Measured Structure Factors Research Articles

Determination of Three Body Correlations in Simple Liquids by RMC Modelling of Diffraction Data. I. Theoretical Tests

Abstract In theory all the information regarding the three dimensional structure of a liquid is contained in the pair distribution function g(r) or, equivalently, the structure factor A(Q), for a system with purely pairwise additive potentials. RMC is a method for modelling the structures of disordered systems based on the experimentally measured structure factor(s). We have performed some theoretical tests of the method, using input data calculated from simulations for various systems, which show that RMC does indeed work successfully if the potentials are pairwise additive. In cases where they are not then the imposition of constraints, e.g. modelling molecular systems with molecules rather than atoms, can enable the three dimensional structure to be determined. When the potentials are very complex, which usually makes the problem unsuitable for MC or MD simulations, RMC with constraints is still a valuable way of distinguishing between various structural possibilities.

Optical structure factor measurements of soot particles in a premixed flame

We have measured the scattered-light intensity as a function of the scattering angle for light scattered from soot particles in a premixed methane/oxygen flame. This yields the optical structure factor for the soot particles. We find that the structure factor shows the soot particles to have a morphology consistent with a fractal interpretation with a fractal dimension in the 1.6 </= D </= 1.8 range and a radius of gyration that increases with the height above the burner. The structure factor may also show the effect of the finite monomer size of the soot clusters.

Effective pair potential reproducing the measured structure factor of liquid Cu near the melting point

An effective pair potential for liquid Co at 1423 K has been obtained from small-angle structure factor data with the help of inverse power potentials to describe the structure factors in the high-wavenumber region. This effective pair potential indicates a marked difference, in position and depth of the first minimum, from the interatomic potential derived from first principles. The experimental structure data, from well below the first peak to high wavenumber, can be reproduced by a molecular dynamics simulation using the present effective pair potential. The molecular dynamics results for the pair distribution function, the specific heat at constant volume, the self-diffusion constant and the shear viscosity coefficient are also presented in comparison with the experimental data.

Structural effects of polydispersity in charged colloidal dispersions

Experimental information on the microscopic structure of charged colloidal dispersions is usually extracted from the intensity of scattered light l(k) which can be linked to the ‘measured structure factor’SM(k). A reinterpretation of given experimental results for SM(k) is given in terms of a size and charge polydisperse model and the systematic analysis of the effects of polydispersity on SM(k). In our model, the interaction between macroions is assumed to be of a Yukawa type and the polydispersity is characterised by histograms with standard deviations from 10 to 40% with up to 10 components. The partial structure factors Sαβ(k) are evaluated by solving the multicomponent Ornstein–Zernike (OZ) equations in connection with the thermodynamically self-consistent closure of Rogers–Young (RY). The accuracy of the RY approximation is demonstrated by comparing the results with simulation data on monodisperse Yukawa systems. The results for SM(k) for polydisperse systems show significant differences from results obtained by treating the systems as being monodisperse. For SM(k) a large increase is found at small k as well as a shift in the main peak. These features are discussed in terms of the fluctuation and scattering abilities of each component of the dispersion. The role played by the charge and the size polydispersity is also analysed by introducing the generalised Bhatia–Thornton structure factors. Finally, SM(k) is compared with scattering data. Quantitative agreement is found for all k values and, in particular, in the range of small k in which all one-component models are particularly inaccurate. The difference between SM(0) for a polydisperse system and the isothermal osmotic compressibility is emphasised.

High Resolution X-Ray Powder Diffraction by the Combination of Synchrotron Radiation and Imaging Plate to Observe Electron Distribution by the Maximum Entropy Method

AbstractA Debye-Scherrer camera (radius 572 mm) was designed for the use at the Photon Factory BL-6A2. It uses 4 pieces of the Imaging Plate to cover 0° ∼ 160° in 2θ. The profiles measured by the camera had 0.083° full width at half maximum (FWHM) and were rather symmetric compared with conventional X-ray source. The observed profiles were well represented by either the split Pearson VII or pseudo-Voigt function. In order to examine the performance of the new camera, the whole powder data of LiF was analyzed by the Maximum Entropy Method(MEM). The MEM analysis of LiF was successfully accomplished using 20 measured structure factors and gave R(Bragg) 0.25%. The density distribution obtained represented a characteristic feature of ionic crystals and was consistent with the theoretical result of the self-consistent LCAO method.

