Finite-size Scaling Law Research Articles

Influence of finite size and wetting on nematic and smectic phase behavior of liquid crystal confined to controlled-pore matrices

The high-resolution calorimetric study was carried out on octylcyanobiphenyl liquid crystal (LC) confined to various controlled-pore glass (CPG) matrices with silane-treated surface. The diameter of the voids cross section ranged between 23.7 and 395 nm. The results are compared to those obtained previously on CPG voids nontreated with silane. We found a striking similarity between the shifts in the isotropic to nematic and nematic to smectic-A phase transition temperatures as a function of the void radius in which order parameter variations at the LC-void interface play the dominant role. Weaker temperature shifts are observed in silane-treated samples, where surface ordering tendency is larger. In nontreated samples, a finite-size scaling law in the maximum value of the heat capacity at the nematic to smectic-A transition was observed for void diameters larger than 20 nm. In silane-treated samples, this behavior is considerably changed by surface wetting interactions.

Finite size effect in epitaxial La 0.7A 0.3CoO 3 (A=Ca, Sr, Ba) thin films

The in-plane and out-of-plane magnetization of La 0.7A 0.3CoO 3 thin films with A=Ca, Sr, or Ba were investigated as a function of the film thickness ranging from about 10 to 1000 monolayers. Nearly lattice matched epitaxial (0 0 1) oriented films have been grown by pulsed laser deposition on (LaAlO 3) 0.3(Sr 2AlTaO 6) 0.7 substrates. Their magnetic properties were characterized by SQUID measurements. Field cooled measurements were carried out with the field parallel or perpendicular to the film surface. All the films showed a finite size scaling law without any sign of a spin reorientation down to at least 10 monolayers. For the critical exponent we obtained λ≈0.9±0.2. For the model parameter No, discussed as the spin–spin coupling length we obtained 2–3 monolayers which compares well with the value measured for Co films.

Thickness-dependent Curie temperature of epitaxial La0.7A0.3CoO3 (A=Ca, Sr, Ba) films

The in-plane and out of plane magnetization of La0.7A0.3CoO3 thin films with A=Ca, Sr, or Ba were investigated as a function of the film thickness ranging from about 10 to 1000 monolayers. Nearly lattice matched epitaxial (0 0 1) oriented films have been grown on (LaAlO3)0.3(Sr2AlTaO6)0.7 substrates. Their magnetic properties were characterized by SQUID measurements. All the films showed a finite size scaling law with a critical exponent of λ=0.9±0.2. We found that the range of the spin–spin coupling obviously decreased when the doping element changed from Ca to Sr or Ba.

Monte Carlo simulation of polymer chain with ferromagnetic Ising interaction

We consider a self-avoiding walking polymer chain with nearest-neighbor ferromagnetic Ising interaction (J>0) on the simple cubic lattice. The magnetic and conformational properties are simulated using Monte Carlo technique with bond-fluctuation model. Spontaneous magnetization is observed at low temperature. We estimate the critical temperature Tc=1.37±0.05(J/kB) and the critical exponents α≈0.42 and ν≈0.80 based on the finite-size scaling law. The values of α and ν are different from those of two- or three-dimensional Ising models, indicating that the phase transition in the Ising polymer chain model belongs to a different universality class. Near Tc, the chain also undergoes a collapse transition from random coil at T>Tc to compacted globule at T<Tc. We conclude that the phase transitions in the Ising polymer chain model result from the cooperation of configurational movement and the ferromagnetic Ising interaction.

Consistency of Microcanonical and Canonical Finite-Size Scaling

Typically, in order to obtain finite-size scaling laws for quantities in the microcanonical ensemble, an assumption is taken as a starting point. In this paper, consistency of such a Microcanonical Finite-Size Scaling Assumption with its commonly accepted canonical counterpart is shown, which puts Microcanonical Finite-Size Scaling on a firmer footing.

