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The practical use of the average and difference intensities of Friedel opposites at different stages of structure analysis has been investigated. It is shown how these values may be properly and practically used at the stage of space-group determination. At the stage of least-squares refinement, it is shown that increasing the weight of the difference intensities does not improve their fit to the model. The correct form of the coefficients for a difference electron-density calculation is given. In the process of structure validation, it is further shown that plots of the observed and model difference intensities provide an objective method to evaluate the fit of the data to the model and to reveal insufficiencies in the intensity measurements. As a further tool for the validation of structure determinations, the use of the Patterson functions of the average and difference intensities has been investigated and their clear advantage demonstrated.

Crystallographic molecular replacement method is the key tool to define an atomic structure of nucleic acids. Frequently nucleic acids are packed forming continuous helices in the crystal. This arrangement of individual molecules in "infinite" pseudo helical structures in crystal may be the reason why the molecular replacement fails to find a unique position of the search atomic model as the method requires. The Patterson function, calculated as a Fourier series with diffraction intensities, has auxiliary peaks for such a molecular packing. Those near the origin peak indicate the orientation of the helices. The coordinates of other peaks are related to the molecular position and the rotation angle between two such "infinite" helices. Thus, the peak analysis allows getting molecular position even without a search model. An intelligent selecting and averaging of the phase sets corresponding to multiple probable positions of the search model again result in a unique solution but in the form of a Fourier synthesis and not a model. This synthesis can be used then to build an atomic model as it is the case for usual phasing methods.

A novel method to correct a partial sampling effect, due to the hexagonal filament array of a statistical superlattice form, on the thick (myosin)-filament-based layer lines in X-ray diffraction patterns from higher-vertebrate striated muscle has been developed using the cylindrically averaged difference Patterson function [ΔQ(r, z)]. The method involves cutting off the inter-filament vector peaks that appear in the radial region beyond ∼32 nm on the ΔQ(r, z) map calculated from the observed layer-line intensities, and then deducing the single-myosin-filament transforms by inverse Fourier transformation of the truncated ΔQ(r, z). The accuracy of the cut-off method was tested using a single-myosin-filament model and a hexagonal filament-array model with a size of one superlattice unit cell. The layer-line intensities calculated from the truncated ΔQ(r, z) of the hexagonal filament-array model showed few sampling peaks, the layer lines being effectively coincident with those from the single-filament model except for the intensities close to the meridian. Some residual differences were caused by the face-to-face inter-crossbridge vectors between closest neighboring filaments, which correspond to ∼27.5% of the total number of crossbridge vectors in the truncated ΔQ(r, z) map, but the face-to-face inter-crossbridge vectors contributed mainly to the intensities close to the meridian. Their remnant off-meridional layer-line intensity components did not significantly affect a search for the optimum azimuthal orientation of myosin crossbridges in the resting state of muscle.

Theoretical concepts and the practical application of the three-dimensional pair distribution function (3D-PDF) method and its variant, the three-dimensional Δ-pair distribution function (3D-ΔPDF) method, are presented. In analogy to traditional Patterson function analysis, advantage is taken of the Fourier transformation either of the full three-dimensional diffraction pattern of a disordered crystal or just of the isolated diffuse scattering, respectively. By the use of three-dimensional information, analysis of disorder becomes straightforward, and it becomes possible to investigate far more complicated structures than is feasible with well established powder diffraction-based PDF analysis. Compared to more traditional modelling techniques, such as Monte Carlo simulation, the 3D-ΔPDF provides direct access to disorder models and allows selective modelling of distinct structural features, which are, in contrast to reciprocal space, well localized in PDF space. The principles of the 3D-ΔPDF approach are exemplified using an analysis of the twofold (∼8 Å) periodic superstructure of a decagonal Al65Cu20Co15quasicrystal. Although analysis of disorder in quasicrystals is far more demanding than in the case of periodic structures, details of the disordered structure could be elucidated. The superstructure is found to be built from columnar units, having a maximum diameter of ∼14.5 Å. The lateral correlation between these columns is weak. Internally, the columns consist of a long-range-ordered alternation of flat and puckered layers. The development of the model and the atomic structure of the columns are described in detail.

trans-2,5-Dibromo-1,4-dithiane crystallizes in the monoclinic system; the space group is P21/c with a = 6.42, b = 7.12, c = 9.01 A, β = 111.6°, and Z = 2. The structure was solved from a three-dimensional Patterson function and refined by the least-squares method using room temperature reflexion data. The dithiane ring has the chair conformation with the bromine atoms in axial positions. Large values (115°) are observed for the valency angles CCS, all other bond angles and bond distances being normal.

