ALBERT

Beschreibung: A theoretical analysis of the parametric X-radiation (PXR) produced by the nonrelativistic electrons in an X-ray tube with a monocrystalline anode is presented. It is shown that PXR can be considered as a source of quasi-monochromatic X-radiation for laboratory diffraction experiments. The intensity of the PXR is comparable with the intensity of the characteristic radiation, but the frequency of the former can be varied in a wide range. Possible applications of the phenomenon for crystallography are discussed.

Beschreibung: The periodic array of very fine cross-overs formed at the exit face of a thin `atomic focuser' crystal, illuminated by a parallel electron beam, may be used to form moiré patterns with a specimen crystal such that the structure of the specimen crystal may be derived with a resolution of better than 0.5 Å. Computer simulations of the moiré pattern formation have been made for the simple idealized case of two parallel gold-like lattices having a 10% difference in lattice constant. Moiré images are shown for the case of a small objective aperture in the viewing electron microscope such that the individual crystal lattices are not resolved and for a larger objective aperture for which the individual crystal lattices are resolved and the intensity is measured at the positions of the atoms of the atomic focuser crystal. The latter case confirms the viability of the scheme for ultra-high-resolution imaging of general specimens by use of a thin-crystal periodic atomic focuser, as previously proposed.

Beschreibung: A new approach to the inversion problem of dynamical transmission electron diffraction is described, based on the method of generalized projections in set theory. An algorithm is described that projects between two sets of constrained scattering matrices. This iterative process can be shown to converge, giving the required structure factors (for some choice of origin) if the sets are convex. For the dynamical inversion problem, the set topology is that of an N 2 torus, the sets are not convex, and traps are therefore sometimes encountered. These can be distinguished from solutions, allowing the algorithm to be restarted until a solution is found. Examples of successful inversion from simulated multiple-scattering data are given, which therefore solve the phase problem of electron diffraction for centrosymmetric or noncentrosymmetric crystal structures. The method may also be useful for the three-beam X-ray diffraction problem.

Beschreibung: The structures of the zirconia-based ceramic oxide superlattice alloy phases Nb2Zr x−2O2x+1 (x ≈ 7–12) are reviewed. It is noted that recent investigations into the alloys' structure suggest they are not strictly crystalline, with consequences for their diffraction properties. It is suggested that several aspects of their behaviour may be interpreted as proper to a frustrated two-dimensional XY model, where the distinctive planes of oxygen atoms in the alloys form a lattice of planar XY `spins'. Evidence is presented for the existence of frustration, which produces superlattice canting in the model including electron-microscope evidence for the existence of domains of staggered Ising-type chirality in the alloys, as predicted for the frustrated two-dimensional XY model by Villain [J. Phys. C (1977), 10, 1717–1734].

Beschreibung: The cell parameters for LiF containing 90.4% 6Li and 50.2% 6Li were determined from a series of divergent-beam photographs taken at approximately 25 K intervals in the ∼15 to 375 K range. The divergent-beam photographs were generated in a Stereoscan S4 scanning electron microscope fitted with a liquid-helium cold finger and a Dingley Kossel camera. Lattice parameters can be determined from the ratios of lines joining conic intersections, and such data, measured on films taken at known temperatures, were correlated with equivalent data from computer-generated divergent-beam patterns using appropriate cell parameters; our cell parameters at 298 K are 6Li90.4%  = 4.0272 (2) and 6Li50.2%.  = 4.0266 (2) Å. If it is assumed that the stated isotopic compositions are correct, linear extrapolation to end-member cell parameters gives 6Li100%  = 4.0273 (2) and 6Li0%  = 4.0259 (2) Å. Since for a cubic crystal only one lattice-parameter measurement is required from each photograph, the usefulness of the ratio approach was tested by comparing values of the lattice parameter deduced from many such ratios obtained from photographs with no symmetry and with mirror symmetry.

Beschreibung: It is usually assumed that high-Tc superconductors have a periodic band structure and a periodic charge density, although amorphous low-Tc superconductors are known. In this paper, it is suggested that the CuO2 conduction planes of cuprate superconductors consist of regions of two different charge densities which do not normally repeat periodically. It is suggested that the pairing of holes occurs in real space in cuprate superconductors. It is proposed that the hole-pairing mechanism is magnetic exchange coupling and the pairing force is strong, the pairing energy being greater than kT at room temperature. The bound hole pair is essentially a bipolaron. A real-space model is very tentatively suggested in which the CuO2 planes of YBa2Cu3O7 contain nanodomains of a 3 × 3 hole lattice surrounded by interfaces one unit cell wide in which the holes are paired. In the superconducting state in this model, the existing hole pairs condense and move coherently and collectively around the insulating nanodomains, like trams running around blocks of houses, with one hole on each tramline. The hole pairs move in an elegant manner with hole pairs hopping from oxygen to oxygen via adjacent copper sites. The model explains the superconducting current being in the ab plane and it also explains the very short coherence lengths. Because the pairing force is strong, the model suggests that room-temperature superconductivity might be possible in carefully designed new oxide materials.

Beschreibung: Green's-function techniques are used to obtain a real-space series solution for the elastic reflection high-energy electron diffraction (RHEED) from a crystalline surface. A renormalized perturbation expansion due to potential self-scattering is developed for the local real-space Green's function. With the Pt (111) surface as an example, numerical results are obtained for reflection coefficients and intensities. In particular, calculations are performed to obtain the local density of states at and near the surface region. Total density-of-states calculations are also performed. These provide a basis for a discussion of the form of resonant electronic Green's functions that can be used to describe the surface resonance phenomenon within RHEED.

Beschreibung: Quantitative convergent-beam electron diffraction is used to determine structure factors and three-phase structure invariants. The refinements are based on centre-disc intensities only. An algorithm for parameter-sensitive pixel sampling of experimental intensities is implemented in the refinement procedure to increase sensitivity and computer speed. Typical three-beam effects are illustrated for the centrosymmetric case. The modified refinement method is applied to determine amplitudes and three-phase structure invariants in noncentrosymmetric InP. The accuracy of the results is shown to depend on the choice of the initial parameters in the refinement. Even unrealistic starting assumptions and incorrect temperature factor lead to stable results for the structure invariant. The examples show that the accuracy varies from 1 to 10° in the electron three-phase invariants determined and from 0.5 to 5% for the amplitudes. Individual phases could not be determined in the present case owing to spatial intensity correlations between phase-sensitive pixels. However, for the three-phase structure invariant, stable solutions were found.

Beschreibung: A simple but sufficiently accurate expression is obtained for the exit wave of a crystal in zone-axis orientation. The exit wave at each atom column can be parametrized with only one parameter, which is a function of the projected `weight' of the column.

Beschreibung: X-ray powder and electron single-crystal diffraction of crystals of Li2TiOSiO4 and Li2TiOGeO4 showed them to be tetragonal, space group P4/nmm unit-cell parameters a = 6.4379,(2), c = 4.40032,(2) Å for Li2TiOSiO4 and a = 6.6110,(8), c = 4.4372,(6) Å for Li2TiGeO4. The compounds are isostructural with their sodium analogues but are considerably compressed along the c axis owing to the smaller size of lithium compared with sodium atoms. Square-pyramidal TiO5 groups are joined in these compounds by tetrahedral SiO4 and GeO4 groups, respectively. 17O nuclear magnetic resonance spectra of the two compounds, isotopically enriched with 17O, showed peaks due to the apical titanyl, Ti—O, and basal, bridging, Ti—O—Si or Ti—O—Ge, oxygen atoms of the title compounds. By comparison with reference spectra, oxygen K edges and titanium L 2,3 edges of electron energy-loss spectra were tentatively assigned.

