KAHIBARO
Table of Contents
Core Idea
The nuclear shell model describes the nucleus as a system in which protons and neutrons occupy discrete energy levels, called shells, somewhat like electrons in atomic orbitals. The analogy is useful, but the physical force is different. Electrons are bound by the electric attraction of the nucleus, while nucleons are bound by the strong nuclear interaction inside an average nuclear potential.
The shell model became important because some nuclei are especially stable when they contain certain numbers of protons or neutrons. These special numbers are called magic numbers, and they strongly suggest that nucleons fill quantized levels in the nucleus.
In this model, each proton and each neutron moves approximately independently in an effective potential produced by all the other nucleons. This is called an independent particle picture. It is not exact, but it captures many important features of nuclear structure.
Why a Shell Structure Appears
Inside a nucleus, the force between individual nucleons is very complicated. A full many body treatment is difficult. The shell model simplifies the problem by replacing the detailed interaction with an average central potential. Each nucleon then moves in this average field.
If nucleons behaved only in a simple central potential, some shell closures would appear, but the experimentally observed magic numbers would not all come out correctly. The crucial improvement is the inclusion of a strong spin orbit interaction. This interaction splits levels with the same orbital angular momentum into states of different total angular momentum.
The result is a level structure that naturally explains the observed stable closed shells.
The key physical idea of the nuclear shell model is this: each nucleon moves in an average potential, and the strong spin orbit coupling rearranges the energy levels so that shell closures occur at the observed magic numbers.
Quantum States of Nucleons
A nucleon state in the shell model is labeled by quantum numbers similar to those used in atomic physics. The most important are the orbital angular momentum $l$, the intrinsic spin $s = \frac{1}{2}$, and the total angular momentum
$$
\vec{j} = \vec{l} + \vec{s}.
$$
For a given $l$, the possible total angular momentum values are
$$
j = l \pm \frac{1}{2}.
$$
Each state with total angular momentum $j$ can hold
$$
2j + 1
$$
nucleons of a given type, meaning either protons or neutrons, corresponding to the different magnetic substates.
Because protons and neutrons are distinct particles in the shell model, they fill their own sets of levels separately. A shell can therefore be closed for protons, for neutrons, or for both.
Pauli Principle in the Nucleus
Nucleons are fermions, so they obey the Pauli exclusion principle. No two identical nucleons can occupy the same quantum state.
This means that protons fill the available proton states from low energy upward, and neutrons fill the neutron states in the same way. The filling pattern determines the structure of the nucleus, especially for the outermost nucleons, called valence nucleons.
Valence nucleons are very important because closed inner shells often contribute little to the low energy behavior, while the few nucleons outside a closed shell largely determine spin, parity, and excited states.
For each nucleon species separately, states are filled from lowest energy upward, and each state can contain only the allowed number of identical nucleons consistent with the Pauli principle.
The Nuclear Potential
The average potential in a real nucleus is not exactly a simple harmonic oscillator and not exactly a Coulomb type potential. A more realistic shape is flatter in the center and drops near the nuclear surface. In advanced treatments, a Woods-Saxon potential is often used.
For learning the shell model, it is enough to know that the potential creates quantized bound states. The spin orbit term then causes an important splitting. States with $j = l + \frac{1}{2}$ are pushed to lower energy than states with $j = l - \frac{1}{2}$, and this large splitting changes the ordering of levels.
This reordering is what allows the shell model to reproduce the observed shell closures.
Role of Spin Orbit Coupling
Spin orbit coupling links a nucleon's orbital motion and intrinsic spin. In nuclei, this effect is unusually strong compared with atomic physics.
Suppose a level has orbital angular momentum $l = 3$. Then two total angular momentum values are possible,
$$
j = \frac{7}{2}, \quad j = \frac{5}{2}.
$$
These two states no longer have the same energy. The $j = l + \frac{1}{2}$ state is lower than the $j = l - \frac{1}{2}$ state. Because high $j$ states can hold many nucleons, large energy gaps can appear after certain fillings, producing closed shells.
Without this strong splitting, the experimentally observed sequence of magic numbers would not be explained correctly.
Level Filling and Shell Closures
As nucleons fill the available levels, large gaps sometimes appear between a filled group of levels and the next higher level. When all levels below such a gap are full, the nucleus has a closed shell.
Closed shell nuclei are especially stable. They often have higher binding energies than neighboring nuclei, and they are usually less easily excited.
A simplified view of shell filling is shown below.
The exact ordering becomes more complicated for heavier nuclei, but the main idea stays the same, quantized levels plus large gaps.
