Showing papers on "Correlation function (statistical mechanics) published in 1973"

Investigation of non-ideal two-phase polymer structures by small-angle X-ray scattering

Abstract: The mean square of the electron-density gradient, 〈|grad η|2〉 in isotropic structures is shown to be proportional to the fourth moment of the SAXS intensity distribution in reciprocal space ∫s4I(s)ds, as well as to the second derivative of the correlation function in the origin. In the case of two-phase structures with unsharp phase boundaries, these relations may be used to find the thickness E of the transition regions. As was shown by Ruland [J. Appl. Cryst. (1971). 4, 70–73] E can also be determined by analysis of the intensity in the tail of the SAXS pattern. This approach is used here to investigate the effect of E on the one-dimensional correlation function. In the application of both methods, separation of the SAXS intensity from the continuous background of liquid scattering constitutes a critical step. A procedure in which the background is represented by a curve of the type a + bsn, where n is an even number, is found to work well for a number of polymers.

Short-Range Correlations and Electron-Gas Response Functions

Abstract: The behavior of interacting electrons is dominated by the Coulomb repulsion in the limit of zero interparticle separation. Hence, in this limit the derivative of the two-particle correlation function is equal to the two-particle correlation function itself divided by the Bohr radius. Because correlation functions can be obtained from linear-response functions, the relation between the correlation function and its derivative can be used as a criterion for the validity of approximate linear-response functions. Some of the more familiar dielectric functions and spin susceptibilities are discussed in this context. For these response functions it is shown that the electric and magnetic response should approach the same ($q\ensuremath{\rightarrow}\ensuremath{\infty}$) asymptotic limit.

Dynamics of Fluids near the Critical Point: Decay Rate of Order-Parameter Fluctuations

Abstract: The decay rate of the order-parameter fluctuations in fluids near the critical point can be determined by measuring the linewidth of the central component in the spectrum of the scattered light, and many such experiments have been reported in the past eight years. In the past two years the dynamical theories developed by Kawasaki and Ferrell to describe the decay rate have been modificed to take into account the anomaly in the shear viscosity and the effect of departures of the correlation function from the Ornstein-Zernike form, and at the same time new accurate measurements of the parameters which enter the theory (the correlation length ξ and the shear viscosity ηs) have been reported. Thus we are able to present here an absolute comparison of the refined mode-mode coupling and decoupled-mode theories with the linewidth data for seven fluids: carbon dioxide, xenon, sulfur hexafluoride, isobutyric acid-water, 3 methylpentane-nitroethane, aniline-cyclohexane, and 2,6 lutidine-water. These linewidth data were obtained in many different laboratories over the past few years; however, in addition to linewidth data previously reported, we also include in our analysis new data which we have obtained for carbon dioxide and xenon and a tabulation and detailed error analysis for all our data for these two fluids. The theories describe only the "critical part" of the decay rate ΓC; hence the nonanomalous background contributions are first substracted from the measured linewidths to obtained ΓC. Then it is shown that the resultant values for a quantity we call the "scaled linewidth," Γ*≡(6πηsΓCkBTq3), are described by a single universal curve as a function of qξ, for all fluids and every thermodynamic path that has been investigated near the critical point. This universal curve is described remarkably well by the modified mode-mode-coupling expression of Kawasaki and Lo and the similar decoupled-mode-theory expression of Perl and Ferrell. The accuracy of this comparison, which involves no adjustable parameters, is limited to ∼10% by the uncertainties in the background corrections, linewidths, viscosities, and correlation lengths, and by the uncertainties in the various modifications to the theories. The two theories differ significantly only in the extreme nonhydrodynamic region (qξ≫1), where the decoupled-mode values for Γ* are ∼10% smaller than those predicted by the mode-mode-coupling theory. Although the available data in the extreme nonhydrodynamic region appear to be described somewhat better by the decoupled-mode theory than the mode-mode-coupling theory, this result is suggestive rather than conclusive since the data in this region are sparse and exhibit considerable scatter.

Exact solution of the mean spherical model for charged hard spheres in a uniform neutralizing background

Abstract: The exact solution of the mean spherical model integral equation is found for a system of charged hard spheres in a uniform neutralizing background. This may be considered as a simple example of a fluid with nonadditive hard sphere diameters. Analytic expressions are given for the direct correlation function and for the Laplace transform of the radial distribution function. These, and the thermodynamic properties of the system, are compared with previous solutions of the mean spherical model.

