Effective proton-neutron interaction near the drip line from unbound states in ${}^{25,26}\mathrm{F}$

Background: Odd-odd nuclei, around doubly closed shells, have been extensively used to study proton-neutron interactions. However, the evolution of these interactions as a function of the binding energy, ultimately when nuclei become unbound, is poorly known. The ${}^{26}\mathrm{F}$ nucleus, composed of a deeply bound $\pi 0d_{5/2}$ proton and an unbound $\nu 0d_{3/2}$ neutron on top of an ${}^{24}\mathrm{O}$ core, is particularly adapted for this purpose. The coupling of this proton and neutron results in a $J^{\pi }=1_1{}^{+}-4_1{}^{+}$ multiplet, whose energies must be determined to study the influence of the proximity of the continuum on the corresponding proton-neutron interaction. The $J^{\pi }=1_1{}^{+},2_1{}^{+},4_1{}^{+}$ bound states have been determined, and only a clear identification of the $J^{\pi }=3_1{}^{+}$ is missing.

Purpose: We wish to complete the study of the $J^{\pi }=1_1{}^{+}-4_1{}^{+}$ multiplet in ${}^{26}\mathrm{F}$, by studying the energy and width of the $J^{\pi }=3_1{}^{+}$ unbound state. The method was first validated by the study of unbound states in ${}^{25}\mathrm{F}$, for which resonances were already observed in a previous experiment.

Method: Radioactive beams of ${}^{26}\mathrm{Ne}$ and ${}^{27}\mathrm{Ne}$, produced at about $440A\mathrm{MeV}$ by the fragment separator at the GSI facility were used to populate unbound states in ${}^{25}\mathrm{F}$ and ${}^{26}\mathrm{F}$ via one-proton knockout reactions on a ${\mathrm{CH}}_2$ target, located at the object focal point of the ${\mathrm{R}}^3\mathrm{B}$/LAND setup. The detection of emitted $\gamma$ rays and neutrons, added to the reconstruction of the momentum vector of the $A-1$ nuclei, allowed the determination of the energy of three unbound states in ${}^{25}\mathrm{F}$ and two in ${}^{26}\mathrm{F}$.

Results: Based on its width and decay properties, the first unbound state in ${}^{25}\mathrm{F}$, at the relative energy of 49(9) keV, is proposed to be a $J^{\pi }=1/2^{-}$ arising from a $p_{1/2}$ proton-hole state. In ${}^{26}\mathrm{F}$, the first resonance at 323(33) keV is proposed to be the $J^{\pi }=3_1{}^{+}$ member of the $J^{\pi }=1_1{}^{+}-4_1{}^{+}$ multiplet. Energies of observed states in ${}^{25,26}\mathrm{F}$ have been compared to calculations using the independent-particle shell model, a phenomenological shell model, and the ab initio valence-space in-medium similarity renormalization group method.

Conclusions: The deduced effective proton-neutron interaction is weakened by about 30–40% in comparison to the models, pointing to the need for implementing the role of the continuum in theoretical descriptions or to a wrong determination of the atomic mass of ${}^{26}\mathrm{F}$.

Figure 1

(a) $Z=9$ transmitted nuclei obtained from the energy losses measured in the DSSSD located just after the target and in the TFW. (b) Mass number $A$ identification of the transmitted $Z=9$ nuclei is obtained from their reconstructed trajectory in the dispersive plane and from their time of flight (see text for details).

Figure 2

Relative energy spectrum of ${}^{25}\mathrm{F}$. The solid black line shows the result of the fit composed of three resonances, marked in different colors, whose energies are written with uncertainties. Corresponding widths are given in Table 1. In the fitting procedure, resonances were folded with the resolution of the LAND detector, enhancing their widths as compared to the intrinsic value.

Figure 3

(a) $\gamma$-ray spectrum obtained from the ${}^{26}\mathrm{Ne}(-1p)$ reaction, in coincidence with ${}^{24}\mathrm{F}$ and the detection of at least one neutron in LAND. (b) Relative energy spectrum for ${}^{25}\mathrm{F}$, gated on the $\sim 510$-keV $\gamma$-ray transition in ${}^{24}\mathrm{F}$.

