KAHIBARO
Table of Contents
Beyond Single-Particle Motion
The collective model describes nuclei by combining two ideas. One idea is that individual protons and neutrons move in shell-like energy levels. The other idea is that the whole nucleus can also move as a single object, changing shape or rotating. In this picture, a nucleus is not always a rigid sphere with nucleons acting independently. Instead, many nucleons can participate together in a shared motion.
This model was developed because some nuclei show behavior that the shell model alone does not explain well. In particular, many nuclei have excited states that look like vibrations or rotations of the entire nuclear shape. The collective model gives a way to understand these patterns.
The Nucleus as a Deformable Object
In the collective model, the nucleus can be thought of as a liquid-like drop with quantum properties. Its surface can deform. A nucleus may be nearly spherical, or it may be stretched slightly into an elongated shape. When the shape changes in a coordinated way, the motion is called collective because many nucleons contribute.
Common possibilities include a spherical shape, a prolate shape, which is stretched like a rugby ball, and an oblate shape, which is flattened like a disk.
A deformed nucleus can rotate, and this rotation produces a characteristic set of energy levels. A nearly spherical nucleus, on the other hand, more often shows vibrational excitations.
Collective Vibrations
If a nucleus is close to spherical, it can oscillate around its equilibrium shape. This is similar in spirit to a vibrating drop. The surface moves inward and outward in a coordinated pattern. These are called vibrational modes.
In a simple picture, the energy levels of a vibrating nucleus come in steps, somewhat like a quantum harmonic oscillator. The excited states correspond to one quantum of vibration, two quanta, and so on. Real nuclei are more complicated than this ideal picture, but the idea helps organize the observed spectrum.
A nucleus with vibrational behavior often has energy levels that are spaced in a way roughly consistent with repeated vibrational excitations.
In the collective model, a vibrational nucleus is one whose excitation is mainly due to oscillations of the whole nuclear shape, not just motion of a single nucleon.
Collective Rotation
If a nucleus has a permanently deformed shape, it can rotate as a whole. This produces a rotational band, which is a sequence of excited states with increasing angular momentum.
For an ideal rotating nucleus, the rotational energy is approximately
$$
E_J = \frac{\hbar^2}{2I} J(J+1),
$$
where $J$ is the nuclear angular momentum quantum number, and $I$ is the moment of inertia of the nucleus.
This formula is very important because it predicts that the energies should grow with $J(J+1)$, not simply in equal steps.
For a rotational band, the characteristic energy pattern is
$$
E_J \propto J(J+1).
$$
This is one of the clearest signatures of collective rotational motion.
For even-even nuclei, which have even numbers of protons and neutrons, the ground state often has $J = 0$, and the rotational band may begin with states such as $2^+$, $4^+$, $6^+$, and so on.
Rotational Bands
A rotational band is a family of levels that belong to the same intrinsic nuclear shape. As the nucleus rotates faster, the angular momentum increases, and higher members of the band appear.
A useful pattern for the lowest levels of an ideal rotational band is shown below.
| State | $J(J+1)$ | Relative energy |
|---|---|---|
| $0^+$ | 0 | 0 |
| $2^+$ | 6 | proportional to 6 |
| $4^+$ | 20 | proportional to 20 |
| $6^+$ | 42 | proportional to 42 |
| $8^+$ | 72 | proportional to 72 |
This means, for example, that if the $2^+$ state has energy $E_{2}$, then in the ideal case
$$
\frac{E_4}{E_2} \approx \frac{20}{6} \approx 3.33.
$$
This ratio helps distinguish rotational nuclei from vibrational ones.
Vibrational Versus Rotational Spectra
The collective model is especially useful because different kinds of collective motion leave different fingerprints in the energy spectrum.
| Type of nucleus | Main motion | Typical level pattern |
|---|---|---|
| Nearly spherical | Vibration | More nearly equal spacing |
| Deformed | Rotation | Energies follow $J(J+1)$ |
For a simple vibrational nucleus, the ratio of the first $4^+$ and $2^+$ excitation energies is often near 2. For a simple rotational nucleus, the ratio is near 3.33. Real nuclei may lie between these limits, which shows that nuclear behavior can be transitional.
A common diagnostic is the ratio
$$
R_{4/2} = \frac{E(4^+)}{E(2^+)}.
$$
Values near 2 suggest vibrational behavior, and values near 3.33 suggest rotational behavior.
Intrinsic Shape and Laboratory States
In the collective model, it is useful to separate two viewpoints. In the intrinsic frame, the nucleus has its own shape, such as spherical or deformed. In the laboratory frame, we observe states with definite energy and angular momentum.
A deformed intrinsic shape does not mean the nucleus points in one fixed direction in space. Quantum mechanically, what we observe are rotational states built from that intrinsic shape. So the deformation is a property of the internal structure, while the measured states reflect allowed quantum rotations.
Electric Quadrupole Transitions
Collective motion often produces strong electric quadrupole transitions, called $E2$ transitions, between energy levels. These transitions are especially important evidence for deformation and collectivity.
If many nucleons contribute coherently to the shape change, the transition probability can be much larger than expected from a purely single-particle picture. Strong $E2$ transitions are therefore a hallmark of collective nuclear motion.
For example, in a rotational band, transitions like $4^+ \to 2^+$ and $2^+ \to 0^+$ are often strong and follow regular patterns.
Large $E2$ transition strengths are strong evidence for collective behavior, especially for deformed nuclei with rotational bands.
How the Collective Model Connects to Other Nuclear Models
The collective model does not replace the shell model or the liquid-drop model. Instead, it borrows ideas from both. From the shell model comes the importance of individual nucleon structure and quantum states. From the liquid-drop picture comes the idea that the nucleus can change shape and behave like a continuous object.
This makes the collective model a bridge between single-particle and bulk nuclear behavior. It is especially successful for medium and heavy nuclei, where many nucleons can act together strongly.
Why Some Nuclei Are Deformed
Deformation happens because the arrangement of nucleons in shells can make a spherical shape less favorable than a slightly distorted one. When this occurs, the nucleus may settle into a deformed equilibrium shape. Once it is deformed, rotational motion becomes possible and often dominates the low-energy spectrum.
Nuclei near closed shells tend to be more spherical and more vibrational. Nuclei far from closed shells are often more deformed and more rotational. This gives a useful connection between shell structure and collective motion.
A Simple Energy Sketch
The difference between vibrational and rotational spectra can be visualized qualitatively.
The exact energies vary from nucleus to nucleus, but the rotational spacing widens according to the $J(J+1)$ rule, while the vibrational case is more nearly step-like.
Limits of the Model
The collective model is an approximation. Real nuclei are quantum many-body systems, and their spectra are not always purely rotational or purely vibrational. Many nuclei show mixed behavior. Some are transitional, meaning they lie between spherical and strongly deformed limits.
Also, the moment of inertia in real nuclei is often not exactly constant, so the simple rotational formula may need corrections.
The collective model is most useful as a framework for recognizing patterns in nuclear spectra, especially vibrational and rotational patterns, rather than as a perfect description of every nucleus.
Summary
The collective model explains how a nucleus can behave through coordinated motion of many nucleons. It is especially important for understanding vibrational excitations in nearly spherical nuclei and rotational bands in deformed nuclei. Its key signature for rotation is the energy law
$$
E_J = \frac{\hbar^2}{2I} J(J+1),
$$
and its key experimental evidence includes regular rotational level sequences and strong $E2$ transitions. In this way, the collective model gives a powerful picture of the nucleus as both a quantum system of individual particles and a deformable object capable of whole-body motion.