4.5 Stability of nuclei

In figure 4.5 we have colour coded the nuclei of a given mass $A=N+Z$ by their mass, red for those of lowest mass through to magenta for those of highest mass. We can see that typically the nuclei that are most stable for fixed $A$ have more neutrons than protons, more so for large $A$ increases than for low $A$. This is the “neutron excess”.

4.5.1 $\beta$ decay

If we look at fixed nucleon number $A$, we can see that the masses vary strongly,

It is known that a free neutron is not a stable particle, it actually decays by emission of an electron and an antineutrino,

The reason that this reaction can take place is that it is endothermic, $m_n{c}^2>m_p{c}^2+m_e{c}^2$. (Here we assume that the neutrino has no mass.) The degree of allowance of such a reaction is usually expressed in a $Q$ value, the amount of energy released in such a reaction,

Generically it is found that two reaction may take place, depending on the balance of masses. Either a neutron “$\beta$ decays” as sketched above, or we have the inverse reaction

For historical reason the electron or positron emitted in such a process is called a $\beta$ particle. Thus in ${\beta }^{-}$ decay of a nucleus, a nucleus of $Z$ protons and $N$ neutrons turns into one of $Z+1$ protons and $N-1$ neutrons (moving towards the right in Fig. 4.6). In ${\beta }^{+}$ decay the nucleus moves to the left. Since in that figure I am using atomic masses, the $Q$ factor is

The double electron mass contribution in this last equation because the atom looses one electron, as well as emits a positron with has the same mass as the electron.

In similar ways we can study the fact whether reactions where a single nucleon (neutron or proton) is emitted, as well as those where more complicated objects, such as Helium nuclei ($\alpha$ particles) are emitted. I shall return to such processed later, but let us note the $Q$ values,