A comparison of correlation and Lyapunov dimensions

This paper investigates the relation between the correlation (D₂) and the Kaplan–Yorke dimension (D_KY) of three-dimensional chaotic flows. Besides the Kaplan–Yorke dimension, a new Lyapunov dimension (D_Σ), derived using a polynomial interpolation instead of a linear one, is compared with D_KY and D₂. Various systems from the literature are used in this analysis together with some special cases that span a range of dimension 2 < D_KY ≤ 3. A linear regression to the data produces a new fitted Lyapunov dimension of the form D_fit = α − βλ₁/λ₃, where λ₁ and λ₃ are the largest and smallest Lyapunov exponents, respectively. This form correlates better with the correlation dimension D₂ than do either D_KY or D_Σ. Additional forms of the fitted dimension are investigated to improve the fit to D₂, and the results are discussed and interpreted with respect to the Kaplan–Yorke conjecture.     

The infinite number of generalized dimensions of fractals and strange attractors

We show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions D_q, q > 0. To this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities D_q. We prove that lim _q→0D_q = fractal dimension (D), lim_q→1D_q = information dimension (σ) and D_q=2 = correlation exponent (v). D_q with other integer q's correspond to exponents associated with ternary, quaternary and higher correlation functions. We prove that generally Dq > D_q for any q′ > q. For homogeneous fractals D_q = D_q. A particularly interesting dimension is D_q=∞. For two examples (Feigenbaum attractor, generalized baker's transformation) we calculate the generalized dimensions and find that D_∞ is a non-trivial number. All the other generalized dimensions are bounded between the fractal dimension and D_∞.     

Evaluation of XPS-data of oxide layers

XPS with variable take-off angle has been applied to the determination of the thickness of thin oxide layers (SiO₂ on Si). The information about D/λ_ox gained by such measurements is strongly influenced by surface roughness. This influence can be demonstrated when the parameters R and D/λ_ox are calculated from sets of the experimental results and for each set the corresponding pair of R and D/λ_ox is plotted in a diagram D/λ_ox = f(R). Having the true value of R it is possible to determine D/λ_ox. With the XPS-results of at least three different oxide layers and their ellipsometric thicknesses D_e one is able to calculate λ_Si, λ_ox/λ_Si and the difference ΔD between D_e and the XPS-thickness D.     

Rovibrational spectra of the Ar–D2O and Kr–D2O van der Waals complexes in the v2 bend region of D2O

Rovibrational spectra of Ar–D₂O and Kr–D₂O complexes are measured in the v₂ bend region of D₂O monomer using a tunable mid-infrared diode laser spectrometer. One para and two ortho bands for both complexes are identified and then analyzed in terms of a nearly free internal rotor model. Molecular constants for the excited vibrational states, including band-origin, rotational and centrifugal distortion constants, and Coriolis coupling constant, are determined accurately. A comparison of the observed band-origins of Ar–D₂O and Kr–D₂O with the previous results of Ne–D₂O shows regular trends of shift from Kr–D₂O to Ne–D₂O.     

Vector analysis vs. regions of integration ratio method for determination of gamma and neutron contributions in a mixed field

The different ratios of the dosimetry peaks and the high temperature peaks after gamma and neutron irradiation can be employed for mixed field dosimetry, after performing proper analysis. In the present work the results obtained by the vector analysis method were compared to those obtained by the regions of integration (ROI) method for TLD600 chips irradiated by different combinations of gamma rays and thermal neutron doses. When applying the ROI method, the calculated neutron to gamma dose ratios (D_n/D_γ) were in good agreement (differences up to about 20%) with the actual experimental values in the D_n/D_γ range from 0.033 to 20. When applying the vector analysis method, the range of the D_n/D_γ ratio with differences up to about 20% is reduced to 0.033–4, and the uncertainties are generally higher. Thus, no advantage is expected when using the vector analysis method compared to the ROI method for evaluation of the dose ratio D_n/D_γ in a mixed field.     

Investigation of proton diffusion in Nafion®117 membrane by electrical conductivity and NMR

The proton dynamics in Nafion ^®117 is investigated by comparison of the diffusion coefficient D_NMR estimated from PFG-NMR with that of D_σ estimated from electrical conductivity. At high water content region, D_σ is about two times higher than D_NMR as a result of Grotthuss mechanism. At low water content region, D_σ and D_NMR are in good agreement with each other. Both of the diffusion coefficients decrease steeply at low water content region. It can be explained as a result of the percolation transition due to the isolation of ion clusters, which is suggested by the recent small angle X-ray scattering data.     