The residual ( / is a very useful function in structure evaluation procedures. Using simple properties of the Patterson function algebraic expressions for the residual are derived. It is shown that the numerical value depends on various factors such as the presence of heavy atoms or pseudosymmetry in the structure. It appears that coincidence of interatomic vectors increases the value of the residual. The Influence of the origin peak in the Patterson function on the residual is described.

Peroxiredoxin II was cloned from mouse B cells into pCold 1 expression vector and produced as a His-tagged recombinant protein in Escherichia coli. A ring form was isolated by gel filtration. A crystal obtained by the sitting-drop vapour-diffusion method diffracted to 1.77 A resolution at 100 K. The crystal belonged to space group P2(1)2(1)2, with unit-cell parameters a = 117.4, b = 133.9, c = 139.1 A. The asymmetric unit is expected to contain six dimers of peroxiredoxin II, with a corresponding solvent content of 39.3%. Peaks in the native Patterson function together with pseudo-systematic absences suggested that the crystals suffered from severe translational pseudosymmetry.

Crystals of a new compound with a superprotonic phase transition Cs 2 (HSeO 4 )(H 2 AsO 4 ), have been synthesized for the first time by a slow evaporation method at room temperature. The structure was solved by a three-dimensional Patterson function and refined to R 1 =0.0309 and WR 2 =0.0482 on the basis of 2062 unique observed reflections using 67 parameters. The structure contains zigzag chains of hydrogen bonded anion tetrahedra that extend in the [010] direction. Each tetrahedron is additionally linked to a tetrahedron neighbouring chain to give a planar structure with hydrogen-bonded sheets lying parallel to ( 10 1 ¯ ) . Thermal-differential analysis of the superprotonic transition in Cs 2 (HSeO 4 )(H 2 AsO 4 ) showed that the transformation to hightemperature phase occurs at 515 K by one-step process. Thermal decomposition of the product takes place at much higher temperatures, with an onset of approximately 760 K. The superprotonic transition was also studied by impedance and modulus spectroscopy techniques. The conductivity in the high temperature phase at 523 K is 2.91×10 −4 Ω −1 cm −1 , and the activation energy for the proton transport is 0.16 eV. The conductivity relaxation parameters associated with the high disorder protonic conduction have been examined from analysis of the M ” / M ” m a x spectrum measured in a wide temperature range. Transport properties in this material appear to be due to proton hopping mechanism.

When gold is deposited onto the Si(553) surface, a highly ordered surface with a quasi-one-dimensional electronic structure can be prepared. This Si(553)-Au surface was investigated by surface X-ray diffraction to determine the surface structure. A structure model was constructed from the Patterson function of the experimental diffracted intensities. There is one gold atom in the primitive unit cell. It substitutes for a top-layer silicon atom on the terraces. The silicon atoms near the step edge reconstruct to form a honey-comb chain. [DOI: 10.1380/ejssnt.2009.533]

Surface x-ray diffraction was employed, in situ, to measure the GaSb(001)-(1 × 5) and (1 × 3) surface phases under technologically relevant growth conditions. We measured a large set of fractional-order in-plane diffraction peaks arising from the superstructure of the surface reconstruction. From the data we calculated two-dimensional (2D) Patterson functions, the peaks of which represent inter-atomic distances weighted by the number of electrons in the individual atoms. For the (1 × 3) phase we obtained good agreement between our data and the β(4 × 3) model proposed in recent experimental and theoretical work. Our measurements on the Sb-rich (1× 5) phase provide evidence that the structure under growth conditions is, in fact, different from that of the models previously suggested on the basis of scanning tunneling microscopy (STM). We discuss reasons for this discrepancy as well as the identified structural elements for these reconstructions, which include surface relaxations and subsurface rearrangement.

