Correlated disorder model

Figure 101 Calculated Poole-Frenkel plots according to the correlated disorder model for different values of o jkT (from top curve downward). The calculations according to the Gaussian disorder model with

This distribution appears whenever g a) is given by a power law in (j, coming from the power law variation of the density of linear cracks g l) with their length 1. In the random percolation model considered here, this does not normally occur (except at the percolation threshold p = Pc)- However, for various correlated disorder models, applicable to realistic disorders in rocks, composite materials, etc., one can have such power law distribution for clusters, which may give rise to a Weibull distribution for their fracture strength. We will discuss such cases later, and concentrate on the random percolation model in this section. 

Figure 40. Mobility (/i, plotted as In//) vs. field strength ( , plotted on a scale proportional to as predicted by the Correlated Disorder Model for DOS of various r.m.s. widths a. The disorder is assumed to arise purely from random charge-dipole interactions. The lattice parameter is a. For a = lOA and = 0.1 eV, eaEja = 1 corresponds to a field strength of 10 V cm. (Reprinted with permission from Ref [63f].)...

By running Monte Carlo (MC) simulations of carriers within such a landscape and by assuming that the probability of a carrier hopping from one site to the other follows the MA expression this model provides us with semi-empirical expressions for the full-temperature and field-dependence of the carrier mobility. Improvements and developments upon this model in common use include the correlated disorder model (CDM) of Novikov and co-workers [28]. Goto and co-workers have also carried out MC simulations using a modified Gay-Berne potential specifically for application to anomalous charge transport in nematic liquid crystals [29]. 

As it was pointed out in the Introduction, the problem of the coexistence of displacive and order-disorder phenomena at the ferroelectric phase transitions of BaTiOs has met growing interest in recent time. Strong support of the order-disorder model comes 30 years ago from EPR measurements performed on Mn" " "-, Cr -, and Fe -doped BaTiOs [218-222] because in the low-temperature rhombohedral phase it was observed that Mn" " ", which substitutes isovalent Ti" " " sites, is displaced off-centre by 0.14 A along <111> directions with a reorientational hopping with correlation times 10 -10 s. 

A special case of static disorder occurs when an atom is present in only a fraction of the unit cells in the crystals. For instance, a ligand might be present in only half of the unit cells, or a side chain might have two or more possible conformations. This type of disorder can be modeled with an occupancy parameter 0, ranging from one (fully present) to zero (absent). It is simply the fraction of unit cells where the atom in question is present. Low occupancy is, however, difficult to distinguish from high B-factor. Indeed B-factors and occupancies are statistically correlated. Therefore, occupancies are normally used only when there are sufficient data (i.e., high resolution) to justify a more complicated disorder model. 

These data may reflect that PDES retains both ordered and disordered phases in the range of -60 to -10°C. Above 25°C, PDES takes only a disordered phase and the molecular motion is in the fast-motion region for the single correlation-time model based on BPP theory [22], because the Si Ti values increase as the temperature is increased from 25 to 125°C. That is to say, the disordered phase (I) is conformationally disordered but shows rudimentary intermolecular packing and reflect a single motional state. 

Mott, N.F., Electrons in disordered structures. Advances in Physics, 1967. 16 p. 49 Moura, F.A.B.F. and M.L. Lyra, Delocalization in the ID Anderson model with long-range correlated disorder. Physical Review Letters, 1998. 81 p. 3735... 

In [221] the X-ray diffraction patterns of PE fibers in the high-pressure hexagonal form have been modeled assuming complete translational disorder along z, rotational disorder of chains around their axes and conformational disorder. The Fourier transforms of disordered structural models were calculated as a function of intra- and inter-molecular parameters related to the presence of conformational disorder and the relative arrangement of close neighboring chains (short-range correlation disorder). The results of the calculations were compared to experimental X-ray diffraction data. 

A comparison between calculated Fourier transform of various disordered models and experimental X-ray diffraction pattern have been reported in [262] and [263]. Structural models characterized by small bundles of parallel chains in 3/1 helical conformation packed with lateral disorder and keeping short-range correlations between neighboring chains similar to those of a form and the hexagonal P form of iPP, have been analyzed [262,263]. Since the crystal structure of the form had not been solved at that time, the hexagonal models built up for Fourier transform calculations in [262] and [263] were different from the now accepted structure of P form [264-266] even though the considered models had hexagonal correlations between chains. 

---