14.5 Magic numbers

In nuclear physics, there are certain special values for the number of protons or the number of neutrons that keep popping up. Those are the values shown by horizontal and diagonal lines in the decay plot figure 14.1:

\begin{displaymath}
\mbox{magic numbers:}\quad
2, 8, 20, 28, 50, 82, 126, \ldots %
\end{displaymath} (14.2)

These numbers were historically found to be associated with unusual stability properties. For example, the magic number of 82 neutrons occurs in 7 stable nuclei, more nuclei than for any other number of neutrons. The runners-up are 20 and 50 neutrons, also both magic numbers, that each occur in 5 stable nuclei.

Nuclei that have a magic number of protons also tend to have unusual stability. For example, the element with the most stable isotopes is tin, with 10 of them. Tin has $Z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 50 protons, a magic number. To be sure, the runner up, Xenon with nine stable isotopes, has $Z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 54, not a magic number, but the heaviest of the nine stable isotopes has a magic number of neutrons.

The last element to have any stable isotopes at all is lead, and its number of protons $Z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 82 is magic. The lead isotope $\fourIdx{208}{82}{}{}{\rm {Pb}}$, with 82 protons and 126 neutrons, is doubly magic, and it shows. It holds the triple records of being the heaviest nucleus that is stable, the heaviest element that is stable, and the highest number of neutrons that is stable.

The doubly magic $\fourIdx{4}{2}{}{2}{\rm {He}}$ nucleus, the alpha particle, is stable enough to be emitted in alpha decays of other nuclei.

Nuclei with magic numbers also have unusually great isotopic presence on earth as well as cosmic abundance. The reason for the magic numbers will eventually be explained through a simple quantum model for nuclei called the shell model. Their presence will further be apparent throughout the figures in this chapter.