Static Critical Properties Research Articles

Neutron scattering investigation of the static critical properties of Rb2CrCl4

We have investigated the static zero-field critical properties of the ionic ferromagnet Rb2CrCl4, using neutron scattering, under conditions such that the quasi-static approximation is valid. A general consideration of the spin Hamiltonian of Rb2CrCl4, with S = 2, shows that it approximates the two-dimensional (2D) XY model, with additional weak perturbations. Rb2CrCl4 orders three dimensionally at Tc = 52.3 K, and very close to Tc there is a regime of three-dimensional critical behaviour. However, both below and above Tc there is a crossover to regimes of two-dimensional critical fluctuations. In these regimes, there is excellent agreement between experiment and theoretical predictions for a finite-sized 2D XY model, suggesting that Rb2CrCl4 is an ideal realization of this system.

Static critical behavior of three-dimensional classical Heisenberg models: A high-resolution Monte Carlo study.

Using both recently developed cluster-algorithm and histogram methods, we have carried out a high-resolution Monte Carlo study of static critical properties of classical ferromagnetic Heisenberg models. Extensive Monte Carlo simulations were performed at several temperatures in the critical region, using an improved cluster-updating scheme, on L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic and body-centered-cubic systems with L\ensuremath{\le}40. Thermodynamic quantities as a function of temperature in the vicinity of the critical point were obtained by an optimized multiple-histogram method, and the critical temperature and static critical exponents were extracted using finite-size scaling. Our best estimates for the inverse critical temperatures are 0.693 035(37) for the simple-cubic system and 0.486 798(12) for the body-centered-cubic system. Estimated static critical exponents for both systems agree with each other within their respective error bars, and the mean estimates \ensuremath{\nu}=0.7048(30) and \ensuremath{\gamma}=1.3873(85) are also in excellent agreement with field-theoretic predictions 0.705(3) and 1.386(4).

Monte Carlo study of critical behavior in the three-dimensional classical Heisenberg ferromagnet

The static critical properties of classical, isotropic Heisenberg ferromagnets have been studied numerically. Using an improved cluster algorithm we performed extensive Monte Carlo simulations on L×L×L simple cubic and body-centered cubic systems with L≤40 over a temperature range covering the critical region. Temperature dependences of thermodynamic quantities were determined with an optimized multiple-histogram method, and both critical temperatures and critical exponents were extracted from finite-size scaling of several different thermodynamic quantities. It was found that within their respective error bars the critical exponents for both ferromagnets agree with each other and with field theoretical predictions.

