Investigating material trends and lattice relaxation effects for understanding electron transfer phenomena in rare-earth-doped optical materials

Rare-earth-doped insulators and semiconductors play an important role in a wide range of modern optical technologies. Knowledge of the relative energies of rare-earth ions’ localized electronic states and the band states of the host crystal is important for understanding the properties of these materials and for determining the potential material performance in specific applications such as lasers, phosphors, and optical signal processing. Current understanding of the systematic variations of electron binding energies in these materials is reviewed with analysis of how lattice relaxation affects the results obtained from different experimental techniques. Detailed examples are presented for rare-earth-doped YAG and LaF₃ material systems. A method for predicting the chemical shift of the 4f electrons of rare-earth impurities from the host crystal’s photoemission spectrum is also demonstrated. Furthermore, a simple model is presented that predicts host-dependent trends in the binding energies of the rare-earth ion states in materials ranging from the elemental metals to the ionic fluorides. By understanding the systematic changes in the relative energies for different states, different ions, and different host materials, insight is gained into electron transfer transitions, valence stability, and luminescence quenching that can accelerate the development of materials for optical applications.     

On the relationship betweenT 1 andT 2 for stochastic relaxation models

We consider the relaxation dynamics of two quantum levels coupled to a stochastic bath. We emphasize that even if the matrix elements of the fluctuating Hamiltonian are Gaussian, a second-order cumulant truncation is not exact. For various stochastic models, including the case of a spin-1/2 particle in a fluctuating magnetic field, we calculate 1/T ₁, the population relaxation rate, and 1/T ₂, the phase relaxation rate, up to fourth order in perturbation theory. We show that unlike the commonly accepted second-order result that 1/T ₂1/2T ₁, when fourth-order terms are included, in some instances 1/T ₂<1/2T ₁.     

Dynamical systems with relaxation time

We define mathematically a class of dynamical systems that exhibit relaxation corresponding to that observed in physical systems, and then show that this class is identical with the class ofK-mixing dynamical systems.     

Statistical mechanics of temporary polymer networks I. The equilibrium theory

In part I of this work (the present article) the equilibrium state of temporary polymer networks is treated in the framework of thermodynamics and statistical mechanics. The network is described as an open system. Thereby we use a modified spring-bead model in which the beads represent junctions that decay and reform thus adding a viscous component to the assumed elastic behaviour of the permanent network. The relevant statistical equation — analogous to Liouville's equation — is solved. The grand-canonical probability density function and two of three equations of state are derived. Explicit formulae are given for several relevant probabilities. For instance the probabilityw (z)dz that a network chain connecting two junctions has a contour length betweenz andz +dz is given by the Wien type formulaw(z) =A z ³ exp {–B z} whereA andB do not depend onz.     

Decay of the remanent magnetization in the asymmetric spin chain

The dynamics of the one-dimensional spin glass with asymmetric interactions between neighboring spins is considered. We confine ourselves to discrete couplings with values ±J. We show that the algebraic decay of the remanent magnetization of the infinite ±J-spin chain at zero temperature is only valid for symmetric couplings. Our analytical investigations as well as computer simulations show stretched exponential decay for any finite concentration of antisymmetric bonds. Thus, the asymmetric ±J-spin chain shows an asymmetry-induced phase transition at zero temperature.     

Statistical properties of aggregation with injection

We study a generalized aggregation process in which charged particles diffuse and coalesce randomly on a lattice. For one-dimensional and mean-field models, we show that there exists a statistically-invariant steady state when randomly charged particles are continuously injected. The steady-state charge distribution obeys a power law with the exponent depending both on the type of the injection and on the spatial dimension. The response of the system to a perturbation (i.e., relaxation) is characterized by either a power law decay (t ^– ,1) or a compressed exponential decay [exp(–t ),>1].     

Frequency domain measurements on Na2B4O7−V2O5 semiconducting glass

Summary Dielectric measurements on Na₂B₄O₇(99.5%)−V₂O₅(0.5%) glass system, in the frequency range 10⁻³ to 10⁴ Hz and temperature range 300 to 500 K, have been carried out. The normalized plots of complex capacitance have shown a single mechanism responsible for conduction for both volume and surface measurements with their close values of activation energies (0.67±0.03) eV and (0.64±0.03) eV, respectively. The low-frequency dispersion (LFD) behaviour has been observed to be perturbed by the presence of more than one competing process. The impedance plots have shown a parallel combination of a capacitor (C) and a resistor (R), with some contribution of a dispersive element due to charge accumulation in the vicinity of the electrodes. The values ofR andC were found to be of the same order of magnitude, for both surface and volume measurements. The observedR has shown a decrease with an increase in temperature due to an increase in mobility of Na⁺ ions, whereasC remains practicaly constant. The complex capacitance surface behaviour is dominated by volume, due to hygroscopy of this glass system.     

An Open Question on Cyclic Relaxation

The problem discussed in this note is highly interesting. It is related to several dual iterative methods, such as the methods proposed by Kaczmarz, Hildreth, Agmon, Cryer, Mangasarian, Herman, Lent, Censor, and others. Cast as row-action methods these algorithms have been proved as useful tools for solving large convex feasibility problems that arise in medical image reconstruction from projections, in inverse problems in radiation therapy, and in linear programming.The question that we want to answer is how these algorithms behave when the feasible region is empty. It is shown that under certain conditions the primal sequence still converges while the dual sequence {y _k } obeys the rule y _k =u _k +k v, where {u _k } is a converging sequence and v is a fixed vector that satisfies A ^T v=0,v0,and,b ^T v>0. It is conjectured that these properties hold whenever the feasible region is empty. However, the validity of this claim remains an open question.This revised version was published online in October 2005 with corrections to the Cover Date.     

On the Global Stability of Solutions of Moment Systems in Nonequilibrium Thermodynamics

In this paper, we study the linearization of the Cauchy problem and the mixed problem for the system of Grad--Hermite moments in nonequilibrium thermodynamics in the neighborhood of the equilibrium state. Stability conditions for solutions of the Cauchy problem are proved as a generalization of the classical Hermite--Biller theorem on stable polynomials. For the mixed problem, we prove an analog of the Vishik--Lyusternik theorem on small singular perturbations of general elliptic problems. The last observation allows us to introduce the Shapiro--Lopatinskii condition, which implies the well-posedness of the mixed problem.     

Two exact algorithms for the capacitated p-median problem

The p-median problem has been widely studied in combinatorial optimisation, but its generalisation to the capacitated case has not. We propose a branch and price algorithm, comparing it with a standard MIP solver and a branch and bound algorithm based on Lagrangean relaxation. We present computational experience, using test instances drawn from the literature and new instances with higher ratio between the number of medians p and the number of nodes N. The branch and price algorithm shows very good performances and computational time robustness in solving problems for any ratio.Received: December 2002, Revised: August 2003AMS classification: 90C10, 90C27