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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].


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


Figure 1: Neutron Beta Decay