Jump motion constant

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

When an electric field is applied, jumps of the ions in the direction of the field are somewhat preferred over those in other directions. This leads to migration. It should be noted that the absolute effect of the field on the ionic motion is small but constant. For example, an external field of 1 V m-1 in water leads to ionic motion with a velocity of the order of 50 nm s 1, while the instantaneous velocity of ions as a result of thermal motion is of the order of 100 ms-1. 

Figure 8 a shows the motionally averaged quadrupole coupling constant, (Cq)/Cq, and asymmetry parameter, ( ), for a two-site jump between axially symmetric equivalent sites. At jump angles of 70° and 109° the principal components (V, Vyy, Vzz) have to be rearranged in order, which leads to the discontinuities in the curve shapes of Fig. 8a. 

The temperature dependent T data are shown in Fig. 9. 7j values decrease from 28 ms at 21°C with increasing temperature, and show a minimum of 6.4 ms at 80° C. These results indicate the presence of the motion with a Larmor frequency of 30 MHz at this temperature. This minimum was found to be attributed to the flipping motion of a phenyl ring from the result of our other experiments discussed in later section.13 The jump rates of the flipping motion were estimated with a two-site jump model that a C-2H bond jumps between two equivalent sites separated by 180°, and that the angle made by the C-2H bond and the rotational axis is 60°. The quadrupole coupling constant of 180 kHz and the asymmetry parameter approximated to zero were used in the calculation. The calculated values for fitting with the... 

The motional dynamics of O J adsorbed on Ti supported surfaces has been analyzed over the temperature range 4.2-400 K in a recent paper by Shiotani et al. (66). Of the several types of 02, a species noted as 02 (III), and characterized by gxx = 2.0025, gyy = 2.0092, g12 = 2.0271 at 4.2 K, exhibited highly anisotropic motion. While gxx and gzz varied with increasing temperature and were accompanied by drastic line shape changes, gyy was found to remain constant. This observation indicates that the molecular motion of this 02 can be described by rotation about the y axis perpendicular to the internuclear axis of 02 and perpendicular to the surface with the notation given in Fig. 4. The EPR line shapes were simulated for different possible models and it was found that a weak jump rotational diffusion gave a best fit of the observed spectra below 57.4 K, whereas some of the models could fit the data above this temperature. The rotational correlation time was found to range from 10 5 sec (below 14.5 K) to 10 9 sec (263 K), while the... 

After the formulation of defect thermodynamics, it is necessary to understand the nature of rate constants and transport coefficients in order to make practical use of irreversible thermodynamics in solid state kinetics. Even the individual jump of a vacancy is a complicated many-body problem involving, in principle, the lattice dynamics of the whole crystal and the coupling with the motion of all other atomic structure elements. Predictions can be made by simulations, but the relevant methods (e.g., molecular dynamics, MD, calculations) can still be applied only in very simple situations. What are the limits of linear transport theory and under what conditions do the (local) rate constants and transport coefficients cease to be functions of state When do they begin to depend not only on local thermodynamic parameters, but on driving forces (potential gradients) as well Various relaxation processes give the answer to these questions and are treated in depth later. 

From the theory of absolute reaction rates it follows4 6,50-561 for a simple barrier model (Fig. 1 a) that the rate constant of the jumps of a particle (motional unit) from site 1 to site 2 is given by... 

The Fick s law diffusion coefficient of a permeating molecule is a measure of the frequency with which the molecule moves and the size of each movement. Therefore, the magnitude of the diffusion coefficient is governed by the restraining forces of the medium on the diffusing species. Isotopically labeled carbon in a diamond lattice has a very small diffusion coefficient. The carbon atoms of diamond move infrequently, and each movement is very small—only 1 to 2 A. On the other hand, isotopically labeled carbon dioxide in a gas has an extremely large diffusion coefficient. The gas molecules are in constant motion and each jump is of the order of 1000 A or more. Table 2.1 lists some representative values of diffusion coefficients in different media. 

Here DR and >, are the diffusion coefficients for the isotropic overall and free internal motions, respectively. Equation 31 assumes a diffusional process for the methyl group. If a jumping process between three equivalent positions separated by 120° is considered,47 the last term becomes C/(6DK + Z), ). Parameters A, B, and C are geometric constants similar to those in Eq. 27, but here the angle is that formed between the methyl C—H vectors and the axis of rotation. Assuming tetrahedral angles, for free internal motion ( >, ) ), 1/7 ,(CH3) is decreased to one-ninth of the value expected for a rigidly attached CH carbon. For slow internal rotation (D, — ) ), l/r1(CH3) becomes one-third of the value of a methine carbon in the same molecule, as predicted by Eq. 16. 

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