Multiwell Potential Research Articles

Self-Ordered Second-Component Nb Clusters in KNbxTa1?xO3 Solid Solutions and Their Physical Properties

Semi-empirical Hartree-Fock calculations using the intermediate neglect of the differential overlap (INDO) method, for self-ordered cubic symmetry clusters of seven Nb ions in KTaO 3 are performed with the aim of verifying the cluster model [1] of second component-induced phase transitions in ferroelectric perovskite matrices. It is shown that a seven-particle cluster in KTaO 3 :Nb has two types of states with different nature. Namely, a state with a dilatation combined with an off-center displacement of the central Nb ion in [111] directions in the cluster, and a state with a full-symmetric compression without any off-center effect. The consequences of such cluster structures on the multi-well potential are discussed. In particular, the resonance tunneling between cluster excited states characterized by low frequency on-center vibrations, on the one hand, and excited states of off-center cluster vibrations, on the other, gives rise to low lying tunnel states. The latter states interact with soft TO modes and form the displacive type ferroelectric phase transition induced by impurities. This new type of displacive-like ordering co-exists with order-disorder effects with development of the central peak. All these results are in agreement with experimental data on KTaO 3 with Nb impurities. The results of the semi-empirical INDO calculations for seven Nb ions forming an impurity cluster in KTaO 3 confirm the key assumptions of the cluster model of induced phase transitions.

Multiwell potentials of endohedral complexes of icosahedral symmetry

We examine multiwell potentials that correspond to the displacements of off-center ions in complexes with icosahedral symmetry (dodecahedrons, icosahedrons, fullerenes, etc.) along symmetry directions (toward vertices, midpoints of edges, and centers of faces). An expression is derived for an effective Hamiltonian, which describes the behavior of endohedral complexes with off-center ions placed in external electric fields of arbitrary strength and orientation. We find the eigenvalues of this Hamiltonian and calculate the intensities of the lines of all possible transitions between tunneling levels. We also predict and analyze the spectra of paraelectric resonances in the absence of an external static electric field (zero-field resonances) and in the presence of such a field. Finally, we provide recommendations for detecting these effects and discuss the specific features of the effects and the possibility of studying them.

Simulation of the effect of reorientation on the electron spin echo decays using the stochastic liouville formalism. Application to interstitial hydrogen in SrF 2 and BaF 2

The thermal evolution of the two pulses (2p) electron spin echo (ESE) decay of interstitial atomic hydrogen formed in hydrolysed SrF 2 and BaF 2 is studied. The influence of the atomic jumps along the (110) directions, that go from the static to the fast motion limit, is included in order to simulate the 2p-ESE decays. An extension of the Stochastic Liouville formalism is developed, which gives explicit expressions for the 2p-ESE decay signals in the case of paramagnetic impurities that can reorient among a discrete number of sites. A comparison between the calculated decays and the experimental ones provides information about the effective displacement of the minima of the multiwell potential in which interstitial hydrogen moves. A discussion of the results taking into account the structure of the host lattices is also given.

Dynamics of interacting clusters and dielectric response in relaxor ferroelectrics

The dielectric response in relaxor ferroelectrics is analyzed in the framework of a model for the polarization dynamics in the presence of polar clusters. We associate the origin of polar clusters with atoms displaced from their centrosymmetrical positions even above ${T}_{c}.$ Their collective hopping in multiwell potentials induced by disorder is analogous with the situation in glasses. The theory explicitly takes into account the distribution of cluster reorientation frequencies and the effect of cluster-cluster interactions in highly polarizable crystals, which we describe in terms of the local field distribution function. The dielectric constant is obtained from an integral master equation for the polarization dynamics in the presence of a time-dependent electric field. The theory is applied to the analysis of the shape of the frequency-dependent permittivity in the typical relaxor ferroelectric PST as a function of temperature. Comparison of theory with experiment shows that in contrast to earlier assumptions, the observed Vogel-Fulcher dependence of the permittivity maximum is a consequence of the Vogel-Fulcher temperature dependence of the cluster-reorientation frequency.

Paraelectric resonance of centers with icosahedral symmetry

The paraelectric resonance spectrum of non-central impurity ions situated in crystal fields of icosahedral symmetry is predicted. A multiwell potential is considered that corresponds to the displacement of the ions from the center of symmetry toward the vertices of the icosahedron, the faces of a dodecahedron, and the pentagonal faces of a fullerene molecule. The effective Hamiltonian of endohedral complexes in external electric fields with arbitrary orientation is obtained, its eigenvalues are found, and the line intensities of different transitions between the tunnelling levels are calculated.

