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A study of 0.71Pb(Mg1/3Nb2/3)O3‐0.29PbTiO3 shows that, when it is doped with 2.5% Sm on the A‐site, in addition to an almost 3‐fold increase in piezoelectric charge coefficient and dielectric permittivity, there is a 2 order of magnitude reduction in conductivity, attributed to a decrease in oxygen vacancy concentration. An analysis of the nonlinearity of permittivity with respect to field amplitude shows that both the reversible and irreversible contributions increase significantly with Sm‐doping, with simple models showing that these changes are consistent with a reduction in the concentration of dipolar defects that can inhibit both polarization rotation and domain wall translation. Contrary to the argument that doping increases heterogeneity, there is little change in the diffuseness of the peak in permittivity as a function temperature, whilst there is a 15% increase in spontaneous polarization with Sm addition. Through comparison of the Rayleigh law parameters with those published for other piezoelectric materials, it is concluded that a significant contribution to the observed increase in piezoelectric performance due to Sm‐doping of PMN‐PT is similar in origin to that seen in soft, donor‐doped PZT and other conventional piezoelectric ceramics.

We numerically investigate sound damping in disordered two-dimensional soft particle packings. We simulate evolution of standing waves of particle displacements and analyze time correlation functions of particle velocities and power spectra. We control the strength of inelastic interactions between the particles in contact to show how the inelasticity affects anomalous sound characteristics of disordered systems: Increasing the strength of inelastic interactions, we find that (i) sound softening vanishes and (ii) attenuation coefficients exhibit a transition from the Rayleigh law to quadratic growth. We also report (iii) how the Ioffe-Regel limit frequencies depend on the strength of inelasticity as useful information for experiments and applications of the sound in disordered media. Our findings suggest that sound damping in soft particle packings is determined by the interplay between elastic heterogeneities and inelasticity.

The extraordinary performances of phase-coexisting ferroelectrics are significantly affected by not only the phase constitution but also the motion of domain walls. The study on the role of phase coexistence in the formation of ferroelectric and ferroelastic domain microstructures is of great importance to explain the enhanced piezoelectric properties. In situ high-energy diffraction and the Rayleigh law are utilized to reveal the interplay of phase constitution and domain configuration to the macroscopic electromechanical coupling effect in the morphotropic phase boundary composition of 0.365BiScO3–0.635PbTiO3 during the application of a weak electrical loading in the present study. It was found that anisotropic phase transition and domain switching occur in polycrystalline ferroelectric ceramics and a phase transition occurs dramatically beyond the coercive field. Taking into account the important role of coupled ferroelectric and ferroelastic domain microstructures, we conceived a configuration of monoclinic domains coexisting with and bridging the tetragonal domains. The existence of bridging domains would provide an insight into the interplay of the phase and domain and explains the piezoelectric performance in the vicinity of morphotropic phase boundaries.

The hysteresis losses and their relation to the parameters of the minor static hysteresis loops were investigated in the nanocrystalline Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy with magnetic-field-induced anisotropy. The analysis of experimental data was performed using a power function, which approximated the dependencies between the hysteresis parameters. It is shown that all the experimental dependences in cores with magnetic-field-induced anisotropy in a weak magnetic field are consistent with the formulas derived from the Rayleigh law. It is also shown, that in a nanocrystalline alloy with uniaxial induced anisotropy, certain empirical relationships obtained earlier for isotropic materials are not met, particularly, for the relation of the hysteresis losses to the remanence and the maximal magnetic induction. It has been found, that samples with longitudinal induced anisotropy demonstrate low initial permeability, and the magnetization process in the Rayleigh region is carried out by the domain walls’ displacement at distances comparable to the correlation length Lex. A magnetic hysteresis mechanism associated with the irreversible magnetization rotation in ferromagnetic clusters of nanocrystalline alloys is proposed. Formulas are proposed to calculate hysteresis losses in a soft magnetic nanocrystalline material with a different magnetic anisotropy.

