---
title: "The Collective Model: Rotations, Vibrations, and Deformed Nuclei"
module: Nuclear Models
moduleNumber: 3
lessonNumber: 4
order: 304
summary: >
  Deformed nuclei rotate with energies proportional to I times I plus one,
  giving the ground-state band its characteristic level ratios, while
  near-spherical nuclei vibrate in quantized surface phonons that build one- and
  two-phonon multiplets. The Nilsson model tracks single-particle levels as the
  well deforms, moments of inertia fall between the rigid and irrotational
  limits, backbending marks the sudden alignment of a broken pair, and giant
  resonances are the bulk dipole and quadrupole modes of the whole nucleus.
topics: [Nuclear Models]
sources:
  - book: Krane
    ref: "Ch. 5 — Nuclear Models; §5.3 Collective Structure"
  - book: Wong
    ref: "Ch. 5 — The Collective Model"
  - book: Povh
    ref: "Ch. 18 — Collective Nuclear Excitations"
draft: false
---

The
[shell model](/nuclear-physics/nuclear-models/shell-model-single-particle)
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](/nuclear-physics/nuclear-models/liquid-drop-collective-coordinates).
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.[^krane-cm]

## 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

$$
E(I) = \frac{\hbar^2}{2\mathcal{I}}\,I(I+1),
$$

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

$$
I^\pi = 0^+,\ 2^+,\ 4^+,\ 6^+,\ \dots
$$

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

$$
\frac{E(4^+)}{E(2^+)} = \frac{4\cdot 5}{2\cdot 3} = \frac{10}{3} \approx 3.33,
\qquad
\frac{E(6^+)}{E(2^+)} = \frac{6\cdot 7}{2\cdot 3} = 7.
$$

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.[^wong-cm]

$$
% caption: The ground-state rotational band of an even-even deformed nucleus:
% level energies grow as I times I plus one, so the spacing widens with each step
% while the ratio of the four-plus to the two-plus energy is fixed at ten thirds.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.2) -- (0,5.8) node[above, black] {energy};
  % levels at heights proportional to I(I+1): 0,6,20,42 scaled
  \foreach \y/\spin in {0.5/{0+}, 1.1/{2+}, 2.4/{4+}, 4.6/{6+}} {
    \draw[very thick] (0.9,\y) -- (3.1,\y);
    \node[black, anchor=west, font=\scriptsize] at (3.2,\y) {$I=\spin$};
  }
  % transition arrows down
  \draw[black, ->] (1.6,1.1) -- (1.6,0.55);
  \draw[black, ->] (1.6,2.4) -- (1.6,1.15);
  \draw[black, ->] (1.6,4.6) -- (1.6,2.45);
  \node[black, anchor=west, font=\scriptsize] at (4.4,3.0) {spacing widens};
\end{tikzpicture}
$$

## Vibrational states

A near-spherical nucleus oscillates about its equilibrium shape. The quadrupole
surface mode built on the
[liquid drop](/nuclear-physics/nuclear-models/liquid-drop-collective-coordinates)
quantizes into phonons of energy $\hbar\omega_2$, angular momentum $2\hbar$, and
even parity. Counting phonons builds a level ladder.

- **Ground state**: zero phonons, $0^+$.
- **One-phonon state**: a single $2^+$ level at $\hbar\omega_2$.
- **Two-phonon states**: two identical bosons of angular momentum 2 couple to a
  nearly degenerate triplet $0^+, 2^+, 4^+$ at $2\hbar\omega_2$.

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 $3.3$.

$$
% caption: The quadrupole vibrational spectrum: one phonon gives a two-plus level
% at the phonon energy, and two phonons give a nearly degenerate zero-plus,
% two-plus, four-plus triplet at twice the phonon energy.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.2) -- (0,5.2) node[above, black] {energy};
  % zero phonon
  \draw[very thick] (0.9,0.6) -- (2.3,0.6);
  \node[black, anchor=west, font=\scriptsize] at (2.4,0.6) {$0^+$, zero phonon};
  % one phonon
  \draw[very thick] (0.9,2.4) -- (2.3,2.4);
  \node[black, anchor=west, font=\scriptsize] at (2.4,2.4) {$2^+$, one phonon};
  % two phonon triplet
  \draw[very thick] (0.9,4.1) -- (2.3,4.1);
  \draw[very thick] (0.9,4.4) -- (2.3,4.4);
  \draw[very thick] (0.9,4.7) -- (2.3,4.7);
  \node[black, anchor=west, font=\scriptsize] at (2.4,4.4) {$0^+ 2^+ 4^+$ triplet, two phonon};
  % phonon energy bracket
  \draw[black, <->] (0.55,0.6) -- (0.55,2.4);
  \node[black, anchor=east, font=\scriptsize] at (0.5,1.5) {one phonon};
\end{tikzpicture}
$$

