Nuclei with magic numbers of protons $Z = 2, 8, 20, 28, 50, 82$ and magic numbers of neutrons $N = 2, 8, 20, 28, 50, 82, 126$ are found to be very stable.
$(i)$ Verify this by calculating the proton separation energy $S_p$ for $^{120}Sn$ $(Z = 50)$ and $^{121}Sb$ $(Z = 51)$. The proton separation energy for a nuclide is the minimum energy required to separate the least tightly bound proton from a nucleus of that nuclide. It is given by $S_p = (M_{Z-1, N} + M_H - M_{Z, N})c^2$. Given:
$^{119}In = 118.9058 \ u, ^{120}Sn = 119.902199 \ u, ^{121}Sb = 120.903824 \ u, ^1H = 1.0078252 \ u$
$(ii)$ What does the existence of magic numbers indicate?

(N/A) $(i)$ For $^{120}Sn$:
$S_p = [m(^{119}In) + m(^1H) - m(^{120}Sn)]c^2$
$S_p = (118.9058 + 1.0078252 - 119.902199) \times 931.5 \ MeV/u \approx 10.64 \ MeV$
For $^{121}Sb$:
$S_p = [m(^{120}Sn) + m(^1H) - m(^{121}Sb)]c^2$
$S_p = (119.902199 + 1.0078252 - 120.903824) \times 931.5 \ MeV/u \approx 5.77 \ MeV$
Conclusion: Since $(S_p)_{Sn} > (S_p)_{Sb}$,$^{120}Sn$ is more stable because $Z=50$ is a magic number.
$(ii)$ The existence of magic numbers indicates that nucleons (protons and neutrons) in a nucleus are arranged in a shell structure,similar to the shell structure of electrons in an atom. It also explains the peaks in the binding energy per nucleon curve.