Radial Correlation Functions Research Articles

Short-range structure of zirconia xerogel and aerogel, determined by wide angle X-ray scattering

The short-range structure of amorphous zirconia, investigated by X-ray diffraction, has been found to bear more resemblance to the monoclinic ZrO 2 structure than it does to the tetragonal structure. This result was achieved in two zirconia samples (xerogel and aerogel) by analyzing both the positions of the maxima of the experimental (Fourier-transformed) ‘reduced’ radial correlation function, G( r), and the experimental radial atom (number) density distribution, p( r). This latter function was simulated as a weighted sum of three partial atom density distributions, ϱ′ ZrO, ϱ′ ZrZr, and τ′ OO, based on the monoclinic or the tetragonal coordination frame and limited inside a sphere of 9 Å in radius. The coordination statistics are described in terms of sums of Gaussian functions into which distortion effects (first-type and second-type disorder) were introduced.

The calculation of activity coefficients of binary mixed electrolytes

Abstract

Large-scale radial velocity correlations as a test of Gaussian initial conditions

view Abstract Citations (13) References (11) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Large-Scale Radial Velocity Correlations as a Test of Gaussian Initial Conditions Landy, Stephen D. ; Szalay, Alexander S. Abstract Here we test for the Gaussian character of the large-scale mass fluctuation spectrum as a function of scale using a relation between higher order velocity correlation functions of Gaussian random fields. An underlying Gaussian distribution should satisfy the following relationship between the two-point second- and fourth-order radial velocity correlation functions <u^2^_1_u^2^_2_>-2<u_1_u_2_>^2^-<u^2^(0)>^2^=0, where u_1_, u_2_ are the radial velocities of mass points in the cosmic microwave background frame separated by a distance r = |r_1_ - r_2_|. In addition, the two-point third-order radial velocity correlation function should equal zero. Our analysis checks the consistency of the hypothesis that the initial fluctuation spectrum can be considered an isotropic Gaussian random field with random phase and that the present distribution has grown linearly as a result of gravitational instabilities. Utilizing a simulated Gaussian random velocity field, we investigate the effects of distance errors, thermal motion, selection functions, Malmquist corrections, and different weighting schemes on the correlation functions. We show a superior fit to the underlying correlation functions using a Malmquist correction derived from the actual distribution of estimated distances and an even weighting scheme. These methods are then applied to a sample of elliptical galaxies. The results show a consistency with the assumption of a Gaussian field; however, the inherent limitations of the data set preclude a definitive answer. Publication: The Astrophysical Journal Pub Date: July 1992 DOI: 10.1086/171556 Bibcode: 1992ApJ...394...25L Keywords: Computational Astrophysics; Elliptical Galaxies; Radial Velocity; Astronomical Models; Correlation Coefficients; Mass Distribution; Monte Carlo Method; Relic Radiation; Velocity Distribution; Astrophysics; COSMOLOGY: THEORY; GALAXIES: DISTANCES AND REDSHIFTS; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE; METHODS: NUMERICAL full text sources ADS |

Chemical and magnetic ordering in amorphous systems.

A statistical method for the analysis of the chemical and magnetic ordering in multicomponent amorphous substances is developed. The model assumes that the amorphous state is that of a crystal with a high level of defects. The Hamiltonian allows for the existence of multiparticle and multispin interaction potentials. The magnetic interactions are of the Ising type. The spin value is not limited. Multiparticle radial distribution functions and multispin correlation functions are used for the description of the atomic and magnetic structure. The internal energy is represented as a low-temperature expansion.

Neutron-diffraction study of magnetic ordering in the pyrochlore series R2Mo2O7 (R=Nd,Tb,Y).

