Jump distance

Ionic transport in solid electrolytes and electrodes may also be treated by the statistical process of successive jumps between the various accessible sites of the lattice. For random motion in a three-dimensional isotropic crystal, the diffusivity is related to the jump distance r and the jump frequency v by [3] ... 

The term Oq now contains n and ze as well as the information on attempt frequency and jump distance. This expression accounts for the fact that ionic conductivity increases with temperature. 

The list below shows the last position reached, in units of the jump step a, during a random walk for 100 atoms, each of which makes 200 jumps. If the jump time is 10-3 s and the jump distance, a, is 0.3 nm, estimate the diffusion coefficient (a) in units of a2 s-1 and (b) in units of m2 s-1 ... 

Figure 6.3 Potential barrier to be surmounted by a diffusing ion in the presence of an electric field schematic. The distance a represents the jump distance between stable sites, and Agm is the average height of the potential barrier.

To get an idea of the probable displacement distribution of the atoms, the jump distances are squared, and the mean of these squares, over the n steps, is computed. Thus, for a single walk ... 

Figure S5.6 Two adjacent planes, 1 and 2, in a crystal, separated by the jump distance of the diffusing atom, a. The number of diffusing atoms on planes 1 and 2 are N1 and N2, respectively.

The deviations from Gaussian behaviour were successfully interpreted as due to the existence of a distribution of finite jump lengths underlying the sublinear diffusion of the proton motion [9,149,154]. A most probable jump distance of A was found for PI main-chain hydrogens. With the model... 

Fig.4.19 Tseif(Q) obtained for a all the protons in PVE empty MD simulations,/ /// NSE, /=0.55) and b the main chain (filled circle, /=0.66) and the side group hydrogens (empty circle, /=0.51), both from the MDS. Dotted lines are expected Q-dependence from the Gaussian approximation in each case. Solid lines are description in terms of the anomalous jump diffusion model. Insets Chemical formula of PVE (a) and distribution functions obtained for the jump distances (b)...

The Q-dependence of the two contributions to Eq. 4.22 (or equivalently, to Eq. 4.26) is displayed in Fig. 4.25 for the case dp=l,5 A. From the oscillation of both contributions with Q the jump distance may be obtained. The associated timescale may be found from the time decay of the inelastic part. 

Fig. 4.26 PB NSE spectra in the -relaxation regime a at 205 K for the Q-values indicated b at Q=1.88 A and c at 2.71 A for the temperatures indicated. Solid lines are the fitting curves obtained in the incoherent approximation for the inelastic part (jump distance dp=1.5 A). (Reprinted with permission from [133]. Copyright 1996 The American Physical Society)...

Fig. 4.33 Q-dependence of the relative quasi-elastic contribution from the -process to the coherent scattering function of PIB for jump distances of 0.5 A and 0.9 A (lines). The static structure factor S(Q) is shown for comparison (filled circle), (Reprinted with permission from [125]. Copyright 1998 American Chemical Society)...

The value of the jump distance in the )0-relaxation of PIB found from the study of the self-motion of protons (2.7 A) is much larger than that obtained from the NSE study on the pair correlation function (0.5-0.9 A). This apparent paradox can also be reconciled by interpreting the motion in the j8-regime as a combined methyl rotation and some translation. Rotational motions aroimd an axis of internal symmetry, do not contribute to the decay of the pair correlation fimction. Therefore, the interpretation of quasi-elastic coherent scattering appears to lead to shorter length scales than those revealed from a measurement of the self-correlation function [195]. A combined motion as proposed above would be consistent with all the experimental observations so far and also with the MD simulation results [198]. 

Ionic Transport, a. Conductivity The specific conductance of the SPS (Na+ form) membranes is shown in Fig. 8, whose data are summarized in Table II, including values of an apparent energy of activation. An exponential increase in ionic conductance together with a decrease in an apparent energy of activation may be related to a decrease in a "jump" distance between ion-exchange sites as a function of lEC. 

Thus, microscopically, the diffusion coefficient may be interpreted as one-sixth of the jumping distance squared times the overall jumping frequency. Since / is of the order 3 x 10 ° m (interatomic distance in a lattice), the jumping frequency can be roughly estimated from D. For D m /s such as Mg diffusion in... 

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