Beschreibung: A form of turbostratic boron nitride containing a large percentage of micrometre-size conically shaped particles was investigated by transmission electron microscopy. Electron diffraction patterns revealed the presence of correlations between adjacent boron nitride layers. The diffraction patterns were also used to determine the cone apex angle of many cone-like objects. The apex angles exhibited a broad distribution from 84 to 130° with an ill defined peak in the 92–95° region. These results could be accounted for by a structure model in which conical boron nitride layers are helically wound about a disclination axis, according to overlap angles that correspond to high densities of coincidental lattice sites between successive layers. The clustering of the apex angles near 93° was attributed to the formation of a square ring during the nucleation stage. This appears to be specific to the boron nitride system.

Beschreibung: The probabilistic formulas relating standard and mixed type reflections (these last show integral and half-integral indices) are derived. It is shown that probabilistic estimates can be obtained by using particular sections of the three-dimensional reciprocal space. The concept of structure invariant is extended to define the wider class of structure quasi-invariant. Their statistical behaviour is briefly discussed with the help of some practical tests.

Beschreibung: The lattice dynamics of 19 compounds with the sodium chloride structure have been investigated using shell models. The models are compared with existing experimentally measured phonon-dispersion curves and refined using a multidimensional downhill simplex method. Debye–Waller factors for these compounds are calculated over the temperature range from 1 to 1000 K where appropriate and the results are fitted analytically using polynomial regression. The results are compared with experimentally measured room-temperature Debye–Waller factors and for most of the compounds the agreement is found to be better than 10%. At lower temperatures, it is expected that these results would be more accurate, since the harmonic approximation, which is crucial to the calculation of the Debye–Waller factors, works better. In choosing the models for particular applications, it is recommended that the model with the smallest standard error sigma is used for fitting the experimentally measured phonon-dispersion curves, or the model that shows best agreement with reliable experimental measurements of more relevant physical quantities, such as Debye–Waller factors in crystallography.

Beschreibung: Tables of modified X-ray scattering factors for neutral elements with Z = 70–100 have been calculated using multiconfiguration Dirac–Fock wave functions. The physical approximation is a next step beyond the usual form-factor approximation in calculating elastic scattering photon intensities. Differences as large as 3–6% compared to previous calculations of the same kind are obtained; differences from the usual form factors are much larger.

Beschreibung: Unfortunately, some printing errors escaped our attention in the proof-reading of our paper [Pilz & Fischer (1998). Acta Cryst. A54, 273–282]. As some of them may cause inconvenience or confusion to readers, they are corrected here: p. 275, equation (6a) should read: Q_i = left[extstyle sum limits_{j = 0}^{i-1}(-1)^{i-j-1}P_{i-j}Q_j ight] igg/i.p. 277 (right column), third line after equation (12) should read:  Using D 1 [from (10a)] …. p. 277 (right column), second line after equation (13) should read:  … namely 2^{m+n-1}. p. 279 (right column), lines 10 and 11 in second paragraph should read:  … σ(gh ) = 0.05gh (or 0.1gh , respectively) for …. p. 279 (right column), fifth line from bottom should read: … example I (Table 2) …. p. 280 (right column), third line before Section 3.2 should read: E.g. one of them systematically ….

Beschreibung: The geometry of a modern imaging diffractometer is discussed in detail. A method to find all relevant instrument parameters from the control single-crystal measurement data is proposed and the limitations of such a procedure are indicated. Optimization of the instrument parameters by the least-squares method is presented.

Beschreibung: Parameterization has been made for the temperature dependence of the Debye–Waller factors of 68 elemental crystals and 17 compounds with the zinc-blende structure. The Debye–Waller factors of 46 elemental crystals were calculated based on the available phonon density of states, and those of the remaining 22 elemental crystals were estimated using the Debye approximation for the phonon density of states. The Debye–Waller factors of the zinc-blende compounds were obtained from the 14-parameter shell-model calculations of Reid [Acta Cryst. (1983), A39, 1–13].

Beschreibung: The possibility of a transmission-type X-ray linear polarizer is investigated using a thin Bragg reflector as a polarizing filter. In this device, the transmitted beam (rather than the Bragg-reflected beam) is the useful output of the device. Consequently, the position and the direction of the transmitted beam are unchanged as the energy is changed, or even when the polarization direction to be filtered out is changed. Theoretical considerations as well as preliminary transmissivity measurements are presented. The use of perfect crystals of silicon and diamond is examined. A polarization ratio, defined as I_H/I_V, higher than 105 was observed in experiments to measure the performance characteristics of the proposed X-ray polarizing reflector. The transmission-type X-ray linear polarizer is well suited for spectroscopic measurements with polarized X-rays.

Beschreibung: The structure of the R phase of the alloy Ti50Ni48.7Co1.3 was determined by the crystallogeometric analysis of martensitic transformations in titanium nickelide. This analysis is based on the concept of cooperative thermal vibrations of the (110)-like planes in the b.c.c. and B2 structures and their freezing (transformation to static displacements) at the point of the phase transition. The space group of the R phase is trigonal P31m with a hexagonal Bravais lattice. The validity of the R-phase structure determination is confirmed by an X-ray diffraction pattern that was obtained at room temperature using Cu Kα radiation. The Rietveld procedure was used to compare the calculated and experimental X-ray diffraction patterns. The proposed structural model of the R phase agrees satisfactorily with the experimental data.

Beschreibung: Four-connected three-dimensional (3D) nets were systematically enumerated by converting some horizontal edges of a vertical stack of three-connected two-dimensional (2D) nets into vertical zigzag chains. 77 out of 131 2D nets were selected for enumeration, and different arrangements of zigzag edges and horizontal edges were investigated. This yielded 138 3D nets of which 19 are represented by known structures: cristobalite; tridymite; MAPO-39 (International Zeolite Association Structure Commission code ATN); svyatoslavite; Li-A(BW) (ABW); cancrinite (CAN); AlPO-31 (ATO); MAPO-36 (ATS); BaFe2O4; `nepheline hydrate' (JBW); bikitaite (BIK); KBGe2O6; CsAlSi5O12 (CAS); UiO-6 (OSI); Theta-1 (TON); ZSM-12 (MTW); ZSM-23 (MTT); AlPO-53C; and CIT-5 (CFI).

Beschreibung: The magnetic subperiodic groups, the 31 magnetic frieze-group types, the 394 magnetic rod-group types and the 528 magnetic layer-group types, are derived and given symbols based on the symbols for the nonmagnetic subperiodic groups in Volume E of International Tables for Crystallography. The symbols are constructed in analogy to the Opechowski–Guccione symbols for magnetic space groups. Tables are given that list one group from each type. Each group is specified not only by its symbol but also by explicitly listing the coset representatives of the coset decomposition of the group with respect to its translational subgroup.

Beschreibung: The zone-axis CBED pattern-matching technique ZAPMATCH [Bird & Saunders (1992). Ultramicroscopy, 45, 241–251] has been applied to low-order structure-factor measurements in nickel and copper. Considerable disagreement exists between previously published results obtained with a variety of solid-state theories and experimental techniques. The nickel ZAPMATCH results confirm previous electron-diffraction critical-voltage measurements and are in excellent agreement with FLAPW (full-potential linearized augmented plane-wave) theory calculations. This is further proof of the accuracy achievable with ZAPMATCH analysis. For copper, however, while the results support the findings of previous experimental measurements, they are consistently higher than those given by a range of solid-state theories, perhaps demonstrating some limitation in the existing theory. Two extensions to the ZAPMATCH technique are also considered. First, rules are developed to determine the number of structure factors that can be refined accurately from a given CBED pattern. Second, the imaginary potential generally introduced to account for the effects of thermal diffuse scattering (TDS) is also refined. It is shown that, while the widely used Einstein model is a useful approximation, the refined values are consistently higher than the model predicts. In addition, the importance of a second-order (real) TDS correction arising from the Einstein model is investigated. Although its effects are limited in this instance, it may prove to be more significant at lower beam energies or for materials of higher atomic number.