${\mathrm{Nd}}_{2}$${\mathrm{Mo}}_{2}$${\mathrm{O}}_{7}$, ${\mathrm{Y}}_{2}$${\mathrm{Mo}}_{2}$${\mathrm{O}}_{7}$, and ${\mathrm{Tb}}_{2}$${\mathrm{Mo}}_{2}$${\mathrm{O}}_{7}$ are all ordered crystalline oxides with the cubic pyrochlore structure. At 53 K, ${\mathrm{Nd}}_{2}$${\mathrm{Mo}}_{2}$${\mathrm{O}}_{7}$ is long-range ordered with a ferrimagnetic structure. A transition temperature of 95 K is determined from small-angle-neutron-scattering (SANS) data. Antiferromagnetic structures are proposed below 20 K that are consistent with previous magnetization results. No magnetic scattering from ${\mathrm{Y}}_{2}$${\mathrm{Mo}}_{2}$${\mathrm{O}}_{7}$ was observed at low temperature, which is believed to be a result of the small moment on ${\mathrm{Mo}}^{4+}$. For ${\mathrm{Tb}}_{2}$${\mathrm{Mo}}_{2}$${\mathrm{O}}_{7}$ strong diffuse scattering from the large ${\mathrm{Tb}}^{3+}$ moments is observed below 25 K. Fourier analysis of this data yields an approximation to the radial correlation function, which gives information concerning spin-spin correlations for the first four coordination shells. The first-, third-, and fourth-neighbor correlations are ferromagnetic, but the second-neighbor correlation is strongly antiferromagnetic. SANS data show magnetic scattering only at large Q, reenforcing the view that antiferromagnetic interactions dominate. The Q-integrated magnetic SANS shows a temperature dependence similar to that for the correlation functions. The long-range order in ${\mathrm{Nd}}_{2}$${\mathrm{Mo}}_{2}$${\mathrm{O}}_{7}$ and lack of long-range order in ${\mathrm{Y}}_{2}$${\mathrm{Mo}}_{2}$${\mathrm{O}}_{7}$ and ${\mathrm{Tb}}_{2}$${\mathrm{Mo}}_{2}$${\mathrm{O}}_{7}$ can be understood, to a certain extent, in terms of Landau-theory results for this highly frustrated lattice.

Electronic properties of random alloys: Special quasirandom structures.

Structural models needed in calculations of properties of substitutionally random ${\mathit{A}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathit{B}}_{\mathit{x}}$ alloys are usually constructed by randomly occupying each of the N sites of a periodic cell by A or B. We show that it is possible to design ``special quasirandom structures'' (SQS's) that mimic for small N (even N=8) the first few, physically most relevant radial correlation functions of an infinite, perfectly random structure far better than the standard technique does. These SQS's are shown to be short-period superlattices of 4--16 atoms/cell whose layers are stacked in rather nonstandard orientations (e.g., [113], [331], and [115]). Since these SQS's mimic well the local atomic structure of the random alloy, their electronic properties, calculable via first-principles techniques, provide a representation of the electronic structure of the alloy. We demonstrate the usefulness of these SQS's by applying them to semiconductor alloys. We calculate their electronic structure, total energy, and equilibrium geometry, and compare the results to experimental data.

Special quasirandom structures.

Structural models used in calculations of properties of substitutionally random ${\mathit{A}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathit{B}}_{\mathit{x}}$ alloys are usually constructed by randomly occupying each of the N sites of a periodic cell by A or B. We show that it is possible to design ``special quasirandom structures'' (SQS's) that mimic for small N (even N=8) the first few, physically most relevant radial correlation functions of a perfectly random structure far better than the standard technique does. We demonstrate the usefulness of these SQS's by calculating optical and thermodynamic properties of a number of semiconductor alloys in the local-density formalism.

Invariants of spherical harmonics as order parameters in liquids: comparison of the structure of a molten and glassy alkali halide

The second order invariants of spherical harmonics for the geometrical characterization of clusters in disordered systems is applied for the comparison of computer simulated configurations of molten and glassy rubidium bromide. The latter has been obtained by the Stillinger method leading to the “inherent” liquid structure. It is shown that the rotational invariants of spherical harmonics characterize the inherent structure even when vibrations are not damped at all. Angular correlation between the positions of closest neighbours proved to be identical in liquid and glassy state, whereas radial pair correlation functions show considerable difference.

The binary system benzene—hexafluorobenzene studied by SSOZ theory and computer simulation. I. The charge symmetry model

The symmetry reduction of the SSOZ equation already developed for pure components is extended to binary mixtures. The general formalism is applied to the binary system C6F6/C6H6, whose “mixing interaction” is described by a simplified “charge symmetry model”. Varying the charge strength in a systematic way the predictions of SSOZ theory are verified by extensive computer simulation (molecular dynamics) studies. Thereby mixtures and pure components are treated in a unified way. Finally, extensions of the “charge symmetry model” are presented and the computed radial excess correlation functions are compared to those measured by neutron and X-ray scattering.

