Disorder parameter

Note that the quantity (1- Q D)) can be thought of as a disorder parameter distinguishing the different regimes of behavior. Numerical measurements of this disorder parameter around the transition point at. which the zigzag pattern collapses spontaneously (i.e. around etc 3.92) suggest a critical-like behavior [kaneko89a] ... 

Figure 12-16. Field dependence of die charge carrier mobility in an undiluted hopping system al various values of the disorder parameter a = a/kT (Kef. [67]).

Eq. (12.14) is recovered. The presence of traps lowers ihe mobility as expected. The essential message of Figures 12-17 and 12-18 is that, to a first order approximation, Eq. (12.14) maintains the icmperalurc dependency of the mobility if one replaces the disorder parameter by an effective disorder parameter ocJj or, equivalently, an effective width of the DOS that depends on both the concentration and the depth of the traps. Deviations from the behavior predicted by Eq. (12.14) become important for ,>0.3 eV, notably at lower temperatures. It is noteworthy, though, that the T- oo intercepts of p(7), if plotted as In p versus 7 2, vary by no more than a factor of 2 upon varying trap depth and concentration. 

Fig. 1 Melting temperatures of polymers k%Tm/Ec) with variable Ev/Ec values. The line is calculated from Eq. 10 and the circles are the simulation results obtained from the onset of crystallization on the cooling curves of disorder parameters, in a short-chain (r = 32) system (occupation density is 0.9375 in a 32-sized cubic box) with a template substrate (Hu and Frenkel, unpublished results)...

Fig. 6 Simulational cooling curves of disorder parameters (solid lines) and mixing parameters (dashed lines) for 32-mers with different sets of energy parameters in a 64-sized cubic box (the concentration is fixed at 0.150). The mixing parameter is defined as the mean fraction of neighboring sites occupied by the solvent for each chain unit [84]... 

Fig. 22 Cooling curves of the disorder parameter for 32-mers in 32-sized cubic lattice with a conventional cooling program (dashed line) and an optimized cooling program (solid line). Polymers have a volume fraction of 0.0625 with B/Ec = 0 and Ep/Ec = 1 [14]...

Let the disorder parameter be z. Then the structural Formula of these olivines can be written as ... 

If the law of mass action applies to Equation (27), the disorder parameter can be estimated to a first approximation by ... 

To analyze the negative field dependence of the mobihty in EHO-OPPE within the Gaussian disorder transport formahsm and to determine the diagonal (energetic) disorder parameter a and the off-diagonal (positional) disorder parameter d, the following relation between the charge mobihty p and the disorder parameters was employed [75] ... 

Off-diagonal (positional) disorder parameter. Diagonal (energetic) disorder parameter at 20 °G. 

Fig. 10 Temperature dependence of the diagonal disorder parameter dfor holes (squares) and electrons (circles). Parameters have been obtained by fitting the experimental data (Fig. 9) to Eq. 1 and Eq. 5. Reproduced with permission from [61]...

Figure 5.55 Mutual dependence of Q i and Q d order parameters. In the upper part of the figure is outlined the T dependence of substitutional disorder Qod for different values of Qdi and, in the lower part, the T dependence of the displacive disorder parameter Qdt for different values of The heavy lines on the surface of local curves represent the solution for thermal equilibrium. From E. Salje and B. Kuscholke, Thermodynamics of sodium feldspar II experimental results and numerical calculations. Physics and Chemistry of Minerals, 12, 99-107, figures 5-8, copyright 1985 by Springer Verlag. Reprinted with the permission of Springer-Verlag GmbH Co. KG.

In an extended version of the hopping concept, positional ( off-diagonal ) disorder in addition to energetic ( diagonal ) disorder has been introduced [54,63]. The simplest ansatz was to incorporate this by allowing the electronic overlap parameter 2ya to vary statistically. Operationally, one splits this parameter into two site contributions, each taken from a Gaussian probability density, and defines a positional disorder parameter I, in addition to the energetic disorder parameter cr. 

We see that the transit signal does not change appreciably with temperature. Also note that at the final stage (t > 1 ), the TOF signal exhibits a power-law decay and appears to be dispersive. According to the Scher-Montroll theory [33], in the case of the dispersive transport process, I(t) should exhibit power-law dependences rd-a) and 7 d+a) fpj. j respectively, where a is the disorder parameter. The... 

