Estimating Jump Distances

The distance to which seed can be dispersed to establish new populations ahead of the migrating species front has been measured in two ways. First, distances have been tabulated between disjunct 

FIGURE 3 Maps showing distances between outlying colonies and the main population at the time the colonies were established, as indicated by the fossil pollen records of beech (a) and hemlock (b) (S. Webb, 1987 Woods and Davis, 1989 Davis et al., 1986 Calcote, 1986 Davis et al, unpublished data). 

Jump distances implied by fossil records of establishment of outlying colonies Beech Fagusgrandifolia) 40 kin (n = 7) 

Average distance to outlying colonies on presettlement range map Beech (Fagiisgrandifolia) 8 km (n = 16) 

FIGURE 4 Maps showing the distribution of beech and hemlock in Wisconsin at the time of the Federal Land Office Survey in the early 19th century (redrawn from Davis et al, 1991). 

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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 ... 

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... 

The Arrhenius equation can be interpreted in terms of the theory of the absolute reaction rates (20). Taking 4.65 A as the jump distance, and estimating the number of charge carriers from the total ion exchange capacity of the zeolite, we can calculate the entropy change of conduction. The values range between —5 and — 20 eu. Freeman and Stamires reported AS values between —2 and —11 eu. Our values have a tendency to decrease with increasing cationic radius. However, all the uncertainties of the measurements accumulate in this factor. Indeed, the influence of the electrode contact caused variations of equal importance as the size of the cations. For this reason, we prefer not to insist on the interpretation of AS values. 

Regarding the distribution of jump lengths, we know that the settlers did not always cover the same distance, and the distribution w x) should include the possibility of different jump lengths. This can be done by fitting the observed data to a continuous distribution. In [64], the jump distances covered by settlers were estimated from individual records obtained from the migrations.org project database, available at http //www.migrations. org. The authors collected 400 individual... 

To compare the predictions from (7.43) with the observed values, we need to estimate the parameters T, D, and r. As in previous sections, T is the mean time elapsed between two successive migrations, i.e., the generation time. It can be estimated as the time needed for a newborn individual to grow into an adult and reproduce. When the adult age is reached, individuals leave the paternal territory and fly to new places. Since the available data for the dispersal process correspond to a histogram of frequencies of jump distances, the diffusion coefficient can be calculated from the discrete version of (7.4) ... 

The obvious disadvantage of this approach is that the mean square jump distance is seldom known with any accuracy and cannot be easily estimated a priori. 

The pulsed field gradient method of self-diffusion measurement was originally developed by Stcjskal and Tanner " for the measurement of diffusion in liquids. The self-diffusivity is measured directly and no prior estimate of jump distance is needed so the results are, in principle, more reliable. The development and application of this technique to the study diffusion in zeolites and zeolitic adsorbents has been achieved largely through the researches of Pfeifer, Karger, and their co-workers at the Karl Marx University in Leipzig,... 

Table 1. Summary of the simulation results for the calculation of the nucleation rate for monodisperse hard sphere coUoids. Here (j) is the volume fraction of the liquid phase. AG is the measured free energy to form a cluster of critical size ric. /Do is the attachment rate of particles to the critical cluster divided by the free diffusion coefficient. / = /

It also seems appropriate to discuss briefly the jump model of diffusion and its effect on linewidth. In this model, the resonant atoms are described as jumping from one lattice site to another by a function h(r), where this function (the correlation function) is the probability of finding the atom at r after a jump from the origin. The average residence time on each site is t,. Since the distance between lattice sites is of the order of the y-ray wavelength, an estimate for the linewidth broadening is... 

Simulations of C NMR lineshapes have shown that experimental spectra that appear to result from a superposition of two different lines (cf. Fig. 15) can be explained by the above-mentioned molecular jump model. Analogous conclusions were drawn from macroscopic sorption kinetic data (82). From the experimental C NMR lineshapes, a mean residence time tj of 20 and 150 p-s for a concentration of six molecules per u.c. at 250 and 200 K, respectively, was derived. Provided that these jumps detected in C NMR spectroscopy are accompanied by a translational motion of the molecules, it is possible to derive self-diffusivities D from the mean residence times. Assuming the diffusion path of a migrating molecule as a sum of individual activated jumps, for isotropic systems the relation (P) = 6Dtj is valid, where (P) denotes the mean square jump length. Following experimental and theoretical studies on the preferential sorption sites of benzene molecules in the MFI framework (83-90), in our estimate the mean distance between adjacent sorption sites is assumed to be 1 nm. 

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