Parabolic Potential Barrier Research Articles

Two mechanisms of conduction in polycrystalline SnO 2

Impedance spectroscopy was carried out on SnO 2 thick films exposed to different atmospheres. Oxygen desorption and adsorption during heating and cooling processes, respectively, onto the grain surface affects the resistance and electrical capacitance of the sensors. Double parabolic Schottky-type potential barriers at grain boundaries were assumed to analyze the tunneling and thermionic contributions to conductivity. In addition, the influence of the temperature on the oxygen diffusion into the grains annihilating oxygen vacancies was studied. It was deduced that in-diffusion substantially affects the conductivity.

Beyond the dynamic density functional theory for steady currents: Application to driven colloidal particles in a channel

Motivated by recent studies on the dynamics of colloidal solutions in narrow channels, we consider the steady state properties of an assembly of noninteracting particles subject to the action of a traveling potential moving at a constant speed, while the solvent is modeled by a heat bath at rest in the laboratory frame. Here, since the description we propose takes into account the inertia of the colloidal particles, it is necessary to consider the evolution of both positions and momenta and study the governing equation for the one-particle phase-space distribution. First, we derive the asymptotic form of its solutions as an expansion in Hermite polynomials and their generic properties, such as the force and energy balance, and then we particularize our study to the case of an inverted parabolic potential barrier. We numerically obtain the steady state density and temperature profile and show that the expansion is rapidly convergent for large values of the friction constant and small drifting velocities. On the one hand, the present results confirm the previous studies based on the dynamic density functional theory (DDFT): On the other hand, when the friction constant is large, it display effects such as the presence of a wake behind the barrier and a strong inhomogeneity in the temperature field which are beyond the DDFT description.

Non-Markovian diffusion over a parabolic potential barrier: Influence of the friction-memory function

The over-passing probability across an inverted parabolic potential barrier is investigated according to the classical and quantal generalized Langevin equations. It is shown that, in the classical case, the asymptotic value of the over-passing probability is determined by a single dominant root of the "characteristic function," and it is given by a simple expression. The expression for the over-passing probability is quite general, and details of dissipation mechanism and memory effects enter into the expression only through the dominant root of the characteristic equation.

Features of light propagation in chiral media

Light propagation in uniaxial chiral liquid crystals with large pitch was studied. Vicinities of turning points were analyzed, and the effect of extraordinary wave transmission through bandgap layers was considered. It was shown that the description of wave transmission through the forbidden zone in the general case requires taking into account the existence of four singular points, two of which are responsible for the interaction of extraordinary rays and two points are responsible for the interaction of ordinary and extraordinary rays. Ray transmission in the case of wide bandgap layers was analyzed, when regions of applicability of the Wentzel-Kramers-Brillouin (WKB) approximation take place between singular points. The case of narrow bandgap layers, where singular points merge, was studied in detail. It was shown that the effect of ray transmission through such a zone and the above-barrier reflection effect is equivalent to the tunneling effect for the parabolic potential barrier. Angular dependences of the intensity of rays subject to internal refraction and transmitted through the bandgap layer were calculated. It was shown that the tunneling effect can be experimentally observed.

Radiation loss of a nanotaper: Singular Gaussian beam model

This paper presents a model of a subwavelength diameter adiabatic microfiber taper (nanotaper), which allows an asymptotically accurate solution of the wave equation. The evanescent field near the nanotaper is expressed through a Gaussian beam having a singularity at the nanotaper axis. For certain values of parameters of the nanotaper, when it has a swell in the middle and narrows down to zero at the infinity, the nanotaper is lossless. For other values, when the nanotaper has a biconical shape, it exhibits an exponentially small radiation loss, which is determined as a tunneling rate through an effective parabolic potential barrier. The latter case represents an exceptional example of the radiation loss being distributed along the length of an adiabatic nanotaper rather than being localized near focal circumferences in the evanescent field region.

Quantum damped oscillator II: Bateman’s Hamiltonian vs. 2D parabolic potential barrier

We show that quantum Bateman’s system which arises in the quantization of a damped harmonic oscillator is equivalent to a quantum problem with 2D parabolic potential barrier known also as 2D inverted isotropic oscillator. It turns out that this system displays the family of complex eigenvalues corresponding to the poles of analytical continuation of the resolvent operator to the complex energy plane. It is shown that this representation is more suitable than the hyperbolic one used recently by Blasone and Jizba.