Theory of transport processes in dense fluids

The projection operator methods of Mori and Zwanzig are used to construct kinetic equations for the single-particle distribution functions of a fluid mixture. The particle interactions are pair-additive sums of hard-core and soft, continuous potentials. The mean field kinetic equations obtained by discarding the multicomponent memory functions are identical in form to van Beijeren and Ernst’s revised Enskog equations for a hard sphere mixture, but incorporate the radial and direct correlation functions characteristic of the composite interaction. The 13-moments methods is used to construct an approximate solution of the mean field kinetic equation for a simple fluid and to relate the associated, zero-frequency transport coefficients to those of the conventional Enskog theory. Our choice of interaction parameters optimally mimicks a Lennard-Jones 12-6 fluid and yields excellent estimates of viscosity.

Role of molecular symmetry and molecular interactions in the Faraday effect of fluids

General expressions are developed for assessing the effects of molecular symmetry and molecular interactions on the rotation of plane polarized light in the presence of a static magnetic field. This is accomplished by writing the Verdet constant V as a sum of four terms : V = VD O + VPO + VDINT + VPINT where the subscripts 0 and INT refer, respectively, to the absence and presence of molecular interactions in the presence of the external magnetic field and the superscripts D and P denote, respectively, diamagnetic and paramagnetic contributions. The results, which are given in terms of c- and i-tensors and radial and radial-angular correlation functions, are analyzed in terms of magnetic point group symmetries and applied to rarefied and dense systems. From the experimental Verdet constants at atmospheric pressure we calculate the mean electromagnetic hyperpolarizability tensor α* 123(ω) for the molecules He, H2, N2, Cl 2, CO2, HCl, CO, NH3 and CH4. We also calculate α*123(ω) for liquid CS2 and C 6H6 and the contribution due to nearest neighbour interaction to the Verdet constant It is found that VDINT/ VDO = 22.8 % for CS2 and 8.4 % for C 6H6. We also list, for all magnetic point groups, the tensors which describe the Faraday effect.

Self-diffusion and shear viscosity of simple fluids. A molecular-dynamics study

Extensive molecular-dynamics, MD, simulations of Lennard-Jones, LJ, and soft-sphere, SS, fluids have been made. The LJ state points agree excellently with the predictions of a recent equation of state for this fluid. Self-diffusion coefficients obtained from the LJ mean-square displacements at equilibrium are fitted to a simple analytic expression, involving temperature and density, which has a density and temperature range of wider applicability than that of Levesque and Verlet. The corresponding shear viscosities in the limit of zero shear rate have been obtained by a new non-equilibrium MD technique, which involves orthogonal longitudinal distortions (eliminating pure expansion or compression terms). The LJ shear rigidity moduli are fitted to better than 1 % by a simple analytic expression. A similar relationship for the shear viscosities is less satisfactory but this merely reflects the greater uncertainty in this collective transport property. The Stokes–Einstein relationship using slip boundary conditions gives an effective ‘flow unit’ diameter which decreases from several molecular diameters to one as the density increases. This suggests that the motion between a molecule and those in its first coordination shell is more cooperative at moderate densities resulting in greater coupling between molecular trajectories. Support for this comes from the direct evaluation of the friction coefficient by non-equilibrium MD, pair radial and pair fluctuation correlation functions.

Structure Investigations on Se1-xTex Melts by Neutron Scattering and X-ray Diffraction

Abstract Neutron scattering results on Se1-xTex melts (x = 0.3, 0.6, 0.8) obtained at 460 °C are combined with earlier X-ray diffraction data. The behaviour of the number and distance of nearest neighbours is discussed with respect to chemical short-range order and transition from twofold to three-fold coordination. For Se70Te30 the radial concentration correlation function 4πr2 ρcc(r) has been evaluated. It shows clearly the preferred formation of unlike pairs (Se-Te) within the chains.

Computer simulation studies of anisotropic systems IV. The effect of translational freedom

We have simulated the properties of 256 cylindrically symmetric particles interacting via a simple anisotropic potential of the form u 2 (r 12 ) P 2 (cos B 12 ) and with a scalar Lennard-Jones 12:6 potential, using the Monte Carlo technique. The simulations were performed for two forms of u2(r12) in the isothermal-isobaric ensemble and yielded values for volume, enthalpy, second-rank orientational order parameter, radial distribution function and second-rank angular correlation function. The specific heat at constant pressure, isothermal compressibility and isobaric expansivity were also obtained but they are subject to considerable error because they were evaluated from fluctuations. The system is found to exhibit a weak, firstorder transition from a nematic to an isotropic phase on increasing the temperature. The isotropic phase possesses short-range spatial and orientational order; it differs from the nematic phase, which has longrange orientational order but only short-range spatial order. The results of these simulations are used to discuss the influence of the range of the anisotropic potential on the behaviour of the nematogen. Previous Monte Carlo simulations of nematic liquid crystals had employed a lattice model with the anisotropic interactions restricted to nearest neighbours. Our results are used to study the effect of these convenient but unrealistic restrictions on the properties of the nematic. The results of our simulations are in reasonable accord with the properties of the nematogen, 4,4'- dimethoxyazoxybenzene, although no attem pt was made to select a pair potential to mimic the behaviour of any substance. Finally, we use the results of our simulations to test the validity of the molecular field approximation, as applied to nematics. This approximation is one of the foundations of the Maier-Saupe theory and its predictions are compared with the behaviour of the simulated nematics. It would appear that this theory provides a better description of our system than the lattice model, with its enforced spatial order and truncated anisotropic pair potential.