MODE-COUPLING APPROACH TO THE SPIN DYNAMICS OF EUROPIUM COMPOUNDS

The results of a numerical solution of the mode-coupling equations for EuO and EuS taking into account the microscopical properties of the compounds are presented. Good agreement with the available neutron scattering data is found and predictions for further experiments on EuS are given. The results of high resolution neutron scattering experiments on insulating ferromagnetic Europium compounds EuO and EuS appeared in literature in recent years [I-41. The particular attention devoted to such compounds, is due to the fact that already in the earlier neutron scattering comprehensive investigation on E n 0 and EuS [I] it was shown that the magnetic behaviour of these systems is very well described by a simple isotropic Heisenberg exchange interaction. The availability of experimental data at and above the ordering temperature Tc has encouraged a renewal of theoretical studies on critical and paramagnetic spin fluctuations in Heisenberg ferromagnets. The most successful theories appear to be those ones based on Renormalization Group (RG) or on the mode-coupling approximation. RG techniques constitute a well established method to study the static and dynamic critical properties of a system; they allow to evaluate critical exponents as well as universal functions which characterize the critical behaviour. Very recently an attempt has been made to extend RG calculations to temperatures above Tc [5], but some problems persist to make possible a quantitative comparison with experimental data. The mode coupling approximation, in spite of its heuristic character, has revealed very useful in many problems of condensed matter physics [6]. For pure Heisenberg magnets, for example, it gives for the critical dynamical scaling ex,ponent z the value z = 171, in a agreement with 2 the experiments. Moreover it has been shown very recently [8] that by taking into account the dipolar macroscopic interaction, the mode-coupling theory can be able to interpret the unexpected simple exponential decay obtained in neutron spin-echo experiment on EuO at T = Tc and very small wave vector [4], which remained until now inexplicable. However the mode-coupling theory appears also suitable to investigate the dynamical behaviour of ferromagnetic systems in all the paramagnetic region and throughout the Brillouin Zone. In this paper we present the results of a numerical solution of the modecoupling equations for EuO and EuS taking into account the microscopical features of these compounds. This allows to show some distinctive characters which distinguish EuS from EuO that cannot be accounted by previous approximate solutions, which were limited to simple cubic nearest neighbours interaction [7] or confined to the continuum limit [6]. Europium compounds can be described by a Heisenberg exchange hamiltonian: I for an FCC lattice of magnetic ions with S = -. The 2 experiments [I] have shown that the interaction is restricted to nearest neighbours (J1) and next nearest neighbours (Jz ) . For EuO: J1 = 1.22' K , J2 = 0.25' K and a = 5.12 A; for EuS: J1 = 0.48' K , J2 = -0.24' K and a = 5.95 A, a being the lattice constant. The final result given by the mode-coupling theory is the following integr~differential equation: for the relaxation function Fq ( t ) , whose Fourier Transform is related to the measured scattering function S (q, w ) by the relation: W (q' W, = 1 exp (-v/KBT) .x (4) F, ( w ) . (3) Not with standing formal differences in the various derivations, behind equation (2) is the following fundamental approximation: correlation and relaxation functions among more than two spin operators have been decoupled. Equation (2) is only one of the forms in which this equation is obtained, however all such forms are equivalent if we use for the static susceptibility that one given by the spherical model: Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19888720 C8 1572 JOURNAL DE PHYSIQUE This susceptibility turns out to be compatible with the dynamical equation (2). The temperature dependent parameter X appearing in equation (4) can be evaluated by using the sum rule: (I/N) C T X ~ = s(S + 1) /3.

Transfer-matrix scaling for anomalous dynamics of a vector spin-glass chain.

The transverse dynamics of a one-dimensional vector spin glass at zero temperature is treated by a new transfer-matrix scaling technique, and also by evaluating static critical properties related to dynamics by a crossover argument. A nontrivial dynamic exponent is found which implies that the system at zero temperature has dynamic critical behavior w\ensuremath{\propto}${k}^{3/2}$, in contradiction to the hydrodynamic behavior normally assumed and to the results of previous work.

Static and dynamic real-space renormalization through series expansions for correlation functions.

The renormalization-group analysis of correlation-function series expansions introduced by Stella et al. for computing the static critical properties of lattice-spin systems is refined by employing an additional interaction variable, and extended to dynamical critical phenomena. The static approach is applied to the square-lattice Ising model with nearest-neighbor and diagonal interactions, with use of the original high-temperature series to ninth order for pair-correlation functions. The critical point, the thermal and magnetic exponent, and the leading correction-to-scaling exponent are computed. Applications of the method to the spin-(1/2) Ising and XY models in two and three dimensions are also reviewed. The possibility of basing a dynamical renormalization approach on this type of analysis is shown. The new dynamical method, which avoids proliferation of interactions and memory effects, is applied to the square-lattice Glauber model. The study of original series for pair relaxation times (to eigth order in the nearest-neighbor interaction and to fourth order in the diagonal one) gives estimates of the dynamic exponent z.

Unified Description of Static and Dynamic Scaling for Kinetic Cluster Formation

A scaling theory is developed for aggregation by means of kinetic clustering of clusters. A global picture of static and dynamic critical properties emerges, whereby the dynamic critical exponent can be related to the fractal dimension. Furthermore, the growth process is described in terms of a purely kinetic model. The scaling predictions agree well with numerical results.

Quantum spin systems: dynamical mean field renormalisation group approach

The mean field renormalisation group approach is applied to the study of the static and dynamical critical properties of the spin-1/2 anisotropic Heisenberg model in a transverse field. The critical surface and estimates of static and dynamical critical exponents are obtained for the one-, two-, and three-dimensional models.

Dynamical renormalisation through classical equations of motion

Classical approximations for critical dynamics are taken as the basis of a new dynamical renormalisation group strategy in real space. The approach is phenomenological and avoids proliferation of interactions, as well as memory effects. For the square and cubic Glauber model, the best determinations of static critical properties come up with values for the dynamic exponent which appear compatible with the most recent estimates by Monte Carlo or series expansion methods.