Relaxation of classical particles in anharmonic multi-well potentials

Using extensive numerical and some exact asymptotic analyses, we argue that the canonical ensemble relaxation of a classical particle in an anharmonic multi-well potential landscape with leading quartic anharmonicity exhibits a 1/ t decay to equilibrium. In addition, we provide numerical evidence of algebraic decay of form 1/ t φ , 0 < φ < 1, for stronger leading anharmonicities. Our results have important consequences for relaxation studies in several problems of considerable interest in condensed matter physics such as in the Krumhansl-Schrieffer model of displacive structural phase transitions, and in various complex systems.

Isotope effect in spin fluctuation models

We have studied the d-wave superconducting transition temperature T c and the isotope effect in the spin fluctuation models. We found that the isotope exponent α is at least one order order of magnitude smaller than typical observed values in the cuprates with models coupling to harmonic phonon degrees of freedom and reasonable limits on coupling strength and phonon spectra. Models with coupling to non-linear tunneling modes in a strongly anharmonic multi-well lattice potential and realistic coupling strengths have effectively larger isotopic mass dependences and may therefore account for the observed magnitudes of α.

Stable forward shooting for eigenvalues and expectation values

Internal differentiation techniques are used to produce a simple but highly accurate forwards shooting method for eigenvalues and expectation values of the Schrödinger equation. A multi-well potential is used as a test case.

Isotope effect and tT c suppression in d-wave superconductors

We have studied the isotope effect on the superconducting transition temperatureTc ofd-wave pairing instabilities due to antiferromagnetic spin fluctuation exchange. We found that models with conventional coupling toharmonic phonon degrees of freedom and reasonable limits on coupling strength and phonon spectra, give values for the isotope exponentα which are at least one order of magnitude smaller than typical observed values ofα in the cuprates. Models with coupling to non-linear tunneling modes in a stronglyanharmonic multi-well lattice potential and realistic coupling strengths have effectively larger isotopic mass dependences andmay therefore account for the observed magnitudes ofα.

Comment on 'Bifurcation behaviour in structural phase transitions with multi-well potentials'

For original work see ibid., vol.2, no.16, p.3737 (1990). Normand and co-workers predicted anomalous temperature dependence of a few thermal averages induced by the bifurcation behaviour or the local breaking of symmetry. Against their prediction, based on the phi 4 atom potential model, the double-quadratic-well (DQW) potential model leads to different temperature dependences on the specific heat and the mean square displacement. It turns out that the anomalous temperature dependence of the phi 4 model arises from the anharmonic character of the potential barrier as well as the bifurcation behaviour. Making a distinction between the two kinds of anomaly, the authors present a plausible temperature variation of the bifurcation behaviour on the basis of the DQW model of the above two quantities.

Bifurcation Phenomena in Sensorymotor Training of a Robot Manipulator

Our framework is grasp oriented, intelligent wrist positioning in the approach phase of a multifingered robot hand. This paper focuses on the performance analysis of trainer sets in a multitarget multiobstacle environment. Targets or attractors (obstacles or repellers) are task relevant (irrelevant) parts which are imperfectly recognized. Due to this uncertainty we model the joint epace as a cellular space and due to the attractor repeller multiplicity the workspace of the robot hand is also modeled as a multiwell potential. Learning performance of a given trainer is analyzed under nonlinear robot dynamics in this cell space and in the parameter space. Bifurcation that causes the system to be structurally unstable is studied from the different perspectives of basin erosion, fractal fingering and pitchfork bifurcations which witness periodicity splitting. The effect of bifurcation in the learning rate is also considered

CHAOS IN THE HERMITIAN ONE-MATRIX MODEL

The recursion coefficients Ri, which appear in the orthogonal polynomial method of solution for the Hermitian one-matrix model, are determined numerically for values of N up to a thousand. For some cases a chaotic behavior appears in some range, preventing a smooth flow from odd to even multicrititical models. This behavior is studied both for single-well and multiwell potentials. For multiwell potentials, the coefficients Ri generically tend toward more than one function in the N→∞ limit, and this structure is analyzed for small i using the Gaussian approximation.