Approach to Scaling in Renormalized Perturbation Theory

Abstract: The consequences of the large magnitude of the bare coupling constant in Wilson's theory of critical phenomena are examined for renormalized field theory in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The scaling behavior of the correlation functions, the relations among critical exponents, and the existence of a scaling equation of state, regular in the temperature around ${T}_{c}$, are then obtained in this framework. Some corrections to the scaling laws are also discussed and are shown to be dependent on another exponent.

Theory of Spin-Dependent Phonon Raman Scattering in Magnetic Crystals

Abstract: A general theory of the spin-dependent phonon Raman scattering in magnetic crystals is developed. The integrated Raman intensity as a function of temperature is expressed as S ( T )=( n 0 +1)| R + M | 2 , where the first term represents the spin-independent part and the second one the spin-dependent part which is proportional to the nearest neighbor spin correlation function . Three types of temperature variation of intensity are predicted in ferromagnetic and antiferromagnetic cases, depending on values of R / M . Two microscopic mechanisms, variation of the d electron transfer with lattice vibrations and that of the non-diagonal exchange, contribute to the spin-dependent part, and it is shown that the former is essentially important in producing the large temperature variation. The theory is applied to explaining the observed temperature variation of various Raman active phonon modes in CdCr 2 S 4 and the experimental results are explained successfully.

Rayleigh scattering: Orientational motion in highly anisotropic liquids

Abstract: Light scattering studies of the depolarized Rayleigh line are reported for liquid benzene over the temperature range +13 to 76°C and for other substituted benzenes. Values of the 〈 P2(o)P2(t)〉 and 〈 Ṗ2(o)Ṗ2(t)〉 correlation functions calculated from these spectra show considerable free rotor behavior (∼ 15° steps) and incomplete loss of memory after collision. Further consideration of the 〈 Ṗ2(o)Ṗ2(t)〉 correlation allows calculation of the angular velocity correlation function. The time between collisions, τBC, is also obtained and found to be in good agreement with a ``moveable wall'' cell model. It is proposed that the ``shoulder'' noted in the wings of many anisotropic liquids is caused by incomplete loss of memory after collision.

Rotational relaxation of macromolecules determined by dynamic light scattering. II. Temperature dependence for DNA

Abstract: Correlation functions have been determined for the fluctuating intensity of the depolarized component of forward-scattered laser light from solutions of DNA. The molecular correlation function of calf thymus DNA (mol wt ∼15 × 106) appears to exhibit a longest relaxation time (τ25,w, ∼ 18 msec) close to what one would predict from the flowdichroism measurements of Callis and Davidson and, in addition, manifests a spectrum of faster times down to tenths of milliseconds. Furthermore, a major fraction of the amplitude of fluctuations in the angular distribution of segment axes is relaxed on a very much shorter time scale (of the order of 20 microseconds) that appears to be relatively insensitive to molecular weight of the DNA, or to near-melting temperatures. The temperature profile of the longest relaxation time has been obtained and found to exhibit a peculiar spike near Tm, which, together with the absence of a corresponding spike in the (high shear) viscosity, has been interpreted as indicative of an increase in the molecular weight of the DNA in a narrow temperature region near Tm. Correlation functions for polarized light scattered at finite angles were obtained in an attempt to determine the temperature dependence of the translational diffusion coefficient. Although the data contain an extremely slow component that does not admit a simple interpretation, there is some indication of a decrease in the translational diffusion coefficient near Tm, thus supporting the notion of an aggregation occurring near Tm. Finally, a “counterion escape” mechanisn is proposed for the apparent aggregation.

Rotational relaxation of macromolecules determined by dynamic light scattering. I. Tobacco mosaic virus

Abstract: The intensity autocorrelation function for the depolarized component of forward-scattered light from a solution of large polymeric molecules is derived in terms of the correlation function for the amplitudes of the Y2,±1(θ,ϕ) fluctuations in the anglar distribution of segments in the solution without any assumptions regarding the statistical properties of the scatterad light field. Effects arising from the use of polychromatic incident light and from the mixing of the scattered and polychromatic incident light beams are examined in detail. Apparatus for observing the depolarized forward-scattered light, digitizing and storing the fluctuating phototube current at rates from 10 to 540,000 times per second, and computing the correlation functions directly in the time-domain is described herein. Correlation functions were obtained for 0.05 mg/ml solution of tobacco mosaic virus at pH 9.1 and also at pH 6. The degree of association of the virus appears to be independent of pH, and the monomer relaxation times (corrected to 25°C) extracted from the data by a least-squares procedure lie in the range 0.44–0.49 msec, also independent of pH. The absence of faster component in the correlation function between 6 μsec and 0.5 msec is used in conjunction with thermal fluctuation theory to infer a lower limit for the effective Young's modulaus of the rod, E ≤ 2.5 × 107 dynes/cm2.