Quantum mechanics of damped systems. II. Damping and parabolic potential barrier

We investigate the resonant states for the parabolic potential barrier known also as inverted or reversed oscillator. They correspond to the poles of meromorphic continuation of the resolvent operator to the complex energy plane. As a by-product we establish an interesting relation between parabolic cylinder functions (representing energy eigenfunctions of our system) and a class of Gel’fand distributions used in our recent paper.

Zero-Energy Flows and Vortex Patin Quantum Mechanics

We show that zero-energy flows appear in many particle systems as well as in single particle cases in 2-dimensions. Vortex patterns constructed from the zero-energy flows can be investigated in terms of the eigenstates in conjugate spaces of Gel'fand triplets. Stable patterns are written by the superposition of zero-energy eigenstates. On the other hand vortex creations and annihilations are described by the insertions of unstable eigenstates with complex-energy eigenvalues into the stable patterns. Some concrete examples are presented in the 2-dimensional parabolic potential barrier case. We point out three interesting properties of the zero-energy flows (i) the absolute economy as for the energy consumption, (ii) the infinite variety of the vortex patterns, and (iii) the perfect stability of the vortex patterns.

Inverse Kramers formula and fusion dynamics of heavy ions

Fusion probability of colliding heavy ions is evaluated by exactly solving a Langevin equation with a parabolic potential barrier. The model, which accommodates dissipation in the approaching phase, leads to an inverse Kramers formula and an expression of the extra-push energy. In the very specific case of an overdamped regime, an Arrhenius-type formula is obtained.

The contribution of tunneling to the diffusion of protons and deuterons in rare gas solids

The stability and diffusion of protons and deuterons in rare gas matrices are reexamined. These are known to be stabilized in rare gas matrices in the form of linear, centrosymmetric Rg2H+ cations. The elementary step in their diffusion, displacement from one Rg–Rg bond to a neighboring one, can be modeled as an isomerization of the triangular Rg3H+ cation. Using an analytic approximation for the thermally averaged transmission coefficients for tunneling through and reflection by a truncated parabolic potential barrier [R. T. Skodje and D. G. Truhlar, J. Phys. Chem. 85, 624 (1981)], we calculate the rate constants for this elementary diffusion step. The calculated rate constants are consistent with all experimental observations and confirm that tunneling makes the dominant contribution to the diffusion of protons and deuterons in rare gas solids. Deuteration reduces the tunneling rates by 5 to 8 orders of magnitude, which agrees with the observation that D+ in rare gas solids is signficantly more stable than H+.

Investigation on anomalous diffusion for nuclear fusion reactions

A simple model for diffusion of a particle with initial velocity coupled to a non-Ohmic environment over a barrier is proposed. An exact expression for the passing probability of the particle over a parabolic potential barrier is obtained. It is shown that in the case of subdiffusion, more than half of the particles cannot overcome the barrier, no matter what the initial velocity is. However, in the case of superdiffusion, a minimum critical velocity needed for passing over the barrier is found, which can lead to a strong friction and make the passing probability increase with kinetic energy, but still slower than in the case of normal diffusion. Effects of quantum diffusion and zero point fluctuation are also discussed. The model is applied to solve a long-standing problem in nuclear fusion reactions, though the mechanism found in this study can have a rather broad application.

Semiclassical approach to states near the potential barrier top

Within the framework of the instanton approach, we present analytical results for the following model problems: (i) particle penetration through a parabolic potential barrier, where the instanton solution practically coincides with the exact (quantum) one; (ii) descriptions of highly excited states in anharmonic potentials of two types: double-well X 4 and decay X 3. For the former potential, the instanton method accurately reproduces not only single-well and double-well quantization, but also a crossover region (in contrast to the standard WKB approach that fails to describe the crossover behavior); for the latter potential, the instanton method allows studying the resonance broadening and collapse phenomena. We also investigate resonance tunneling that plays a relevant role in many semiconducting devices. We show that the instanton approach gives exact (quantum) results in a broad range of energies. Applications of the method and of the results are applicable to various systems in physics, chemistry, and biology exhibiting double-level behavior and resonance tunneling.

