  This is NOT what the nucleus is doing. |
 Rotational spectra are insanely complex and properly the study of so-called “nuclear chemistry.” We will not say much about the details. |
 Deformed nuclei occupy the regions of “half-filled” shells, in other words the regions around halfway between “magic numbers.” |
The Dy isotopes offer a gorgeous example of rotational symmetry breaking... as more and more pairs of neutrons are added, the spectrum shifts from pure vibrational to pure rotational! The nuclear many-body Hamiltonian is rotationally invariant, meaning it does not depend on the orientation of the coordinate axes. However, the mean field solution (the lowest-energy configuration) can still be non-spherical. This happens because the nuclear potential and density distribution can find an energy minimum in a deformed shape rather than a sphere. [The mean field method approximates the total interaction by an average potential that each nucleon experiences, derived from the collective field due to all other nucleons.]
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Nuclear deformation in practice means that the intrinsic ground state (or a low-lying state) has a non-spherical shape, for example, an ellipsoid. This breaks the full rotational symmetry of the Hamiltonian. However, the nuclear many-body Hamiltonian still has intrinsic rotational invariance, in the sense that the kinetic energy and interactions are invariant under rotations of the coordinate system. The deformation is a spontaneous symmetry breaking: the lowest-energy solution is not spherical, but the laws governing the system remain rotationally symmetric.
| A surprisingly good AI summary: In even-even nuclei, the pairing of nucleons results in a Jπ=0+ ground state with an intrinsic angular momentum projection K = 0. Symmetry requirements: For axially symmetric nuclei, a 180° rotation perpendicular to the symmetry axis requires that states with K = 0 must have even angular momentum J. Combined effect: The combination of the K = 0 intrinsic structure and the symmetry requirement restricts the ground-state rotational band to only even values of J, giving rise to the characteristic 0+, 2+, 4+ etc., sequence of states. |  |
 Irrotational! |
 In reality! |
Rigid! |
For low-lying states, the extracted rotational inertia for deformed nuclei is neither that of a rigid ellipsoid, nor that of an “irrotational” superfluid, but about halfway between. The importance of pairing in nuclear matter makes the nucleus very much like a superfluid or superconductor, in general, however. Drastic things happen to the rotational bands when the excitation is great enough to break a significant number of pairs.
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Walter Greiner (1935 -
2016) was the best-known German theoretical physicist
of the last half of the 20th Century. In his long and
active career, he worked in just about every field of
theoretical physics, but particularly in nuclear
theory and elementary particle theory. He also
co-wrote a long series of textbooks covering pretty
much all of theoretical physics at the graduate level.
One
of his mid-1960 papers had a major impact on my own
research! Tamura and I called the method SMART. |
Sabine Hossenfelder, famous for her weekly YouTube series Science Without the Gobbledygook, was one of Greiner's graduate students.