Magnetic properties of the mixed spin-5/2 and spin-3/2 Blume-Capel Ising system on the two-fold Cayley tree

We use exact recursion relations to study the magnetic properties of the half-integer mixed spin-5/2 and spin-3/2 Blume-Capel Ising ferromagnetic system on the two-fold Cayley tree that consists of two sublattices A and B. Two positive crystal-field interactions Δ₁ and Δ₂ are considered for the sublattice with spin-5/2 and spin-3/2 respectively. For different coordination numbers q of the Cayley tree sites, the phase diagrams of the model are presented with a special emphasis on the case q = 3, since other values of q reproduce similar results. First, the T = 0 phase diagram is illustrated in the (D _A = Δ₁/J,D _B = Δ₂/J) plane of reduced crystal-field interactions. This diagram shows triple points and coexistence lines between thermodynamically stable phases. Secondly, the thermal variation of the magnetization belonging to each sublattice for some coordination numbers q are investigated as well as the Helmoltz free energy of the system. First-order and second-order phase transitions are found. The second-order phase transitions become sharper and sharper when D _A or D _B increases. The first-order transitions only exist for some appropriate non-zero values of D _A and/or D _B . The corresponding transition lines never connect to the second-order transition lines. Thus, the non-existence of tricritical points remains one of the key features of the present model. The magnetic exponent β ₀ of the model is estimated and found to be ¼ at small values of D _A = D _B = D and β ₀ = ½ at large values of D. At intermediate values of D, there is a crossover region where the magnetic exponent displays interesting behaviours.     

Critical properties of the biaxial Ising model with both the longitudinal crystal field and the transverse dilution crystal field

Within the effective field theory (EFT), the critical properties of the biaxial Ising model with both longitudinal crystal field and transverse dilution crystal field are investigated for a simple cubic lattice. The tricritical point (TCP) and its trajectory are discussed in T-D_x and T-D_z space. A new phenomenon of two TCPs is found in T-D_x space. There exists a second-order line between two first-order lines, separated by two TCPs. The change of dilution concentration leads to a complex relation of the trajectory of the TCP. The degenerate patterns at the ground state appear by changing the longitudinal crystal field. The range of the ordered phase for transition lines labelled as a positive or (negative) value of D_x/J becomes larger or (smaller) with the decrease of t_x in T-D_z space. Some results have not been revealed in previous works.     

Ab initio determination of spectroscopic parameters for ethane-like molecules in the ground vibrational state

The quadratic rotational constants A and B and torsional barrier V₃, distortion parameters D_J, D_K, and D_JK, torsional distortion parameters D_m, D_Jm, and D_Km, and barrier-dependence parameters F_3J, F_3K, and F_3m have been determined for CH₃CH₃, CH₃CD₃, CD₃CD₃, and CH₃SiH₃ from the results of ab initio calculations done at the CCSD(T) level. Calculated values for the first six parameters are consistently within about 1% of experimental values, while the relative errors for D_m, D_Km, F_3J, and F_3K are generally less than 20%. Calculation of the parameters D_Jm and D_sJ is found to be more problematic, even with the application of vibrational averaging in the harmonic oscillator approximation. There is evidence that this is due to the influence of vibrational contact transformations in the experimental values.     

Effects of competition between the anisotropy and the spin quantum number on the magnon energy gap in a four-layer ferromagnetic superlattice

The magnon energy bands are studied for a four-layer ferromagnetic superlattice, with regard to the effects of the competition between the anisotropy and the spin quantum number. A special attention is also paid on the effects of the symmetry of the system. It is found that three modulated energy gaps exist in the magnon energy band along K_x direction perpendicular to the superlattice plane. The magnetic anisotropy affects significantly the magnon energy gaps. The zero energy gap Δω₂₃ correlates with the conditions between anisotropy constants, D₁+D₃=D₂+D₄ and D₁=D₃ (or D₂=D₄), while the disappearance of the magnon energy gaps Δω₁₂ and Δω₃₄ corresponds to a translational symmetry of x-direction in a unit cell. When the parameters of the system deviate from these conditions, the energy gaps Δω₁₂, Δω₂₃ and Δω₃₄ become larger. There is a competition effect of the anisotropy and the spin quantum number on the magnon energy gaps Δω₁₂ and Δω₂₃. When the symmetry of the system is higher, the competition can achieve a balance to cause the zero energy gap.     

