From spin to pseudospin symmetry: The origin of magic numbers in nuclear structure

Abstract

Magic numbers lie at the heart of nuclear structure, reflecting enhanced stability in nuclei with closed shells. While the emergence of magic numbers beyond 20 is commonly attributed to strong spin-orbit coupling, the microscopic origin of the spin-orbit potential remains elusive, owing to its dependence on the resolution scale and renormalization scheme of nuclear forces. Here, we investigate the evolution of shell structure with varying momentum resolution in nuclear interactions derived from chiral effective field theory, using the similarity renormalization group to link different scales. We uncover a novel transition from spin symmetry to pseudospin symmetry as the resolution scale decreases, during which magic numbers emerge naturally. A similar pattern is found in calculations using relativistic one-boson-exchange potentials, underscoring the robustness of the phenomenon. This establishes a direct connection between realistic nuclear forces with a high resolution scale and effective nuclear forces at coarse-grained scales, offering a first-principles explanation for the origin of magic numbers and pseudospin symmetry in nuclear shell structure, and new insights into the structure of exotic nuclei far from stability.

Figures (3)

• Figure 1: Nuclear shell structure in [132]Sn for the chiral NN+3N(lnl) interaction.a, Schematic illustration of the nuclear potentials in the ${}^3S_1$ channel in both momentum space and coordinate space, with the momentum resolution scale varying from high to low regimes. b, The ESPEs of protons as functions of $\lambda$, relative to the $1s_{1/2}$ state. The leftmost column shows Nilsson model results with empirical values for comparison. The labels $\tilde{d}$ and $\tilde{f}$ indicate states with $\tilde{\ell}=2$ and $3$, respectively. c,d, Strength parameters $(\hbar\omega_0\kappa, \mu)$ of the Nilsson model, determined by mapping to the results of the chiral NN+3N(lnl) interaction at different energy scales. The error bars represent the statistical uncertainty in the fitting process. See the main text for details.
• Figure 2: Decomposition of the ESPEs and the SO splittings in [132]Sn using the EM family interaction as functions of the energy scale $\lambda$.a, The ESPE of the $0f_{5/2}$ state. b, The energy splitting of the SO doublets $(0f_{5/2}, 0f_{7/2})$. c, Contributions of different terms in the $3N$ interaction to the SO splitting shown in b.
• Figure 3: Evolution of energy splittings $\Delta\varepsilon$ for SO and pseudospin doublets of protons in [132]Sn with the OBEP$\Lambda$ potentials.a, The energy splitting of the SO doublets ($0f_{7/2}, 0f_{5/2}$) and pseudospin doublets ($1p_{3/2}, 0f_{5/2}$) as functions of the momentum cutoff $\Lambda$. b, Evolution of the radial wave functions of the lower component of the Dirac spinors for the pseudospin doublets ($1p_{3/2}, 0f_{5/2}$). The strong oscillations in the wave functions for the hardest potential ($\Lambda=\infty$) arise from the unbound nature of the states.