures that yield the same set of intensities. The number of homometric structures has not been rigorously analyzed owing to the complexity of this problem. By using the method of elementary symmetric polynomials with a new origin definition, one-dimensional crystal structures of a small number of identical atoms (N < 5), determined from a minimum (N - 1) of the lowest-resolution intensities, are enumerated. It is demonstrated that such a structure is unique for N ≤ 3. Interestingly, for N = 4, the structure can be determined either uniquely or twofold ambiguously, depending on the intensity values. These results suggest that, even for larger structures, a minimum set of (or not many more) accurately measured intensities can yield a unique structure.

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Solving Crystal Structures without Fourier Mapping. I. Centrosymmetric Case

Cited by 6 publications

References 5 publications

Solving crystal structures without Fourier mapping. II. Non-centrosymmetric case

Solving crystal structures without Fourier mapping. II. Non-centrosymmetric case

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

Conditional ambiguity of one-dimensional crystal structures determined from a minimum of diffraction intensity data

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