Question

(c) Explain what is the significance of the magic numbers (N 2, 8, 20, 28, 50, 82 and 126). Explain how do the magic numbers depend on the spin-orbit term of the nuclear potential: Where the total angular momentum j is given by the sum of the orbital angular momentum i and the nucleon spin, 2 Derive the expression of Vor for j-1+and for j 7 marks]

Answer 1

1. The nuclei with these numbers (magic numbers) of proton or neutron are both with magic numbers it was found that those nuclei are more stable. Eg: He⁴ (2protons and 2 neutrons) is more stable than B⁸ (5protons and 3 neutrons). He⁴ contains the magic number of both protons and neutrons and because of that, it is very stable.

The significance of Magic number: These numbers sheds light on the structure of the nucleus. The nuclei with these magic numbers were more stable, just like that of fully filled outer shell atoms like Ne. So, these magic numbers suggested that even the nucleus has a shell-like structure and each shell is filled according to that of Pauli's exclusion principle.

2. Dependence of the magic numbers on the spin-orbit term: If you take any potential for a nucleus and try to get the energy states in 3D-radial coordinates you will get some grouping of the energy states(shells). Eg: If you consider the nuclear potential to be of square-well, the shells got are 1s,1p,1d,2s,1f,2p,1g,2d and so on. So if we see the number of nucleons in each shell based on Pauli's exclusion principle the numbers turn out to be 2,8,20,34,58,92,138. Which are not magic numbers seen in experiments. When one adds this spin-orbit term this spin-orbit coupling will redistribute the energy levels (by changing the effective potential offered) depending on the l and s value and thus give exact magic numbers. This effect can be seen by the terms which you are going to derive in the next section.

3.

V_tot = V (r) + V_So (r) (l^^rightarrow middot s^rightarrow) We have to calculate (l^rightarrow middot b^rightarrow) We know that j^rightarrow = l^rightarrow + b^rightarrow \r\n Square on both sides. (j^rightarrow)^2 = (l^rightarrow + b^rightarrow)^2 (j^rightarrow)^2 = (l^rightarrow)^2 + (s^rightarrow)^2 + 2. l^rightarrow middot s^rightarrow j^rightarrow^2 = j (j + 1) h^2 s^rightarrow^2 = s (s + 1) h^2 l^rightarrow^2 = l (l + 1) h^2 j (j + 1) h^2 = l (l + 1) h^2 + s (s + 1) h^2 + 2. (l^rightarrow middot s^rightarrow) 2 (l^rightarrow middot b^rightarrow) = j (j + 1) h^2 - l (l + 1) h^2 - s (s + 1) h^2 (l^rightarrow middot b^rightarrow) = 1/2 h^2 [j (j + 1) - l (l + 1) - s (s + 1)] - (1) If j = l + 1/2 s = 1/2 (l^rightarrow middot s^rightarrow)_j = l + 1/2 = 1/2 h^2 [(l + 1/2) (l + 1/2 + 1) - l (l + 1) - 1/2 (1/2 + 1)] = 1/2 h^2 [(l + 1/2) (l + 3/2) - l^2 - l - 1/4 - 1/2] (l . b)_j = l + 1/2 = 1/2 [l^2 + 3/2 l + 1/2 l + 3/4 - l^2