Langevin Function Research Articles

AIMD study and experimental verification of Ge-doped Fe80P13C7 amorphous soft magnetic alloys

Fe80P13-xC7Gex (x = 0, 2, 4) amorphous alloys were synthesized by rapid solidification technique and studied using Ab-initio molecular dynamics (AIMD) and first-principles calculations based on density-functional theory (DFT) for their electronic structures. Brillouin function was used to calculate saturation magnetization (µ0MS) at elevated temperatures utilizing experimentally measured Curie temperature (TC) and µ0MS at 0 K calculated by DFT.To verify the calculated µ0MS, ingots of Fe80P13-xC7Gex (x = 0, 1, 2, 3, 4, 5, 6) alloy were prepared by melt-spinning. An increase in experimental µ0MS with Ge concentration agrees with the calculated µ0MS. TC also increases with Ge concentration. Partial substitution of Ge for P enhances the µ0MS to 1.63 T (x = 4) from 1.46 T (x = 0), which indicated that the Fe80P13-xC7Gex (x = 4) shows optimum soft magnetic properties.

Tuning the magnetocrystalline anisotropy of rare-earth free L10-ordered Mn1-xTMxAl magnetic alloy (TM = Fe, Co, or Ni) with transition elements

Tuning of the magnetocrystalline anisotropy of MnAl was studied by substituting Mn of MnAl with transition elements (Fe, Co, or Ni). The Brillouin function and semi-empirical Callen and Callen relation predicted the thermal behaviors of saturation magnetization and magnetocrystalline anisotropy energy. First-principles calculations based on density functional theory (DFT) were performed to calculate the electronic structures of Mn0.5TM0.5Al, where TM = Mn, Fe, Co, and Ni. The estimated total magnetic moment of Mn0.5TM0.5Al decreases as the number of valence electrons (n) of TM (e.g., 7 for Mn (3d5 4 s2), 8 for Fe (3d6 4 s2), 9 for Co (3d7 4 s2), and 10 for Ni (3d8 4 s2)) increases. Ni-substituted MnAl becomes ferrimagnetic, while other TM-substituted MnAl retain a ferromagnetic state. Curie temperature rapidly decreases with increasing the valence electrons from 685 K for MnAl to 20 K for Ni-substituted MnAl. Thermomagnetic behaviors of Mn0.5TM0.5Al (TM = Mn, Fe, Co, or Ni) are reported. Our magnetocrystalline anisotropy energy (MAE) calculations demonstrate that the magnetocrystalline anisotropy changes to the in-plane from the out-of-plane (uniaxial) direction for Co– and Ni-substituted MnAl. The K reaches a maximum of 2.98 MJ/m3 at n = 8, i.e., Fe substitution.

Schröder-Based Inverse Function Approximation

Schröder approximations of the first kind, modified for the inverse function approximation case, are utilized to establish general analytical approximation forms for an inverse function. Such general forms are used to establish arbitrarily accurate analytical approximations, with a set relative error bound, for an inverse function when an initial approximation, typically with low accuracy, is known. Approximations for arcsine, the inverse of x − sin(x), the inverse Langevin function and the Lambert W function are used to illustrate this approach. Several applications are detailed. For the root approximation of a function, Schröder approximations of the first kind, based on the inverse of a function, have an advantage over the corresponding generalization of the standard Newton–Raphson method, as explicit analytical expressions for all orders of approximation can be obtained.

Table-like magnetocaloric effect due to field-induced inter-cluster interactions in Eu0.5Sr0.5MnO3–EuMnO3 composites

The table-like behavior of the magnetic entropy change is generally observed in rare-earth-based alloys (e.g. La(Fe,Si)13, Gd50Co45Fe5, Gd56Ni15Al27Zr2, GdCo x Al y etc). Here, we report similar table-like behavior in a bulk perovskite Eu0.5Sr0.5MnO3–EuMnO3 composite system with a working temperature span of ∼80 K for a field change of ∼70 kOe. AC susceptibility and time-dependent DC magnetization relaxation measurements confirm the presence of interacting ferromagnetic clusters with a Vogel–Fulcher freezing temperature of 42.7 K. The Langevin function fittings of the isothermal magnetization curves show a non-monotonous variation of cluster size with temperature. We propose a model for the high-field dependency of the ferromagnetic cluster coupling in the composite system. This widens the transition temperature region of long-range order formation and hence gives rise to the table-like magnetocaloric effect in Eu0.5Sr0.5MnO3–EuMnO3 composite systems.

