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.

Evaluated nucleus ${}^{166}\mathrm{Er}$

First excited level in model $80.6\,\mathrm{keV}$

Predicted higher-to-lower energy ratio $3.33$

Comparison-level RMS residual $34.0\,\mathrm{keV}$

One nucleus, different explanations

Which spacing survives the comparison?

Drag the handle to compare predicted level spacings.

Moment of inertia relative to reference

\mathcal I/\mathcal I_0

1.00

Models, data and keyboard controls

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.

The rotor and vibrator share the evaluated first excited energy. The oxygen spin-orbit strength is calibrated to the two d levels. Residuals on other levels test the model; calibration points are labelled in the table.

Focus the scene and use left or right to change the active parameter. In the vibration comparison the spectrum stays fixed; the two-phonon multiplet also contains zero- and two-spin states, listed below.

Model assumptions

Evaluated data · ideal model predictions

The spacing is the test

Evaluated levelsRotor / single-neutron modelHarmonic 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 · Frozen data and provenance

Study nuclear structure, scattering and stellar rates

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 misconceptionEvery higher nuclear level means that one neutron moved to a higher orbital.

Better mental modelSeveral 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 misconceptionThe rotor lines nearly coincide with erbium, so the model reproduces the evaluated spectrum exactly.

Better mental modelThe 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

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.
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.
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.

Nuclear Spectra and Collective Motion

The spacing is the test