General decay equations

The solution to the general decay equations is often given in textbooks (e.g., Faure 1986). However, this solution is given for initial abundances of the daughter nuclides that are equal to zero. In the most general cases, the initial abundances of the daughter nuclides are not equal to zero. For example, in many geological examples, we make the assumptions that the decay chain is in secular equilibrium. The solutions of these equations can also be used to solve simple box models of U-series nuclides where first order kinetics are assumed. 

Radioactive Decay Equations General Decay Equations... 

GENERAL SOLUTIONS OF U-SERIES DECAY EQUATION USING LAPLACE TRANSFORMS... 

Because there is a double decay involved in parent P disintegration, the decay equation corresponding to the general formula of equation 11.83 reads as follows ... 

Although the above /tth-order expressions are quite simple they are general enough to embrace many of the decay equations used to date [see Szepe and Levenspiel (1968)]. 

It can be seen from Table II that unless experimental conditions are carefully chosen, the contribution from emission to the over-all decay of X(np5 2Py2) can usually be neglected as more rapid kinetic processes result in the removal of the excited atoms. Further, as the mean radiative lifetime is long, it is possible to detect these atoms by absorption spectroscopy. (See Section VII.B, general kinetic equation for the removal of 2Pyt atoms.)... 

Thus, for Hz we obtain an equation which does not depend on the other components and which describes the general decay of Hz and defines the quantity —8Hz/dz = q(x,y,z,t). Consequently, the two-dimensional vector H2 can be represented only as the sum of the vortical and potential components... 

On the other hand, if the position r(n) of the particle is smaller than the decay length of the interaction potential, the external foroe experienced by the particle during the time interval t < t < cannot be assumed constant because the interaction potential is a strong function of position. In order to discretize the generalized Langevin equation, the time interval t < t < t +i is further subdivided into K subintervals such that the... 

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

The relationship between the decay constant X and the half-life tll2 can be derived from the general radioactive decay equation... 

Whereas the correlation function S(Q, t) decays to zero in the liquid state, this is no longer the case below Tc, where the system becomes non-ergodic. Such scenario results from a generalized oscillator equation with non-linear damping. The slowing down of the molecular dynamics creates an enhanced damping, which in turn slows down the correlation function etc. The density is taken as a control parameter. In Fig. 38a, /(the Q dependence is omitted in the most simple approach) is displayed... 

The rubidium-strontium geochronometer used in the Rb-Sr geochronological method is based on the radioactive 3 -decay of Rb to Sr. The growth of radiogenic Sr in a Rb rich mineral can be described by the following Equation (9.6). In the rubidium-strontium age dating method, the radioactive Rb isotope with a natural isotope abundance of 27.85 % and a half-hfe of 4.88 X 10 ° years is fundamental to the 3 decay to the isobar Sr. The equation for the Rb-Sr method can be derived from the general equation of radioactive decay (Equation 8.8 in Section 8.8) ... 

According to Eq. (50), each resonance state contributes independently to the survival probability. Equation (50) can be shown to provide an appropriate approximation even for NO2 [HO] in which the resonances overlap. An interesting discussion of a more general decay function for overlapping resonances can be found in Ref. 274. 

By superposition, the batch-decay equation can be further generalized for the case of arbitrary initial amounts Nf of any of the radionuclides in the chain ... 

Later Rubinson (1949) generalized the equations to include transmutation of nuclei by neutron capture in addition to spontaneous decay. The governing factor in this case is where (T is the neutron capture cross section (cm ) of species n, and 4> is the neutron flux (n cm s ). 

In general, an equation can be written for each isotope in the reactor, with some combination of these terms involving decay and transmutation to and from the isotope ... 

I LEARNING CHECK 10.3 Write a general balanced equation for the nuclear process that occurs during positron emission. How do the atomic numbers of daughter nuclei compare with those of the decaying nucleus ... 

In general, these equations are susceptible only to numerical treatment, which has been given by Graham (1978), Sabin and Graham (1983). However, if viscoelastic effects are small, they can be regarded as a perturbation of the exact elastic theory, and some approximate formulae may be given. One way in which viscoelastic effects may be small is when the total collision time is short compared with the decay times of the material, so that we can approximate the viscoelastic... 

Generalization of the one-dimensional instanton theory to two and higher dimensions has been naturally tried by many authors [17,40-42]. Here the two-dimensional extension is briefly explained according to the work by Benderskii et al. [17]. First, let us consider the decay of a metastable state. The direct two-dimensional (Qi, Q2) generalization of Equations (2.142) and (2.143) is given by... 

It is generally believed that the relaxation of the phase coherence (dephasing) occurs on a shorter time scale than the time scale for the decay of excited state populations. For times much longer than the time scale for the loss of phase coherence, the generalized master equation is assumed to take the form of the kinetic equation... 

In order to integrate Eqs. (3.1) and (3.2), it is necessary to know the distribution function fu. In the absence of external forces, the rate of decay of a nonequilibrium velocity distribution to an equilibrium one is given by the generalized Boltzmann equation... 

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