Disordered systems model

J. M. Ziman, Models of Disorder the Theoretical Physics of Homogeneously Disordered Systems. Cambridge Cambridge University Press, 1979. 

B. Derrida, Random-energy model an exactly solvable model of disordered systems. Phys. Rev. B 24, 2613-2626 (1981). 

The photoelectron wave-vector k is evaluated using = 2m(E — E ) where E is the energy of the X-ray photon, , a reference energy and m, the mass of the electron. x(k) is multiplied by k"(n = 2 or 3 usually) to magnify the faint EXAFS at large k (Lytle et al, 1975) /c"x(k) is Fourier transformed to yield the RSF, < (R). In the model compound, the first peak at a distance Rj represents the distance to the nearest-neighbour shell and may be compared to R[, the known distance. We can then define a as (R — Rj), which represents the experimentally determined phase correction. In principle, 2a should be equal to the theoretically estimated k-dependent part of /k), viz. if the identity of the scatterer environment has been correctly assumed. It must be emphasized that wherever scatterer identities are obscure (e.g. in several covalently bonded and disordered systems) use of a (and not j) is advisable. Further, the k-dependence of < /k) introduces an intrinsic limitation to its quantitative accuracy. 

Fig. 1.20 Density of states in a disordered system (a) in the Anderson model (Fig. 1.17) (i) shows the behaviour without disorder, simple cubic lattice, and (ii) that with disorder ...

The percolation model, which can be applied to any disordered system, is used for an explanation of the charge transfer in semiconductors with various potential barriers [4, 14]. The percolation threshold is realized when the minimum molar concentration of the other phase is sufficient for the creation of an infinite impurity cluster. The classical percolation model deals with the percolation ways and is not concerned with the lifetime of the carriers. In real systems the lifetime defines the charge transfer distance and maximum value of the possible jumps. Dynamic percolation theory deals with such case. The nonlinear percolation model can be applied when the statistical disorder of the system leads to the dependence of the system s parameters on the electrical field strength. 

The fundamental vibrations have been assigned for the M-H-M backbone of HM COho, M = Cr, Mo, and W. When it is observable, the asymmetric M-H-M stretch occurs around 1700 cm-1 in low temperature ir spectra. One or possibly two deformation modes occur around 850 cm l in conjunction with overtones that are enhanced in intensity by Fermi resonance. The symmetric stretch, which involves predominantly metal motion, is expected below 150 cm l. For the molybdenum and tungsten compounds, this band is obscured by other low frequency features. Vibrational spectroscopic evidence is presented for a bent Cr-H-Cr array in [PPN][(OC)5Cr-H-Cr(CO)5], This structural inference is a good example of the way in which vibrational data can supplement diffraction data in the structural analysis of disordered systems. Implications of the bent Cr-H-Cr array are discussed in terms of a simple bonding model which involves a balance between nuclear repulsion, M-M overlap, and M-H overlap. The literature on M-H -M frequencies is summarized. 

Small solute atoms in the interstices between the larger host atoms in a relaxed metallic glass diffuse by the direct interstitial mechanism (see Section 8.1.4). The host atoms can be regarded as immobile. A classic example is the diffusion of H solute atoms in glassy Pd8oSi2o- For this system, a simplified model that retains the essential physics of a thermally activated diffusion process in disordered systems is used to interpret experimental measurements [20-22]. 

In this book we wish to discuss partially disordered systems and cannot thus always expect true long range order to exist. A proper treatment of the statistical mechanics of a partially ordered system is extremely difficult, though some success has been achieved by numerical modelling... 

While many researchers believe that the above mentioned three ingredients for rl behavior to appear are more or less independent, we have argued since long [6] that the primary cause of RL behavior is the lattice disorder, which is at the origin of the occurrence of polar nanodomains and their fluctuations within the highly polarizable lattice. In order to describe disordered systems and to explore their basic thermodynamic behavior simple spin models are frequently used. The model Hamiltonian... 

We consider the hamiltonian (4.26) for a binary system (its extension to an arbitrary number of partners or to continuous disorder creates no difficulty in principle). The Schrodinger solution associated with this hamiltonian describes a coupling of very general occurrence, since it is encountered in conduction and transport phenomena, as well as in kinetics models of disordered systems ... 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

When analyzing experimental EPR spectra of spin probes in micellar phase of complexes we used the model "Microscopic Order and Macroscopic Disorder" (MOMD) [33], This model is often used for description of EPR spectra of spin probes in micelles, dispersions, vesicles and other microscopically ordered but macroscopically disordered systems [34, 35],... 

In 1991, Bonny and Leuenberger [40] explained the changes in dissolution kinetics of a matrix controlled-release system over the whole range of drug loadings on the basis of percolation theory. For this purpose, the tablet was considered a disordered system whose particles are distributed at random. These authors derived a model for the estimation of the drug percolation thresholds from the diffusion behavior. 

The use of the percolation model to analyze the d.c. conductivity in hydrated lysozyme powders (Careri et al., 1986, 1988) and in purple membrane (Rupley et al, 1988) introduces a viewpoint from statistical physics that is relevant to a wide range of problems originating in disordered systems. Percolation theory is described in the appendix to this article, for readers unfamiliar with it. Here, we discuss the significance of percolation specihcally for protein hydration and function. 

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