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A method for locating in an unknown one-dimensional structure (or structure projection) a resonant scattering atom from reflection intensities measured at two wavelengths is presented in terms of the parameter space concept. Even with rather few one-dimensional data the method permits high resolution (well beyond that of Fourier or Patterson calculations). In presence of some hundred ‘light’ atoms, a resonant scattering ‘heavy’ atom can be retrieved from a small number of proposed positions, even if reflection intensities with maximum relative errors of up to 20 or 30% are used. The same applies to structures composed of up to about 50 ‘normal’ atoms with scattering powers similar to that of the resonant scatterer. Further, the method allows for combining the results from sub-sets of a harmonic reflection data sequence, and there is no need to include the lowest reflection orders. Results for some hypothetical model structures both with and without inversion centre are presented and discussed. Also, the advantage of the technique compared to conventional one-dimensional difference Patterson calculations is illustrated and possible applications of the method and extensions to several resonant scatterers are adressed.

For a centric crystal structure represented by m equal point scatterers at rest, absolute scaling of a small number n of reflection data reduced to relative geometrical structure amplitudes g′( h ) = K · |∑ cos (2πi hr j )|, j = 1, …, m; K = scaling factor) is obtained by dividing each amplitude through the r.m.s. average of the amplitudes to be considered. For the same batch of reflections, the resulting values e( h , n) are proportional to the well known normalized structure amplitudes |E( h )| in Direct Methods. Choosing a set of n harmonic reflections of a central reciprocal lattice row, the e( h , n) serve to determine the m independent coordinates of the point scatterers projected onto the corresponding direct space direction, e.g. h00-reflections for coordinates xj , hh0-reflections for (x + y)j (j = 1, …, m), etc. This is achieved by applying the concept of an m-dimensional parameter space P m with asymmetric part A m containing (m – 1)-dimensional iso-surfaces E( h , n; e) determined by the values e( h , n), which define boundaries between forbidden and permitted solution regions (the latter containing test structure vectors Xt ) based on observed inequalities, e.g. e( h , n) < e( k , n). Due to the spatial resolution potential of the concept even less than m data suffice to yield in A m tractable amounts of such test structure vectors ready for conventional least-squares refinement based on n > m data in order to obtain the “best” solution of the considered one-dimensional structure projection. The refined coordinates of various different projections can then be combined for reconstructing the three-dimensional structure. Properties of the e( h , n) and their iso-surfaces E( h , n; e) are discussed and determinations of two very small structures (centric and acentric) as well as of a centric 15-atom structure are presented as examples for the applicability of the method.

Using for a one-dimensional centrosymmetric structure of m equal point scatterers an m-dimensional coordinate parameter space P m and the (n – 1) independent ratios q(h, k) = [I′(h)/I′(k)]1/2 = g′(h)/g′(k) of n > 1 structure amplitudes observed on relative scale it is shown that the determination of the structure or possibly homometric structures is equivalent to finding the common intersection(s) of (n – 1) ≥ m independent isosurfaces Q[h, k; q] of dimension (m – 1).A fast and efficient reduction of the parameter space to the fraction(s) of P m that contain(s) the solution(s) can already be achieved on the basis of the qualitative inequalities between the observations. Therefore, emphasis is put on showing that each q(h, k) < 1, i.e. g′(h) < g′(k) (or vice versa) rules out all the regions in P m that contain point scatterer arrangements incompatible with q(h, k). For small structures, solution strategies are discussed and an estimate is given on the number of data necessary for solving even ‘tough’ problems.

A concept is presented for determining a one-dimensional structure ofmindependent point scatterers by mapping into anm-dimensional spacePmat leastmobservations as (m– 1)-dimensional so-called “isosurfaces” defined bys(h) ·g(h) org(h) alone, wheres(h) andg(h) are sign and modulus, respectively, of the geometrical structure factor. Values ofg(h) are derived from the observed |F(h)|. The “solution vector(s)”, (x1, …,xm), representing the coordinatesxjof the point scatterers is (are) found from the “common” intersection(s) ofn≥mdifferent isosurfaces. Homometric and quasi-homometric structures can thus safely be detected from multiple solutions, the latter mainly arising from the experimental uncertainties of theg(h). Spatial resolution is by far higher than that offered by a corresponding Fourier transform. Computer time can be drastically reduced upon consideration of both the symmetry ofPmand the anti-symmetry and self-similarity properties of the isosurfaces, which allow for tayloring of the intersection search routines and/or applying linear approximations. The basic principles of the method are illustrated by various two- and three-atom structure examples and discussed in view of the application potential to real structure problems.

