Exact solutions for spectra and Green's functions in random one-dimensional systems

For chains of harmonic oscillators with random masses a set of equations is derived, which determine the spatial Fourier components of the average one-particle Green's function. These equations are valid for complex values of the frequency. A relation between the spectral density and functions introduced by Schmidt is discussed. Exact solutions for this Green's function and the less complicated characteristics function-the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions-are derived for exponential distributions of the masses. For some cases the characteristic function is calculated numerically. For gamma distributions the equations are cast in the form of ordinary, higher order differential equations; these have been solved numerically for determining the characteristic function. For arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the Green's function are discussed.The method is also applied to chains where the spring constants and/or the masses have random values. Also for these systems exact solutions are discussed; for exponential distributions, e.g., of both masses and spring constants the characteristic function is expressed in Bessel functions. The relation with certain random relaxation models is shown. Finally, X-Y Hamiltonians with random exchange constants and/or magnetic fields-or, equivalently, tight-binding electron models with diagonal and/or off-diagonal disorder-are considered. Here the Green's function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin. New solutions for the characteristic function and Green's function are derived for a number of cases, including exponentially distributed magnetic fields and power law distributed exchange constants.     

Quantum localization in one-dimensional quasi-random systems and magnetic breakdown

The magnetic breakdown in metals is shown to cause the appearance of a new class of one-dimensional quasi-random “incommensurable” systems where the electrons are localized due to quantum interference effects. At this time both absolute localization and phase transition of “metal- dielectric” type can be realized.     

Random walks on sparsely periodic and random lattices: I. Random walk properties from lattice bond enumeration

We have developed a new technique for calculating certain asymptotic random walk properties on sparsely periodic and related random lattices in two and three dimensions. This technique is based on an ansatz which relates the number of lattice bonds in “irreducible lattice fragments” to the number of steps along these bonds. We show that certain random walk properties can be calculated very simply on the basis of this ansatz and that they depend only on the density of bonds and not on the arrangement of the bonds within the lattice. The random walk properties calculated here (mean square displacements, number of distinct sites visited, probability of return to the origin) are in complete agreement with results obtained earlier via generating function techniques. A subsequent paper contains generating function calculations which verify a number of new results presented here, such as mean occupation frequency of lattice sites, and a proof of our basic assumption on the relation between the number of lattice bonds and random walk steps.     

Localization in general one dimensional random systems,I. Jacobi matrices

We consider random discrete Schrödinger operators in a strip with a potentialV (n, ) (n a label in and a finite label across the strip) andV an ergodic process. We prove thatH ₀+V has only point spectrum with probability one under two assumptions: (1) Theconditional distribution of {V (n,)} _n=0,1;all conditioned on {V } _n0,1;all has an absolutely continuous component with positive probability. (2) For a.e.E, no Lyaponov exponent is zero.Research partially supported by USNSF grant MCS-81-20833     

Elastic wave localization in two-dimensional phononic crystals with one-dimensional random disorder and aperiodicity

The band structures of in-plane elastic waves propagating in two-dimensional phononic crystals with one-dimensional random disorder and aperiodicity are analyzed in this paper. The localization of wave propagation is discussed by introducing the concept of the localization factor, which is calculated by the plane-wave-based transfer-matrix method. By treating the random disorder and aperiodicity as the deviation from the periodicity in a special way, three kinds of aperiodic phononic crystals that have normally distributed random disorder, Thue-Morse and Rudin-Shapiro sequence in one direction and translational symmetry in the other direction are considered and the band structures are characterized using localization factors. Besides, as a special case, we analyze the band gap properties of a periodic planar layered composite containing a periodic array of square inclusions. The transmission coefficients based on eigen-mode matching theory are also calculated and the results show the same behaviors as the localization factor does. In the case of random disorders, the localization degree of the normally distributed random disorder is larger than that of the uniformly distributed random disorder although the eigenstates are both localized no matter what types of random disorders, whereas, for the case of Thue-Morse and Rudin-Shapiro structures, the band structures of Thue-Morse sequence exhibit similarities with the quasi-periodic (Fibonacci) sequence not present in the results of the Rudin-Shapiro sequence.     

