Allowed, Forbidden, Fermi, and Gamow-Teller Beta Decay Transition

Last edited: 2024-11-05 13:47:27

The selection rules are important for classifying beta decay's transitions. They are based on the conservation of angular momentum and parity during decay.

Beta + vs Beta - Decay

There exist different kinds of beta decay, two of them being beta + ($\beta^+$) and beta - ($\beta^-$) decay. In $\beta^+$ decay, a proton $p^+$ decays into a neutron $n$, a positron $e^+$, and an electron neutrino ${\nu}_e$:

$$p^+ \rightarrow n + e^+ + \nu_e.$$

In $\beta^-$ decay, a neutron $n$ decays into a proton $p^+$, an electron $e^-$, and an electron antineutrino $\bar{\nu}_e$:

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

(p. 272).

Fermi and Gamow-Teller Transitions

For $\beta$ decay, the total spin quantum number $S$ is obtained by summing the spins of the emitted particles. Since both the electron and the neutrino have a spin of 1/2, the sum will therefore be either 0 (antiparallel) or 1 (parallel), depending on the orientation of the spins. $S=0$ is called a Fermi transition, while $S=1$ is called a Gamow-Teller transition.

If we take a look at the change in angular momentum of the nucleus

$$\Delta I = |I_i - I_f|,$$

where $I_i$ is the angular momentum of the initial nuclear state and $I_f$ is the angular momentum of the final nuclear state. We see that $\Delta I$ does not change ($\Delta I = 0$) for a Fermi transition since $S=0$. When the electron and neutrino spins are parallel (Gamow-Teller decay) they carry a total angular momentum of 1 unit so the conversation of angular looks like this in vector form:

$$\vec{I_i} = \vec{I_f} + \vec{1}.$$

This is only possible if $\Delta I = 0$ or $1$. However, it is not possible when $I_i = I_f = 0$ ($0 \rightarrow 0$), in that case only the Fermi transition can contribute (p. 289-292).

Allowed and Forbidden Transitions

The orbital angular momentum $L_\beta$ can take on all positive integers and also dictates the type of transition. When $L_\beta=0$ the decay is called allowed, while the remaining values are called forbidden (because they are generally less probable than allowed decays), with $L_\beta=1$ being the first-forbidden decay, $L_\beta=2$ being the second-forbidden, and so on. The allowed and all the forbidden decays have each a Fermi and Gamow-Teller type, and the allowed Fermi decay is sometimes called superallowed (p. 289-292).

Parity in Beta Decay

$L_\beta$ also determines the parity change of the wave function for beta decay according to $\Delta \pi=(-1)^{L_\beta}$. So for example, allowed transitions' parity does not change.

First-forbidden Decay

First-forbidden decays, $L_\beta = 1$, like allowed decays, have both a Fermi and Gamow-Teller type. For the Fermi transition type, we couple $S=0$ with $L_\beta = 1$ to get a total of 1 unit of angular momentum carried by the beta decay so the change in angular momentum becomes $\Delta I = 0$ or $1$ (but no $0 \rightarrow 0$ transition). For the Gamow-Teller type $S=1$ combined with $L_\beta = 1$ gives a total of 0, 1, or 2 units of total angular momentum so $\Delta I = 0$, $1$, or $2$. Since $L_\beta = 1$ is an odd number the parity changes so $\Delta \pi=(-1)^{L_\beta} =$ yes (p. 291).

Second-forbidden Decay

Using a similar methodology as the first-forbidden decay if we couple $L_\beta = 2$ and $S=0$ we in principle get $\Delta I = 0$, $1$, or $2$ for the Fermi type, and for the Gamow-Teller type ($L_\beta = 2$ and $S=1$) we in principle get $\Delta I = 0$, $1$, $2$, or $3$. However, since the parity does not change ($\Delta \pi=(-1)^{L_\beta} =$ no) the $\Delta I = 0$ and $1$ cases fall within the allowed decay's selection rules so we can exclude them since the allowed variations are much more probable to occur (p. 292).

Beta Decay Transition Rules

A table consolidating all the transition rules for beta decay mentioned above is shown below.