The 9Li(d,p) reaction, a specific probe of 10Li, paradigm of parity--inverted nuclei around N=6 closed shell

TL;DR

This paper investigates the structure of $^{10}$Li using nuclear field theory and reaction models, revealing the presence of a virtual $1/2^+$ state and explaining parity inversion phenomena in light nuclei.

Contribution

It demonstrates that the parity inversion in $^{10}$Li is due to the same mechanism as in $^{11}$Be and $^{11}$Li, providing a unified explanation for these phenomena.

Findings

Presence of a virtual $1/2^+$ state in $^{10}$Li

Parity inversion mechanism similar to $^{11}$Be and $^{11}$Li

Consistent level sequence in $N=7$ isotones

Abstract

We show, within the framework of renormalized nuclear field theory and of the induced reaction surrogate formalism, that the highly debated $^{10}$Li structure, observed in a recent $^9$Li(d,p)$^{10}$Li one--neutron transfer experiment is consistent with or better, requires the presence of a virtual $1/2^+$ state of similar single--particle strength than that of the $1/2^-$ resonance at 0.45$\pm$ 0.03 MeV. Based on continuum spectroscopy self-energy techniques, we find that the physical mechanism responsible for parity inversion in $^{10}_3$Li is the same as that at the basis of the similar phenomenon observed in $^{11}_4$Be and to that needed in $^{11}$Li to have an important $s$--wave ground state component. Furthermore, it is also consistent with the (normal) sequence of the $1p_{1/2}$ and $2s_{1/2}$ levels in the $N=7$ isotones $^{12}_5$B and $^{13}_6$C.

Figures

Full text

Supplemental material

The bare mean field potential

The parameters of the bare mean field adopted in the various nuclei are listed in Table 1. They have been fitted in the even nuclei, minimising the difference between the experimental energies of the $1/2^{+},1/2^{-}$ and $5/2^{+}$ states with the theoretical results obtained including the many-body renormalisation effects. In the case of the odd-odd nucleus 12B we have fitted the centroids of the multiplets (see below). In the case of 10Li, the fit has been obtained extrapolating the potentials obtained for the other nuclei and then adjusting the parameters so as to reproduce the shape of the strength function obtained in the transfer reaction 9Li(d,p)10Li by Cavallaro et al. [1], as discussed in the main text. We note that in individual nuclei, fits of similar quality can be obtained with different families of parameters. We have privileged solutions having relatively large radii and diffusivities, and evolving smoothly from one isotope to the next.

Deformation parameters

The renormalization of the neutron single-particle states due to the coupling to the low-lying collective quadrupole vibration is an essential ingredient of our calculations of the structure of the $N=7$ isotones ${}_{3}^{10}$Li, ${}_{4}^{11}$Be,${}_{5}^{12}$B,${}_{6}^{13}$C. The properties of such dressing effects are determined by the energy $\hbar\omega_{2^{+}}$ and by the deformation parameter $\beta_{n}$. The experimental data associated with the $N=6$ even-even cores ${}_{2}^{8}$He, ${}_{4}^{10}$Be,${}_{6}^{12}$C are indicated in Fig. 1 ($\hbar\omega_{2^{+}}$ ) and Fig. 2 ($\beta_{n}$) by filled circles. In the case of ${}_{4}^{10}$Be,${}_{6}^{12}$C the value of $\beta_{n}$ is inferred from the value of the electromagnetic $B(E2)$ and of the deformation parameter obtained by inelastic (p,p’) excitation. In the case of 8He, the value of $B(E2)$ is not known and we have derived it from the values of $M_{n}/M_{p}$ and $M_{n}$ reported in ref. [3].

In our calculations the values of the neutron deformation parameter have been fitted in each calculated N=7 isotone so as to optimize the properties of the calculated energy spectrum as compared to experiments; they are shown in Fig. 2 by empty circles. It is reassuring that the values adopted for ${}_{4}^{11}$Be, and ${}_{6}^{13}$C lie close to the experimental values of the corresponding ${}_{4}^{10}$Be,${}_{6}^{12}$C cores, and that the values adopted for the odd-odd nuclei ${}_{3}^{10}$Li and ${}_{5}^{12}$B interpolate smoothly between them. In our calculation of ${}_{4}^{11}$Be and ${}_{6}^{13}$C we have used the experimental values of the $2^{+}$ energy in ${}_{4}^{10}$Be and ${}_{6}^{12}$C. The values of $\hbar\omega_{2^{+}}$ in the odd-odd isotones ${}_{3}^{10}$Li and ${}_{5}^{12}$B have instead been fitted. Experimental and theoretical values are reported in Figs. 1 and 2.

Treatment of the neutron-proton interaction.

This section presents a simple but quantitative treatment of the effects related to the presence of the odd proton in ${}^{12}_{5}$B, that can be directly extrapolated to case of ${}^{10}_{3}$Li.

The lowest states observed in the energy spectrum of the odd-odd nucleus 12B are two doublets with angular momentum and parity ($1^{+},2^{+}$) and ($1^{-},2^{-}$) which can be naturally interpreted as arising from the coupling of the odd proton lying on the $1p_{3/2}$ orbital with the odd neutron lying on the $1p_{1/2}$ ($\pi_{1p_{3/2}}\otimes\nu_{1p_{1/2}}$ ) or on the $2s_{1/2}$ ($\pi_{1p_{3/2}}\otimes\nu_{2s_{1/2}}$ ) weakly bound orbitals.