Bonding and composition diagrams for sputtered a-Si : H

The combined influence of H partial pressure (p_H) and deposition rate (R_D) on Si-H bonding and total H content in diode-sputtered a-Si : H is presented in two simple graphs for the case of substrate temperature (T_s) equal to 225°C. Similar to a phase diagram, the graphs predict the H content and Si-H bonding that will result if a deposition is carried out with any prescribed pair of values (p_H, R_D), where 0.04 < p_H < 10 Paand 0.01 < R_D < 1 nm/sec. Well defined regions of Si-H bonding represented by dominant infrared stretching absorptions at 2000, 2090 and 2150 cm^?1 are obvious in the bonding diagram. The absorption at 2090 cm^?1 is the most commonly observed and is obtainable over a wide range of intermediate values of p_H and R_D. The absorption at 2000 cm^?1 is dominant only for the lowest p_H and the highest R_D. The absorption at 2150 cm^?1 is dominant in films deposited at high p_H and low R_D. The composition diagram shows that highest total H content is obtained for low R_D and high p_H, and lowest total H content results for high R_D and low p_H     

Symmetry and lattice-dynamic aspects of structural phase transitions in (CH3NH3)2MnCL4 and related compounds

A group theoretical analysis of the second-order structural phase transition in (CH₃NH₃)₂MnCL₄ at 394°K and of similar transitions in ethyl and propyl compounds (D¹⁷_4h→D¹⁸_2h) was performed. The soft mode transforms according to the τ^x₅-irreducible representation at the X-point of the Brillouin-zone boundary and its eigenvector is discussed. The transition is of the order-disorder type and is caused by a slowing down of the hindered rotation of NH₃-groups. Knowing the symmetry of the order parameter, a thermodynamic potential expansion was constructed and expected anomalies in material constants around the transition temperature are briefly discussed. The high temperature phase transitions in analogous copper compounds are explained as a sequence D¹⁷_4h→D¹⁸_2h→D¹⁵_2h. The second of these phase transitions is driven by a soft mode transforming as the τ^Y₇-representation at the D¹⁸_2h Brillouin zone boundary.     

Correlations and accuracy of estimation of spectroscopic constants and term values

A comparison and evaluation of the various methods for reducing spectroscopic data to spectroscopic constants or term values is made with the aid of an analysis of a number of “synthetic” Σ-Σ bands generated from fixed sets of constants with random noise superimposed on the line positions. It is shown that the strong correlations that exist between the upper-state constants B′, D′ and the lower-state constants B₀, D₀ can be effectively broken up by using the difference constants ΔB = B′ - B₀ and ΔD = D′ - D₀, along with ν₀, for representing the upper states. The lower state constants B₀ and D₀ and their standard errors calculated from the combination differences Δ₂F″(J) are shown to be as good as those obtained from direct polynomial fits. If data for a number of bands originating in the same lower state are available, a considerable increase in accuracy of estimating the lower-state constants can be attained by analyzing the bands simultaneously, e.g., using combination differences, provided the data are free from systematic errors. The dependence of the accuracy of determining the constants B₀, D₀, ν₀, ΔB, and ΔD on the extent of the band analyzed was investigated by varying the minimum and maximum J-values. The plots showing this dependence for both the actual errors and standard errors can be used e.g., to assess the band size necessary to attain a desired accuracy for a given constant.Åslund's term-value method is cast in a form which permits simple derivation of explicit formulas for the correlation coefficients connecting all the upper- and lower-state term values, and of explicit relations between the term values and the combination differences. Modifications of the term-value method suitable for the case where one or more transitions originate in an unperturbed state are explored. It is shown that the “difference term values” referred to the lower (unperturbed) state with the same value of J, T_i(J)-T₀(J), are essentially uncorrelated to the lower state constants B₀ and D₀. Since these quantities can be expressed directly as the eigenvalues of the energy matrix for a given J in the presence of perturbations, their use for representing the energies of perturbed rovibronic states is recommended.     

The effect of spin-spin contributions to the zero field splitting of Cr3+ in trigonal symmetry

The local spin-spin contributions to the zero field splitting D_ss of Cr³⁺ in D_3d symmetry have been calculated in a range of trigonal fields. Each value of D_ss is compared with spin-orbit contributions D_so of Gregorio.     

