Mean square displacement for

Fig. 6.10 Mean squared displacement for Li ions in Li N for motion parallel (xy) and perpendicular (z) to the LijN layers [Wolfet al. 1984],...

Fig. 3.1.4 Anisotropic self-diffusion of water in and filled symbols, respectively). The horizon-MCM-41 as studied by PFG NMR. (a) Depen- tal lines indicate the limiting values for the axial dence of the parallel (filled rectangles) and (full lines) and radial (dotted lines) compo-perpendicular (circles) components of the axi- nents of the mean square displacements for symmetrical self-diffusion tensor on the inverse restricted diffusion in cylindrical rods of length temperature at an observation time of 10 ms. / and diameter d. The oblique lines, which are The dotted lines can be used as a visual guide, plotted for short observation times only, repre-The full line represents the self-diffusion sent the calculated time dependences of the...

Another noteworthy example is x-ray absorption fine structure (EXAFS). EXAFS data contain information on such parameters as coordination number, bond distances, and mean-square displacements for atoms that comprise the first few coordination spheres surrounding an absorbing element of interest. This information is extracted from the EXAFS oscillations, previously isolated from the background and atomic portion of the absorption, using nonlinear least-square fit procedures. It is important in such analyses to compare metrical parameters obtained from experiments on model or reference compounds to those for samples of unknown structure, in order to avoid ambiguity in the interpretation of results and to establish error limits. 

One of the first applications of CMD to a realistic and important system was to study the quantum dynamical effects in water. It was found that, even at 300 K, the quantum effects are remarkably large. This finding, in turn, led us to have to reparameterize the flexible water model (called the SPC/F2 model) in order to obtain good agreement with a variety of experimental properties for the neat liquid. An example of the large quantum effects in water can be seen in Fig. 3 in which the mean-spared displacement correlation function, ( x(t) - x(0) 2) is plotted. (These are new results which are better converged than those in Ref 34.) Shown are the quantum CMD and the classical MD results for the SPC/F2 model. The mean-squared displacement for the quantized version of the model is 4.0 X 10- m s-f while the classical value is 4.0 x 10 m s-f The error in these numbers is about 15%. These results suggest that quantum effects increase the diffusivity of liquid water by a factor of two. 

P(i, /, N) is the probability of finding the adatom, initially at site i, at site j after N jumps in an unrestricted random walk. The mean square displacement for a particular starting position, site i, is given by... 

The mean square displacements for the center of mass of the diatomic cluster in a series of walks starting from configurations 0 and 1 are, respectively, given by,... 

For long periods, such that t> l/(k+ + k ), the mean square displacement for the center of mass becomes... 

The calculated root-mean-square displacement for a general sequence of jumps has two terms in Eq. 7.31. The first term, NT(r2), corresponds to an ideal random walk (see Eq. 7.47) and the second term arises from possible correlation effects when successive jumps do not occur completely at random. 

The mean squared displacement for the FFPE (19) in the absence of a force can be calculated similarly ... 

Figure 5. Mean squared displacement for the fractional (a = 1 /2, full line) and normal (dashed) Omstein-Uhlenbeck process. The Brownian process shows the typical proportionality to t for small times it approaches the saturation value much faster than its subdiffusive analogue, which starts off with the r /2 behavior and approaches the thermal equilibrium value by a power law, compare Eq. (61)...

The mean square displacement for N steps can easily be calculated with the help of (A.2)... 

Teller systems the opposite will apply, Table 2 (c), and for the pseudo compressed Jahn-Teller systems, Table 2(d), which involve a static distortion along Cu-N(l), but a dynamic distortion along Cu-N(2) and Cu-N(3), smaller root-mean-square displacements are predicted to lie along the Cu-N bonds for N (1), and larger root-mean-square displacements for N(2) and N(3). These predictions are clearly born out for A2Pb-[Cu(N02)6], [A Cs, Rb], but only partially for the less accurate data for K2Pb[Cu(N02)6] (276 K). 

Estimate the root-mean-square displacement for a 2-p.m silica dust particle (p = 2.65 g/cm3) over a 10-min period. 

