# A Brief Introduction to Beta Decay

## Introduction

Beta decay is the most common mode of nuclear decay. The weak interaction mediates this nuclear transformation. Beta decay lifetimes can be much longer than decays associated with other interactions. To contrast, gamma-ray emission from an excited nuclear state typically occurs in picoseconds. We can summarize beta decay with two simple expressions:

$$^A_ZX \rightarrow ^A_{Z+1}X’+e^-+\bar{\nu}_e$$

$$^A_ZX \rightarrow ^A_{Z-1}X’+e^++\nu_e$$

Where the first equation represents $\beta^-$ decay. The second equation represents $\beta^+$ decay. As we can see this process does not change the atomic number of the nucleus, but does change the element.

## Types of Decays

We can break beta decays into two distinct classes: allowed and forbidden. Gamow-Teller (GT) transitions[efn_note] Gamow, G. and Teller, E.; Selection Rules for the β-Disintegration; Phys. Rev., American Physical Society, 1936, Vol. 49, pp. 895-899 [/efn_note], are the most common form of decay. The total angular momentum and isospin of the parent and daughter nuclei can change by $\pm$1 or 0 units. The orbital angular momentum and parity of the nucleus remains unchanged. We call these transitions “allowed”.

During forbidden decays[efn_note] Konopinski, E. J. and Uhlenbeck, G. E.; On the Fermi Theory of β-Radioactivity; Phys. Rev., 1935, Vol. 48, pp. 7-12 [/efn_note],[efn_note] Konopinski, E. J. and Uhlenbeck, G. E.; On the Fermi Theory of β-Radioactivity. II. The “Forbidden” Spectra; Phys. Rev., American Physical Society, 1941, Vol. 60, pp. 308-320 [/efn_note]leptons carry angular momentum away from the nucleus. This yields a smaller decay probability than allowed transitions. The most common form of forbidden decay is the first forbidden (FF) transition. It involves changes in orbital angular momentum and parity change. The total angular momentum can change by 0, 1, or 2 units. Parity change occurs when the emitted electron and neutrino couple to an odd value of angular momentum with respect to the nucleus[efn_note]Krane, K. S.; Introductory Nuclear Physics; John Wiley & Sons, Inc., 1988[/efn_note].

$$T_{1/2}^{-1}= \sum^{E_i \leq Q_\beta}_{E_{i}\geq 0}S_\beta(E_i) \times f(Z,Q_\beta – E_i)$$
$Q_\beta$ represents the energy difference between parent and daughter ground states. $E_i$ is the energy of the final state in the daughter. Z is the number of protons in the decay daughter. $f(Z,Q_\beta – E_i)$ is the Fermi integral. The Fermi integral represents the kinematic phase space, and derives from the relativistic Dirac equation. As $Q_\beta$ increases, the Fermi integral quenches decays to excited final states. Finally, $S _\beta(E)$ is the beta decay strength function. It contains reduced transition probabilities for the decay. They are the matrix elements for the GT operator[efn_note]Grotz, K. and Klapdor, H.; The Weak Interaction in Nuclear, Particle, and Astrophysics; IOP Publishing, 1990[/efn_note].