Symbolic Addition and Multisolution Methods

Ladd, M. F. C.; Palmer, Rex A.

doi:10.1007/978-1-4613-2979-4_3

Cited by 4 publications

(5 citation statements)

References 33 publications

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“…On the other hand, there is no guarantee that direct methods must be successful. For example, in a discussion by Ladd and Palmer (1980) on the application of direct methods to x-ray data sets, one finds a subsection: 'Success is not Guaranteed'. Sometimes, for inexplicable reasons, 'traditional' direct methods will not lead to a useful outcome.…”

Section: Direct Phase Determination In Crystallographymentioning

confidence: 99%

Applications of direct methods for structure determination to problems in electron and x-ray fibre diffraction of polymers

Dorset

2003

Rep. Prog. Phys.

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In this review, the prospect of determining polymer crystal structures directly from single crystal electron or x-ray powder/fibre diffraction intensity data is explored. When care is taken to collect the best data set from a polymer, i.e. sampling the most intense regions of the three-dimensional reciprocal lattice, structure analyses can be carried out effectively by the conventional direct phasing methods employed in small molecule crystallography, even though the data sets are relatively small. This means that constraints imposed by multiple beam interactions in electron diffraction and reflection overlap in x-ray diffraction often do not limit the structure analysis.Several three-dimensional single crystal electron diffraction structure analyses are presented to illustrate the problems encountered.These include: polyethylene, poly(ε-caprolactone), isotactic poly(butene-1) form III, isotactic polypropylene β-form, poly(p-oxybenzoate) and poly(ethylene sulfide). Techniques for sampling the complete reciprocal lattice are discussed. If an adequate number of data is collected, it is also possible to refine the structure by constrained least squares. In any case Fourier refinement is always useful.From powder x-ray diffraction the crystal structure of polyethylene has been determined by direct methods; from fibre x-ray intensities, this review reports the direct analyses of poly(ε-caprolactone), poly(ethylene terephthalate) and poly(tetramethyl-p-silphenylene siloxane). Rietveld methods might also be useful for the structure refinement.Direct structure analyses are advantageous in that they circumvent a priori assumptions about the molecular model and also the linkage bonds to base conformational refinement. Instead, individual atomic positions or molecular envelopes are observed directly in potential or electron density maps, to which the model can then be fit, a situation analogous to current practice in protein crystallography.

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Section: Direct Phase Determination In Crystallographymentioning

confidence: 99%

Applications of direct methods for structure determination to problems in electron and x-ray fibre diffraction of polymers

Dorset

2003

Rep. Prog. Phys.

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“…Identification of familiar patterns of atomic peaks in a density map, where there are also false ones, is a type of pattern recognition that leads to a successful conclusion in small molecule crystallography. However, it has been stated emphatically (Ladd, Palmer, 1980) that`success is not guaranteed'. This pattern recognition approach, of course, cannot be applied to protein crystallography at low resolution because there are no conservative`bonding' rules for the assembly of globular subunits.…”

Section: Discussionmentioning

confidence: 95%

Direct structure analysis in protein electron crystallography via the density histogram: deoxycholate-treated purple membrane

Dorset¹

2000

Zeitschrift Für Kristallographie - Crystalline Materials

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The crystal structure of bacteriorhodopsin in purple membrane treated with sodium deoxycholate (plane group p3, a 57.3 A) was solved ab initio in projection from previously published electron diffraction amplitudes to 6.2 A by direct methods for crystallographic phase determination. The best solution from the Sayre equation was identified by minimizing the skew properties of the cross-correlation of the experimental density histogram with the expected histogram. After Fourier refinement, the overall mean phase error is 43.3 for all 35 unique reflections or 16.9 for the 14 most intense maxima. For six clearly identified helix sites, the mean deviation of positions, when compared to the original model, was 0.7 A

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“…For crystals containing an inversion centre (centrosymmetric crystals), the problem reduces to determining only the sign of each structure factor since the Fourier coefficients are real quantities. Direct methods 12 can yield a solution to the phase problem, and centrosymmetric crystals containing racemic mixtures of chiral macromolecules like proteins have been grown to this end. 13–15 In principle however, experimental observation of phases from centrosymmetric crystals may one day become possible to exploit resonant inelastic X-ray scattering, 16,17 or by using magnetic reference layers as in neutron scattering.…”

Section: Introductionmentioning

confidence: 99%

Riemann Zeros and the Inverse Phase Problem

Tourigny

2013

Mod. Phys. Lett. B

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Finding a universal method of crystal structure solution and proving the Riemann hypothesis are two outstanding challenges in apparently unrelated fields. For centrosymmetric crystals however, a connection arises as the result of a statistical approach to the inverse phase problem. It is shown that parameters of the phase distribution are related to the non-trivial Riemann zeros by a Mellin transform.

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Symbolic Addition and Multisolution Methods

Cited by 4 publications

References 33 publications

Applications of direct methods for structure determination to problems in electron and x-ray fibre diffraction of polymers

Applications of direct methods for structure determination to problems in electron and x-ray fibre diffraction of polymers

Direct structure analysis in protein electron crystallography via the density histogram: deoxycholate-treated purple membrane

Riemann Zeros and the Inverse Phase Problem

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