Note that in Eq. 6.34 the mean-square displacement is used, rather than the root-mean-square displacement. For a one-dimensional random walk, the mean-square displacement is given by 2Dt, and for a two-dimensional random walk, 4Dt. Since the jump distance (a vector) is A, if the jump frequency is now defined as F = n/t (the average number of jumps per unit time), then on combining Eq. 6.33 and 6.34 gives ... 

From the coordinate data stored in the trajectory files, the mean-square displacements for all tiie movable atoms were calculated, and the diffusion coefficient, D sd, computed using Einstein s relation... 

Fig. 31. Averaged mean square displacements for water and protein atoms of myoglobin. 9, All backbone NCCO atoms O, all side-chain atoms , 160 water molecules found crystallographically -------, linear regression — —, linear regression, neglecting the val-...

Both p and m can be expressed in terms of the basic quantities p (the encounter diameter), o (the root mean square displacement for relative diffusion motion), and v (the frequency of relative diffusion displacements) as [3]... 

Figures 42 and 43 show scanning electron micrographs of two ZSM-5 samples of different configurations coffin-shaped crystals and polycrystalline grains. To emphasize the relationship between sample dimensions and diffusion paths followed during the PFG NMR experiment, magnifications are referenced against typical root mean square displacements for methane and propane molecules during typical PFG NMR observation times.

Figure The time dependence of the mean square displacements for adsorbed methane at T = 300 K in the direction parallel to the pore axis is shown for a range of time sufficient for the squares of the molecular displacement to become linear functions of time. NP denotes the total number of molecules in the pore. The slopes of the linear portions of the plots give the self-diffusion constant D = 2x slope.

Here fl,- is the force constant for atom i and is the thermally averaged mean-square displacement for atom i in the protein the latter quantity is proportional to the crystallographically determined Debye-Waller factor if static disorder is neglected (see Chapt. VI). To simplify the treatment, average mean-square displacements can be used to represent the different types of atoms. The factor 5(r,) is an empirical scaling function that accounts for the interatomic screening of particles which are away from the RZ-RR boundary, 108 it varies from 0.5 at the reaction zone boundary to zero at the reaction region (see Fig. 8). 

Figure 51. Solvent mean-square displacement for water in the cleft region of lysozyme. The mean-square displacement is plotted versus time (ps) for water within a 6.0- A sphere around the apolar atoms (a) Asp-48 C3 (b) Ser-72 0s (c) Asn-46 C3 (d) Gly-71 C . The dashed line indicates the linearly extrapolated diffusional motion.

The mean square displacement for the field component of frequency e> will be... 

In this case and under these conditions (no cell is fllled to capacity) the movement of any particle between the cells is independent of the presence of the other particles. The coefficient of diffusion for the migration of a single component [(case (a)] must then be the same as the diffusion coefficient of this component in the presence of another sorbate [binary or self-diffusion, case (j8)]. In both cases the mean square displacement for i > 1/r is... 

Figure 7. Mean-squared displacement for = 5A and (a) K = and (b) K = at several temperatures = 0.156 (dotted line), 0.625 (dashed line), 2.5 (dash-dotted...

Figure 16. Plot of the mean-squared displacement for a quantum particle solvated in a classical Lennard-Jones fluid. The solid line is the CMD/PIMC algorithm described in Section III.C.l, while the dashed line is for classical MD.

Fig. 4.8. The ratio of the mean square displacements for attached and free poly(methylene) chains (1) normal to the interface (2) parallel to the interface (after Feigin and Napper, 1979).

Fig. 7 Temperature dependence of mean square displacement ( ) for 400 A films of PS of different molecular weights...

Figure 7. Mean square displacement for Si and O ions in bulk silica glass at 6000 K.

Figure 29. Mean square displacement for O ions on the snrface at 6000 K. Data calculated from O ions that remain above the cut-off in Z shown in the inset for the last 90 ps of the 100 ps run over which the data were collected. The cnt-offs pertain to the density profile at 6000 K shown in Figure 26. (see text)...

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