Search

Surfaces in i-Al68Pd23Mn9 as observed with STM and LEED experiments show atomic terraces in a Fibonacci spacing. We analyze them in a bulk tiling model due to Elser which incorporates many experimental data. The model has dodecahedral Bergman clusters within an icosahedral tiling T^*(2F) and is projected from the 6D face-centered hypercubic lattice. We derive the occurrence and Fibonacci spacing of atomic planes perpendicular to any 5fold axis, compute the variation of planar atomic densities, and determine the (auto-) correlation functions. Upon interpreting the planes as terraces at the surface we find quantitative agreement with the STM experiments., 30 pages, see also http://homepages.uni-tuebingen.de/peter.kramer/ to be published in J.Phys. C

The crystal structure of bovine heart cytochrome c oxidase has been determined at 2.8 A resolution by the multiple isomorphous replacement (MIR) method with three heavy-atom derivatives. An asymmetric unit of the crystal has a molecular weight of 422 kDa. Eight heavy atoms as main sites of a CH3HgCl derivative were clearly located by solving the difference Patterson function. The electron density obtained by the MIR method was refined by density modification, consisting of solvent flattening, histogram matching and non-crystallographic symmetry averaging. The enzyme exhibits a dimeric structure in the crystal. Out of 3606 amino-acid residues in 26 subunits in the dimer, 3560 residues were located in the electron-density map. The structure was refined by X-PLOR. The final R factor and the free R factor were 0.199 and 0.252 at 2.8 A resolution, respectively. One monomer in the dimeric structure with a stronger packing interaction has a lower averaged temperature factor than the other, by 16 A2. The region \pm12 A from the centre of the transmembrane part is almost 100% \alpha-helix, despite the glycine residue content being as high as 7.1% in the transmembrane region. The residues around haem a of animals have evolved away from those of bacteria in contrast with the residues of the haem a3. The hierarchy of the structural organization of the enzyme complex has been proposed on the basis of intersubunit interactions.

In protein crystallography, the use of low-resolution reflections is important in defining the molecular mask and polypeptide backbone. However, in Laue data collection, the loss of low-resolution reflection data (>2dmin) can be as high as 40–50%, even after the deconvolution of multiples. To estimate the reflection intensities that are not recorded in data collection, a new method is presented based on maximizing the entropy of the Patterson function subject to the constraints imposed by the intensities of the observed reflections. The method has been tested with Laue diffraction data from hen egg-white lysozyme. All unobserved reflections within 5 A resolution were estimated, and their inclusion in the electron-density-map calculation significantly improved the connectivity. This method could also be applied to improve the completeness of monochromatic data.

The symmetry elements detected by the self-rotation and the Patterson functions, associated with strong correlations between the positions of the molecules in the asymmetric unit, are used to reduce the effective number of independent bodies to be located by the molecular replacement method. A distinction is made between `frustrated' crystallographic symmetries, i.e. those that are almost crystallographic ones, and `standard' non-crystallographic symmetries, which are taken into account by specific techniques. These have been successfully applied to many-body macromolecular crystal structures, with important savings in time and computational effort.

The positivity of the Patterson function is used as prior information for the decomposition of a powder diffraction pattern. An automatic procedure is described which, integrated with the Le Bail method, is able to provide estimates of |F| values that are often remarkably better than those obtained by application of standard techniques. The procedure, implemented in a computer program, has been applied to a set of test structures (experimental data); the results show that, particularly for structures with some heavy atoms, the new estimates of the structure-factor moduli are far better (on average) than those obtained via the default application of EXTRA [Altomare et al. (1995). J. Appl. Cryst. 28, 842–846]. 1. Symbols and abbreviations The papers by Altomare, Carrozzini et al. (1996), Altomare, Foadi et al. (1996) and Carrozzini et al. (1997) are referred to as papers I, II and III, respectively.

The indium-induced Si(001)-(4 × 3) reconstruction has been investigated by surface X-ray diffraction (SXRD) measurements with synchrotron radiation and scanning tunneling microscopy (STM). The Patterson function analysis enables us to exclude In dimers as a structural element in this reconstruction. We present a new structural model which includes 6 In atoms threefold coordinated to Si atoms and 5 displaced Si atoms per unit cell. Relaxations down to the sixth layer were determined. ‘Trimers’ made up of In Si In atoms are a key structural element.

