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.

Single-particle shell gap $0$

Neutron rms radius $0$

Density at the probe $0$

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

Occupied neutron states

Unoccupied states

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

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 misconceptionThe visible neutrons travel along classical circular paths.

Better mental modelThe displayed cloud is a stationary quantum probability distribution with no assigned trajectories.

A model gap is not a measured separation energy

Typical misconceptionThe readout gives a precision energy for removing a neutron from a real isotope.

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

Run the experiment

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.
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.
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.