Complex Decay Schemes

The explanation along this line is usually made in most textbooks. However, the ideal conditions are seldom achieved in any practical counting system, and some modifications of the fundamental equations are required in order to correct the possible effects which may disturb the ideal conditions. For example, the 47t P- proportional counter has an appreciable sensitivity to y-rays. Furthermore the y-transition is detected by the p-detector through the internal conversion process, if any. Besides, because a coincidence mixer has a finite resolving time, false accidental coincidences are inevitably produced by chance. In addition to this problem, further consideration must be given when a nuclide with a complex decay scheme is measured. Taking account of all of these effects the coincidence equation becomes... 

For many nuclei, more than one mode of decay is positive. Users of radioisotopic sources need information about particles emitted, energies, and probabilities of emission. Many books on atomic and nuclear physics contain such information, and the most comprehensive collection of data on this subject can be found in the Table of Isotopes by Lederer and Shirley." Figure 3.12 shows an example of a complex decay scheme taken from that book. 

Figure 3.12 A complex decay scheme. For complete explanation of all the symbols and numbers see Ref. 4. Half-life is given for each element s ground state, and energy of each level is given at intermediate states. Q is the neutron separation energy. Transition probabilities are indicated as percentages (from Ref. 4).

Such selection rules are very useful in the interpretation of complex decay schemes. They apply only in so far as the strong coupling model is a good approximation in effect, their application is limited to the extent to which the nuclear wavefunction can be separated into a part which depends exclusively on the particle configuration and another which depends on the rotation. When this approximation is no longer sufficient, the transition probabilities are more complex. We shall discuss some applications of this selection rule in Sect. 82. 

It can be expected whenever nuclides with a complex decay scheme are measured. 

Decay Schemes. Eor nuclear cases it is more useful to show energy levels that represent the state of the whole nucleus, rather than energy levels for individual atomic electrons (see Eig. 2). This different approach is necessary because in the atomic case the forces are known precisely, so that the computed wave functions are quite accurate for each particle. Eor the nucleus, the forces are much more complex and it is not reasonable to expect to be able to calculate the wave functions accurately for each particle. Thus, the nuclear decay schemes show the experimental levels rather than calculated ones. This is illustrated in Eigure 4, which gives the decay scheme for Co. Here the lowest level represents the ground state of the whole nucleus and each level above that represents a different excited state of the nucleus. 

As a result of slow (thermal) neutron irradiation, a sample composed of stable atoms of a variety of elements will produce several radioactive isotopes of these activated elements. For a nuclear reaction to be useful analytically in the delayed NAA mode the element of interest must be capable of undergoing a nuclear reaction of some sort, the product of which must be radioactively unstable. The daughter nucleus must have a half-life of the order of days or months (so that it can be conveniently measured), and it should emit a particle which has a characteristic energy and is free from interference from other particles which may be produced by other elements within the sample. The induced radioactivity is complex as it comprises a summation of all the active species present. Individual species are identified by computer-aided de-convolution of the data. Parry (1991 42-9) and Glascock (1998) summarize the relevant decay schemes, and Alfassi (1990 3) and Glascock (1991 Table 3) list y ray energy spectra and percentage abundances for a number of isotopes useful in NAA. 

Lower), as reported earlier, but also rises more rapidly than kp (Fig. 13 Upper). This result immediately requires a more complex kinetic scheme than that of Scheme I. Excellent self-consistent fits to the time evolution of [1] (t) are obtained with an expression that is the sum of three kinetic phases, all having a common rate constant for triplet decay, kp, but with differing values of the rate constants for the decay of 1(3000 s , 40s , 5s ). We have further seen that complexes with different Cc show similar behavior, but with the fractional contribution of these multiple phases varying with species. 

In contrast, at [H+] > 0.1 M, the same reaction results in two or three new EPR signals (glso 1.974, 1.971, and 1.966) in addition to the one already mentioned feso= 1.979).68,75 These EPR signals turn out to be consistent with six-coordinated oxo-Cr(V) species. In this situation, the relative intensity of the EPR signal is pH dependent but is independent of the aldohexose/Cr(VI) ratio. In fact, six-coordinated species are dominant at [H+] > 0.75 M. In addition, both species [six- and five-coordinated oxo-Cr(V) complexes] decay at the same rate, meaning that they are in a rapid equilibrium. Scheme 5 shows the complexation chemistry and the observed Cr(V)-sugar redox processes. 

Scheme 1 predicts a complex decay for the singlet-oxygen phosphorescence. However, there are three simple, limiting situations. For simplicity, let us assume only two pseudophases, the lipidic and the aqueous pseudophases. The three above-mentioned situations are as follows ... 

