Thomas Welberry

Interpretation of diffuse X-ray scattering via models of disorder

While Bragg diffraction data can only reveal average one-body information, such as atomic positions, thermal ellipsoids and site occupancies, diffuse scattering contains two-body information and is thus potentially a rich source of information on how atoms and molecules interact. The two major mathematical approaches that have been used to understand and analyse this diffuse part of the diffraction pattern are outlined and the difficulties and limitations which arise from the necessary complexity of these mathematical descriptions for any but the simplest of systems are discussed. Further, the progress that has been made in an alternative approach to understanding the local atomic and molecular arrangements in disordered materials, which overcomes some of these difficulties, is reviewed. The method consists of comparing diffraction patterns calculated from a computer model of the disordered structure with measured X-ray diffuse intensities. The advantage of the method is that it can be applied generally to all systems regardless of their complexity or size of the atomic displacements that might be present. Examples are taken from a variety of material systems and include oxygen–vacancy ordering and metal-atom displacements in yttria-stabilized cubic zirconia, disorder in the molecular crystal p-chloro-N-(p-methylbenzylidene)aniline, C14H12ClN (MeCl), and orientational disorder in crystals of 1,3-dibromo-2,5-diethyl-4,6-dimethylbenzene, C12H16Br2 (BEMB2). The computer models are generated using both real-space Monte Carlo methods, which employ near-neighbour effective interactions, and a reciprocal-space synthesis approach. Diffraction patterns are computed in three dimensions using realistic atomic form factors so direct comparison can be made with mathematical descriptions of the diffuse scattering.

An electron diffraction and lattice-dynamical study of the diffuse scattering in β-cristobalite, SiO2

Strong diffuse scattering has been observed on electron diffraction photographs of β-cristobalite. This is observed to be in the form of diffuse planes normal to the six (110) directions. A simple Born–von Karman force model including nearest-neighbour Si–O, O–O and Si–Si interactions has been used to calculate lattice-dynamical properties for β-cristobalite assuming the structure to be the simple Wyckoff [Z. Kristallogr. (1925), 62, 189–200] C9 structure. The force-model parameters used were obtained by fitting to the infrared and Raman frequencies of α-cristobalite and are very similar to those obtained by Barron, Huang & Pasternak [J. Phys. C (1976), 9, 3925–3940] for quartz. Calculated thermal diffuse scattering patterns obtained are in good agreement with the observed patterns, and may be interpreted in terms of low-lying relatively dispersionless branches of the phonon dispersion curve diagrams. These branches correspond to motions of the structure involving coupled rotations of the tetrahedra in columns along the (110) directions, the motion in neighbouring columns being virtually uncoupled.

The structure and microstructure of α-cristobalite and its relationship to β-cristobalite

Recent transmission electron microscopy (TEM) electron diffraction investigation of β-cristobalite has revealed a strong characteristic diffuse intensity distribution. In this work a similar, but less intense, diffuse scattering is reported for synthetic α-cristobalite. This material also displays both tetragonal twinning and characteristic striations correspond to planar boundaries which may exist on any {1, 0, l}tetragonal plane.

An optical transform and Monte Carlo study of the disorder in β-cristobalite SiO2

A previously reported lattice-dynamical model of β-cristobalite [Hua, Welberry, Withers & Thompson (1988). J. Appl. Cryst. 21, 458–465], which assumed the Wyckoff C9 structure but gave a satisfactory qualitative description of observed diffuse intensity in electron diffraction patterns, is shown to give atomic displacements which are much too small to be consistent with the 147° Si—O—Si angle observed in other forms of silica. Monte Carlo simulations of a modified model in which this angle is constrained to be close to 147° have been carried out and optical diffraction patterns obtained from the resulting lattice realisations. The diffraction patterns are very similar in appearance to those calculated for the simple C9 model, but the displacements are now consistent both with the expected geometry and also with mean-square atomic displacements derived from X-ray powder studies. In this simulation the oxygen atoms are distributed uniformly around an annulus which encircles the 16(c) sites of the Fd3m C9 structure. The hypothesis of Wright & Leadbetter [Philos. Mag. (1975). 31, 1391–1401], that the oxygen atoms preferentially occupy the six 96(h) sites which occur around this annulus, has been tested and it is concluded that if such ordering exists it is not very pronounced.

Diffuse scattering study of aspirin forms (I) and (II).

Full three-dimensional diffuse scattering data have been recorded for both polymorphic forms [(I) and (II)] of aspirin and these data have been analysed using Monte Carlo computer modelling. The observed scattering in form (I) is well reproduced by a simple harmonic model of thermally induced displacements. The data for form (II) show, in addition to thermal diffuse scattering (TDS) similar to that in form (I), diffuse streaks originating from stacking fault-like defects as well as other effects that can be attributed to strain induced by these defects. The present study has provided strong evidence that the aspirin form (II) structure is a true polymorph with a structure quite distinct from that of form (I). The diffuse scattering evidence presented shows that crystals of form (II) are essentially composed of large single domains of the form (II) lattice with a relatively small volume fraction of intrinsic planar defects or faults comprising misoriented bilayers of molecular dimers. There is evidence of some local aggregation of these defect bilayers to form small included regions of the form (I) structure. Evidence is also presented that shows that the strain effects arise from the mismatch of molecular packing between the defect region and the surrounding form (II) lattice. This occurs at the edges of the planar defects in the b direction only.