Understanding crystal bonding: The contribution of electron diffraction

Some of the most incisive information about bonding mechanisms in materials can be obtained from accurate X-ray crystal structure factors and Debye-Waller factors. This has long been known by diffraction workers in all fields, and in the last twenty years intensive efforts have been made to accurately measure structure factors by a variety of means. At the same time theoreticians have made progressively more sophisticated band structure calculations of structure factors (usually of cubic elements). Three experimental methods have emerged as being able to determine structure factors with the required accuracy for crystal bonding studies:- these are (i) X-ray Pendellosung methods (see e.g. 1 and 2) (ii) Gamma-ray diffraction (see e.g. 3) and various electron diffraction techniques (see e.g. 4 to 6). The best accuracy possible with all three methods is around 0.1%.The potential of electron diffraction for the accurate measurement of low-angle structure factors was first recognised in the late sixties by Goodman and Lehmfuhl (convergent beam-rocking curve method), Gjonnes and Hoier (intersecting Kikuchi line, IKL, method) and Nagata and Fukuhara (critical voltage, Vc, technique).

High-accuracy structure-factor measurements in germanium.

Structure factors ${\mathit{F}}_{\mathit{h}}$ of five reflections in the range 0.15 \ensuremath{\le}sin\ensuremath{\theta}/\ensuremath{\lambda}\ensuremath{\le}1.07 A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}1}$ were measured in germanium. An accuracy of \ensuremath{\le}20 millielectron per atom was achieved for all values except one. This represents 1 order of magnitude improvement over previous measurements for the three highest-order ${\mathit{F}}_{\mathit{h}}$. Measurements were done using the W K${\mathrm{\ensuremath{\alpha}}}_{1}$ line (E=59.3 keV) and a monolithic thin-crystal Laue-case diffractometer, the rocking curves of which were computer fitted to yield ${\mathit{F}}_{\mathit{h}}$. Good agreement is obtained with previous measurements, where available. The bonding contribution to ${\mathit{F}}_{111}$ is found to be larger than predicted by ab initio calculations. No evidence is found for an anharmonic contribution to the atomic potential, within the accuracy of the measurements. This is in keeping with x-ray results for silicon, but in contrast with neutron and x-ray ``forbidden'' reflection measurements. The Debye parameter B=(0.5661\ifmmode\pm\else\textpm\fi{}0.0026) A${\mathrm{\r{}}}^{2}$ derived from our data is in excellent agreement with older results.

Pair potentials for liquid sodium near freezing from electron theory and from inversion of the measured structure factor.

An effective pair potential for liquid sodium near freezing has been calculated from electron theory using the density-functional method. The main features of the potential extracted by Reatto, Levesque, and Weis [Phys. Rev. A 33, 3451 (1986)] by inverting the measured structure factor of Greenfield, Wellendorf, and Wiser [Phys. Rev. A 4, 1607 (1971)] are faithfully reflected by electron theory. To obtain precise agreement between the two methods will evidently require further progress in setting up nonlocal exchange and correlation functionals.

Accurate structure analysis by the maximum-entropy method

The electron-density map of an Si crystal is drawn by the maximum-entropy method (MEM) with 30 structure factors which were determined very accurately on an absolute scale by the Pendellosung method [Saka & Kato (1986). Acta Cryst. A42, 469-478]. It is shown that the following three beneficial points arise from the use of the MEM for accurate structure analysis. Firstly, a precise electron-density map can be obtained; the existence of bonding electrons is clearly visible in the maximum-entropy map, even though no forbidden reflections are included in the analysis. The structure factors calculated from the maximum-entropy map for the 222 and the 442 forbidden reflections show good agreement with those measured by different experiments. The final R factor was 0.05% as a consequence of the high accuracy of the measured structure factors. Secondly, there is no need to seek a better structural model which could be a complicated and time-consuming process in accurate structure analysis. Thirdly, the resolution of the maximum-entropy map is much higher than that of the map drawn by conventional Fourier transformation. It is also found in the case of silicon that phase information is not absolutely necessary because exactly the same density-distribution map can be obtained without any phase information as the map drawn with all the phase information.

Structure of liquid equiatomic KSn and CsSn.