Void structure in textiles may be characterized by a variety of methods, including many based on nuclear magnetic resonance (NMR). The NMR techniques are based on the measurement of fluid distribution or diffusion within porous media, and are therefore particularly relevant for characterizing the void space that governs fluid interactions and movement within textile substrates. In this article, we describe the use of NMR imaging, also known as magnetic resonance imaging, to measure average void dimensions in textile fabrics. In addition to direct visualization using NMR images, void space may be analyzed by calculating the fluid density autocorrelation function. The advantages of both types of data analysis are discussed. These NMR techniques are applied to a water-saturated nylon fabric and the results compared to void sizes measured from optical microscopy.

We have investigated the atomic arrangement of a barium (Ba)-induced Si(111)-3×2 surface by using surface-sensitive synchrotron X-ray scattering. In-plane fractional-order reflections obtained from three rotational domains of 3×1 symmetry have been used to examine several competing structural models including the honeycomb-chain-channel (HCC) and the double-π-bonded chain (DπC) models. By fitting two different data sets, both the goodness-of-the fits χ2 and the residuum R-factor strongly favor the HCC over the DπC model. We conclude that a Ba-paired HCC structure with a \(\frac{1}{3}\) monolayer of Ba coverage best describes the Ba-induced 3×2 surface with a Ba-Ba interatomic spacing d = 1.93 A.

The adsorption of cyanogen ( C 2 N 2) on Cu (001) has been investigated using helium atom scattering. Adsorption at 263 K produces a two domain structure with periodicity c(10 × 6) and symmetry c2 mm. The principle axes of the two domains in the c(10 × 6) pattern lie at right angles to one another. Adsorption at lower temperatures (163 K) produces a disordered structure with no observable diffraction features other than the specular. Annealing the disordered phase to 263 K does not produce an ordered structure. Cooling the ordered structure from 263 K to 163 K produces an enhancement of the diffraction features consistent with the Debye–Waller effect but no disruption of the c(10 × 6) periodicity. Annealing the ordered structure shows that the structure remains stable at temperatures up to 373 K. Analysis of the in-plane diffraction features suggests a reconstruction of the Cu (001) substrate consistent with the 2D Frenkel–Kontorova model for overlayer strain relief.

The problem of diffraction pattern deconvolution in Helium Atom Scattering (HAS) has been solved directly within the Eikonal approximation. A formulation based on the Patterson construction has been developed which reduces the problem to the solution of a set of simultaneous equations in {ζj}, where ζj are the corrugation parameters for the surface. The formulation returns real space structures that satisfy unitarity, a secondary problem in HAS diffraction image inversion. Numerical benchmarks of this procedure are presented.

The technique of Patterson analysis has proven to be a powerful tool in the deduction of structures, both in 2- and 3-D. Using the standard formulation of the Patterson function, the process has been benchmarked for the Helium Atom Scattering (HAS) experiment. Analytically expanding the Patterson function reveals a straightforward Fourier series, whose coefficients are explicit functions of the surface corrugation and the momentum exchange during scattering. Examples based on this formulation reveal that in surface scattering experiments, the Patterson formulation may not be directly interpreted as a map of the inter-nuclear distances, but as a function of surface corrugation and perpendicular momentum exchange.

