Linear Correlation Function Research Articles

Use of SAXS and linear correlation functions for the determination of the crystallinity and morphology of semi-crystalline polymers. Application to linear polyethylene

The use of correlation functions to obtain the morphological parameters of crystalline-amorphous two-phase lamellar systems is critically reviewed and extended. It is shown that processing of the experimental SAXS-patterns only significantly affects the curvature of the autocorrelation triangle and that the parameters of the corresponding ideal two-phase structure can be determined independently of the data processing procedure. The methods to be used depend on the normalization of the correlation function. The validity of the formulation is illustrated for a sample of linear polyethylene, cooled and heated at 10°C per min. Crystallite thickening during crystallization and surface melting during heating are observed. The overall crystallinity and the fraction of semi-crystalline stacks during crystallization and melting are determined quantitatively as a function of temperature using the total scattering power of the corresponding ideal two-phase structure, correlation functions, and a scaling procedure. Absolute intensities are not required. The SAXS results are confirmed by independent techniques (DSC, WAXD, and SALLS). During crystallization, amorphous regions are present outside the semi-crystalline regions because growing spherulites do not fill space completely. During melting, larger amorphous regions develop in the spherulites because of the complete melting of stacks. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 1715–1738, 1999

Solvation and catalyst–substrate superstructure of a tungsten tris(dithiolene) complex dissolved in water–acetone A molecular dynamics model calculation

The intermolecular interactions of the catalyst tris(1,2-ethylenedithiolate-S,S′)tungsten, W(S2C2H2)3, with the molecules of a 25:75 water–acetone mixed solvent, are examined by statistical mechanical methods, and specifically by a molecular dynamics (MD) technique, using charge distributions obtained by extended Huckel calculations. The results are presented in the form of pair correlation functions (PCF), and show that an average of up to three water molecules come close to the sites of the catalyst, whereas the acetone molecules form an open cage at a somewhat longer distance. The supramolecular structure around the catalyst is asymmetric, and is examined in the two characteristic geometries corresponding to the D3h and C2v symmetries which the ‘tris(dithiolene)’ molecule transiently assumes during its fluxional metamorphoses in solution. At large distance (large values of the correlation parameter r) the system is homogeneous. Yet, at nanometer distances the symmetry breaks down, and the system becomes highly asymmetric with differentiation (selectivity) in space. The mobility of the water and acetone molecules, close to the catalyst, was also studied by estimating the translational self-diffusion coefficients Dwat and Dacet from the center of mass linear velocity correlation functions. The results show that the water molecules are more mobile than the acetone molecules, which is the opposite of what happens without the catalyst.

Phase defects and spatiotemporal disorder in traveling-wave convection patterns

Spatiotemporal disorder is studied in traveling-wave convection in ethanol-water mixtures. Spectral measures of disorder, linear correlation functions, and mutual information are used to characterize the patterns, and are found to give a weak indication of the level of disorder. The calculation of the complex order parameter for experimental patterns is described. It is found that the ordering of the patterns is accompanied by a dramatic change in the topological structure of the order parameter. Specific arrangements of defects are found to be associated with the elements of traveling-wave patterns, and the net charge and total number of defects is introduced as a measure of disorder in the patterns. The coarsening of the patterns is marked by an accumulation of net charge and a dramatic decrease in the number of defects. The physical significance of the defects is discussed, and it is shown that the phase velocity of the waves is lower in the vicinity of the defects. The defect-defect correlation functions are calculated for the convection patterns. It is shown that the ordering of the patterns is closely related to the apparent defect-defect interactions. {copyright} {ital 1997} {ital The American Physical Society}

Kinetic study and crystal structure of tetrakis(1,4-thioxane)palladium(II). A comparison between platinum(II) and palladium(II) reactivity

