Obtaining diffuse scattering patterns from computer simulations - a retrospective.

The paper describes how the calculation of diffuse scattering from atomistic model crystals has developed over the last approximately 50 years. Not only has the quality of observed diffuse X-ray scattering data improved immensely with the advent of electronic area detectors and synchrotron radiation but the enormous increase in computer power has enabled patterns, of comparable quality to the observations, to be calculated from a Monte Carlo model.

Dynamic Displacement Disorder of Cubic BaTiO_{3}.

The three-dimensional distribution of the x-ray diffuse scattering intensity of BaTiO_{3} has been recorded in a synchrotron experiment and simultaneously computed using molecular dynamics simulations of a shell model. Together, these have allowed the details of the disorder in paraelectric BaTiO_{3} to be clarified. The narrow sheets of diffuse scattering, related to the famous anisotropic longitudinal correlations of Ti ions, are shown to be caused by the overdamped anharmonic soft phonon branch. This finding demonstrates that the occurrence of narrow sheets of diffuse scattering agrees with a displacive picture of the cubic phase of this textbook ferroelectric material. The presented methodology allows one to go beyond the harmonic approximation in the analysis of phonons and phonon-related scattering.

Interpretation of diffuse scattering in the high-Tc superconductor HgBa2CuO4+δ.

Published diffuse X-ray scattering from the high-Tc superconductor HgBa2CuO4+δ has been reexamined with a view to developing a model that more satisfactorily accounts for the observed patterns. The present work agrees with the previous conclusion that the doping oxygen atoms form long and isolated interstitial chains that run in both tetragonal directions. However, a distinctly different model is proposed for the accompanying displacement patterns of the atoms surrounding these linear defects. In this new model it is proposed that it is the correlated shifts of the Ba atoms along the length of the defect chains that are the primary source of the observed scattering, and that the variations of intensity in the generated diffuse streaks of scattering originate from lateral shifts of both Hg and Ba atoms away from defects. The new model yields diffraction patterns that are in much more convincing agreement with the observations than the original model.

Diffuse scattering and partial disorder in complex structures.

The study of single-crystal diffuse scattering (SCDS) goes back almost to the beginnings of X-ray crystallography. Because SCDS arises from two-body correlations, it contains information about local (short-range) ordering in the sample, information which is often crucial in the attempt to relate structure to function. This review discusses the state of the field, including detectors and data collection and the modelling of SCDS using Monte Carlo and ab initio techniques. High-quality, three-dimensional volumes of SCDS data can now be collected at synchrotron light sources, allowing ever more detailed and quantitative analyses to be undertaken, and opening the way to approaches such as three-dimensional pair distribution function studies (3D-PDF) and automated refinement of a disorder model, powerful techniques that require large volumes of low-noise data.

Stacking faults and superstructures in a layered brownmillerite.

Single crystals of Ca(4)Fe(2)Mn(0.5)Ti(0.5)O(9) have been synthesized using a flux method. The structural characterization using single-crystal X-ray diffraction revealed the space group Amma and unit-cell dimensions of a = 5.3510 (6), b = 26.669 (3), c = 5.4914 (6) Å. The structure is isotypic with Sr(3)NdFe(3)O(9) [Barrier et al. (2005). Chem. Mater. 17, 6619-6623] and exhibits separated brownmillerite-type layers. One-dimensional diffuse scattering shows that the unit cell is doubled along c by alternating the intra-layer order of tetrahedral chains, causing stacking faults along the b direction. A computer simulation was performed, proving that the observed intensity variations along the diffuse scattering rods originates from two different local structures depending on the configuration of the tetrahedral chains. Selected-area electron diffraction experiments exhibit well ordered regions characterized by satellite reflections corresponding to two different superstructures. Both superstructures can be described using the superspace group A2(1)/m(0ßγ)0s, with γ = 0.5 and ß ≃ 0.27 or ß = 0.

Diffuse scattering resulting from macromolecular frustration.