Phase transitions in the six-state vector Potts model in two dimensions

The static critical properties of the six-state vector Potts model (clock model) on L×L square lattices are studied in zero field using Monte Carlo simulations. Data are obtained for L≤48 keeping between 2.4 and 4.0×104 MCS (Monte Carlo steps/site) for computing averages. The data are analyzed using a block distribution technique in which the critical temperature and exponents may be estimated from the size dependence of the cumulants of the order parameter. We see clear evidence for two phase transitions but our estimates for the transition temperatures are different from results obtained through other methods. For the lattice sizes considered the intermediate phase does not appear to be XY-like.

Static and Dynamic Critical Properties of Dielectric Constant in Ferroelectric CsH2PO4 and CsD2PO4

Monodispersive dielectric dispersion has been observed in CsH 2 PO 4 and CsD 2 PO 4 in the frequency range between 1 MHz and 1 GHz. Critical anomalies found in the static dielectric constant and the relaxation time, e H s and τ H for CsH 2 PO 4 and e D s and τ D for CsD 2 PO 4 , respectively, show pronounced deviations from the Curie-Weiss behavior. Simple relations of e D s / e H s =1.3 and τ D /τ H =5.2 for the values at the same T - T c , where T c is the transition temperature of each material, are found in wide temperature ranges above T c . It is hard to explain these results by the tunneling model.

Critical Behavior of a Micellar Solution

The static and dynamic critical properties of a nonionic amphiphile aqueous solution showing a lower consolution point are investigated by laser-light scattering. The osmotic isothermal compressibility and the correlation length of the concentration fluctuations diverge with mean-field-theory critical exponents. The behavior of the mass diffusion constant in the hydrodynamic region is in good agreement with the predictions of the mode-mode-coupling theory.

Renormalization group recursion relations involving dynamics for a superconductor in a magnetic field

A generalized model is proposed for the treatment of both static and dynamic critical properties near the transition point of a superconductor. Renormalization group recursion relations for this model are derived in the first-order ϵ-approximation. The dynamic critical exponent is found to the same order.

Dynamics of the roughening transition in one dimension

In one dimension, the roughening transition occurs only at zero temperature. In that case it is well known that the static critical properties exhibit an essential singularity. We present results for the dynamics at both finite and low temperatures. We found that the diffusion coefficient decreases to a finite value as the temperature approaches zero similar to the two-dimensional situation; however, the rate of decrease is now given by a functional form $D(T)={D}_{0}+\ensuremath{\alpha}{e}^{\ensuremath{-}\frac{\ensuremath{\beta}}{T}}$. This result is discussed and compared with that of the one-dimensional kinetic Ising model studied by Glauber.

Local critical properties near an isolated impurity

The effect of an isolated impurity on local static critical properties is examined by extrapolating exact high-temperature series. In accord with results of alternative methods no change in the critical exponents is found. The amplitudes of the singularities change in general, but a special choice of the nature of the impurity may even leave the amplitudes unchanged. This provides a convenient situation for numerical tests.

Effect of inelastic scattering on resistive anomalies at magnetic critical points

Theoretical treatments of resistive anomalies at magnetic critical points have been based on the quasielastic approximation according to which the scattering system of localized spins is essentially static on the relevant time scale. This approximation is justified by thermodynamic slowing down of spin fluctuations near the critical point. In the case of ferromagnets, for example, this view is most obviously valid for long-wavelength spin fluctuations. These, however, are irrelevant for the resistivity. In this paper, we give an exact evaluation of the lowest-order corrections to the resistivity due to inelastic scattering for both ferromagnets and antiferromagnets. These corrections are numerically significant only for very low spin (e.g., \ensuremath{\sim} 30% for $S=\frac{1}{2}$) and vary as $\frac{1}{S(S+1)}$. Their temperature dependence reflects directly that of the internal energy of the spin system so that static critical properties will continue to be a useful guide in interpreting resistive anomalies.

Shear viscosity and order parameter dynamics of fluids near the critical point

We present a simple modification of the mode-coupling equations which relate the shear viscosity to the decay rate of order parameter fluctuations of simple fluids and binary mixtures near their critical points. By including the background Rayleigh linewidth in these equations, both the critical shear viscosity and the critical Rayleigh linewidth can be calculated with no adjustable parameters, using only background transport data and static critical properties. Our predictions are found to be in good agreement with presently available experimental data.