The idea of using suitably modified Patterson maps for the purpose of extrapolating X-ray diffraction amplitudes to higher resolution is not new. It was first proposed by Karle and Hauptman [1] in 1964 as a means of refining E-values to obtain values which were more consistent with the point atom approximation upon which direct methods probability relationships are based. What K&H proposed is that one could compute a sharpened, origin-removed Patterson function using |E|2-1 as coefficients, and then modify the density to remove unexpected negative regions, as well as peaks which represented vectors which would place molecules closer than a van der Waals contact to proper symmetry elements such as rotation axes, inversion centers and mirror planes. When one back transformed the modified map, one not only obtained refined estimates for the observed |E|2-1 amplitudes, but also obtained values for terms which were not included in the original map. Each cycle of refinement included calculation, modification, and back transformation of the map, and renormalization of the extrapolated ||E|2-1|> values to fit the expected distribution, and insure that negative IEI2 values were reset to zero as their smallest possible value. One must also eliminate reflections corresponding to space group extinctions, as the modified Patterson will not respect this constraint. Only one subsequent publication [2] appears to have used this Patterson refinement scheme to improve the ‘accuracy’ of the measured E-values to solve a structure.

A method based on using the Fourier transform of finite functions has been developed to reconstruct the distribution function of interatomic vectors in a nanoparticle from x-ray diffuse scattering data. This distribution is similar to the Patterson function used in the structure analysis of single crystals. The method for determining the structure has been developed and verified for an actual cluster consisting of 36 atoms. The atomic cluster structure was previously determined in single crystals with the fluorite structure (Cd0.90Tb0.10F2.10. The vectors between heavy atoms are localized on the distribution of interatomic vectors and are used to construct superpositional synthesis. This synthesis yields the positions of the atoms in the nanoparticle. The method is also applicable to the nanocrystals with a limited number of unit cells.

The crystal structure of the sublimated form (m.p. = 424 K) of tris(3,5-dimethylpyrazol-l-yl)methane has been solved by a Patterson search method from laboratory X-ray powder diffraction data. Crystal data: trigonal symmetry with the unit-cell parameters a = 16.152 (1) and c = 5.353 (1) Å, space group P3, C16H22N6, Z = 3, 293 K. After indexing the powder pattern by two methods, the unit-cell parameters found were refined by a least-squares technique. A whole pattern-fitting program was used to extract the integrated intensities. The structure was solved taking a related compound as a search model and the final Rietveld refinement converged to R wp = 0.077 and R p = 0.059. This study is one of the first examples of Patterson search structure determination from an hemihedral space group using powder data. The complexity of the structural determination is increased by the presence of three molecules in the asymmetric unit.

A new method is developed for interpreting the difference Patterson function in surface X-ray structure analysis. The difference Patterson function has negative parts which arise from subtracting the integral Patterson function from the true Patterson function. This suggests, that by adding a judiciously chosen function one may utilize the information contained in the positivity constraint for the true Patterson function. We formulate conditions on this function and find a method to construct this function. It is shown that in many cases this function coincides with the integral Patterson function of the so-called “complementary structure”. The proposed new Patterson function obtained by adding the so constructed function to the difference Patterson function is easier to interpret than the conventional difference Patterson function.

Refined structure of (√3 × √3) surface reconstruction induced by Sb adsorbed on Si(111) was obtained from X-ray diffraction, using a Patterson function and a least-squares fit. A wide range of reciprocal space was probed to get information for the lateral and the vertical displacements of adsorbate and the top layers of Si. The Patterson function confirms the “milk stool” trimer model, which, according to a least-squares fit, is centered on the T 4 site. The thermal vibration amplitudes of Sb ad-atoms and those of surface layers of Si were obtained and comparison with the values in other surfaces are made.