The Tin dioxide (SnO2) and Titanium dioxide (TiO2) are very promising materials in Material science. The SnO2 is commonly used as a gas sensor while the TiO2 is used as a catalyst in many reactions. Despite of the usefulness of these two substances, their surface structures lack detail investigations in the previous years. The Low Energy Electron Diffraction (LEED) technique is commonly used to characterize surfaces in the past 40 years, it is a mature system that many researches rely on its result. However, structural analysis in LEED requires comparison with computational results based on pre-defined structure models, which is a time-consuming method and the results are not guaranteed to be found. The direct determinations of structure by Patterson function inversion methods introduced by Huasheng Wu and S. Y. Tong could provide a different path to search for surface structure. In the Patterson function, each maximum in the function corresponds to a relative position vector of atomic pairs. Multiple-angle-incident LEED has to be performed to obtain an artifact-free Patterson function. Serveal SnO2 and TiO2 surfaces have been characterized by LEED and Patterson function inversion. SnO2 (110), (100), (101) , Rutile TiO2 (110), Anatase TiO2 (110) have been prepared by argon ion sputtering and annealing cycles and the cleanness has been checked by Auger Electron Spectroscopy and LEED. Reconstruction is observed based on the study of the LEED patterns. SnO2 (110) surface shows a 4 x 1 reconstruction in UHV environment while it gives 1 x 1 under annealing in oxygen and C(2 x 2) at higher annealing temperature afterward. SnO2 (100) , (101) and Rutile TiO2 (110) surfaces show 1 x 1 reconstruction in UHV environment and the reconstruction persists for further annealing. The Anatase TiO2 (110) surface shows a 3 x 4 reconstruction in UHV environment. The 3 x 4 reconstruction of Anatase TiO2 (110) surface would raise research interests as it is quite a special reconstruction. Multiple-angle-incident LEED has been performed on the SnO2 (100), (101) and Rutile TiO2 (110) surfaces. Patterson function inversion is performed on the surfaces SnO2 (100) and Rutile TiO2 (110) . Only LEED is performed on SnO2 (110) , (101) and Anatase TiO2 (110) surfaces. From Patterson functions analysis, the surface atoms positions are determined for the surface SnO2 (100) and Rutile TiO2 (110). The results show that their reconstructions are negligible, but they have obvious relaxations.

The scattering of electrons off nuclei is one of the best methods of probing nuclear structure. In this paper we focus on electron scattering off nuclei with spin and isospin zero within the Skyrme model. We consider two distinct methods and simplify our calculations by use of the Born approximation. The first method is to calculate the form factor of the spherically averaged Skyrmion charge density; the second uses the Patterson function to calculate the scattering intensity off randomly oriented Skyrmions, and spherically averages at the end. We compare our findings with experimental scattering data. We also find approximate analytical formulae for the first zero and first stationary point of a form factor., Comment: 24 pages, 38 figures

© 2015. Ab initio phasing of macromolecular structures, from the native intensities alone with no experimental phase information or previous particular structural knowledge, has been the object of a long quest, limited by two main barriers: structure size and resolution of the data. Current approaches to extend the scope of ab initio phasing include use of the Patterson function, density modification and data extrapolation. The authors' approach relies on the combination of locating model fragments such as polyalanine -helices with the program PHASER and density modification with the program SHELXE. Given the difficulties in discriminating correct small substructures, many putative groups of fragments have to be tested in parallel; thus calculations are performed in a grid or supercomputer. The method has been named after the Italian painter Arcimboldo, who used to compose portraits out of fruit and vegetables. With ARCIMBOLDO, most collections of fragments remain a 'still-life', but some are correct enough for density modification and main-chain tracing to reveal the protein's true portrait. Beyond -helices, other fragments can be exploited in an analogous way: libraries of helices with modelled side chains, β-strands, predictable fragments such as DNA-binding folds or fragments selected from distant homologues up to libraries of small local folds that are used to enforce nonspecific tertiary structure; thus restoring the ab initio nature of the method. Using these methods, a number of unknown macromolecules with a few thousand atoms and resolutions around 2 - Å have been solved. In the 2014 release, use of the program has been simplified. The software mediates the use of massive computing to automate the grid access required in difficult cases but may also run on a single multicore workstation (http://chango.ibmb.csic.es/ARCIMBOLDO-LITE) to solve straightforward cases., This work was supported by grants BFU2012-35367 and IDC-20101173 (the Spanish Ministry of Economy and Competitiveness) and Generalitat de Catalunya (2009SGR-1036)

