Question

Magic numbers are associated with protons and neutrons in a nucleus. State the
magic numbers for protons and neutrons, and give two evidences to support the
existence of these magic numbers

Answer 1

Magic numbers represent a shell being completely filled so additional nucleons have to go into a higher shell. A higher shell involves a greater separation from the other nucleons and lower interaction energy. The conventional magic numbers are $(2,8,20,28,50,82,126)$ .These numbers were found in the case of protons by comparing the number of stable isotopes for different proton numbers. For neutrons the magic numbers were found by comparing the number of stable nuclides with the same neutron numbers.

A stronger indication of magicality of a number is in terms of the incrimental binding energy $(IBE)$ .

Let$BE(n,p)$ indicate the binding energy of a nuclide with n neutrons and p protons. The incremental binding energy of the n-th neutron in that nuclide is$IBE(n=BE(n,p)-BE(n-1,p)$ .For example, the IBEn for the isotopes of Strontium are $38,50,56,61,64$ .

The sharp drop after $50$ neutrons is evidence of a shell being filled. There are $50$ neutrons in all of the shells up to that point. The odd-even sawtooth pattern is an indication of the formation of neutron-neutron spin pairs. The amplitude of the fluctuation associated with the formation of neutron-neutron spin pairs also includes the effect of the adjustment to the spin pair. The sharp drop after $38$ . $38$ being the atomic number of Strontium, is a result of there not being any additional formation of neutron-proton spin pairs after $38$ neutrons.