Pairing Effects in Coulomb Energies and the Radii of Mirror Nuclei

TLDR

Coulomb energy differences between mirror nuclei exhibit an odd‑even alternation with Z that reflects short‑range nuclear pairing, and this effect is interpreted within the shell model as spherically symmetric pairing of lowest‑seniority states. The authors employ a harmonic‑oscillator jj‑coupling model, fixing a single parameter to within ±1.5 % by fitting experimental data across the p1/2 and d5/2 shells, to compute Coulomb energies (including exchange) and the rms charge‑distribution radius for lowest‑proton‑seniority states. Second‑difference analysis shows the odd‑even alternation persists up to Z = 15 with additional shell‑structure irregularities, while the model predicts a smooth decline of the charge‑radius parameter r0 from 1.34 to 1.20 between C13 and Al27, fails for A ≤ 11 and shows inconsistencies beyond A ≥ 31, and recent data suggest r0 may drop to 1.1–1.15 near A ≈ 39.

Abstract

The Coulomb energy difference between the nuclei of a mirror pair exhibits an odd-even alternation with $Z$ that is presumed to reflect the well-known pairing property of the short-range nuclear forces. By taking second differences of Coulomb energy (differences between successive mirror pairs), the alternation is seen to continue to at least $Z=15$, and additional irregularities appear that may be shell-structure effects. The analysis of Feenberg and Goertzel is discussed from the point of view of the shell model, and the pairing of spins is extended to the spherically symmetric pairing characteristic of the state of lowest seniority. A harmonic oscillator model with $\mathrm{jj}$ coupling is used to calculate the Coulomb energy, including exchange effects, in the state of lowest proton seniority. The single parameter of the model is determined by comparison with experimental data and remains constant to \ifmmode\pm\else\textpm\fi{}1.5 percent through the ${p}_{\frac{1}{2}}$ and ${d}_{\frac{5}{2}}$ shells. The rms radius of the nuclear charge distribution is calculated by the same model. Between ${\mathrm{C}}^{13}$ and ${\mathrm{Al}}^{27}$, the equivalent ${r}_{0}$ decreases fairly smoothly from 1.34 to 1.20. For $A\ensuremath{\le}11$ the model is not satisfactory, and for $A\ensuremath{\ge}31$ there are some serious inconsistencies in the data. The most recent data indicate that ${r}_{0}$ may decrease to the range 1.1 to 1.15 for $A\ensuremath{\cong}39$.