By taking into account the effects of domain structures and X-ray absorption, the superstructure of PbZrO3, lead zirconate, has been determined at room temperature. The space group is Pbam with a unit cell of a = 5.8884 (19), b = 11.771 (4) and c = 8.226 (3) Å, with Z = 8. The intensity data were collected using short-wavelength synchrotron X-rays of 0.350 Å; this reduces the linear absorption coefficient to 11.93 mm−1. The structure refinement was performed using only the data of superlattice reflections which are free from ambiguity and resulting from the domain structure; the final R value is 0.047 for 335 unique superlattice reflections. Zr atoms show the antiphase-type displacement along the z axis; oxygen octahedra show tilt of the type a − a − c 0 using Glazer's [Acta Cryst. (1972), B28, 3384–3392; Acta Cryst. (1975), A31, 756–762] notation.

The first experimental evidence for orientational dispersion of an absorption tensor in the X-ray regime is presented. For two amphibole minerals, edenite and hastingsite, the X-ray absorption tensors were determined at the Fe K edge by means of polarized fluorescence spectroscopy. Analysis of the energy-dependent tensor elements revealed anisotropic anomalous scattering and, in analogy to visible light optics, energy dependence of the respective absorption tensor orientations. Hence, the experiments provide further evidence for the optical model developed earlier describing the polarized near-edge absorption.

A sequence of steps for determining a crystal structure, possibly without ambiguities, is presented. The prerequisites are: centrosymmetry (at present) and two different anomalous scatterers, a1, a2. Their partial structure amplitudes |Fa1(hkl)| and |Fa2(hkl)| are separated by multiple-wavelength measurements (MAD). The core part of the method is a recursive algebraic technique applied to the geometrical part of these structure amplitudes from central reciprocal-lattice rows. At least m + 1 reflections are necessary at each row if 2m atoms of e.g. a1 are in a unit cell with space group P{\bar 1}. For each partial structure of a1 and a2 atoms, respectively, the algebra finds all homometric and pseudohomometric solutions and presents the corresponding signs for each Fa(hkl) used. Regions of confidence for atomic coordinates are given. Five reciprocal-lattice rows (or more) suffice for a `tomographic' location of all atoms a1 and a2 in three dimensions. The two independently determined partial structures for a1 and a2 are then aligned to the same origin and moduli plus signs of the remaining partial structure factors of the non-resonant atoms are determined. Various aspects of the method are discussed by application to the Cu3SbSe3 structure, an example exhibiting partial pseudosymmetry.

Intensity measurements of several superlattice reflections of lead zirconate, PbZrO3, have been made on single-crystal specimens as a function of incident X-ray energy around the Pb L III and Zr K absorption edges. Anomalies in the squared structure factor ∣F(hkl)∣2 of superlattice reflections with l odd, and of those with l even, are observed at both the absorption edges. This indicates that both Pb and Zr atoms show an out-of-phase type of displacement, not only on the xy plane but also along the z axis in the orthorhombic cell. The results of the Zr displacement are in contradiction with the models in the literature. It is shown that the use of X-ray anomalous dispersion is effective to obtain information on the atomic displacement and to investigate the superstructure.

A new instrument for X-ray diffraction offers the use of “white” synchrotron radiation on a small sample producing a Laue pattern and selecting parts of the spectrum for sending reflections into two energy dispersive detectors. Integrated Bragg intensities of up to 20 “harmonics” are obtained simultaneously. The sample is mounted on a Kappa goniometer with a lot of angular space around it for controlling special environments. Possibilities are provided for energy dependent polarization monitoring, for fine adjustment of the diffractometer, and for installing different monochromators and auxiliary equipment for limiting unwanted fluorescence and other background. The EDL diffractometer is controlled via CAMAC by a PC which also handles collecting as well as reduction of data. A number of applications are mentioned and partly demonstrated by test results.

In continuation of an earlier study on monoclinic and orthorhombic space groups, the effects of Anisotropic Anomalous Scattering (AAS) on the systematically extinct (‘forbidden’) axial reflections of all tetragonal space groups are investigated under the limiting conditions of purely σ-polarized radiation and one generally positioned ‘edge atom’ in the asymmetric unit. The compilation of the results, Polarized Structure Factors and ensuing characteristic intensity functions, I([unk],Ψ), offers a concise survey over the different cases of relevant symmetry, and contributes to the understanding of FRED (Forbidden Reflection near Edge Diffraction). One interesting application of FRED is partial structure determination, i.e. the derivation of possible ‘edge atom’ coordinates, from the intensity variations I([unk],Ψ), of a small number of ‘forbidden’ axial reflections. The method is invariant against the normally unknown AAS so that nothing but the occurrence of AAS and a few intensity measurements at selected azimuthal settings are required. Characteristic cases for both h00 and 00l reflections are discussed.

