Question

(a) Which of the following nuclides are magic:${\mathbf{}}^{\mathbf{122}}\mathbf{Sn}\mathbf{,}\mathbf{}{\mathbf{}}^{\mathbf{132}}\mathbf{Sn}\mathbf{,}{\mathbf{}}^{\mathbf{198}}\mathbf{AU}\mathbf{,}\mathbf{}{\mathbf{}}^{\mathbf{208}}\mathbf{pb}$? (b) Which, if any, are doubly magic?

Short answer

1. The nuclides that are magic nuclides:${}^{132}\mathrm{Sn},{}^{98}\mathrm{Cd},{}^{208}\mathrm{Pb}$
2. The nuclides that are doubly nuclides: ${}^{132}\mathrm{Sn},{}^{208}\mathrm{Pb}$

Step-by-step solution

The given data:

The given nuclides are ${}^{122}\mathrm{Sn},{}^{132}\mathrm{Sn},{}^{98}\mathrm{Au},{}^{208}\mathrm{Pb}$.

Understanding the concept of magic numbers:  

In nuclear physics, magic numbers are several nucleons such that the nucleons of the nuclide are arranged into complete shells within the atomic nucleus. Thus, their nuclei are more stable than in comparison to other nuclides. They are 2, 8, 20, 28, 50, 82, 126, and so on.

In some cases there the isotopes can consist of magic numbers for both protons and neutrons and those are called double magic numbers. They occur at the heavier nuclides. The magic numbers are: 2, 8, 20, 28, 50, 82, and 114.

a) Calculation of the nuclides with magic numbers:

Any nuclide whose proton number Z or neutron number N has one of these values turns out to have a special stability that may be made apparent in a variety of ways are called magic nuclides.

Thus, the neutron number of ${}^{132}\mathrm{Sn}$is

$132-50=82$

Whichis a magic number.

Again, the neutron number of ${}^{98}\mathrm{Cd}$is,

$98-48=50$

Which also is a magic number.

Again, the neutron number of ${}^{208}\mathrm{Pb}$is,

$208-82=126$

Which also is a magic number.

Hence as per the concept, the nuclides with the magic numbers are ${}^{132}\mathrm{Sn},{}^{98}\mathrm{Cd},{}^{208}\mathrm{Pb}$.

(b) Calculation of the nuclides with double magic numbers:

Any nuclide whose proton number Z and neutron number N has both of the values turns out to have a special stability that may be made apparent in a variety of ways are called doubly magic nuclides.

The nuclide ${}^{208}\mathrm{Pb}$is called “doubly magic” because they contain both filled shells of protons and filled shells of neutrons.

Thus, the neutron number of ${}^{132}\mathrm{Sn}$is

$132-50=82$

Which is a magic number and also the proton number 50 is magic number.

Hence as per the concept, the nuclides with the double magic numbers are ${}^{132}\mathrm{Sn},{}^{208}\mathrm{Pb}$.