Beschreibung: Scattering of electrons is affected by the distribution of valence electrons that participate in chemical bonding and thus change the electrostatic shielding of the nucleus. This effect is particularly significant for low-angle scattering. Thus, while chemical bonding effects are difficult to measure with small-unit cell materials, they can be substantial in the study of proteins by electron crystallography. This work investigates the magnitude of chemical bonding effects for a representative collection of protein fragments and a model ligand for nucleotide-binding proteins within the resolution range generally used in determining protein structures by electron crystallography. Electrostatic potentials were calculated by ab initio methods for both the test molecules and for superpositions of their free atoms. Differences in scattering amplitudes can be well over 10% in the resolution range below 5 Å and are especially large in the case of ionized side chains and ligands. We conclude that the use of molecule-based scattering factors can provide a much more accurate representation of the low-resolution data obtained in electron crystallographic studies. The comparison of neutral and ionic structure factors at resolutions below 5 Å can also provide a sensitive determination of charge states, important for biological function, that is not accessible from X-ray crystallographic measurements.

Beschreibung: A coordinate-space multislice description of the scattering of high-energy electrons is constructed from consecutions of differential operators acting upon atomic potentials. It is used to find expressions for the intensity distribution in high-resolution electron-microscope images of crystals whose atoms are periodically displaced relative to a reference lattice according to a modulation wave. Both static correlated displacements, such as occur in modulated structures, and time-dependent correlated displacements, as are generated by phonons, are considered. Two aspects of the image are examined in detail; its translational symmetry and its dependence upon the correlations between the atomic displacements. It is shown that the intensity distribution due to scattering from static correlated displacements has the translational symmetry of the modulated structure in that projection, as determined by the component of the modulation wavevector perpendicular to the incident beam, whereas that due to scattering from phonons has the translational symmetry of the reference lattice in that projection. The former is a consequence of higher-order Laue-zone interactions. The intensity distribution due to scattering from static displacements depends upon the absolute phase of the displacement at each scattering atomic site whereas that due to scattering from phonons depends only upon the relative phase of the displacements between different scattering sites, both within the same atomic column parallel to the beam and in adjacent columns. In both cases, the influence of the component of the correlation wavevector parallel to the incident beam is different to that perpendicular to the beam; the former affects the intensity mostly at the atomic sites whilst the latter affects the intensity mostly between the atomic sites. It is also observed that, as a consequence of the periodic nature of the polarization-vector function, the interference terms are small, both relative to the non-interference term and in an absolute sense, particularly for phonon scattering. For this reason, the contribution to the image due to scattering from correlated atomic displacements will have greater and sharper atomic contrast than that due to scattering from the reference structure without displacements. In addition, this component of the intensity distribution will not exhibit strong contrast reversal when the objective-lens defocus is changed.

Beschreibung: It is shown that the small-angle scattering of X-rays or neutrons by dislocations within a deformed metal, which are partially ordered into wall-like structures, is characterized by several factors. Principally these are associated with: (i) a single dislocation or dipole; (ii) the dislocation configuration in the plane of the wall; and (iii) the distribution of dislocations across the wall thickness. With the assumption of isotropic elasticity, small-angle scattering will be sensitive only to the edge components of the dislocations. The scattered intensity is dominated by scattering from dislocations that lie perpendicular to the scattering vector, f q, and reaches a maximum when f q is normal to the slip plane of these dislocations. Above a particular | f q |, the scattered intensity is sensitive only to the total edge dislocation content of the scattering dislocations (i.e. scattering is incoherent), while, below this value, the scattering is dominated by how the dislocations are distributed in walls. For walls normal to their slip planes, the configuration factor will reflect the dislocation distribution in the plane of the wall, while, for walls parallel to their slip planes, the distribution in the thickness direction will be visible. Therefore, even though a deformed material is composed of complicated dislocation structures, only those segments conforming to these rather strict prescriptions will be singled out for scattering, and, by adjusting the beam/slip system geometry, many parameters of the microstructure can be determined experimentally.

Beschreibung: Quantitative CBED techniques, such as the ZAPMATCH zone-axis pattern-matching method [Bird & Saunders (1992). Ultramicroscopy, 45, 241–251], have been applied with great success to the accurate refinement of low-order structure factors. The major limitation on the accuracy of the structure-factor measurements is uncertainty in the Debye–Waller factors describing the temperature-dependent atomic vibrations. While X-ray and neutron diffraction techniques are both capable of accurate measurements of Debye–Waller factors, the frequent use of liquid-nitrogen-cooled samples in CBED experiments means that previous measurements are rarely available at the temperatures required. This has prompted attempts to determine Debye–Waller factors from electron diffraction data obtained under experimental conditions that match those used for the quantitative CBED work. In this paper, the possibility of extracting accurate Debye–Waller factors from the low-order reflections of a zone-axis CBED pattern is investigated. In this way, the Debye–Waller factors and structure factors could be obtained from the same data set. With this new approach, it is shown that errors lower than ±0.02 Å2 can be obtained for the measurement of Debye–Waller factors from room- and liquid-nitrogen-temperature nickel and copper 〈110〉 zone-axis data. The results obtained are compared with previous measurements and theoretical predictions.

Beschreibung: Scattering of a quasi-monochromatic electron beam by a crystal with a defect is described with the use of the mutual coherency function and the formalism of quasi-Bloch waves. An expression correlating the mutual intensity on the exit and entrance surfaces of the crystal in terms of the scattering matrix has been found. The matrix elements are determined by a system of integro-differential equations, which have been obtained without using the column approximation. It has been shown that calculations of the matrix elements can be significantly simplified when the approximation of the small-angle scattering of quasi-Bloch waves by the defect displacement field is satisfied. Such an approximation can be applied in many cases, e.g. to a crystal with a dislocation. The mutual intensity on the crystal entrance surface has been found for the general case of defocused illumination. As an example of applying the new approach, expressions for the intensity in convergent-beam electron diffraction (CBED) and large-angle CBED (LACBED) patterns have been obtained. The LACBED patterns of a crystal with a dislocation have been simulated. It has been shown that the developed approach allows a more exact simulation of the LACBED than do the conventional approaches using the column approximation and the approximation of independent plane waves filling the illumination cone.

Beschreibung: The unique ability of tessellation methods to characterize atomic structures is highlighted, with the Voronoi tessellation, by virtue of its freedom from constraining assumptions, emphasized as pre-eminent among these methods. It is demonstrated how to extract information from an atomic structure by constructing its Voronoi tessellation and analysing the results. In so doing, insight is gained into the coordination patterns of the atoms and how their positions are influenced by neighbours. As an example of such a procedure, typical Voronoi volumes are calculated for 72 atom types found in a set of 249 binary intermetallic and ionic compounds. Whereas the various atom types have broadly characteristic volumes, these are also sensitive to the other atom types with which they are combined in a given compound, i.e. whether the latter behaves as a metal, semimetal, semiconductor or insulator. Further, smaller variations in volume occur for different atom types within these four categories. Thus, structural information obtained from the Voronoi tessellation may possibly be correlated directly with atomic properties.

Beschreibung: The probability distribution of the four-phase structure invariants (4PSIs) involving four pairs of structure factors is derived by integrating the direct methods with isomorphous replacement (IR). A simple expression of the reliability parameter for 16 types of invariant is given in the case of a native protein and a heavy-atom derivative. Test calculations on a protein and its heavy-atom derivative using experimental diffraction data show that the reliability for 4PSI estimates is comparable with that for the three-phase structure invariants (3PSIs), and that a large-modulus invariants method can be used to improve the accuracy.

