The Collective Model: Rotations, Vibrations, and Deformed Nuclei

The shell model describes nuclei near closed shells, where one valence nucleon carries the observed properties. Mid-shell nuclei, with many valence nucleons, behave differently: they deform into stable spheroids and show low-lying excited states in regular sequences no single-particle scheme predicts. These are collective motions of the nucleus as a whole, the rotations and vibrations of the charged liquid drop. Bohr and Mottelson unified the two pictures by letting a nucleon move in a deformed well that itself rotates and vibrates, and the resulting collective model accounts for the rotational bands, phonon multiplets, and giant resonances that dominate the spectra away from closed shells.¹

Rotational bands

A nucleus with a permanent quadrupole deformation has a body axis, and rotation about an axis perpendicular to it costs energy. Quantizing a rigid rotor of moment of inertia $\mathcal{I}$ gives

where $I$ is the total angular momentum. For an even-even deformed nucleus the ground state is $0^{+}$ and reflection symmetry admits only even spins, so the ground-state band is

The energy ratios are fixed by the $I(I+1)$ law alone, independent of $\mathcal{I}$:

A measured ratio $E(4^{+})/E(2^{+})\approx 3.3$ is the signature of a good rotor. The rare-earth and actinide nuclei (${}^{170}\mathrm{Er}$, ${}^{238}\mathrm{U}$) reach it closely. The absolute spacing fixes the moment of inertia: a large $\mathcal{I}$ compresses the band, so rotational levels lie only tens to hundreds of $\mathrm{keV}$ apart, far below single-particle energies.²

Vibrational states

A near-spherical nucleus oscillates about its equilibrium shape. The quadrupole surface mode built on the liquid drop quantizes into phonons of energy $\hslash {\omega }_2$, angular momentum $2\hslash$, and even parity. Counting phonons builds a level ladder.

• Ground state: zero phonons, $0^{+}$.
• One-phonon state: a single $2^{+}$ level at $\hslash {\omega }_2$.
• Two-phonon states: two identical bosons of angular momentum 2 couple to a nearly degenerate triplet $0^{+},2^{+},4^{+}$ at $2\hslash {\omega }_2$.

The vibrational signature is $E(4^{+})/E(2^{+})\approx 2$, distinct from the rotor's $3.3$. Adding an octupole ($\lambda =3$) phonon of odd parity produces a low-lying $3^{-}$ state. Real nuclei interpolate between the vibrational and rotational limits as the deformation grows, and the ratio $E(4^{+})/E(2^{+})$ moves continuously from $2$ toward $3.3$.

The Nilsson model of the deformed well

When the nucleus deforms, the spherical shell-model levels split. A nucleon in a spheroidal well no longer has a good orbital angular momentum, only the projection $\Omega$ of its angular momentum on the symmetry axis is conserved. Each spherical $j$-level of degeneracy $2j+1$ splits into $j+\frac{1}{2}$ pairs labelled by $\Omega =\frac{1}{2},\frac{3}{2},\dots ,j$, one for each pair of $\pm \Omega$ states. Nilsson computed these levels as functions of the deformation: the diagram of single-particle energy against deformation is the map on which deformed nuclei are read.

Orbitals whose density lies along the symmetry axis are lowered by a prolate deformation, those lying in the equatorial plane are raised, and the crossing levels reorganize the magic gaps. A nucleus deforms when it can lower the energy of its valence nucleons by more than the surface energy cost, so the interplay of the Nilsson levels with the drop's deformation energy fixes the equilibrium shape.

Moments of inertia and backbending

The measured moment of inertia falls between two theoretical limits. A rigid body of the nuclear mass and deformation gives ${\mathcal{I}}_{\text{rig}}$; a frictionless irrotational flow, where only the surface shape rotates while the interior stays still, gives ${\mathcal{I}}_{\text{irr}}$, several times smaller. Observed moments lie between them, closer to irrotational, because nucleon pairing makes the nuclear interior partly superfluid.

At high spin the band deviates from the smooth $I(I+1)$ law. Plotting the moment of inertia against the square of the rotational frequency, a nucleus follows a smooth curve and then jumps abruptly to a larger moment of inertia. This backbending occurs when the Coriolis force of the rotation breaks a nucleon pair and aligns the two nucleons' angular momenta with the rotation axis. The aligned pair adds angular momentum at little energy cost, so the nucleus rotates faster without climbing the band, and the effective moment of inertia rises sharply toward the rigid-body value.

Giant resonances

The whole nucleus has bulk collective modes at high excitation, the giant resonances, seen as broad peaks in photoabsorption and inelastic scattering. The giant dipole resonance is a coherent oscillation of all the protons against all the neutrons, an isovector $E1$ mode. Its centroid follows

so it sits near $15\mathrm{MeV}$ for heavy nuclei and near $20\mathrm{MeV}$ for light ones, with a width of a few $\mathrm{MeV}$. In a deformed nucleus the peak splits into two, one for oscillation along each principal axis, and the splitting measures the deformation directly. The giant quadrupole resonance, an isoscalar $E2$ mode near $63A^{-1/3}\mathrm{MeV}$, is the dynamic partner of the static quadrupole deformation. These modes exhaust most of the classical dipole and quadrupole sum rules, confirming that they are motions of essentially the entire nucleus.

The unified picture

The collective model completes the division of labor among the nuclear models. Each captures a different regime, and a full description of a mid-shell nucleus combines single-particle motion in a deformed well with the rotation and vibration of that well.

The rotational and vibrational bands are the low-energy face of the same surface dynamics that governs the fission barrier, where the quadrupole coordinate carries the nucleus all the way to scission rather than oscillating about equilibrium.