Types Of Potential Wells Research Articles

Quantum nature of proton transferring across one-dimensional potential fields* *Project supported by the National Natural Science Foundation of China (Grant No. 11474285).

Proton transfer plays a key role in the applications of advanced energy materials as well as in the functionalities of biological systems. In this work, based on the transfer matrix method, we study the quantum effects of proton transfer in a series of one-dimensional (1D) model potentials and numerically calculate the quantum probability of transferring across single and double barriers (wells). In the case of single barriers, when the incident energies of protons are above the barrier height, the quantum oscillations in the transmission coefficients depend on the geometric shape of the barriers. It is found that atomic resonant tunneling (ART) not only presents in the rectangular single well and rectangular double barriers as expected, but also exists in the other types of potential wells and double barriers. For hetero-structured double barriers, there is no resonant tunneling in the classical forbidden zone, i.e., in the case when the incident energy (E i) is lower than the barrier height (E b). Furthermore, we have provided generalized analysis on the characteristics of transmission coefficients of hetero-structured rectangular double barriers.

Externally driven molecular ratchets on a periodic potential surface: a rate equations approach.

The long time dynamics of molecular ratchets on a 1D periodic potential energy surface (PES) subjected to an external stimulus is studied using the rate equation method. The PES consisting of repeated waveforms made of two peaks is considered as an example of a spatially symmetric or asymmetric PES. This PES may, for example, correspond to diffusion of a bipedal molecule that moves along an atomic track via an inchworm walk mechanism [Raval et al., Angew. Chem., Int. Ed., 2015, 54, 7101]. Generalisation to a PES consisting of an arbitrary number of peaks of various heights is straightforward. Assuming the validity of the transition state theory (TST) for the calculation of the transition rates between neighbouring potential wells, the probability of occupying each type of potential well on the PES is obtained analytically, and then the net current for the molecules to move preferentially in a particular direction under application of external fields over a long time is derived. Note that different to methods based on solving numerically the corresponding Fokker-Plank equation, our method is entirely analytical in the limit of weak external fields. The results of the analytical calculations are compared with the exact numerical solution of the derived rate equations. The following external stimuli are considered: constant, sinusoidal and shifted sinusoidal fields due to either a spatially uniform thermal gradient or an electrostatic field. The possible applications of the method for extracting energy from the Brownian motion under load and separating molecules of different chiralities on the surface are also discussed.

Neutron Scattering Reveals the Dynamic Basis of Protein Adaptation to Extreme Temperature

To explore protein adaptation to extremely high temperatures, two parameters related to macromolecular dynamics, the mean square atomic fluctuation and structural resilience, expressed as a mean force constant, were measured by neutron scattering for hyperthermophilic malate dehydrogenase from Methanococcus jannaschii and a mesophilic homologue, lactate dehydrogenase from Oryctolagus cunniculus (rabbit) muscle. The root mean square fluctuations, defining flexibility, were found to be similar for both enzymes (1.5 A) at their optimal activity temperature. Resilience values, defining structural rigidity, are higher by an order of magnitude for the high temperature-adapted protein (0.15 Newtons/meter for O. cunniculus lactate dehydrogenase and 1.5 Newtons/meter for M. jannaschii malate dehydrogenase). Thermoadaptation appears to have been achieved by evolution through selection of appropriate structural rigidity in order to preserve specific protein structure while allowing the conformational flexibility required for activity.

Coherent transport in linear arrays of quantum dots: The effects of period doubling and of quasi-periodicity

We evaluate the phase-coherent transport of electrons along linear structures of varying length, which are made from two types of potential wells set in either a periodic or a Fibonacci quasi-periodic sequence. The array is described by a tight-binding Hamiltonian and is reduced to an effective dimer by means of a decimation-renormalization method, extended to allow for connection to external metallic leads, and the transmission coefficient is evaluated in a T-matrix-scattering approach. Parallel behaviors are found for the energy dependence of the density of electron states and of the transmittivity of the array. In particular, we explicitly show that on increasing its length the periodic array undergoes a metal–insulator transition near single occupancy per dot, whereas prominent pseudo-gaps emerge away from the band center in the Fibonacci-ordered array.