Beschreibung: The connection between the crystallographic phase problem and the feasible set approach is explored. It is argued that solving the crystallographic phase problem is formally equivalent to a feasible set problem using a statistical operator interpretable via a log-likelihood functional, projection onto the non-convex set of experimental structure factors coupled with a phase-extension constraint and mapping onto atomic positions. In no way does this disagree with or dispute any of the existing statistical relationships available in the literature; instead it expands understanding of how the algorithms work. Making this connection opens the door to the application of a number of well developed mathematical tools in functional analysis. Furthermore, a number of known results in image recovery can be exploited both to optimize existing algorithms and to develop new and improved algorithms.

Beschreibung: Multiple scattering of neutrons by the inhomogeneities responsible for small-angle neutron scattering (SANS) during the passage of the beam through the specimen can be used to provide valuable information about the shape of the objects and the absolute value of the contrast between the scattering particles and the matrix. The neutrons emerging from the specimen are classified into those that have been scattered n times. The index n ranges from zero to infinity. The remnant of the incident beam is the group of neutrons for which n equals zero. Each group contributes separately to the scattering profile. The small-angle scattering cross section is independent of the neutron wavelength for n = 1 only. Thus collection of data as a function of specimen thickness and of neutron wavelength will provide a number of different profiles describing the same physical situation. Simultaneous analysis of these profiles provides absolute values of the cross section for scattering into the small-angle region and of the cross section for removal of neutrons from the small-angle region. So that the method can be used generally, a profile function that is a very good approximation to those in the literature is introduced. The implications for time-of-flight SANS are discussed.

Beschreibung: The X-ray standing wave has been studied when the real part of the scattering factor is zero. In the symmetric Laue case, the phase of the standing wave advances by π when the deviation parameter W changes from −1 to 1, which is the same variation as in the usual symmetric Bragg case when only the real part of the scattering factor exists. However, the phase in the former case is different from that in the latter by pi/2. By using the standing waves, the origins of the anomalous transmission and anomalous absorption effects reported by Fukamachi & Kawamura [Acta Cryst. (1993), A49, 384–388] have been analysed. The standing wave in the Laue case can give rise to a more accurate method of site determination of a specified impurity atom as well as a wider range of applications than a conventional standing-wave approach.

Beschreibung: Off-axis electron holography has been used to determine the mean inner potential of germanium using cleaved 90° wedge samples, where the wedge thickness profiles were checked by weak-beam dark-field extinction fringes. Dynamical contributions to the phase of the image were minimized by tilting to weakly diffracting conditions, as confirmed by reference to convergent-beam electron diffraction patterns. Small residual corrections were determined using multislice calculations. From a total of 18 separate measurements, it is concluded that the value of the mean inner potential is 14.3 (2) V, which agrees with recent theoretical calculations to within experimental error.

Beschreibung: Ab initio crystal structure prediction can proceed by minimization of the packing energy of Z independent molecules per cell, or alternatively by energy minimization taking one molecule as independent and Z-1 molecules as related by assumed space-group symmetry. In the former method, a large number of positional variables must be considered. In the latter method, a large number of space groups must be considered. An alternative, more efficient, procedure is proposed, where it is recognized that values of Z and the number of molecules in the asymmetric unit, Z′, impose restrictions on possible space groups. Examples of application of this method to crystal structure prediction are given.

Beschreibung: Following a preliminary revisitation of the nomenclatures in use for mica polytypes, the properties of the periodic intensity distribution (PID) function, which represents the Fourier transform of the stacking sequence, are analysed. On the basis of the relative rotations of neighbouring layers, mica polytypes are classified into three types; for each type, the PID exists in different subspaces of the reciprocal space. A revised procedure to compute the PID, in which further restrictions on the structural model orientation are introduced, is presented. A unifying terminology based upon the most common symbols used to describe mica polytypes (RTW, Z and TS) is derived; these symbols represent the geometrical basis for the computation of the PID. Results are presented for up to four layer polytypes and are compared with the reflection conditions derived by means of Zvyagin's functions. Both the PID values and the reflection conditions are expressed in suitable axial settings and compared with previous partial reports, revealing some errors in previous analyses. A computer program to compute PID from the stacking symbols is available.

Beschreibung: The broken-bond model for the surface free energy of crystals is considered. The consequences are derived for the morphology of periodic crystals, quasicrystals, incommensurately modulated crystals and intergrowth compounds. It is found that low-index facets on periodic crystals and quasicrystals are the result of finding an optimized position along the vector normal to the surface of the surface plane. The same principle explains the normal facets on modulated crystals and intergrowth crystals. The so-called satellite facets are the result of surface pinning of the phase of the modulated wave. On intergrowth compounds, facets may be found that are stabilized by a combination of both mechanisms. It is shown that the most stable facets on intergrowth crystals are the normal facets that are common to the subsystems, independent of the details of the structure.

Beschreibung: Algebraic expressions for projection operators and symmetry-adapted functions (SAFs) of the icosahedral group for spinor (double-valued) representations are found by using the double-induced technique and eigenfunction method. The SAFs are functions of the angular momentum j, the quantum numbers lambda, u, mu of the group chain I supset D_5 supset C_5, and the multiplicity label ar m. By this procedure, SAFs for the group I are provided once for all instead of one j value at a time.

Beschreibung: It is well known that the crystallography of approximants is directly related to that of the parent quasicrystal, once its unit-cell vectors are identified as parallel projections of certain N-dimensional lattice nodes { f A}^{i}. Derived here are explicit simple relations for calculating the shear matrices { oldvarepsilon} and the related crystallographic properties of the corresponding approximants, including diffraction indexing and the determination of the lattice in perpendicular space. Applied to low-dimensional approximants, the derivation shows that the systematic `accidental' extinction rules observed in the pentagonal phases are generic extinctions that are due to the geometrical properties of the projected 1D lattice and are independent of the actual model of the quasicrystal.

Beschreibung: From analysis of the elastic neutron scattering data of Dolino et al. [J. Phys. (Paris) (1984), 45, 361–371], it is shown that, besides the well identified components u_x and u_y of the acoustic displacements in the incommensurate (IC) phase of quartz, there also exists a strong component of the u_z vector of the modulation. The existence of the large u_z is not consistent with the currently accepted model for the IC transition in quartz, since the long-period IC modulation observed in quartz cannot induce any noticeable acoustic component u_z. The need for a new model is keenly felt in order to understand the origin of the IC modulation in quartz.

Beschreibung: In the present work, the positive-definite conditions of the elastic constants of two-dimensional (2D) quasicrystals with non-crystallographic symmetries have been derived. These results are necessary for studying elastic and mechanical properties of two-dimensional quasicrystals both in theory and in experiment.

Beschreibung: The electrostatic potential is a multicenter property that can be expressed as a sum of the contributions of electric moments located at each atomic site of a molecule. Independently of the model used to generate the electrostatic potential around the system, these atomic moments can be accurately obtained by the fit of this physical property outside the van der Waals envelop. However, the larger the system, the greater the number of parameters. In this study a way is proposed to reduce the number of centers in the representation of the electrostatic potential which becomes a sum of fragment contributions rather than atomic ones. A sample of six water molecules in different crystal environments was chosen to discuss the derived values of the electric moments referred to the molecular center of mass.

Beschreibung: The probability distribution of the structure factors with non-integral indices is derived in P1¯. For integral values of at least one of the indices, the intensity distribution coincides with that provided by Wilson's statistics, but may strongly differ when the indices are (or are close to) half-integers. The deviations are stronger when the integral part of the indices is small, and increase with the size of the structure. In favourable circumstances, moduli and phases of the reflections may be accurately estimated.