Room temperature magnetic ordering and in vitro antibacterial potency of Nd3+ doped CeO2 against the pathogenic bacterial strains

In this study, 5% Nd3+ doped Cerium oxide (Ce0.95Nd0.05O2) was prepared by a modified solid-state reaction method. Crystallite size and lattice strain were determined from the Williamson-Hall (W-H) plot. The four distinct peaks in the photoluminescence (PL) spectra in the visible region signify the structural defects within the crystal lattice. Room temperature ferromagnetism (RTFM) was observed from the hysteresis loop, and the mathematical fitting of the hysteresis loop suggests ferromagnetic dominance (∼83%) in the magnetic response. The virgin M-H loop was fitted by the ferromagnetic Brillouin function, which suggests possible domain formation in the sample. The in vitro antibacterial activity of 5% Nd3+doped CeO2 was evaluated against two clinically isolated multidrug-resistant pathogenic bacterial strains, namely MLD 2 (Escherichia coli) and MLD 4 (Staphylococcus aureus), utilizing the minimum inhibitory concentration (MIC), minimum bactericidal concentration (MBC), disc agar diffusion (DAD) assays, etc. Notably, compared to the MLD 4 bacterial strain, the synthesized material revealed comparable better antibacterial results against the Gram-negative MLD 2 and in comparison, to the MLD 2 bacterial strain, a relatively higher concentration of the sample is required to completely eradicate the bacterial population of MLD 4. This implies that Ce0.95Nd0.05O2 can destroy the growth of the two tested bacterial strains. In the context of cytotoxicity, up to a certain concentration, the synthesized particle demonstrated no appreciable toxicity towards healthy PBMCs.

Magnetocrystalline anisotropy of interstitially and substitutionally Sn-doped MnBi for high temperature permanent magnet applications

First-principles calculations were performed to calculate the electronic structures of low temperature phase (LTP) MnBi (Mn50Bi50) and substitutionally and interstitially Sn-doped MnBi [Mn50Bi25Sn25, (Mn0.5Bi0.5)66.7Sn33.3]. Brillouin function predicts the temperature dependence of saturation magnetization M(T). Sn substitution for Bi in MnBi (Mn50Bi25Sn25) changes the magnetocrystalline anisotropy constant (Ku) from −0.202 MJ/m3 (the in-plane magnetization) for LTP MnBi to 1.711 MJ/m3 (the out-of-plane magnetization). In comparison, the Ku remains negative but slightly decreases to −0.043 MJ/m3 when Sn is interstitially doped in MnBi [(Mn0.5Bi0.5)66.7Sn33.3]. The Curie temperature (TC) decreases from 716 K for LTP Mn50Bi50 to 445 K for Mn50Bi25Sn25 and 285 K for (Mn0.5Bi0.5)66.7Sn33.3. Mn50Bi25Sn25 has a lower magnetic moment of 5.034 μB/f.u. but a higher saturation magnetization of 64.2 emu/g than (Mn0.5Bi0.5)66.7Sn33.3 with a magnetic moment of 6.609 μB/f.u. and a saturation magnetization of 48.2 emu/g because the weight and volume of the substitutionally Sn-doped MnBi are smaller than the interstitially Sn-doped MnBi. The low Curie temperature and magnetization for Sn-doped MnBi are attributed to the high concentration of Sn. Thus, future study needs to focus on low Sn-concentrated MnBi.

Optical solitons of the complex Ginzburg-Landau equation having dual power nonlinear form using $\varphi^{6}$-model expansion approach

This paper employs a novel $\varphi ^{6}$-model expansion approach to get dark, bright, periodic, dark-bright, and singular soliton solutions to the complex Ginzburg-Landau equation with dual power-law non-linearity. The dual-power law found in photovoltaic materials is used to explain nonlinearity in the refractive index. The results of this paper may assist in comprehending some of the physical effects of various nonlinear physics models. For example, the hyperbolic sine arises in the calculation of the Roche limit and the gravitational potential of a cylinder, the hyperbolic tangent arises in the calculation of the magnetic moment and the rapidity of special relativity, and the hyperbolic cotangent arises in the Langevin function for magnetic polarization. Frequency values, one of the soliton's internal dynamics, are used to examine the behavior of the traveling wave. Finally, some of the obtained solitons' three-, two-dimensional, and contour graphs are plotted.