Finite element method versus finite difference methods : theoretical investigations of some models of potential of biological interest

In view of evaluating the specific advantages of the finite difference methods and the finite element method (FEM) in the resolution of the Schrödinger nuclear equation, we performed accurate investigations of some one-dimensional multi-well potentials. Two finite difference methods were used : an Adams method and the powerful Numerov-Cooley method. The FEM investigations were carried out with quadratic interpolation functions and the shooting method was used in place of the usual algorithms.

New theorem on action integrals and oscillation periods of motion in polynomal multi-well potentials

The present theoretical investigation shows that the oscillation periods and the corresponding classical action integrals, associated with motion of a given energy in the local wells of a multi-well anharmonic potential, are linearly dependent. The proof makes use of the analytic behaviour of the local momentum of the particle in the complex position variable. The theorem applies to polynomial potentials of arbitrary even degree.

Superconductivity and dissipation in small-diameter Pb-In wires.

Superconductivity in very small-diameter Pb-In wires has been studied through measurements of the resistance in the low-current limit, and the voltage-current characteristics. These properties were measured as functions of temperature and magnetic field, and in the presence of microwave radiation. The results are analyzed in terms of several different theoretical models. Predictions based on possible granularity and the Coulomb blockade, on the effects of mesoscopic fluctuations, and on the phenomena of thermal and quantum phase slip, are all considered. It appears that over most of the range which has been studied the behavior is best described by phase slip, and evidence for both thermally activated and quantum phase slip is found. The behavior during a phase-slip event is closely analogous to the motion of a particle in a multiwell potential. In this picture, thermally activated phase slip occurs when the particle is thermally activated over the potential barrier which separates adjacent wells, while quantum phase slip corresponds to tunneling of the particle between wells. We have also observed behavior which implies the existence of discrete energy levels of the particle within a well. While most of the results can be understood at least qualitatively with use of the phase-slip picture, certain aspects remain unexplained.

Bifurcation behaviour in structural phase transitions with multi-well potentials

A model solid with a double-well potential at each site and intersite coupling J is considered, where J is sufficiently small that kTC is much less than the depth kT0 of the well. The effect of the local breaking of symmetry that occurs around T0 is investigated for a single site. Anomalies are particularly noticeable in the oscillation frequencies, mean squared displacement and heat capacity. There are examples of materials and computer simulations that display such behaviour.

An algorithm to study tunneling in a wide class of one-dimensional multiwell potentials, II

We describe a simple algorithm to determine the behaviour of the ground state wave function and to compute the lowest part of the spectrum of the Schrödinger operator in the semiclassical limit, when the potential has many absolute minima. This approach may be useful in the study of complex systems, like long molecular chains or superlattice structures. As an application we determine the number of the energy levels in the first band if the potential is a binary sequence of two types of barriers and give a method to handle more general cases. We estimate statistically the versatility of our algorithm, with the aid of a computer program that implements it. It turns out that ∼99% of potentials in some general classes are solvable with our method.

An algorithm to study tunnelling in a wide class of one-dimensional multiwell potentials. I

The author proves a theorem which gives an algorithmic solution to the problem of finding the logarithmic derivative of the ground-state wavefunction of one-dimensional systems. By means of this quantity, as is well known, one can determine the lowest part of the spectrum of the Hamiltonian by probabilistic methods. The author shows that, in some natural classes of potentials, the complexity of the algorithm is less than N3, where N is the number of the absolute minima of the potential. The author's approach allows a systematic treatment of cases of much greater complexity than those analysed so far in the literature and it can be useful in the study of physical systems like, for example, long molecular chains or superlattice structures.

Tunnelling instability via perturbation theory

The authors study the semiclassical limit of low lying states in a multiwell potential by rigorous perturbative techniques. In particular they obtain in a simple way tunnelling instability and localisation of wavefunctions under deformations of symmetric potentials.

Numerical solutions for brownian motion of particles in a periodic potential

The Kramers equation for one dimensional motion of brownian particles in a multiwell cosine potential is solved by a series expansion method followed by numerical solution of the resulting set of linear differential equations to obtain angular velocity and orientational autocorrelation functions (a.c.f.s) in the Laplace domain. The former, inverted by analytic function fitting, take the correct forms at zero and large (harmonic limit) potentials and intermediate behaviours compare well with those found using other numerical techniques. Solutions evaluated using delta function initial conditions are of interest in the study of pendula or of chemical dissociation, while solutions evaluated under equilibrium initial conditions and with imaginary Laplace variable iw are important in dielectric spectroscopy. Application of the theory both to superionic conductivity and dielectric-far-I.R. absorption of rotator phases and liquids is demonstrated.