Since nonpiezoelectric interfacial layers even at the nanoscale significantly affect the performance of lead-free piezoelectric thin films, the quantitative characterization of property changes of thin films due to interfacial layers is of great importance and should be precisely undertaken for piezoelectric microelectromechanical system (MEMS) and nanoelectromechanical system (NEMS) devices. In contrast to widely accepted concepts for interfacial layer thickness estimation based on the existing series capacitor model, we find that the interfacial layer thickness at the top and the bottom interfaces is clearly different in chemical solution deposition (CSD)-derived (K0.5,Na0.5)(Mn0.005,Nb0.995)O3 (KNMN) thin films. Interestingly, the thickness of the bottom interface increases linearly with increasing thin-film thickness, while the thickness of the top interface is constant regardless of the thin-film thickness. In this work, nanointerfacial layer effects of CSD-derived KNMN thin films are theoretically and experimentally addressed in a combinatorial way using a modified series capacitor model. The obtained information is used to envisage the origins and the mechanisms of nonpiezoelectric interfacial layers and associated dielectric and ferroelectric properties of KNMN thin films. Our research connects macroscopic properties with microscopic origins and is greatly facilitated by separating intrinsic and extrinsic contributions to phenomenological behaviors, as well as engineering interface-related properties of the films. We believe these studies to be crucial for the further development and applications of KNN-based lead-free piezoelectric devices, which also open the door to future studies on other lead-free piezoelectric material systems for practical MEMS and NEMS applications.

A method of dimensionless quantities for the analysis of hysteresis phenomena is proposed. Dimensionless hysteresis quantities of zero, first, second and third orders are introduced. Analytic expressions are obtained for dimensionless hysteresis quantities in a weak magnetic field. In the Rayleigh region, the order of dimensionless hysteresis quantities coincides with the exponent of the power function that relates these quantities with a hysteresis factor κr. It is shown that the scaling in the Rayleigh region extends over the entire region of the magnetic field in which permeability increases. Analytical expressions for the hysteresis losses and permeability are obtained, which are confirmed experimentally.

Based on the example of analyzing the Rayleigh equations for ascending and descending branches of the magnetic hysteresis loop, the dimensionless hysteresis quantities Br/Bmax, Wh/Wmax, Hc/Hmax, and BrHc/Wh, which have the same functional dependence on the external magnetic field Hmax, have been found. Hysteresis quantities proportional to the permeability μ or hysteresis loss Wh have been obtained. An analysis of the magnetically soft amorphous alloy Co66Fe3Cr3Si15B13 with very high initial permeability showed that the similarity of dimensionless hysteresis quantities takes place not only in the Rayleigh region, but also in a wider region of the magnetic field.

The nonlinear dielectric responses of both unpoled and poled Mn-doped PMS-PZT ceramics as a function of hydrostatic pressure were investigated. Based on the application of Rayleigh law analysis, the contribution of irreversible domain wall motion to the dielectric response under hydrostatic pressure was quantitatively evaluated. Initial increased irreversible extrinsic contribution to the dielectric responses were observed in both unpoled and poled samples which can be ascribed to the pressure induced domain wall pinning relaxation effect. In addition, the pressure induced depolarization effect may also facilitate domain related extrinsic contribution to dielectric response in poled samples. The present study may shed further light on understanding the domain wall motion dynamic in piezoelectric ceramics under the application of hydrostatic pressure and is quite helpful for the design and optimization of piezoelectric devices used in underwater areas.

In the work, (1- x ) (K 0.48 Na 0.52 ) (Nb 0.96 Sb 0.04 )O 3 - x (Bi 0.5 K 0.5 ) 0.7 Ca 0.3 ZrO 3 [(1- x )KNNS– x BKCZ] lead-free piezoceramics are prepared to investigate location and associated impacts of non-equivalent ionic dopants on phase transition and property enhancement. Results of XRD refinement and analysis of transmission electron microscopes (TEM) strongly prove that Rhombohedral-Tetragonal (R-O-T) phase coexistence for 0 ≤ x ≤ 0.04. According to lattice parameters, R phase should be ascribed to lattice deformation subjected to shear stress owing to the cooperation of multiple ionic dopants in A sites and B sites. Based on the variation of temperature-dependent capacitances with the frequencies and atomic-resolution HR-O-TEM image, much more oxygen vacancies are investigated because of major substitution of Ca 2+ in B site, which leads to an obvious decrease of T C in (1- x )KNNS– x BKCZ besides c / a of O phase. Additionally, domain wall motion is demonstrated to become easier through calculating intrinsic piezoelectric response using Rayleigh law and consequently, piezoelectric properties of (1- x )KNNS– x BKCZ are greatly enhanced ( d 33 ∼440 pC/N and d* 33 –500 pm/V) in R-O-T phase coexistence region.