Synthesis and characterization of pH-triggered doxorubicin-conjugated polydopamine-coated cobalt ferrite nanoparticles for in-vitro/in-vivo studies in liver cancer therapy

Polydopamine-coated cobalt ferrite nanoparticles (PDA-coated CF-NPs) were synthesized and characterized for their potential in drug delivery evaluation. Small-angle X-ray scattering (SAXS) showed post-coating spherical shape of magnetic PDA NPs. The scattering intensity at zero angle (I(0)) of PDA-coated CF (0.01) had higher scattering intensity than CF-NPs (0.49 × 10-3). PDA increased the flexibility degree of unfolding on Kratky plot shifting from 1.05 to 1.87 at same qRg values. The pair-distance distribution function (P(r)) of PDA-coated CF-NPs indicated a more homogeneous and compact scattering structure. GNOM indicated larger size for CF, while PDA-coated CF (a narrower and taller peak) displayed a more compact and uniform scattering profile. PDA reduced crystallinity size to 2.51 nm with amorphous peaks at 38.12°, 44.28°, and 64.43° degree and infrared spectrum confirmed PDA incorporation onto CF-NPs. PDA-coated CF had 1.5-fold larger pore size and enhanced surface area. CF and PDA-coated CF exhibited “S” shape behavior in Langevin function without hysteresis. The kinetic models suggested Fickian diffusion and sustainable release for the erosion-dominated release of the PDA-coated CF. In vitro, doxorubicin (DOX)-carried PDA-coated CF (D-P@CF) showed low toxicity to HepG2 cells; in vivo, liver tumor inhibition indicated medication delivery potential for cancer treatment.

Fractional Maxwell viscoelastic model to explain dynamic magneto-viscoelastic properties of an isotropic magnetorheological elastomer containing flake-shaped magnetic particles

ABSTRACT A modified fractional Maxwell viscoelastic model was developed for the dynamic magneto-viscoelastic behavior of an isotropic magnetorheological elastomer (MRE) having flake-shaped iron particles. This model can capture the low-frequency and high magnetic field static friction contribution with frequency, both in storage and loss modulus. The magnetic dipole-dipole interaction model is used to find the influence of the external magnetic field on viscoelastic behavior. The variation of field-induced modulus with the magnetic field is well described using the Langevin function. The predicted and experimental measured storage and loss modulus agree with the present model with an accuracy of ≥99%. The present model is compared with the results of spherical particle-based MRE.

Magnetic properties of graphene oxide decorated with magnetite nanoparticles

The present work reports a study on the magnetic properties of graphene oxide decorated with magnetite nanoparticles (GO·Fe3O4), at room temperature. A total of ten samples were prepared via co-precipitation method, controlling the magnetite incorporation on the GO surface. Magnetic characterization was performed using a vibrating sample magnetometer. Transmission electron microscopy and X-ray diffraction were employed for structural investigation. All systems presented magnetisation loops that resemble a Langevin function at the high field region, but also have non-zero coercivity and remanent magnetisation. These features are attributed to the size distribution of the magnetic nanoparticles. A portion of the magnetite particles is small enough to present superparamagnetic behaviour, while the bigger ones are ferrimagnetic. Using the in-field δMR interaction-plots technique, it was observed that demagnetising interactions are predominant. Magnetic parameters like coercivity and remanent magnetisation, however, are mainly governed by average particle size, and not by such interactions.

Modified Jiles-Atherton Model for Dynamic Magnetization in X-Space Magnetic Particle Imaging.

Magnetic particle imaging (MPI) is a promising medical modality that can image superparamagnetic iron-oxide nanoparticle (SPIO) concentration distributions safely and with high sensitivity. In the x-space reconstruction algorithm, the Langevin function is inaccurate in modeling the dynamic magnetization of SPIOs. This problem prevents the x-space algorithm from achieving a high spatial resolution reconstruction. We propose a more accurate model to describe the dynamic magnetization of SPIOs, named the modified Jiles-Atherton (MJA) model, and apply it to the x-space algorithm to improve the image resolution. Considering the relaxation effect of SPIOs, the MJA model generates the magnetization curve via an ordinary differential equation. Three modifications are also introduced to further improve its accuracy and robustness. In magnetic particle spectrometry experiments, the MJA model shows higher accuracy than the Langevin and Debye models under various test conditions. The average root-mean-square error is 0.055, 83% and 58% lower than the Langevin and Debye models, respectively. In MPI reconstruction experiments, the MJA x-space improves the spatial resolution by 64% and 48% compared to the x-space and Debye x-space methods, respectively. The MJA model shows high accuracy and robustness in modeling the dynamic magnetization behavior of SPIOs. By integrating the MJA model into the x-space algorithm, the spatial resolution of MPI technology was improved. Improving the spatial resolution by using the MJA model helps MPI have a better performance in medical fields, including cardiovascular imaging.