Beschreibung: Without invoking anomalous dispersion and heavy-atom derivatives, it is demonstrated that it is possible to directly determine the phases of a large number of reflections collected in a short time from macromolecular crystals using a stereoscopic oscillation-crystal imaging technique, in a multibeam diffraction geometry, where two crystallographic axes in opposite directions are employed as the rotation axes. The intensity profiles (distributions) of the diffraction spots versus the varying tilt Bragg angle of the rotation axis in the two stereoscopically related images yield quantitative phase information. Many multiple diffraction profiles of tetragonal lysozyme and an unknown protein structure are obtained at the rate of 100 profiles per 30 min of X-ray exposure.

Beschreibung: Charge densities and crystal structures can be determined routinely from X-ray diffraction as X-ray scattering is relatively weak and single scattering can be assumed. The strong dynamical diffraction of high-energy electrons has prevented electron diffraction from being used in the same way. Dynamical diffraction describes both the propagation of the Bragg diffracted wave in the crystal and the scattering by the crystal potential. The balance between these two processes changes as a function of voltage due to relativistic effects. The difference in diffracted intensities recorded at two voltages is shown to be directly proportional to the crystal potential. This is confirmed by calculations using first-order perturbation theory which show negligible differences compared to exact calculation. It should therefore be possible to use differences in intensity measured as a function of voltage to determine the crystal potential directly. If the full complex wave function is available, then there is a particularly simple procedure to recover the potential, even under dynamical conditions.

Beschreibung: The use of error-correcting codes as a source of efficient designs of phase permutation schemes is described. Three codes are used, all taken from the Bricogne BUSTER program [Bricogne (1993). Acta Cryst. D49, 37–60]: the Hamming [7, 4, 3], the Nordström–Robinson (16, 256, 6) and the Golay [24, 12, 8] or its punctured [23, 12, 7] form. These are used in a maximum-entropy–likelihood phasing environment to carry out phase permutation of basis-set reflections instead of the usual quadrant permutation or magic integer approaches. The use of codes in this way inevitably introduces some errors in the phase choices, but for most structures this is not significant especially when the gain in sampling efficiency is considered. For example, the Golay [24, 14, 8] allows the permutation of 24 centric phases in such a way that only 4096 phase sets are produced instead of 224 = 16 777 216, and one of these sets has, at most, only four wrong phases. The method is successfully applied to three powder diffraction data sets of increasing complexity, and with increasing degrees of overlap {Mg3BN3, Sigma-2 ([Si64O128],cdot,4C10H17N) and the NU-3 zeolite}, a sparse electron diffraction data set for buckminsterfullerene, C60, and the small protein molecule crambin at 3 Å resolution where 42 reflections are phased with a U-weighted mean phase error of 58.5°.

Beschreibung: The non-crystallographic symmetry of d-dimensional (dD) quasiperiodic structures is incompatible with lattice periodicity in dD physical space. However, dD quasiperiodic structures can be described as irrational sections of nD (n 〉 d) periodic hypercrystal structures. By appropriate oblique projection of particular hypercrystal structures onto physical space, discrete periodic average structures can be obtained. The boundaries of the projected atomic surfaces give the maximum distance of each atom in a quasiperiodic structure from the vertices of the reference lattice of its average structure. These maximum distances turn out to be smaller than even the shortest atomic bond lengths. The metrics of the average structure of a 3D Ammann tiling, for instance, with edge lengths of the unit tiles equal to the bond lengths in elemental aluminium, correspond almost exactly to the metrics of face-centred-cubic aluminium. This is remarkable since most stable quasicrystals contain aluminium as the main constitutent. The study of the average structure of quasicrystals can be a valuable aid to the elucidation of the geometry of quasicrystal-to-crystal transformations. It can also contribute to the derivation of the physically most relevant Brillouin (Jones) zone.

Beschreibung: A special kind of three-periodic minimal surface has been studied, namely surfaces that are generated from disc-like-spanned skew polygons and that intersect themselves exclusively along straight lines. A new procedure for their derivation is introduced in this paper. Several properties of each such surface may be deduced from its generating polygon: the full symmetry group of the surface, its orientability, the symmetry group of the oriented surface, the pattern of self-intersections, the branch points of the surface, the symmetry and periodicity of the spatial subunits demarcated by the surface, and the Euler characteristics both of the surface and of the spatial subunits. The corresponding procedures are described and illustrated by examples.

Beschreibung: Symmetry restrictions on the intensities and polarization properties of main reflections and their satellites are found for incommensurately modulated crystals in the case of anisotropic anomalous X-ray diffraction near absorption edges. It is shown that the modulation becomes a source of additional anisotropy for each resonant scatterer and induces a modulated behaviour of the susceptibility tensor. The four-dimensional approach is used to calculate the set of possible reflections. It is found that additional (`forbidden') reflections may appear both in the system of main reflections and in the system of satellites. The anisotropy also results in complex azimuthal and polarization properties of each reflection. The displacive modulation is discussed in detail. The ATS reflections corresponding to the resonant X-ray diffraction near the K-edge of iron in pyrrhotite-5.5C are considered.

Beschreibung: In the past few years, a number of strategies have been outlined to resolve the SAS phase ambiguity given that unique estimates omega(h, k) of the triple invariants are available. A new least-squares method is described that can in principle resolve the phase ambiguity to determine macromolecular phases provided that omega(h, k) estimates are unbiased. Limitations of the method in practical applications are discussed. An example is given where the correct solution can be identified by use of the SAS tangent formula in the instance that traditional SAS phasing methods have lead to an incorrect heavy-atom substructure.

Beschreibung: A dynamical calculation scheme that employs Cartesian coordinates with a z axis normal to the crystal surface to define polarization unit vectors and wavefields is applied to interpret the intensity distribution of crystal truncation rods for surfaces and interfaces. A comparison between this calculation scheme and the asymptotic iteration approach using the conventional presentation of the polarization components of the wavefields, with the σ and π components perpendicular to the wave-vectors, is presented. It is found that the use of Cartesian coordinate systems can provide correct boundary conditions in determining the wavefield amplitudes, thus leading to a rigorous and general calculation scheme for dynamical diffraction from surfaces and interfaces.

Beschreibung: The algebraic expressions for the reduced projection operators wp^{(lambda) ar{mu}}_{mu} = sum_{i = 1}^4 u_i hat{ eta_i} for the irreducible representation (irrep) lambda of the icosahedral group I are found by using the double-induced technique and eigenfunction method, where hat{ eta}_i are the double-coset generators of I with respect to the cyclic subgroup C_5. Simple algebraic expressions are derived for the symmetry-adapted functions (SAF's) by applying the reduced projection operators wp^{(lambda) ar{mu}}_mu to Y_{l ar m}. The SAF's are functions of the angular momentum l, the quantum numbers lambda, mu of the group chain I supset C_5 and the multiplicity label ar m. In this way, the SAF problem of the group I is solved once for all instead of for one angular momentum l each time.

Beschreibung: In the paper by Dorset et al. [Acta Cryst. (1999), A55, 901–907], the last line of page 904 should read `…the S atoms. However, the autocorrelation function…'.

Beschreibung: An off-axis ring pattern was observed in the langle 1 ar{1}1 angle zone-axis CBED pattern of an NiO single crystal. The pattern was interpreted as resulting from anisotropic dispersion of the band structure of the two most strongly excited free Bloch waves and strong absorption of the most tightly bound 1s Bloch state. Comparison has also been made with the case of MgO which has the same structure as NiO but weaker scattering power. The tightly bound 1s Bloch state in MgO was found to play an important role together with a strongly excited free Bloch state and a more conventional zone-axis ring pattern similar to that observed in the langle 001 angle zone-axis CBED pattern of silicon was observed.

