We previously assumed spherical nucleons however there are deformations which lead to changes in their potential.
For nucleus with lots of valence nucleons they appear to rotate and/or vibrate in unison.
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The collective model: takes into consideration the shell model and the deformations
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💼 Case: isotope far from magic numbers
In this case we make the distinction between
Nucleons that form closed shells
Treat as a solid “core”
Valence nucleons
Treat as a fluid surface
💼 Case: consider $^{164}_{\;\,66}\rm Dy$ where there are $16$ valence protons and $16$ valence neutrons
Assume all $32$ valence nucleons rotate together with energy
$$ E_L=\frac{L(L+1)\hbar^2}{2\mathcal L} $$
where if we approximate the moment of inertia $\mathcal L\approx 32M_N R^2$ and $R\approx R_0 A^{1/3}$ we get $E_2\approx 93 \, \rm keV$ which is sim to experimental result of $73 \, \rm keV$ given our simple model
🗒️ Note: end of examinable content