# Beta Decay

## Single Beta Decay

Beta decay is a nuclear transition, where the atomic number Z of the nucleus changes by one unit, while atomic mass A remains the same. This results in three possible decay modes:

β − decay

$(Z,A) \quad \rightarrow \quad (Z+1,A) \quad + \quad e^{-} \quad + \quad \bar{\nu}_{e} \qquad (1)$

β+ − decay

$(Z,A) \quad \rightarrow \quad (Z-1,A) \quad + \quad e^{+} \quad + \quad \nu_{e} \qquad (2)$

Electron Capture

$e^{-} \quad + \quad (Z,A) \quad \rightarrow \quad (Z-1,A) \quad + \quad \nu_{e} \qquad (3)$

The basic underlying mechanism for (1) is given by

$n \quad \rightarrow \quad p \quad + \quad e^{-} \quad + \quad \bar{\nu}_{e} \qquad \qquad \text{or} \qquad \qquad d \quad \rightarrow u \quad + \quad e^{-} \quad + \bar{\nu}_{e} \qquad (4)$

on the quark level respectively, see Figure 1. The other decay modes are understood in an analogous way.

The corresponding decay energies are given by the following relations, where $m(Z,A)$ denotes the mass of the neutral atom (not the nucleus) [1]:

β - decay:

$\begin{eqnarray} Q^{-} \quad & = & \quad [m(Z,A) \quad - \quad Zm_{e}]c^{2} \quad - \quad [(m(Z+1,A) \quad - \quad (Z+1)m_{e}) \quad + \quad m_{e}]c^{2} \\ & = & \quad [m(Z,A) \quad - \quad m(Z+1, A)]c^{2} \end{eqnarray} \qquad (5)$

The Q-value corresponds exactly to the mass difference between the mother and the daughter atom. It represents the available energy in a nuclear transition.

β+ - decay:

$\begin{eqnarray} Q^{+} \quad & = & \quad [m(Z,A) \quad - \quad Zm_{e}]c^{2} \quad - \quad [(m(Z-1,A) \quad - \quad (Z-1)m_{e}) \quad + \quad m_{e}]c^{2} \\ & = & \quad [m(Z,A) \quad - \quad m(Z-1, A) \quad - \quad 2m_{e}]c^{2} \end{eqnarray} \qquad (6)$

Because all masses are given for atoms, this decay requires the rest mass of two electrons. Therefore, the mass difference between both has to be larger than $2m_{e} c^{2}$ for β+ -decay to occur.

Electron capture:

$\begin{eqnarray} QEC \quad & = & \quad [m(Z,A) \quad - \quad Zm_{e}]c^{2} \quad + \quad m_{e}c^{2} \quad - \quad [(m(Z-1,A) \quad - \quad (Z-1)m_{e})]c^{2} \\ & = & \quad [m(Z,A) \quad - \quad m(Z-1, A)]c^{2} \end{eqnarray} \qquad \text{(7)}$

The Q-values of the last two reactions are related by

$Q^{+} \quad = \quad QEC \quad - \quad 2m_{e}c^{2} \qquad \qquad \text{(8)}$

If Q is larger than $2m_{e}c^{2}$, both electron capture and β+-decay are competitive processes, because they lead to the same daughter nucleus. For smaller Q-values only electron capture will occur. Obviously, for any of the modes to occur the corresponding Q-value has to be larger than zero [1].

### References

[1] Kai Zuber, Neutrino Physics, Institute of Physics (2004)

Figure 1: Neutron Beta Decay