---
title: "Nuclear Shell Filling Interactive Lab | Sandbox Physics"
description: "How does spin–orbit splitting open a shell gap at twenty-eight neutrons? Spin–orbit interaction lowers the high-angular-momentum branch of a multiplet. In this neutron model, eight states separate from the higher oscillator shell and fill after twenty neutrons. Filling those eight opens a gap at twenty-eight; a partly filled level has no such gap."
canonical_url: "https://sandboxphysics.com/labs/nuclear-shell-filling/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/nuclear-shell-filling/"
last_modified: "2026-10-24"
---

E-X3 · Fill the neutron states

# Nuclear Shell Filling

**How does spin–orbit splitting open a shell gap at twenty-eight neutrons?** Spin–orbit interaction lowers the high-angular-momentum branch of a multiplet. In this neutron model, eight states separate from the higher oscillator shell and fill after twenty neutrons. Filling those eight opens a gap at twenty-eight; a partly filled level has no such gap.

Fill a spherical neutron cloud and turn on spin–orbit splitting. Compare its occupied levels, shell gap and radial density, then drag a probe through the cloud.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

This stationary neutron-only model uses spherical harmonic-oscillator states with a phenomenological negative spin–orbit term. Proton fields, residual interactions, pairing and deformation are omitted. The spin–orbit coefficient reaches 0.12 oscillator spacings multiplying twice orbital-dot-spin in reduced-Planck units. Levels are filled from low energy with their full angular-momentum degeneracy. A partially filled multiplet equally occupies magnetic substates, producing a spherical ensemble density. The same normalized radial wavefunctions determine the displayed density and rms radius. The radial probe is a visualization cursor. Level gaps are model single-particle gaps, not measured separation energies. The range compares light closures at two, eight, twenty, twenty-eight and forty; realistic heavier magic numbers need a more complete Hamiltonian. The cloud shows probability density, with schematic spatial magnification; its particles do not orbit classically.

Specified quantum model

## The states filled by the same neutron cloud

Default state

> Interactive data plots; numerical values can be exported.

Occupied neutron states

Unoccupied states

The radial curve integrates to the neutron count. Partial multiplets are spherically averaged; all clouds are stationary.

Physics tutorial

## A split multiplet changes where a shell closes

**Background** A spherical average field permits discrete neutron states. The exclusion principle limits how many neutrons fill each multiplet.

**Why it matters** A generic potential well gives levels, but nuclear shells require the right level order. Spin–orbit interaction can change that order.

Start with the essentials

- **Focus question**: How does spin–orbit splitting open a shell gap at twenty-eight neutrons?
- **One-sentence intuition**: Lowering the higher-angular-momentum branch isolates eight states above the twenty-neutron closure.

### Core mathematical model

#### The declared oscillator and spin–orbit energies

$$
E_{nlj}=\hbar\omega\left[2n+l+\frac32-0.12g_{\rm so}\left(j(j+1)-l(l+1)-\frac34\right)\right]
$$

The radial quantum number starts at zero. The adjustable term lowers the higher total-angular-momentum branch.

#### Capacity of one multiplet

$$
d_j=2j+1
$$

Neutrons fill the available magnetic substates. Partial filling is spread equally across them in this spherical ensemble.

#### Radial neutron count density

$$
P(r)=\sum_a n_a r^2|R_a(r)|^2,\qquad\int_0^\infty P(r)\,dr=N
$$

The same normalized oscillator wavefunctions generate both the cloud and radial plot.

### Common difficulties

#### A cloud is not a set of orbits

**Typical misconception** The visible neutrons travel along classical circular paths.

**Better mental model** The displayed cloud is a stationary quantum probability distribution with no assigned trajectories.

#### A model gap is not a measured separation energy

**Typical misconception** The readout gives a precision energy for removing a neutron from a real isotope.

**Better mental model** It is a gap in this specified one-particle Hamiltonian. Real separation energies also depend on correlations, pairing and other fields.

### Run the experiment

1. 01

   #### Compare the oscillator with a split shell

   Choose Unsplit oscillator, then Gap at twenty-eight.

   **What to observe:** The unsplit oscillator closes at forty. Splitting separates eight states, creating a gap after twenty-eight.

2. 02

   #### Leave a level partly filled

   Choose Partly filled shell and change neutron number one at a time.

   **What to observe:** An unfilled multiplet has available states at the same energy, so the single-particle gap is zero.

3. 03

   #### Probe the density

   Drag the radial cursor through the cloud and compare the radial plot below.

   **What to observe:** The cloud is a stationary spherical density. The radial count plot includes the increasing volume of spherical shells.

Physical conclusion

Count each level’s degeneracy and inspect the gap to the next available state. A density cloud and an energy diagram describe the same occupancy.

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)