Crystallographic Problem Research Articles

Fast quantum crystallography

AbstractTypical contemporary X‐ray crystallography delivers the geometries and, at best, the electron densities of molecules or periodic systems in the crystalline phase. Energies, electron momentum densities, and information relating to the pair density such as electron delocalization measures—all crucial to chemistry—are simply missed. Quantum crystallography (QCr) is an emerging line of research aimed at filling this gap by solving the crystallographic problem under the constraints of quantum mechanics. In this way, not only geometries and electron densities become experimentally accessible but also the entire panoply of quantum mechanical properties that are in the output of any quantum chemical software package. However, QCr remains limited to smaller systems (small molecules or small unit cells) due to the exponential bottleneck that plagues quantum mechanical calculations. When combined with a fragmentation technique, termed the “kernel energy method (KEM)”, QCr's reach to larger molecules is extended considerably to almost “any size”, that is, systems of up to many hundreds of thousands of atoms. KEM has made this doable with any chemical model and is capable of providing the entire quantum mechanics of large molecular systems. The smallness of theR‐factor adjudicates the accuracy of the quantum mechanics extracted from the crystallography.

Benchmark Problems for Phase Retrieval

In recent years, the mathematical and algorithmic aspects of the phase retrieval problem have received considerable attention. Many papers in this area mention crystallography as a principal application. In crystallography, the signal to be recovered is periodic and comprised of atomic distributions arranged homogeneously in the unit cell of the crystal. The crystallographic problem is both the leading application and one of the hardest forms of phase retrieval. We have constructed a graded set of benchmark problems for evaluating algorithms that perform this type of phase retrieval. The data, publicly available online from https://github.com/veitelser/phase-retrieval-benchmarks, is provided in an easily interpretable format. We also propose a simple and unambiguous success/failure criterion based on the actual needs in crystallography. Baseline runtimes were obtained with an iterative algorithm that is similar but more transparent than those used in crystallography. Empirically, the runtimes grow exponentially with respect to a new hardness parameter: the sparsity of the signal autocorrelation. We also review the algorithms used by the leading software packages. This set of benchmark problems, we hope, will encourage the development of new algorithms for the phase retrieval problem in general, and crystallography in particular.

Determination of atomic vacancies in InAs/GaSb strained-layer superlattices by atomic strain.

Determining vacancy in complex crystals or nanostructures represents an outstanding crystallographic problem that has a large impact on technology, especially for semiconductors, where vacancies introduce defect levels and modify the electronic structure. However, vacancy is hard to locate and its structure is difficult to probe experimentally. Reported here are atomic vacancies in the InAs/GaSb strained-layer superlattice (SLS) determined by atomic-resolution strain mapping at picometre precision. It is shown that cation and anion vacancies in the InAs/GaSb SLS give rise to local lattice relaxations, especially the nearest atoms, which can be detected using a statistical method and confirmed by simulation. The ability to map vacancy defect-induced strain and identify its location represents significant progress in the study of vacancy defects in compound semiconductors.

AgBiI4 as a Lead-Free Solar Absorber with Potential Application in Photovoltaics

AgBiI4 powder, crystals, and polycrystalline films were synthesized by sealed tube solid state reactions, chemical vapor transport (CVT), and solution processing, respectively, and their structural, optical and electronic properties are reported. The structure of AgBiI4 is based unambiguously upon a cubic close packed iodide sublattice, but it presents an unusual crystallographic problem: we show that the reported structure, a cubic defect-spinel, cannot be distinguished from a metrically cubic layered structure analogous to CdCl2 using either powder or single crystal X-ray crystallography. In addition, we demonstrate the existence a noncubic CdCl2-type polymorph by isolation of nontwinned single crystals. The indirect optical band gap of AgBiI4 is measured to be 1.63(1) eV, comparable to the indirect band gap of 1.69(1) eV measured for BiI3 and smaller than that reported for other bismuth halides, suggesting that structures with a close-packed iodide sublattice may give narrower band gaps than those with...

Uniqueness of the macromolecular crystallographic phase problem.