Crystals of a new silver-gadolinium, AgGd(PO 3 ) 4 , have been synthesized for the first time by a flux method. The structure [monoclinic, P 2 1 / n , a = 7.2378(2) A, b = 13.0913(5) A, c = 9.8136(4) A, β = 90.494(1)°, V = 929.83(6) A 3 and Z = 4] was solved by a three-dimensional Patterson function and refined to R 1 = 0.0399 ( w R 2 = 0.0885 ) on the basis of 3270 unique observed reflexions using 164 parameters. AgGd(PO 3 ) 4 is isostructural to NaLa(PO 3 ) 4 , AgLa(PO 3 ) 4 and AgNd(PO 3 ) 4 . The structure is based upon long chain polyphosphate running along the shortest unit-cell direction and made up of PO 4 tetrahedra (with a period of four tetrahedra) sharing two corners, linked to the polyhedra of gadolinium and silver by common oxygen atoms to the chains. This type of organisation gives rise to a three-dimensional framework. Gadolinium atoms have an eight-fold coordination while silver atoms have approximately six oxygen neighbours. Infrared spectrum of AgGd(PO 3 ) 4 at room temperature, investigated in the frequency range 400–1500 cm −1 , confirms the atomic arrangement within the structure. The electrical conductivity of AgGd(PO 3 ) 4 has been measured on pellets of the polycrystalline powder and evaluated as a function of temperature. The conductivity of this phase was quite low and the values of 4.01 × 10 −7 and 1.10 × 10 −6 Ω −1 cm −1 were obtained at 803 and 873 K, respectively.

In this paper the systematic method of dealing with the arbitrary decorations of quasicrystals is presented. The method is founded on the average unit cell formalism and operates in the physical space only, where each decorating atom manifests itself just by an additional component of the displacement density function in the average unit cell. Such approach allows us to use almost all classical crystallography algorithms for structure refining based on experimental data and may meaningly decrease the number of parameters which have to be fit. Further help for such analysis may be the use of proposed recently average Patterson function, here applied to decorated sets. As an example we present a description of a class of decorated quasicrystals based on Sturmian sequence of two interatomic spacings: we calculate explicitly structure factor, the shape of average Patterson function and give an algorithm for pattern analysis.

The maximum-likelihood method is applied to direct methods to derive a more general probability density function of the triple-phase sums which is capable of predicting negative values. This study also proves that maximization of the origin-free modulus sum function S yields, within the limitations imposed by the assumed approximations, the maximum-likelihood estimates of the phases. It thus represents the formal theoretical justification of the S function that was initially derived from Patterson-function arguments [Rius (1993). Acta Cryst. A49, 406-409].

The atomic structure of thin iron silicide film, grown epitaxially on the Si(111) surface, has been analyzed by means of the three-dimensional RHEED Patterson function analysis. The iron-silicide-terminated surface with (2 × 2) periodicity has been prepared by a solid-phase epitaxy method. 2 ML of Fe were deposited on the Si(111)-(7 × 7) surface and annealed at 500°C. Three-dimensional Patterson function was calculated from series of φ-scanned RHEED intensity distributions converted to the k-space. The resulting model of γ-FeSi2 structure consists of two silicide layers faulted to each other with three relaxed Si adatoms above the H3 site.

Covering the two-dimensional Penrose atomic structure with different kite clusters consisting of four, seven and 17 atoms is discussed. These clusters are compared to the well-known 33-atom Gummelt's cluster. The number and concentrations of different atoms decorating the clusters were found. An analytical formula for the structure factor of decorated clusters was derived by a statistical approach.

A statistical approach was used for the modulated structures and some universal behaviours of probability distributions and their autocorrelation functions (average Patterson functions) were found. This method was also used for the analysis of the Fibonacci chain regarded as a multi-q modulated structure.

The usefulness of fully performed constant-momentum-transfer-averaging (CMTA) in low-energy electron diffraction (LEED) has been demonstrated for a single-domain Si(111)4×1-In surface. In the fully performed CMTA, thousands of I(V) curves were measured for the single-domain Si(111)4×1-In surface and CMTA intensity profiles were obtained for more than three hundred reciprocal rods in order to directly reconstruct three dimensional (3D) atomic positions of the surface. In the direct reconstruction, the minimum function method, a standard method in X-ray crystallography of the Patterson function analysis, was used. The 3D Si positions so obtained were in excellent agreement with the known heavily-reconstructed surface Si positions of 4×1. Indium atoms were not ‘ visible’ in the present case due to large thermal vibration amplitude of indium atoms for the room-temperature sample, which is not an inherent shortcoming of the fully performed CMTA method. [DOI: 10.1380/ejssnt.2006.661]

The reaction center-light-harvesting 1 (RC-LH1) core complex is the photosynthetic apparatus in the membrane of the purple photosynthetic bacterium Rhodopseudomonas viridis. The RC is surrounded by an LH1 complex that is constituted of oligomers of three types of apoproteins (alpha, beta and gamma chains) with associated bacteriochlorophyll bs and carotenoid. It has been crystallized by the sitting-drop vapour-diffusion method. A promising crystal diffracted to beyond 8.0 A resolution. It belonged to space group P1, with unit-cell parameters a = 141.4, b = 136.9, c = 185.3 A, alpha = 104.6, beta = 94.0, gamma = 110.7 degrees. A Patterson function calculated using data between 15.0 and 8.0 A resolution suggested that the LH1 complex is distributed with quasi-16-fold rotational symmetry around the RC.