Beschreibung: A new diffraction method has been developed in which the intensity of Bragg reflections is measured while changing continually the wavelength of the radiation over a range in the vicinity of the absorption edge of an atom contained in the crystal. It is shown that the intensity gradient with respect to the wavelength of the hkl reflection is in a simple relation to the real and imaginary parts of the structure factor of that reflection and, if the positions of the anomalously scattering atoms are known or properly assumed, the phase of F(hkl) can be derived by solving simultaneous linear equations. The procedure is particularly simple when the crystal is centrosymmetric. The method, called the wavelength-modulated diffraction (WMD) method, is free from the problem of intensity scaling encountered in other methods of phase determination. Synchrotron radiation is most suited to WMD measurements. Suggestions on how to measure the intensity gradient are given and the possible errors involved are discussed.

Beschreibung: Fundamental notions concerning absolute structure and absolute configuration, and their determination from single-crystal diffraction measurements, are presented and reviewed. A glossary of terms with definitions useful in this field is provided. For absolute structure and its determination, the separate but interacting influences of the structure and the inversion-distinguishing power of an X-ray diffraction experiment with dispersive scatterers are examined. Important experimental and algorithmic details of the current methods used for absolute-structure determination are provided. Characterization of crystals for absolute-structure determination and of molecules for absolute-configuration determination are treated. Attention is given to the analysis of absolute structure and absolute configuration in twinned crystals.

Beschreibung: The intermolecular interaction energy in crystalline urea has been calculated both from diffraction data and from the Hartree–Fock crystalline electron-density distribution, using a modified atom–atom approximation scheme. The electrostatic part of this energy has been calculated from the atomic multipole moments, obtained by adjustment of the multipole model to experimental X-ray and to theoretical Hartree–Fock structure amplitudes. To obtain the induction energy, multipole moments were calculated from structure amplitudes for the crystalline electron density and from those that refer to the electron density of a superposition of isolated molecules. This worked well for the calculation of the interaction energy from Hartree–Fock data (6% difference from the sublimation-energy value), but not for the interaction energy from experimental data, where the moments of the superposition have to come from Hartree–Fock calculations: the two sets of multipole moments are far too different. The uncertainty of the phases of the structure amplitudes, combined with systematic errors in the theoretical data and noise in the experimental values, may account for the discrepancies. The nature of the different contributions to intermolecular interactions for urea is examined.

Beschreibung: A new reverse Monte Carlo (RMC) method for modelling both lattice and magnetic disorder in powder crystalline materials by direct calculation of the structure factor has been developed. The method, the program and the basic theory are described in some detail. Initial results from modelling the lattice and magnetic structure of MnO around the Néel temperature are also presented.

Beschreibung: The optimization procedure with the greatest power of convergence is the full-matrix method. This method has not been utilized to a great extent in macromolecular refinement because of the great cost of both calculating and inverting the `normal' matrix. This paper describes an algorithm that can calculate this matrix in a relatively short amount of computer time. The procedure requires two Fourier transforms, which can be performed with the fast-Fourier transformation (FFT) algorithm, as well as a large number of simple function products.

Beschreibung: Pattern simulations for three-beam and six-beam X-ray diffraction are presented using multislice calculations based on Moodie & Wagenfeld's formulation of the X-ray equations, which factorize Maxwell's equations into Dirac format, using circular-polarization bases. The results are presented in three forms: one-dimensional rocking curves, Pendellösung thickness fringes, and convergent/divergent-beam patterns of single-diffraction orders, using experience gained from CBED (convergent-beam electron diffraction) and LACBED (large-angle CBED) techniques developed for high-voltage electron diffraction transmission patterns. This latter and quite new technique displays the results in the most compact form. The acronym DBXRAD (divergent-beam X-ray diffraction) is used for these patterns. The optics required for these patterns has only recently become available for radiations up to Mo Kα1 in energy and for limited angular divergences, but with capillary focusing currently undergoing rapid development these limits are likely to be extended. However, these simulations define critical angular ranges within reach of current designs. Simulations for light- and heavy-atom structures belonging to the enantiomorphic space-group pair P3121 and P3221 provide clear evidence of chiral interaction between radiation and structure, highlighting divergences from the use in structure analysis of the well studied CBED pattern symmetries. Mo Kα1 and Ta Kα1 wavelengths were used to minimize absorption for the two structures studied, an important factor owing to the large thicknesses (up to 20 mm) required.

Beschreibung: X-ray diffraction topographs of wafers produced by separation by implanted oxygen (SIMOX) show moiré fringes in both reflection and transmission geometry. These fringes reveal deformations of the order of 10−6 to 10−8 between the layer and the substrate of the SIMOX material. A new method for a quantitative analysis of moiré fringes is developed and allows reconstruction with a high sensitivity of the three components of the relative displacement field between layer and substrate directly from a set of topographs. This method is used for the interpretation of moiré topographs of entire 4 in SIMOX wafers and of regions around crystal defects. Finally, the capabilities of an analysis of moiré fringes are compared with those of the usual diffraction topography.

Beschreibung: Strong violations of Friedel symmetry are observed in hk0 electron diffraction patterns from lamellar crystals of poly(tert-butylethylene sulfide) obtained at 120 kV. These deviations are largely explained by a multislice dynamical scattering calculation based on the crystal structure model. Further improvement is found when a secondary scattering component is added, in keeping with a perfect crystallite thickness less than that of the lamellar thickness. Despite the multiple-scattering perturbations, the frustrated chain packing can still be determined by direct methods followed by Fourier refinement. However, the Friedel-related intensities must be averaged before calculation of normalized structure factors.

Beschreibung: The theory of moiré fringes on X-ray diffraction topographs of bicrystals is derived from the dynamical theory of X-ray diffraction for the reflection (Bragg) and the transmission (Laue) case. The influence on the moiré fringes of the diffraction geometry, of the geometry of the sample, of its optical properties and of the topographic method is investigated. The perfect-crystal theory is also expanded to weakly deformed bicrystals.

Beschreibung: The crystal structure of beef liver catalase was determined ab initio in projection to 9 Å resolution using electron diffraction data at room temperature from hydrated specimens maintained in an environmental chamber in the electron microscope. A conservative combination of symbolic addition with maximum entropy and likelihood led to a model with a Patterson correlation coefficient C = 0.89 to the observed data. This independent solution could then be compared favorably to a previous 23 Å analysis of electron micrographs from frozen hydrated preparations. Prediction of the higher-resolution structure by extension of the lower-resolution image-based phase basis set also gave a good match to the direct-methods solution, particularly for the most intense reflections.

Beschreibung: A 3D quasiperiodic pattern by projection from an nD lattice can be defined by an orthonormal n × n lattice matrix which produces basis vectors in pattern space with a prescribed arrangement and basis vectors in perpendicular or test space satisfying the quasicrystallographic condition. A 16 × 16 lattice matrix is derived which produces basis vectors in pattern space as a combination or mixing of dually positioned dodecahedral star and icosahedral star. It is shown that the mixed star constitutes a eutactic star. Since the module generated by the mixed eutactic star is totally irrational, patterns generated by projection using the lattice matrix are quasicrystallographic and the equilateral pattern is generated for 1: 1 mixing.

Beschreibung: In order to understand the induced electron-density response of covalently bonded materials to an externally applied electric field, the change of structure-factor phases of several weak reflections of GaAs has been measured by three-beam interferences. Using a modulation technique, phase variations of the order of 1° for a field strength of 1 kV mm−1 were determined. Although the statistics of these first experiments are rather poor, the experiment verifies qualitatively the predictions of a semi-empirical bond-charge model. The measured phase variation is the smallest value determined up to now.

Beschreibung: The error in the molecular dipole moment as derived from accurate X-ray diffraction data is shown to be origin dependent in the general case. It is independent of the choice of origin if an electroneutrality constraint is introduced, even when additional constraints are applied to the monopole populations. If a constraint is not applied to individual moieties, as is appropriate for multicomponent crystals or crystals containing molecular ions, the geometric center of the entity considered is a suitable choice of origin for the error treatment.