The quasiclassical Langevin equation and its application to the decay of a metastable state and to quantum fluctuations

We report on investigations on the consequences of the quasiclassical Langevin equation. This Langevin equation is an equation of motion of the classical type where, however, the stochastic Langevin force is correlated according to the quantum form of the dissipation-fluctuation theorem such that ultimately its power spectrum increases linearly with frequency. Most extensively, we have studied the decay of a metastable state driven by a stochastic force. For a particular type of potential well (piecewise parabolic), we have derived explicit expressions for the decay rate for an arbitrary power spectrum of the stochastic force. We have found that the quasiclassical Langevin equation leads to decay rates which are physically meaningful only within a very restricted range. We have also studied the influence of quantum fluctuations on a predominantly deterministic motion and we have found that there the predictions of the quasiclassical Langevin equations are correct.

Ionic conductivity in glasses: A Monte-Carlo study of ordered and disordered one-dimensional models

Ionic conductivity in ordered and disordered one-dimensional lattice gas models has been investigated using a Monte-Carlo simulation technique. Two one-dimensional lattices have been investigated: (a) an ordered array of deep and shallow potential wells and (b) a disordered array of the same two types of potential wells. An interesting influence of disorder on activation has been observed based upon the Arrhenius plots of the ionic conductivity. The activation energy for the disordered arrangement of sites is lower than that for the ordered one. This result is in agreement with the experimentally-observed increase of activation energy for β-eucryptite upon proceeding from the glassy to the crystalline state, and arises from decreased ion-ordering in the disordered potential. Effects of ion-ion correlations in one-dimensional systems are also discussed and these provide a simple physical interpretation of the charge correlation factor, f c.

Pyroelectricity and piezoelectricity are not true volume effects

The total polarization of a crystal is sensitive to the state of the surface, even for very large volumes. This point was emphasized by Larmor and later became the basis of more detailed discussions of pyroelectricity and piezoelectricity, by Woo and by the author, questioning the usual textbook theorems. The earlier arguments are summarized and supplemented by a new model: a discussion of electronic redistribution in one-dimensional ordered periodic array of two types of potential wells.

Approximate Solutions to the One-Dimensional Schrödinger Equation by the Method of Comparison Equations

The method of "comparison equations" for obtaining approximate solutions to the one-dimensional Schr\"odinger equation is discussed. In this method, one Schr\"odinger equation, ${\ensuremath{\psi}}^{\ensuremath{'}\ensuremath{'}}(x)+{k}^{2}(x)\ensuremath{\psi}(x)=0$, is transformed into another, ${v}^{\ensuremath{'}\ensuremath{'}}(z)+{K}^{2}(z)v(z)=0$, by a simultaneous change of independent and dependent variables, $x\ensuremath{\rightarrow}z$, $\ensuremath{\psi}\ensuremath{\rightarrow}v$. Then $\ensuremath{\psi}(x)$ and $v(z)$ are related by $\ensuremath{\psi}(x)=v(z){(\frac{\mathrm{dz}}{\mathrm{dx}})}^{\ensuremath{-}(\frac{1}{2})}$ whenever ${k}^{2}(x)$ and ${K}^{2}(z)$ satisfy the relation ${K}^{2}(z){(\frac{\mathrm{dz}}{\mathrm{dx}})}^{2}={k}^{2}(x)\ensuremath{-}(\frac{1}{2}) 〈z;x〉$. Here $〈z;x〉=\frac{{z}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}}{{z}^{\ensuremath{'}}}\ensuremath{-}(\frac{3}{2}){(\frac{{z}^{\ensuremath{'}\ensuremath{'}}}{{z}^{\ensuremath{'}}})}^{2}$ is the Schwarzian derivative of $z$ with respect to $x$ and the prime indicates $\frac{d}{\mathrm{dx}}$. A set of "best" criteria for the transformed potential ${K}^{2}(z)$ is obtained, where by "best" is meant that we can completely neglect $〈z;x〉$ in first approximation and yet not have the turning-point problems that plague the WKB method (which is a special case of the comparison-equation method). The WKB method sets ${K}^{2}(x)\ensuremath{\equiv}1$ and neglects $〈z;x〉$; the result is that the transformation $x\ensuremath{\rightarrow}z$, $\ensuremath{\psi}\ensuremath{\rightarrow}\ensuremath{\nu}$ is singular at the turning points, where ${k}^{2}(x)=0$. We choose ${K}^{2}(z)$ to have the proper asymptotic behavior far from the turning points, so that ${z}^{\ensuremath{'}}\ensuremath{\approx}1$; hence $〈z;x〉\ensuremath{\approx}0$ in these regions, and we match the zeroes of ${k}^{2}(x)$ and ${K}^{2}(z)$ in order to keep the transformation regular. This method is applied to various potentials with one and two turning points. Transmission and reflection coefficients $T$ and $R$ and transmitted and reflected phase shifts $\ensuremath{\mu}$ and $\ensuremath{\nu}$ are calculated for potentials with one turning point and potential barriers, and expressed in terms of the energy $E$ and the quantity $W=|\ensuremath{\int}{{x}_{1}}^{{x}_{2}}k\mathrm{dx}|$, where ${x}_{1,2}$ are the possibly complex turning points, ${k}^{2}({x}_{1,2})=0$. Quantization rules, in terms of the classical action, are derived for various types of potential wells.

