Self-correlation Function Research Articles

Computer simulation of defect motion in model normal and ‘fast ion’ conductors II. SrCl 2

The molecular dynamics method has been applied to the simulation of the fast-ion conductor SrCl 2 . Calculations have been done at increasing temperatures from low temperatures through the normal and high-conductivity regions into the melt. Because of the periodic boundary conditions imposed and the size of the simulation sample (324 ions), the minimum defect concentration of anion Frenkel defects that can exist in the steady state is 1/216 defects per anion lattice site. At low temperatures Frenkel defects introduced into the initial configuration are eliminated, but at higher temperatures Frenkel defects form spontaneously on the anion sublattice. Anions move primarily in <1 0 0> directions by the vacancy mechanism, although more rarely <1 1 0> jumps also occur. At intermediate temperatures interstitials were observed to make non-collinear interstitialcy jumps, but at higher temperatures the motion of interstitials makes little contribution to mass transport. The vacancies appear to be very mobile and interstitials have little or no chance to move before being annihilated by a wandering vacancy. The picture is thus one of a highly dynamic system in which individual defects do not survive for long. However, it must be recognized that the rigid-ion potential used has its limitations and may overemphasize to some extent the vacancy mobility. There is so much motion on the anion sublattice at temperatures close to melting that various indicators such as the anion–anion radial distribution function, the individual ion displacements in a fixed time, and the anion self-correlation function are quite similar below and above the melting temperature. In this sense therefore, the applied description of sublattice melting does not seem to be inappropriate. However, periodic structure is still recognizable on the anion sublattice until it also disappears on the cation sublattice. The small sample molecular dynamics simulation does not predict defect clustering; this is probably because of limitations of the method and so our results are not necessarily in conflict with recent quasi-elastic neutron scattering results.

Computer simulation of defect motion in model normal and ‘fast-ion’ conductors I. NaCl and AgCl

The molecular dynamics simulation technique has been used to study Frenkel defects in crystalline NaCl and AgCl. Partial radial distribution functions, mean square displacements and the velocity autocorrelation function with its Fourier transform have been calculated for cation and anion species. Because the minimum Frenkel defect concentration that can be introduced into the molecular dynamics sample is 1/108, the initial configurations represent a non-equilibrium state and their evolution in time provides an opportunity to study mechanisms of defect migration that are involved in the recombination of Frenkel defects. Direct interstitial jumps, collinear interstitialcy migration, and [110] vacancy jumps have all been observed. The time dependence of the mean square displacements, and particularly the ratio of the square of second moment to the fourth moment of the self-correlation function, are sensitive indicators of the occurrence of defect jumps. Plots of the individual ion displacements in various time windows reveal which ions are in motion, while the projections of the cation and anion sublattices in the simulation cell give a detailed picture of the motion of the defects.

Time correlation function for a Cu-Mn spin glass

The self-correlation function xi (t) for a Cu-3 at.% Mn alloy has been determined with good statistical accuracy using conventional time-of-flight as well as the more sophisticated backscattering technique. Around the low-frequency (10 Hz) freezing temperature Tf the correlation function xi (t) shows a power-law decay, i.e. xi (t)=at-v with nu approximately 0.25 above Tf, but nu =0.1+or-0.02 and just below Tf. At lower temperatures (T/Tf approximately=0.75) xi (t) is best expressed as xi (t)=at-v+b log(t) and yields nu =0.4+or-0.1. At still lower temperatures (T<or approximately=0.5Tf) xi (t) can be expressed as xi (t)=c+at-v, with a value of nu which is consistent with nu approximately 0.5, predicted by recent mean-field theories as well as computer simulations. An important result of the measurements is that the mean correlation time tau 1e/ obeys the Vogel-Fulcher law suggesting a glass-like transition with Tg approximately=0.8 Tf(10 Hz).

Exact solution of continued fraction for tracer diffusion in solids

The continued fraction expansion of the self-correlation function is solved exactly for a lattice gas with hard-core interactions. Successive steps of the continued fraction provide correlation effects from the different jump steps of the tracer in the whole concentration range. The results are compared with the exact single-vacancy results of Pruthi (1981) in a square planar lattice.

A theory of self-correlation functions in simple liquids

A microscopic theory of the self-current correlation function is presented by deriving its memory function which consists of two parts. The first is the kinetic term whose time decay is expressed in terms of the velocity autocorrelation function and the self-intermediate scattering function. The second is the potential term which is expressed in terms of the interatomic potential, the radial distribution function, the self-intermediate scattering function and another quantity which describes collective effects in the liquid. For the wavenumber value k=0, this memory function is identical with the memory function of the velocity autocorrelation function which has been derived by Chiakwelu and Gaskell (1976). A preliminary calculation with liquid rubidium shows that the memory function is rapidly damped with increasing values of k. This latter behaviour is presumably due to the strong modulating effect of the self-intermediate scattering function.

