Independent decays

The exponent Up in Eq. (9.2) was found to be 0.65 for our samples [31 ]. With this value we attempted to model the experimental curve in Figure 9-17 by Eq. (9.8). We obtained 0.12 and 0.37 eV for the energies W, and W2, respectively. These numbers mean that the 0.12 eV process reaches the magnitude of the temperature-independent decay rale, E, at 170 K, while the 0.37 eV process reaches this level at 200 K and becomes the dominant decay channel above 220 K (see Fig. 9-17). 

Figure 11.8A shows the radioactivity emitted by a mixture of two independently decaying species 1 and 2, detected with detection coefficients Cj and C2 (independent and not necessarily identical, but identical here, for the sake of simplicity). 

As shown in figure 11.8A, in the case of independently decaying species, composite decay curve a attains the limiting behavior imposed by the long-lived nuclide. 

Figure 11,8 Composite decay curves for (A) mixtures of independently decaying species, (B) transient equilibrium, (C) secular equilibrium, and (D) nonequilibrium, a composite decay curve b decay curve of longer-lived component (A) and parent radio nuclide (B, C, D) c decay curve of short-lived radionuclide (A) and daughter radionuclide (B, C, D) d daughter radioativity in a pure parent fraction (B, C, D) e total daughter radioactivity in a parent-plus-daughter fraction (B). In all cases, the detection coefficients of the various species are assumed to be identical. From Nuclear and Radiochemistry, G. Friedlander and J. W. Kennedy, Copyright 1956 by John Wiley and Sons. Reprinted by permission of John Wiley and Sons Ltd.

The ko. A, and Fa parameters obtained for a few alkanes are collected in Table 3. kg is around 10 sec A 10 to 10 sec and Fa 10 to 20 kJ mol h In principle, the decay of excited states may involve Si- Sx-type internal conversion transitions [IC, where Sx is some singlet state that gives the product(s) of chemical decomposition] and Si T -type intersystem crossing processes (ISC). The temperature-independent decay was attributed, on the basis of the size of the rate parameter (ko 10 sec ), to Si T -type intersystem crossing. At the same time the temperature-activated decay with a frequency factor of 10 to 10 sec was attributed to an internal conversion process that takes place by overcoming a barrier of Fa 10-20 kJ mol and leads finally to some... 

The process is simply a combination of the mutually independent decay processes of the individual nuclei. Let w be the probability for a single nucleus to survive at time tx. Even before computing w one may state that... 

We now complete the discussion of Section XII-C by taking into account the independent decay of the initial state, 5>, due to interaction with the radiation field. Let Ftot(t)> denote the wave function which describes the state of our system at time t, in the case where both the perturbations F and HiDt are considered. Then for the overall probability of finding the molecule in the initial state, S>, at time t > 0, we have... 

According to the Simmons model, in the low bias regime the tunneling current is dependent on the barrier width d as J a (1/d) exp(-(30 d), where PQ is bias-independent decay coefficient ... 

If the bridge B consists of several identical repeating components (e.g., phenylene or methylene groups) then the bias-independent decay constant fj was estimated by McConnell32 [19] from... 

A new experimental approach to the problem has been used by Hunter and Stock (231) involving measurement of the frequency dependence of the acoustic signal resulting from transformation of electronic excitation from vibrational to translational energy, consequent to excitation of the B2u state. This technique was used to measure the decay of the benzene triplet over the range 0 to 6 torrs, and it was found, in agreement with what could be inferred from some of the earlier work, that decay of the triplet depends upon benzene concentration. The pressure-independent decay process was found to have a rate constant of (1.7 0.1)... 

The U (uranium)-Th (thorium)-Pb (lead) isotopic system represents three independent decay schemes and is a powerful but complex tool with which to unravel the history of the Earth s mantle (Text box 3.2). During planetary accretion U and Th are refractory, lithophile elements and will reside in the mantle. Pb on the other hand is a volatile and chalcophile/ siderophile element and may in part, be stored in the core. Initial U and Th concentrations are derived from chondritic meteorites, and initial Pb isotope compositions are taken from the iron-sulfide troilite phase in the Canyon Diablo meteorite. The initial bulk Earth U/Th ratio was 4.0 0.2 (Rocholl Jochum, 1993). 

FIG. 4.10. Decay diagram of a mixture of two independently decaying nuclides with half-lives of 4 and 24 h. 

Although the emission spectra that originate from excited states of complexes have structure, the vibrational bandwidths exceed the splittings of the electronic levels. There is little sharpening at low temperature, and incorporation into a lattice does not appear to improve the resolution (15). Nonetheless, it is possible to obtain the level splittings from the decay data. If one assumes a manifold of excited states in thermal equilibrium at all temperatures, each decaying with temperature-independent decay constants, one arrives at an analytical expression for the temperature dependence of the mean decay time of the ensemble (8, 9, 10) ... 

If the sample contains more than one radionuclide, the method of activity measurement can still be used with the following modifications. The total (gross) activity of a sample containing several independently decaying radionuclides is... 

In general one can decide how many independent decay components are needed to describe the observed kinetics by closely inspecting the residuals (difference between measured and fitted curves). The singular value decomposition (SVD) method is a statistical tool to determine the maximum number of components that can be extracted with confidence from the experimental traces. 

The luminescence characterized by temperature-independent decay is called fluorescence. However, according to modern conventions, fluorescence refers to emission of relatively short persistence of the order of s. In fluorescence, the chemical substrate undergoes internal energy transi-... 

In other situations the decay law is, however, non-exponential. It should be stressed that such simplified models, if they describe the basic physics correctly, may introduce some artifacts which disappear if one uses more realistic assumptions (e.g. random spacings, random coupling ). Assuming independent decay into many channels a>, one may sum equation (63) to obtain an RRKM-type formula (65), where W is the number of accessible dissociation continue or open... 

After about 30 minutes, when the fuel temperatures have subsided, the small capacity independent decay heat removal plant is introduced and the boilers are allowed to dry out and are isolated. 

Transient radionuclide generator kinetics for the system Mo/ " Tc. A, independent activity of the parent nuclide B, growth of cumulative parent and daughter activity in a pure parent fraction C, growth of daughter activity in a pure parent fraction D, independent decay of the separated pure daughter fraction at maximum of generated activity... 

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