The structure factors of liquid equiatomic KSn and CsSn alloys have been determined by neutron diffraction. A distinct peak at about 1 A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}1}$, the first sharp diffraction peak (FSDP), occurs in all measured structure factors. The temperature dependence of the structure has been studied for KSn: apart from the FSDP, whose position shifts toward smaller wave vector and magnitude decreases with increasing temperature, the structure factor remains essentially unaltered. The radial distribution function clearly shows three maxima in the first coordination shell. The simultaneous presence of the FSDP and several maxima in the first coordination shell are indications of formation of polyatomic species in the liquid. Application of two simple structural models, taking into account the presence of structural units, yields reasonable agreement with the measured structure factors.

Thermal motion of atoms in crystalline silicon: Beyond the Debye theory.

Evidence is presented for nonrigid thermal vibrations of crystal-bound silicon atoms, using an extensive set of measured structure factors ${F}_{m}$ accurate to 0.1%. A model allowing for a different Debye-Waller factor (DWF) for each electronic shell is found to fit all ${F}_{m}$ well with four such factors. DWF's refined from ${F}_{m}$ using calculated shell-by-shell partial structure factors yield good agreement with earlier results derived from low-order reflections but not with ab initio shell-model calculations. Also, no evidence is found for an anharmonic term in the atomic potential.

Pair potential in liquid lead as a test case for the modified hypernetted-chain approximation.

The interionic pair potential derived from the measured structure factor S(Q) of liquid Pb at 613 K using the modified hypernetted-chain approximation was tested in the molecular-dynamics simulation and found to be inadequate. It is shown that the same simple model of the potential with three parameters, having been optimized in an iterative procedure involving the large-scale molecular-dynamics simulation, provides a very good fit to the original structure.

Structures of [001] twist boundaries in gold. I. Measurement and use of absolute boundary x-ray structure factors

An x-ray-diffraction method for measuring absolute grain-boundary structure factors is described. The method is based on ratioing appropriate integrated scattered intensities from the grain-boundary region and from the adjoining perfect-crystal region in a bicrystal specimen containing a flat grain boundary. Common unknown factors then cancel out, and an expression for the absolute structure factor is obtained in which all quantities are either known, can be measured, or can be calculated with acceptable accuracy. A practical experimental technique for making the necessary measurements, which employs a four-circle diffractometer, is described. The technique is applied to an x-ray diffraction study of the atomistic structures of a series of [001] twist boundaries in gold in the following paper of our work. The results obtained there clearly demonstrate that the measurement of absolute structure factors, rather than relative structure factors, provides important information in diffraction studies of the structures of grain boundaries, particularly in cases where a limited number of structure factors is measured.

Structure of liquid K-Pb alloys

The structure factors of liquid Pb, K0.25Pb0.75, K0.50Pb0.50, K0.65Pb0.35 and K0.80Pb0.20 have been measured by means of neutron diffraction. The main peaks of the structure factors of the alloys are preceded by a pre-peak which is most pronounced in K0.50Pb0.50. This pre-peak indicates that there is considerable ordering in the alloys, although there is also evidence for a tendency towards phase separation in at least one of the alloys. From inspection of the radial distribution functions it follows that Pb clusters are formed, the concentrations of which increase when the equi-atomic composition is approached. A simple model of the structure of K0.50Pb0.50 based on the reference interaction site model proves to be in excellent agreement with the measured structure factor.

Crystal structure determination by convergent-beam electron diffraction

In recent years, significant progress has been made towards a solution for the general problem of crystal structure determination by convergent beam electron diffraction (CBED). Even if we consider only perfectly ordered, periodic crystals defined by one of the conventional space groups, diffraction methods based on a focussed sub-micron beam of electrons are applicable to several related sets of structural problems that are not accessible to conventional X-ray or neutron diffraction techniques. We assume here that the space group either is known or has been determined from CBED patterns and that phases and amplitudes for some subset of the structure factors are required. Two limiting cases have been explored in some detail. For crystals where the atomic parameters and Debye-Waller factors are known accurately from high quality X-ray data, information on the charge redistribution for bonding electrons is available from precise measurements of the low order structure factors. Following the original research of Kambe, some recent work has demonstrated that accurate structure amplitudes and three-beam phase invariants can be extracted from the dynamical intensity distribution in CBED reflections. In principle, this approach is completely general but considerable labour would be required to extract sufficient data to solve the structure of an unknown crystal, whereas a large set of kinematic intensities is acquired from a single X-ray pattern.

