Disorder dynamic

Zwanzig R 1990 Rate processes with dynamical disorder 4cc. Chem. Res. 23 148-52... 

Only one tautomer is present in the crystal, no dynamic disorder... 

That the most likely coarse velocity is equal to the most likely terminal velocity can only be true in two circumstances either the system began in the steady state and the most likely instantaneous velocity was constant throughout the interval, or else the system was initially in a dynamically disordered state, and x was large enough that the initial inertial regime was relatively negligible. These equations are evidently untrue for x —> 0, since in this limit the most... 

In summary, following a fluctuation the system is initially dynamically disordered, and the flux is zero. If the flux were constrained to be zero for an intermediate time, the second entropy would be less than the first entropy of the fluctuation, and it would have decreased at a constant rate,... 

The static probability places the subsystem in a dynamically disordered state, Ti so that at x = 0 the flux most likely vanishes, x(ri) = 0. If the system is constrained to follow the adiabatic trajectory, then as time increases the flux will become nonzero and approach its optimum or steady-state value, x(x) —> L(x, +l)Xi, where xj =x(Ti) and X] = X r1]). Conversely, if the adiabatic trajectory is followed back into the past, then the flux would asymptote to its optimum value, x(—r) > —L(xi, — 1 )Xj. 

The atomic temperature factor, or B factor, measures the dynamic disorder caused by the temperature-dependent vibration of the atom, as well as the static disorder resulting from subtle structural differences in different unit cells throughout the crystal. For a B factor of 15 A2, displacement of an atom from its equilibrium position is approximately 0.44 A, and it is as much as 0.87 A for a B factor of 60 A2. It is very important to inspect the B factors during any structural analysis a B factor of less than 30 A2 for a particular atom usually indicates confidence in its atomic position, but a B factor of higher than 60 A2 likely indicates that the atom is disordered. 

Effect of diagonal dynamic disorder (DDD). Fluctuations of the polarization and the local vibrations produce the variation of the positions of the electron energy levels eA(Q) and eB(C ) to meet the requirements of the Franck-Condon principle. 

Effect of off-diagonal dynamic disorder (off-DDD). The interaction of the electron with the fluctuations of the polarization and local vibrations near the other center leads to new terms VeP - V P, Vev - Vev and VeAp - VAPd, VA - VAd in the perturbation operators V°d and Vfd [see Eqs. (14)]. A part of these interactions corresponding to the equilibrium values of the polarization P0l and Po/ results in the renormalization of the electron interactions with ions A and B, due to their partial screening by the dielectric medium. However, at arbitrary values of the polarization P, there is another part of these interactions which is due to the fluctuating electric fields. This part of the interaction depends on the nuclear coordinates and may exceed the renormalized interactions of the electron with the donor and the acceptor. The interaction of the electron with these fluctuations plays an important role in processes involving solvated, trapped, and weakly bound electrons. 

Effect of diagonal-off-diagonal dynamic disorder (D-off-DDD). The polarization fluctuations and the local vibrations give rise to variation of the electron densities in the donor and the acceptor, i.e., they lead to a modulation of the electron wave functions A and B. This leads to a modulation of the overlapping of the electron clouds of the donor and the acceptor and hence to a different transmission coefficient from that calculated in the approximation of constant electron density (ACED). This modulation may change the path of transition on the potential energy surfaces. 

Additional effect of diagonal dynamic disorder. The variations of the electron densities near the centers A and B due to polarization fluctuations and local vibrations lead to changes in the interaction of the electron with the medium and, hence, to changes in the shape of the potential energy surfaces Ut and Uf as compared... 

Effects of Diagonal and Off-Diagonal Dynamic Disorder in Reactions Involving Transfer of Weakly Bound Electrons (A Configurational Model)... 

To take into account the additional effect of diagonal dynamic disorder in the improved Condon approximation it was suggested in Ref. 16 that fluctuations of the polarization of the type... 

In addition to the dynamic disorder caused by temperature-dependent vibration of atoms, protein crystals have static disorder due to the fact that molecules, or parts of molecules, do not occupy exactly the same position or do not have exactly the same orientation in the crystal unit cell. However, unless data are collected at different temperatures, one cannot distinguish between dynamic and static disorder. Because of protein crystal disorder, the diffraction pattern fades away at some diffraction angle 0max. The corresponding lattice distance <7mm is determined by Bragg s law as shown in equation 3.7 ... 

The term exp(-2k2c ) in (6-9) accounts for the disorder of the solid. Static disorder arises if atoms of the same coordination shell have slightly different distances to the central atom. Amorphous solids, for instance, possess large static disorder. Dynamic disorder, on the other hand, is caused by lattice vibrations of the atoms, as explained in Appendix 1. Dynamic disorder becomes much less important at lower temperatures, and it is therefore an important advantage to measure spectra at cryogenic temperatures, especially if a sample consists of highly dispersed particles. The same argument holds in X-ray and electron diffraction, as well as in Mossbauer spectroscopy. 

To separate the effects of static and dynamic disorder, and to obtain an assessment of the height of the potential barrier that is involved in a particular mean-square displacement (here abbreviated (x )), it is necessary to find a parameter whose variation is sensitive to these quantities. Temperature is the obvious choice. A static disorder will be temperature independent, whereas a dynamic disorder will have a temperature dependence related to the shape of the potential well in which the atom moves, and to the height of any barriers it must cross (Frauenfelder et ai, 1979). Simple harmonic thermal vibration decreases linearly with temperature until the Debye temperature Td below To the mean-square displacement due to vibration is temperature independent and has a value characteristic of the zero-point vibrational (x ). The high-temperature portion of a curve of (x ) vs T will therefore extrapolate smoothly to 0 at T = 0 K if the sole or dominant contribution to the measured (x ) is simple harmonic vibration ((x )y). In such a plot the low-temperature limb is expected to have values of (x ) equal to about 0.01 A (Willis and Pryor, 1975). Departures from this behavior indicate more complex motion or static disorder. 

The dynamic disorder in the interlayers of LDHs is also confirmed by single crystal XRD, since the interlayer anions and water molecules generally have occupancy factors less than unity and elevated temperature/displace-ment factors [172],... 

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