The compound [Pd(1,4-thioxane) 4](BF 4) 2·4CH 3NO 2 crystallizes in the monoclinic space group P2 1/ c with a=9.474(2), b=9.529(5), c=21.945(9) Å, β=99.77(3)°, Z=2; final R=0.051 for 4325 observed reflections (Pd, C, S and O anisotropically refined). The thioxane ligands around Pd form a square-planar arrangement, with two nitromethane molecules at a distance of 3.115(2) Å occupying sites perpendicular to the square-plane formed by the sulfur-bonding thioxanes giving an overall tetragonal bipyramidal coordination around Pd. Intermolecular exchange of 1,4-thioxane on [Pd(1,4-thioxane) 4](BF 4) 2 in deuterated nitromethane and intramolecular interconversion in the bound ligand have been studied as a function of temperature and pressure by 1H NMR lineshape analysis. The first-order rate constant and activation parameters obtained for the intramolecular process, assigned to sulfur inversion, are as follows: k i 298=294±26 s −1, Δ H i ≠=61.9±1.9 kJ mol −1, Δ S i ≠=+10.2±6.5 J K −1 mol −1, Δ V i ≠=+2.7±0.2 cm 3 mol −1. The intermolecular process yielded the following second-order rate constant and activation parameters: k 2 298=920±40 m −1 s −1, Δ H 2 ≠=39.5±0.9 kJ mol −1, Δ S 2 ≠=−55.6±2.8 J K −1 mol −1, Δ V 2 ≠=−9.5±0.3 cm 3 mol −1. Second-order kinetics and negative values for the entropy and volume of activation indicate an associative Ia or A exchange mechanism. The reactivity of the analogous Pt 2+ compound is estimated using the proposed empirical linear correlation function Δ G ≠(Pt)=−32.1+1.92Δ G ≠(Pd). The calculated second-order rate constant and free energy of activation are 0.3 m −1 s −1 and 76 kJ mol −1, respectively. This relationship indicates also that the well established reactivity order Pd 2+≫Pt 2+ could be reversed for better π-accepting ligands than MeNC and that, in this case, five-coordinated species on Pt(II) are more stable than the Pd(II) analogue.

Modelling the evolution of correlation functions in gravitational clustering

Padmanabhan has suggested a model to relate the non-linear two-point correlation function to the linear two-point correlation function. In this paper, we extend this model in two directions: (1) by averaging over the initial Gaussian distribution of density contrasts, we estimate the spectral dependence of the scaling between nonlinear and linear correlation functions; (2) by using a physically motivated Ansatz, we generalize the model to N-ppoint correlation functions and relate the non-linear, volume averaged N-point correlation function with the linearly extrapolated volume averaged two-point correlation function evaluated at a different scale. We compare the point of transition between the different regimes obtained from our model with that from numerical simulations, and show that the spectral dependence of the scaling relations seen in the simulations can be easily understood. A comparison of the calculated form of with the simulations shows reasonable agreement. We discuss several implications of the results.

Modelling the non-linear gravitational clustering in the expanding Universe

The gravitational clustering of collisionless particles in an expanding universe is modelled using some simple physical ideas. I show that it is indeed possible to understand the nonlinear clustering in terms of three well defined regimes: (1) linear regime (2) quasilinear regime which is dominated by scale-invariant radial infall and (3) nonlinear regime dominated by nonradial motions and mergers. Modelling each of these regimes separately I show how the nonlinear two point correlation function can be related to the linear correlation function in heirarchical models. This analysis leads to results which are in good agreement with numerical simulations thereby providing an explanation for numerical results. The ideas presented here will also serve as a powerful anlytical tool to investigate nonlinear clustering in different models. Several implications of the result are discussed.

A computer simulation study of the disorder in ammonium perrhenate

Ammonium scheelites (of which ammonium perrhenate is an example) show several anomalies in their behaviour (including thermal contraction of the a lattice parameter, large thermal expansion of the c lattice parameter, enhanced nuclear quadrupolar relaxation, increased specific heat) which has been attributed to the onset of orientational disorder of the ammonium ions. Unlike in the ammonium halides there is no actual phase transition. Extensive molecular dynamics simulations of this material are reported. The model shows anomalies in the thermal expansion which are associated with a broadening and shift of the orientational distribution of the ammonium ions. There is no evidence of more than one preferred orientation. The ammonium ions also begin to jump between the twelve equivalent orientations, and we present evidence that this is primarily by rotation about one of the bonds. The activation energy for this agrees well with experiment. Angular and linear velocity-time correlation functions for the ammonium ion show rapid dephasing at higher temperatures, which is consistent with the observed broadening of the corresponding lines in the Raman spectrum. The origin of the anomalous thermal expansion is discussed in terms of various contributions to the entropy of the crystal.

