Ab initio calculation of the potential bubble nucleus ${}^{34}\mathrm{Si}$

Background: The possibility that an unconventional depletion (referred to as a “bubble”) occurs in the center of the charge density distribution of certain nuclei due to a purely quantum mechanical effect has attracted theoretical and experimental attention in recent years. Based on a mean-field rationale, a correlation between the occurrence of such a semibubble and an anomalously weak splitting between low angular-momentum spin-orbit partners has been further conjectured. Energy density functional and valence-space shell model calculations have been performed to identify and characterize the best candidates, among which ${}^{34}\mathrm{Si}$ appears as a particularly interesting case. While the experimental determination of the charge density distribution of the unstable ${}^{34}\mathrm{Si}$ is currently out of reach, $(d,p)$ experiments on this nucleus have been performed recently to test the correlation between the presence of a bubble and an anomalously weak $1/2^{-}\text{−}3/2^{-}$ splitting in the spectrum of ${}^{35}\mathrm{Si}$ as compared to ${}^{37}\mathrm{S}$.

Purpose: We study the potential bubble structure of ${}^{34}\mathrm{Si}$ on the basis of the state-of-the-art ab initio self-consistent Green's function many-body method.

Methods: We perform the first ab initio calculations of ${}^{34}\mathrm{Si}$ and ${}^{36}\mathrm{S}$. In addition to binding energies, the first observables of interest are the charge density distribution and the charge root-mean-square radius for which experimental data exist in ${}^{36}\mathrm{S}$. The next observable of interest is the low-lying spectroscopy of ${}^{35}\mathrm{Si}$ and ${}^{37}\mathrm{S}$ obtained from $(d,p)$ experiments along with the spectroscopy of ${}^{33}\mathrm{Al}$ and ${}^{35}\mathrm{P}$ obtained from knock-out experiments. The interpretation in terms of the evolution of the underlying shell structure is also provided. The study is repeated using several chiral effective field theory Hamiltonians as a way to test the robustness of the results with respect to input internucleon interactions. The convergence of the results with respect to the truncation of the many-body expansion, i.e., with respect to the many-body correlations included in the calculation, is studied in detail. We eventually compare our predictions to state-of-the-art multireference energy density functional and shell model calculations.

Results: The prediction regarding the (non)existence of the bubble structure in ${}^{34}\mathrm{Si}$ varies significantly with the nuclear Hamiltonian used. However, demanding that the experimental charge density distribution and the root-mean-square radius of ${}^{36}\mathrm{S}$ be well reproduced, along with ${}^{34}\mathrm{Si}$ and ${}^{36}\mathrm{S}$ binding energies, only leaves the ${\mathrm{NNLO}}_{\text{sat}}$ Hamiltonian as a serious candidate to perform this prediction. In this context, a bubble structure, whose fingerprint should be visible in an electron scattering experiment of ${}^{34}\mathrm{Si}$, is predicted. Furthermore, a clear correlation is established between the occurrence of the bubble structure and the weakening of the $1/2^{-}\text{−}3/2^{-}$ splitting in the spectrum of ${}^{35}\mathrm{Si}$ as compared to ${}^{37}\mathrm{S}$.

Conclusions: The occurrence of a bubble structure in the charge distribution of ${}^{34}\mathrm{Si}$ is convincingly established on the basis of state-of-the-art ab initio calculations. This prediction will have to be reexamined in the future when improved chiral nuclear Hamiltonians are constructed. On the experimental side, present results act as a strong motivation to measure the charge density distribution of ${}^{34}\mathrm{Si}$ in future electron scattering experiments on unstable nuclei. In the meantime, it is of interest to perform one-neutron removal on ${}^{34}\mathrm{Si}$ and ${}^{36}\mathrm{S}$ in order to further test our theoretical spectral strength distributions over a wide energy range.

Figure 1

ADC(2) ground-state energy of ${}^{34}\mathrm{Si}$ as a function of the harmonic oscillator spacing $\hslash \omega$ and for increasing size $N_{\text{max}}$ of the single-particle model space.

Figure 2

ADC(2) ground-state rms charge radius of ${}^{34}\mathrm{Si}$ as a function of the harmonic oscillator spacing $\hslash \omega$ and for increasing size $N_{\text{max}}$ of the single-particle model space.

Figure 3

ADC(2) ground-state point-proton density distribution of ${}^{34}\mathrm{Si}$ for (a) different model space dimensions at $\hslash \omega =20$ MeV and (b) different harmonic oscillator frequencies at $N_{\text{max}}=13$. Panels (c) and (d) show the same distributions respectively of (a) and (b) but with a logarithmic vertical scale.

Figure 4

Point-proton and point-neutron density distributions of ${}^{34}\mathrm{Si}$ and ${}^{36}\mathrm{S}$ computed at the ADC(3) level.

Figure 5

Natural orbital partial-wave decomposition ${\rho }_p^{\ell j}\left(r\right)$ of the point-proton density distribution computed at the ADC(3) level for ${}^{34}\mathrm{Si}$ (a) and ${}^{36}\mathrm{S}$ (b).