Beschreibung: The simple cosine function used in the formulation of the traditional minimal principle and the related Shake-and-Bake algorithm is here replaced by a function of exponential type and its expected value and variance are derived. These lead to the corresponding exponential minimal principle and its associated Exponential Shake-and-Bake algorithm. Recent applications of the exponential function to several protein structures within the Shake-and-Bake framework suggest that this function leads, in general, to significant improvements in the success rate (percentage of trial structures yielding solution) of the Shake-and-Bake procedure. However, only in space group P1 is it presently possible to assign optimal values a priori for the exponential-function parameters.

Beschreibung: The joint probability distribution function method has been developed in P1¯ for reflections with rational indices. The positional atomic parameters are considered to be the primitive random variables, uniformly distributed in the interval (0, 1), while the reflection indices are kept fixed. Owing to the rationality of the indices, distributions like P(F p 1 , F p 2 ) are found to be useful for phasing purposes, where p 1 and p 2 are any pair of vectorial indices. A variety of conditional distributions like P(|F p 1 | | |F p 2 |), P(|F p 1 | |F p 2 ), P(varphi_{{ f p}_1}|,|F_{{ f p}_1}|, F_{{ f p}_2}) are derived, which are able to estimate the modulus and phase of F p 1 given the modulus and/or phase of F p 2 . The method has been generalized to handle the joint probability distribution of any set of structure factors, i.e. the distributions P(F 1, F 2,…, F n+1), P(|F 1| |F 2,…, F n+1) and P(varphi1| |F|1, F 2,…, F_{n+1}) have been obtained. Some practical tests prove the efficiency of the method.

Beschreibung: A model is proposed that explains the origin of the bright contrast of dislocation walls consisting of edge dislocation dipoles in electron channelling contrast images (ECCI) of fatigued crystals, when the incident beam is parallel to the edge dislocations. The model is based on the assumption that the contrast arises from the dislocation segments terminating the dipoles. These are modelled as screw-type kinks which scatter electrons. Scattering by randomly distributed kinks leads to the randomization of phase of transmitted and diffracted beams and suppresses the anomalous transmission of electrons. The predicted behaviour of electron-channeling contrast images agrees well with experimental observations.

Beschreibung: A theoretical investigation of diffraction in a rectangular times l crystal for nonsymmetrical coplanar scattering has been undertaken. The asymmetry in the scattering geometry, measured by an angle gamma, causes different weights for the mixed Laue–Bragg contributions to the integrated power. Primary extinction and ordinary absorption are only moderately affected when the value of the geometrical parameter zeta = (t/l)anheta_{oh} ,lt, 1, heta_{oh} being the Bragg angle. For increasing values of zeta and gamma the surface integration set-up for the Laue and Bragg regions is extensively changed, leading to pronounced effects. Analytical results for the normal absorption factor are presented.

Beschreibung: The moduli and flow diagram of the program SIR99 are described. New phasing algorithms are proposed working both in direct and in reciprocal space. Their cooperative work is able to solve the structures of both small and large molecules. In particular, small proteins can be solved ab initio without any use of prior information and any user intervention. The efficiency of the various algorithms employed by SIR99 has been tested, and the role of the tangent formula clarified. The user is also provided with some practical information concerning the computer power needed.

Beschreibung: A low-resolution Fourier synthesis is thought to show a molecule as a compact region of a high electron density. As a consequence, the number of such regions, chosen at a proper cut-off level, should be equal to the number of molecules in the unit cell. This hypothesis may be used as a basis for selection criteria in multisolution ab initio phasing procedures. However, when working with a small number of reflections, this hypothesis may break down. The suggested Fourier-synthesis decomposition explains some reasons for failure and provides a connectivity-based procedure for the determination of macromolecular position in the crystal unit cell and the phasing of several low-resolution reflections. The simplest decomposition consists in separating the reflections into two sets according to whether their phases do or do not depend on a permitted origin shift. It is shown that the partial Fourier syntheses corresponding to these subsets are simply a half-sum and a half-difference of the initial electron-density distribution with its shifted copy. Therefore, they display the true images overlapped with the shifted ones (or with shifted and additionally flipped copies for the latter synthesis). The paper generalizes the decomposition for the case of a finite subgroup of the group of permitted origin shifts and reveals the role of one-phase seminvariants.

Beschreibung: X-ray and neutron diffraction data of a single crystal of ammonium dihydrogen phosphate have been used for the determination of the electron density using multipolar expansion of the density around each nucleus. As the ammonium group was found to be nearly neutral from unconstrained multipole refinement, constrained refinements have been performed with the charge of the ammonium group ranging from zero to one. On the other hand, the expansion of the radial functions of the phosphorus atom was varied. All refinements led to almost the same agreement factors and residual densities. The consequences of such uncertainties on the topology of the electron density are discussed, namely the topology of the P—O bond critical point.

Beschreibung: The phenomenon of chirality is described in terms of symmetry and via symmetry operators.

Beschreibung: The goal of this work is to analyse whether the generalized likelihood criterion can be used to find the best spherical envelope of a macromolecule in a unit cell. A family of spherical envelopes is ranged in accordance with their likelihood values calculated by means of a Monte Carlo-type computer procedure. Two kinds of envelope families were tested. The first one was composed of spherical envelopes of fixed radius but different positions in the unit cell. In the second case, the sphere radii were linked to their centre position so that the total volume occupied by all symmetry-related spheres was roughly equal to the total volume occupied by the real molecule. The experiments showed that when using the first type of envelope the level of the signal for the right solution is higher than the one obtained with the straightforward R-factor-based single-Gaussian-atom search, but spurious maxima (usually placed on the symmetry axes) may still exist. The use of the second type of envelope family reduces the level of the spurious maxima.

Beschreibung: Equations derived from a statistical model and valid for all space groups give estimates of the elements of the matrix for least-squares refinement of atomic coordinates and isotropic thermal parameters for a large crystal structure with many diffraction data. The elements are functions of the lengths and directions of Patterson vectors and the distribution of weights as a function of Bragg angle. For a large data set, this matrix, or a portion of it, can be calculated in a time that is approximately proportional to the number of elements calculated and independent of the number of reflections.

Beschreibung: The modular approach was applied to analyse the structure of complex cuprates. The modular series of various types were distinguished in the family of layered cuprates making use of the classification scheme in a fragment approximation developed earlier. The polysomatic series were revealed in the groups of cuprates which were expected to possess the properties of both salt and oxide. The structure of the ladder-type cuprates was analysed in more detail. The present study has shown that the polysomatic model can be efficiently applied to the structural analysis of complex cuprates.

Beschreibung: The results of the Kato spherical-wave approach to the dynamical theory for perfect crystals are obtained by a simple and straightforward method based on the multiple-scattering expansion.

Beschreibung: Various topological approaches to mathematical analysis, classification and enumeration of four-connected three-dimensional (3D) nets are listed. Four-connected 3D nets are being generated systematically by conversion of edges of a vertical stack of congruent three-connected two-dimensional (2D) nets into chains. This paper describes 57 3D nets obtained by converting into crankshaft chains all edges of the simpler 2D nets in the catalog of the Consortium of Theoretical Frameworks. Atomic coordinates are given for distance-least-squares modeling in the highest space group. Nine nets occur in known structures: tridymite, Zn2P2O8 cdot organic, aluminophosphates AlPO-5 (International Zeolite Association Structure Commission code AFI), AlPO-8 (AEI), AlPO-11 (AEL), AlPO-25 (ATV), AlPO-41 (AFO), AlPO-54 (VFI) and AlPO-H2 (AHT).

