Decay functions

Diffusion and Mass Transfer During Leaching. Rates of extraction from individual particles are difficult to assess because it is impossible to define the shapes of the pores or channels through which mass transfer (qv) has to take place. However, the nature of the diffusional process in a porous soHd could be illustrated by considering the diffusion of solute through a pore. This is described mathematically by the diffusion equation, the solutions of which indicate that the concentration in the pore would be expected to decrease according to an exponential decay function. 

A mathematical treatment of the kinetic model shown in Scheme 2 gives a decay function as... 

X 10 years old, this implies that the content of the reservoir today is about half of what it was when the Earth was formed. The probability density function of residence time of the uranium atoms originally present is an exponential decay function. The average residence time is 6.5 x 10 years. (The average value of... 

Figure 8. Rate of carbon monoxide oxidation on calcined Pt cube monolayer as a function of temperature [27]. The square root of the SFG intensity as a function of time was fit with a first-order decay function to determine the rate of CO oxidation. Inset is an Arrhenius plot for the determination of the apparent activation energy by both SFG and gas chromatography. Reaction conditions were preadsorbed and 76 Torr O2 (flowing). (Reprinted from Ref. [27], 2006, with permission from American Chemical Society.)...

Figure 3.18. Time dependence of the peak position of the 1570 cm Raman band of Sj trans-stilbene in chloroform solution (filled triangle). The time dependence of the anti-Stokes/Stokes intensity ratio is also shown with open circles. The best fit of the peak position change with a single-exponential function is shown with a solid curve, while the best fit of the anti-Stokes/Stokes intensity ratio is shown with a dotted curve. The obtained lifetime for both single-exponential decay functions was 12ps. (Reprinted with permission from reference [78]. Copyright (1997) American Chemical Society.)...

Fig. 1.15 Left propagator for unrestricted self- tained in an experiment, 5(q), plotted semi-diffusion. The propagator P(R, A) is shown for logarithmically over q2. In this representation, increasing encoding times A and becomes the slope of the decaying function is equal to broader with increasing A, while its intensity at (4 jt)2AD, so that the diffusion coefficient D zero displacement is reduced due to the re- can be obtained directly by comparing at least quirement that the area remains normalized to two measurements taken at different values unity. Right signal function as would be ob- of q.

Exponential decay often occurs in measurements of diffusion and spin-relaxation and both properties are sensitive probes of the electronic and molecular structure and of the dynamics. Such experiments and analysis of the decay as a spectrum of 7i or D, etc., are an analog of the one-dimensional Fourier spectroscopy in that the signal is measured as a function of one variable. The recent development of an efficient algorithm for two-dimensional Laplace inversion enables the two-dimensional spectroscopy using decaying functions to be made. These experiments are analogous to two-dimensional Fourier spectroscopy. 

In the case of emulsions, the oil phase is confined in a spherical geometry (droplets), and the random motion of the oil molecules is restricted to the droplet boundary. In this case, the signal decay function S/S0, assuming a Gaussian phase... 

Equation (75) shows that (u(t) is an exponentially decaying function for long times with a decay constant /p. For very massive B particles M N mN with M/mN = q = const, the decay rate should vary as 1 /N since p = mNq/ (q + 1). The time-dependent friction coefficient (u(t) for a B particle interacting with the mesoscopic solvent molecules through repulsive LJ potentials... 

It is obvious that (21) is equivalent to a stretched exponential decay function of the general form... 

Summarizing, we stress that the anisotropy and the fluorescence decay functions change in a complex way as a function of target concentration. Species that fluoresce more intensely contribute disproportionably stronger to the measured parameters. Simultaneous measurements of steady-state intensities allow accounting this effect. 

It is interesting to note that the fast emission dynamics of Ag(0) (shown in Figure 21.7) differs from that of Au(0) j [101]. The decay curve for (Au(0) j could be reasonably fit by a two-exponential decay function with time constants of 74 fs, 5.5 ps and relative amplitudes 0.95, 0.05, respectively (best fit curve shown in Figures 21.5 and 21.7). The Ag nanoparticles initial (71 fs with 0.91 amplitude) and final (5.3 ps with 0.01 amplitude) decay components were similar to those of gold however, an additional component of 650 fs (with 0.08 ampli-... 

