Richardson plot

Figure 12.7. Topography of the surface of chocolate as determined by laser scanning microscopy. Fractal dimensions were calculated from a Richardson plot using SURFRAX . A completely flat surface would have a.FD = 2.00.

Temperature dependent I-V measurements in the temperature range 300-500 K, which allowed determination of the activation energy, were also performed, and the results are listed in Table 6.2. The barrier heights calculated from the Richardson plot using Equation (6.2) are consistent with the room temperature values for each sample. However, the calculated effective Richardson constant is much smaller than the theoretically expected value which is commonly reported in the literature for GaN [2]. As we can see in Table 6.2, the effective Richardson constant is also related to the crystal quality, and obviously further work is required to shed light on the discrepancy between the measured and theoretical values endemic to GaN. 

Figure 9.3 Concepts of fractal geometry (a) profile of aggregate (b) structured walk explorations (c) Richardson plot of a series of explorations of the aggregate...

A detailed comparison of the random walk and the Markov chain models using experimental data for migrating endothelial cells has shown [148] that aU five locomotory parameters defined above affect the speed S and persistence time P estimated by the persistent random walk model. The persistent random walk model can provide a measure of the speed of locomotion S, appropriately weighted to account for the frequency and duration of ceU stops (or slow-downs). The persistence time P, however, is a composite measure of all five locomotory parameters mentioned above. Hence, it may not be always possible to extract all the information necessary to describe cell locomotion from the two parameters of the persistent random walk model. The Markov chain approach, however, also has its drawbacks. Perhaps the most serious drawback is that the values of ceU speeds computed according to Equation 35.5 depend on the time interval A t between ceU observations. Accurate estimations of locomotion speeds with Equation 35.5 are possible only when (a) ceU trajectories are not very tortuous and (b) the cell positions are observed at short time intervals At. This time interval must be carefuUy chosen using Richardson plots to minimize errors in the computation of ceU speed [148]. 

Figure 9.14. The fractal front created in a sandstone by an invading fluid, carries information on the pore structure of the rock. This information may be useful in methods used in secondary recovery of oil. a) Simple experimental setup for examining fractal fronts in sandstone, b) Trace of a front created by the method of (a), c) Richardson plot for the data generated for the front in (b).

FKJUKK 2(1. ) Profiles of simulated particle ag lomeiaics and their Richardson plots. After Kaye ( yjJ9), keprinted with pennission of Wilcy-VCH VcrlagsBessicHschaft. Weioheim, Germany,... 

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