Structural correlation functions in the coexistence region from molecular dynamics

We present the radial correlation function and structure factor for two partially condensed states of a monatomic fluid in the coexistence region obtained by molecular-dynamics simulation. A comparison is made with the normal liquid structure.

Molecular dynamics calculation of the dielectric constant

Molecular dynamics simulation of a sample of a two-dimensional fluid of Stockmayer molecules (i.e. particles interacting via a central Lennard-Jones interaction plus a point dipole interaction) are reported. The dipolar interaction adopted is that required by two-dimensional electrostatics, so that the convergence problem is conserved. The sample (≈13 molecular diameters in size) is kept in vacuo by a steep circular potential barrier. It is first shown that for distances greater than 3 to 5 molecular diameters the macroscopic laws of electrostatics apply, by checking that the mean squared moment of an inner disc, as a function of the diameter of the disc, can be fitted with a single dielectric constant, which is thus determined. The Kirkwood correlation factor for an infinite sample is then evaluated. For highly polar systems, it is greater than unity. Also the radial vector correlation function h Δ(r), which describes the weight of in an expansion of the angle-dependent pair distribution f...

Equilibrium properties of a semiclassical fluid with square-well plus hard core potential

A cluster expansion theory, in which the quantum hard sphere system is taken as a reference system and the attractive interactions as a perturbation, is applied to calculate the equilibrium properties of the square-well fluid in the semiclassical limit. The radial distribution function and direct correlation function are obtained using the exponential approximation. The isothermal compressibility is also evaluated.

Neutron-scattering studies of bulk polyethylene at intermediate Q values

We report SANS studies of bulk polyethylene (PE) samples crystallized from the melt at different rates. All samples contained both protonated and deuterated PE. Substantial isotopic concentration fluctuations are present in slow-cooled samples. This is seen as large differences in the low-angle (low-Q) SANS. At larger Q values, the SANS data from both materials is approximately identical. We interpret this in terms of an exponential radial correlation function for composition fluctuation whose correlation length is comparable to lamellar dimensions. Numerical calculations show that concentration fluctuations have minimal effect in the intermediate Q region.

A Molecular Dynamics Study of Aqueous Solutions. VII. Improved Simulation and Comparison with X-Ray Investigations of a NaCl Solution

The molecular dynamics simulation has been improved by introducing the Ewald summation for ion-ion interactions. On the basis of the Pauling radii new Lennard-Jones parameters have been derived for the halide ions. The results of a simulation of a system consisting of 200 water molecules, 8 sodium and 8 chloride ions are compared with x-ray investigations of a 2 molal NaCl solution through the structure functions. The first neighbor model commonly used in the x-ray analysis of multicomponent liquids is fitted to the experimental structure function and to the structure function calculated from the radial pair correlation functions of the simulation. The parameters of the two fits and the radial pair correlation functions calculated from the fitted first neighbor model are compared. On the basis of this comparison the usefulness and the limitations of the first neighbor model are discussed

Structure determination of liquid argon by x-ray diffraction

X-ray diffraction measurements and subsequent data analyses have been carried out on liquid argon at five states in the density range of 0.91–1.35 g/cc and temperature range of 127–143 °K. Duplicate measurements were made on all states. These data yielded radial distribution and direct correlation functions, which were then used to compute the pair potential using the Percus–Yevick and CHNC theories. The potential minima are in the range of −105 to −120 °K, and appear to substantiate current theoretical estimates of the effective pair potential in the presence of a weak three-body force. Numerical experiments on the data have been carried out to study the subsidiary peak phenomenon in the radial distribution function and a distortion phenomenon. The distortion phenomenon for an experiment of the type conducted partially explains why the earlier data by Mikolaj–Pings yielded argon pair potential well depths from the Percus–Yevick equation that were too shallow. The derivation of the correction procedure for distortion is presented.