Exploring the halo character and dipole response in the dripline nucleus ${}^{31}\mathrm{F}$

Background: Lying at the lower edge of the “island of inversion,” neutron-rich fluorine isotopes (${}^{29\text{–}31}\mathrm{F}$) provide a curious case to study the configuration mixing in this part of the nuclear landscape. Recent studies have suggested that a prospective two-neutron halo in the dripline nucleus ${}^{31}\mathrm{F}$ could be linked to the occupancy of the $pf$ intruder configurations.

Purpose: Focusing on configuration mixing, matter radii, and neutron-neutron ($nn$) correlations in the ground state of ${}^{31}\mathrm{F}$, we explore various scenarios to analyze its possible halo nature as well as the low-lying electric dipole ($E1$) response within a three-body approach.

Method: We use an analytical, transformed harmonic oscillator basis under the aegis of a hyperspherical formalism to construct the ground-state three-body wave function of ${}^{31}\mathrm{F}$. The $nn$ interaction is defined by the Gogny-Pires-Tourreil potential that includes the central, spin-orbit, and tensor terms, while the different two-body potentials to describe the core $+n$ subsystems are constrained by the different possible scenarios considered.

Results: The ${}^{31}\mathrm{F}$ ground-state configuration mixing and its matter radius are computed for different choices of the ${}^{30}\mathrm{F}$ structure coupled to the valence neutron. The admixture of $p_{3/2}$, $d_{3/2}$, and $f_{7/2}$ components is found to play an important role, favoring the dominance of inverted configurations with dineutron spreads for two-neutron halo formation. The increase in matter radius with respect to the core radius, $\mathrm{\Delta }r\gtrsim$ 0.30 fm and the dipole distributions along with the integrated $\mathrm{B}\left(E1\right)$ strengths of $⩾2.6$ $e^2{\mathrm{fm}}^2$ are large enough to be compatible with other two-neutron halo nuclei.

Conclusion: Three-body results for ${}^{31}\mathrm{F}$ indicate a large spatial extension in its ground state due to the inversion of the energy levels of the normal shell model scheme. The increase is augmented by and is proportional to the extent of the $p_{3/2}$ component in the wave function. Additionally, the enhanced dipole distributions and large $\mathrm{B}\left(E1\right)$ strengths all point to the two-neutron halo character of ${}^{31}\mathrm{F}$.

Figure 1

The Jacobi-$Y$ and Jacobi-$T$ coordinate representations for a ${}^{29}\mathrm{F}+n+n$ system comprising the three-body composite ${}^{31}\mathrm{F}$.

Figure 2

Phase shifts for ${}^{30}\mathrm{F}$ corresponding to sets A, D, and AH of case I in panels (a)–(c), respectively. The phase shift curves for the $f_{7/2}$ state are shown by the (red) solid line while the $p_{3/2}$ orbital is represented by the (blue) dashed line. The dotted (black) line corresponds to $\pi /2$ radians.

Figure 3

Phase shifts for ${}^{30}\mathrm{F}$ corresponding to sets A and D of case II in panels (a) and (b), respectively. The phase shift curves for the $f_{7/2}$ orbital are shown by the (red) solid line while the $d_{3/2}$ orbital is represented by the (green) dash-dotted line. The dotted (black) line corresponds to $\pi /2$ radians.

Figure 4

Phase shifts for ${}^{30}\mathrm{F}$ corresponding to sets A, B, C, D, AH, and AH2 of case III in panels (a)–(f), respectively. The phase shift curves for the $f_{7/2}$ orbital are shown by the (red) solid line while the $d_{3/2}$ and $p_{3/2}$ orbitals are represented by the (green) dash-dotted and the (blue) dashed lines, respectively. The dotted (black) line corresponds to $\pi /2$ radians.

Figure 5

Variation of matter radius of fluorine isotopes with mass number $A$. The experimental values, shown in black circles, for ${}^{20\text{–}25}\mathrm{F}$ isotopes were taken from Ref. [63], while those for ${}^{27,29}\mathrm{F}$ were extracted from Ref. [5]. The solid red line corresponds to the $R_0{A}^{1/3}$ fit for $A=20\text{–}29$, while the shaded area gives the region of 99.99% confidence level in the fit. Note the increase in the radii for ${}^{24}\mathrm{F}$ and ${}^{29}\mathrm{F}$, as both are confirmed halos [5, 6, 64]. The squares represent case I, diamonds show case II, and triangles display case III for the different values of ${}^{31}\mathrm{F}$ radius resulting from the different schemes under consideration. The various colors portray the different sets, with blue depicting set A, violet for set B, magenta for set C, red for set D with the inverted schemes, green for antihalo set AH, and orange for AH2. Almost all configurations of ${}^{31}\mathrm{F}$ point to a two-neutron halo nucleus.

Figure 6

The contribution by different energy orbitals in the different configurations considered for the g.s. of ${}^{31}\mathrm{F}$. The dominance of $p_{3/2}$, $d_{3/2}$, and $f_{7/2}$ subshells is evident for the various schemes.

Figure 7

The ground state probability density distribution (in units of ${\mathrm{fm}}^{-2}$) of ${}^{31}\mathrm{F}$ for all the configurations considered, as a function of $r_{nn}$ and $r_{c\text{−}nn}$.

Figure 8

The $d\mathrm{B}\left(E1\right)/d\mathrm{E}$ distribution for sets A (black dotted line), D (red solid line), and AH (green dashed line) of the open shell configuration (case III) considered in this study. Note the dominance of the inverted configuration, i.e., set D, in the amplitude.

Figure 9

The convergence of the ground state two-neutron separation energy of ${}^{31}\mathrm{F}$ for $S_{2n}=0.15$ MeV with variation in hyperspherical momentum $K_{\max }$ and the basis size, $i_{\max }$. Convergence is reached around $K_{\max }=40$ and $i_{\max }=20$. Calculations are done for Case I set A, case II set A, and case III set B.

Figure 10

The $d\mathrm{B}\left(E1\right)/d\mathrm{E}$ distribution with variation in the basis size for the second state ($1^{-}$ in our case). The calculations converge at $i_{\max }\left(2\right)=35$. The width of the Poisson distribution ($w$) used to generate these was fixed at 30, while the hypermomentum ($K$) used was 40. The calculations shown are done for case III, set D.