Brownian motion in a periodic potential: Application to dielectric relaxation

The relaxational dynamics of a planar rotator in anM-fold cosine potential subject to a random torque is investigated in detail. For the case of a periodic potential with large barrier height, the numerical results of the relaxation dynamics are in complete agreement with an approximate analytical solution. The latter, is derived on assuming a harmonic potential at the bottom of the potential minima and a large time-scale separation between the short-time libration inside each potential minima and a long-time hopping phenomenon over the potential barriers. ForM2, the hopping phenomenon is found to be the dominant feature of the orientational autocorrelation function. The average hopping time is explained satisfactorily in terms of the Kramers activation rate theory. In particular a complete agreement is found between the numerical results of the escape rate and those obtained from the modified Kramers' predictions valid for low friction coefficient. The cosine model is applied to the study of dielectric spectroscopy. The particle mobility and the complex permitivity, of a dielectric material are calculated by numerical solutions for rotational velocity and orientational auto-correlation functions, respectively. The main features of the experimental observables are determined analytically and compared to the corresponding numerical results. The applicability of the plane rotator model to dielectric spectroscopy is also discussed.From Jan. 1^st 1985: Institut für Theoretische Physik, RWTH, Aachen, FRG     

Time behaviour of non-linear stochastic processes in the presence of multiplicative noise: From Kramers' to Suzuki's decay

The diffusional regime of a Brownian particle in a double-well potential in the presence of both additive and multiplicative noise is explored. As a relevant effect of the multiplicative noise, the escape rate from a well is shown to change from the small value of the Kramers theory into the large relaxation rate of the Suzuki regime. It is shown, furthermore, that the time required to get equilibrium in a well after sudden application of multiplicative noise (the activation time) is very much shorter than the Kramers relaxation time. We envisage therefore an operational scheme making available multiplicative noise for a short interval of time (for example using a light pulse) as an efficient tool to get a fast process of escape from a well. These results are obtained by using a continued-fraction algorithm which makes it possible even to successfully deal with the decay of an unstable state at the critical point.     

Tunneling as a Classical Escape Rate Induced by the Vacuum Zero-point Radiation

We make a brief review of the Kramers escape rate theory for the probabilistic motion of a particle in a potential well U(x), and under the influence of classical fluctuation forces. The Kramers theory is extended in order to take into account the action of the thermal and zero-point random electromagnetic fields on a charged particle. The result is physically relevant because we get a non-null escape rate over the potential barrier at low temperatures (T → 0). It is found that, even if the mean energy is much smaller than the barrier height, the classical particle can escape from the potential well due to the action of the zero-point fluctuating fields. These stochastic effects can be used to give a classical interpretation to some quantum tunneling phenomena. Relevant experimental data are used to illustrate the theoretical results.     

Derivative analyticity relations and asymptotic energies

Starting from the general principles of the S-matrix theory theorems are derived showing that derivative analyticity relations analogous to those of Bronzan, Kane and Sukhatme hold at asymptotic energies if the high-energy limits of certain physical quantities exist.     

Preparation,electric conductivity and dielectrical properties of Cu6PS5I-based superionic composites

Composite samples based on microcrystalline Cu₆РS₅I superionic conductors, embedded in matrices of polyvinylacetate, epoxy and conducting glue, are produced. Temperature and frequency studies of complex electric conductivity and dielectric permittivity as well as dielectric loss of the obtained composites are carried out. The effect of differences in the composite production technology on the temperature of a second-order phase transition in the superionic phase, values of electrical conductivity, activation energy and dielectric permittivity is shown.     

Exact analytic relation between quantum defects and scattering phases with applications to Green’s functions in quantum defect theory

The relation between the quantum defects, , and scattering phases, , in the single-channel Quantum Defect Theory (QDT) is discussed with an emphasis on their analyticity properties for both integer and noninteger values of the orbital angular momentum parameter . To derive an accurate relation between and for asymptotically-Coulomb potentials, the QDT is formally developed for the Whittaker equation in its general form “perturbed” by an additional short-range potential. The derived relations demonstrate that is a complex function for above-threshold energies, which is analogous to the fact that is complex for below-threshold energies. The QDT Green's function, , of the “perturbed” Whittaker equation is parameterized by the functions and for the continuous and discrete spectrum domains respectively, and a number of representations for are presented for the general case of noninteger . Our derivations and analyses provide a more general justification of known results for nonrelativistic and relativistic cases involving Coulomb potentials and for a Coulomb plus point dipole potential. Received 25 June 1999     

Kramers–Kronig transform for the surface energy loss function

A new pair of Kramers–Kronig (KK) dispersion relationships for the transformation of surface energy loss function Im[−1/(ɛ + 1)] has been proposed. The validity of the new surface KK transform is confirmed, using both a Lorentz oscillator model and the surface energy loss functions determined from the experimental complex dielectric function of SrTiO₃ and tungsten metal. The interband transition strength spectra (J_cv) have been derived either directly from the original complex dielectric function or from the derived dielectric function obtained from the KK transform of the surface energy loss function. The original J_cv trace and post-J_cv trace overlapped together for the three modes, indicating that the new surface Kramers–Kronig dispersion relationship is valid for the surface energy loss function.     