Electron spin resonance studies of methyl radicals stabilized on porous VYCOR glass: various surface interactions, second-order splitting, and a linewidth temperature study

Electron spin resonance studies have been performed on methyl radicals stabilized on the surface of porous VYCOR glass. As previously reported, two very different methyl radicals, denoted Me and Me′, were seen. Results indicate that the radical Me′ is most probably weakly bound to the boroxane group (=B—O—B=) which is prominent on the surface of high temperature (700–900 °C) pretreated porous VYCOR glass. Four satellite lines about each Me line, denoted previously as radical X, were observed when the surface was pretreated at lower temperatures (400–500 °C). In an earlier publication radical X was attributed to the interaction between methyl radicals and the isotopic surface species 11B. In this study two additional satellite lines about each Me line are reported as well as a reassignment of the origin of all satellite lines. Four of the six satellite lines have been assigned to forbidden "spin–flip" transitions while the other two have been assigned to direct interaction between methyl radicals and the surface hydroxyl protons. Support for the irreversibility of high temperature dehydroxylation of porous VYCOR glass is also presented.A partial resolution of the second-order splitting of each of the central pair of lines was achieved for the normal methyl radical. The value of the second-order splitting was between 220 and 230 mG. The linewidths of the two central lines were measured at 77 °K. The samples were then allowed to warm up to various temperatures and then recooled to 77 °K. The linewidths were smaller after completing this cycle. Storage of the sample at 77 °K allowed the linewidths to return to their original values. These linewidths effects are explained by postulating the existence of two different types of potential wells on the surface.

Configuration Mixing and the Effects of Distributed Nuclear Magnetization on Hyperfine Structure in Odd-ANuclei

The theory of Blin-Stoyle and of Arima and Horie, in which the deviations of the nuclear magnetic moments from the single-particle model Schmidt limits are ascribed to configuration mixing, is used as a model to account quantitatively for the effects of the distribution of nuclear magnetization on hyperfine structure (Bohr-Weisskopf effect). A diffuse nuclear charge distribution, as approximated by the trapezoidal Hofstadter model, is used to calculate the required radial electron wave functions. A table of single-particle matrix elements of ${R}^{2}$ and ${R}^{4}$ in a Saxon-Woods type of potential well is included. Explicit formulas are derived to permit comparison with experiment. For all of the available data satisfactory agreement is found. The possibility of using hyperfine structure measurements sensitive to the distribution of nuclear magnetization in a semiphenomenological treatment in order to obtain information on nuclear configurations is indicated.

Electronic Structure of Disordered One-Dimensional Chains

The positions of the electronic levels in a variety of finite one-dimensional models of disordered alloys and of liquids have been evaluated numerically over a wide range of energies. The disordered alloys consist of two types of potential wells thrown together in a random sequence, but in equal proportions. In the disordered case the tightly bound states give rise to narrow bands in which the distribution of states cannot be described by a smooth function. The distribution of states can be interpreted in terms of an approximation in which each electron is confined to its own group of adjacent identical atoms. At high energy levels the disordered alloys obey perturbation theory, at least in a qualitative fashion, and the ordered alloy differs from the disordered one because of the effect of Bragg reflections from the superlattice. At high energies the distribution of states can be described by a reasonably smooth density function. Between the high energy range and the tightly bound levels, in some of the cases a transition region occurs, in which the levels are distributed irregularly, but without well-defined gaps and without being susceptible to the simple explanation of localized states. The total number of states (not counting the two spin possibilities as separate states) up to a well-pronounced energy gap is always an integral multiple of one-half the total number of atoms in the chain. The liquids consist of a series of identical wells separated by a variable well-to-well space. In a liquid the forbidden energy ranges are either smaller than in the strictly periodic crystal, or else nonexistent. The extent to which an energy gap in the strictly periodic crystal will be affected by the introduction of dispersion in the spacing depends on the original gap width as well as on the sensitivity of the gap position, in the strictly periodic case, to uniform dilations. In both the liquid cases and the disordered alloy models a distinction must be made between energy ranges in which the density of states is very low and ranges in which it is strictly zero.