Uniqueness of the phase problem in macromolecular crystallography, and its relationship to the case of single particle imaging, is considered. The crystallographic problem is characterized by a constraint ratio that depends only on the size and symmetry of the molecule and the unit cell. The results are used to evaluate the effect of various real-space constraints. The case of an unknown molecular envelope is considered in detail. The results indicate the quite wide circumstances under which ab initio phasing should be possible.

NDTB‐1: A Supertetrahedral Cationic Framework That Removes TcO4− from Solution

A cubic thorium borate possesses a porous supertetrahedral cationic framework with extraframework borate anions. These anions are readily exchanged with a variety of environmental contaminants, especially those from the nuclear industry, including chromate and pertechnetate.

The uniqueness of determination of the space group symmetry by means of the vibrational spectroscopy methods

Vibrational spectroscopy methods are used to solve the most fundamental crystallographic problem—determination of crystal symmetry. It is shown that if X-ray diffraction methods do not make it possible to unambiguously choose the symmetry group, it can be successfully done on the basis of joint analysis of the vibrational spectroscopy and X-ray diffraction data. Irreducible representations of vibrations for all systems are systematized, and the cases are indicated where the ambiguity in assignment of crystal structures to the space groups with unified diffraction extinction rules can be avoided.

Affinity tags can reduce merohedral twinning of membrane protein crystals

This work presents a comparison of the crystal packing of three eukaryotic membrane proteins: human aquaporin 1, human aquaporin 5 and a spinach plasma membrane aquaporin. All were purified from expression constructs both with and without affinity tags. With the exception of tagged aquaporin 1, all constructs yielded crystals. Two significant effects of the affinity tags were observed: crystals containing a tag typically diffracted to lower resolution than those from constructs encoding the protein sequence alone and constructs without a tag frequently produced crystals that suffered from merohedral twinning. Twinning is a challenging crystallographic problem that can seriously hinder solution of the structure. Thus, for integral membrane proteins, the addition of an affinity tag may help to disrupt the approximate symmetry of the protein and thereby reduce or avoid merohedral twinning.

The decomposition of an orthogonal transformation as a product of reflections

In this work, an algorithm to decompose a given orthogonal transformation as a product of reflections through hyperplanes is presented. This in fact constitutes a constructive proof of a Cartan theorem, valid over any field K=Q, R or C. Clifford algebras are used to explicitly calculate the reflections that decompose a given orthogonal transformation. Our algorithm may have application in fields such as computer graphics or crystallography, and can also play an important role in orthogonal elimination and the solution of equations systems. An explicit example is provided and we apply our results to the crystallographic problem of coincidence lattices.

JECP/SP: a computer program for generating stereographic projections, applicable to specimen orientation adjustment in TEM experiments

1. The crystallographic problem Stereographic projection (SP) is an important crystallographic tool in carrying out electron diffraction experiments and in the analysis of crystal structures and defects. There are several commercial or public-domain computer programs available that allow users to draw stereographic projections of direct and reciprocal space. While these programs are often sufficient for regular use, there are always situations when the user will need to perform a certain operation that is not a feature of any of the existing programs. In the present work, a computer program was written for generating stereographic projections and was also extended as an application for specimen orientation adjustment in TEM experiments. Computer-assisted specimen orientation adjustment in TEM experiments would be especially useful under certain circumstances, for example, when (i) a severely stained specimen is observed and the visibility of Kikuchi line pairs is thus poor; (ii) a beam-sensitive specimen is examined, and thus it is essential to shorten the time spent on specimen tilting when the electron beam is illuminating the area of interest; (iii) a smallgrain-size specimen is investigated, for which even a slight orientation adjustment may cause the corresponding diffraction pattern to disappear due to a lateral sample shift; and (iv) a crystalline specimen is used in acquiring electron diffraction intensities of reachable zone-axis patterns for structure determination. In these cases, under the guidance of this software, the microscope can be kept in the image mode when the specimen is tilted. The holder is tilted to the computer-predicted angles and the illuminated area of the specimen is kept in the area of interest or in some neighboring area until the target tilt angles have been obtained, afterwhich it is moved to the area of interest.

