Displacement distribution function analysis

The factor R may be called a diffusion anisotropy factor for the surface. For diffusion of a W on the W (110), Tsong Casanova find a diffusion anisotropy factor of 1.88 from a set of data taken at 299 K, and of 2.14 from a set of data taken at 309 K. The average is 2.01, which agrees with the theoretical value, 2, to within 0.5%. A more detailed general analysis has since then been reported,137 and diffusion anisotropy on the W (110) surface has also been observed in field emission experiments.138 It should be noted, however, that the same ratio can be expected if an adatom jumps instead in the [001] and [110] directions with an equal frequency. Thus a measurement of surface diffusion anisotropy factor alone does not establish uniquely the atomic jump directions. The atomic jump directions can, of course, be established from a measurement of the displacement distribution function in two directions as discussed in the last section. Such measurements can only be done with atomic resolution field ion microscopy. 

Figure 5 Sequences using stimulated echoes for flow velocity imaging (A) and tor the measurement of the displacement distribution function without imaging (B). The use of stimulated echoes often proves very convenient since the NMR signal can be observed for longer delays A, limited only by the T, value. The pair of gradient pulses, when exactly matched, produce no phase shift for nonmoving spins, and for moving spins they induce a phase shift A(j> equal to the product of the wavevector q = yGd, by the displacement (r(A) - r(0)). For a uniform velocity field v, (r(A) - r(0)) = Av everywhere in space and v can be found from the phase-shift measurement at a given value of q (A). In more complex flow situations, the full displacement distribution can be obtained from a Fourier transform analysis of data acquired with incremented G, and thus q, values (B).

Fig. 4) for relatively long chains37. However, under conditions of molecular orientation, the distribution function is usually displaced towards higher fi and the analysis of the crystallization process should be carried out over a wide range of fi values. 

Diffusion Analysis Based on Distribution Function of Displacements... 

DDF analysis gives precise diffusion coefficients of target molecules. In this analysis, molecule displacements during a short lag-time are derived from trajectories in order to make the distribution function, which is used to obtain the diffusion coefficients. This method does not require any fitting processes in order to construct the DDF and hence is suitable for precise mobility analysis of target molecules on the membrane. Here we describe the DDF analysis method for cARl, which has one diffusion state. This analysis can be applied to other molecules that exhibit more complicated behaviors some molecules exhibit two different types of diffusion states and state transition between them. 

Nj and Rj are the most important structural data that can be determined in an EXAFS analysis. Another parameter that characterizes the local structine aroimd the absorbing atom is the mean square displacement aj that siunmarizes the deviations of individual interatomic distances from the mean distance Rj of this neighboring shell. These deviations can be caused by vibrations or by structural disorder. The simple correction term exp [ 2k c ] is valid only in the case that the distribution of interatomic distances can be described by a Gaussian function, i.e., when a vibration or a pair distribution function is pmely harmonic. For the correct description of non-Gaussian pair distribution functions or of anhar-monic vibrations, different special models have been developed which lead to more complicated formulae [15-18]. This term, exp [-2k cj], is similar to the Debye-Waller factor correction used in X-ray diffraction however, the term as used here relates to deviations from a mean interatomic distance, whereas the Debye-Waller factor of X-ray diffraction describes deviations from a mean atomic position. 

The particular array of chemical shifts found for the nuclei of a given polymer depends, of course, on such factors as bond orientation, substituent effects, the nature of nearby functional groups, solvation influences, etc. As a specific example, derivatives of the carbohydrate hydroxyl moieties may give rise to chemical shifts widely different from those of the unmodified compound, a fact that has been utilized, e.g., in studies (l ) on commercially-important ethers of cellulose. Hence, as illustrated in Fig, 2, the introduction of an 0-methyl function causes (lU,15) a large downfield displacement for the substituted carbon. This change allows for a convenient, direct, analysis of the distribution of ether groups in the polymer. Analogously, carboxymethyl, hydroxyethyl and other derivatives may be characterized as well... 

Eqn (2.92) is the culmination of our efforts to compute the displacements due to an arbitrary distribution of body forces. Although this result will be of paramount importance in coming chapters, it is also important to acknowledge its limitations. First, we have assumed that the medium of interest is isotropic. Further refinements are necessary to recast this result in a form that is appropriate for anisotropic elastic solids. A detailed accounting of the anisotropic results is spelled out in Bacon et al. (1979). The second key limitation of our result is the fact that it was founded upon the assumption that the body of interest is infinite in extent. On the other hand, there are a variety of problems in which we will be interested in the presence of defects near surfaces and for which the half-space Green function will be needed. Yet another problem with our analysis is the assumption that the elastic constants... 

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