In a Laue diffraction pattern, 10–20% of the spots result from the exact superposition of two or more reflections that are `harmonics'; a high proportion of these are low-resolution reflections. For the solution of large or difficult structural problems, the intensities of the remaining 80–90% of the reflections, measurable as singles, may not be sufficient and thus the evaluation of the intensities of the components of the multiple spots is important. A new method for this deconvolution is presented that is based on maximizing the entropy of the Patterson function subject to the constraints imposed by the observed intensities of single and overlapping reflections. This method does not require data redundancy and therefore is of particular interest for time-resolved studies on a short time scale. A new computer program (ME) was implemented and tested with Laue diffraction data from hen egg white lysozyme. The R-factor between the deconvoluted reflection intensities from Laue multiple spots and observed intensities from monochromatic data was 0.116.

A reciprocal space technique for determining the location of symmetry elements in Patterson superposition maps has been developed. The overlap of the superposition map with a copy of the map generated by the placement of the symmetry element at an arbitrary position is calculated using the Fourier transform coefficients of the original map. When the overlap is maximized, the symmetry element has been placed in the correct position. An alternate expression, which minimizes the phase differences in the Fourier coefficients, is also described. Expressions are developed for constraining the positions of different symmetry elements to meet space group constraints. For superposition maps containing predominantly one image of the structure, the phases of the Fourier coefficients can be modified, using the locations of the symmetry elements, to provide approximate phases for the observed structure factors.

The crystal structures of pure L-carvone [(R)-(−)-2-methyl-5-(1-methylethenyl)-2-cyclohexen-1-one, C10H14O] and the equimolar mixture DL-carvone (RS) have been determined by Patterson-search methods at low resolution from laboratory X-ray powder diffraction data (218 K). Crystal data: (L) a = 6.8576 (3), b = 6.8831 (5), c = 19.988 (2) Å, P212121 space group, Z = 4; (DL) a = 6.9744 (3), b = 6.8094 (6), c = 20.038 (7) Å, Pcmn space group, Z = 4. The L-carvone structure has been refined by the Rietveld method as a rigid body, allowing the rotation of the isopropenyl group (R\rho, = 0.030 and R wp = 0.043). Although the structure of DL-carvone could be unambiguously established, the Rietveld refinement was not possible due to the existence of preferred orientation in the sample and the difficulty in modelling the disorder. The molecular packing is essentialy the same for both compounds and can be explained as a stacking of two different molecular layers in the [001] direction. In each layer the molecules are placed with their long axis perpendicular to the layer plane, in a head-to-tail manner. The great similarity between the molecular shapes of L and D enantiomers favours the positional disorder in DL-carvone. This result confirms the mixed crystal formation for the chiral carvone system as proposed in recent thermodynamic studies. The DL-carvone crystal must be considered as a pseudo-racemate, since both enantiomers are randomly distributed over all the lattice sites.

The structural bases on the metal/semiconductor interfaces, such as gold trimers on the [Formula: see text] R30° surface and antimony trimers on the [Formula: see text] R30° surface, can be imaged directly with a simple inversion of low-energy (>600 eV) Kikuchi-electron patterns (Kikuchi-electron holography—KEH). The relative positions of the building blocks (trimers) on the adsorbates to the substrate atoms are also determined. This short-range-order KEH tool, which provides the 3D Patterson function, can be viewed as a twin of grazing-incidence X-ray diffraction. Using direct structural information obtained by KEH, one can greatly reduce the tested models in a complete trial-and-error structural-determination process.

Anomalous dispersion of X-ray diffraction from bovine trypsin has been measured at three wavelengths near the K-absorption edge of sulphur (λK = 5.02 Å) which are characteristic of sulphur in methionine and the cystine disulphide bridges. The feasibility of phasing by means of the dispersion analysis is demonstrated in two steps: 1. Scaling of the experimental data and calculation of the anomalous difference Patterson map, showing the correlation between sulphur atoms. 2. Using the measured dispersion to calculate phases based on the known sulphur sites of the refined trypsin model. Although the resulting electron density map of 5 Å resolution is calculated from 600 unique reflections only, there is a reasonable agreement with the superimposed molecular model.