CALCIPOW: a program for calculating the diffraction by disordered lamellar structures

1. The crystallographic problem A general feature of materials built of layers is the variety of the stacks; the phyllosilicate family is a good example of such diversity. In their crystals, the layers can be translated in different ways and rotated, and mixed layering can occur. As a consequence, depending on the nature and proportion of stacking faults, the reflections are more or less broadened and/or shifted in comparison with those expected for ordered crystals. The determination of the stacking parameters then requires a modelling of the experimental diffraction patterns. Concerning the mixed-layering problem, a theoretical description of the diffraction was first presented by Kakinoki & Komura (1965). The general formalism for the diffraction by crystals containing translational or rotational stacking faults was described by

Tcl/Tk-based programs. IV.CALCRYS: crystallographic calculator

1. The crystallographic problem Crystallographers need to make a number of simple algebraic calculations in order to obtain lengths, angles, define relations between molecules, transformations of the coordinates, etc. These calculations can be performed by general means, e.g. either with a pocket calculator or with a computer using standard mathematical packages. Another possibility is to have a specialized tool, tuned particularly to these calculations, easy to call and to use, and fast to address the problem.

PowderX: Windows-95-based program for powder X-ray diffraction data processing

The crystallographic problem: A Windows-95-based program (PowderX) has been written for powder X-ray data processing and analysis. It can be used for plotting X-ray patterns, data smoothing, background subtraction, 2 elimination, peak search, indexing and zero-angle error correction. PowderX can also be used for data format conversions to prepare the input data for Rietveld re®nement and structure determination programs such as DBWS, FULLPROF, GSAS, SIMPRO and EXPO.

PROFVAL: functions to calculate powder-pattern peak profiles with axial-divergence asymmetry

The crystallographic problem: In a powder diffractometer, the curvature of the Debye---Scherrer rings causes an asymmetry in the peak profiles as the Bragg angle approaches either 0 or 180 °. For accurate Rietveld refinement, this asymmetry, which is known as axial divergence, must be modeled correctly. Traditionally, empirical ad hoc functions such as a split-Pearson-VII function or multiple overlapping Gaussian peaks have been employed. Finger e t a / . (1994) extended the method of van Laar & Yellon (1984) for convolution of the ring shape with an intrinsic peak profile a procedure that describes extremely asymmetric profiles and involves only physically meaningful parameters. To date, however, only the program GSAS (Larson &von Dreele, 1990) has incorporated this method. This contribution makes available tested code for the convolution of the asymmetry function with the intrinsic line function in the hope that more Rietveld codes will incorporate this method.

Comment on "Filling three-dimensional space with tetrahedra: A geometric and crystallographic problem"

We discuss the filling of three-dimensional Euclidean space with tetrahedra having two possible kinds of vertex atoms and three kinds of edge lengths (interatomic distances) as described by K. Y. Szeto and J. Villain [Phys. Rev. B 36, 4715 (1987)]. We identify certain periodic structures corresponding to the authors' third solution with members of the family of tetrahedrally close-packed (tcp) alloy structures [including Frank-Kasper (FK) phases] and discuss factors limiting the generation of additional tcp alloy structure types with the procedure described.

Filling three-dimensional space with tetrahedra: A geometric and crystallographic problem

Exact, nonperiodic, close-packed structures with randomness that can fill three-dimensional space are found. We find many solutions for distances between atoms that satisfy the necessary conditions of filling three-dimensional space with tetrahedra formed with two kinds of atoms. Only three solutions that also satisfy the sufficient condition of filling three-dimensional space are discussed. They all involve periodic as well as nonperiodic structures resulting from the random stacking of layers. One solution corresponds to the NaCl structure. Another solution exhibits both tetragonal and hexagonal symmetry, which violates crystallography. A third solution has a unit cell whose surface exhibits distorted pentagonal symmetry and whose elementary unit is a 44-face polyhedron. Suggestions for a possible growth model in three dimensions with tetrahedra are discussed.