Diffusion in thermotropic liquid crystals

Diffusion coefficients in mesophases of thermotropic liquid crystals have been measured to a considerable extent only during the last twelve years. Some theories on diffusion in mesophases have been developed too. Measurements in nematic, smectic A, smectic B, and smectic C mesophases have been carried out by mass transport techniques, detected by radio-tracers or optically, by NMR spin-echo techniques, and by quasi-elastic neutron scattering. The diffusion is anisotropic in most of the cases. In nematics self-diffusion parallel to the molecular director (D₆) is somewhat faster than perpendicular (D_⊥). Both diffusion coefficients show about the same activation energy. Impurity diffusion (small molecules dissolved in mesophases) demonstrates the same behaviour with less anisotropy. In smectics A and C the diffusion coefficients D₆ and D_⊥ can be nearly equal. The anisotropy is now shown in their respective activation energies with E₆>E_⊥. This effect is more pronounced in impurity than in self-diffusion. In addition impurity diffusion shows a strong anisotropy of the diffusion coefficients (D_⊥?D₆). In smectics B the activation energies E₆ and E_⊥ of self-diffusion seem again to be equal and the diffusion coefficients show a small anisotropy (D_⊥≥D₆).     

Fluorescence ratio of the two D sodium lines in flames for D1 and D2 excitation

The ratio of the fluorescence intensities from the two lines of the yellow sodium doublet has been investigated in several hydrogen-based and acetylene-based flames and with various types of lasers as excitation sources. The systematic deviation from 2 for the ratio D₂/D₁ when either D₁ or D₂ are used as exciting transitions can be simply explained by the relative difference in the values of the collisional mixing constants between the excited P-levels and the collisional quenching constants transferring atoms from both P-levels to the ground S-level. Results obtained in flames characterized by different values of the quantum efficiency follow the behavior predicted by the fluorescence expressions obtained by a simple, steady-state theoretical approach.     

Quantum phase transition in the anisotropic one-dimensional XY model with long-range interactions

In this paper we study the one-dimensional XY model with single ion anisotropy and long-range interaction that decay as a power law. The model has a quantum phase transition, at zero temperature, at a critical value D_c of the anisotropy parameter D. For values of D below D_c we use a self-consistent harmonic approximation. We have found that the critical temperature increases with D for small values of this parameter. For values of D above D_c we use the bond operator technique and calculate the gap as a function of D, at zero temperature.     

X-Ray photoelectron study of the reaction of water with cerium

The interaction of D₂ O with a polycrystalline cerium surface, successfully cleaned by heavy Ar⁺ bombardment and annealing, was studied at 120 were observed at BE 530.3 (Ce₂O₃) and 532.7eV (adsorbed OD). When clean Ce at 120 K was exposed to D₂ O, the O(1s) spectra were initial eV (adsorbed D₂ O). For exposures greater than 10 Langmuir (L), a multilayer of ice grows and the O(1s) spectra become dominated by a peak at 5The results of interaction with D₂ O are compared with oxidation by O₂. The significant differences are: (1) the absence of Ce(IV) when oxidati relatively small extent of oxidation that occurs when Ce is exposed to D₂ O at 120 K, and (3) the larger chemical-shift of the Ce(III)-derived specThe XPS studies of the interaction of D₂ O with Ce reported here may be summarized as follows:(1) Exposure at 300 K gives rise to O(1s) features characteristic of oxide and hydroxide, while the Ce(3d) spectra indicate Ce(III), but no CE((2) Exposure at 120 K gives O(1s) features characteristic of adsorbed OD, chemisorbed D₂ O, a multilayer of ice, and a small amount of oxide. T are characteristic of clean Ce except for slight broadening.(3) Exposure at 120 K followed by warming to 240 and 300 K gives spectra characteristic of hydroxide and oxide surface-species. Between 240 and 300 K, O(1s) intensity.(4) At 300 K, a relatively thick layer of oxide forms, and after an exposure of 50 L the features characteristic of metallic Ce are no longer observabl(5) As compared to the case for O₂, exposure to D₂ O gives rise to different satellite-splittings in the Ce(3d) spectra, suggesting that di formed in the two cases.(6) The spectra observed for Ce exposed to D₂ O are in excellent accord with those found for the heavier lanthanides [4].     

Effect of correlated noises on Brownian motor

The Brownian motor operating between two correlated Gaussian white noises was investigated. The expressions of the current and the energy conversion efficiency of the Brownian motor were analytically derived by exploiting adiabatic approximation. The results indicates that: (i) Regulating the correlation strength λ between the two noises and the ratio D₂/D₁ of the two noise intensities can change the rotational direction of the motor; (ii) For the smaller D₂/D₁, an optimal λ can make the positive current and the efficiency be maximal, and for the smaller λ, an optimal D₂/D₁ also let the positive current be maximal. The results were explained from a viewpoint of modified potentials. The study is of important significance in the aspect of controlling the operation of the Brownian motor.