Ornstein-Zernike Form Research Articles

Spin correlations and Haldane gap in the S = 2 antiferromagnetic Heisenberg chain

Haldane's conjecture for the S = 2 antiferromagnetic Heisenberg chain is investigated with a quantum Monte Carlro method. The ground-state spin-spin correlation function and an excitation gap immediately above the ground state are calculated for chains up to 512 spins. The large-separation correlations are well fitted to the two-dimensional Ornstein-Zernike form. The correlation length and the Haldane gap in the unit of the exchange constant are estimated to be 61 −6 +8 and 0.074 ± 0.016, respectively.

The effect of shear-flow on critical correlations in colloidal systems: Microstructure, turbidity, and dichroism

On the basis of the Smoluchowski equation, ‘‘the Liouville equation for colloidal systems,’’ shear flow distortion effects on the structure factor for temperatures and densities close to the spinodal, c.q. the critical point, are considered. The results are valid in the classical region where the equilibrium structure factor attains the Ornstein–Zernike form. From the structure factor distortion we derive scaling relations for the shear-rate dependent turbidity and flow dichroism for various flow geometries. The dimensionless group which expresses the effect of the shear flow is essentially λ=Pe0(γ̇)/(ξRV)4, with Pe0(γ̇)=γ̇R2V/2D0 (γ̇ is the shear rate, RV the range of the pair-interaction potential, ξ−1 the correlation length, D0 is the Stokes–Einstein diffusion coefficient). As a consequence, very small shear rates (small values of Pe0) have very large effects close to the spinodal, c.q. the critical point, since then ξRV is a small number. Flow dichroism can be many decades larger than for systems far into the stable region of the phase diagram. Relaxation of turbidity and flow dichroism as a shear flow is turned off is also considered. The temperature, density, shear rate, and time dependence of the relaxation is described by scaling functions depending on the two dimensionless groups λ and γ̇t (t is the time lapse after switching off the shear flow).

Mean-field Ising crossover and the critical exponents gamma, nu, and eta for a polymer blend: d-PB/PS studied by small-angle neutron scattering.

The critical behavior of the polymer blend d-PB/PS was investigated by small-angle neutron scattering experiments. 3D Ising behavior was clearly observed with the critical exponents \ensuremath{\gamma}=1.26\ifmmode\pm\else\textpm\fi{}0.01, \ensuremath{\nu}=0.59\ifmmode\pm\else\textpm\fi{}0.01, and \ensuremath{\eta}=0.047\ifmmode\pm\else\textpm\fi{}0.004. The crossover to mean-field behavior occurs at ${\mathit{T}}^{\mathrm{*}}$=${\mathit{T}}_{\mathit{c}}$+5.4 K. This is compared with the results of other experiments and the Landau-Ginzburg criterion. The Q dependence of the structure factor S(Q) follows the Ornstein-Zernike form in both regimes.

Angular intensity of nonequilibrium interfacial dynamic light scattering: Succinonitrile and naphthalene.

We have investigated the phenomenon of intense dynamic light scattering at the nonequilibrium crystal-melt interface in succinonitrile and naphthalene, in order to resolve the ongoing controversy over its origin. Of the several models that have been proposed to explain this phenomenon, the microbubble model of H. Z. Cummins et al. [Solid State Commun. 60, 857 (1986)] and the mesophase model proposed by J. Bilgram and co-workers [P. Boni, J. H. Bilgram, and W. Kanzig, Phys. Rev. A 28, 2953 (1983)] are the only two still considered to be consistent with most of the experimental observations. In these experiments the angular dependence of the scattered light was investigated. In the mesophase model the angular dependence of the scattered light is described by the Ornstein-Zernike form I(q)=${\mathit{I}}_{0}$(1+${\mathit{q}}^{2}$${\ensuremath{\xi}}^{2}$${)}^{\mathrm{\ensuremath{-}}1}$, whereas light scattered by bubbles can be modeled by the Mie scattering theory. The data for both materials were found to be incompatible with the Ornstein-Zernike form, but could be reasonably well fit by the Mie theory. The behavior of the onset of scattering was also investigated, and it was found that the product ${\mathit{R}}_{0}$${\mathit{t}}_{0}$${\mathit{v}}_{\mathit{g}}^{2}$ was a constant, where ${\mathit{R}}_{0}$ is the onset radius, ${\mathit{t}}_{0}$ the onset time, and ${\mathit{v}}_{\mathit{g}}$ the crystal growth velocity. This result is consistent with the analysis of Mesquita et al. [Phys. Rev. B 38, 1550 (1988)], in which the onset of the scattering was modeled by considering the rate of buildup of dissolved gas at the advancing crystal-melt interface. The time taken for the disappearance of the scattering after growth was terminated was also investigated. Lastly, the gases dissolved in our samples of succinonitrile were identified by mass spectroscopy and found to have a composition similar to air.

