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Critical Exponent (critical + exponent)
Selected AbstractsParametric Rietveld refinement for the evaluation of powder diffraction patterns collected as a function of pressureJOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 3 2010Ivan Halasz Under the assumption that the structural parameters of a crystalline phase change `smoothly' with increasing pressure, the evolution of the parameters can be parameterized as a function of pressure using continuous monotonic functions. Four different approaches to determine the structural evolution of As2O5 with increasing pressure from a set of powder diffraction patterns collected over the pressure range from 2.5 to 19.5,GPa have been investigated. Approach (A) was the common sequential refinement of atomic coordinates with restraints on the geometry and was compared with three parameterization approaches. Approach (B) used direct parameterization by low-order polynomials of each crystallographically distinct atomic coordinate, (C) described the atoms of the asymmetric unit as a rigid body and allowed the internal degrees of freedom of the rigid body to vary with the change in pressure using rigid unit modes, and (D) described the crystal structure as a distortion of the higher-symmetry structure of As2O5 (which is here also a high-temperature phase) by using symmetry-adapted distortion modes. Approach (D) offers the possibility to directly introduce an order parameter into Rietveld refinement through an empirical power law derived from Landau theory and thus to obtain the value of the critical exponent. In contrast, the rigid-body approach did not fit the data as well. All parameterizations greatly reduce the number of required parameters. [source] Critical behavior of KDCO3 from 2H and 39K single crystal NMRMAGNETIC RESONANCE IN CHEMISTRY, Issue 1 2008Christophe Odin Abstract Potassium hydrogenocarbonate KDCO3 presents an order/disorder phase transition at Tc, 353 K. The critical behavior of this phase transition was studied by single crystal 2H and 39K NMR. The evolution of the order parameter as a function of temperature is quantified, and the critical exponent was determined, indicating a transition close to a tricritical point. The 2H Zeeman relaxation rate is strongly increased near the transition temperature. By calculating the noncritical contribution to the Zeeman relaxation rate, we show that the observed relaxation rate clearly presents a pseudo-divergent behavior near Tc, with a logarithmic singularity. The nature of the phase transition is discussed in the light of these results. Copyright © 2007 John Wiley & Sons, Ltd. [source] A critical exponent in a degenerate parabolic equationMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2002Michael Winkler We consider positive solutions of the Cauchy problem in for the equation $$u_t=u^p\,\Delta u+u^q,\quad p\geq1,\; q\geq 1$$\nopagenumbers\end and show that concerning global solvability, the number q = p + 1 appears as a critical growth exponent. Copyright © 2002 John Wiley & Sons, Ltd. [source] Fluctuation conductivity analysis on the Bi-based superconductors processed under same conditionsPHYSICA STATUS SOLIDI (C) - CURRENT TOPICS IN SOLID STATE PHYSICS, Issue 9 2006F. Ben Azzouz Abstract We report electrical conductivity fluctuation measurements on different Bismuth-based granular samples synthesized under same processing conditions.Using the fluctuation conductivity ,, as a function of the reduced temperature , in the range ,7 < ln , < 1, we identified Gaussian and critical fluctuation conductivity in (Bi,Pb)-2223 and Bi-2212 samples. Within the mean field region, samples show predominately two dimensional (2D) behaviour with exponent , = ,1. Closer to critical temperature TC, we have observed a crossover of , from ,2/3 to ,1/3 in the critical region. The obtained exponents are consistent with 3D-XY model predictions. The regime with the critical exponent ,2/3 is dominate in the case of (Bi,Pb)-2223 sample. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Disorder dependence of phase transitions in a Coulomb glassPHYSICA STATUS SOLIDI (C) - CURRENT TOPICS IN SOLID STATE PHYSICS, Issue 1 2004Michael H. Overlin Abstract We have performed a Monte Carlo study of a three dimensional system of classical electrons with Coulomb interactions at half filling. We systematically increase the positional disorder by starting from a completely ordered system and gradually transitioning to a Coulomb glass. The phase transition as a function of temperature is second order for all values of disorder. We use finite size scaling to determine the transition temperature TC and the critical exponent ,. We find that TC decreases and that , increases with increasing disorder. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] On the critical exponents of random k -SATRANDOM STRUCTURES AND ALGORITHMS, Issue 2 2002David B. Wilson Abstract There has been much recent interest in the satisfiability of random Boolean formulas. A random k -SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known that when the number of variables n is large, there is a sharp transition from satisfiability to unsatisfiability; in the case of 2-SAT this happens when m/n , 1, for 3-SAT the critical ratio is thought to be m/n , 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called , = ,k (the smaller the value of , the sharper the transition). Experiments have suggested that ,3 = 1.5 ± 0.1. ,4 = 1.25 ± 0.05, ,5 = 1.1 ± 0.05, ,6 = 1.05 ± 0.05, and heuristics have suggested that ,k , 1 as k , ,. We give here a simple proof that each of these exponents is at least 2 (provided the exponent is well defined). This result holds for each of the three standard ensembles of random k -SAT formulas: m clauses selected uniformly at random without replacement, m clauses selected uniformly at random with replacement, and each clause selected with probability p independent of the other clauses. We also obtain similar results for q -colorability and the appearance of a q -core in a random graph. