Partial decay constant

Corresponding to each partial decay constant there is a partial half-life t j/2 where... 

The theoretical description of a emission relies on calculating the rate in terms of two factors. The overall rate of emission consists of the product of the rate at which an a particle appears at the inside wall of the nucleus times the (independent) probability that the a particle tunnels through the barrier. Thus, the rate of emission, or the partial decay constant ka, is written as the product of a frequency factor,/, and a transmission coefficient, T, through the barrier ... 

X and Xy are the partial decay constants (probabilities) of conversion (electron emission) and y-ray emission. The conversion coefficient a is the sum of the partial conversion coefficients of the K shell, L shell,... 

It should be pointed out that it is the total decay constant that is used by Eqs. 3.69 and 3.73a, and not the partial decay constants. 

Each mode of decay in branching may be treated separately the decay in an individual branch has a half-life based on the partial decay constant. Since only the total decay constant (the rate with which the mother nuclide, 2X in (4.50), decays) is observable directly, partial decay constants are obtained by multiplying the observed total decay constant by the fraction of parent decay corresponding to that branch. Cu decays 43 % by electron capture, 38% by negatron emission, and 19% by positron emission. The observed total decay constant is equal to 0.0541 h based on the half-life of 12.8 h. The partial constants are ... 

These partial decay constants correspond to partial half-lives of 29.7 h for electron capture decay, 33.6 h for decay, and 67.5 h for positron decay. 

Note that if the nucleus emits both ot and P particles, the total decay constant is the sum of the partial decay constants > totai =... 

The additive property of the partial decay constants indicated above follows from the same property of the respective partial decay rates. In the case of K, e.g., one can write... 

Thus, it turns out that the atomic ratio of Ar to is equal to the ratio of the partial decay constants called branching ratio at any moment ... 

Therefore, the atomic fraction of either product is indicative of the contribution of the respective partial decay constant to the total 1. For instance, in the case of Ar produced by electron capture (EC) one obtains... 

It should be noted that a different value of X belongs to each different value of Ea- Thus the effective A (as well as the effective T1/2) for an a-emitting nucleus is determined by the sum of partial decay constants XA,-. (Note that this is a special type of branching decay consisting of a branches only.)... 

When one takes into consideration that only 98.62% of the decay of Ac goes to Th and that 1.38% goes to Fr, one finds that the maximum fractional activities for Th and Fr are 0.975 and 0.0138, respectively. Note that the Fr reaches maximum sooner (when the production rate is nearly equal to the initial production rate) it meets the criterion for secular equihbrium. A subtle point is that one must distinguish between total and partial decay constants. The total describes the rate of disappearance of a species, while the partial decay constants describe the rate of production of the product of a particular branch. These points are developed in more detail in the next section. 

One can consider the terms in the Bateman equations to have three parts a product of A s in the numerator, a product of differences in A s in the denominator, and an exponential factor. For branched decay of component i, the 1,- in the numerator is the partial decay constant, and the 1/s in the denominator and the exponential for i are the total decay constants. Another way to view the use of partial decay constants in the numerator is to multiply the entire equation by the fraction for that particular branch. This is not a simplification, because chains with different branches have different numerators and some different exponential factors. 

Mathematical treatments of this equation and for n > 4 members of a decay chain were developed (Bateman 1910). Moreover, for many decay chains, the case of branching decay needs to be considered. When a nuclide A can decay by more than one mode (e.g., by P decay generating the radionuclide B and by a decay generating the radionuclide C) the two partial decay constants As and Ac must be considered, cf. the generator systems with the Bi parent. The considerations for branching decay, of course, are valid as well for a single, i.e., non-decay-chain decay. 

Because of differences in chemical properties of U, Th, and Pa, the elements are fractionated in many geochemical processes, such as sedimentation, mantle partial melting, and coral precipitation from water. With fractionation, the nuclide activities of Th, and Pa do not equal one another. Define the time of disturbance to be time zero. Use hi, A2, and A3 to denote the decay activity of Th, and Pa, respectively, and Xy X2, and X3 to denote the decay constants of Th, and Pa. Start from the full evolution equation for Pa in Box 2-6,... 

The previous four chapters deal with the fractionation of stable trace elements during partial melting. In this chapter, we study the behaviors of radioactive uranium decay series during partial melting. Since quantitative models for uranium-series disequilibria need to include additional parameters in decay constants and are thus more complicated, for simplicity, we assume that the partition coefficients remain constant during partial melting. Thus, we only present modal dynamic melting. 

