Nuclear Shell Model

The nuclear shell model is a fundamental framework for understanding the structure of atomic nuclei. It explains nuclear properties by treating nucleons (protons and neutrons) as moving independently in an average potential, analogous to how electrons occupy atomic orbitals.

Historical Motivation

Experimental observations revealed that certain nuclei are particularly stable. These "magic numbers" of nucleons—2, 8, 20, 28, 50, 82, 126—correspond to nuclei with:

• Unusually high binding energies
• Abundant isotopes or isotones
• Spherical ground states
• Large energy gaps to excited states

These properties suggested that nucleons fill discrete energy levels, similar to the electronic shell structure in atoms.

Basic Assumptions

The shell model makes several key assumptions:

1. Independent particle motion: Each nucleon moves in an average potential created by all other nucleons
2. Mean-field approximation: The many-body problem is reduced to a one-body problem
3. Single-particle orbitals: Nucleons occupy discrete energy levels (shells)
4. Pauli exclusion principle: Each orbital can hold at most two nucleons (one spin-up, one spin-down)

The difficulty in reading a shell model diagram

In the casse of the shell model there are a few effects that you need to be aware of. There are a few biases you get from studying atomic systems which we need to correct for the nuclear shell model. Spin orbit coupling is strong. In chemistry there is this nomenclature of the "principal quantum number". In nuclear physics there is no such thing. This number dictates the electron shell in atomic physics. In nuclear physics, the spin-orbit coupling can send you into a completely different shell. Similarly, there are not restrictions that state l must be less than n. Your preconcieved notions from atomic quantum systems must be discarded. The shells of nuclear physics are clusters of levels that are not dictated by a single quantum number. Atomic physics places much emphasis on level splitting, but these level splittings from magnetic or electric fields do not usually re-order the energy levels as much as spin orbit coupling the nuclear shell model.

Let's count up the nuclear shell levels

1. 0s, n=0, l=0 s + 1/2 = 1/2 so we have a 0s1/2 level, l - 1/2 is not allowed. We have two levels because 2*1/2 + 1 = 2
2. 0p, n=0, l=1 p + 1/2 = 3/2 so we have a 0p3/2 level, now l - 1/2 = 1/2 so we have a 0p1/2 level. We have six levels.

A simple example which is quite pathological

Where do I even start. When I figured this out I was ready to throw my laptop out the window. Let's imagine you wanted to peform a shell model calculation. The first thing you would need are single particle energy levels. This is after all Full Configuration Interaction calculation. Now, there is very clearly the option of the harmonic oscillator, or the woods saxon potential. From these potentials single particle energy levels can be calculated. Then you go to solve the shell model and all you see are m-scheme this, j-scheme that. What is even going on!?!?! Oh I'll tell you what's going on. The problem I have ran into while understanding this is that the shell model is often solved in a SUBSPACE. This subspace MAY NOT EVEN DEPEND on the single particle levels. If you have a 0s7/2 subspace that you are working in you basically have a SPIN-ONLY MATRIX. For two particles you need to find a subspace with "good total m" which means "find pairs where the m is the same"

Building a single-particle m-scheme basis

So remember that example where we counted up the number of levels for different quantum numbers? 0s1/2, 0p3/2, 0p1/2. We will be doing that a few more times. Let's look at the 0d5/2 space. Basically you have |0, 2, 5/2, -5/2> |0, 2, 5/2, -3/2> |0, 2, 5/2, -1/2> |0, 2, 5/2, 1/2> |0, 2, 5/2, 3/2> |0, 2, 5/2, 5/2> These are the single particle states which will be used to build the many particle states. Now you can do 6 choose 2 and count the total number of states, which happens to be 15.

