Many nuclides have more than one decay mode. Consider a nuclide in which there are two decay modes. The probability that an atom will decay by process 1 in time dt is . Similarly the probability that it will decay by process 2 in time dt is . Hence the equation governing the radioactive decay can be written as
The total decay constant for the decay of the parent nuclide is . Hence, the branching ratios for modes 1 and 2 are defined as
In general, the branching ratio for a particular decay mode is defined as the ratio of the number of atoms decaying by that decay mode to the number decaying in total, i.e.
Alternatively, given the total decay constant, the “partial” decay constant is given by
and similar relations exist for the partial decay constant and the partial half-life, i.e.
where the subscript i refers to the particular mode of decay.
For the nuclide Ra-226, what is the partial half-life cluster decay?
The half-life of Ra-226 is 1600 y. The branching ratio for cluster decay is 2.6x10-11.
It follows that the partial half-life for cluster decay is (1600 y) / 2.6x10-11 = 6.15x1013 y
Wikipedia on Branching Ratio
J. Magill and J. Galy, Radioactivity Radionuclides Radiation, Springer Verlag, 2005.