Mode coupling in nonequilibrium granular flow systems

In this paper we examine the role of mode coupling in nonequilibrium granular flow systems and derive mode coupling equations for equal-time multi-linear correlation functions in general nonequilibrium systems. By applying N-ordering analysis and projection operator techniques, the hierarchy of equations obtained is simplified which permits the equal-time multilinear averages to be calculated systematically. The linear and equal-time bilinear nonequilibrium correlation functions are calculated explicitly for a steady-state system characterized by a small linear shear flow and uniform number and internal energy densities.

Conservation equations for hot carriers—II. Dynamic features

The general dynamic features of the velocity and energy balance equations are investigated. For the small-signal response an analytical theory based on the linear response correlation functions is developed. For the nonlinear and large-signal response the direct numerical solving of the balance equations has been used. For both responses good agreement with the Monte Carlo simulation of the small- and large-signal characteristics of hot electrons in n-InP are obtained. The eigenvalue spectrum for hot electrons in n-InP is analyzed using the analytical consideration of the balance equations. The minimum frequency of the intrinsic relaxation processes of the hot carrier system is shown to be about 5–10 THz at high electric fields E ⩾ 30 kV/cm. Possibilities of using of this high frequency in the high-speed devices are discussed.

Linearized cloudpoint curve correlation for ternary systems consisting of one polymer, one solvent and one non-solvent

A linear correlation function is found for cloudpoint composition curves of ternary systems consisting of one polymer, one solvent and one non-solvent. The conditions for validity of this correlation function appear to be that the polymer is strongly incompatible with the non-solvent, and that only liquid-liquid demixing occurs. The linearized cloudpoint (LCP) curve is interpreted in terms of the various parameters occurring in the Flory-Huggins theory. The slope of the LCP line appears to be only dependent on the molar volumes of the components. Information about the binary Flory-Huggins interaction parameters and their concentration dependence can be obtained from the intercept of the linearized curve. Cloudpoints induced by crystallization do not follow the correlation. This gives an opportunity to distinguish between crystallization and liquid-liquid demixing without any additional experiments.

Cascaded GMW sequences

Pseudorandom binary sequences with high linear complexity and low correlation function values are sought in many applications of modern communication systems. A new family of pseudorandom binary sequences, cascaded GMW sequences, is constructed. These sequences are shown to share many desirable correlation properties with the GMW sequences of B. Gordon, W.A. Mills, and L.R. Welch (1962)-for example, high-shifted autocorrelation values and, in many cases, three-valued cross-correlation values with m-sequences. It is shown, moreover, that in many cases the linear complexities of cascaded GMW sequences are far greater than those of GMW sequences.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stochastic trajectory models for turbulent diffusion: Monte Carlo process versus Markov chains

Abstract Turbulent diffusion of passive scalars and particles is often simulated with either a Monte Carlo process or a Markov chain. Knowledge of the velocity correlation generated by either of these stochastic trajectory models is essential to their application. The velocity correlation for Monte Carlo process and Markov chain was studied analytically and numerically. A general relationship was developed between the Lagrangian velocity correlation and the probability density function for the time steps in a Monte Carlo process. The velocity correlation was found to be independent of the fluid velocity probability density function, but to be related to the time-step probability density function. For a Monte Carlo process with a constant time step, the velocity correlation is a triangle function; and the integral time scale is equal to one-half of the time-step length. When the time step was chosen randomly with an exponential pdf distribution, the resulting velocity correlation was an exponential function. Other time-step probability density functions, such as a uniform distribution and a half-Gaussian distribution, were also tested. A Markov chain, which presumes one-step memory, has a piecewise linear velocity correlation function with a finite time step. For a Markov chain with a short time step, only an exponential velocity correlation function can be realized. Thus, a Monte Carlo process with random time steps is more versatile than a Markov chain. Direct numerical calculation of the velocity correlation verified the analytical results. A new model which combines the ideas of the Monte Carlo process and the Markov chain was developed. By examining the long-time mean square dispersion, we found an exact solution for the Lagrangian integral time scale of the new model in terms of the intercorrelation parameter and the mean and the variance of the time steps. Using this new model, we can generate any velocity correlation, including one with a negative tail. Two approximate solutions that give upper and lower bounds for the Lagrangian velocity correlation are proposed.

