Read a nuclear spectrum one neutron at a time

An extra neutron gives the core a handle

The oxygen-16 ground state has spin zero. In an inert-core approximation the extra neutron therefore carries the total angular momentum of oxygen 17. The label $0d_{5/2}$ specifies zero radial nodes, orbital angular momentum two and total angular momentum five halves. The letters s, p and d denote orbital angular momenta zero, one and two.

A neutron has intrinsic spin $s=1/2$. Adding it to orbital angular momentum gives $j=\ell\pm1/2$, except that an s orbital has only $j=1/2$. Spatial inversion multiplies an orbital wavefunction by $(-1)^\ell$. This parity is positive for both s and d orbitals.

An orbital assignment is an approximate configuration, with possible admixtures from core excitation. Transfer reactions and spectroscopic strengths are needed to assess how strongly a real state follows this one-neutron picture.

Source: TUNL · M. Goeppert Mayer

The d orbital splits into two partners

Add $-\kappa\mathbf L\!\cdot\!\mathbf S/\hbar^2$ to a spherical central potential, with $\kappa>0$. The negative sign adopts the nuclear ordering in which the larger-j partner lies lower. Squaring $\mathbf J=\mathbf L+\mathbf S$ expresses the interaction through three angular-momentum squares.

For a d orbital the $j=5/2$ shift is $-\kappa$ and the $j=3/2$ shift is $3\kappa/2$. Their separation is $5\kappa/2$. Each j has $2j+1$ magnetic substates: six and four here. Their combined ten states preserve the original spin-times-orbital state count.

Source: M. Goeppert Mayer

Worked example: fit two levels and test the third

TUNL Table 17.10, evaluation 1993TI07, sets the $5/2^+$ ground-state excitation to zero. The $1/2^+$ level is $870.73\pm0.10\,\mathrm{keV}$ and the $3/2^+$ level is $5084.8\pm0.9\,\mathrm{keV}$. The source lists energies in MeV but errors in keV, so the two columns need different conversions.

In a harmonic-oscillator central potential, $0d$ and $1s$ share one major shell. After subtracting the lower d energy, the three predicted excitation energies are $0,\ \kappa,\ 5\kappa/2$. Matching the d separation fixes $\kappa=2033.92\,\mathrm{keV}$.

The s level now tests the model. Its evaluated value is $870.73\,\mathrm{keV}$, the model gives $2033.92\,\mathrm{keV}$, and model minus evaluation is $+1163.19\,\mathrm{keV}$. This discrepancy greatly exceeds the energy uncertainty. The radial central potential and many-body correlations need improvement; the calibrated d separation cannot test them.

The upper $3/2^+$ level is also neutron unbound, with evaluated width $96\pm5\,\mathrm{keV}$. Only its centroid enters this phenomenological calibration. A resonance width differs from an uncertainty on its central energy. Calculating the resonance itself requires an outgoing-neutron boundary condition.

Source: TUNL

A photon must carry angular momentum too

An electric quadrupole photon has multipole order two, abbreviated E2. The initial spin, final spin and multipole order must form an angular-momentum triangle, and parity stays unchanged.

The transition $2^+\rightarrow0^+$ passes. A $4^+\rightarrow0^+$ transition needs a larger multipole. A $0^+\rightarrow0^+$ transition fails the other side of the triangle even though its spin difference is zero.

These rules identify possible matrix elements. Their sizes still require nuclear wavefunctions and the quadrupole operator. At high excitation a neutron channel may compete with gamma decay. Allowed arrows therefore do not specify measured branching strengths. Neglecting small nuclear recoil, the photon energy is approximately the level difference.

Source: M. Hjorth-Jensen