Prospects for the accurate structure determination of heavy-atom-containing organic crystals were evaluated with electron diffraction data from perchloro and perbromo derivatives of copper phthalocyanine. While the extensive overlap of experimental Patterson maps (from 1200kV intensities) with respective crystal autocorrelation functions explains the success of previous direct structure analyses, it is clear that multiple-scattering perturbations still evident at high voltage will frustrate the determination of accurate bond distances and angles. If, however, the result obtained after direct structure analysis and Fourier refinement is used to position an idealized molecular model (i.e. with chemically reasonable bonding parameters), the correct structure can then be justified by a rotational search coupled with a multislice dynamical calculation. Even though dynamical scattering is not the only major perturbation to such data sets, the resolution-limited correction is still sufficient to identify the correct molecular orientation in the unit cell. Alternatively, an acceptable unconstrained structure refinement can be carried out via a procedure proposed by Huang, Liu, Gu, Xiong, Fan & Li [Acta Cryst. (1996), A52, 152-157]. A phenomenological adjustment of observed intensities, based initially on the heavy-atom positions found in a high-resolution electron micrograph, will permit all light atoms to be observed near their ideal positions in the ensuing Fourier refinement.

The program DISCUS is a versatile tool for the analysis of diffuse scattering and for defect structure simulations. The model structure can be created from an asymmetric unit of a unit cell or a complete structure can be read from a file. A Fortran77 style interpreter that includes IF statements and various loops combined with predefined defect types like thermal displacements, waves and microdomains allows one to create all sorts of defect structures. The Fourier-transform segment of the program allows one to calculate neutron as well as X-ray intensities including isotropic temperature factors and anomalous scattering. The calculation of the inverse and difference Fourier transform as well as the Patterson function is also implemented. A model structure can be `fitted' to observed diffuse scattering data by reverse Monte Carlo (RMC) simulations. The RMC segment allows one to model displacive as well as occupational disorder. The program is completely written in Fortran77 and the source code is available via the World Wide Web.

The ClpP component Clp protease fromEscherichia colihas been crystallized and examined by X-ray crystallography and self-rotation function calculations. The crystal belongs to the monoclinic space groupP21with unit cell dimensions ofa=196.9 A,b=104.3 A,c=162.4 A and β=98.3°. The X-ray diffraction pattern extends at least to 2.5 A Bragg spacing when exposed to CuKα X-rays. Self-rotation function analyses indicate that the ClpP oligomer has 72-point group symmetry. This symmetry suggests that the ClpP oligomer is a tetradecamer, (ClpP)14, consisting of two heptamers, (ClpP)7stacked on top of each other in a head-to-head fashion. The measurement of crystal density indicates that two independent copies of the ClpP oligomers are present in the asymmetric unit, giving a crystal volume per protein mass (VM) of 2.73 A3/Da and a solvent content of 54.9% (v/v). Self-rotation function calculations are consistent with the presence of two ClpP tetradecamers in the asymmetric unit. The Patterson function suggests that a translation ofx=0.5 andy=0.5 relates a pair of ClpP oligomers in one asymmetric unit to another pair in the other asymmetric unit. And the two independent tetradecamers in one asymmetric unit are related by a relative rotation of about 18° around the 7-fold axis.

The Genetic Algorithm (GA) and Simulated Annealing (SA), two techniques for global optimization, were applied to a reduced (simplified) form of the phase problem (RPP) in computational crystallography. Results were compared with those of “enhanced pair flipping” (EPF), a more elaborate problem-specific algorithm incorporating local and global searches. Not surprisingly, EPF did better than the GA or SA approaches, but the existence of GA and SA techniques more advanced than diose used in this study suggest that these techniques still hold promise for phase problem applications. The RPP is, furthermore, an excellent test problem for such global optimization methods.

Horse-spleen apofemtin crystallizes in two different space groups: cubic F432 and tetragonal P42(1)2 while its iron-containing analogue is known to present a cubic and an orthorhombic form. Up to now, only the structure of the cubic form has been fully investigated by X-ray diffraction, although some information concerning the molecular packing of the two other forms was deduced from analysis of X-ray photographs. While growing cubic crystals of horse-spleen apoferritin with Pt-mesoporphyrin IX, we obtained one crystal, with a diffraction limit of 2.4 A, belonging to the orthorhombic P2(1)2(1)2 space group, with unit-cell dimensions a = 181.6, b = 128.9, c = 128.9 A. The orientation of the non-crystallographic axes of the molecule was determined by self-rotation Patterson function and the structure was determined by the molecular-replacement method. The asymmetric unit consists of half an apoferritin molecule. Refinement of the structure is in progress, some preliminary results of the molecular packing are given.

