Is the structure of 42Si understood?

TL;DR

This study tests nuclear force models in 2Si by comparing experimental reaction data with shell-model predictions, revealing better agreement with the SDPF-MU interaction and insights into shape coexistence and excited states.

Contribution

It provides a detailed experimental validation of shell-model interactions in 2Si, especially distinguishing between SDPF-MU and SDPF-U-Si models.

Findings

Better agreement of data with SDPF-MU predictions

Identification of a 2Si state as the second 0+ level

Insights into shape coexistence in 2Si

Abstract

A more detailed test of the implementation of nuclear forces that drive shell evolution in the pivotal nucleus \nuc{42}{Si} -- going beyond earlier comparisons of excited-state energies -- is important. The two leading shell-model effective interactions, SDPF-MU and SDPF-U-Si, both of which reproduce the low-lying \nuc{42}{Si}($2^+_1$) energy, but whose predictions for other observables differ significantly, are interrogated by the population of states in neutron-rich \nuc{42}{Si} with a one-proton removal reaction from \nuc{43}{P} projectiles at 81~MeV/nucleon. The measured cross sections to the individual \nuc{42}{Si} final states are compared to calculations that combine eikonal reaction dynamics with these shell-model nuclear structure overlaps. The differences in the two shell-model descriptions are examined and linked to predicted low-lying excited $0^+$ states and shape coexistence. Based on the present data, which are in better agreement with the SDPF-MU calculations, the state observed at 2150(13)~keV in \nuc{42}{Si} is proposed to be the ($0^+_2$) level.

Full text

42Si, GRETINA, in-beam $\gamma$-ray spectroscopy

Modeling the nuclear landscape with predictive power, including the most exotic nuclei near the limits of nuclear existence, is an overarching goal driving 21st century nuclear science. This quest thrives through the interplay of experiment and theory, whereby observables measured for very neutron-proton asymmetric nuclei reveal isospin-dependent aspects of the nuclear force. They also identify benchmark nuclei, critical for understanding and for quantitative extrapolations toward the shortest-lived rare isotopes – many outside of the reach of laboratory studies but whose properties underpin the modeling of nucleosynthesis processes, for example. Over the few decades of rare-isotope research, certain nuclei defying textbook expectations have emerged as pivotal – they are typically located in regions of rapid structural change or at the extremes of weak binding where open quantum systems properties are exhibited. The $Z=14$ isotope 42Si28 is one such nucleus.

At present, the most neutron-rich Si isotope known to exist is 44Si, with neutron number $N=30$ Tar07 , and the most neutron-rich $N=28$ isotone with known spectroscopic information is 40Mg Cra19 . This places their even-even neighbor 42Si ($Z=14,N=28$) at the frontier of nuclear experimentation. A description of 42Si has challenged nuclear structure physics for a long time. Early on, the $\beta$-decay half-life of 42Si Gre04 and the particle stability of 43Si Not02 were interpreted as indicators that the $N=28$ magic number had broken-down, but that a pronounced $Z=14$ sub-shell closure may prevent 42Si from being well deformed Cot98 ; Cot02 ; Fri05 ; Fri06 . These speculations were resolved by the first successful spectroscopy of 42Si Bas07 , revealing a surprisingly low-lying first $2^{+}$ state, at $E(2^{+}_{1})=770(19)$ keV, the onset of collectivity, and the breakdown of the $N=28$ magic number in 42Si.

Reproducing this evolution, (a) along the Si isotopic chain, starting from doubly-magic 34Si20, with the rapid increase in collectivity or deformation at $N=28$, and (b) along the isotone line from doubly-magic 48Ca28 towards Si, has been a formidable challenge for the nuclear shell model. Two shell-model effective interactions, SDPF-U Now09 and SDPF-MU Uts12 , succeeded to reproduce a low-lying $2^{+}_{1}$ state in 42Si footnote . The mechanism underlying the collapse of the $N=28$ shell gap was attributed to: (i) the filling of the neutron $0f_{7/2}$ orbit reducing the $Z=14$ gap relative to 34Si, and, in concert, (ii) the removal of protons from the $0d_{3/2}$ orbit reducing the $N=28$ gap relative to 48Ca, both the result of the proton-neutron monopole parts of the tensor force Ots13 . $\Delta l,j=2$ quadrupole correlations, reaching across the so-narrowed $Z=14$ and $N=28$ gaps, then mutually enhance one another leading to deformation, as argued within the context of an SU(3)-like scheme Bas07 ; Now09 or a nuclear Jahn-Teller effect  Uts12 . While both shell-model interactions reproduce the low energy first-excited 42Si($2^{+}$) state, their predictions for the level density and energies of states beyond the first $2^{+}$ differ dramatically. This demands confrontation with additional experimental data to validate these different implementations of the suspected drivers of rapid shell evolution in this benchmark region Rod02 ; Gau10 ; Sor13 ; Cau14 ; Gad16a ; Ots19 , where the spectrum of the near-dripline nucleus 40Mg turned out to be surprising Cra19 .

It required half a decade and a new-generation accelerator facility for spectroscopy beyond the 42Si first excited state to be performed Tak12 . There, the 12C(44S,42Si$+\gamma$)X two-proton removal reaction was used to populate excited states in 42Si. The first $4^{+}$ state was suggested at 2173(14) keV, with the ratio $R_{4/2}=E(4^{+}_{1})/E(2^{+}_{1})$ close to the rotational limit, as one may expect for a well-deformed nucleus Tak12 . However, a direct reaction model analysis, using the SDPF-U/SDPF-U-Si and SDPF-MU shell-model two-nucleon amplitudes Tos13 , could not reconcile the $\gamma$-ray spectra and assignments reported in Tak12 , indicating that 42Si was not understood within the current shell-model picture after all; one-proton removal to 42Si was proposed to clarify the situation Tos13 .