Enhanced localization of waves in one-dimensional random media due to nonlinearity: Fixed input case

We study the influence of nonlinearity on wave localization in one-dimensional random media. Using a discrete nonlinear Schrödinger equation with a random on-site energy term, we calculate the localization length in a numerically exact manner. Unlike in many previous works, we fix the intensity of the incident wave and calculate quantities as a function of other parameters. We find that localization is enhanced due to nonlinearity for the focusing and defocusing nonlinearities. For small nonlinearity, the localization length is a decreasing function of nonlinearity. For sufficiently large nonlinearity, however, the localization length is found to approach a saturation value.     

Anderson localization in different one-dimensional systems with off-diagonal disorder and spin-dependence

The localization properties of certain spin-dependent, one-dimensional electronic systems with only off-diagonal disorder are studied. In higher dimensions (d=2,3) the models considered would correspond to different universality classes, whereas ford=1 no qualitative difference is found: ForE=0, all eigenstates are exponentially localized, whereas forE0 the localization length diverges logarithmically, such that exactly atE=0 the geometric average of the transmission coefficient would decay with increasing chain lengthL as exp (-const. ·L ^1/2), instead of the usual, exponential decay.ForE=0, in the interior of the band, the localization lengthr ₀ diverges W ₂ ^–2 in the limit of weak disorder (W ₂0), whereas just at the band edge one has roughlyr ₀W ₂ ^–2/3. A universal recursion relation, depending only on the energy and on certain randomly distributed determinants, determines the localization length and the density of states for all systems considered.     

Photoexcited processes at metal and alloy surfaces: electronic structure and adsorption site

Photoexcited processes of NO and CO at photon energies ranging from 2.3 to 6.4 eV are investigated on Pt(111), Ni(111) and Pt(111)---Ge surface alloys by reflection-absorption infrared spectroscopy and resonance-enhanced multiphoton ionization. The branching between three competitive processes of desorption, recapture and dissociation upon laser irradiation is dramatically changed on the three surfaces. On Pt(111), NO is either photodesorbed or photodissociated depending on the coverage, while NO is exclusively photodissociated on Ni(111). UV-photon irradiation of NO on Pt(111)---Ge, on the other hand, induces only desorption of NO. Desorption of CO bound at the on-top site of Pt(111) is induced by laser irradiation. The electronic mechanism for photodesorption and competitive branching is discussed in terms of the electronic structure of the substrate and the adsorbate.     

Highly resolved electronic absorption and luminescence spectra of some phenanthrolines in Shpolskii matrices part I. Phosphorescence spectra

In n-hexane matrices, at 77 K, the phosphorescence spectra of 1,9-, 1,8-, 1,7- and 4,7- phenanthrolines show quasi-linear structure (Shpolskii effect). Highly resolved spectra of isometric phenanthrolines allow measuring slight differences between 0-0 wave numbers of respective isomers and detecting characteristic features of the vibronic structure of the spectrum of each isomer. Vibrational analysis of the spectra show that in T₁ → S₀ electronic transitions of the phenanthrolines examined there participates, besides the totally symmetric ones, at least one out-of-plane vibration (a₂ or a″). The presence of this vibration and its combination with the totally symmetric ones seem to explain the shape of phosphorescence polarization spectra of phenanthrolines in polar glasses, described in the literature.     

Kinetics of nonlinear systems with random and regular disturbances. Exact results versus Fokker-Planck equation

Using exact solutions of some nonlinear ordinary differential (kinetic) equations with time-dependent coefficients, it is found that different types of regular and random disturbances may have diverse effect on macroscopic kinetics. Among others, regular impulses nonlinearly coupled to linear (smooth) relaxation kinetics can lead to chaotic behaviour.     

Coherent electronic dynamics and absorption spectra in an one-dimensional model with long-range correlated off-diagonal disorder

In this work we study an one-dimensional Anderson model with long-range correlated off-diagonal disorder. We numerically demonstrate the presence of extended states and an anomalous optical absorption spectrum for high degrees of correlation. We also show that the electric field biased electronic wave packet dynamics shows Bloch-like oscillations.     