The DIRDIF program system for solving crystal structures using Patterson procedures and direct methods applied to difference-structure factors, uses a variety of X-ray intensity scaling procedures based on the formulae of Wilson (1942), and when part of the structure is known, Parthasarathy (1966), and Gould, Van den Hark and Beurskens (1975). Applications of these procedures to various heavy atom structures are discussed. A new scaling procedure is presented which is based on the relative frequency of calculated partial structure factors being larger than the observed structure factors on an absolute scale. This procedure is especially useful for structures with superstructure effects.

The viability of solving the structure type of zeolitic and layered materials applying multisolution direct methods to low-resolution (~2.2 A) powder diffraction data is shown. The phases are refined with the tangent formula derived from Patterson-function arguments [Rius (1993). Acta Cryst. A49, 406–409] and the correct phase sets are discriminated with the conventional figures of merit. The two test examples presented are (a) the already known tetragonal zeolite ZSM-11 (space group 1{\bar 4}m2) at 2.3 A resolution and (b) the hitherto unknown layer silicate RUB-15 (Ibam) at 2.2 A resolution. In both cases, the tetrahedral Si units appear as resolved peaks in the Fourier maps computed with the phases of the highest-ranked direct-methods solutions.

The three-dimensional configuration of crystallized structures is obtained by reading off partial information about the Fourier transform of such structures from diffraction data obtained with an X-ray source. We consider a discrete version of this problem and discuss the extent to which `intensity only' measurements as well as `higher-order invariants' can be used to settle the reconstruction problem. This discrete version is an extension of the study undertaken by Patterson in terms of `cyclotomic sets', corresponding to arrangements of equal atoms that can occupy positions on a circle subdivided into N equally spaced markings. This model comes about when the usual three-dimensional Fourier transform is replaced by a one-dimensional discrete Fourier transform. The model in this paper considers molecules made up of atoms with possibly different (integer-valued) atomic numbers. It is shown that information of order six suffices to determine a structure uniquely.

It is well known that the structure analysis with the Patterson function provides a powerful tool in X-ray crystallography and many researchers have applied it for various carbon materials. In the present study, the effect of variation of sampling distance in the measurement of 002 diffraction profiles on the Patterson function in relation to the reliability of the analysis was investigated using the sampling theorem and the structural change of mesocarbon microbeads (MCMB) by heat treatment was analysed on this basis. Thereby, the Patterson functions calculated from 002 diffraction profiles with sampling distance of Δ (2θ) ≤0.5° are found to contain many Fourier ripples and aliasing effect, whereas very few ripples and aliasing effects in those obtained with Δ (2θ) ≤0.1° A 30 A band was observed for MCMBs heat-treated below 800°C, giving rise to the modulation in oscillation curves of the Patterson functions. Such a result is in agreement with that reported by Hirsch for coals.

Unravelling the Patterson function (the auto-correlation function of the crystal structure) (A.L. Patterson, Phys. Rev. 46 (1934) 372) can be the only way of solving crystal structures from neutron and incomplete diffraction data (e.g. powder data) when direct methods for phase determination fail. The negative scattering lengths of certain isotopes and the systematic loss of information caused by incomplete diffraction data invalidate the underlying statistical assumptions made in direct methods. In contrast, the Patterson function depends solely on the quality of the available diffraction data. Simpson et al. (P.G. Simpson et al., Acta Crystallogr. 18 (1965) 169) showed that solving a crystal structure with a particular superposition of origin-shifted Patterson functions, the symmetry minimum function, is advantageous over using the Patterson function alone, for single-crystal X-ray data. This paper describes the extension of the Patterson superposition approach to neutron data and powder data by (a) actively using the negative regions in the Patterson map caused by negative scattering lengths and (b) using maximum entropy Patterson maps (W.I.F. David, Nature 346 (1990) 731). Furthermore, prior chemical knowledge such as bond lengths and angles from known fragments have been included. Two successful structure solutions of a known and a previously unknown structure (M. Hofmann, J. Solid State Chem., in press) illustrate the potential of this new development.