Conventional decay-schemes studies do not seem appropriate, because of the complexity of the decay schemes, the errors to which they are subject (cf. the discussion of 8 Br above) and the large amount of time needed to carry them out. Direct measurement of the p-strength functions themselves, utilizing total-absorption y spectrometry and, where relevant, delayed-neutron-gamma coincidence techniques, promises to provide a means of producing the necessary information in a reasonable time. 

Complementary nanosecond experiments ratify the photosensitization of the triplet features in 5 and 6 (Fig. 8.15). In the absence of molecular oxygen multiexponential decay kinetics point to a fairly complex deactivation scheme. 

A significant amount of work has been carried out on complexes that are analogous to 8 in that they contain pyridinium acceptors directly coordinated to a photoactive metal center [82-85]. As noted above, in these complexes electronic coupling between the metal center and the pyridinium acceptor is comparatively large, and as a result the dynamics of photoinduced forward and back ET are best considered by using excited state decay theory [86]. In any event, these complexes have figured prominently in the study of ET in metal-organic dyads and some of the important discoveries made with them are briefly reviewed in this section. The Re(I) complex 10a (Scheme 5) has been featured in much of this work... 

Excimers are excited state complexes which consist of two identical species, one of which is in the excited state prior to complexation (See Scheme I). The subject has been thoroughly reviewed for polymers in a recent article by Semerak and Frank ( ). Briefly (Scheme I), an excited monomer species M combines with an identical ground state molecule M to produce an excimer E. Both excited species M and E may undergo the normal processes for deactivation of excited states, i.e., non-radiative decay, radiative decay, or product formation. 

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). 

It is clear that a straightforward physical meaning cannot be attached to the monomer amplitudes nor to the excimer amplitudes A2. For a more complex kinetic scheme such as Scheme (II), the expressions for the amplitudes in the triple-exponential decays similarly are functions of all the rate constants involved (25). Therefore, also in this case, a simple physical meaning cannot be attributed to these amplitudes, see Section 4.3.3. 

The main features of this experiment which require an explanation are the complex shape of the luminescence decay curves, especially at 93°K., and the temperature dependence of the emission. On a simple excitation theory which will be considered initially, only triplet states remain 1 /xsec. or longer after irradiation as all the singlets produced will have returned to the ground state or undergone intersystem crossing to triplet states. In addition to the direct phosphorescence, Ti — S0, a number of different pathways are available for the removal of the remaining triplets. A complete decay scheme must include the following processes. 

The first excited state of I can be populated by decay of 33-day Te or 70-minute Te. Both parents are conveniently produced by the Te(n, y) reaction. The decay scheme (shown in simplified form in Fig. 15.17a) is very complex, and there is some divergence of opinion regarding the details. We have adapted the recent work of Berzins et al. [69]. Early I measurements were made using inaccurate values for some of the relevant nuclear constants. The currently accepted value for the I MOssbauer y-ray energy is = 27-72(6) keV [70]. The excited-state lifetime is = 16-8(2) ns, giving a natural width of 0-59 mm s" [70]. The y-ray has nearly pure Ml multipolarity, and the nuclear spin states are and /g =... 

The 84-2-keV resonance of Pa was first reported in 1968 and also requires complicated experimentation [2]. The 25 5-hour precursor of Th is troublesome to prepare and has to be separated from impurities and fission products following an ( , y) reaction on separated °Th. The decay scheme is complex and the relevant details are shown in simplified form in Fig. 18.1. The 84-2-keV level is probably the third excited state of Pa. This isotope is itself radioactive and decays by a-emission with a half-life of 3-25 x 10 y. It is only recently that quantities of this isotope adequate for preparing chemical compounds have become available. 

Eq. (9.5) becomes complicated (NCRP 1985b) when some beta particles are counted in the gamma-ray detector or vice versa, or the decay scheme is more complex than for a beta particle followed by a single gamma ray. For example, conversion electrons may be in coincidence with beta particles and X rays, or... 

Complex kinetic schemes cannot be handled easily, and, in general, a multidimensional search problem must be solved, which can be difficult in practice. This general problem has been considered for first-order reaction networks by Wei and Prater [13] in their now-classical treatment. As described in Ex. 1.4-1, their method defines fictitious components, B , that are special linear combinations of the real ones, Aj, such that the rate equations for their decay are uncoupled, and have solutions ... 

The reactivity of [Rh(triphos)GO]PF6 triphos=bis(2-diphenylphosphinoethyl)phenyl phosphine with aryloxides ArO Ar=G6H5, G6H4-/>-GH3, G6H4-/>-OGH3) allowed the formation of a series of aryloxycarbonyl complexes Rh(triphos)(GO)(OAr), which were observed by FTIR spectroscopy. These aryloxycarbonyl complexes decayed via different pathways involving the/a - and r-isomers of Rh(triphos)(GO)(OAr) (Scheme... 

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