Distinctive diffuse scattering in the form of diffuse rings around Bragg positions has been observed in the diffraction patterns of a crystal of the N-terminal fragment of the Gag protein from Feline Foamy Virus. It is shown that these are caused by geometric frustration as molecules try to pack on the triangular b-c mesh of the space group P6(1)22. In order to explain the strong diffuse scattering it is necessary for the crystal to contain occupational disorder such that each unit cell contains one or other of two different molecular arrangements, A and B. The frustration arises because the nearest-neighbour packing prefers neighbouring cells to be AB or BA, which cannot be achieved on all three sides of a triangle simultaneously. To explain the observation that reciprocal sections hk5n, where n = integer, contain only Bragg peaks it is necessary that A and B are identical molecular arrangements differing only by a translation of 0.2c. The implications of the disorder for solving the structure of the protein by conventional techniques as well as the possibility of using the diffuse scattering for this purpose are discussed.

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.

Precursor effects of the orthorhombic to monoclinic phase transition in benzocaine form (II) revealed by X-ray diffuse scattering.

We described the development of a Monte Carlo computer model for the room-temperature form (II) polymorph of benzocaine that incorporates, on a local scale, structural features derived from the low-temperature form (III) polymorph. The introduction of this extra information convincingly reproduces those observed diffraction features that an earlier harmonic model was unable to achieve. In both form (I) and form (II) the hydrogen-bonded chains of molecules that extend along the respective a axes tend to slide backward and forward along their lengths. While in form (I) the motion is well modelled by a harmonic potential in form (II) there is a degree of anharmonicity that means that some intermolecular contact vectors, which are identical in the average structure, are distributed bimodally with either longer or shorter vectors being preferred to the average. Moreover there is a tendency for these deviations from average to be correlated to give short-range ordered domains that are the precursors of the two twinned variants of the long-range ordered low-temperature form (III) structure.

Single-crystal diffuse scattering studies on polymorphs of molecular crystals. I. The room-temperature polymorphs of the drug benzocaine.

The drug benzocaine (ethyl 4-aminobenzoate), commonly used as a local anaesthetic, is a bimorphic solid at room temperature. Form (I) is monoclinic P2(1)/c, while the metastable form (II) is orthorhombic P2(1)2(1)2(1). Three-dimensional diffuse X-ray scattering data have been collected for the two forms on the 11-ID-B beamline at the Advanced Photon Source (APS). Both forms show strong and highly structured diffuse scattering. The data have been interpreted and analysed using Monte Carlo (MC) modelling on the basis that the scattering is purely thermal in origin and indicates the presence of highly correlated molecular motions. In both forms (I) and (II) broad diffuse streaks are observed in the 0kl section which indicate strong longitudinal displacement correlations between molecules in the 031 directions, extending over distances of up to 50 A. Streaks extending between Bragg peaks in the hk0 section normal to [100] correspond to correlated motions of chains of molecules extending along a that are linked by N-H...O=C hydrogen bonds and which occur together as coplanar ribbon pairs. The main difference between the two forms is in the dynamical behaviour of the ribbon pairs and in particular how they are able to slide relative to each other. While for form (I) a model involving harmonic springs is able to describe the motion satisfactorily, as simple excursions away from the average structure, there is evidence in form (II) of anharmonic effects that are precursors of a phase transition to a new low-temperature phase, form (III), that was subsequently found.

Neutron diffuse scattering in deuterated para-terphenyl, C(18)D(14).

Neutron diffuse scattering is used to explore the short-range order (SRO) in deuterated para-terphenyl, C(18)D(14). The crystal shows SRO because the central of the three phenyl groups of each molecule can twist positively or negatively and these twists are correlated over the local scale. The presence of incipient Bragg peaks at [Formula: see text] at 200 K shows that these flips are negatively correlated along the a direction (nearest neighbour correlation coefficient of ∼-0.3) and b direction (nearest neighbour correlation coefficient of ∼-0.87) and appear essentially uncorrelated along c. Diffuse peak anisotropy indicates that the range of the correlations along b is found to be ∼3 times that along a. These correlations persist, although weaker, at room temperature. A Monte Carlo simulation was used to impose a correlation structure on the population of central ring twists that was deduced from Bragg scattering. By then allowing displacive relaxation of the structure, the observed diffuse scattering was well reproduced. Modelling the displacive motions of molecules showed that the positions of nearest ab-plane neighbour molecules are strongly positively correlated, particularly for motions approximately parallel to a, while the displacive correlations are weaker between molecules stacked along c. The apparent contradiction that the displacements are most strongly correlated along a while the occupancies are most strongly correlated along b is explained in terms of the connectivity of molecular interactions.