Figure 6

Proton natural orbitals occupations $n_{n\ell j}$ computed at the ADC(3) level in ${}^{34}\mathrm{Si}$ and ${}^{36}\mathrm{S}$.

Figure 7

Radial part of proton $0s_{1/2},1s_{1/2}$, and $2s_{1/2}$ natural single-particle wave functions calculated at the ADC(3) level in ${}^{34}\mathrm{Si}$ and ${}^{36}\mathrm{S}$.

Figure 8

Same as Fig. 6 for neutrons.

Figure 9

Point-proton density distributions of ${}^{34}\mathrm{Si}$ and ${}^{36}\mathrm{S}$. (a) Results from Dyson ADC(1), ADC(2), and ADC(3) calculations. (b) Results from Gorkov and Dyson ADC(2) calculations.

Figure 10

Proton natural orbitals occupations $n_{n\ell j}$ in ${}^{34}\mathrm{Si}$. Results are displayed at the ADC(1) and ADC(3) levels.

Figure 11

Radial part of selected natural single-particle wave functions ${\phi }_{n\ell jm}\left(\vec{r}\right)$ in ${}^{34}\mathrm{Si}$. Results are displayed at the ADC(1) and ADC(3) levels.

Figure 12

Partial-wave contributions $\delta {\rho }_p^{\ell j}\left(r\right)$ to the difference between point-proton density distributions of ${}^{34}\mathrm{Si}$ computed at the ADC(3) and ADC(1) levels.

Figure 13

Charge and proton densities of ${}^{34}\mathrm{Si}$ and ${}^{36}\mathrm{S}$ at the ADC(3) level. The experimental charge density of ${}^{36}\mathrm{S}$ is taken from Ref. [61].

Figure 14

Angular dependence of the form factor obtained for 300 MeV electron scattering on ${}^{34}\mathrm{Si}$ and ${}^{36}\mathrm{S}$. Results from both ADC(2) and ADC(3) calculations are displayed.

Figure 15

Charge density distributions computed at the ADC(2) level with four different (2N+3N) interactions for ${}^{34}\mathrm{Si}$ (a) and ${}^{36}\mathrm{S}$ (b). The experimental charge density of ${}^{36}\mathrm{S}$ is also shown [61].

Figure 16

Charge density distributions of ${}^{34}\mathrm{Si}$ and ${}^{36}\mathrm{S}$ computed at the ADC(2) level with and without 3N interactions. (a) ${\mathrm{NNLO}}_{\text{sat}}$ Hamiltonian. (b) NN+3N400(1.88) Hamiltonian. The experimental charge density of ${}^{36}\mathrm{S}$ is also shown [61].

Figure 17

One-nucleon addition and removal spectral strength distribution along with associated effective single-particle energies in ${}^{34}\mathrm{Si}$. Left panel: neutrons. Right panel: protons. Dashed lines denote the Fermi energies and separate the addition and removal parts of the spectra.

Figure 18

Same as Fig. 17 for ${}^{36}\mathrm{S}$.

Figure 19

One-neutron addition energies to the lowest-lying states above the Fermi energy from ADC(3) SCGF calculations based on the ${\mathrm{NNLO}}_{\text{sat}}$ Hamiltonian. Left panel: ${}^{34}\mathrm{Si}$ (final states in ${}^{35}\mathrm{Si})$. Right panel: ${}^{36}\mathrm{S}$ (final states in ${}^{37}\mathrm{S})$. Experimental energies obtained via $(d,p)$ reactions are taken from Refs. [19, 20, 21].

Figure 20

One-proton removal energies to the lowest-lying states below the Fermi energy from ADC(3) SCGF calculations based on the ${\mathrm{NNLO}}_{\text{sat}}$ Hamiltonian. Left panel: ${}^{34}\mathrm{Si}$ (final states in ${}^{33}\mathrm{Al})$. Right panel: ${}^{36}\mathrm{S}$ (final states in ${}^{35}\mathrm{P})$. Experimental energies obtained via knock-out reactions on ${}^{34}\mathrm{Si}$ $\left({}^{36}\mathrm{S}\right)$ are taken from Ref. [24] (Refs. [22, 23]). In both cases, the larger the excitation energy of a given state, the more negative the associated one-proton removal energy.

Figure 21

(a) Reduction of the $E_{1/2^{-}}\text{−}{E}_{3/2^{-}}$ spin-orbit splitting going from ${}^{37}\mathrm{S}$ to ${}^{35}\mathrm{Si}$ against the charge depletion factor $F_{\mathrm{ch}}$ in ${}^{34}\mathrm{Si}$ for all presently available SCGF calculations. (b) Same as the upper panel but for the neutron ESPE spin-orbit splitting $e_{1p_{1/2}}\text{−}{e}_{1p_{3/2}}$ going from ${}^{36}\mathrm{S}$ to ${}^{34}\mathrm{Si}$. (c) Change of rms charge radius between ${}^{36}\mathrm{S}$ and ${}^{34}\mathrm{Si}$ against the charge depletion factor $F_{\mathrm{ch}}$ of the latter for all presently available SCGF calculations. Open symbols correspond to ADC(1), i.e., HF, calculations.