In this chapter we shall consider direct (phase probability) methods of solving the phase problem, and certain other techniques which are usually involved in the overall investigation of crystal and molecular structure.

In this chapter we wish to draw together, by means of actual examples, material presented earlier in the book. It may be desirable for the reader to refer back to the previous chapters for descriptions of the techniques used, since we shall present here mainly the results obtained at each stage.

This chapter provides an introduction to noncrystallographic symmetry. Topics covered include: the definition of noncrystallographic symmetry; the use of the Patterson function to interpret noncrystallographic symmetry; the interpretation of generalized noncrystallographic symmetry where the molecular structure is partially known; and the power of noncrystallographic symmetry in structure analysis. Keywords: Patterson functions; cross-rotation function; cross-translation function and noncrystallographic symmetry; molecular replacement; noncrystallographic symmetry; rotation functions; structure determination; translation functions

A probabilistic approach is described that is able to estimate triplet phase invariants when prior information on interatomic vectors and interatomic triangles is available. The conclusive formula is compared with the vector-interaction formula derived by Hauptman & Karle [Acta Cryst. (1962), 15, 547-550] and with a probabilistic formula obtained via maximum-entropy methods.

Crystals of macromolecules often have two or more molecules per asymmetric unit, or contain domains of a macromolecule or a macromolecular complex that are structurally independent. In such cases the conventional molecular-replacement method attempts to determine the position of each structural unit independently. Typically, some parts of the structure can be determined more easily or more reliably than other parts. Methods are proposed whereby information from a part of a crystal structure that has been determined can be used to help determine the structure of the remainder. Two different strategies are discussed, 'subtraction' and 'addition'. With 'subtraction' strategy the Patterson function of the known part of the structure is subtracted from the 'observed' Patterson. This approach is found to be most effective in the context of the rotation function in that it eliminates peaks that are irrelevant to the desired solution. With 'addition' strategy the structure factors of the known component are added to those of the search model. This procedure is most effective in the context of the translation function because it brings the structure factors calculated from the search model closer to those observed. Methods of applying the fast Fourier transform to facilitate these calculations are described. A number of examples are provided including structures of mutants of T4 lysozyme that might not have been solved without recourse to the proposed methods. A method of including information from a heavy-atom derivative in a translation function is also developed and shown to be superior in some situations to the conventional translation function.

A novel procedure has been developed for locating heavy-atom positions in crystals of macromolecules. This method used genetic algorithms (GA's) to search for heavy-atom sites that are consistent with an observed difference Patterson function. The procedure is straightforward to apply, space-group independent, and particularly powerful for cases involving non-crystallographic symmetry of multiple heavy atoms in the asymmetric unit. In this paper, we introduce how GA's are used for determining the heavy-atom positions and show how this method is more efficient than a sequential search.

The technique of electron-emission holography (EEH) is reviewed with respect to its potential as a direct local structural probe. Direct inversion of the measured Kikuchi electron patterns is emphasized. The description of how the multiple scattering effects are eliminated by the integral-energy phase-summing method is presented. High-fidelity, artifact-free three-dimensional atomic images obtained by inverting simulated diffuse LEED and photoelectron diffraction patterns are shown. Direct inversion of the measured Kikuchi patterns shows clear images of the neighboring atoms within the range of the electron mean free path, thus yielding structural information with high resolution (~1 Å) in all directions. Special attention is paid to the correct role of background subtraction in removing artifacts, and to the selection of minimal experimental data in the practical application of EEH. Moreover, direct inversion of diffraction patterns from many different local emitters is found to yield a three-dimensional Patterson function of the near-surface structure. This leads to direct surface structural determination by inverting Kikuchi electron patterns. Finally, the future of this direct method based upon inverting Kikuchi electron patterns, including the comparison with diffuse LEED and photoelectron holography, is discussed.

