Branching ratio - Aetherwisp

In nuclei that decay in more than one mode, branching ratios (BR) are numbers that determine how likely one decay mode is compared to the others. They are numbers between 0 and 1 and as such may be interpreted as the Probability of a given decay mode occurring. Over an entire material, they are the fraction of the material that decays in that mode. The total of all branching ratios for a nuclide must equal one:

\text{BR}_{1}+\ldots+\text{BR}_{N}=1\quad\text{where}\quad\text{BR}_{i}\in[0,1]

Branching ratios are useful to determine the partial decay constants and partial half-lives, since they are just the product of the total decay constant or half-life by the appropriate branching ratio:

\lambda _\text{partial}=\lambda _\text{total}\cdot\text{BR},\quad t_{1/2,\text{partial}}=t_{1/2,\text{total}}\cdot\text{BR}

Actinium-226

For example, actinium-226 alpha decays 0.006% of the time, beta minus decays 83% of the time, and captures an electron 17% of the time. The branching ratios are

0.00006 ( $\alpha$ )
0.83 ( $\beta^{-}$ )
0.17 (EC)

Rounding prevents an precise sum to one, but are otherwise consistent. Branching ratios can be used to find the partial decay constants and half-lives from the total decay constant. The half-life of actinium-226 is $t_{1/2}=29\text{ h}$ , so the total decay constant is $\lambda_{t}=6.6\times10^{-6}\text{ s}^{-1}$ . The partial constants will therefore be

\lambda_{\beta^{-}}&=0.83\lambda_{t}=5.5\times10^{6}\text{ s}^{-1} \\ \lambda_{\text{EC}}&=0.17\lambda_{t}=1.1\times10^{-6}\text{ s}^{-1} \\ \lambda_{\alpha}&=6\times10^{-5}\lambda_{t}=4\times10^{-10}\text{ s}^{-1} \end{align}

> and the partial half-lives > $$\begin{align} > t_{1/2,\beta^{-}}&=\frac{\ln2}{\lambda_{\beta}}=1.3\times10^{5}\text{ s}=35\text{ h} \\ > t_{1/2,\text{EC}}&=\frac{\ln2}{\lambda_{\text{EC}}}=6.1\times10^{5}\text{ s}=170\text{ h} \\ > t_{1/2,\alpha}&=\frac{\ln2}{\lambda_{\alpha}}=1.7\times10^{9}\text{ s}=55\text{ yr} > \end{align}

The term also applies to particles and particle decay modes.