On the Urbach rule in sulphur-implanted Cu6PS5I superionic conductors

Cu₆PS₅I superionic crystals, grown using chemical vapour transport, were implanted by sulphur ions. The ion implantation effect on the phase transitions is studied by temperature isoabsorption investigation of the optical absorption edge. For the implanted crystals the optical absorption edge shape is studied in the temperature range 77-320 K, the parameters of exciton-phonon interaction, resulting in the Urbach behaviour of the optical absorption edge, are determined, the temperature dependences of the optical pseudogap and Urbach energy are obtained. The implantation effect on the ordering-disordering processes in Cu₆PS₅I superionic conductors is studied.     

On an application of the causality principle to the theory of ion transport processes

We suggest an approach for ion transport processes that is motivated by analogies with the S-matrix theory. By using causality-based analyticity assumptions, conductivity frequency pre-factors are expressed in terms of the transfer amplitude. This provides a means of dealing with a unified scheme that includes ionic oscillations in wells and classical-like escaping. As an application, we use this technique to reproduce results obtained in terms of the Kramers problem without invoking the Kramers’ Fokker-Planck equation or details of the transfer mechanism in solid electrolytes.     

Analytical and numerical approaches to calculating the escape function for the emission of medium-energy electrons from uniform specimens

A program for the simulation of electron transport by the Monte Carlo method has been developed. This program implies the model of single scattering and dielectric approach (to calculate the characteristics of an inelastic interaction). The escape functions for aluminum, germanium, and gold in the 0.012–20 keV energy range were calculated. The characteristic lengths determining the electron dynamics (the elastic and inelastic mean free paths, isotropization length, and integral path) were calculated using the differential cross sections for both elastic and inelastic interactions for these elements. The analysis of the relations between the characteristic lengths made it possible to determine the applicability range of the analytical expressions for the emission functions obtained in [1]. The comparison of the results obtained analytically and numerically confirmed the conclusion of [1] about the form of the analytical approximation of the emission function for electrons of various energies and showed the validity of the obtained analytical expressions for the escape lengths of electrons.     

On the Role of the Results Obtained by the Analytical Theory of Elementary Particles

To prove that it is possible to obtain many results (such as the asymptotic relations of Pameranchuk type, Froissart inequality, Martin theorem, etc.) of the analytical theory of elementary particles in any theory where the ordinary analyticity of dispersion amplitude is replaced by a certain generalized analyticity (such as the p-analyticity of Poloji, the generalized analyticity of Vekua). Hence we cannot consider these results as criteria for the validity of the analyticity at high energies. Therefore, in order to check the analyticity we ought to establish the relations which can not be deduced from any generalized analyticity, except ordinary analyticity.     

Semiconductor dielectric function,excitons and the Penn model

Improved computation of the dielectric function considering excitonic effects and long wavelength is performed and compared with the nearly free electron band approximation, similarly with the Penn’s model case. New expressions for the real and imaginary part of the dielectric function are presented and the real part compared with the Penn’s result. The obtained functions satisfy the Kramers–Krönig relations, in contrast with earlier results in the literature. In addition, our improved dielectric function presents a coefficient of 2/3 for small-gap approximation (different from the value of 1 in the original Penn model) is very close to the value 0.62 obtained in [Can. J. Phys. 53 (1975) p.2549] from pure numerical procedures. The obtained dielectric function also is used in a rough and stimative analysis of the metal–insulator transition in molecular hydrogen being the critical densities determinated near the experimental values for the hydrogen coming from other approach. The approximated expressions and critical values are given and the usefulness of the rough methods involved in the determination of the critical points briefly discussed.     

Lattice-assisted proton hopping in oxides at low temperatures

Stimulated diffusion of protons in oxides such as ABO₃ crystals and rutile TiO₂ is discussed in the context of quantum Brownian motion. A self-consistent lattice-assisted proton hopping (LAPH) model is developed by going from white noise (characteristic of the standard stochastic theory of superionic conduction) to colored noise in the Markovian limit. This model differs from the commonly used ion jump models in that the hydrogen diffusion rate prefactor is identified as a quantity proportional to the frequency of phonon assistance. Application of the quantum fluctuation–dissipation theorem suggests that the dynamic activation energy for diffusion is a function of a bath-mode frequency. The LAPH model can predict enhanced rates of barrier jumping at room temperature compared to thermally activated proton diffusion. This indicates that low-temperature solid oxide devices are potential candidates for use in hydrogen energy research. The LAPH model offers a valid explanation for the mechanism of high protonic mobility recently observed for TiO₂ in a picosecond transient pump-probe experiment. This unexpected dominant lattice relaxation channel must be considered as a new classical-like (but low-temperature) proton transfer mechanism. For vibration-assisted protonic jumps to occur at low temperature, the phonon assistance must be classified as a low-frequency vibration specific to each lattice.     

