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Beta Decay

Sunday, April 3, 2022

The Beta Decay Process

In a beta decay process, a neutron in the nucleus changes into a proton (or vice versa). This means that both $Z$ and $N$ change, but $A$ stays constant. Beta particles, emitted during the transition are electrons, meaning beta decay is written as $\text{n}\rightarrow\text{p}+\text{e}$ (plus another particle; see below). The electron was not present in the atom (or nucleus) before the decay, but instead was produced by the decay itself (assuming the rest energy difference was at least $m_ec^2$).

Beta decay initially appeared to violate two of the fundamental principles of the universe: first, the decay $\text{n}\rightarrow\text{p}+\text{e}^-$ appears to violate conservation of angular momentum due to the spin of the particle. Additionally, the electrons emitted were measured to have different kinetic energies (violating conservation of energy), which would be impossible if the same process occurred repeatedly. In fact, the electrons had a continuous distribution of kinetic energies instead of just one value.

Using the differences in masses of the particles, we see the $Q$ value of the decay should be

$$Q=\left(m_n-m_p-m_e\right)c^2=0.782~\text{MeV}$$

However, the electrons had a continuous energy from $0$ to $0.782~\text{MeV}$.

Neutrinos

The solution to the above two issues was found in 1930 by Wolfgang Pauli: a third particle emitted by the decay process. The new particle would have no electric charge (since the proton and electron already balance each other), would have spin $1/2$ (to fix the angular momentum problem), and a very small mass (since it changes the electron's kinetic energy by only $0.782~\text{MeV}$).

This particle is called the neutrino and has the symbol $\nu$. The neutrino's antiparticle is called the antineutrino, $\overline{\nu}$. Therefore, the complete beta decay process is

$$\text{n}\rightarrow\text{p}+\text{e}^-+\overline{\nu}$$

or, in a nucleus,

$$^A_ZX_N\rightarrow~^A_{Z+1}X'_{N-1}+\text{e}^-+\overline{\nu}$$

The $Q$ value for the decay is therefore

$$Q=\left[m\left(^AX\right)-m\left(^AX'\right)\right]c^2$$

This energy appears as the energy of the neutrino and the kinetic energies of the particles:

$$Q=E_\nu+K_e+K_{X'}\approxeq E_\nu+K_e$$

Another beta decay process is when a proton decays into a neutron, emitting an antielectron (positron) and neutrino:

$$\text{p}\rightarrow\text{n}+\text{e}^++\nu$$

This process is described as

$$^A_ZX_N\rightarrow~^A_{Z-1}X'_{N+1}+\text{e}^++\nu$$

and

$$Q=\left[m\left(^AX\right)-m\left(^AX'\right)-2m_e\right]c^2$$

Electron Capture

Another beta decay process is called electron capture, and it is when a proton captures an atomic electron from its orbit and converts into a neutron and neutrino:

$$\text{p}+\text{e}^-\rightarrow\text{n}+\nu$$

The electrons used for electron capture must be inner electrons, and the shell that the capture electron comes from identifies the capture process: $K$-shell capture, $L$-shell capture, etc.

In the nuclei, the electron capture process is described as

$$^A_ZX_N+e^-\rightarrow~^A_{Z-1}X'_{N+1}+\nu$$

and

$$Q=\left[m\left(^AX\right)-m\left(^AX'\right)\right]c^2$$

In electron capture, the small initial kinetic energy of the electron and the small recoil energy of the nucleus are negligible, meaning the neutrino takes all of the final energy (called a monoenergetic neutrino):

$$E_\nu=Q$$