Radioactive decay equations secular equilibrium

The experimentally measured rates of decay of radioactive atoms show that the decay is first order, where the number of atoms decomposing in a unit of time is proportional to the number present—this can be expressed in the following equation dN/dt = —X N. Another term used for characterizing rate of decay is half-life (t 1/2), the time required for half of the initial number of atoms to decay. Isotopes are considered to be in secular equilibrium, when the rate of decay of the parent is equal to that of its daughter. 

These equations are the same as those derived for radioactive equilibrium between mother and daughter nuclide (eqs. (4.23) and (4.24)) i.e. in secular equilibrium the relations in section 4.4 are not only valid for the directly succeeding daughter nuclide, but also for all following radionuclides of the decay series. This has already been applied in the examples given in section 4.4. 

The equations and solutions for closed-system radioactive decay chains have been known since Bateman (1910). To understand the behavior of these systems, however, it is useful to express them as a linear system of ordinary differential equations and use some basic results from linear algebra to discuss the general solutions. This treatment helps to elucidate the ideas of secular equilibrium and relaxation to equihbrium. 

Equation (2.8) describes the decay of a radioactive parent with ingrowth of its radioactive daughter. The three types of ingrowth relations—secular, transient, and no equilibrium—are discussed in Section 2.2.4 and shown in Figs. 9.10-9.12. 

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