Beschreibung: Algorithms are presented for three-dimensional crystallographic space groups, handling tasks such as the generation of symmetry operations, the characterization of symmetry operations (determination of rotation-part type, axis direction, sense of rotation, screw or glide part and location part), the determination of space-group type [identified by the space-group number of the International Tables for Crystallography (Dordrecht: Kluwer Academic Publishers)] and the generation of structure-seminvariant vectors and moduli. The latter are an algebraic description of allowed origin shifts, which are important in crystal structure determination methods or for comparing crystal structures. The space-group type determination produces a change-of-basis matrix which transforms a given space-group representation to the standard one according to the International Tables for Crystallography. The algorithms were implemented and tested using the SgInfo library. The source code is free for non-commercial applications.

Beschreibung: A general formalism, based on the Takagi–Taupin equations, for calculating rocking curves in perfect times l crystals is presented. It includes nonsymmetrical scattering, refraction, and ordinary and anomalous absorption. t and l may be varied independently. In the limit of a semi-infinite crystal, the standard results from the fundamental theory are retrieved. For crystal dimensions less than the extinction length, the theory converges to the kinematical limit. Simulations for germanium and silicon show significant influence of crystal finiteness. When dynamical effects are prominent, the curves exhibit various degrees of asymmetry and the full width at half-maximum is generally larger than the corresponding Darwin width. This is attributed to combined Laue and Bragg contributions which are shifted with respect to each other owing to refraction.

Beschreibung: A three-repeat saw (s) chain has each vertical edge separated by a tooth composed of two tilted edges zig and zag. Some horizontal (h) edges from a parallel stack of three-connected two-dimensional (2D) nets can be converted into an s chain. Each resulting four-connected vertex in the three-dimensional (3D) net may be part of either one, two or three s chains. The first type of (h,s)* 3D net is related by a sigma-type mirror plane to a (h,z)* net listed in paper II [Han & Smith (1998). Acta Cryst. A55, 342–359]. The second type does not have an (h,z)* relative. Using the same three-connected 2D nets as in paper II, 174 four-connected 3D nets were selected from the first two types, including six in known structures: `nepheline hydrate' (International Zeolite Association Structure Commission code JBW), AlPO4-12-TAMU (ATT), offretite (OFF), Linde Type L (LTL), SUZ-4 (SZF) and ZSM-10 (ZST). The third type with three back-to-back s chains is represented by edingtonite (EDI), and systematic enumeration is in progress. The geometrical and topological properties of the 3D nets are given. Idealized unit-cell data and atomic coordinates for tetrahedral bonding were obtained for 40 selected 3D nets by distance-least-squares (DLS) refinement.

Beschreibung: Highly monochromatic (Deltalambda/lambda=10^{-6}) short-wavelength (lambda=0.03 Å) γ radiation has been used in the study of the diffraction processes in real single crystals. Dislocation-free silicon crystals and quartz crystals with small concentrations of chaotically distributed dislocations were investigated. The experimental results were processed with use of statistical dynamical theories of diffraction, both fundamental and advanced. It is shown that the advanced version is more reliable in both cases. However, the values of the average lattice phase E, describing the long-range distortions of the lattice, can differ markedly from the value of 1 for quartz crystals, and parameter L=-{ m ln(E) has a square dependence on the extinction length.

Beschreibung: The method of the joint probability distribution functions of structure factors has been extended to reflections with rational indices. The most general case, space group P1, has been considered. The positional parameters are the primitive random variables of our probabilistic approach, while the reflection indices are kept fixed. Quite general joint probability distributions have been considered from which conditional distributions have been derived: these proved applicable to the accurate estimation of the real and imaginary parts of a structure factor, given prior information on other structure factors. The method is also discussed in relation to the Hilbert-transform techniques.

Beschreibung: The use of X-ray and neutron scattering as a tool to study phase transitions is well established. As techniques improve and experiments are made under successively higher resolution, the need to consider the role of both the distribution of diffracting length scales and the incident-beam coherence volume is emphasized. The interplay of diffracting length scales and the beam coherence volume no longer permits calculation of diffraction profiles in terms of the sample intensity response convolved with an instrumental resolution function. Rather, the probe and sample now enter the calculation on an equal footing at the level of the scattering amplitudes. Under these conditions, it is found that the summation of coherent scattering amplitudes leads to characteristic profiles in wave-vector and, in the case of resonant X-ray scattering, energy space. In this latter case, in the vicinity of strong absorption edges, as used for example in resonant magnetic X-ray diffraction, the energy dependence of diffraction profiles may uniquely allow spatial localization of the scattering volume below the sample surface. This observation may considerably augment the range and power of resonant X-ray scattering.

Beschreibung: When the electron density in a crystal or a quasicrystal is reconstructed from its Fourier modes, the global minimum value of the density is sensitively dependent on the relative phases of the modes. The set of phases that maximizes the value of the global minimum corresponds, by positivity of the density, to the density having the minimum total charge that is consistent with the measured Fourier amplitudes. Phases that minimize the total electronic charge (i.e. the average electron density) have the additional property that the lowest minima of the electron density become exactly degenerate and proliferate within the unit cell. The large number of degenerate minima have the effect that density maxima are forced to occupy ever smaller regions of the unit cell. Thus, by minimization of the electronic charge, the atomicity of the electron density is enhanced as well. Charge minimization applied to simulated crystalline and quasicrystalline diffraction data successfully reproduces the correct phases starting from random initial phases.

Beschreibung: The structure of the insulating manganites Nd1−x Ca x MnO3 (0.3 ≤ x ≤ 0.5) has been studied against temperature using electron diffraction and neutron diffraction. At room temperature, the three compounds with x = 0.3, 0.4 and 0.5 exhibit the orthorhombic GdFeO3-type structure (a ≈ 21/2 ap , b ≈ 2ap , c ≈ 21/2 ap ). Below the charge-ordering temperature of ∼250 K, a doubling of the lattice is observed along the a axis which could be interpreted as an ordering of the d z 2 Mn3+ orbitals. The low-temperature polymorph has orthorhombic symmetry for x = 0.5 and monoclinic symmetry for x = 0.4 and 0.3. The low-temperature structure of the x = 0.4 sample was refined using neutron diffraction data. The HREM study at room temperature confirms the average GdFeO3-type structure and indicates the existence of defective areas with a lower symmetry.

Beschreibung: The probabilistic formulae [Giacovazzo, Siliqi & Fernández-Castaño (1999). Acta Cryst. A55, 512–524] relating standard and half-integral index reflections are modified for practical applications. The experimental tests prove the reliability of the probabilistic relationships. The approach is further developed to explore whether the moduli of the half-integral index reflections can be evaluated in the absence of phase information; i.e. by exploiting the moduli of the standard reflections only. The final formulae indicate that estimates can be obtained, even though the reliability factor is a constant.

Beschreibung: The compliance fourth-rank tensor related to crystalline or other anisotropic media belonging to the monoclinic crystal system is spectrally decomposed for the first time, and its characteristic values and idempotent fourth-rank tensors are established. Further, it is proven that the idempotent tensors resolve the stress and strain second-rank tensors into eigentensors, thus giving rise to a decomposition of the total elastic strain-energy density into non-interacting strain-energy parts. Several examples of representative inorganic crystals of the monoclinic system illustrate the results of the theoretical analysis. It is also proven that the essential parameters required for a coordinate-invariant characterization of the elastic properties of a crystal exhibiting monoclinic symmetry are both the six characteristic values of the compliance tensor and seven dimensionless parameters. These material constants, referred to as the eigenangles, are shown to be accountable for the orientation of the stress and strain eigentensors, when represented in a stress coordinate system. Finally, the restrictions dictated by the classical thermodynamical argument on the elements of the compliance tensor, which are necessary and sufficient for the elastic strain-energy density to be positive definite, are investigated for the monoclinic symmetry.