---
title: "Nuclear Spectra and Collective Motion Interactive Lab | Sandbox Physics"
description: "Can nuclear level spacings tell rotation from a single-neutron excitation? A rotational band has level spacings that grow with angular momentum."
canonical_url: "https://sandboxphysics.com/labs/nuclear-spectroscopy/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/nuclear-spectroscopy/"
---

FR07-A · Read the spacing between nuclear levels

# Nuclear Spectra and Collective Motion

**Can nuclear level spacings tell rotation from a single-neutron excitation?** A rotational band has level spacings that grow with angular momentum. A neutron outside a closed core instead reveals orbital energies and spin-orbit splitting. Comparing both the energies and the allowed transitions helps identify the motion.

Compare evaluated oxygen and erbium levels with a one-neutron model, a rotor and a vibrator. Drag the nucleus to change the inertia and watch the predicted band move.

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

Model assumptions

Evaluated energies are a frozen TUNL 1993 oxygen-17 excerpt and ENSDF March 2008 erbium-166 ground band. The rotor has constant inertia; the harmonic vibrator shows only stretched states. The oxygen model uses an inert oxygen-16 core, degenerate oscillator sd orbitals and a constant spin-orbit term. The higher oxygen d state is an unbound resonance. Pairing, core excitation, realistic radial potentials, widths and transition strengths are not calculated. Shapes and angular-momentum arrows are schematic; stationary states are not trajectories.

Evaluated data · ideal model predictions

## The spacing is the test

Evaluated levels Rotor / single-neutron model Harmonic vibrator

Compare evaluated levels with two ideal spectra. Arrows mark selected E2-allowed transitions.

Energies in keV; residual is model minus evaluated.

| State | Evaluated | Model | Residual | Use |
| --- | --- | --- | --- | --- |
| $0^{+}$ | $0\pm0$ | $0.00$ | $0.00$ | reference |
| $2^{+}$ | $80.5776\pm0.002$ | $80.6$ | $0.00$ | calibration |
| $4^{+}$ | $264.99\pm0.003$ | $269$ | $3.60$ | comparison |
| $6^{+}$ | $545.454\pm0.004$ | $564$ | $18.6$ | comparison |
| $8^{+}$ | $911.208\pm0.006$ | $967$ | $55.7$ | comparison |

Transitions and vibrational multiplets

A quadrupole phonon carries angular momentum two. Two identical phonons combine into spin zero, two and four at the same harmonic energy. The chart follows only the stretched sequence; actual nuclei can split and mix these multiplets.

Arrows connect selected E2-allowed downward transitions. Their widths carry no transition-strength information. The rotor and vibrator share the evaluated first excited energy at reference inertia.

$n=0:\ 0^+;\quad n=1:\ 2^+;\quad n=2:\ 0^+,2^+,4^+$

[ENSDF 166Er · March 2008 · adopted levels, pp. 1–2](https://www.nndc.bnl.gov/ensnds/166/Er/adopted.pdf) · [Frozen data and provenance](https://sandboxphysics.com/data/nuclear-bridge/spectroscopy/evaluated-levels.json)

[Study nuclear structure, scattering and stellar rates](https://sandboxphysics.com/stories/nuclear-bridge/)

Physics tutorial

## Nuclear Spectra and Collective Motion

**Background** A nucleus releases a definite energy when it falls between two quantum levels. A sequence of such energies can reveal whether the motion involves one nucleon or the nucleus as a whole.

**Why it matters** Erbium 166 has a closely spaced low-energy band. Oxygen 17 has one neutron beyond a closed oxygen-16 core. These two cases give different tests of a nuclear model.

Start with the essentials

- **Focus question**: Can nuclear level spacings tell rotation from a single-neutron excitation?
- **One-sentence intuition**: Collective rotation predicts a growing spacing, whereas a one-neutron spectrum tests orbital energies and the spin-orbit force.

### Core mathematical model

#### The rotational band at fixed inertia

$$
E_J=\frac{\hbar^2}{2\mathcal I}J(J+1),\qquad \frac{E_4}{E_2}=\frac{10}{3}
$$

The even-even, axially symmetric ground band has even angular momenta beginning at zero. Energies are excitation energies in keV. The reference inertia is fixed by the evaluated first excited level.

#### Spin-orbit splitting of one orbital

$$
\begin{aligned}H_{\rm so}&=-\kappa\,\mathbf L\cdot\mathbf S/\hbar^2\\\Delta E_j&=-\frac{\kappa}{2}\left[j(j+1)-\ell(\ell+1)-\frac34\right]\end{aligned}
$$

A positive strength lowers the aligned partner. For orbital angular momentum two, the two d levels separate by two and a half times the strength. The two partners together still contain ten magnetic substates.

#### When an electric quadrupole transition is allowed

$$
|J_i-J_f|\leq2\leq J_i+J_f,\qquad \pi_i=\pi_f
$$

The angular-momentum triangle and unchanged parity are necessary conditions for an E2 transition. A transition between two zero-spin states fails the triangle. Allowed does not determine a transition probability.

### Common difficulties

#### A band is not an orbital list

**Typical misconception** Every higher nuclear level means that one neutron moved to a higher orbital.

**Better mental model** Several low-lying levels can share an intrinsic configuration and differ in collective angular momentum. Single-particle excitations change the orbital configuration.

#### A near match can still miss the data precision

**Typical misconception** The rotor lines nearly coincide with erbium, so the model reproduces the evaluated spectrum exactly.

**Better mental model** The evaluated uncertainties are only a few electronvolts. The rotor residuals reach tens of kiloelectronvolts. The model explains the pattern at a much coarser scale.

### Run the experiment

1. 01

   #### Predict the fourth level

   Choose An erbium rotational band. Use the evaluated first excited level to predict the higher levels, then compare the three columns.

   **What to observe:** The rotor captures the increasing spacing much better than equal phonon steps. Its residuals still grow with angular momentum, so constant inertia is an approximation.

2. 02

   #### Increase the inertia

   Predict whether the band will stretch or compress. Drag the bright handle to the right, then try A larger moment of inertia.

   **What to observe:** A larger inertia lowers every rotational excitation. Doubling inertia halves the energies; the evaluated points remain fixed.

3. 03

   #### Test the one-neutron model

   Choose One neutron outside oxygen. Compare the two d levels used for calibration with the independent s level, then reduce Spin-orbit strength to zero.

   **What to observe:** The d splitting can be matched, but the s excitation is overpredicted by about 1.16 MeV. At zero strength the oscillator sd levels become degenerate. The missing radial structure matters.

Physical conclusion

Use level spacings and transition rules together, then test the levels that did not set the model parameters.

## Related experiments and articles

Nuclear physics · Reading path

### From nuclear binding to stellar reaction rates

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a finite nuclear well selects discrete levels**](https://sandboxphysics.com/stories/nuclear-bridge/finite-wells-and-spin-orbit/)
2. [Article **Read a nuclear spectrum one neutron at a time**](https://sandboxphysics.com/stories/nuclear-bridge/spectra-and-selection/)
3. [Article **Paired nucleons in a rotating nucleus**](https://sandboxphysics.com/stories/nuclear-bridge/pairing-and-collectivity/)
4. [Article **Why shells and tails change near the neutron drip line**](https://sandboxphysics.com/stories/nuclear-bridge/shell-evolution-and-halos/)

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)