Generalized theory of neutron scattering from hydrogen in metals

A recent analysis by the author of inelastic scattering of neutrons from dilute hydrogen in metals in terms of coherent itinerant-proton energy bands is generalized to include incoherent processes such as the spontaneous decay of the proton from excited-oscillator states to the ground state, as well as incoherent hopping among excited local-oscillator states centered at neighboring interstitial occupancy sites. Similarly, the analysis of Chudley and Elliott of quasielastic neutron scattering and its extension by Rowe, Sk\old, Flowtow, and Rush are generalized to include coherent hopping (band transport) in the self-correlation function describing motion of the proton among neighboring oscillator ground states (and, when applicable, among excited states). The general formalism developed here encompasses quasielastic and inelastic scattering and allows for the coexistence of coherent and incoherent processes. At each level of complexity, the expressions obtained for the cross sections are shown to reduce to earlier results in the limits when either the coherent or the incoherent contributions to the neutron bandwidths can be ignored.

Van hove self-correlation function of a hard-disk fluid: Enskog theory and computer simulation

The van Hove self-correlation function in a hard-disk fluid is analyzed using the Lorentz-Enskog kinetic equation and the kinetic model method of solution. Numerical convergence of the model solutions is demonstrated and accurate model results are used to interpret molecular dynamics simulation data at finite wave numbers. It is found that at about 60% of freezing density the error in the Enskog theory can be mainly attributed to an underestimate of the effective self-diffusion coefficient, but at 90% freezing density a theory which treats correlated collisions is needed to describe the width behavior of the singleparticle density fluctuation spectrum.

Dynamic Theory of Spin Glasses

Here is presented a dynamic theory of spin glasses based on the extended mean-field theory of Thouless, Anderson, and Palmer. The kinetic equations for the magnetization and the spin correlation function are derived, from which dynamical linear and nonlinear susceptibilities are calculated and their low-frequency behavior is mainly investigated. The spin self-correlation functions have different relaxation times near above and below T c –the former is proportional to | T - T c | -2 , while the latter to | T - T c | -4 –which shows the asymmetric critical slowing down. A qualitative comparison of the present results with experimental data is presented.

Computer Simulation of Molecular Dynamics of Anisotropic Fluids

Abstract Molecular dynamics calculations have been performed for fluid systems containing ellipsoidal particles which interact pair-wise with a modified Lennard-Jones potential. Assuming molecular parameters close to those of the CB7 compound, we have calculated both static and dynamic properties such as the internal energy, the entropy and specific heat changes at T NI, the compressibility factor, the temperature dependence of the orientational order parameters, the reorientational angular momentum and velocity time-dependent self-correlation functions, and the diffusion coefficients in both the nematic and isotropic phases. The thermodynamical data show a transition between isotropic and nematic phases and the results obtained are in qualitative agreement with the experimental data. The influence of a decenterd dipole along the molecular axis has been studied and partially bilayered smectic phases have been obtained.

Evaluation of diffusion broadened Mössbauer spectra in single crystals of FCC metals

The self-correlation functions for long-range diffusion via the monovacancy mechanism in FCC lattices have been calculated within the encounter approximation varying parameters of the five-frequency model. The validity of this approach to the existing single crystal Mossbauer data has been discussed. The so-called high temperature limit has been outlined and its applicability considered. A simple method for calculating the experimental solid angle correction has been introduced for diffusion-broadened Mossbauer spectra from the single crystal sources.

Velocity autocorrelation function of a 2D classical electron liquid

Expressions for the first six frequency moments of the self-correlation function have been obtained appropriate for a 2D classical electron liquid interacting with a 1/r potential. The results of the moments and the Mori-Zwanzig formalism have been used to analyse the recent molecular dynamics data for the velocity autocorrelation function. It is found that a phenomenological form of the relaxation kernel with two relaxation times provides a good agreement with the molecular dynamics data.

Orientational disorder in plastic molecular crystals - III. — Infrared and Raman spectroscopy of internal modes

The formalism of symmetry adapted functions for molecular orientations introduced previously [5, 6] is here applied to the analysis of the Raman and infrared spectroscopy of internal modes in orientationally disordered molecular crystals. In some favorable cases, the contribution to the Raman and infrared lineshapes arising from the rotational dynamics of the molecules can be disentangled from other contributions arising from the vibrational lifetime or various coupling effects. Here, the rotational lineshapes are analysed in terms of independent, symmetry adapted rotational self-correlation functions. Furthermore, it is shown that the integrated intensity of internal modes provides a measurement of the first symmetry independent coefficients in the development of the orientational probability density function.

Random walk on a random walk

The authors investigate the random walk of a particle on a one-dimensional chain which has been constructed by a random-walk procedure. Exact expressions are given for the mean-square displacement and the fourth moment after n steps. The probability density after n steps is derived in the saddle-point approximation, for large n. These quantities have also been studied by numerical simulation. The extension to continuous time has been made where the particle jumps according to a Poisson process. The exact solution for the self-correlation function has been obtained in the Fourier and Laplace domain. The resulting frequency-dependent diffusion coefficient and incoherent dynamical structure factor have been discussed. The model of random walk on a random walk is applied to self-diffusion in the concentrated one-dimensional lattice gas where the correct asymptotic behavior is found.

