Prolate-oblate shape competition and impact on charge radii in Bk isotopes

Abstract

The nuclear charge radius provides a fundamental probe of nuclear structure, yet experimental data remain rare in the actinide region. Using the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) with the PC-PK1 functional, we carry out a systematic investigation of prolate-oblate shape competition in odd-$A$ Bk isotopes. Deformation is found to play an important role in the description of charge radii $r_c$ by extending the density distribution. Notably, $r_c$ exhibits a distinct shape dependence: for a given absolute quadrupole deformation $|β_2|$, oblate shapes yield larger charge radii than their prolate counterparts in well-deformed nuclei near the mid-shell region, where the empirical formula $r_c(β_2) = \left(1 + \frac{5}{4π}|β_2|^2\right) r_c(0)$ fails to capture the observed behavior. This enhancement is attributed to a central depression (or ``bubble" structure) in the proton density, which microscopically originates from the non-occupation of the spherical $3s_{1/2}$($Ω=1/2$) orbital in oblate minima. These findings establish a clear microscopic connection between nuclear shape, single-particle occupancy, and nuclear size.

Figures (5)

• Figure 1: (Color online) Evolutions of the potential energy curves (PECs) denoted by solid and dashed lines of $^{227,235,\cdots,355}$Bk isotopes with $\Delta N=8$ obtained from the constrained DRHBc calculations in steps of $\Delta\beta_2=0.05$. All curves are scaled to their respective ground-state energies and shifted upward by $1.0$ MeV per additional neutron. Ground states (red solid circles) and second local minima (blue open squares) from unconstrained DRHBc calculations are also shown, for all odd-$A$ isotopes.
• Figure 2: (Color online) Charge radii $r_{\rm c}$ (a) and rms radii for neutrons $r_{\rm n}$, protons $r_{\rm p}$, and total nuclear matter $r_{\rm m}$ (b) as functions of neutron number $N$ in Bk isotopes. Results from DRHBc calculations with PC-PK1 are shown for ground states (red solid circles) and local minima (blue open squares), while RCHB results ADNDT2018Xia_121_1 (grey open triangles) are also included for comparison.
• Figure 3: (Color online) Differences between local minima and ground states for radii $\Delta r$ and squared quadrupole deformation $\Delta\beta_2^{2}$ of Bk isotopes as functions of the neutron number. Panel (a) displays the charge radii $r_{\rm c}$ and proton quadrupole deformation $\beta_{2,{\rm p}}$; panel (b) presents rms radii for neutrons $r_{\rm n}$, protons $r_{\rm p}$, and all nucleons $r_{\rm m}$ together with the corresponding quadrupole deformation $\beta_{2,{\rm n}}$, $\beta_{2,{\rm p}}$, and $\beta_{2}$. The yellow-shaded region denotes nuclei in prolate deformations.
• Figure 4: (Color online) Angle-averaged proton density distributions $\rho_{\rm p}(r)$ for the ground states (solid lines) and local minima (dashed lines) of $^{315}$Bk and $^{339}$Bk. The ground state of $^{315}$Bk is prolate ($\beta_{2,{\rm p}}=0.284$) while its local minimum is oblate ($\beta_{2,{\rm p}}=-0.277$). Conversely, $^{339}$Bk has an oblate ground state ($\beta_{2,{\rm p}}=-0.132$) and a prolate local minimum ($\beta_{2,{\rm p}}$=0.141).
• Figure 5: (Color online) Proton single-particle levels around the Fermi energy $\lambda_{\rm p}$ as functions of total quadrupole deformation $\beta_2$ ($-0.4\leq\beta_2\leq 0.6$), obtained by the constraint DRHBc calculations for $^{315}$Bk (a) and $^{339}$Bk (b). Red and blue vertical lines mark the deformations of the ground states and local minima, respectively.