Decay, exponential

Integration of the differential equation with time-mdependent/r leads to the familiar exponential decay ... 

One advantage of the photon counting teclmique over the phase-shift method is that any non-exponential decay is readily seen and studied. It is possible to detect non-exponential decay in the phase-shift method too by making measurements as a fiinction of tlie modulation frequency, but it is more cumbersome. 

The exponential decay of the A population corresponds to a Lorentzian line shape for the absorption (or emission) cross section, a, as a fiinction of energy E. The lineshape is centred around its maximum at E. The fiill-width at half-maximum (F) is proportional to... 

The vibrational echo experiments yielded exponential decays at all temperatures. The Fourier-transfonn of the echo decay gives the homogeneous lineshape, in this case Lorentzian. The echo decay time constant is AT, where is... 

One may also observe a transition to a type of defect-mediated turbulence in this Turing system (see figure C3.6.12 (b). Here the defects divide the system into domains of spots and stripes. The defects move erratically and lead to a turbulent state characterized by exponential decay of correlations [59]. Turing bifurcations can interact with the Hopf bifurcations discussed above to give rise to very complicated spatio-temporal patterns [63, 64]. 

The change in current as a function of time in controlled-potential coulometry is approximated by an exponential decay thus, the current at time t is... 

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. 

The region of the gradual potential drop from the Helmholtz layer into the bulk of the solution is called the Gouy or diffuse layer (29,30). The Gouy layer has similar characteristics to the ion atmosphere from electrolyte theory. This layer has an almost exponential decay of potential with increasing distance. The thickness of the diffuse layer may be approximated by the Debye length of the electrolyte. 

Figure 2 Exponential decay of a hypothetical radionuclide (N) to a stable daughter (D) as measured in half-lives (ti/,)-Note that as t approaches CO, N approaches 0 and D approaches Nq (Adapted from Faure" )...

This equation demonstrates the exponential decay of the rate of formation of products in a first-order reaction widr time. When... 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

Coupling to these low-frequency modes (at n < 1) results in localization of the particle in one of the wells (symmetry breaking) at T = 0. This case, requiring special care, is of little importance for chemical systems. In the superohmic case at T = 0 the system reveals weakly damped coherent oscillations characterised by the damping coefficient tls (2-42) but with Aq replaced by A ft-If 1 < n < 2, then there is a cross-over from oscillations to exponential decay, in accordance with our weak-coupling predictions. In the subohmic case the system is completely localized in one of the wells at T = 0 and it exhibits exponential relaxation with the rate In k oc - hcoJksTY ". 

The ohmic case is the most complex. A particular result is that the system is localised in one of the wells at T = 0, for sufficiently strong friction, viz. rj > nhjlQo. At higher temperatures there is an exponential relaxation with the rate Ink oc (4riQllnh — l)ln T. Of special interest is the special case t] = nhl4Ql. It turns out that the system exhibits exponential decay with a rate constant which does not depend at all on temperature, and equals k = nAl/2co. Comparing this with (2.37), one sees that the collision frequency turns out to be precisely equal to the cutoff vibration frequency Vo = cojln. 

Scavenging may also be considered as an exponential decay process ... 

Spin-spin relaxation is the steady decay of transverse magnetisation (phase coherence of nuclear spins) produced by the NMR excitation where there is perfect homogeneity of the magnetic field. It is evident in the shape of the FID (/fee induction decay), as the exponential decay to zero of the transverse magnetisation produced in the pulsed NMR experiment. The Fourier transformation of the FID signal (time domain) gives the FT NMR spectrum (frequency domain, Fig. 1.7). 

This predicts an exponential decay of stress as shown in Fig. 2.42. 

This predicts an instantaneous recovery of strain followed by an exponential decay. 

As pointed out by Flory [16], the principle of equal reactivity, according to which the opportunity for reaction (fusion or scission) is independent of the size of the participating polymers, implies an exponential decay of the number of polymers of size / as a function of /. Indeed, at the level of mean-field approximation in the absence of closed rings, one can write the free energy for a system of linear chains [11] as... 

MWDs p x) are plotted against reduced size x = 1/ L) of the chains for a number of densities (f) and are seen to collapse nicely on a single master curve, Fig. 5(a). The exponential decay, expected from Eq. (16) at high densities, is clearly observed in contrast to the indicated exp(-7x) behavior. This finding is in agreement with the simulations in Id [62], but it contradicts the predictions of Gujrati [15] according to whom the Shulz distribution, Eq. (16b), holds independently of the overlap. 

FIG. 10 Micelle size distribution for H2T2 surfactants within the Larson model. The dashed lines show fits to the expected form for spherical micelles (main peak) and cylindrical micelles (tail). Inset shows the tail of the distribution on a semi-logarithmic plot to demonstrate the exponential decay predicted for the cylindrical micelles. (From Nelson et al. [120].)... 

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