Pig. 4.— Variation of the observed first order decay constants with partial pressure of NOj. The gas mixtures contained 2.5 Torr Ha and wwe made up to a total pressure of 30 Torr by adding He. 

Fig. 7.—Variation of the observed first order decay constants partial pressure of NO. The gas mixtures contained O, —0.125 Torr NO2+2.5 Torr H2 , —0.20 Ton NO2+2.5 Torr H2 , —0.125 Torr NO2 + I5 Torr H2. In all cases the total pressure was made up to 30 Torr by...

Decay constants and partial a half-lives conq>uted from (11.42) are normally within a factor 4 of the measured values for even-even nuclei. The half-lives of even-odd, odd-even, and odd-odd nuclides are olt longer than predicted by equations like (11.42), even after... 

Aj is referred to as the disappearance constant of nuclide i, while K is the partial formation constant, and a and X the partial reaction cross-sections and decay constants for nuclide i + 1 a, a, X and X may be equal to zero in some cases. The exact meaning of A and A is explained below with examples. Introducing these abbreviations for the formation of the Xj j species of the constant (A + l)-chain, and dropping the second index, which is = 1 in the left vertical row in Figure 15.2, we obtain the equation ... 

When a polymer relaxes at a constant anodic potential, the relaxation and partial opening of the polymeric structure involve a partial oxidation of the polymer. Once relaxed, the oxidation and swelling of the relaxed polymer goes on until total oxidation is reached this is controlled by the diffusion of the counter-ions through the film from the solution. This hypothesis seems to be confirmed by the current decay after the chronoam-perometric maximum is reached. We will focus now on the diffusion control. 

In addition, data are available for a study on the degradation of dimethyl-, dibutyl-, and dioctyltin chlorides in soil (Terytze et al, 2000). It is of note that the results of the degradation testing indicate that the diorganotin compounds are partially degraded to the corresponding monosubstituted compounds by way of example, dioctyltin concentrations were observed to decrease from 40 to 12 ng/1 over a 3-month period while the monooctyltin concentration stayed relatively constant at around 2 ng/1. It is therefore likely that only a fraction of dioctyltin decayed to monooctyltin and/or the... 

The current efficiencies for the different reaction products CO2, formaldehyde, and formic acid obtained upon potential-step methanol oxidation are plotted in Fig. 13.7d. The CO2 current efficiency (solid line) is characterized by an initial spike of up to about 70% directly after the potential step, followed by a rapid decay to about 54%, where it remains for the rest of the measurement. The initial spike appearing in the calculated current efficiency for CO2 formation can be at least partly explained by a similar artifact as discussed for formaldehyde oxidation before, caused by the fact that oxidation of the pre-formed COacurrent efficiency. The current efficiency for formic acid oxidation steps to a value of about 10% at the initial period of the measurement, and then decreases gradually to about 5% at the end of the measurement. Finally, the current efficiency for formaldehyde formation, which was not measured directly, but calculated from the difference between total faradaic current and partial reaction currents for CO2 and formic acid formation, shows an apparently slower increase during the initial phase and then remains about constant (final value about 40%). The imitial increase is at least partly caused by the same artifact as discussed above for CO2 formation, only in the opposite sense. 

Reactions of eh with H and OH were once considered diffusion-controlled see, however, Elliot et al. (1990). The rate constants, 2.5—3.0 x 1010 M-1s 1 (see Table 6.6), are high. In both cases, a vacancy exists in the partially filled orbitals of the reactants into which the electron can jump. Thus, hydrogen formation by the reaction eh + H may be visualized in two steps (Hart and Anbar, 1970) eh + H—H, followed by H + H20— OH" This reaction has no isotope effect, which is consistent with the proposed mechanism. The rate of reaction with OH is obtained from the eh decay curve at pH 10.5 in the absence of dissolved hydrogen or oxygen, where computer analysis is required to take into account some residual reactions. At higher pH (>13), OH exists as O- and the rate of eh + O—"02 has been measured as 2.2 x 1010 M-1s-1. 

The structure constants of the SU(3) group responsible for this transition are equal to zero /391 = /3i0i = /39s = /3108 = 0. It should be noted that the diagrams 2 and 3 do not contribute to the partial width of the 0 — con0 decay channel which is obvious also due to the hadronic flavor conservation principle. According to the expression (2),also the anomalous diagram (FIG.4.) does not contribute to the partial width of the 0 —> W7T° decay because... 

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