Acta Crystallographica Section A最新文献

A new two-step algorithm is developed for reconstructing the three-dimensional diffraction intensity of a globular biological macromolecule from many experimentally measured quantum-noise-limited two-dimensional X-ray laser diffraction patterns, each for an unknown orientation. The first step is classification of the two-dimensional patterns into groups according to the similarity of direction of the incident X-rays with respect to the molecule and an averaging within each group to reduce the noise. The second step is detection of common intersecting circles between the signal-enhanced two-dimensional patterns to identify their mutual location in the three-dimensional wavenumber space. The newly developed algorithm enables one to detect a signal for classification in noisy experimental photon-count data with as low as ~0.1 photons per effective pixel. The wavenumber of such a limiting pixel determines the attainable structural resolution. From this fact, the resolution limit due to the quantum noise attainable by this new method of analysis as well as two important experimental parameters, the number of two-dimensional patterns to be measured (the load for the detector) and the number of pairs of two-dimensional patterns to be analysed (the load for the computer), are derived as a function of the incident X-ray intensity and quantities characterizing the target molecule.

The topological complexity of a crystal structure can be quantitatively evaluated using complexity measures of its quotient graph, which is defined as a projection of a periodic network of atoms and bonds onto a finite graph. The Shannon information-based measures of complexity such as topological information content, I(G), and information content of the vertex-degree distribution of a quotient graph, I(vd), are shown to be efficient for comparison of the topological complexity of polymorphs and chemically related structures. The I(G) measure is sensitive to the symmetry of the structure, whereas the I(vd) measure better describes the complexity of the bonding network.

The results of a high-resolution study of the (002, 113, 11 ̅1) four-beam diffraction in Si are presented. The incident synchrotron radiation beam was highly monochromated and collimated with a multi-crystal arrangement in a dispersive setup in both vertical and horizontal planes, in an attempt to experimentally approach plane-wave incident conditions. The Renninger scheme was used with the forbidden reflection reciprocal-lattice vector 002 normal to the crystal surface. The azimuthal and polar rotations were performed in the crystal surface plane and the vertical plane correspondingly. The polar angular curves for various azimuthal angles were measured and found to be very close to theoretical computer simulations, with only a small deviation from the plane monochromatic wave. The effect of the strong two-beam 002 diffraction was observed for the first time with the maximum reflectivity close to 80%. The structure factor of the 002 reflection in Si was experimentally determined as zero.

Two phasing equations based on the Fourier syntheses δ(P) = T(-1)[(E(2) - )exp(iφ)] and δ(M) = T(-1)[(E - )exp(iφ)] were recently described [Rius (2012). Acta Cryst. A 68, 77-81] (E is the quasi-normalized structure factor and is the average over all reflections). These equations were found by comparison with the direct methods origin-free modulus sum function and constitute the core of the `δ recycling' phasing procedure. The derivation of these phasing equations from the minimization of a residual (R(P)) between two differently calculated density functions (one of them including the positivity constraint) is shown.

A central problem in crystallography is crystal structure determination directly from diffraction intensities. For structures of small molecules, this problem has been addressed by probabilistic direct methods that allow one to obtain the structure coordinates with a high degree of certainty given a sufficiently large set of intensities. In contrast, deterministic algebraic methods that could guarantee a solution and may be applicable to macromolecules have not yet emerged. In this study a basic algebraic question is posed: how many crystal structures can be obtained from a given set of intensities? Recently, by using a new origin definition and the method of elementary symmetrical polynomials, all small (N ≤ 4 atoms) one-dimensional crystal structures that could be obtained from the minimum set of N - 1 lowest-resolution intensities were enumerated. Here, by using methods of modern algebraic geometry the maximum number of one-dimensional crystal structures that can be determined from the minimum set of intensities for N > 4 is obtained. It is demonstrated that this ambiguity increases exponentially with the increasing number of atoms in the structure N (~4(N)/N(3/2) for N >> 1) and includes non-homometric structures. Therefore, a minimum set of intensities, even in principle, is insufficient for structure determination for all but very small structures.

ELMAM2 is a generalized and improved library of experimentally derived multipolar atom types. The previously published ELMAM database is restricted mostly to protein atoms. The current database is extended to common functional groups encountered in organic molecules and is based on optimized local axes systems taking into account the local pseudosymmetry of the molecular fragment. In this approach, the symmetry-restricted multipoles have zero populations, while others take generally significant values. The various applications of the database are described. The deformation electron densities, electrostatic potentials and interaction energies calculated for several tripeptides and aromatic molecules are calculated using ELMAM2 electron-density parameters and compared with the former ELMAM database and density functional theory calculations.

Polysynthetic Brazil twinning in α-quartz, which occurs commonly in amethyst, is interpreted in the literature as having its composition planes parallel to one of the faces of the major rhombohedron r. It is shown that, instead, the composition planes are parallel to one of the faces of the minor rhombohedron z. The proposed translation 0.4547a between neighbouring lamellae leads to binding distances and binding angles across the composition plane that differ less from their bulk values than for the translation 0.5a proposed in the literature. Density functional calculations show that the energy of the unrelaxed polysynthetic twin is lower for the proposed translation. They also show that relaxation increases the thickness of the polytwin by 4 pm per composition plane.

Three decades after their discovery, the unique long-range structure of quasicrystals still poses a perplexing puzzle. The fact that some ancient Islamic patterns share similar quasi-periodic symmetries has prompted several scientists to investigate their underlying geometry and construction methods. However, available structural models depend heavily on local rules and hence they were unable to explain the global long-range order of Islamic quasi-periodic patterns. This paper shows that ancient designers, using simple consecutive geometry, have resolved the complicated long-range principles of quasi-periodic formations. Derived from these principles, a global multi-level structural model is presented that is able to describe the global long-range translational and orientational order of quasi-periodic formations. The proposed model suggests that the position of building units, locally and globally, is defined by one framework, and not tiled based on local rules (matching, overlapping or subdividing). In this way, quasi-periodic formations can grow rapidly ad infinitum without the need for any defects or mismatches. The proposed model, which presents a novel approach to the study of quasi-periodic symmetries, will hopefully provide a deeper understanding of the structure of quasicrystals at an atomic scale, allowing scientists to achieve improved control over their composition and structure.

A new Fourier cycling phasing method is proposed based on the mathematical principle of the global minimization. In reciprocal space, the Fourier coefficient is of a mixed form of the normalized structure factors (2E(o)(2) - E(c)(2))E(c), while in direct space the Fourier map is modified with a peak-picking procedure. This method does not use any preliminary information and does not rely on any critical parameter; it can start with either randomly assigned phases or fixed phases (all zeros). This method performs significantly better than the commonly used forms of Fourier cycling.

This article quantitatively reconciles crystallographic and mechanics approaches to lattice refinement as part of X-ray diffraction procedures. The equivalence between the refinement based on unit-cell parameters to that based on a lattice deformation tensor is established from a fixed reference configuration. Justification for the small strain assumption, commonly employed in X-ray diffraction based stress analysis, is also derived. It is shown that relations based on infinitesimal strains are correct to within an error of quadratic order in strain. This error may be important to consider for high-precision or high-strain experiments. It is hoped that these results are of use for facilitating communication and collaboration between crystallography and experimental mechanics communities, for studies where X-ray diffraction data are the fundamental measurement.