NMR dynamics in disordered magnets

The excitation dynamics of site diluted magnets can be described at low energies (long length scales) by magnons, and above a crossover frequency, ωc, (short length scales) by fractons. The density of fracton states is given by $$N(\omega )\alpha \omega ^{\bar d - 1} $$ , where $$\bar \bar d$$ is the fracton dimensionality. Dilution gives rise to a characteristic length ξ∝(p−p c)ν, wherep c is the critical concentration for (magnetic) percolation. The crossover frequency ωc is proportional to ξ-1[1+(θ/2)], where θ is the rate at which the diffusion constant decays with distance for diffusion on an equivalent network. A fractal dimensionD describes the density of magnetic sites on the infinite network, and $$\bar \bar d = 2D/(2 + \theta )$$ . For percolating networks, $$\bar \bar d \simeq {4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}$$ for all dimensions ≥2. Neutron scattering structure factor measurements by Uemura and Birgeneau compare well with calculations using fracton concepts. Magnons are extended at low energies, while the fracton states are geometrically localized, with a wave function envelope proportional to exp $$\{ - [{r \mathord{\left/ {\vphantom {r {\Lambda (\omega )}}} \right. \kern-\nulldelimiterspace} {\Lambda (\omega )}}]^{d_\phi } \} $$ . Here, $$\Lambda (\omega ) \propto \omega ^{{{ - \bar d} \mathord{\left/ {\vphantom {{ - \bar d} D}} \right. \kern-\nulldelimiterspace} D}} $$ is the fracton length scale at frequency ω. The exponentd ϕ lies between 1 andd min, the chemical length index (of the order of 1.6 in three dimensions). The localization of the magnetic excitations causes a spread in the NMR relaxation rates. A given nuclear moment will experience only a limited set of fracton excitations, resulting in an overall non-exponential decay of the NMR relaxation signal. When strong cross-relaxation is present, the relaxation will be exponential, but the temperature dependence will be strongly altered from the concentrated result.

Structure Factors of Molten CsCl at LowQ

Abstract The structure factors for three samples of molten CsCl with different isotopic chlorine compositions have been measured accurately in the region 0.2 > Q > 3.0 A−1 using the D1B diffractometer at the Institut Laue Langevin. A rise in the measured structure factors for Q > 0.5 A−1 is explained as being due to inelastic scattering from overdamped acoustic modes.

Adhesive hard-sphere colloidal dispersions. A small-angle neutron-scattering study of stickiness and the structure factor

Small-angle neutron-scattering structure factor measurements were made on sterically stabilized silica spheres dispersed in benzene up to volume fractions of 0.30. Benzene is only a marginal solvent for the stabilizing layer on the surface of the particles. The particles are made attractive by lowering temperature. This attraction is modeled by a square well potential, the depth of which varies with temperature. At the highest temperature studied, our experimental system behaved effectively as an assembly of hard spheres, whereas at the lowest temperature the system approaches a spinodal. Using Baxter's theory we were able to evaluate the interaction parameters and to calculate the structure factor. Experimental structure factors were satisfactorily reproduced over the entire temperature range studied.

Reverse Monte Carlo Simulation: A New Technique for the Determination of Disordered Structures

Abstract We have developed a new technique, based on the standard Monte Carlo simulation method with Markov chain sampling, in which a set of three dimensional particle configurations are generated that are consistent with the experimentally measured structure factor. A(Q), and radial distribution function, g(r), of a liquid or other disordered system. Consistency is determined by a standard χ2 test using the experimental errors. No input potential is required, we present initial results for liquid argon. Since the technique can work directly from the structure factor it promises to be useful for modelling the structures of glasses or amorphous materials. It also has other advantages in multicomponent systems and as a tool for experimental data analysis.

The split in the second peak in the structure factor of binary colloidal suspensions: glass-like order

Structure factors of dilute binary colloidal suspensions of various compositions, particle diameter ratios and ionic impurity concentrations are investigated using polarised light scattering. For the first time, a split in the second peak of the measured structure factor is observed, suggesting the existence of glass-like order in the binary suspensions. The near-forward-scattered intensity from the colloidal suspension exhibiting a split second peak fluctuates much more slowly than that from the liquid-like ordered phase, corroborating the existence of glass-like order in the binary mixtures. The time evolution of the structure indicates that a liquid-like order continuously evolves into glass-like order as the strength of inter-particle interaction is increased by reducing the impurity concentration. The glass-like order is unstable towards formation of the crystalline state above a critical volume fraction (which depends on the impurity concentration) of the particles.