Zero-energy solutions and vortices in Schrödinger equations

All two-dimensional Schr\"{o}dinger equations with symmetric potentials \break $(V_a(\rho)=-a^2g_a \rho ^{2(a-1)/2} {with} \rho=\sqrt{x^2+y^2} {and} a\not=0)$ is shown to have zero energy states contained in conjugate spaces of Gel'fand triplets. For the zero energy eigenvalue the equations for all $a$ are reduced to the same equation representing two-dimensional free motions in the constant potential $V_a=-g_a$ in terms of the conformal mappings of $\zeta_a=z^a$ with $z=x+iy$. Namely, the zero energy eigenstates are described by the plane waves with the fixed wave numbers $k_a=\sqrt{mg_a}/\hbar$ in the mapped spaces. All the zero energy states are infinitely degenerate as same as the case of the parabolic potential barrier (PPB) shown in ref. \cite{sk4}. Following hydrodynamical arguments, we see that such states describe stationary flows round the origin, which are represented by the complex velocity potentials $W=p_a z^a$, ($p_a$ being a complex number) and their linear combinations create almost arbitrary vortex patterns. Examples of the vortex patterns in constant potntials and PPB are presented.

Statistical mechanics for unstable states in Gel'fand triplets and investigations of parabolic potential barriers.

Free energies and other thermodynamical quantities are investigated in canonical and grand canonical ensembles of statistical mechanics involving unstable states which are described by the generalized eigenstates with complex energy eigenvalues in the conjugate space of Gel'fand triplet. The theory is applied to the systems containing parabolic potential barriers (PPB's). The entropy and energy productions from PPB systems are studied. An equilibrium for a chemical process described by reactions A+CB <==>AC+B is also discussed.

Entropy burst from parabolic potentials

The change of the ground-state energy is investigated in a thermodynamical process by using the model described by one-dimensional harmonic oscillator + two-dimensional isotropic parabolic potential barrier. When two systems are suddenly mixed, the lowest energy changes through the mixing and the entropy burst occurs.

The dislocation origin and model of excess tunnel current in GaP p-n structures

An excess tunnel current in GaP epitaxial nondegenerate p-n junctions on GaP and Si substrates was studied. An important experimental result is that the slope of exponential current-voltage (I–V) characteristic (in lnI–V coordinates) is independent of the width of the space-charge region, i.e., on n-and p-region doping levels. This fact is unexplained by existing models. A dislocation shunt model based on multihop tunneling through a dislocation line, which may be considered as a chain of parabolic potential barriers, is proposed. The density of dislocations predicted by this model is in agreement with the transmission electron microscopy (TEM) observations.

Stationary flows of the parabolic potential barrier in two dimensions

In the two-dimensional isotropic parabolic potential barrier $V(x, y)=V_0 -m\gamma^2 (x^2+y^2)/2$, though it is a model of an unstable system in quantum mechanics, we can obtain the stationary states corresponding to the real energy eigenvalue $V_0$. Further, they are infinitely degenerate. For the first few eigenstates, we will find the stationary flows round a right angle that are expressed by the complex velocity potentials $W=\pm\gamma z^2/2$.

Operator methods of the parabolic potential barrier

The one-dimensional parabolic potential barrier dealt with in an earlier paper [T. Shimbori, T. Kobayashi, Nuovo Com B 115 (2000) 325] is re-examined from the point of view of operator methods, for the purpose of getting generalized Fock spaces.

Role of inertial and non-Markovian effects on activated barrier crossing dynamics for charge transfer reactions in solution

Charge transfer processes in a polar solvent, modeled as an activated crossing over a parabolic potential barrier, are studied here through the Grote–Hynes (G–H) theory using the recent microscopic approaches to the dielectric friction based on mode coupling theory. The calculated frequency-dependent friction consisting of the short-range as well as the long-range contributions is used to obtain the barrier crossing rates based on the G–H theory, and for comparison also Kramers' theory using the zero-frequency value of the friction. The effect of considering the inertial and non-Markovian motion of the solvent molecules on the calculated friction as well as the barrier crossing rates is investigated. The inertial component of solvent motion is found to be an important factor.

Dwell Time of Particles in Tunneling Barriers of Arbitrary Shape

For a one-dimensional barrier of arbitrary shape, an expression of the dwell time of tunneling particles has been obtained by subdividing it into infinitesimal rectangular barrier elements and then summing the dwell times spent by particles inside the barrier elements, and the dwell times in a parabolic potential barrier were calculated. The results show that the dwell times are almost independent of the barrier width as it is greater than a characteristic width Wc. This can be ascribed that in the width range beyond Wc, the tunneling probabilities through the barrier elements are very small, thus the times taken to tunnel through them make little contribution to the summation of the dwell times in all the barrier elements.