Extensive simulations for longest common subsequences

: Given two strings X and Y of N and M characters respectively, the Longest Common Subsequence (LCS) Problem asks for the longest sequence of (non-contiguous) matches between X and Y. Using extensive Monte-Carlo simulations for this problem, we find a finite size scaling law of the form for the average LCS length of two random strings of size N over S letters. We provide precise estimates of for .We consider also a related Bernoulli Matching model where the different entries of an array are occupied with a match independently with probability 1/S. On the basis of a cavity-like analysis we find that the length of a longest sequence of matches in that case behaves as where r=M/N and . This formula agrees very well with our numerical computations. It provides a very good approximation for the Random String model, the approximation getting more accurate as S increases. The question of the “universality class” of the LCS problem is also considered. Our results for the Bernoulli Matching model show very good agreement with the scaling predictions of [#!HwaLassig96_PRL!#] for Needleman-Wunsch sequence alignment. We find however that the variance of the LCS length has a scaling different from Var in the Random String model, suggesting that long-ranged correlations among the matches are relevant in this model. We finally study the “ground state” properties of this problem. We find that the number of solutions typically grows exponentially with N. In other words, this system does not satisfy “Nernst's principle”. This is also reflected at the level of the overlap between two LCSs chosen at random, which is found to be self averaging and to approach a definite value q S <1 as .

Ground State Properties of One DimensionalS=1/2Heisenberg Model with Dimerization and Quadrumerization

The one dimensional S=1/2 Heisenberg model with dimerization and quadrumerization is studied by means of the numerical exact diagonalization of finite size systems. Using the phenomenological renormalization group and finite size scaling law, the ground state phase diagram is obtained in the isotropic case. It exhibits a variety of the ground states which contains the S=1 Haldane state, S=1 dimer state and S=1/2 dimer state as limiting cases. The gap exponent $\nu$ is also calculated which coincides with the value for the dimerization transition of the isotropic Heisenberg chain. In the XY limit, the phase diagram is obtained analytically and the comparison is made with the isotropic case.

Critical behavior of magnetic films in the Ising model

The Curie temperature ${T}_{C}(l)$for Ising spin films calculated analytically as a function of $l,$ the number of monolayers in the film, is compared with available data. When the exchange coupling between nearest neighboring pairs is assumed to be uniform, the normalized ${T}_{C}(l)$ is generally higher than what is observed, especially in the dimensionality crossover region in which ${T}_{C}(l)$ increases with $l$ much too fast. To reproduce the experimental data measured by various groups on ultrathin Ni films in (001) orientation, we calculate the normalized transition temperature ${T}_{C}{(l)/T}_{C}$(bulk) to fourth order in the variational cumulant expansion for a face-centered-cubic lattice with variable anisotropic exchange coupling ${J}_{\mathrm{ij}}.$ It is shown that the data can be very well described by adjusting ${J}_{\mathrm{ij}}$ for a few monolayers near the surfaces. The results are consistent with the finite-size scaling law outside the region of dimensionality crossover, providing a clue to the number of monolayers influenced by the presence of surfaces.

Mean Field and Corrections for the Euclidean Minimum Matching Problem

Consider the length $L_{MM}^E$ of the minimum matching of N points in d-dimensional Euclidean space. Using numerical simulations and the finite size scaling law $< L_{MM}^E > = \beta_{MM}^E(d) N^{1-1/d}(1+A/N+... )$, we obtain precise estimates of $\beta_{MM}^E(d)$ for $2 \le d \le 10$. We then consider the approximation where distance correlations are neglected. This model is solvable and gives at $d \ge 2$ an excellent ``random link'' approximation to $\beta_{MM}^E(d)$. Incorporation of three-link correlations further improves the accuracy, leading to a relative error of 0.4% at d=2 and 3. Finally, the large d behavior of this expansion in link correlations is discussed.