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 182,195, 2002 [source] Order,disorder transition in monoclinic sulfur: a precise structural study by high-resolution neutron powder diffractionACTA CRYSTALLOGRAPHICA SECTION B, Issue 6 2006W. I. F. David High-resolution neutron powder diffraction has been used in order to characterize the order,disorder transition in monoclinic cyclo-octasulphur. Rapid data collection and the novel use of geometrically constrained refinements has enabled a direct and precise determination of the order parameter, based on molecular site occupancies, to be made. The transition is critical and continuous; with a transition temperature, Tc = 198.4,(3),K, and a critical exponent, , = 0.28,(3), which is indicative of three-dimensional ordering. Difficulties encountered as a consequence of the low thermal conductivity of the sample are discussed. [source] Scaling law and critical exponent for ,0 at the 3D Anderson transitionANNALEN DER PHYSIK, Issue 12 2009L.J. Vasquez Abstract We use high-precision, large system-size wave function data to analyse the scaling properties of the multifractal spectra around the disorder-induced three-dimensional Anderson transition in order to extract the critical exponents of the transition. Using a previously suggested scaling law, we find that the critical exponent , is significantly larger than suggested by previous results. We speculate that this discrepancy is due to the use of an oversimplified scaling relation. [source] A fluorescence study on critical exponents during sol-gel phase transition in complex monomeric systemsMACROMOLECULAR SYMPOSIA, Issue 1 2003Demet Kaya Abstract Methyl methacrylate (MMA), ethyl methacrylate (EMA) and various combinations of MMA with EMA were used during FCC experiments. Pyrene (Py) was introduced as a fluorescence probe and fluorescence lifetimes from its decay traces were measured during sol-gel phase transitions. The fast transient fluorescence (FTRF) technique was used to study the critical exponents during sol-gel phase transition in free-radical crosslinking copolymerization (FCC). The results were interpreted in the view of percolation theory. The critical exponents of gel fraction, , and weight average degree of polymerization, , were measured near the point of gel effect and found to be around 0.37 ± 0.015 and 1.69 ± 0.05 in all systems studied respectively. [source] Two-step mean-field renormalization group results for the large square Ising clustersPHYSICA STATUS SOLIDI (B) BASIC SOLID STATE PHYSICS, Issue 2 2003G. Kamieniarz Abstract A transfer matrix approach has been worked out to test the predictions of the improved three-step mean-field renormalization group approach to square Ising clusters with linear size up to L = 11. Performing the asymptotic analysis, the convergence of the finite-size critical couplings and the critical exponents towards the exact values is shown. [source] On the critical exponents of random k -SATRANDOM STRUCTURES AND ALGORITHMS, Issue 2 2002David B. Wilson Abstract There has been much recent interest in the satisfiability of random Boolean formulas. A random k -SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known that when the number of variables n is large, there is a sharp transition from satisfiability to unsatisfiability; in the case of 2-SAT this happens when m/n , 1, for 3-SAT the critical ratio is thought to be m/n , 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called , = ,k (the smaller the value of , the sharper the transition). Experiments have suggested that ,3 = 1.5 ± 0.1. ,4 = 1.25 ± 0.05, ,5 = 1.1 ± 0.05, ,6 = 1.05 ± 0.05, and heuristics have suggested that ,k , 1 as k , ,. We give here a simple proof that each of these exponents is at least 2 (provided the exponent is well defined). This result holds for each of the three standard ensembles of random k -SAT formulas: m clauses selected uniformly at random without replacement, m clauses selected uniformly at random with replacement, and each clause selected with probability p independent of the other clauses. We also obtain similar results for q -colorability and the appearance of a q -core in a random graph. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 182,195, 2002 [source] Scaling law and critical exponent for ,0 at the 3D Anderson transitionANNALEN DER PHYSIK, Issue 12 2009L.J. Vasquez Abstract We use high-precision, large system-size wave function data to analyse the scaling properties of the multifractal spectra around the disorder-induced three-dimensional Anderson transition in order to extract the critical exponents of the transition. Using a previously suggested scaling law, we find that the critical exponent , is significantly larger than suggested by previous results. We speculate that this discrepancy is due to the use of an oversimplified scaling relation. [source] Marginal relevance of disorder for pinning modelsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2010Giambattista Giacomin The effect of disorder on pinning and wetting models has attracted much attention in theoretical physics. In particular, it has been predicted on the basis of the Harris criterion that disorder is relevant (annealed and quenched models have different critical points and critical exponents) if the return probability exponent ,, a positive number that characterizes the model, is larger than ½. Weak disorder has been predicted to be irrelevant (i.e., coinciding critical points and exponents) if , < ½. Recent mathematical work has put these predictions on firm ground. In renormalization group terms, the case , = ½ is a marginal case, and there is no agreement in the literature as to whether one should expect disorder relevance or irrelevance at marginality. The question is also particularly intriguing because the case , = ½ includes the classical models of two-dimensional wetting of a rough substrate, of pinning of directed polymers on a defect line in dimension (3 + 1) or (1 + 1), and of pinning of an heteropolymer by a point potential in three-dimensional space. Here we prove disorder relevance both for the general , = ½ pinning model and for the hierarchical pinning model proposed by Derrida, Hakim, and Vannimenus, in the sense that we prove a shift of the quenched critical point with respect to the annealed one. In both cases we work with Gaussian disorder and we show that the shift is at least of order exp(,1/,4) for , small, if ,2 is the disorder variance. © 2009 Wiley Periodicals, Inc. [source] | ||||||||||||||||