Theory of the effective Debye–Waller factor in neutron scattering from high frequency molecular modes

A theory is given of the incoherent scattering cross section from high frequency intramolecular vibrational modes (ωλ) in a molecular crystal. It is assumed that ωλ≫ωD, where ωD is the maxium frequency of the low frequency lattice phonon continuum. It is shown that the scattering from a ωλ mode is proportional to a Debye–Waller factor arising only only from the internal molecular modes, multiplied by Ss(K,ω−ωλ), where Ss(K, ω) is the self-correlation function of the molecular lattice. In the case when the Debye–Waller factor due to the lattice phonons is negligibly small, we show that Ss(K, ω−ωλ) is a Gaussian in ω−ωλ with Doppler broadening due to translational and rotational motion. Our theory is a development of the short-time approximation and explains the unexpectedly strong intensity of the C–H stretching and bending modes recently observed by Jobic, Ghosh, and Renouprez in hexamethylenetetramine (HMT). The present paper gives the theoretical basis for the quantitative interpretation of the high energy intramolecular vibrational spectrum measured by inelastic neutron scattering.

Diffusion in concentrated lattice gases. Self-diffusion of noninteracting particles in three-dimensional lattices

The dynamical process of the diffusion of tagged particles in concentrated lattice gases is investigated. The lattice is fcc and the particles are noninteracting except that double occupancy is forbidden. The self-diffusion coefficient and the waiting-time distributions for two consecutive jumps of a tagged particle are calculated numerically by Monte Carlo methods. The time-dependent correlations between consecutive jumps are analyzed in detail, especially the initially increased rate for backward jumps of the tagged particle. The self-correlation function of a diffusing particle is also calculated by a correlated-jump model, which represents a generalization of the usual treatment of self-diffusion to frequency dependence and to finite concentrations. The resulting diffusion coefficient agrees well with the direct numerical calculations. Some other analytical predictions are also tested. The incoherent dynamical scattering function shows the effect of correlations at small frequencies and mean-field behavior at high frequencies.

Two-state random walk model of lattice diffusion. 1. Self-correlation function

Diffusion with interruptions (arising from localized oscillations, or traps, or mixing between jump diffusion and fluid-like diffusion, etc.) is a very general phenomenon. Its manifestations range from superionic conductance to the behaviour of hydrogen in metals. Based on a continuous-time random walk approach, we present a comprehensive two-state random walk model for the diffusion of a particle on a lattice, incorporating arbitrary holding-time distributions for both localized residence at the sites and inter-site flights, and also the correct first-waiting-time distributions. A synthesis is thus achieved of the two extremes of jump diffusion (zero flight time) and fluid-like diffusion (zero residence time). Various earlier models emerge as special cases of our theory. Among the noteworthy results obtained are: closed-form solutions (ind dimensions, and with arbitrary directional bias) for temporally uncorrelated jump diffusion and for the ‘fluid diffusion’ counterpart; a compact, general formula for the mean square displacement; the effects of a continuous spectrum of time scales in the holding-time distributions, etc. The dynamic mobility and the structure factor for ‘oscillatory diffusion’ are taken up in part 2.

Dynamic Light Scattering Study of Concentrated W/O Microemulsions

Light beating spectroscopy experiments were performed on concentrated W/O microemulsions located in the vicinity of the gel phase on the phase diagram. New results are reported: the most interesting feature is the non-exponential behaviour of the self-correlation function g (1)(K, τ) of the scattered field. The initial decay rate of g (1)(K, τ) is not directly proportional to K 2; whereas the long time decay is found fairly exponential.

Calculations of correlation functions for two-dimensional square ice

Two approximation methods, the Monte Carlo method and the series expansion method, are used to calculate correlation functions, including the Kirkwood correlation function g, in two-dimensional square ice. The approximation methods are tested on the exact values of the polarization correlation function G and the self-correlation function g (o). Both approximate methods are reasonably precise and both methods give values of G and g (o) in good agreement with the exact values. Using both methods we estimate that g = 1.5 ± 0.3. Just as for three-dimensional ice this value of g is significantly smaller than G, thus supporting the assumption of dimensional similarity in the behavior of the correlation functions. Our results strongly indicate that g is a discontinuous function in the ideal ice limit as the Bjerrum fault concentration is varied. From these results, previous results for three-dimensional ice and general considerations relating g and G it is concluded that for real ice in three dimensions with a non-zero concentration of Bjerrum faults the values of g and G are both close to three.

Random walk model with correlated jumps: Self-correlation function and frequency dependent diffusion coefficient

Random walk model with correlated jumps: Self-correlation function and frequency dependent diffusion coefficient

Self-diffusion beyond Fick's law

The Van Hove self-correlation function, the intermediate incoherent scattering function and its Laplace transform are determined asymptotically for a one component fluid in equilibrium, using the mode coupling theory. The results reproduce in the hydrodynamic limit the predictions from Fick's law. The corrections to Fick's law are consistent with a long time tail in the velocity correlation function and with a diverging super Burnett coefficient in the linear diffusion equation.