(K,Na)NbO3 is a leading lead-free piezoelectric system, but its thermal instability of high piezoelectricity is still one of its shortcomings compared to its lead-containing counterparts. This work found that incorporating perovskite compounds with Zr ions at the B-site can lead to the simultaneous enhancement of piezoelectricity and thermal stability. The piezoelectric coefficient d33 increased to ∼450 pC N−1, and such a high d33 value can be maintained within 10% change over a wide temperature range of 22–80 °C. The piezoelectric strain also reached 0.13% strain with a small variation (

Object and purpose of research. This paper analyses general formulation and solution of the boundary problem for magnetization of thin-walled infinitely long cylindrical shell, its material being magnetized as per Rayleigh law. The purpose of the study is to determine magnetization of this shell.Materials and methods. The study uses magnetization theory of thin-walled shell, the method of solving infinite systems of equations, as well as numerical solution methods for non-linear equations.Main results. The paper provides general formulation of the boundary problem for magnetization of thin-walled infinitely long cylindrical shell with its material magnetized as per Rayleigh law. This problem is expressed as an infinite system of non-linear equations for the coefficients in the Fourier series representing this magnetization. It is shown that sufficient solution to this system can be obtained through the truncation method. The study also provides an example of calculating the magnetization of thin-walled infinitely long cylindrical shell. Conclusion. The results of this work can be used in the practical calculations dealing with magnetization of oblong thin- walled cylindrical shells, taking into account that magnetic parameters of their materials in the Rayleigh domain.

Effects of quenching process on dielectric, ferroelectric and piezoelectric properties of 0.71BiFeO3-0.29BaTiO3 ceramics with Mn modification (BF-BT-xmol%Mn) were investigated. The dielectric, ferroelectric and piezoelectric properties of BF-BT-xmol%Mn were improved by quenching, especially to the BF-BT-0.3mol%Mn ceramics. The dielectric loss tanδ of quenched BF-BT-0.3mol%Mn ceramics was only 0.28 at 500 °C, which was half of the slow cooling one. Meanwhile, the remnant polarization Pr of quenched BF-BT- 0.3mol%Mn ceramics increased to 21 μC/cm2. It was notable that the piezoelectric constant d33 of quenched BF-BT-0.3mol%Mn ceramics reached up to 191 pC/N, while the TC was 530 °C, showing excellent compatible properties. The BF-BT-xmol%Mn system ceramics showed to obey the Rayleigh law within suitable field regions. The Rayleigh law results indicated that the extrinsic contributions to the dielectric and piezoelectric responses of quenched BF-BT-xmol%Mn ceramics were larger than the un-quenched ceramics. These results presented that the quenched BF-BT-xmol%Mn ceramics were promising candidates for high-temperature piezoelectric devices. This article is protected by copyright. All rights reserved.

Mn-Zn ferrites prepared by different sintering schedules at 1325 °C, 1340 °C, and 1360 °C, have been characterized from the structural, electrical, and magnetic viewpoint. Magnetic losses and complex permeability have been, in particular, measured and analyzed from quasi-static excitation up to 1 GHz. It is observed that lower sintering temperatures and shorter treatment times lead to more homogeneous grain structure and better soft magnetic response at all frequencies. It is shown, however, that, once the contribution by eddy currents is singled out, the energy losses tend to coincide beyond a few MHz in the differently treated samples. The interpretative approach consists in separating the contributions by the domain wall displacements and the magnetization rotations to complex permeability and losses as a function of frequency. This can be accomplished in a relatively simple way in the low induction region described by the Rayleigh law, where these quantities can be quantitatively related and the linear Landau-Lifshitz-Gilbert equation applies, account being taken of the distribution in amplitude and orientation of the local anisotropy fields.

