When studying nuclei it is said that spherical nuclei do not rotate, instead rotations are considered for deformed nuclei only. I do not understand why is that. If one can write the hamiltonian of deformed nuclei as $H=H_{rot}+H_{intrinsic}$, with the same coordinate transformation one could do the same for spherical nuclei. The only difference is that now $I_1=I_2=I_3 \equiv I$ and the rotational spectrum is $E_{rot} = \frac{\hbar^{2}}{2I}J(J+1)$.
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This is colloquial to say that spherical nuclei do not have a rotational spectra.
The nucleus could have in principle rotational energy but we’d have no way of measuring energy differences through quadrupole transitions since all quadrupole moments are $0$. The rotational energy would then appear as a constant offset in any energy calculation.
(Nota: since rotational transitions have the selection rule $\Delta L=\pm 2$, they must arise as a result of quadrupole transitions.)
answered Apr 25, 2022 at 13:05