Electroelasticity of copolymer networks

We introduce a novel statistical mechanics-based analysis of electroactive copolymer networks. This enables to express the coupled response of a multi-species polymer network in terms of a Gibbs-like energy-density function. Following the common affine deformation assumption we derive the expressions for the network response in terms of the species’ properties and molar fraction. Next, we specialize the results to the class of monomers with spontaneous electric dipoles and determine the monomers’ orientational distribution in terms of a generalized Langevin function. We reveal that the coupled response of the copolymer depends on the contrast between the intensities of the dipolar moments of the species and their fraction of the chain end-to-end length. In the long chains limit we obtain closed-form expressions for the electrical and mechanical fields. Next, we consider two-species copolymers and find that the long chains approximation is valid for actuation stretch ratios up to 40% of the locking stretch. We also demonstrate that the maximal susceptibility is attained when the contour lengths of the two species are equal. Moreover, the overall electroelastic response is enhanced if the monomers with the weaker dipole are longer than those with the stronger one. Finally, we show that an appropriate choice of actual monomers may lead to copolymers with relative dielectric constants of more than 30.

Influence of the hierarchical architecture of multi-core iron oxide nanoflowers on their magnetic properties

Magnetic properties of superparamagnetic iron oxide nanoparticles are controlled mainly by their particle size and by their particle size distribution. Magnetic properties of multi-core iron oxide nanoparticles, often called iron oxide nanoflowers (IONFs), are additionally affected by the interaction of magnetic moments between neighboring cores. The knowledge about the hierarchical structure of IONFs is therefore essential for understanding the magnetic properties of IONFs. In this contribution, the architecture of multi-core IONFs was investigated using correlative multiscale transmission electron microscopy (TEM), X-ray diffraction and dynamic light scattering. The multiscale TEM measurements comprised low-resolution and high-resolution imaging as well as geometric phase analysis. The IONFs contained maghemite with the average chemical composition gamma-Fe_{2.72pm 0.02}O_4. The metallic vacancies located on the octahedral lattice sites of the spinel ferrite structure were partially ordered. Individual IONFs consisted of several cores showing frequently a specific crystallographic orientation relationship between direct neighbors. This oriented attachment may facilitate the magnetic alignment within the cores. Individual cores were composed of partially coherent nanocrystals having almost the same crystallographic orientation. The sizes of individual constituents revealed by the microstructure analysis were correlated with the magnetic particle sizes that were obtained from fitting the measured magnetization curve by the Langevin function.

First-order phase transition in high-performance La(Fe,Mn,Si)13H despite negligible hysteresis

Optimizing the performance of magnetocaloric materials is facilitated by understanding the thermomagnetic transitions they undergo, including the order of these transitions and their strength. Those exhibiting strong first-order phase transitions (FOPT) are accompanied by large heating and cooling responses but with relatively small cyclic responses, while materials with second-order (SOPT) character exhibit moderate heating and cooling responses. However, the lack of hysteresis could partially compensate for the lower magnitudes with a more cyclic response. One way to effectively maximize the cyclic response, combining the advantages of FOPT and SOPT, is to fine tune the transition towards the borderline of FOPT-SOPT, which can minimize hysteresis. For the well-known La(Fe,Si)13 family, it is challenging to identify and/or evaluate the critical point where FOPT crossovers to SOPT based on conventional techniques. To address these ambiguities, in this work, we apply the field dependence exponent n criteria to a series of lowly hysteretic and high-performance La(Fe,Mn,Si)13H magnetocaloric materials with compositions close to the critical one. Even if the sample with the lowest hysteresis resembles characteristics of SOPT, it is evidently identified as undergoing FOPT from the n criteria: (1) existence of n > 2 overshoot and (2) n at the transition temperature, ntransition, is 0.37. This proximity to the critical composition (ntransition=0.4) further explains the low hysteresis observed. This FOPT character of the series is confirmed by temperature-dependent 57Fe Mössbauer spectrometry studies, fitting the hyperfine field to the Bean-Rodbell model instead of the usual Brillouin function. As it is a zero-field method, the confirmation by Mössbauer spectrometry gives further strength to the n-criterion.

