# Radioactive Source - Mathematics of Radioactive Decay - Universal Law of Radioactive Decay - Chain-decay Processes

Chain-decay Processes

Chain of two decays

Now consider the case of a chain of two decays: one nuclide A decaying into another B by one process, then B decaying into another C by a second process, i.e. A → B → C. The previous equation cannot be applied to a decay chain, but can be generalized as follows. The decay rate of B is proportional to the number of nuclides of B present, so again we have:

but care must be taken. Since A decays into B, then B decays into C, the activity of A adds to the total number of B nuclides in the present sample, before those B nuclides decay and reduce the number of nuclides leading to the later sample. In other words, the number of second generation nuclei B increases as a result of the first generation nuclei decay of A, and decreases as a result of its own decay into the third generation nuclei C. The proportionality becomes an equation:

adding the increasing (and correcting) term obtains the law for a decay chain for two nuclides:

The equation is not

since this implies the number of atoms of B is only decreasing as time increases, which is not the case. The rate of change of NB, that is dNB/dt, is related to the changes in the amounts of A and B, NB can increase as B is produced from A and decrease as B produces C.

Re-writing using the previous results:

The subscripts simply refer to the respective nuclides, i.e. NA is the number of nuclides of type A, NA0 is the initial number of nuclides of type A, λA is the decay constant for A - and similarly for nuclide B. Solving this equation for NB gives:

Naturally this equation reduces to the previous solution, in the case B is a stable nuclide (λB = 0):

as shown above for one decay. The solution can be found by the integration factor method, where the integrating factor is eλBt. This case is perhaps the most useful, since it can derive both the one-decay equation (above) and the equation for multi-decay chains (below) more directly.

Chain of any number of decays

For the general case of any number of consecutive decays in a decay chain, i.e. A1 → A2 ··· → Ai ··· → AD, where D is the number of decays and i is a dummy index (i = 1, 2, 3, ...D), each nuclide population can be found in terms of the previous population. In this case N2 = 0, N3 = 0,..., ND = 0. Using the above result in a recursive form:

The general solution to the recursive problem are given by Bateman's equations:

 Bateman's equations