The fluctuating elasticity (FE) model, introduced phenomenologically and developed by Schirmacher [J. Non-Cryst. Solids 357, 518 (2011)], is today the only theoretical framework available to analyze low-temperature elastic acoustic scattering in glasses. Its existing formulations, which neglect the tensorial nature of elasticity and exclude long-range disorder correlations, predict that the acoustic damping coefficients obey the standard Rayleigh scaling law: $\mathrm{\ensuremath{\Gamma}}\ensuremath{\sim}{k}^{d+1}$, with $k$ the acoustic wave vector, in dimension $d$. However, recent numerical data, supported by the analysis of existing experimental results, show that $\mathrm{\ensuremath{\Gamma}}$ does not obey this scaling law but $\mathrm{\ensuremath{\Gamma}}\ensuremath{\sim}\ensuremath{-}{k}^{d+1}\mathrm{ln}k$. Here we analyze in detail how a fully tensorial FE model can be constructed as a long wavelength approximation of the elastic response of the discrete, atomistic, problem. We show that, although it incorporates all long-range correlations, it fails to capture the observed damping in two respects: (i) it misses the anomalous scaling, and predicts the standard Rayleigh law; (ii) it grossly underestimates the amplitude of scattering by about 2 orders of magnitude. This brings clear evidence that the small scale nonaffine displacement fields, although not simply reducible to local defects, play a crucial role in acoustic wave scattering and hence cannot be ignored.

The Rayleigh law describes the linear dependence of the permittivity of a ferroelectric on the applied ac electric field amplitude due to irreversible motions of domain walls. We show that this gives rise to a new equivalent-circuit element predestined to fit the impedance spectra of ferroelectrics based on an accepted physical model. Such impedance spectroscopy is a powerful tool to obtain a dielectric and resistive representation of the entire sample structure. The superiority of the Rayleigh analysis based on impedance spectroscopy compared to the common single-frequency approach is demonstrated for a ferroelectric Si : HfO₂ thin film

Lead–free thin films with excellent high recoverable energy density (Wrec) and efficiency (η) are promising for application in miniature power devices. In this work, Na0.5Bi0.5(Ti,Fe)TiO3 (NBFT) and Ba(Ti,Sr)O3 (BST) systems are chosen for fabricating a series of 7NBFT/nBSxT (where 7, n and x mark the spin–coating cycles of NBFT, BSxT, and Sr2+ doping content, respectively) (n = 0, 2, 3, 4, and x = 0, 5, 10, 15, 20 mol. %) multilayer structures on Pt/Ti/SiO2/Si substrates via sol–gel process to explore the effects from key parameters of BSxT buffer layer on 7NBFT/nBSxT, including thickness (by modulating number of spin–coating cycles) and Sr2+ doping content. The highest Wrec ~ 56.5 J cm−3 and η ~ 66.6% are achieved for the relaxor 7NBFT/3BS0.15T multilayer, which are highly relevant to the enhanced homogeneous breakdown strength (EBD) of 2814 kV cm−1, large gap between the maximum and remnant polarization (Pmax–Pr) of 61.8 μC cm−2, originated from the highest αe and αeE0/einit values as interpreted according to the Rayleigh Law. Besides, the existence of polar nano–regions (PNRs) contributes by lowering the energy for domain back–switching. These provide evidence that NBT–based multilayer thin film is of considerable potential for lead–free energy storage application.

Textured (K0.44Na0.52Li0.04) (Nb0.86Ta0.10Sb0.04)O3 thick film was fabricated by the screen printing method with plate-like NaNbO3 particles as template. Thick film with 75% grain orientation was prepared. Remnant polarization and coercive field observed from the P–E loops of textured thick film were 3.6 μC cm−2 and 21 kV cm−1, respectively. Textured (K0.44Na0.52Li0.04) (Nb0.86Ta0.10Sb0.04)O3 thick film exhibited diffusion behaviour by analysing the temperature dependence of permittivity and loss tangent. The result of leakage current density showed a conduction mechanism of Schottky emission. Piezoelectric (PZT) properties of the thick film were characterized by the relationship of unipolar strain and applied electric field and the PZT constant $\phantom {\dot {i}\!} {d}_{{33}}^{{\ast }}$ of textured thick film reached to 150 pm V−1. Nonlinear PZT property of the thick film was investigated by Rayleigh law.

We analyze basic approaches to the mathematical modeling of motion of the domain walls in ferromagnetics under the influence of external magnetic fields. According to the results of synthesis of the literature data, we present an equation for the displacement of a domain wall with regard for the Brownian distribution of the field. We also describe the available mathematical models constructed on the basis of the dependences of the thickness of domain walls on mechanical stresses, which enable us to determine the maximum displacements of a domain wall and estimate the amplitude of magnetoelastic acoustic emission.