Heat conduction in herbertsmithite: Field dependence at the onset of the quantum spin liquid regime

We report thermal conductivity measurements on single crystals of herbertsmithite, over a wide range of temperatures (0.05-120 K) in magnetic fields up to 15 T. We also report measurements of the thermal Hall effect, found to be vanishingly small. At high temperatures, in the paramagnetic regime, the thermal conductivity has a negligible field dependence. Upon cooling and the development of correlations, the onset of a clear monotonic field dependence below about 20 K signals a new characteristic temperature scale that may reflect the subtle crossover towards the quantum spin liquid regime. Deconfined spinons, if present, are not detected and phonons, as the main carriers of heat, are strongly scattered by the intrinsic spin excitations and the magnetic defects. In view of the colossal fields required to affect the intrinsic spins, most of the field-induced evolution is attributed to the progressive polarization of some magnetic defects. By elaborating a phenomenological model, we extract the magnetization of these main scattering centers which does not resemble the Brillouin function for free spins 1/2, requiring to go beyond the paradigm of isolated paramagnetic spins. Besides, the onset of a nonmonotonic field dependence below about 2 K underlines the existence of another characteristic temperature scale, previously highlighted with other measurements, and sheds new light on the phase diagram of herbertsmithite down to the lowest temperatures.

Homogenization of ferrofluid flow models in porous media with Langevin magnetization law

The paper is concerned with the homogenization of the equations describing the flow of a ferrofluid through a heterogeneous porous medium Ω in the presence of an applied magnetic field. We discuss two models where the magnetization M is parallel to the magnetic field H. In the first one M and H satisfy the relation M=λ01ΩfH in Ω, where λ0 is a positive constant and 1Ωf is the characteristic function of Ωf (the pore space). In the second model, M and H satisfy the Langevin magnetization law M=MsL(b1|H|)|H|1ΩfH, where L is the Langevin function given by L(x)=1tanh⁡x−1x, Ms is the saturation magnetization and b1 is a positive physical constant. The velocity and the pressure satisfy the Stokes equation with a Kelvin magnetic force. We perform the homogenization of the equations of each of the two models. Using the two-scale convergence method, we rigorously derive the homogenized equation for the magnetic potential and determine the asymptotic limit of the magnetization. Then we rigorously derive a Darcy law.

Ferromagnetism and dielectric properties in Zn0.95−xNdxTM0.05O (TM=Co, Fe) nanocrystals: Collective role of grain boundaries and oxygen vacancies

Herein, Zn0.95−xNdxTM0.05O (TM = Co, Fe) (x = 2%, 6%) nanocrystals with spherical shape, wurtzite-like hexagonal structure, and average crystallite size of 35 nm were prepared by sol-gel method, those exhibit ferromagnetic (FM) behavior at room temperature. X-ray photoelectron spectroscopy (XPS) results confirmed the successful incorporation of dopants ions such as Nd3+ and Co2+/Fe2+ into the host ZnO lattice. Size-strain analysis based on the Williamson-Hall plot was performed to evaluate the crystallite size and lattice strain. High-resolution transmission electron microscope characterization shows that the fine nanocrystals are separated by grain boundaries (GBs). The calculated specific grain boundary area (SGB) of 60 × 106m2m-3 in the co-doped samples is above the threshold value of 5.3 × 107 m2m-3. The UV–Visible analysis shows that the energy band gap decreases with increasing Nd3+ concentration. The photoluminescence spectra (PL) show a broad emission band in the visible region corresponding to the various structural defects and oxygen vacancies in the prepared nanocrystals responsible for the activation of bound magnetic polarons (BMPs). Fitting the experimental magnetization data with the BMP model based on the Langevin function shows that the density of magnetic polarons (1020cm−3) in the doped samples is equal to the typical percolation threshold. The dielectric constant and dielectric loss was observed to increase at lower frequency due to the presence of GBs and oxygen vacancies, and this behavior was explained based on Koop's phenomenological theory. Based on these results, it was demonstrated that the FM response and the dielectric properties in studied samples are strongly associated with GBs and oxygen vacancies.