Small angle scattering on reentrant spin glass Fe 1− xAl x

Small Angle Neutron Scattering (SANS) on a Reentrant Spin Glass (RSG) Fe 1− x Al x was measured in a very wide Q range from 0.008 to 1.0 Å −1, which gives semi-macroscopic and local structural information simultaneously. The scattering profiles at small Q region is well described with Ornstein-Zernike form representing thermal magnetic fluctuation, while in the high Q region Porod law is well applicable, suggesting the existence of clusters 30 Å diameter. The clusters appear much above T g, dividing the ferromagnetic long-range order into small segments. We found that the RSG transition is induced by a cooperative freezing of the cluster spins. The segmentation of the ferromagnetic order may be due to possible surface fractal behavior represented by Q −3 dependence at the medium Q region.

Light-scattering study of dynamical behavior of antiferromagnetic spins in the layered magnetic semiconductor FePS3.

Critical phenomena were studied by Raman scattering in the two-dimensional Ising-type antiferromagnet ${\mathrm{FePS}}_{3}$. Below ${\mathit{T}}_{\mathit{N}}$=118 K the Brillouin-zone-boundary phonon Raman peaks were observed because of the formation of a magnetic superstructure. Above and below ${\mathit{T}}_{\mathit{N}}$ a low-frequency broad band was observed, and it was interpreted in terms of Raman scattering from transverse acoustic (TA) phonons throughout the Brillouin zone, which was induced by the spin disorder. The spectrum reflects the two-dimensional one-phonon density of states of the TA phonons. The temperature dependence of the Raman intensity shows a maximum at ${\mathit{T}}_{\mathit{N}}$ and agrees with the Ornstein-Zernike form. A quasielastic component due to the magnetic critical scattering was first observed by light scattering. The intensity tends to diverge at ${\mathit{T}}_{\mathit{N}}$. As soon as the spin-disorder-induced Raman spectrum was quenched with decreasing temperature, a one-magnon Raman spectrum began to emerge. The Raman process was discussed using a phenomenological spin-dependent Raman theory.

Evidence of large cluster aggregates in potassium oleate microemulsion by elastic light scattering measurements

Quasielastic light scattering measurements give evidence that in concentrated microemulsions of water in oil a strong interaction among the constitutive droplets is present. This interaction is attractive and therefore can originate aggregation processes. In the frame of the Ornstein-Zernike form the analysis of the scattered intensity profiles furnishes the dimension of such aggregates (∼ 2800 Å).