Diffuse X-ray scattering from benzil, C14H10O2: analysis via automatic refinement of a Monte Carlo model

A recently developed method for fitting a Monte Carlo computer-simulation model to observed single-crystal diffuse X-ray scattering has been used to study the diffuse scattering in benzil, diphenylethanedione, C(6)H(5)-CO-CO-C(6)H(5). A model involving 13 parameters consisting of 11 intermolecular force constants, a single intramolecular torsional force constant and a local Debye-Waller factor was refined to give an agreement factor, R = [summation operator omega(Delta I)(2)/summation operator omega I(obs)(2)](1/2), of 14.5% for 101,324 data points. The model was purely thermal in nature. The analysis has shown that the diffuse lines, which feature so prominently in the observed diffraction patterns, are due to strong longitudinal displacement correlations. These are transmitted from molecule to molecule via a network of contacts involving hydrogen bonding of an O atom on one molecule and the para H atom of the phenyl ring of a neighbouring molecule. The analysis also allowed the determination of a torsional force constant for rotations about the single bonds in the molecule. This is the first diffuse scattering study in which measurement of such internal molecular torsion forces has been attempted.

Analysis of Single-Crystal Diffuse X-ray Scattering via Automatic Refinement of a Monte Carlo Model

Although diffuse scattering from disordered crystals has been known since the earliest days of X-ray diffraction its interpretation and analysis is still far from being a routine process and quantitative studies are relatively rare. The most extensive and quantitative methods have been developed in the field of alloys and simple oxides where the number of parameters needed to define a disordered structure is small and the diffracted patterns can be attributed unambiguously to basic interatomic interactions’. For structures containing more than a few atoms, however, this type of detailed analysis is simply not feasible, and alternative approaches must be adopted (see e.g. the reviews by Welberry & Butlersand Frey6’7). For large (macromolecular) structures interpreting diffuse scattering, when the structure contains so many parameters, would appear to be a quite intractible problem; but even here useful information can be extracted8-11. The present work forms part of a long-term study to develop methods that will allow routine analysis of diffuse scattering for a diverse range of materials.

Single-crystal neutron diffuse scattering and Monte Carlo study of the relaxor ferroelectric PbZn1/3Nb2/3O3 (PZN)

Full three-dimensional diffuse neutron scattering data have been recorded from a single crystal of Pb(Zn 1/3 Nb 2/3 )O 3 (PZN) at 300 K using the time-of-flight Laue technique on the SXD single-crystal instrument at ISIS. The data show a series of diffuse rods of scattering oriented parallel to each of the six (11 0) crystal directions. Monte Carlo simulation has been used to demonstrate that the diffuse rods are caused by planar nanodomains oriented normal to the (1 1 0) directions. Within these domains, there are correlated displacements of the atoms away from their average site positions. In order to explain the systematic absence of some rods of scattering in the (h k 1) data but the presence of all rods in the (h k 0) data, it is necessary that the displacement of an O atom is of opposite sign to that of its neighbouring Pb atoms. This is explained in terms of a model based on the fact that Pb 2+ possesses a lone pair of electrons, giving the Pb ion directionality.

Diffuse scattering in yttria-stabilized cubic zirconia

Abstract We have used a position-sensitive detector (PSD) system to make measurements of the diffuse X-ray scattering on a cubic Y 2 O 3 -stabilized zirconia, Zr 0.61 Y 0.39 O 1.805 , in far greater detail than has hitherto been reported. In addition to the fairly prominent diffuse peaks visible in 〈1 1 0〉 sections that have been the center of discussion in many previous studies we see a number of other characteristic diffraction features. We report results of some Monte Carlo simulations in which we attempt to find the origins of these various features. We believe one feature, in the form of sets of dark planes normal to 〈1 1 0〉, occurs because of size-effect -like strains induced along 〈1 1 0〉 intermetal vectors, and is only visible because of the similarity of the scattering factors for Zr and Y. A second feature, in the form of bow-tie -shaped regions of scattering, originates from the same basic strains but the symmetry of these requires that displacements in the [1 1 0] direction are out of phase with those in the [1 – 1 0], implying a distortion of the basic metal coordination such that an expansion along [1 1 0] is accompanied by a contraction along [1 – 1 0].

Analysis of diffuse scattering via the reverse Monte Carlo technique : a systematic investigation

The viability of using the reverse Monte Carlo (RMC) method for quantitative analysis of the diffuse X-ray or neutron scattering from single crystals of disordered materials is investigated. The method has been applied to a number of two- and three-dimensional model examples in which both occupational and displacement disorder are present separately and in combination. While occupational or displacement disorder are each separately handled well, with the RMC simulation reproducing the input correlation structure of the models quite effectively, more difficulty was encountered when the different types of disorder were present in combination. In this case, it was necessary to employ a strategy where the occupation shifts and displacement shifts were carried out alternately with no more than 10% of the crystal being visited before switching between the two modes. It was also found to be advantageous to exclude all high-angle diffraction data when assessing occupation shifts. Furthermore, a new method of modelling distortions by swapping the displacements of two atom sites rather than shifting each atom individually was found to produce chemically more realistic bond-length distributions.