Growth of Radioactive Products

Because in nature the ground state of a stable nuclide is often attained by decay chains involving intermediate species decaying at different rates, it is worth evaluating the implications of the relative magnitudes of the various decay constants on the isotopic composition of the element. 

If we call Ni the number of nuclides of the first species, decaying at rate Aj, the number of disintegrated nuclides of type 1 per increment of time dt is 

The population of type 2 nuclides produced by decay of is affected both by the rate of production (corresponding to the decay rate of species 1—i.e., eq. 11.26 changed in sign) and by its own rate of disintegration  

Several substitutions yield the following equation, relating the population of the second decaying species to elapsed time t  

We can now distinguish three general cases, depending on whether the first decaying species has a longer, a much longer, or a shorter half-hfe than that of the daughter nuclide. These three cases are transient equilibrium, secular equilibrium, and nonequilibrium. 

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