The first successful multi-wavelength anomalous diffraction (MAD) experiments using ultra-high-energy X-rays (∼35 keV) were performed for iodine and xenon derivatives of hen egg-white lysozyme crystals. The beamline BL41XU of SPring-8 enabled the collection of high-quality MAD data, which led to the calculation of anomalous or dispersive difference Patterson maps that determined the positions of iodine and xenon atoms. The electron density maps obtained by the density modification method for both cases proved to be of sufficient quality for building molecular models. I-MAD and Xe-MAD phasing are now available at SPring-8, and the utilization of ultra-high-energy X-rays will make a significant contribution to the solution of the phase problem in protein crystallography.

Using a statistical approach, the average unit cells have been constructed for modulated structures with different types of modulation, from simple sinusoidal to square-wave functions. The obtained unit cells fully describe diffraction intensities of the main reflections and their satellites. A universal distribution, valid for different lengths of the modulation vector, has been found. Average Patterson functions have been constructed and used to distinguish between different types of modulation.

An automated Patterson deconvolution technique, based on the minimum superposition function, has been implemented into theSIR2002package, a program for theab initiodetermination of small and large molecules. The procedure, combined with the direct-space phase refinement routines present in theSIR2002program, has been designed to solve proteins diffracting at atomic resolution. The procedure has been validated using a large set of test structures, containing either heavy or intermediate atoms (e.g.P, S and Cl) and with structural complexity up to 1500 non-hydrogen atoms in the asymmetric unit. The success, obtained in a completely automatic way, requires a very short computing time and indicates that Patterson deconvolution techniques are a valid alternative to current direct-methods approaches.

Diffuse inter-layers from a decagonal quasicrystal with nominal composition Al 70 Co 12 Ni 18 were investigated at 1120 K, 1070 K and 300 K. Patterson maps calculated from the inter-layers are interpreted such that the main units for correlated displacements along a 5 are structure motifs (‘clusters’) having a diameter of about 15 A. At 1120 K, displacements of the clusters are uncorrelated along quasiperiodic directions, while they form about 42 A-sized super-clusters at lower temperatures. The arrangement but not the inner structure of the super-clusters differs significantly at 1070 K and 300 K. Further, a first approach to the atomic structure of the 15 A cluster is presented.

The average unit cells and the average Patterson functions have been constructed for 1D quasicrystals in statistical approach. A simple formula for the structure factor of decorated Fibonacci chain has been derived. Although the used method operates in the physical space only, its equivalence to the higher-dimensional analysis has been proved. Applications of the analysis to different decorated structures have been also discussed.

Using a statistical approach, the average unit cells and the average Patterson functions have be constructed for modulated structures with different types of modulations. Universal probability distributions and autocorrelation functions for different lengths of the modulation vector have been found. Such universal behaviour allows to distinguish between differently modulated structures directly from the experimentally obtained average Patterson function.

The theory for Fourier transform of X-ray diffraction intensity proposed by Hirsch was applied to the microstructure analysis of mesocarbon microbeads. The precision of the result of Fourier transform was improved by the consideration of measurement and integration conditions, and the distribution of the interlayer-spacing of carbon was roughly estimated.

It is shown here that the interfacial profile between two bonded wafers can be directly determined using X-ray reflectivity without resorting to standard model-fitting of the data. The phase problem inherent to any structure determination by scattering technique is solved in this case using a known silicon/silicon oxide interface, which acts as a phase reference for the reflected signals.