Theory of oxidation/reduction-induced valence transformations of metal ion dopants in oxide crystals mediated by oxide-vacancy diffusion: II. Kinetic analysis

We consider theoretically valence transformations of doping metal ions in oxide crystals induced by oxidation and reduction obtained by changes in the ambient oxygen partial pressure. Three types of oxygen vacancies are assumed to mediate transformations: neutral, singly ionized, and doubly ionized. In the companion part I paper we provide thermodynamic analyses yielding concentration relations among the oxygen vacancy, metal ions, holes and electrons, as functions of the ambient oxygen pressure. In the present companion part II paper we provide time dependent concentration profiles of the various species and reaction rate profiles. The diffusion exhibits a complex behavior; under some conditions, it may be described by a constant diffusivity, and is symmetric with respect to oxidation and reduction. However, under a wide range of conditions, the ionic state changes are highly asymmetric with respect to oxidation and reduction. For example, in the case of a neutral vacancy, a very narrow reaction front may establish during reduction. In the inverse (oxidation) process, however, the different species' profiles are quite smooth.     

色散关系的简单证明

在此短文中,我们给色散关系一个简单的但不严格的证明。证明的方法为将因果振幅对中间态展开,分别研究能量分母及相应的分子的解析性。能量分母的解析性是较显然的。至于分子的解析性的证明,我们先研究μ²换为—p²时的相应量(μ为介子质量,p为核子在Breit坐标系中的动量,介子核子散射为我们所考虑的具体对象),研究其解析性,通过一个变数的变换而达到我们证明的目的。在此方法中,p²可以大至M²—μ²,M为核子质量。我们也考虑了在位场散射中相位移η(k)的解析开拓问题,证明了如果位能在r→∞处形如e^-αr(a>0),则η(k)可以开拓至|Imk|<1/2α的区域。     

Energy diffusion controlled reaction rate in dissipative Hamiltonian systems

In this paper the energy diffusion controlled reaction rate in dissipative Hamiltonian systems is investigated by using the stochastic averaging method for quasi Hamiltonian systems. The boundary value problem of mean first-passage time (MFPT) of averaged system is formulated and the energy diffusion controlled reaction rate is obtained as the inverse of MFPT. The energy diffusion controlled reaction rate in the classical Kramers bistable potential and in a two-dimensional bistable potential with a heat bath are obtained by using the proposed approach respectively. The obtained results are then compared with those from Monte Carlo simulation of original systems and from the classical Kramers theory. It is shown that the reaction rate obtained by using the proposed approach agrees well with that from Monte Carlo simulation and is more accurate than the classical Kramers rate.     

Ionic pseudopotential for semiconductors without cut-off parameter for core-repulsion with application to Si

An ionic pseudopotential for semiconductors is proposed, which consists of a set of continuous exponential functions. Introduced damping and amplitude parameters into the pseudopotential are to be treated as adjustable. The most important features of the proposed pseudopotential is that (1) it has no sharp cut-off parameter for the core-repulsion and (2) it is continuous and has continuous derivatives to arbitrary order. The proposed pseudopotential is applied to Si and the adjustable parameters are determined so as to be consistent with the Si crystal empirical pseudopotential of high quality by taking a valence electron dielectric screening effect into account. The effectiveness of the proposed ionic pseudopotential is discussed by (1) comparing the calculated ionic energy levels of Si with experiments, (2) checking the consistency between the ionic and crystal pseudopotentials for Si, and so on.     

Escape by diffusion from a square well across a square barrier

The problem of escape of a particle by diffusion from a square potential well across a square barrier is studied on the basis of the one-dimensional Smoluchowski equation for the space- and time-dependent probability distribution. For the model potential the Smoluchowski equation is solved exactly by a Laplace transform with respect to time. In the limit of a high barrier the rate of escape is given by an asymptotic result similar to that derived by Kramers for a curved well and a curved barrier. An approximate analytic formula is derived for the outward time-dependent probability current in terms of the width and depth of the well and the width and height of the barrier. A similar expression holds for the complete probability distribution.     

Sum rules for the vertex function from local commutativity (I) Relations for the vacuum expectation values (the coordinate space vertex function)

A set of sum rules is derived based upon the analyticity properties of the Källén-Wightman holomorphy domain, i.e. essentially the assumptions of local commutativity properties for the occuring field operators, together with assumptions on boundedness properties of the vertex functions. The results are expressed as relations between the vacuum expectation values of the theory.