Conductance distributions in random resistor networks. Self-averaging and disorder lengths

The self-averaging properties of the conductanceg are explored in random resistor networks (RRN) with a broad distribution of bond strengthsP(g)g ^–1. The RRN problem is cast in terms of simple combinations of random variables on hierarchical lattices. Distributions of equivalent conductances are estimated numerically on hierarchical lattices as a function of sizeL and the distribution tail strength parameter . For networks above the percolation threshold, convergence to a Gaussian basin is always the case, except in the limit 0. Adisorder length _D is identified, beyond which the system is effectively homogeneous. This length scale diverges as _Dµ^–v ( is the regular percolation correlation length exponent) when the microscopic distribution of conductors is exponentially wide (0). This implies that exactly the same critical behavior can be induced by geometrical disorder and by strong bond disorder with the bond occupation probabilityp. We find that only lattices at the percolation threshold have renormalized probability distributions in aLevy-like basin. At the percolation threshold the disorder length diverges at a critical tail strength µ_c as µ–^–z withz3.2±0.1, a new exponent.Critical path analysis is used in a generalized form to give the macroscopic conductance in the case of lattices abovep _c.     

The evolution of electronic structure in lanthanide metal clusters: Threshold photoionization spectra of and

The photoionization spectra of Pr₂-Pr₂₁ and Ce₂-Ce₁₇ have been measured near threshold. The ionization potentials (IPs) of and vary discontinuously with size, but trend downward toward the work function of the bulk metals. In general, the IPs of cerium clusters display more variation than those of praseodymium clusters. The sudden discontinuities observed in the IPs of both and is akin to that displayed by clusters of transition metal atoms, suggesting that as in transition metal clusters, the rapid evolution in geometric structure with size is the source of these discontinuities. Received: 2 January 1998 / Accepted: 10 March 1998     

Diffusion: A Layman's approach and its applications to one-dimensional random systems

Diffusion characteristics are reduced to the solution of a simple stationary equation. In classical one-dimensional random systems, time-irreversability (provided, e.g., by a magnetic field) implies L ∝ $\text{l}$nt, L is a diffusion length, t is time, while local space assymetry (provided by random fields) implies L ∝ $\text{l}$n²t; both cases are related to the localized particle distribution.     

The electronic structure of dinuclear transition-metal complexes containing metal—metal interactions: II. He(I) and He(II) photoelectron spectra of hexacarbonyldi-irondisulfur and related complexes

The He(I) and He(II) photoelectron spectra of a series of Fe₂(CO)₆LL¹-type complexes (L = L¹ = S, (i-propyl)S; L,L¹ = t-but of an all-electron, ab initio SCF MO calculation on Fe₂(CO)₆S₂ and of extended CNDO calculations on related molecules. Assignments given ar He(I)/He(II) intensity differences, and on comparison with related molecules.The coordination of the bridging ligands to the metal centres and the nature of the metal—metal interactions are discussed.     

Reaction,trapping, and multifractality in one-dimensional systems

In the first part of this paper, we present two variants of the A+AA and A+AP reaction in one dimension that can be investigated analytically. In the first model, pairs of neighboring particles disappear reactively at a rate which is independent of their relative distance. It is shown that the probability density (x) for a nearest neighbor distance equal tox approaches the scaling form(x) c exp(–cx/2)/(cx)^1/2 in the long-time limit, withc being the concentration of particles. The second model is a ballistic analogue of the coagulation reaction A+A A. The model is solved by reducing it to a first-passagetime problem. The anomalous relaxation dynamics can be linked in a direct way to the fractal time properties of random walks. In the second part of this paper, we discuss the complications that arise in systems with disorder. We present a new approach that relates first-passage-time characteristics in a one-dimensional random walk to properties of random maps. In particular, we show that Sinai disorder is a borderline case for the appearance of multifractal properties. Finally, we apply a previously introduced renormalization technique to calculate the survival probability of particles moving on the line in the presence of a background of imperfect traps.     

Dielectric susceptibility and correlation length of one-dimensional CDW in disordered systems. I. strong forward scattering

The dependence of dielectric susceptibility χ of one-dimensional charge, density wave, interacting with a lattice and with strong forward scattering impurities, on incommensurability parameter h has been found. At sufficiently large $\text{h}{}_{\mathrm{\chi }}\text{h}\text{?}{}^{\mathrm{4/3}}$.