Effects of isotopic substitution on the critical behavior of aqueous solutions of n-dodecylhexaoxyethylene glycol monoether (C12E6)

The critical behavior of a nonionic surfactant, n-dodecylhexaoxyethylene glycol monoether (C12 E6) in D2O, is examined using static and dynamic light scattering and small angle neutron scattering (SANS). Modification of the short-range attractive interactions responsible for phase separation by substitution of D2O for H2O has no effect on the measured universal critical exponents describing the divergence of the correlation length (ν=0.6±0.05) and the isothermal compressibility (γ=1.2±0.1) although such substitution results in a reduction of both the nonuniversal amplitudes and critical temperatures. The measured exponents are the same as those of binary molecular fluids. However, in contrast to binary fluids, significant deviations from the Ornstein–Zernike form are observed at intermediate length scales in SANS measurements near the critical point. Isotopic substitution also preserves the dynamic scaling of the Kawasaki form for the measured Rayleigh linewidth, as found previously for H2 O (Ref. 1). However, this scaling is restricted to a more limited range of distance from the critical point than the static scaling. A maximum in compressibility at the critical isochore is reflected in the concentration dependence of both the compressibility and diffusion coefficient even large distances (∼25 °C) from the coexistence curve. Estimates of micelle size, shape, and polydispersity are obtained.

Experimental Study of New Type Phase Transition in Triangular Lattice Antiferromagnet VCl2

Critical properties of a Heisenberg antiferromagnet on a triangular lattice are investigated experimentally using a quasi-two-dimensional antiferromagnet VCl 2 by means of neutron scattering. It is found that VCl 2 shows characteristic critical behaviors of two and three dimensional models. A little above T N , the line shape of χ( Q ) is found to be represented by the Ornstein-Zernike form (κ 2 + q 2 ) -1 . This suggests the existence of magnetic point defects which were predicted in the two dimensional model. The measurecl critical exponents are β=0.20±0.02, γ=1.05±0.03, and ν=0.62±0.05, in agreement with the theoretical work for three dimensional SO(3) systems. Successive phase transitions due to a small Ising anisotropy were found at T N1 =35.88±0.01 K and T N2 =35.80±0.01 K. The spin structure and the dispersion relation of the spin wave well below T N are also determined.

An experimental study of the critical fluctuations in a two-dimensional Ising model

Neutron scattering measurements have been made of the critical fluctuations in K2CoF4 at the Brookhaven High Flux Beam reactor. The critical fluctuations were studied for both temperatures above and below Tc. Above Tc the scattering is well described by the Ornstein-Zernike form and the exponents and amplitude of kappa (the inverse correlation length) are in excellent accordance with theory. Below Tc the Ornstein-Zernike form gives a reasonable description of the results, but the resulting kappa are very different from those expected theoretically. The approximate form developed by Tarko and Fisher (1975) gives a better description of the results and the resulting parameters are in accordance with theoretical expectations. The authors conclude that the critical fluctuations in the d=2 Ising model are not described by the Ornstein-Zernike form. The amplitude ratios have been measured for the correlation length, the susceptibility, and the ratio connecting the Bragg scattering to the susceptibility by using two-scale-factor universality. The results in each case agree with theory to within the experimental error which varies between 10 and 15%.

Neutron cloud poinds and concentration fluctuations of polymer blends

Cloud points were observed in the blends of deuterated polystyrene (PSD) and hydrogeneous poly (vinyl methyl ether) (PVME) by means of temperature scanning Small Angle Neutron Scattering (SANS) technique. The scattering function in the miscible region can be described by the random phase approximation results calculated by de Gennes. This scattering function can also be expressed in the Ornstein-Zernike form in the small q region. A correlation length and spinodal point can then be determined from this critical fluctuation approach.

Spin-spin correlation function in the high-temperature Villain model

Considers the spin-spin correlation function in the high-temperature regime of the generalised Villain model (1975). Using the 'Debye-Huckel' approximation plus systematic corrections, the authors show that this correlation function has the expected Ornstein-Zernike form (1914) and they calculate approximately the high-temperature behaviour of the correlation length. The methods presented should prove useful in matching calculations of thermodynamic quantities in the two-dimensional XY model, and in 'Debye-Huckel' analyses of a variety of two-dimensional Coulomb gas problems.