## 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 $j + \tfrac12$ pairs
labelled by $\Omega = \tfrac12, \tfrac32, \dots, j$, 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.

$$
% caption: A Nilsson diagram: each spherical single-particle level splits with
% deformation into components labelled by the angular-momentum projection on the
% symmetry axis, the lines fanning out as the well elongates.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.4) -- (7.0,0.4) node[right, black!70] {deformation};
  \draw[->, black] (0.6,0.2) -- (0.6,4.4) node[above, black!70] {single-particle energy};
  % degenerate spherical level at deformation 0
  \fill[black] (0.9,2.4) circle (1.8pt);
  \node[black, anchor=east, font=\scriptsize] at (0.8,2.4) {spherical};
  % fan out
  \draw[very thick] (0.9,2.4) -- (6.4,3.9);
  \draw[very thick] (0.9,2.4) -- (6.4,3.0);
  \draw[very thick] (0.9,2.4) -- (6.4,1.9);
  \draw[very thick] (0.9,2.4) -- (6.4,0.9);
  \node[black, anchor=west, font=\scriptsize] at (6.5,3.9) {high projection};
  \node[black, anchor=west, font=\scriptsize] at (6.5,0.9) {low projection};
\end{tikzpicture}
$$

## 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.

$$
% caption: Backbending: the moment of inertia rises smoothly with the square of
% the rotational frequency, then jumps back to a larger value when a nucleon pair
% breaks and its angular momentum aligns with the rotation axis.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.4) -- (7.0,0.4) node[right, black!70] {rotational frequency squared};
  \draw[->, black] (0.6,0.2) -- (0.6,4.4) node[above, black!70] {moment of inertia};
  % smooth rise then S-shaped backbend
  \draw[acc, very thick] (0.9,1.0)
    .. controls (2.0,1.5) and (2.8,2.0) .. (3.4,2.4)
    .. controls (3.1,2.9) and (3.0,3.3) .. (3.7,3.6)
    .. controls (4.6,3.9) and (5.6,4.0) .. (6.5,4.1);
  \node[black, anchor=west, font=\scriptsize] at (3.7,2.7) {pair breaks};
  \node[black, anchor=west, font=\scriptsize] at (1.0,1.4) {ground band};
\end{tikzpicture}
$$

## 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

$$
E_{\text{GDR}} \approx 79\,A^{-1/3}\ \mathrm{MeV},
$$

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 $63\,A^{-1/3}\ \mathrm{MeV}$, 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.

$$
% caption: The giant dipole resonance appears as a broad peak in the photonuclear
% cross section near 79 over the cube root of A in MeV, the collective
% oscillation of the proton fluid against the neutron fluid.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0.4) -- (7.2,0.4) node[right, black!70] {photon energy};
  \draw[->, black] (0.6,0.2) -- (0.6,4.4) node[above, black!70] {cross section};
  % broad resonance peak
  \draw[acc, very thick] (0.9,0.6)
    .. controls (2.2,0.7) and (2.9,3.7) .. (3.7,3.8)
    .. controls (4.5,3.9) and (5.2,0.9) .. (6.8,0.6);
  % centroid marker
  \draw[black, dashed] (3.7,0.4) -- (3.7,3.8);
  \node[black, anchor=north, font=\scriptsize] at (3.7,0.38) {centroid};
  \node[acc, anchor=west, font=\scriptsize] at (4.3,3.4) {giant dipole};
\end{tikzpicture}
$$

## 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.

| Excitation | Spectrum | Signature | Nuclei |
| --- | --- | --- | --- |
| Rotation | $E \propto I(I+1)$ | $E(4^+)/E(2^+) \approx 3.3$ | deformed rare earths, actinides |
| Vibration | phonon ladder $n\hbar\omega$ | $E(4^+)/E(2^+) \approx 2$ | near-spherical, mid-shell |
| Nilsson levels | split single-particle | $\Omega$ labelling | deformed odd-$A$ |
| Giant resonance | broad photoabsorption peak | $E \approx 79 A^{-1/3}\ \mathrm{MeV}$ | all nuclei |

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

[^krane-cm]: Krane, "Introductory Nuclear Physics," Wiley (1988), §5.3. Level
    schemes for deformed rotors from NNDC,
    [https://www.nndc.bnl.gov/](https://www.nndc.bnl.gov/).
[^wong-cm]: Wong, "Introductory Nuclear Physics," 2nd ed., Wiley-VCH (1998),
    Ch. 5; Povh et al., "Particles and Nuclei," Springer, Ch. 18.