Diffusion, coefficients lattice

For an ion to move through the lattice, there must be an empty equivalent vacancy or interstitial site available, and it must possess sufficient energy to overcome the potential barrier between the two sites. Ionic conductivity, or the transport of charge by mobile ions, is a diffusion and activated process. From Fick s Law, J = —D dn/dx), for diffusion of a species in a concentration gradient, the diffusion coefficient D is given by... 

The process requires the interchange of atoms on the atomic lattice from a state where all sites of one type, e.g. the face centres, are occupied by one species, and the cube corner sites by the other species in a face-centred lattice. Since atomic re-aiTangement cannot occur by dhect place-exchange, vacant sites must play a role in the re-distribution, and die rate of the process is controlled by the self-diffusion coefficients. Experimental measurements of the... 

The diffusion coefficient corresponding to the measured values of /ch (D = kn/4nRn, is the reaction diameter, supposed to be equal to 2 A) equals 2.7 x 10 cm s at 4.2K and 1.9K. The self-diffusion in H2 crystals at 11-14 K is thermally activated with = 0.4 kcal/mol [Weinhaus and Meyer 1972]. At T < 11 K self-diffusion in the H2 crystal involves tunneling of a molecule from the lattice node to the vacancy, formation of the latter requiring 0.22 kcal/mol [Silvera 1980], so that the Arrhenius behavior is preserved. Were the mechanism of diffusion of the H atom the same, the diffusion coefficient at 1.9 K would be ten orders smaller than that at 4.2 K, while the measured values coincide. The diffusion coefficient of the D atoms in the D2 crystal is also the same for 1.9 and 4.2 K. It is 4 orders of magnitude smaller (3 x 10 cm /s) than the diffusion coefficient for H in H2 [Lee et al. 1987]. 

Lateral density fluctuations are mostly confined to the adsorbed water layer. The lateral density distributions are conveniently characterized by scatter plots of oxygen coordinates in the surface plane. Fig. 6 shows such scatter plots of water molecules in the first (left) and second layer (right) near the Hg(l 11) surface. Here, a dot is plotted at the oxygen atom position at intervals of 0.1 ps. In the first layer, the oxygen distribution clearly shows the structure of the substrate lattice. In the second layer, the distribution is almost isotropic. In the first layer, the oxygen motion is predominantly oscillatory rather than diffusive. The self-diffusion coefficient in the adsorbate layer is strongly reduced compared to the second or third layer [127]. The data in Fig. 6 are qualitatively similar to those obtained in the group of Berkowitz and coworkers [62,128-130]. These authors compared the structure near Pt(lOO) and Pt(lll) in detail and also noted that the motion of water in the first layer is oscillatory about equilibrium positions and thus characteristic of a solid phase, while the motion in the second layer has more... 

Fig. 7.37 Diffusion coefficients for some impurities in NiO grain boundaries compared with the corresponding lattice diffusivities (the grain boundary width is assumed to be I nm) (after...

Lord Rayleigh [310] modeled transport in a homogeneous suspension of spheres placed in a square lattice. In terms of diffusion coefficients, his solution was... 

Referring to the Pick Equations of diffusion, let us now reexamine 1/T, the jump frequency, so as to relate diffusion processes to lattice vibration processes. The reciinrocal of the time of stay is the jump frequency and is related to the diffusion coefficient by ... 

Calibration of the MC time step in the simulation on the 2nnd lattice can be achieved by comparison of rr or DN with the results from a conventional MD simulation (as in the second and third columns of Table 4.8), or via comparison with a translational diffusion coefficient obtained from experiment with a... 

NMR signals are highly sensitive to the unusual behavior of pore fluids because of the characteristic effect of pore confinement on surface adsorption and molecular motion. Increased surface adsorption leads to modifications of the spin-lattice (T,) and spin-spin (T2) relaxation times, enhances NMR signal intensities and produces distinct chemical shifts for gaseous versus adsorbed phases [17-22]. Changes in molecular motions due to molecular collision frequencies and altered adsorbate residence times again modify the relaxation times [26], and also result in a time-dependence of the NMR measured molecular diffusion coefficient [26-27]. 

For Schottky defects in an ionic crystal with a cubic lattice, the diffusion coefficient is given by the relationship, e.g. for a cation,... 

Zi(Air, x) and 7)(N2, x) are spin-lattice relaxation times of nitroxides in samples equilibrated with atmospheric air and nitrogen, respectively. Note that W(x) is normalized to the sample equilibrated with the atmospheric air. W(x) is proportional to the product of the local translational diffusion coefficient D(x) and the local concentration C(x) of oxygen at a depth x in the membrane, which is in equilibrium with the atmospheric air ... 

The diffusion coefficients of palladium in a Pd-Ag alloy and silver in a range of Pd-Ag alloys are known, and the diffusion of palladium and silver atoms in a 20% Pd-Ag alloy was calculated (30) for t = 3600 sec representing the film preparation time. At temperatures of 100°, 200°, 300°, and 400°C, silver atoms would diffuse in this time distances of 3 X 10-4, 0.15, 9, and 150 A, respectively whereas at the corresponding temperatures, palladium atoms would diffuse 26, 460, 3000, and 11,000 A. Palladium atoms can thus penetrate the alloy lattice at moderate temperatures, whereas silver atoms have a probability of diffusing distances equivalent to a few unit cells only when the substrate temperature is greater than 300°-400°C. 

When a crystal is heated, lattice members become more mobile. As a result, there can be removal of vacancies as they become filled by diffusion. Attractions to nearest neighbors are reestablished that result in a slight increase in density and the liberation of energy. There will be a disappearance of dislocated atoms or perhaps a redistribution of dislocations. These events are known to involve several types of mechanisms. However, the diffusion coefficient, D, is expressed as... 

It is clear that the diffusion of H through semiconductors is a very complicated issue not only does the motion itself involve complex interactions between the impurity and the lattice, but the calculation of a diffusion coefficient requires the inclusion of different charge states plus the interaction of H with itself (molecule formation) and with other defects and impurities in the crystal. Chapter 10 of this volume discusses this problem in more detail. 

D diffusion coefficient of oxygen species in the solid catalyst lattice (m2/sec)... 

It is found that the relaxation parameter T p as a function of temperature does not follow an increase with chain length, as the square of the number of methylene carbons. Nor is it linear with N, the number of methylene carbons, which should be true if relaxation to the lattice were rate controlling. Rather, it shows a temperature-induced increase of the minimum value of Tjp with about the 1.6 of N. So, both spin diffusion and spin lattice coupling are reflected. For a spin diffusion coefficient D of approximately 2 x 10 12 cm.2/sec., the mean square distance for diffusion of spin energy in a time t is the ft1 = 200/T A, or about 15A on a Tjp time scale. 

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