A preceding paper reported how to obtain a priori quantitative information on the lack of isomorphism (LOI), considered as noise corrupting the heavy-atom signal in a derivative data set. This related paper initially examines how additional a priori information can be drawn from the knowledge of the level of LOI. First, a corrected estimate of the coefficients necessary for a difference Patterson synthesis is derived. An estimate of their accuracy is also obtained. Then, individual and, independently, shell-averaged figures of merit that can be expressed in terms of the phasing power obtained in the preceding paper are determined. These afford an early estimate of the probable phase error on the heavy-atom structure factor. In a second and independent part of the paper, a correlation/translation function is proposed for the localization of the heavy-atom site(s). The results, bearing on both test and real cases, show that this method can be helpful in many situations.

A procedure for Patterson search, or molecular replacement, is described in which the criteria of fit are based on matching the asymmetric unit of the entire Patterson function. In rotation search, the Patterson function is compared with the self-Patterson of the search model; in translation search, the comparison is with the full Patterson function of the search model. Significant features of the method are: (1) all overlaps of vector sets of neighboring molecules are taken into account; (2) all overlaps of the search model with neighboring copies are detected and the evaluation bypassed; and (3) the criteria of fit are flexible and can be expressed in either Patterson or reciprocal space.

Archived PDF is a preprint.; International audience; This paper defines homometry in the rather general case of locally-compact topological groups, and proposes new cases of its musical use. For several decades, homometry has raised interest in computational musicology and especially set-theoretical methods, and in an independent way and with different vocabulary in crystallography and other scientific areas. The link between these two approaches was only made recently, suggesting new interesting musical applications and opening new theoretical problems. We present some old and new results on homometry, and give perspective on future research assisted by computational methods. We assume from the reader's basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform.

Archived PDF is a preprint.; International audience; This paper describes phase-retrieval approaches in music by focusing on the particular case of the cyclic groups (beltway problem). After presenting some old and new results on phase retrieval, we introduce the extended phase retrieval for a generalized musical Z-relation. This concept is accompanied by mathematical definitions and motivations from computer-aided composition. We assume from the reader basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform.

Some theoretical and practical aspects of the application of transmission microdiffraction (µXRD) to thin sections (≤30 µm thickness) of samples fixed or deposited on substrates are discussed. The principal characteristic of this technique is that the analysed micro-sized region of the thin section is illuminated through the substrate (tts-µXRD). Fields that can benefit from this are mineralogy, petrology and materials sciences since they often requirein situlateral studies to follow the evolution of crystalline phases or to determine new crystal structures in the case of phase transitions. The capability of tts-µXRD for performing structural studies with synchrotron radiation is shown by two examples. The first example is a test case in which tts-µXRD intensity data of pure aerinite are processed using Patterson-function direct methods to directly solve the crystal structure. In the second example, tts-µXRD is used to study the transformation of laumonite into a new aluminosilicate for which a crystal structure model is proposed.

A method is presented to determine the helix orientation starting from X ray diffraction data. Our method is based on the periodicity properties of alpha helix and on the analysis of the properties of the corresponding Patterson function, obtained directly from raw crystallographic intensities. Knowledge of helix orientation is a useful information within the structure solution process of proteins. © 2011 IEEE.

Image analysis uses either Fourier transforms (FTs) or cross-correlations (CCs) and FTs are based on CCs, which are usually calculated with FTs. This chapter introduces the concept of correlation function, referring to situations in molecular biology. Correlation is used in sequence comparisons. Cross-correlation function can be considered from two different viewpoints. Although they give identical answers, they differ in their intuitive significance, and one viewpoint may be more appropriate than another for certain applications. Convolution is discussed as a process of replacing every element of one function by an entire copy of the other function. Correlation of a function with itself gives its auto-correlation function (ACF). The ACF is always an even function, because the later stages of overlap exactly mirror the early stages, except that the top and bottom curves are exchanged. In electron microscopy, this should apply to two exact copies of a particle in the same orientation. In X-ray crystallography the ACF is called the Patterson function, which can be calculated directly from the experimental intensity data, without the arduous and uncertain search for the phases of the diffraction pattern. Consequently, much effort has gone into finding its uses.