Figure 4

Left: Illustrative picture of the expected configurations populated in ${}^{25}\mathrm{F}$ from the ${}^{26}\mathrm{Ne}$($-1p$) reaction. Right: Experimental level scheme of ${}^{25}\mathrm{F}$ compared to shell-model calculations performed using the phenomenological USDA [32] and WBP [51] interactions and with ab initio valence-space Hamiltonians derived from IM-SRG [23, 24]. The unbound states above $S_n=4270\left(100\right)$ keV were obtained in the present work, while the bound states were studied in Ref. [55]. Gray rectangles shown in the experimental spectrum and in the IM-SRG predictions, with a $J^{\pi }$-dependent horizontal widths, correspond to uncertainties on the energy centroids of the states. These uncertainties on calculated energies overlap between 4 and 7 MeV, where many resonances are present. The bound states come mainly from ${\left(\pi 0d_{5/2}\right)}^1$ [case (a)] and ${\left(\pi 0d_{5/2}\right)}^1{\left(\nu 1s_{1/2}\right)}^{-1}{\left(\nu 0d_{3/2}\right)}^1$ [case (b)] configurations. We propose that unbound states come mainly from ${\left(\pi 0p_{1/2}\right)}^{-1}{\left(\pi 0d_{5/2}\right)}^2$ [case (c)] and ${\left(\pi 0d_{5/2}\right)}^1{\left(\nu 1s_{1/2}\right)}^{-2}{\left(\nu 0d_{3/2}\right)}^2$ [case (d)] configurations.

Figure 5

Relative energy spectrum for ${}^{26}\mathrm{F}$. The solid black line shows the result of a Breit-Wigner fit using two $\ell =2$ resonances whose centroid values are given in the figure. These resonances are folded with the resolution of the LAND detector. (a) With the energy shift ${\mathrm{\Delta }}_{\ell }\left(E_{\mathrm{rel}}\right)$ defined as in Eq. (5). (b) Assuming ${\mathrm{\Delta }}_{\ell }\left(E_{\mathrm{rel}}\right)=0$.

Figure 6

Left: Illustrative picture of the expected configurations populated in ${}^{26}\mathrm{F}$ from the ${}^{27}\mathrm{Ne}$($-1p$) reaction. Right: Experimental level scheme of ${}^{26}\mathrm{F}$ compared to shell-model calculations performed using the phenomenological USDA interaction [32] and with ab initio valence-space Hamiltonians derived from IM-SRG [23, 24]. The energies of unbound states, above $S_n=1071\left(130\right)$ keV, were newly measured in this work, while those of the bound states are taken from Refs. [3, 17, 19]. Gray rectangles shown in the experimental spectrum and in the IM-SRG predictions correspond to uncertainties on the energies centroids of the states and of the $S_n$. The bound states $J^{\pi }=1_1{}^{+},2_1{}^{+},4_1{}^{+}$ as well as the unbound $J^{\pi }=3_1{}^{+}$ state are proposed to come from ${\left(\pi 0d_{5/2}\right)}^1{\left(\nu 0d_{3/2}\right)}^1$ [case (a)] configuration, while the second unbound state could come from ${\left(\pi 0d_{5/2}\right)}^1{\left(\nu 1s_{1/2}\right)}^{-1}{\left(\nu 0d_{3/2}\right)}^2$ [case (b)] and/or ${\left(\pi 1s_{1/2}\right)}^1{\left(\nu 0d_{3/2}\right)}^1$ [case (c)] configurations.

Figure 7

Experimental interaction energies corresponding to the $\pi 0d_{5/2}\times \nu 0d_{3/2}$ coupling in ${}^{26}\mathrm{F}, \mathrm{Int}{\left(J\right)}^{\exp }$ (green circles), are plotted as a function of $J(J+1)$ and compared to calculations using (a) a $\delta$ interaction without ($▾$) or with ($\blacksquare$) $J$-dependent radial corrections (see text for details), (b) the IM-SRG procedure, and (c) the USDA interaction. Fitted parabolas are drawn to guide the eye. Extracted experimental and calculated monopole values $V_{pn}$ are given in each panel. All values of $\mathrm{Int}\left(J\right)$ and $V_{pn}$ are given in Table 3.