Reconstructing the primordial spectrum of fluctuations of the universe from the observed nonlinear clustering of galaxies

It was discovered that the nonlinear evolution of the two point correlation function in N-body experiments of galaxy clustering with Omega = 1 appears to be described to good approximation by a simple general formula. The underlying form of the formula is physically motivated, but its detailed representation is obtained empirically by fitting to N-body experiments. In this paper, the formula is presented along with an inverse formula which converts a final, nonlinear correlation function into the initial linear correlation function. The inverse formula is applied to observational data from the CfA, IRAs, and APM galaxy surveys, and the initial spectrum of fluctuations of the universe, if Omega = 1.

Dynamics of an ammonium ion in water: Molecular dynamics simulation

The translational and rotational dynamics of an aqueous ammonium ion are examined using an NVE molecular dynamics computer simulation and a rigid model of water. The linear momentum, angular momentum and reorientational autocorrelation functions of the ammonium ion have been measured. From these, the memory functions for the linear and angular momentum correlation functions are calculated. Reorientation of the ammonium ion about its center of mass shows unambiguous departure from Debye rotational behavior. The translational and rotational diffusion constants have been calculated.

A method of detection of the correlation function and frequency power spectrum for random noise or vibration with amplitude limitation

A method for determining the linear correlation function and the frequency power spectrum for random noise or vibration of which the amplitude fluctuation is limited by upper and/or lower levels is proposed, for an idealized case of a Gaussian random process. The standard method based on the definition of the linear correlation function and the FFT procedure has frequently been employed for linear correlation and spectrum analyses. These usual methods, however, cannot be adopted for actual noise or vibration signals with the above amplitude limitations. In this paper, a simplified method which can determine the correlation function through the conditional average is used. First, an explicit expression for the probability density function for random noise or vibration signals with amplitude limitations is introduced by using the well known Gaussian distribution and Dirac's delta functions. Next, by use of this expression, an original linear correlation function can be determined through the conditional average calculated by using the first and second order moments of this limited observed data. Then, the power spectrum latent in the actual noise or vibration signals with no amplitude limitations can be determined through Fourier transformation of the linear correlation function based on the well known Wiener-Khintchine theorem. Finally, the effectiveness of the proposed method is confirmed experimentally not only by means of the digital simulation technique but also by applying it to actual noise and vibration signals observed at a hydroelectric plant.

On a characteristic point at the base of the F 2 layer

The behaviour of the point of inflection of the electron density profile below the peak of the F 2 layer, considered as a “characteristic point”, is analyzed using a large number of profiles calculated from composite ionograms obtained at low and middle latitudes southern hemisphere stations. The diurnal, seasonal and latitudinal variations of the electron density, height and height gradient at that point are discussed. It is also shown that these parameters are linearly correlated with the electron density and height of the peak of the F 2 layer. Linear correlation functions between the coordinates of the point and the values of f 0F 2 and M(3000)F 2 read in the ionograms have been found. These functions can be used to calculate the electron density, height and height gradient of the “characteristic point” in the electron density profile, from routinely scaled ionospheric parameters.

Computer simulation and group theoretical statistical mechanics of liquid water and methyl chloride: Part 1, Neumann's Principle.

Neumann's Principle is applied to the molecular dynamics of liquids in the laboratory frame (X, Y, Z) and molecule fixed frame (x, y, z). In the absence of symmetry breaking influences, and in an isotropic medium of diffusing molecules, it states that the totally symmetric irreducible representation is the key to whether a thermodynamic ensemble average exists in either frame. In liquid water, with molecules of C 2ν symmetry, and liquid methyl chloride, with molecules of C 3ν symmetry, the Neumann Principle, evaluated with the help of group theory, is found to be in agreement with computer simulations of components of linear and angular momentum auto correlation functions in both frames.