The doublets display a similar energy splitting equal to approximately 0.95 MeV. The $1^{+}$ and the $2^{-}$ states lie lowest in the two multiplets. The relative position of the centroids of the $(1^{+},2^{+})$ ($E_{cent}=0.59$ MeV) and $(1^{-},2^{-})$ doublets ($E_{cent}=2.02$) in 12B indicates a difference in the position of the $1p_{1/2}$ or on the $2s_{1/2}$ neutron states equal to $\Delta_{ps}$ = -1.43 MeV, to be compared with with the value $\Delta_{ps}$ = -3.09 MeV associated with the relative position of the states $1/2^{-}$ (g.s.) and $1/2^{+}$ (3.09 MeV) in 13C.

To calculate these splittings use has been made of the schematic two-body interaction $V_{np}=F_{0}\delta(\vec{r}_{1}-\vec{r}_{2})(1-\alpha+\alpha\sigma_{1}\cdot\sigma_{2})$, originally introduced in order to interpret Nordheim’s rule for the spin of the ground state of odd-odd nuclei [4, 5, 6]. The magnitude of the splittings within each $\pi(p_{3/2})\otimes\nu(nl_{j})$ multiplet is determined in first order by the diagonal matrix element of $V_{np}$ in the neutron-proton wavefunction, and the relative ordering of the levels is determined by the parameter $\alpha$. For values of the parameter $\alpha\approx 0.5$, one reproduces the experimental ordering of the states in 12B within the two doublets (the states $1^{+}$ and 2- are the lowest ones).

The action of the two-body interaction we have adopted is in general different, according to whether the neutron and proton levels involved are of hole- or of particle-character. In the case of one-hole and one-particle states, the shift caused by the spin-independent part changes sign, while the $\sigma_{1}\cdot\sigma_{2}$ part does not change. However, the ordering within the $\pi(p_{3/2})\times\nu(2s_{1/2})$ and $\pi(1p_{3/2})\times\nu(1p_{1/2})$ doublets in 10Li is predicted to be the same as within the $\pi(1p^{-1}_{3/2})\times\nu(2s_{1/2})$ and $\pi(1p^{-1}_{3/2})\times\nu(1p_{1/2})$ doublets in 12B. In fact ,the $1p_{1/2}$ and $2s_{1/2}$ states can be considered as particles or as holes (one particle in a two-level orbital), and for symmetry the contributions of the spin-independent part does not produce any splitting, but only a level shift which is taken into account by the mean field. As a consequence, the splittings caused by these interaction are expected to be the same, apart from the difference arising from the different radial form factors. This is a very peculiar characteristic of the multiplets based on $s_{1/2}$ or $p_{1/2}$ orbitals. In general the situation is different , as for example in the case of the multiplets based on the $1d_{5/2}$ orbital, which are predicted to show inverted splittings in 10Li and 12B.

For a more quantitative calculation, the interaction should be diagonalized in a space of good total angular momentum $J$ generated by all the possible combinations of the odd proton and the odd neutron orbitals that may couple to $J$. We will make two simplifying assumptions. Our first approximation will be to freeze the proton in the $1p_{3/2}$ orbital, which is justified due to its large binding energy. Our second approximation will be to neglect the mixing between different multiplets, an approximation which is justified by their relatively large energy separation. In particular, we will not mix the $2^{-}$ state produced by coupling the proton with the neutron moving in the $2s_{1/2}$ orbital, with the $2^{-}$ states obtained from the coupling with the neutron moving in the $1d_{5/2}$ or in the $1d_{3/2}$ orbitals. With these two approximations, the problem becomes equivalent to a $J(lj)$-dependent one-body problem. The neutron wavefunction may be obtained solving the Schrödinger equation adding a $J(lj)$-dependent potential given by $\Delta V_{J(lj)}=F_{0}|\phi_{p_{3/2}}(r)|^{2}<(1-\alpha+\alpha\sigma_{1}\cdot\sigma_{2})>_{J(lj)}$ to the bare potential used to fit the centroid, making use of the radial proton wave function $\phi_{p_{3/2}}(r)$.

For simplicity we have substituted the radial form factor $\phi_{p_{3/2}}(r)$ by a Wood Saxon potential having the same geometry as the bare potential used to fit the centroid, determining its depth so as to reproduce the experimental value of the position of the $1^{+}$ level, leading to $\Delta V_{1(s_{1/2})}=$ -2.6 MeV. The depth of the potentials used for the other $J(lj)$ states have been scaled according to the value of $<(1-\alpha+\alpha\sigma_{1}\cdot\sigma_{2})>_{J(lj)}$. We have adopted the same procedure in the case of 10Li, but starting from the bare WS potential with parameters extrapolated from those determined in 13C and 11Be. The comparison with the data of the reaction discussed in the main text favours somewhat smaller values of $\Delta V_{1(s_{1/2})}$. We have adopted $\Delta V_{1(s_{1/2})}=$ -1.8 MeV which leads to a rather good reproduction of the data both in in 12Be and 10Li.