The structure of the (√3 × √3)R30° reconstruction induced by the adsorption of 1 3 of a monolayer of In on the Si(111) surface has been determined using surface X-ray diffraction. The fractional order Patterson function obtained from structure factor intensities at zero perpendicular momentum transfer (l = 0), indicates lateral displacement in the silicon surface atoms. Intensity profiles of fractional order rods and one integer order rod gives information concerning displacements normal to the surface of the silicon substrate. The indium adatoms are shown to occupy the 4-fold coordinated T4 sites above the second layer silicon atoms. Keating elastic strain energy minimisation has been used to determine relaxations down to the sixth layer of the bulk.

The crystal structure of magnesium boron nitride in a high-pressure phase, Mg 3 NB 3 (H), has been determined from X-ray powder diffraction data. The strategy used was: powder pattern indexing, extraction of integrated intensity data by peak decomposition, Patterson function calculation, trial-and-error model building and Rietveld refinement. The cell is orthorhombic (space group Pmmm, Z=1) with a=3.0933 (2), b=3.1336 (2) and c=7.7005 (5) A. The structure contains linear NBN groups with an observed B-N interatomic distance of 1.34 A which is 2% shorter than in the low-pressure form of the material, Mg 3 BN 3 (L)

A new method has been developed for the improved intensity assignment of severely or exactly overlapping Bragg reflections in a powder diffraction pattern. This fast iterative Patterson squaring (FIPS) method addresses, in particular, the systematic lack of small and large normalized intensities (|E| values) in severely overlapping powder data, which causes (a) the intensity statistics to be strongly acentric (even when the structure itself is centrosymmetric) and (b) the under-estimation of strong structure-invariant relationships. Direct methods for structure determination are more likely to succeed with FIPS-improved data than with conventionally used equipartitioned data (ratio of |E| values for overlapping reflections set to 1.0). In the case of the molecular sieve AFR (in which 65% of the reflections severely overlap), the ab initio structure solution was only possible after a redistribution of the intensities by the FIPS method.

In a Laue diffraction pattern, 10–20% of the spots result from the exact superposition of two or more reflections that are 'harmonics', e.g. hkl, 2h, 2k, 21. .... For the solution of large or difficult structures, the intensities of the remaining 80–90% of the reflections, measurable as singles, may not be sufficient and thus evaluation of the intensities of the components of the multiple spots is important. A procedure for this deconvolution is given, based on the assumption of non-negativity and nonoverlapping peaks in the Patterson function. It has been tested with Laue diffraction data from an organic crystal, C25H20N2O2, where it allowed 275 reflection intensities to be evaluated from multiple spots, 140 of them with |F|2 > 3σ(|F|2). For these 140 reflections, agreement with Fcalc is reasonable (R = 0.14) and their addition to the 1129 singles made structure solution (by direct methods) significantly easier.

The most widespread application of the rotation function is the determination of the relative orientation of a given search fragment in the unit cell of an unknown crystal structure [Rossmann & Blow (1962). Acta Cryst. 15, 24–31]. Here a modification is presented of the rotation function for this specific application, which exploits the information of the intensity data more effectively, thus leading to a higher signal size with the same computing cost.

Most Patterson-search methods are based on functions measuring the coincidence of the observed Patterson function with the model Patterson function calculated as a function of some variables, e.g. the rotation angles in the case of the rotation function. In parallel with such methods, a new phase-refinement function based on the maximization of ZR(Φ) = ΣH(EH − 〈EH〉) GH(Φ) as a function of the collectivity Φ of phases of the reflections with large E values is presented, where EH − 〈EH〉 and GH(Φ) are the Fourier coefficients of two Patterson-type functions, the first with the origin peak removed and the second taking into account the atomicity condition. It will be shown how ZR(Φ) can be maximized by a new tangent formula actively using the small E values and not possessing any weighting factor. The utility of this formula is illustrated on the basis of two representative test examples.

Six-dimensional Patterson functions as measured for AlCuFe and AlPdMn perfect quasi-crystals show only two correlation features located at lattice sites and body centres of a cubic unit cell. They have been interpreted in terms of a CsCl-like 6-dim structure. In the present work, the uniqueness of this interpretation has been addressed. Another structure with weak mid-edge decoration happens to be also consistent with the observed Patterson functions. A significant 5% density difference should make a selection of the good model possible.