14.4 Draft: 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 14.2:

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

These magic 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 sta­ble 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 these 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 im­por­tance will fur­ther be ap­par­ent through­out the fig­ures in this chap­ter.