Cluster size distribution for the Ornstein–Zernike correlation

In the light scattering of microemulsions and micellar solutions, the Fourier transforms of the density correlation functions of scattering centers often satisfy the Ornstein–Zernike form, const⋅ (k2+ξ−2)−1 to large values of ξ2k2 (sometimes up to ξ2k2?10), where k is the momentum conjugate to the coordinate of the correlation function and ζ the correlation length. In this paper I study the cluster size distributions which give rise to the Ornstein–Zernike type correlations. The clusters are assumed to be rods in one dimension, circular disks in two dimensions and spheres in three dimensions, and their positions are completely random.

Structure of the BBGKY hierarchy near phase transition

A nonperturbative method, as opposed to diagrammatic expansions, is used to study critical phenomena in a fluid with a small hard core and a weak, long-ranged attractive potential. Using the natural small parameter related to the inverse of the range of the attractive potential, spatially uniformly valid asymptotic estimates are made for the magnitudes of all correlations (which are defined as the excess from the generalized superposition approximation) in a region near phase transition in arbitrary number of dimensions. It is shown that if the dimension of the space is larger than four, the correlation hierarchy truncates at the three-body level. The pair correlation satisfies a linear equation. The solution is precisely of Ornstein-Zernike form. For dimensions smaller than four, the hierarchy is still an infinite chain, but considerably simpler than the BBGKY hierarchy. In this case, at the critical point, the correlations are shown to satisfy a scaling law which is the same as that for S4 spin systems.

Resistance anomaly of the order-disorder systemFe3Al

The electrical resistance of the alfenol alloy of nominal composition ${\mathrm{Fe}}_{3}$Al has been measured to high resolution in a broad temperature interval which embraces the second-order, order-disorder transition at ${T}_{c}=813.6$ K. For $T\ensuremath{\gtrsim}{T}_{c}$, the temperature derivative of the resistivity exhibits specific-heat-like behavior, which is consistent with the predictions of scaling arguments. For $T\ensuremath{\gg}{T}_{c}$, the critical resistivity exhibits an unusual temperature dependence which is interpreted in terms of the classical Ornstein-Zernike form of the correlation function describing the concentration fluctuations.

Magnetic short-range order in Gd

The magnetic short-range order in a ferromagnetic, isotopically enriched $^{160}\mathrm{Gd}$ metal single crystal has been investigated by quasielastic scattering of 81-meV neutrons. Since Gd behaves as an $S$-state ion in the metal, little anisotropy is expected in its magnetic behavior. However, the data show that there is anisotropic short-range order present over a large temperature interval both above and below ${T}_{C}$. The data have been analyzed in terms of an Ornstein-Zernike Lorentzian form with anisotropic correlation ranges. These correlation ranges as deduced from the observed data behave normally above ${T}_{C}$ but seem to remain constant over a fairly large interval below ${T}_{C}$ before becoming unobservable at lower temperatures. These observations suggest that the magnetic ordering in Gd may be a more complicated phenomenon than first believed.

Master equations, Langevin equations, and the effect of diffusion on concentration fluctuations

An anomalous prediction of the master equation formalism is not obtained when concentration fluctuations in chemical reactions are described by the fluctuation-dissipation theory. The known connection between these two theories shows that the discrepancy may be eliminated by taking the properly scaled thermodynamic limit of the master equation prior to achieving steady state. The effect of diffusion on fluctuations is also considered, and exact results are obtained for several reaction mechanisms. At steady states away from equilibrium nonlocal correlations of the Ornstein–Zernike form are found. This agrees with the continuum limit of the master equation formalism used by Gardiner et al.