Decrease properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems

Abstract: We present and discuss some physical hypotheses on the decrease of truncated correlation functions and we show that they imply the analyticity of the thermodynamic limits of the pressure and of all correlation functions with respect to the reciprocal temperature β and the magnetic fieldh (or the chemical potential μ) at all (real) points (β0,h0) (or (β0, μ0)) where they are supposed to hold. A decrease close to our hypotheses is derived in certain particular situations at the end.

Two frequency radar interferometry applied to the measurement of ocean wave height

Abstract: A technique is developed for measuring the rms wave height averaged over an area of the sea that is much greater than any horizontal scale of the surface waves. The method involves a nadir looking radar which transmits and receives two monochromatic signals simultaneously. Signal processing at the receiver involves the computation of the correlation between the two returning signals as a function of theft variable frequency separation. The cross correlation between the amplitude and phase functions of the individual returning carriers depends on the distribution of discrete scatterers along the direction of propagation. This information can be used to determine the rms surface elevation (about the mean); it does not depend on the temporal or spatial frequency spectrum of the wave height or slope. Under conditions which are typical for a microwave signal being normally incident and reflected by the sea, the two frequency correlation function R(\Delta k) is seen to be equal to the characteristic function of the surface elevation of specular points. Laboratory measurements have been conducted on wind driven waves, and the measured correlation function compares favorably with the theory developed here.

Spatial structure and thermodynamic properties of a classical fluid of hard spheres with attractive square wells

Abstract: The spatial structure and the thermodynamic properties of a classical fluid of hard spheres with attractive square wells are analyzed, based on the Percus‐Yevick equation reformulated by Baxter. The equation of state, the isothermal compressibility, the internal energy, the direct correlation function, the radial distribution function, and the structure factor are evaluated over a wide range of temperatures and densities when the width of the attractive square well is half the diameter of the hard core. The temperature‐ and the density‐dependent behaviors of the obtained quantities are discussed. The equation of state is compared with theoretical results obtained by other authors. The pressures calculated via the compressibility equation are in satisfactory agreement with the pressures obtained from computer experiments both at the supercritical temperature kT/e=3.333 and at the subcritical temperature kT/e=1.17. Attempts are made to fit the pressures calculated via the compressibility equation to the exp...

Higher-Order Correlation Properties of a Laser Beam

Abstract: The intrinsic third-order intensity correlation function of a laser beam has been measured by a digital correlation technique, for a He: Ne laser operating near threshold. The results are compared with some recent calculations of Cantrell, Lax, and Smith and are found to be in good agreement.

Dielectric Response Function of Electron Liquids. II Static Properties

Abstract: The static properties of the dielectric response function for an electron liquid, obtained previously by one of the authors, is investigated. It is shown that the resulting pair correlation function has the correct long-range and short-range behaviors so that the correlation energy calculated therefrom reproduces the known exact values to the order of e2, when an expansion with respect to the plasma parameter E is carried out. The compressibility sum rule is satisfied to the order of e2 ln E. It is found that within the accuracy stated above the short-range behavior of the pair correlation function is not ,sensitive to the form of the ternary correlation function.

Raman band shape of the N2 molecule dissolved in liquids

Abstract: We have studied the vibrational Raman band of the N2 molecule dissolved in various solvents. The band shape of isotropic and anisotropic scattering is discussed and compared with the one obtained from N2 gas. A collisional narrowing effect has been observed for the isotropic scattering. For N2 dissolved in SF6 the rotational correlation function is analyzed.

Rigorous Bounds for Time-Dependent Correlation Functions

Abstract: Rigorous upper and lower bounds are determined for time-dependent correlation functions, of the type used in statistical mechanics and spectroscopy. The input data are the values of any finite number of initial time derivatives of the correlation function. As an example, bounds are found for the classical velocity correlation function for a lattice vibration problem. The bounds are found to be much more accurate than the Taylor series based on the same time derivatives.