Typically, a series of several 2D spectra are recorded with various relaxation delays, ranging from very short to the longest delays, which usually correspond to 1.5-2 relaxation times. The delay values are usually selected so that they are uniformly distributed over this time interval or so that the signal values are uniformly spread. Another sparse sampling strategy proposed in [12] is based on an optimal sampling scheme for a monoexponential decay function a five-point variant of this strategy uses one measurement at a very short relaxation delay and four measurements at 1.3 T2 (or 7 i). 

Once the best fit parameters are obtained, the quality of the fitted decay function must be judged using statistical and graphical criteria. 

The curves were fitted using the following decay function, which must be considered as a purely mathematical model ... 

A variety of extrapolation algorithms have been applied to the sequences generated by the correlation-consistent cc-pVnZ basis sets [12, 51-55], Dunning and his colleagues had initially suggested fitting their calculations to an exponentially decaying function [12, 51, 52],... 

Mathematically these are radically different functions. Du Di, and D3 are all double exponential decays, but their preexponential factors deviate radically and the lifetimes differ noticeably. The ratio of preexponentials for the fast and slow components vary by a factor of 16 D has comparable amplitudes, while D2has a ratio of short to long of 4, and D3 has a ratio of short to long of 1/4. D4 is a sum of three exponentials. All five functions vary from a peak of about 104 to 25, and all four functions, if overlaid, are virtually indistinguishable. To amplify these differences, we assume that the Gaussiandistribution, Da, is the correct decay function and then show the deviations of the other functions from Do. These results are shown in Figure 4.10. The double exponential D fits the distribution decay essentially perfectly. Even Dj and Ds are a very crediblefit. >4 matches Do so well that the differences are invisible on this scale, and it is not even plotted. 

There should exist a correlation between the two time-resolved functions the decay of the fluorescence intensity and the decay of the emission anisotropy. If the fluorophore undergoes intramolecular rotation with some potential energy and the quenching of its emission has an angular dependence, then the intensity decay function is predicted to be strongly dependent on the rotational diffusion coefficient of the fluorophore.(112) It is expected to be single-exponential only in the case when the internal rotation is fast as compared with an averaged decay rate. As the internal rotation becomes slower, the intensity decay function should exhibit nonexponential behavior. 

In the case that the macromolecule is nonspherical or sidechain or segmental motions occur, then the anisotropy will decay as a sum of exponential functions. The work of Kinosita etal. deals with the case in which there are restricted motions. The anisotropy decay function becomes... 

The rotational relaxation of DNA from 1 to 150 ns is due mainly to Brownian torsional (twisting) deformations of the elastic filament. Partial relaxation of the FPA on a 30-ns time scale was observed and qualitatively attributed to torsional deformations already in 1970.(15) However, our quantitative understanding of DNA motions in the 0- to 150-ns time range has come from more accurate time-resolved measurements of the FPA in conjunction with new theory and has developed entirely since 1979. In that year, the first theoretical treatments of FPA relaxation by spontaneous torsional deformations appeared. 16 171 and the first commercial synch-pump dye laser systems were delivered. Experimental confirmation of the predicted FPA decay function and determination of the torsional rigidity of DNA were first reported in 1980.(18) Other labs 19 21" subsequently reported similar results, although their anisotropy formulas were not entirely correct, and they did not so rigorously test the predicted decay function or attempt to fit likely alternatives. The development of new instrumentation, new data analysis techniques, and new theory and their application to different DNAs in various circumstances have continued to advance this field up to the present time. 

FPA measurements on poly(dA-dT) were also undertaken/146) but the excited-state decay function S(t) contained additional intermediate compo-... 

In a final RTD experiment, a sheet of dye was frozen as before and positioned in the feed channel perpendicular to the flight tip. The sheet positioned the dye evenly across the entire cross section. After the dye thawed, the extruder was operated at five rpm in extrusion mode. The experimental and numerical RTDs for this experiment are shown in Fig. 8.12, and they show the characteristic residence-time distribution for a single-screw extruder. The long peak indicates that most of the dye exits at one time. The shallow decay function indicates wall effects pulling the fluid back up the channel of the extruder, while the extended tail describes dye trapped in the Moffat eddies that greatly impede the down-channel movement of the dye at the flight corners. Moffat eddies will be discussed more next. Due to the physical limitations of the process, sampling was stopped before the tail had completely decreased to zero concentration. 

As already discussed, modelling this multiple exponential decay function with the numerical cell model gives valuable information about cell morphology and membrane permeability. Similar information is available... 

We notice that the generating and decay functions characterize the nonequilibrium process in the steady state and, consequently, have a general dependence on the affinities which play the role of nonequilibrium parameters. 

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