Diffusing droplet model for Rayleigh linewidth studies on critical fluids

Measurements of the intensity autocorrelation of light scattered from diffusing micropheres suspended in a normal fluid show that if the particle radii satisfy the condition $l\ensuremath{\lesssim}{\ensuremath{\lambda}}_{0}$ where ${\ensuremath{\lambda}}_{0}$ is the wavelength of the incident light, and if the suspension is polydisperse (a finite-width particle-size distribution), the experimental diffusion constant displays a dependence upon the scattered wave number $k$ which is qualitatively the same as that for a critical fluid system in the nonhydrodynamic ($k\ensuremath{\xi}\ensuremath{\gtrsim}1$) regime. These experimental observations have led us to propose a model of a critical fluid in which the order-parameter fluctuations are considered as spherical molecular clusters, or droplets, performing Brownian motion in a normal host fluid. The droplets are assumed to have a Gaussian index of refraction spacial profile, and a particle-size distribution $N(l)$ characterizing the suspension determined from the requirement that the scattered light intensity be of the modified Ornstein-Zernike form. In a first approximation, the droplets are assumed to diffuse without changing size for times of the order of the characteristic diffusion time. The intensity-autocorrelation function of light scattered by our model system of diffusing droplets is evaluated, and the effective Rayleigh linewidth is found to agree to within a constant factor of order 1 with the ansatz for the critical part of the Rayleigh linewidth chosen by Perl and Ferrell. The resulting line-shape function is nearly Lorentzian, except in the wings where experimental detection of a departure from Lorentzian behavior becomes extremely difficult. Our droplet-size distribution is very similar to that which results from the static droplet model of Fisher which has been applied successfully to describe static critical phenomena in fluids below the critical temperature. In the diffusing droplet model, the $k$ dependence of the diffusion constant extracted from light-scattering measurements on critical fluids is seen to be an artifact introduced by the light-scattering process, and introduces no new physical information concerning the critical fluid behavior which is not contained in the droplet-size distribution function.

Scaling function for two-point correlations with long-range interactions to order 1/η

The two-point correlation function Ĝ (q, ξ) is calculated in the critical region of momentum space q in terms of a suitable correlation lenght ξ, by means of perturbation expansion to order 1/n, for an n-vector system with long-range interactions decaying as |R/a|−(d + σ), for |R/a| å 1, where a is the spacing on a d-dimensional lattice, σ < d < 2σ and 0 < σ ⩽ 2 − ηSR. The calculations are done in zero field for T ⩾ Tc. Explicit expansions for long-range propagators are developed for σ « 1 and for the neighborhood of σ ⪅ 2 − ηSR, in terms of which a universal, cut-off independent scaling function is obtained over the whole range of x = |q| ξ, and it is shown that the amplitude of the correlation-length dependence of the susceptibility becomes a universal parameter. Both the exponents and the coefficients of the expansion for fixed q as (T − Tc)Tc→0 are calculated explicitly. The former are shown to require the validity of the operator-product expansion and explicit logarithmic correction terms are obtained for d = d∗ = 3σ/2. For these and other dimensionalities, the coefficients are shown to be finite functions of d and σ. The correction to the Ornstein-Zernike form is given explicitly, with non-integer powers of x that have finite coefficients.

Double-scattering correction in the interpretation of Rayleigh scattering data near the critical point of a binary liquid

Double-scattering contributions to the intensity of Rayleigh-scattered light near the critical point of a binary liquid are calculated for 90\ifmmode^\circ\else\textdegree\fi{} scattering. We consider a family of sample geometries, namely cylinders with arbitrary length-to-radius ratio and cuboids with square cross sections and arbitrary length-to-side ratio. For binary liquids with nearly matched refractive indices, these terms can be sufficiently large that they affect the interpretation of data with respect to the value of the critical exponent $\ensuremath{\eta}$ and the deviation of the correlation function from Ornstein-Zernike form, yet small enough that higher-order contributions can be neglected. Numerical results are presented for all temperatures above ${T}_{c}$ together with analytical results in certain limiting cases. Near the critical point, the double-scattered intensity is comparable with that lost through extinction and these two effects cancel in leading order. Some analytical results are also obtained for the depolarization ratio. For example, if the height of the sample seen by the detector is small, the depolarization ratio is proportional to the differential cross section and to the sample height. The proportionality constant is $\frac{\ensuremath{\pi}}{4}$. Numerical results are presented for more general cases.