Structural and magnetocrystalline anisotropy contributions to the blocking temperatures of NixCo(1−x)O exchange couples (abstract)

The simple spin and crystal structure of the antiferromagnetic (AF) monoxide NixCo1−xO facilitates structural characterization and modeling of its magnetic properties NixCo1−xO is also receiving technological attention for biasing magnetoresistive sensors as a robust alternative to FeMn. In this work, we studied the thickness dependence of the AF layer on the exchange field created through interfacial exchange coupling between NixCo1−xO and Ni81Fe19 films. The NixCo(1−x)O films were reactively sputtered on silicon substrates and capped with a 300 Å Ni81Fe19 layer. An additional silver layer was deposited as an oxidation barrier. An exchange field was observed on all the exchange couples with the NixCo(1−x)O thickness ranging from 25 to 1000 Å. The exchange couples were characterized using transmission electron microscopy (TEM) to yield information on the orientation, structure, and size of the crystallites. The films were polycrystalline with columnar grains. For oxide thicknesses below 300 Å, the film was randomly oriented but developed a texture with a preferred orientation for thicker films. We found that the blocking temperature, at which the exchange field goes to zero, decreases with decreasing oxide thickness. The blocking temperature is relatively thickness independent for higher thicknesses but decreases rapdily below a certain thickness. The decrease in the blocking temperature does not appear to be consistent with a finite size scaling law. Instead we modeled this behavior to take into account the temperature dependence associated with the changing microstructure of the AF layer with thickness, in particular the size of the crystallites. In thick oxide films, the blocking temperature corresponds to the AF ordering temperature, the Néel temperature. The decrease of the blocking temperature below the Néel temperature suggests at the competing magnetic and thermal energies necessary for exhibiting unidirectional anisotropy. The thickness sensitivity of the blocking temperatures of the AF monoxides are found to be strongly dependent on their magnetocrystalline anisotropy.

Magnetic Behavior of Ultrathin Films of Pseudomorphic Binary Alloys

AbstractIn this work we compare the ferromagnetic behavior of ultrathin FexNi(1-x) and CoxNi(1-x) films grown on Cu(100) epitaxially with varying stoichiometry. The thickness regime chosen was 1 to 5 ML over a wide range of alloy composition. Using a finite-size scaling law we proposed for the Curie temperature vs. film thickness measurements, we extrapolate and plot the bulk fee Curie temperature as a function of composition. The results suggest that the growth of these films is pseudomorphic with the films adapting a fee structure and the Cu lattice constant. Besides, the Invar effect is not observed in these ultrathin films of FexNi(1-x) alloys and the magnetic phase of Fe atoms is the low-spin ferromagnetic phase. The CoxNi(1-x) films show the expected monotonic decrease in Tc with increasing Ni content.

Magnetic properties of ultrathin Ni/Cu(100) films determined by a UHV-FMR study

We have performed temperature dependent ferromagnetic resonance (FMR) measurements in ultrahigh vacuum on ultrathin Ni films, epitaxially grown on Cu(100). The thickness dependence of anisotropy fields has been determined from measurements with parallel and perpendicular field orientation. Separating the resulting anisotropy constants in volume and surface contributions, we get K eff v ≈ 3.5(7) × 106 erg/cm 3 and K eff s = -0.17(3) erg/cm 2. K eff v is explained within a model of homogeneous strain due to pseudomorphic growth below 7 ML. In addition very small FMR linewidths of ∼ 200 G have been observed, indicating that the films consist of large magnetic homogeneous domains. The increase of the linewidth at the phase transition allows us to determine the Curie temperature, whose thickness dependence is explained by a finite size scaling law.

Transient pattern dynamics and domain growth

Abstract General results for transient pattern dynamics in the context of spinodal decomposition as well as domain-wall formation in the Freedericksz transition in nematics are reviewed. The question of wavelength selection and the relevance of finite-size effects in this issue are considered in a general model of transient pattern dynamics. A finite-size scaling law for domain growth is shown to be satisfied by this model.

Ordering and finite-size effects in the dynamics of one-dimensional transient patterns.