We present a complete formal analysis of crystallographic point and space group symmetries in the local one- and two-body density ρ(x) and ρ 2(x 1,x 2) of crystals and from there develop the mathematical theory of generalized Patterson functions. The two-body density ρ 2(x 1,x 2) and ρ(x 1) ρ(x 2) may be written as symmetrized lattice Fourier series over all reciprocal lattice vectors K with Fourier transforms that are functions of the relative vector r = x 1 – x 2. In the new formalism derived here the usual Patterson function turns out to be the lowest-lying K = 0 lattice Fourier transform in the Fourier series representation of the uncorrelated product ρ(x 1)ρ(x 2) of one-body densities. All other K ≠ 0 Fourier transforms in the Fourier series of ρ(x 1)ρ(x 2) may be regarded as generalized Patterson functions. In complete analogy to these generalized Patterson functions the Fourier transform functions in the Fourier series of the fully correlated two-body density ρ 2(x 1,x 2) may be regarded as K = 0 and K ≠ 0 fully correlated generalized Patterson functions. It is shown that the former generalized Patterson functions represent the uncorrelated long-range limits for large interparticle distances of the latter, fully correlated, generalized Patterson functions. Both types of generalized Patterson functions may be cast in the form of symmetry-adapted trigonometric series with expansion functions that depend on the relative distance r = |r| and on the polar angle of r. The symmetrized trigonometric series representations of the generalized Patterson functions are given here explicitly in specific applications of the new formalism to crystallographic space group P63/mmc. In exact path integral Monte Carlo case studies of novel high-temperature/high-pressure 4He quantum crystals in the hexagonal close-packed structure we present and discuss exact numerical results for the one-body density ρ(x) and for expansion functions, for both types of generalized Patterson functions, in the symmetrized trigonometric series representations of the generalized Patterson functions. The numerical results for the two types of generalized Patterson functions are compared to each other.

Although often an object of controversy, electron crystallography has emerged as a useful technique for characterization of the microcrystalline state, capable of elucidating crystal structures of unknown substances. Despite the complicated multiple scattering perturbations to diffracted intensities, experimental conditions can be adjusted to favor data collection where the experimental Patterson function still resembles the autocorrelation function of the actual crystal structure. Satisfying this condition is often sufficient to permit structure solution from such data by direct methods. While the application to organic structures may seem obvious, there are surprising successes with data sets from inorganic materials. The account given in this paper, in part, portrays work leading to the A. L. Patterson Award to the author from the American Crystallographic Association.

A series of structure analyses during 1994-2001 by electron crystallographic techniques applied to phases in aluminum alloys are reviewed. Methods for structure solution employ electron diffraction intensity data collected by the precession technique, by selected area micro-diffraction and by the convergent-beam technique. High-resolution electron microscope images (HRTEM) are treated by different kind of processing, including exit wave reconstruction. Crystallographic calculations are performed either by direct method or Patterson and Fourier procedures, assuming kinematical scattering, or by refinement from models derived from HRTEM images. Dynamical scattering calculations can be introduced in the refinement stage or as a correction procedure applied to part of the intensity data. The phases studied include primary Al-Fe-(Si) particles, Al-Mn-Si dispersoids, Al-Co quasicrystals and two types of precipitate phases in age-hardening Al-Mg-Si and Al-Zn-Mg alloys.

The penthylenediammonium pentachlorothallate(III) salt is orthorhombic with the following unit cell dimensions: a = 7.696(3) A, b = 13.2890(17) A, and c = 13.503(18) A, space group P212121 with Z = 4. The structure was solved by Patterson methods and refined to a final R value of 0.0387 for 1991 observed reflections. The structure consists of penthylenediammonium cations and polynuclear anions in which slightly distorted [TlCl6]3− octahedral sharing two vertices are interconnected into chains. These chains are themselves interconnected by means of the N–H⋅⋅⋅Cl bonds originating from the [NH3(CH2)5NH3]2+ entities forming a three-dimensional network.

