Nuclear Shell Model

 

Nuclear shell model explains the “nuclei magic numbers”, in which nuclei with number of protons \(Z\) or neutrons \(N\) of 2, 8, 20,28,50, 82, 126 have high binding energies. Nuclei energy states are characterised by three quantum numbers \(n,\ell ,j\) (principal, angular, total angular). \(n=1,2,3,\ldots ;\ell =0,1,2,\ldots =s,p,d,f,g,h,i,j,k,\ldots ;j=s+\ell\)   with \(j\geq 0\)   and \(s\)   is spin, \(s=-\frac{1}{2},\frac{1}{2}\)  . Here I present how to obtain nuclei magic numbers.

Step 1 : calculate \(f(n,\ell )=2 n+\ell -2\)   for \(n=1,2,3,4\)   and \(\ell =0,1,\ldots ,8\), sort by \(n\)   then \(\ell\)

Step 2 : sort by \(f(n,\ell )\)   then \(n\)   then \(\ell\)

Step 3 : calculate \(g(\ell ,s)=\left\{\begin{array}{cc}-\frac{\ell }{2} & s=\frac{1}{2} \\\frac{\ell +1}{2} & s=-\frac{1}{2} \\\end{array}\right.\) for \(f(n,\ell )=1,2,\ldots ,6\)

Step 4 : sort by \(f(n,\ell )\)   then \(g(\ell ,s)\)

Step 5 : calculate maximum number of protons \(Z=2 j+1\) or neutron \(N=2 j+1\)

Step 6 : summarise \(Z\) or \(N\), then the magic numbers appear

Reference

R Gautreau & W Savin, Schaum’s Outline of Theory and Problems of Modern Physics, 2^nd ed, McGraw-Hill, 1999, p. 182-192.