Radioactive equilibria

This situation, when the activity of the higher atomic number nuclide, the parent, is equal to the activity in the next step in the chain, the daughter, is known as radioactive equilibrium (also referred to as secular equilibrium). Thus, secular equilibrium between a parent and a daughter implies an activity ratio of 1. 

Figure 1. (a) Schematic representation of the evolution by radioactive decay of the daughter-parent (N2/N1) activity ratio as a function of time t after an initial fractionation at time 0. The initial (N2/Ni)o activity ratio is arbitrarily set at 2. Time t is reported as t/T2, where T2 is the half-life of the daughter nuclide. Radioactive equilibrium is nearly reached after about 5 T2. (b) Evolution of (N2/N1) activity ratios for various parent-daughter pairs as a function of time since fractionation (after Williams 1987). Note that the different shape of the curves in (a) and (b) is a consequence of the logarithmic scale on the x axis in (b). 

This equation shows that on a plot of (N2/Ni)g versus e , the volcanics erupted from this reservoir should define a straight line whose y intercept (at e = 0, i.e., in the future at 0 = -00) is 1 (radioactive equilibrium, cf Fig. 3). A linear relationship is also obtained if isotope ratios, like ( °Th7 Th)e, are reported versus e (see Fig. 11 in Condomines et al. 1988). This latter diagram is similar to the well known isotope... 

It may be worth mentioning the analogy between the laws of radioactive equilibrium in decay chains (Ni/Ti = N2/T2 =... Ni/Ti) and those of successive magma reservoirs at steady state (i.e., with the same input and output rates O, where Mi/xi = M2/X2 =... Mi/xi ), as illustrated in Figure 5. 

Figure 4. Evolution of the (N2/N1) ratio in a reservoir in the two cases of closed system evolution (as a function of t/T2, where t is the time since fractionation), or in an open-system, steady-state reservoir (the steady-state (N2/N1) ratio is plotted as a function of x/ T2, where x is the residence time of the magma in the reservoir). Initial fractionation results in an arbitrarily chosen ratio of 2, which is kept constant for the iirfluent magma in the continuously replenished reservoir. The diagram shows that radioactive equilibrium is reached sooner in a closed system evolution. It also illustrates the fact that the radioactive parent-daughter pair should be chosen such as T2 is commensmate with the residence time of the magma in the reservoir (e.g., x/ T2 between 0.1 and 10). If T2 is much longer than the residence time x, then the (N2/N1) ratio will remain close to the initial value (here 2). If T2 is much shorter than x, equilibrium will be nearly established in the reservoir.

Figure 5. A schematic representation of superposed steady-state reservoirs of constant volumes Vi (fractional crystallization is omitted in this schema). At steady-state, Vi/xi=V2/x2=..., where x is the residence time. This is analogous to the law of radioactive equilibrium between nuclides 1 and 2 Ni/Ti=N2/T2=...A further interest of this simple model is to show that residence times by definition depend on the volume of the reservoirs.

Figure 6. Schematic representation of the model used by Williams et al. (1986) to calculate the age of the Oldoinyo Lengai (Tanzania) caibonatite magma. The model assumes an instantaneous Ra-Th fractionation produced by the exsolution of a carbonatite melt from a nephelinite parental magma in radioactive equilibrium for both Ra-Th pairs. The existence of Ra- Th disequihbria indicates that the fractionation occurred shortly before eruption, and thus the ( Tla/ °Th) ratios have not significantly changed since the exsolution. By assuming the same Ra-Th fractionation for both pairs, the ( Ra/ °Th) in the carbonatite gives the ( Ra/ h) ratio just after the exsolution, and its age can then be calculated from the equation ...

Figure 16. Schematic representation of a degassing magma reservoir in a physical steady-state (mass M of magma constant). ( ) and [Ik] denote fluxes and radionuclide Ik concentrations, respectively. Indices 0, L, G, E, I, R, refer to deep undegassed magma (in radioactive equilibrium), degassed lava, gas phase, and erupted, intended, or recycled degassed magma, respectively (after Gauthier and Condomines 1999).

While it is expected that the source rocks for the radionuclides of interest in many environments were deposited more than a million years ago and that the isotopes of uranium would be in a state of radioactive equilibrium, physical fractionation of " U from U during water-rock interaction results in disequilibrium conditions in the fluid phase. This is a result of (1) preferential leaching of " U from damaged sites of the crystal lattice upon alpha decay of U, (2) oxidation of insoluble tetravalent " U to soluble hexavalent " U during alpha decay, and (3) alpha recoil of " Th (and its daughter " U) into the solute phase. If initial ( " U/ U).4 in the waters can be reasonably estimated a priori, the following relationship can be used to establish the time T since deposition,... 

Steady state vs. non-steady state profiles. Determination of " Th deficits is relatively simple. Production is balanced against decay and export, with the expectation that in the absence of the latter, " Th should be in radioactive equilibrium with and the flux is given by the integral term in Equation (9) ... 

Therefore, at radioactive equilibrium, the amount of different radio elements present will be inversely proportional to their decay constant, or directly proportional to their half- or average-life periods. 

Secular equilibrium It is limiting case of radioactive equilibrium, in which the half-life of the parent is many times greater than the half-life of the daughter) usually by a factor of 10 or greater). 

Several organizations (e.g., NIST, NRC-Canada, and IAEA) provide sediment reference materials containing radionuclides, many of which are only certified for artificial radionuclides ( Cs, Sr, Am, and Pu). Certain specific radionuclides have no certified natural matrix materials, including ocean, lake, and river sediments. Although these sediments are certified for a few naturally occurring and artificial radionuclides, the extent of radioactive equilibrium of the uranium and thorium decay series in these environmental materials is not provided. NIST currently offers an ocean sediment Standard Reference Material (SRM 4357) in... 

Because plants take in carbon dioxide as long as they live, any carbon-14 lost to decay is immediately replenished with fresh carbon-14 from the atmosphere. In this way, a radioactive equilibrium is reached where there is a constant ratio of about 1 carbon-14 atom to every 100 billion carbon-12 atoms. When a plant dies, replenishment of carbon-1.4 stops. Then the percentage of carbon-14 decreases at a constant rate given by its half-life, but the amount of carbon-12 does not change because this isotope does not undergo radioactive decay. The longer a plant or other organism is dead, therefore, the less carbon-14 it contains relative to the constant amount of carbon-12. 

Two nongaseous isotopes, A and B, are involved in the same radioactive series, which (over the past eons) has established radioactive equilibrium. The half-life of A is 106yrs an

The naturally occurring radioactive elements at the upper end of the periodic table of elements form a number of series, the elements of each series existing m radioactive equilibrium, unless individual elements are separated chemically away horn the series. These series include... 

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