Shell Model Notation
Single particle states are often written in the form
$$
n l_j.
$$
Here, $n$ is a radial label, $l$ is the orbital angular momentum, and $j$ is the total angular momentum. The letter for $l$ follows the usual convention:
| $l$ | Letter |
|---|---|
| 0 | $s$ |
| 1 | $p$ |
| 2 | $d$ |
| 3 | $f$ |
| 4 | $g$ |
| 5 | $h$ |
So, for example, $1d_{5/2}$ means a state with radial label 1, orbital angular momentum $l=2$, and total angular momentum $j=5/2$.
The capacity of a subshell is $2j+1$. For example:
| State | Capacity for protons | Capacity for neutrons |
|---|---|---|
| $1s_{1/2}$ | 2 | 2 |
| $1p_{3/2}$ | 4 | 4 |
| $1p_{1/2}$ | 2 | 2 |
| $1d_{5/2}$ | 6 | 6 |
Closed Shells and Nuclear Stability
A nucleus with a closed proton shell or a closed neutron shell tends to be more stable than nearby nuclei. A nucleus with both closed proton and closed neutron shells is called doubly magic.
These nuclei often have distinctive properties. They are commonly more spherical, more tightly bound, and have relatively high first excited energies.
Examples of doubly magic nuclei include $^4\text{He}$, $^{16}\text{O}$, $^{40}\text{Ca}$, and $^{208}\text{Pb}$.
The shell model explains this by saying that all low energy states up to a large gap are filled, so promoting a nucleon to the next available state requires relatively large energy.
A closed shell corresponds to a filled set of nuclear energy levels separated by a large gap from the next level. Closed shell nuclei are unusually stable.
Ground State Spin and Parity
One of the shell model's most useful predictions is the ground state spin and parity of many nuclei.
For even even nuclei, meaning nuclei with even proton number and even neutron number, the nucleons usually pair off so that their angular momenta cancel. This often gives ground state
$$
J^P = 0^+.
$$
For nuclei with one unpaired nucleon outside closed shells, the total nuclear spin and parity are usually determined mainly by that nucleon.
The parity of a state comes from the orbital angular momentum:
$$
\pi = (-1)^l.
$$
So if the unpaired nucleon is in an $f$ state, where $l=3$, the parity is negative. If it is in a $d$ state, where $l=2$, the parity is positive.
Thus, if the last unpaired nucleon occupies a $p_{3/2}$ state, the nucleus often has
$$
J^P = \frac{3}{2}^-.
$$
Pairing of Nucleons
In real nuclei, nucleons have a tendency to form pairs with opposite angular momentum projections. This pairing lowers the energy.
Because of pairing, nuclei with even numbers of protons and even numbers of neutrons are especially common and stable. Odd nucleon numbers tend to leave an unpaired particle, which strongly affects the observed spin and magnetic properties.
The shell model is often used together with this pairing idea. Closed shells plus pairing explain a great deal of the regular behavior of nuclei.
Successes of the Shell Model
The shell model successfully explains several important facts about nuclei. It accounts for magic numbers, predicts many ground state spins and parities, and helps describe low lying excited states near closed shells.
It also provides a natural language for talking about valence nucleons and configuration structure. For example, one may say that a nucleus has two neutrons outside a closed core, or one proton hole below a closed shell.
This particle and hole description is very useful in nuclear spectroscopy.
Particle and Hole Picture
If a shell is almost empty, it is natural to describe the nucleus in terms of a few particles in that shell. If a shell is almost full, it can be simpler to describe it in terms of a few holes, meaning missing nucleons in an otherwise filled shell.
A hole behaves in many ways like an active object with angular momentum and parity. This makes calculations and qualitative reasoning much easier for nuclei near closed shells.
For example, one missing neutron in a filled subshell can determine the total angular momentum much like one extra neutron in an otherwise empty subshell.
Limitations of the Shell Model
The shell model is powerful, but it is not a complete picture of all nuclei. Some nuclei are strongly deformed rather than spherical, and collective motion of many nucleons becomes important. In such cases, models based on rotation or vibration may work better.
Also, the independent particle idea is only approximate. Nucleons do interact with one another beyond the average field, and these residual interactions can shift energies and mix configurations.
So the shell model is best seen as a foundational model, especially successful for nuclei near closed shells.
The shell model is an approximate independent particle model. It works especially well near closed shells, but residual interactions and collective effects can become important.
Visualizing Filled Shells
The picture below shows the idea of shells being filled from low to high energy, separately for protons and neutrons.
Summary
The nuclear shell model treats protons and neutrons as moving in quantized energy levels inside an average nuclear potential. The Pauli principle forces them to fill these states from low energy upward. A strong spin orbit interaction splits and reorders levels, creating large gaps that explain shell closures and the special stability of magic nuclei.
The model is especially useful for understanding closed shells, valence nucleons, ground state spin and parity, and the difference between especially stable nuclei and their neighbors. Even though it is approximate, it is one of the central ideas in nuclear structure physics.