Improved data quality now makes it feasible to exploit the weak anomalous signal derived only from the sulfurs inherent to the protein or in particular from halide ions incorporated by soaking. The latter technique requires the location of a high number of partially occupied halide sites. This number appears to be roughly proportional to the exposed protein surface. This paper explores the application of dual-space ab initio methods as implemented in the program SHELXD to the location of substructures of sulfur in SAD experiments, bromide in SAD and MAD experiments and iodide using SAD and SIRAS to determine the anomalous-atom substructure. Sets of atoms consistent with the Patterson function were generated as a starting point for the dual-space recycling procedure in SHELXD. The substructure is then expanded to the full structure by maximum-likelihood phasing with SHARP and density modification with the program DM. Success in the location of the substructures and subsequent phasing depends critically on the quality of the data and on the extent of the anomalous signal. This varies with each crystal and soak, but for the same crystal the significance of the anomalous signal was found to be highly sensitive to the redundancy of the intensity measurements, which in some cases made all the difference. This is illustrated by the determination of the previously unknown structure of repeat 11 of the human mannose-6-phosphate/insulin-like growth factor II receptor (Man6P/IGFII-receptor), with 310 amino acids in the asymmetric unit, which was phased by soaking the crystals in a cryoprotectant solution containing halide anions.

ACORN is a flexible and general procedure for the ab initio solution of protein structures when atomic resolution data are available. It has two components — ACORN-MR that locates a single atom or a fragment of the structure and ACORN-PHASE that refines the structure starting with the phase estimates from the fragment. ACORN-MR depends heavily on a correlation coefficient (CC) between the observed normalized structure factors and those from the fragment to find the fragment‘s correct orientation and/or position. ACORN-PHASE has three components, a Patterson-superposition sum function, real-space Sayre-equation refinement and, most importantly, Dynamic Density Modification (DDM). The concepts that underline DDM are explained by a theoretical approach. ACORN has been shown in trials to be able to solve structures with up to 5000 independent non-hydrogen atoms in the asymmetric unit. Several previously unknown structures have also been solved. At lower resolution, if anomalous scattering or isomorphous replacement data are available, then substructures can be determined using the ACORN approach. The quality of the final maps from ACORN, without further processing, enables automatic interpretation in terms of structure to be readily made.

An automatic structure determination method based on an interpretation of the Patterson function has been improved by utilizing information on superposition pseudosymmetry. This pseudosymmetry can be detected using MIF (multiple implication function) groups. A new method of structure determination based on an enantioselective cross-vector function and on a sum atomic minimum superposition is introduced. This new method seems to be suitable for equal-atom structures. The reliability and speed of the structure determination has been increased.

Double potassium indium and rubidium indium phosphates K3In(PO4)2 (I) and Rb3In(PO4)2 (II) are synthesized by solid-phase sintering at T = 900°C. The compounds prepared are characterized by X-ray powder diffraction (I and II), X-ray single-crystal diffraction (II), and laser-radiation second harmonic generation. Structure I is solved using the Patterson function and refined by the Rietveld method. Both compounds crystallize in the monoclinic crystal system. For crystals I, the unit cell parameters are as follows: a = 15.6411(1) A, b = 11.1909(1) A, c = 9.6981(1) A, β = 90.119(1)°, space group C2/c, Rp = 4.02%, and Rwp = 5.25%. For crystals II, the unit cell parameters are as follows: a = 9.965(2) A, b = 11.612(2) A, c = 15.902(3) A, β = 90.30(3)°, space group P21/n, R1 = 4.43%, and wR2 = 10.76%. Structures I and II exhibit a similar topology of the networks which are built up of { In[PO4]2} (I) and { In2[PO4]4} (II) structural units.

We review the symmetrized data acquisition method for producing artifact-free Patterson functions from low-energy electron diffraction?(LEED) IV spectra. The differences between LEED Patterson functions and LEED holograms are discussed. This article was originally intended to be part of the special issue on holographic and other direct methods for surface structures using electrons and photons which was published as issue 47, volume 13.

This paper presents a completely new method for the calculation of expectations (and thus joint probability distributions) of structure factors or phase invariants. As an example, a first approximation of the expectation of the triplet invariant (up to a constant) is given and acomplexnumber is obtained. Instead of considering the atomic vector positions or reciprocal vectors as the fundamental random variables, the method samples over all functions (distributions) with a given number of atoms and given Patterson function. The aim of this paper was to explore the feasibility of the method, so the easiest problem was chosen: the calculation of the expectation value of the triplet invariant inP1. Calculation of the jointprobabilitydistribution of the triplet is not performed here but will be done in the future.