A room-temperature X-ray diffuse scattering study of form (II) of the trimorphic molecular system p-(N-methylbenzylidene)-p-methylaniline.

Three-dimensional X-ray diffuse scattering data have been collected at room temperature for form (II) of the trimorphic molecular system p-(N-methylbenzylidene)-p-methylaniline. Although this polymorph has been reported to have a perfectly normal ordered average structure, strong and highly structured diffuse scattering was observed, indicating that substantial thermal disorder is present. A diffuse scattering analysis has been carried out using Monte Carlo simulation techniques. Narrow streaks of intensity extending between Bragg peaks in the h0l section were found to arise from planes of diffuse scattering in three dimensions. These are caused by highly correlated molecular displacements along chains of end-to-end disposed molecules running in the a-c direction, corresponding to methyl-methyl intermolecular interactions. A second significant feature--rods of diffuse scattering running in the b* direction--indicates that molecular layers normal to b have a tendency to undergo lateral displacements with little correlation between layers. Finally, the internal flexibility of the molecule is required for a best fit. Changes in the two dihedral angles are found to be strongly correlated and show large excursions (> +/- 20 degrees) from the average values. All of these features suggest possible mechanisms for the way in which form (II) might transform to other polymorphs.

X-ray diffuse scattering from HMTA: analysis via a Monte Carlo model.

Hexamethylenetetramine (HMT, C(6)H(12)N(4), also referred to as urotropin) and azelaic acid [A, HOOC-(CH(2))(7)-COOH] form a co-crystal or adduct (HMTA, also referred to as urotropin azelate) which exhibits several structural phases as a function of temperature. At room temperature, the structure is orthorhombic, but shows substantial disorder. Here, this disorder is explored by analyzing the diffuse scattering from single crystals of HMTA via Monte Carlo simulation. The disorder is in part occupational, with two orientations of azelaic acid occurring, and in part thermally induced, which is to say dynamic. The occupational disorder can be thought of as a combination of limited-range in-plane (bc plane) negative correlations combined with effectively zero correlation between planes (along a), rather like stacking faults. Size effect, the cross-correlation between molecular orientation and displacement from average position, is required to reproduce the observed diffuse scattering.

The interpretation and analysis of diffuse scattering using Monte Carlo simulation methods.

Studies of diffuse scattering had a prominent place in the first issue of Acta Crystallographica 60 years ago at a time when conventional crystallography (determination of the average structure from Bragg peaks) was in its infancy. Since that time, conventional crystallography has developed enormously while diffuse-scattering analysis has seemingly lagged well behind. The paper highlights some of the extra difficulties involved in the measurement, interpretation and analysis of diffuse scattering and plots the progress that has been made. With the advent of the latest X-ray and neutron sources, area detectors and the ever-increasing power of computers, most disorder problems are now tractable. Two recent contrasting examples are described which highlight what can be achieved by current methods.

The molecular conformation of Ibuprofen, C13H18O2, through X-ray diffuse scattering.

A study of the three-dimensional distribution of the X-ray diffuse scattering from the pharmaceutical Ibuprofen (2-(4-isobutylphenyl)propionic acid, C(13)H(18)O(2)) has been undertaken. The most important aspects of the molecular flexibility have been isolated, as have key correlations between variables within a single molecule. Hence, aspects of the conformational space of the molecule within a crystalline environment have been outlined. For example the correlations between torsional twists on different bonds in the molecule (corresponding to the torsional angles on atoms C3, O1 and C11) have been established. Even though the atoms are on opposite sides of the phenyl group, there is a strong negative correlation between the torsional angle on C3 and that on C11. This shows that the ability to predict the conformation a fragment of a molecule will adopt based on the conformations of the fragment found in other molecules may be limited, as aspects of molecular geometry far from the fragment itself will have impact on its conformation. Between molecules, the positional coordinate components which coincide with the direction of propagation of strong intermolecular contacts, particularly the -COOH-HOOC- dimerising interactions, are positively correlated. On the other hand, motions perpendicular to such directions, and rotations about the torsional angles, do not propagate strongly from molecule to molecule.