The parameter-space concept for solving crystal structures from reflection amplitudes (without employing or searching for their phases) is described on a theoretically oriented basis. Emphasis is placed on the principles of the method, on selecting one of three types of parameter spaces discussed in this paper, and in particular on the structure model employed (equal-atom point model, however usually reduced to one-dimensional projections) and on the system of `isosurfaces' representing experimental `geometrical structure amplitudes' in an orthonormal parameter space of as many dimensions as unknown atomic coordinates. The symmetry of the parameter space as well as of the imprinted isosurfaces and its effect on solution methods is discussed. For point atoms scattering with different phases or signs (as is possible in the case of X-ray resonant or of neutron scattering) it is demonstrated that the `landscape' of these isosurfaces remains invariant save certain shifts of origin known beforehand (under the condition that all atomic scattering amplitudes have been reduced to 1 thus meeting the requirement of the structure model above). Partly referring to earlier publications on the subject, measures are briefly described which permit circumventing an analytical solution of the system of structure-amplitude equations and lead to either a unique (unequivocal) approximate structure solution (offering rather high spatial resolution) or to all possible solutions permitted by the experimental data used (thus including also all potential `false minima'). A simple connection to Patterson vectors is given, also a first hint on data errors. References are given for practical details of various solution techniques already tested and for reconstruction of three-dimensional structures from their projections by `point tomography'. We would feel foolish if we tried to aim at any kind of `competition' to existing methods. Having mentioned `pros and cons' of our concept, some ideas about potential applications are nevertheless offered which are mainly based on its inherent resolution power though demanding rather few reflection data (use of optimal intensity contrast included) and possibly providing a result proven to be unique.

Determination of a crystal structure without Fourier calculation of the scattering density (thus also avoiding the phase problem) is achieved in a fractional coordinate parameter space of dimension 3m where m is the number of independent atoms, reduced to equal point scatterers at rest. For demonstration of the basic ideas, two-dimensional parameter spaces (representing, e. g., one-dimensional two-atom structures) are used. "Central reciprocal lattice row" reflections allow for solving one-dimensional projections of the structure, each requiring less reflections and simultaneously providing better resolution than does a corresponding Fourier summation. The projection solution can be obtained either from the common intersection of the hyper-faces in the m-dimensional parameter space defined by the chosen scattering amplitudes or by exploring the permitted "solution region(s)" that follow from the mere ranking of these amplitudes. All possible solutions satisfying the data are found, including "false minima". The reconstruction of a hypothetical three-dimensional 11 atom structure from the solutions of one-dimensional projections is illustrated in an example based on "theoretical", i. e. error-free data. Since most of the theoretical background is laid down in two former, refereed publications, emphasis is put on different options to cope with the computing demands in practical applications. Advantages and shortcomings of the concept are discussed.

Accurate synchrotron-radiation X-ray transmission measurements have been carried out on hexagonal LiNbO3 in the vicinity of the Nb K-absorption edge (E = 18.986 keV). The experiments were performed on the two-axis diffractometer at HASYLAB in dedicated mode of DORIS II (3.7 GeV). Single-crystal wafers cut perpendicular to [10.0] were rotated around the monochromatized beam using an experimental set up analogous to the optical polarizing- microscope. Both the horizontally and vertically polarized components of the transmitted radiation were recorded at the same time and analysed in terms of a classical optical model derived from the Jones calculus. Fits to the observations yielded agreement indices between 0.013 and 0.052 supporting the applicability of the model to X-ray energies. X-ray dichroism and birefringence are proved to occur at resonance energies, and all findings show the possibility of simultaneous assessment of anisotropies of the real and imaginary parts of the refractive index in a uniaxial crystal. They also indicate that the polarization of the transmitted radiation (of suitable energy) can be varied considerably in dependence of crystal thickness and orientation with respect to the polarization direction of the incident radiation. For special structures like LiNbO3, where the tensorial scattering factor of the edge atom (Nb) is invariant against the symmetry operations of the space group, the method yields also the energy-dependent anisotropy of the anomalous-dispersion corrections, f′ and f′′, in the same experiment.

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[\textstyle \sum \limits_{j = 0}^{i-1}(-1)^{i-j-1}P_{i-j}Q_j\right]\bigg/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 ….

Die Verfeinerung des atomaren Streuvermogens mittels des Verfahrens der kleinsten Quadrate wird auf die Untersuchung von Si-Al-Verteilungen angewandt. Der geringe Unterschied in den f-Kurven beider Atomarten gestattete im untersuchten Falle einer geordneten Si-Al-Verteilung brauchbare Aussagen.

By appropriate selection of two or three wavelengths, intensity differences can be used for obtaining directly the electron density distribution (i.e. the arrangement of atoms) for parts of a crystal structure. Application to macro­molecules and amorphous binary substances appear fea­sible.

The paper reports the observation of significant "resonance peaks" in the intensity vs. energy curves around the Cu-K-absorption edge for forbidden reflections in centro-symmetric cubic Cu2O. A qualitative interpretation is given in terms of an anisotropic anomalous dispersion model based on ƒ′, ƒ″-tensors compatible with the symmetry at the Cu-site.