Note on occupation numbers and electron configurations

You cannot think of this state counting like you do when you fill electron oribitals. Let's play out the scenario where we have two electrons in a 5/2 subspace. Your natural instinct is to start counting all the electron configurations with up and down that can fit on each of the levels. You cannot think of second quantization like this. Second quantization algebraically handles all of occupancies. The rule is just "no particle may occupy the same state". But can't I have a spin up and a spin down? You can't think of it like this. It is an abstract state which does not correspond directly to an electron at that level. This is just me relaying my confusion. It's fine if you don't understand right now, just trust that this second quantization algebra gets the job done.

Single-Particle Hamiltonian

In the simplest shell model, each nucleon is described by a single-particle Hamiltonian:

where $V(\mathbf{r})$ is the average potential. The most common choice is the harmonic oscillator potential or the Woods-Saxon potential.

Harmonic Oscillator Potential

The harmonic oscillator potential provides a good starting point:

This yields energy levels:

where $n$ is the radial quantum number and $l$ is the orbital angular momentum. The degeneracy of each level is $2(2l+1)$ (accounting for spin and magnetic quantum number).

Woods-Saxon Potential

A more realistic potential is the Woods-Saxon form:

where $R=r_0{A}^{1/3}$ is the nuclear radius, $a$ is the surface diffuseness, and $V_0$ is the well depth. This potential better reproduces experimental magic numbers when combined with spin-orbit coupling.

Spin-Orbit Coupling

A crucial ingredient for reproducing magic numbers is the spin-orbit interaction:

This interaction splits levels with the same $n$ and $l$ but different total angular momentum $j=l\pm 1/2$. The level with higher $j$ is lowered in energy, creating larger energy gaps that explain magic numbers like 28, 50, 82, and 126.

Magic Numbers

Magic numbers arise when a shell is completely filled, creating a large energy gap to the next shell. The sequence of filled shells gives:

Configuration and Many-Body States

For a nucleus with $A$ nucleons, the many-body wavefunction is constructed from single-particle orbitals. In the simplest approximation, the ground state is a Slater determinant of the lowest-energy orbitals:

This ensures antisymmetry under exchange of any two nucleons, as required by the Pauli exclusion principle.

Beyond the Independent Particle Model

While the independent particle model explains many nuclear properties, it has limitations:

• Residual interactions: Nucleons interact beyond the mean field, leading to configuration mixing
• Collective effects: Some nuclei exhibit collective motion (rotations, vibrations) not captured by single-particle motion
• Correlations: Short-range correlations between nucleons are not fully described

Modern shell model calculations include:

• Configuration interaction: Mixing of many Slater determinants
• Effective interactions: Two-body matrix elements fitted to experimental data
• Valence space: Restricting calculations to active orbitals near the Fermi surface

Key Predictions

The shell model successfully explains:

• Magic numbers: Enhanced stability at closed shells
• Nuclear spins and parities: Ground state quantum numbers from unpaired nucleons
• Magnetic moments: From single-particle $g$-factors
• Quadrupole moments: Deformation from partially filled shells
• Energy spectra: Excited states as particle-hole excitations

Connection to Other Models

The shell model complements other nuclear models:

• Liquid drop model: Describes bulk properties (binding energy, fission)
• Collective model: Describes rotational and vibrational spectra
• Mean-field models: Self-consistent Hartree-Fock calculations

Core Idea

The nuclear shell model treats nucleons as independent particles moving in an average potential. Magic numbers arise from closed shells, and the model provides a foundation for understanding nuclear structure, spectra, and properties.

References

• Mayer, M. G. (1949). On Closed Shells in Nuclei. Physical Review, 75(12), 1969-1970.
• Haxel, O., Jensen, J. H. D., & Suess, H. E. (1949). On the "Magic Numbers" in Nuclear Structure. Physical Review, 75(11), 1766.
• Talmi, I. (1993). Simple Models of Complex Nuclei: The Shell Model and Interacting Boson Model. Harwood Academic Publishers.
• Heyde, K. (1994). The Nuclear Shell Model. Springer-Verlag.