In the absence of dynamic and static disorder, all partially filled band systems would exhibit coherent transport over long distances. With static and dynamic disorder, the modulation of the simple molecular orbital or band structure by nuclear effects entirely dominates transport. This is clear both in the Kubo linear response formulation of conductivity and in the Marcus-Hush-Jortner formulation of ET rates. The DNA systems are remarkable for the different kinds of disorder they exhibit in addition to the ordinary static and dynamic disorder expected in any soft material, DNA has the covalent disorder arising from the choice of A, T, G, or C at each substitution base site along the backbone. Additionally, DNA has the characteristic orientational and metric (helicoidal) disorder parameters arising from the fundamental motif of electron motion along the r-stack. 

Economou et al. (1985) and Soukoulis et al. (1985, 1986, 1987) have used somewhat similar methods to calculate both the density of states, the mobility edge and the conductivity as a function of energy for the case of diagonal disorder their work is limited to disorder parameters V0 less than one-fifth of the bandwidth B, and is therefore relevant to the band tail... 

Anderson type (though affected of course by long-range interaction). Until recently it was supposed by the present author that the former is the case. We must now favour, however, the latter assumption for many-valley materials (e.g. Si and Ge), the Hubbard gap opening up only for a value of the concentration n below nc. The first piece of evidence comes from a calculation of Bhatt and Rice (1981), who found that for many-valley materials this must be so. The second comes from the observations of Hirsch and Holcomb (1987) that compensation in Si P leads to localization for a smaller value of nc than in its absence. As pointed out by Mott (1988), a Mott transition occurs when B = U (B is the bandwidth, U the Hubbard intra-atomic interaction), while an Anderson transition should be found when B 2 V, where V is some disorder parameter. Since U e2/jcuH, where aH is the hydrogen radius, and K e2/jca, and since at the transition a 4aH, if the transition were of Mott type then it should be the other way round. 

Comparing this with eqn (1.57), it immediately follows that Na = Nj, eV + eV = fip. An intrinsic disorder parameter C , as a measure of the disorder of metal atoms at the stoichiometric composition, is defined as (by use of... 

The Allen-Cahn equation applies to the kinetics of a diffuse-interface model for a nonconserved order parameter—for example, the order-disorder parameter r](f,t)... 

The crystal size in polyphenylene, as determined from x-ray peak widths, is of the order of 5 nm476) with a disorder parameter g = 0.026 nm. Compression at up to 12kB decreased the (/-spacing perpendicular to the chains, decreased the peak size and increased the disorder slightly. Annealing at temperatures above 250 °C increases the crystal size and perfection 472). The spin concentration increases above 300 °C, but unlike those in polyacetylene, these spins are not mobile477. The crystallinity has variously been estimated as 80% 327) and 20 to 30% 478). It seems to depend on the catalyst used in the Kovacic method. Polyphenylene produced by the precursor route has a crystallinity from 60-80% dependent on the conversion conditions 252). 

This method uses the factor for the "apparent" crystallinity, which is, itself, a function of the disorder parameter, ),... 

Table II shows effects of the disorder parameter on the calculated crystallinity of cotton, nylon 66 (PA 66) and PET fiber samples. When k = 0, no correction for distortion is made during calculation of crystallinity. Accordingly, values of X become smaller as the integration intervals increase. On the nomogram of K values, it is possible to determine the disorder parameter value that maintains as approximately constant this crystallinity when using the different intervals the deviation from the constancy is used, in the computing program to determine the best value of k and to estimate the errors of these analyses. The disorder parameter k is higher in cotton and PA 66 than in all PET fibers. Besides, one can see that apparent crystallinity values (disorder parameter not considered) are lower than the true ones.

Paradoxically, such a mechanical fatigue apparently acts as a treatment for relaxation of stress and allows flaws, located at the stress areas to be partly dissipated. Also, mechanical fatigue treatments can eliminate effects of a previous thermal treatment. Two samples that were initially different became more similar with regard to their crystallinity after 50 x 103 cycles. They had a medium level of crystallinity characterized by disorder parameter values that are particularly low. 

Kevlar Fibers. Table IX shows results obtained with Kevlar 950 fibers. The crystallinity of this aromatic polyamide is only slightly higher than that for the aliphatic polyamide samples that we studied (Table VI). But in Kevlar, as for PET fibers, the disorder parameter k is smaller than in PA 66 fibers. Thermal treatment at 220°C for 1 hour in slack conditions does not substantially increase the average crystallinity of the Kevlar 950 fibers and produces more distortions. Consequently, we conclude that the behavior of this Aramid fiber during our annealing treatment is similar to PA 66 as far as crystallinity is concerned. Kevlar 950 is also similar to PET fiber with regard to the behavior of the disorder parameter. 

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