Simultaneous fitting of X-ray and neutron diffuse scattering data.

Conventional crystallographic refinement uses the Bragg-peak intensities and gives the single-site average crystal structure. Information about short-range order and local order is contained in the diffuse scattering that is distributed throughout reciprocal space. Models of the short-range order in materials can now be automatically refined. The complementarity of X-ray and neutron diffraction data, and the value of simultaneously refining a structural model against both types of data, has long been known. This paper presents the first refinement of a short-range-order model against comprehensive X-ray and neutron diffuse scattering data simultaneously. The sample is the organic molecular crystal benzil, C(14)H(10)O(2) (for neutron work H is replaced by D). The technique gives new insights into local order in crystalline materials, including the dynamic correlation structure indicative of the dynamics of molecules in the crystalline state, and successfully overcomes limitations of using only the X-ray data set.

Diffuse X-ray scattering from 4,4'-dimethoxybenzil, C16H14O4: 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 4,4'-dimethoxybenzil, C16H14O4. A model involving only nine parameters, consisting of seven intermolecular force constants and two intramolecular torsional force constants, was refined to give an agreement factor, omegaR = [sigma omega(deltaI)2/sigma omegaI2(obs)](1/2), of 18.1% for 118 918 data points in two sections of data. The model was purely thermal in nature. The analysis has shown that the most prominent features of the diffraction patterns, viz. diffuse streaks that occur normal to the [101] direction, are due to longitudinal displacement correlations along chains of molecules extending in this direction. These displacements are transmitted from molecule to molecule via contacts involving pairs of hydrogen bonds between adjacent methoxy groups. In contrast to an earlier study of benzil itself, it was not found to be possible to determine, with any degree of certainty, the torsional force constants for rotations about the single bonds in the molecule. It is supposed that this result may be due to the limited data available in the present study.

High-energy diffuse scattering on the 1-ID beamline at the advanced photon source.

This paper reports on experiments in which high-energy (65.35 keV) X-rays were used to record the detailed diffuse diffraction patterns of a number of ceramic materials. The methodology has enabled a greater q-range to be explored (up to sintheta/lambda approximately 0.97) than is possible with laboratory-based experiments, with better q-space resolution and increased sensitivity, thus allowing previously unseen detail in diffraction patterns to be recorded. In all, 11 sections of data have been collected for Ca-CSZ, eight for Y-CSZ and six for wüstite.

Diffuse X-ray scattering and strain effects in disordered crystals.

It is shown in this paper that a feature that has been observed in the diffuse scattering patterns of a wide variety of different materials - a diffuse 'ring' or 'doughnut'-shaped region of scattering - can be understood in terms of a simple model that has been borrowed from the field of sol-gel science. In this, it is supposed that there is a balance between the local attractive forces that are trying to make a particular structure and a rather longer range repulsive force. In the present context, it is believed that this latter force has its origin in the strain that builds as the preferred local structure tries to fit into the average crystal lattice. Simple Monte Carlo (MC) computer simulations are described that demonstrate this principle for three example materials: cubic stabilized zirconia, the p-didecylbenzene/urea inclusion compound and the pure molecular compound 1,3-dibromo-2,5-diethyl-4,6-dimethylbenzene, C12H16Br2 (BEMB2).

Diffuse X-ray scattering from benzil, C(14)H(10)O(2): 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.

Printed circuit microelectrodes and their application to honeybee brain.

The application of printed circuit technology to the production of a new type of multi-channel microelectrode is described. 2. Recordings have been obtained from the the protocerebrum of the honeybee Apis mellifera L. using printed circuit microelectrodes in both restrained and free-moving preparations. 3. These recordings are compared with those previously obtained from conventional probe microelectrodes and are found to have similar characteristics of transient voltage changes superimposed on an undifferentiated high frequency background. 4. A wide range of development possibilities for the printed circuit microelectrode are discussed.