Karl F. Fischer

Superstructure determination of PbZrO3

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) A, with Z = 8. The intensity data were collected using short-wavelength synchrotron X-rays of 0.350 A; 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−c0 using Glazers [Acta Cryst. (1972), B28, 3384–3392; Acta Cryst. (1975), A31, 756–762] notation.

X-Ray birefringence and dichroism in lithium niobate, LiNbO3

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.

Structure determination without Fourier inversion. Part I. Unique results for centrosymmetric examples

Abstract A concept is presented for determining a one-dimensional structure of m independent point scatterers by mapping into an m-dimensional space Pm at least m observations as (m – 1)-dimensional so-called “isosurfaces” defined by s(h) · g(h) or g(h) alone, where s(h) and g(h) are sign and modulus, respectively, of the geometrical structure factor. Values of g(h) are derived from the observed |F(h)|. The “solution vector(s)”, (x1, …, xm), representing the coordinates xj of the point scatterers is (are) found from the “common” intersection(s) of n ≥ m different isosurfaces. Homometric and quasi-homometric structures can thus safely be detected from multiple solutions, the latter mainly arising from the experimental uncertainties of the g(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 of Pm and 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.

Use of X-ray Anomalous Dispersion: The Superstructure of PbZrO3

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 LIII 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.

Representation and tabulation of spherical atomic scattering factors in polynomial approximation

I t is recommendable to use an analyt ical expression to compu te the a tomic scat ter ing factor . Ou t of several possible analyt ica l no ta t ions one is chosen where In / (s) is represented by a polynomial expression. F o r three mos t common rad ia t ions (Cu, Mo, Ag) the avai lable d a t a of t he In t e rna t iona l Tables were converted into seven coefficients per ion. A list of these coefficients is provided.

Structure determination without Fourier inversion. Part II: The use of intensity ratios and inequalities

Abstract Using for a one-dimensional centrosymmetric structure of m equal point scatterers an m-dimensional coordinate parameter space Pm 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 Pm 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 Pm 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.

Polarized X-ray absorption. Evidence of orientational dispersion in hornblende minerals

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.

Structure determination without Fourier inversion. Part IV: Using quasi-normalized data

Abstract 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 (2πihrj)|, 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 Pm with asymmetric part Am 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 Am 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.

A single-crystal X-ray diffractometer for white synchrotron radiation with solid state detectors: Energy dispersive Laue (EDL) instrument at HASYLAB, Hamburg/Germany

Abstract 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.

Solving crystal structures without Fourier mapping. I. Centrosymmetric case. Erratum.

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) = [ summation operator(j = 0)(i-1) (-1)(i-j-1)P(i-j)Q(j)]/i. p. 277 (right column), third line after equation (12) should read: Using D(1) [from (10a)] ellipsis. p. 277 (right column), second line after equation (13) should read: ellipsis namely 2(m+n-1). p. 279 (right column), lines 10 and 11 in second paragraph should read: ellipsis sigma(g(h)) = 0.05g(h) (or 0.1g(h), respectively) for ellipsis. p. 279 (right column), fifth line from bottom should read: ellipsis example I (Table 2) ellipsis. p. 280 (right column), third line before Section 3.2 should read: E.g. one of them systematically ellipsis.