Health effects of Chernobyl.

<b>2244</b> <h3><b>Objectives</b></h3> To estimate the error in ⁹⁰Y microsphere delivery vial exposure rate measurement, and assess its impact on the percent of activity delivered calculation. <h3><b>Methods</b></h3> Linear correlation analysis of background-corrected exposure rate (mR/h) at 30 cm from the vial (V), measured with a calibrated ion chamber survey meter, versus activity (GBq) from a dose calibrator calibrated for assaying ⁹⁰Y vials, was performed using the data from our first fifty-seven hepatocellular carcinoma (HCC) ⁹⁰Y microspheres liver brachytherapy treatments. Employing the mR/h computed from the linear correlation function as the correct value, the difference between measured and true (ΔV) was calculated. The resulting difference (Δ%D) in estimated percent of activity delivered (%D=100 × [V - residual waste jar mR/h] / V), using measured versus true V, was then computed. A p value ≤ 0.05 was considered statistically significant. <h3><b>Results</b></h3> Excellent correlation of mR/h vs. GBq was observed (y=1.682x, r=0.993, p&lt;0.0001), over a range of GBq from 0.51 to 6.98 and corresponding mR/h from 0.90 to 11.97. The ΔV was not significant (Mean %ΔV: -1.42±6.70, Range: -34.12 to 10.11, p=0.34). The corresponding Δ%D was also not significant (Mean: -0.06±0.27, Range: -1.43 to 0.73, p=0.094), over the range of %D (Mean: 96.8±6.4, Range: 64.0 to 100%). <h3><b>Conclusions</b></h3> Based on measured - fit residuals, the ⁹⁰Y microspheres vial activity mR/h "assay" appears to be within 10% of true. Since the estimated %D tends to be very high in HCC patients (e.g., 49 of 57 ≥ 95% and 53 of 57 ≥ 90% in this cohort), the equation approaches V/V most of the time, and thus the effect of an error in V is generally negligible. On the other hand, the one ΔV outlier (-34.1%) suggests a large error is possible. If the true %D was 80 for that treatment, the error would have resulted in a %D of 70 and thus an apparent misadministration (&gt; 20% difference from prescribed), suggesting that minimizing inaccuracy in V is prudent (e.g. by minimizing variation in the measurement geometry)

A simplified method for detecting information on linear correlation latent in random noise or vibration waves

It is well-known that linear correlation information is very useful in establishing the statistical properties of random noise or vibration waves. As the amount of data increases, however, calculating the linear correlation function by the usual multiplicative operations for large amounts of data can be very troublesome, and so some simplified practical methods of detecting the correlation function are desirable. In this paper, a practical method of detection of a linear correlation function latent in random noise or vibration waves is proposed, based on use of the conditional probability distribution. First, a general expression for the bivariate joint probability distribution for discrete level sampling is introduced in the form of a series expansion in orthonormal functions. The expansion coefficients give principally the information on linear and non-linear correlations, so that a general expression for the correlation function can be derived by conditional averaging. A new simplified version of this detection method is then presented, which does not require higher order statistical information. Finally, the effectiveness of the method is confirmed experimentally by applying it to some actual street noise in Hiroshima City.

Molecular dynamics of the rough sphere fluid. III. The dependence of translational and rotational motion on particle roughness

Molecular dynamics studies of two generalized forms of the rough sphere model are reported. The generalization permits us to vary the coupling between rotational and translational motion through a stick parameter λ. The autocorrelation functions show deviations from the Enskog theory for all values of λ studied—indicating that collective effects are involved. It appears that the angular velocity correlation function is the sum of two exponentials. A simulation of a mixture of disparate sized rough spheres was also performed. It is found that the rotational motion of the small particle is not significantly altered upon introduction of the large particles, but the linear velocity correlation function of the small particle differes considerably from that of the neat fluid (in the time regime from 5–15 mean collision time units).