Lattice-Distortion-Induced Change in the Magnetic Properties in Br-Defect Host CsPbBr3Perovskite Quantum Dots

Herein, we report temperature- and field-induced magnetic states in CsPbBr3 perovskite quantum dots (PQDs) attributed to Br defects. We find that temperature-dependent structural distortion is the main source of various temperature-induced magnetic states in Br-defect host CsPbBr3 PQDs. Comprehensively examined magnetization data through Arrott plots, Langevin and Brillouin function fitting, and structural analysis reveal the presence of various oxidation states (i.e., Pb0, Pb+, Pb2+, and Pb3+) yielding different magnetic states, such as diamagnetic states above 90 K, paramagnetic states below ≈90 K, and perhaps locally ordered states between 58 and 90 K. It is realized from theoretical fits that paramagnetic ions exist (i.e., superparamagnetic behavior) due to Br defects causing Pb+ (and/or Pb3+ ions) in the diamagnetic region. We anticipate that our findings will spur future research of the development of spin-optoelectronics, such as spin light-emitting diodes, and spintronics devices based on CsPbBr3 PQDs.

Magnetization of Zn1−xCoxO nanoparticles: single-ion anisotropy and spin clustering

The magnetization of Zn1−xCoxO (0.0055 ≤ x ≤ 0.073) nanoparticles has been measured as a function of temperature T (1.7 K ≤ T ≤ 10 K) and for magnetic field up to 65 kOe using a SQUID magnetometer. Samples were synthesized by three different growth methods: microwave-assisted hydrothermal, combustion reaction and sol-gel. For all studied samples, the magnetic properties derive from the antiferromagnetic (AF) spin clustering due to the Co2+ nearest-neighbors. The magnetization curves have been fitted to the Brillouin function (BF). We have used the normalized root mean square error (NRMSE) between the experimental and calculated BF curves as goodness of fit indicator. For all samples, the magnetization of the Co2+ ions has a BF behavior at T ≥ 6 K with NRMSE≈ 0.004 typically, but below 6 K, it shows a notable deviation. The magnitude of the deviation is found to increase with decreasing temperature and reaches its maximum value at T = 1.7 K with NRMSE ≈ 0.026. The observed deviation as well as the magnetization curve shape have been successfully explained by a model based on single-ion anisotropy (SIA) with uniaxial symmetry for the Co2+ ions and the assumption of randomly oriented nanoparticles. Fits of the magnetization curves have been performed by using this model. The axial-SIA parameter D and the effective Co concentration x¯ corresponding to the technical saturation value of the magnetization were used as adjustable parameters. The quality of the fit using this model is clearly improved with NRMSE around 0.007 at T = 1.7 K. The obtained value of the axial-SIA parameter D (typically D = 4.4 K) is slightly larger that the bulk value D = 3.97 K. No significant change of D has been observed as a function of the Co concentration or the growth process. Comparison of the concentration dependence of the obtained values of x¯ with predictions based on the nearest-neighbors cluster model has been made. The result shows an enhancement of the AF spin clustering independent of the growth method which can be ascribed to a clamped non-random distribution of the cobalt ions in the nanoparticles. The approach of the local concentration (xL) has been used to quantify the observed deviation from randomicity. We obtained xL = 1.4 x. Assuming the picture of a ZnO core / Zn1−xCoxO shell nanoparticle, the thickness of the shell has been estimated from the ratio xL∕x to be one fourth of the diameter of the core.

Nonlinear response theory for Markov processes. IV. The asymmetric double-well potential model revisited.

The dielectric response of noninteracting dipoles is discussed in the framework of the classical model of stochastic reorientations in an asymmetric double-well potential (ADWP). In the nonlinear regime, this model exhibits some pecularities in the static response. We find that the saturation behavior of the symmetric double-well potential model does not follow the Langevin function and only in the linear regime are the standard results recovered. If a finite asymmetry is assumed, then the nonlinear susceptibilities are found to change the sign at a number of characteristic temperatures that depend on the magnitude of the asymmetry, as has been observed earlier for the third-order and fifth-order responses. If the kinetics of the barrier crossing in the ADWP model is described as a two-state model, then we can give analytical expressions for the values of the characteristic temperatures. The results for the response obtained from a (numerical) solution of the Fokker-Planck equationfor the Brownian motion in a model ADWP behaves very similarly to the two-state model for high barriers. For small barriers no clear-cut timescale separation between the barrier crossing process and the intrawell relaxation exists and the model exhibits a number of timescales. In this case, the frequency-dependent linear susceptibility at low temperatures is dominated by the fast intrawell transitions and at higher temperatures by the barrier crossing kinetics. We find that for nonlinear susceptibilities the latter process appears to be more important and the intrawell transitions play only a role at the lowest temperatures.