On force-force correlation functions and the resistivity

Abstract: A method of calculating the resistivity of disordered systems by a force- force correlation function is criticized as being inexact This correlation function is the zero-frequency limit of a certain response function directly related to the conductivity, and should vanish in the zero-frequency limit However at least in the case of a dilute system of scatterers, an approximate treatment gives the conventional result for the resistivity An attempt is made to analyse why this should be so

Molecular theory of the translational Stokes-Einstein relation

Abstract: A molecular‐statistical derivation is presented for the Stokes‐Einstein expression for the translational diffusion constant for a Brownian (large) particle. The friction constant is expressed in terms of a correlation function of the forces on the particle, and this correlation function is related to the correlation function of pressure fluctuations in a volume equal to that of the particle. Whereas the forces are those on the particle surface, the pressure fluctuations are a property of the liquid solvent. The amplitude of the pressure fluctuations can be obtained from fluctuation theory and the time dependence can be obtained from hydrodynamics, which in turn can be derived from molecular‐statistical considerations. Although the exact numerical factors cannot be obtained, the linear dependence of the friction constant on the radius of the Brownian particle is derived. Finally, a discussion of the diffusion constant for small diffusing particles is presented.

Light scattering from independent particles-nonGaussian correction to the clipped intensity correlation function

Abstract: The authors consider Rayleigh scattering of laser light from an assembly of independent particles. They allow the particle number M within the scattering volume to fluctuate around a certain mean value (M). They derive a general formula for the intensity moments and find the nonGaussian correction to the single-clipped-at-zero photocount correlation function to order 1/(M)2.

Field Theory Description of Continuous Phase Transitions

Abstract: We present a formalism of a scalar, classical, and time-independent field theory of the type proposed by Ferrell for the treatment of continuous phase transitions. The formalism is developed along lines similar to those of many-body theory. All physical quantities, e.g., susceptibility, correlation length, and free energy, are expressed as functionals of the two-point time-independent correlation function and the order parameter. This is done both in the ordered and in the disordered phase. We obtain renormalized equations and diagram expansions of all quantities and self-consistent approximation schemes arc presented. It is shown that near the transition temperature, which is defined within the theory, no weak coupling limit exists. The generalization to more complicated field symmetries is straight-forward.

Fluctuations in guiding center plasma in two dimensions

Abstract: Statistical mechanics is developed for a two‐dimensional guiding center plasma. Because there is no kinetic energy associated with guiding center motion, this development is unconventional. Thermal equilibrium is discussed, and an interesting limiting case is noted. Using the random phase assumption, a kinetic equation for the density fluctuations is obtained, which has thermal equilibrium and its limiting form as the only stationary states. However, despite the phase averaging this kinetic equation is reversible and when disturbed the system oscillates about equilibrium. Similar oscillatory behavior appears in the microscopic correlation function of the fluctuations, and the oscillation frequencies are obtained explicitly; however, these oscillations do not significantly change the macroscopic diffusion coefficient derived earlier by McNamara and Taylor.

Scattering of electromagnetic waves by rough perfectly conducting circular cylinders

Abstract: A numerical method is proposed for computing the normalized correlation functions of the real and imaginary parts of the field scattered from a statistically rough perfectly conducting circular cylinder. The deviation of the surface from its mean radius is assumed to be small. The correlation function of the far-field is related to the correlation function of the scattering object by an integral equation. Far-field correlation functions are found for two types of surface correlation functions: the delta function and a periodic exponential function.

Ion diffusion and momentum correlation functions in dilute electrolytes

Abstract: The convergent cluster expansion of the ion-ion momentum time correlation functions in a dilute electrolyte, recently derived by Friedman, is applied here for the calculation of the limiting law for ion self-diffusion. Considering only the non-chain cluster terms leads to a result analogous to the Debye-Huckel limiting law for ionic conductivity. Taking into account the additional chain terms leads to exactly the Onsager result. We also use the cluster expansion to determine the long-time behaviour of the ion momentum time auto-correlation function, which we find goes as t -3/2, with a coefficient proportional to the square root of the solute concentration. Similar results are given for the associated memory function and the ion-ion momentum time correlation function.