# Branching ratio

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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 $k_1dt$. Similarly the probability that it will decay by process 2 in time dt is $k_2dt$. Hence the equation governing the radioactive decay can be written as $dN/dt = â��(k_1 + k_2)N$

The total decay constant for the decay of the parent nuclide is $k = k_1 + k_2$. Hence, the branching ratios for modes 1 and 2 are defined as $BR_1 = k_1/k, BR_2 =k_2/k$

In general, the branching ratio $(BR)$ 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. $BR_i =k_i/(k_1 + k_2 + . . . k_i + . . .) = k_i/k$

Alternatively, given the total decay constant, the “partial” decay constant is given by $k_i = BR_i Â· k$

The relation between the decay constant and the half-life is given by $\tau = ln2/k$ $0.693/k$

and similar relations exist for the partial decay constant and the partial half-life, i.e. $\tau_i = ln2/k_i$ $0.693/k_i$ = $\tau/{BR}_i$

where the subscript i refers to the particular mode of decay.

Example

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

References:

See half-life

Wikipedia on Branching Ratio

J. Magill and J. Galy, Radioactivity Radionuclides Radiation, Springer Verlag, 2005.