We introduce and analyze a general one-dimensional model for the description of transient patterns which occur in the evolution between two spatially homogeneous states. This phenomenon occurs, for example, during the Fr\'eedericksz transition in nematic liquid crystals. The dynamics leads to the emergence of finite domains that are locally periodic and independent of each other. This picture is substantiated by a finite-size scaling law for the structure factor. The mechanism of evolution towards the final homogeneous state is by local roll destruction and associated reduction of local wave number. The scaling law breaks down for systems of size comparable to the size of the locally periodic domains. For systems of this size or smaller, an apparent nonlinear selection of a global wavelength holds, giving rise to long-lived periodic configurations which do not occur for large systems. We also make explicit unsuitability of a description of transient pattern dynamics in terms of a few Fourier mode amplitudes, even for small systems with a few linearly unstable modes.

Finite-temperature phase transition of SU(3) gauge theory on Nt=4 and 6 lattices.

The deconfining finite-temperature transition of SU(3) gauge theory is studied on the dedicated parallel computer QCDPAX. Monte Carlo simulations are performed on 12{sup 2}{times}24{times}4, 24{sup 2}{times}36{times}4, 20{sup 3}{times}6, 24{sup 3}{times}6, and 36{sup 2}{times}48{times}6 lattices with 376 000 to 1 112 000 iterations. The finite size scaling behavior of the first-order transition is confirmed both on the {ital N}{sub {ital t}}=4 and {ital N}{sub {ital t}}=6 lattices and clear two-phase structures are observed on spatially large lattices (24{sup 2}{times}36{times}4 and 36{sup 2}{times}48{times}6). The latent heat at the deconfining transition is estimated both by a direct measurement of the gap on the spatially large lattices and by applying a finite-size scaling law. The results obtained by these two independent methods are remarkably consistent with each other on both the {ital N}{sub {ital t}}=4 and 6 lattices. The latent heat for {ital N}{sub {ital t}}=6 is much smaller than that for {ital N}{sub {ital t}}=4 and is about 1/3 of the Stefan-Boltzmann value 8{pi}{sup 2}/15. The details of the data and the error analysis are presented.

Deconfining transition of SU(3) gauge theory on NT = 6 lattices

Abstract We report a Monte Carlo study of finite temperature SU (3) gauge theory simulated on the parallel computer QCD-PAX. The study of the deconfining transition on N t =4 lattices, reported at the Lattice '90, is extended to the case of N t = 6 using 20 3 × 6, 24 3 × 6 and 36 2 × 48 × 6 lattices. Clear two phase structure is observed on large lattices. The finite size scaling we found is consistent with a first-order phase transition. The latent heat is estimated both by a direct measurement of the gap on large lattices and by applying a finite size scaling law. The two results are mutually consistent both on N t = 4 and 6 lattices. We find that the latent heat for N t = 6 is much smaller than the value previously obtained on a smaller lattice.

Study of the phase transition line and of its endpoint in the SU(2) higgs model

A recently developed multi-histogram method was used to study the order of the Higgs phase transition line in the SU(2) Higgs model at λ = 0.5 close to its endpoint at low β. The critical exponents α and ν could be estimated from the finite size scaling laws for the maximum of the specific heat and of the first zero of the complex partition function. The results indicate that the exponent ν might be smaller than the Gaussian value ν = 0.5.

Intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators

A theory of intrinsic fluctuations is developed of a phase ordering parameter for large populations of weakly and uniformly coupled limit-cycle oscillators with distributed native frequencies. In particular it is shown that the intensity as well as the correlation time of fluctuations exhibit power-law divergence at the onset of mutual entrainment with critical exponents which depend on whether the coupling strength approaches the threshold from below or above. This peculiar feature is demonstrated by numerical simulations mainly through finite-size scaling analyses. In the course of exploring its origin, we encounter a new concept termed a “correlation frequency” which provides a natural interpretation of the finite-size scaling laws. A comment is given on a recent theory by Kuramoto and Nishikawa to clarify why it contradicts our results.

Static and Dynamic Finite-Size Scaling Theory Based on the Renormalization Group Approach

Fisher's static finite-size scaling law is derived on the basis of the renormalization group theory and it is extended to dynamic critical phenomena in a finite system. This dynamic finite-size scaling law yields a cross-over effect with respect to the size and time-region. This effect is useful in analyzing computer simulations and also in studying the scaling property of the Kondo effect near the absolute zero temperature.