(1-x)K0.5Na0.5Nb0.965Sb0.035O3-x (Bi0.9Al0.1)0.5Na0.5Zr0.42Hf0.58O3 [abbreviated as (1-x)KNNS-xBANZH] ceramics were synthesized by conventional solid reaction method to investigate combined effects of Al and Hf on electrical properties of sodium potassium niobate-based ceramics. The results showed that rhombohedral-tetragonal (R-T) phase boundary was constructed at x = 0.04–0.05 through co-doping, which was demonstrated by refined X-ray diffraction patterns, Raman spectra, transmission electron microscope analysis and temperature-dependent dielectric constants. Consequently, the ceramics with x = 0.045 exhibited significantly improved piezoelectric properties (d33∼451 pC/N, d∗33–404 p.m./V and kp∼0.46). According to calculated intrinsic piezoelectric constants (dinit) by Rayleigh law, reversible non-180° domain motion and efficient domain wall motion were of essential importance for x = 0.045, which also contributed a lot to d∗33. Together with scanning electron microscope analysis, R-T phase coexistence and fine grains accounted for the property enhancement. As a result, the addition of (Bi0.9Al0.1)0.5Na0.5Zr0.42Hf0.58O3 [abbreviated as BANZH] is an effective method to promote electrical properties of K0.5Na0.5Nb0.965Sb0.035O3 [abbreviated as KNNS] piezoceramics and the ceramics with optimized electrical properties can be treated as promising candidates for Pb-based piezoceramics in some usage fields.

Stress is one of the principal external factors affecting the magnetization of materials. A new and improved equation for modeling the magnetomechanical effect has been developed based on extension of the previous equation to include the Rayleigh law. Accordingly the previous theory of the magnetomechanical effect has been refined by including a new linear term in the original model equation.

This paper is the last in a series of three investigating the friction laws and their parametrisation in idealised gravity currents in a rotating frame. Results on the dynamics of a gravity current (Wirth, 2009) and on the estimation of friction laws by data assimilation (Wirth and Verron, 2008) are combined to estimate the friction parameters and discriminate between friction laws in non-hydrostatic numerical simulations of gravity current dynamics, using data assimilation and a reduced gravity shallow water model. I demonstrate, that friction parameters and laws in gravity currents can be estimated using data assimilation. The results clearly show that friction follows a linear Rayleigh law for small Reynolds numbers and the estimated value agrees well with the analytical value obtained for non-accelerating Ekman layers. A significant and sudden departure towards a quadratic drag law at an Ekman layer based Reynolds number of around 800 is shown, in agreement with classical laboratory experiments. The drag coefficient obtained compares well to friction values over smooth surfaces. I show that data assimilation can be used to determine friction parameters and discriminate between friction laws and that it is a powerful tool in systematically connecting models within a model hierarchy.

In this paper, a method for the assessment of magnetic properties of ferromagnetic materials at low magnetic field values is presented. The knowledge of these properties is crucial for the design of magnetically shielded rooms that must be effective against low-frequency external magnetic field disturbances on the order of 0.1– $1~\mu \mathrm {T}$ . The considered shielding structure is composed by a number of layers made of different high-permeability ferromagnetic sheets as grain-oriented electric steel and nickel–iron soft magnetic alloy. The ferromagnetic materials in this region follow the Rayleigh law that is a quadratic law between magnetic flux density and the applied field. Two material-dependent parameters of this law are obtained by a characterization of the material behaviour performed on a scaled prototype of a magnetically shielded room. The procedure is validated on a multilayer shield structure.

Piezoelectric responses of Bi 0.5 (Na 0.82 K 0.18 ) 0.5 TiO 3 (NKBT) ceramic films with thickness of 10–100 μm have been studied in a wide range of ac electric field and frequency. By the Rayleigh law fit, the typical irreversible Rayleigh coefficient, α , for piezoelectric response is 39.69±0.41×10 −18 m 2 V −2 , which is much larger than that of the piezoelectric thin films, indicating significantly enhanced non-180° domain wall motions in NKBT thick films. The logarithmic frequency dependence of the piezoelectric response is associated with domain wall pinning in NKBT thick films, and the frequency dispersion increases with increasing ac field amplitude, which is attributed to the relaxation of non-180° domains.