Here, we report this first high-resolution in-beam $\gamma$-ray spectroscopy of 42Si in the direct one-proton removal reaction 9Be(43P,42Si$+\gamma$) using GRETINA gretina ; Wei17 . The measured partial removal cross sections are compared to direct reaction calculations combining eikonal dynamics and shell-model spectroscopic factors. We probe the different implementations of the drivers of shell evolution on the valence single-particle levels through the theoretical spectroscopic factors from the SDPF-MU and SDPF-U-Si shell-model calculations. The stark differences in observables (other than the $2^{+}_{1}$ energy) predicted by the two shell-model descriptions of 42Si reveal that this key nucleus is not yet sufficiently understood.

The secondary beam of 43P was produced by fragmentation of a 140 MeV/u stable 48Ca beam, delivered by the Coupled Cyclotron Facility at NSCL Gad16b , impinging on a 1363 mg/cm2 9Be production target and separated using a 150 mg/cm2 Al degrader in the A1900 fragment separator  a1900 . The momentum acceptance of the separator was set to transmit $\Delta p/p=3$%, yielding rates of typically 45 43P/second. About 20% of the secondary beam composition was 43P, with 42P and 44S as the most intense other components.

The secondary 9Be reaction target (476 mg/cm2 thick) was located at the target position of the S800 spectrograph. Reaction products were identified on an event-by-event basis in the S800 focal plane with the standard focal-plane detector systems s800 . The inclusive cross section for the one-proton knockout from 43P to 42Si was measured to be $\sigma_{inc}=3.4(2)$ mb.

The $\gamma$-ray detection system GRETINA gretina ; Wei17 , an array of 40 high-purity Ge crystals that are each 36-fold segmented, was used to detect the prompt $\gamma$ rays emitted by the reaction residues. The ten detector modules – with four crystals each – were arranged in two rings, with four modules located at 58*∘* and six at 90*∘* with respect to the beam axis. Online signal decomposition provided $\gamma$-ray interaction points ($xyz$) for event-by-event Doppler reconstruction of the photons emitted in-flight at $v/c\approx 0.4$. The information on the momentum vector of projectile-like reaction residues, as reconstructed through the spectrograph, was incorporated into the Doppler correction. Figure 1 shows the Doppler-reconstructed $\gamma$-ray spectrum for 42Si with nearest-neighbor addback included Wei17 . It is apparent that only little cross section is carried by excited states beyond the $2^{+}_{1}$ level. Nevertheless, the remarkable peak-to-background ratio allows for spectroscopy at such modest levels of statistics and, as shown in the inset of Fig. 1, weak peak structures at 1413(10), 2037(10), 2351(10), and 2743(10) keV are visible, in addition to the strong $2^{+}_{1}\rightarrow 0^{+}_{1}$ transition at 737(8) keV. The lowest three of these higher-energy $\gamma$ rays likely correspond to the 1431(11), 2032(9), and 2357(15) keV transitions reported in Tak12 .

In spite of the low statistics at high excitation energy, a coincidence analysis provides some limited guidance for the placement of the transitions in the level scheme. Figure 2(a) shows the projection of the $\gamma\gamma$ coincidence matrix and the coincidences with the $2^{+}_{1}\rightarrow 0^{+}_{1}$ transition (inset). In comparison to the $\gamma$-ray singles spectrum of Fig. 1, the projection of the coincidence matrix shows a significantly increased number of counts at $E>800$ keV relative to the 737-keV peak counts, indicating that the high-energy region bears coincidences. Due to the low statistics, no peaks are expected in the coincidence spectrum (inset) but groups of counts appear to cluster where, with more statistics, the peaks and/or Compton edges of the transitions reported here would occur. Turning the analysis around and showing the sum of cut spectra coincident with the 1.4, 2.0, 2.3 and 2.7 MeV photopeaks returns the $2^{+}_{1}\rightarrow 0^{+}_{1}$ transition at about the right intensity for all higher-lying transitions to be coincident with it. The inset shows a coincident spectrum to the broad energy region of $0.8<E<3.5$, now including, in addition to the photopeaks, also the Compton continua. The number of counts in the $2^{+}_{1}\rightarrow 0^{+}_{1}$ is increased by a factor of about three as one would expect from the peak-to-Compton ratio of GRETINA at these energies. We, therefore, tentatively propose that all of the higher-lying transitions reported here feed the first $2^{+}$ state. All the resulting excited states lie below the (rather uncertain) neutron separation energy of $S_{n}=3721(747)$ keV.

The photopeak efficiency of GRETINA was calibrated with standard sources and corrected for the Lorentz boost of the $\gamma$-ray distribution emitted by the residual nuclei moving at almost 40% of the speed of light and addback factors from GEANT simulations geant . Partial cross sections to the specific final states were determined from the efficiency-corrected $\gamma$-ray peak areas, with discrete feeding subtracted, relative to the number of incoming 43P projectiles and the number density of the target.

One-nucleon removal is a direct reaction with sensitivity to single-particle degrees of freedom. The cross sections for the population of individual states in the reaction residue depend sensitively on the overlap, and spectroscopic factor, of the projectile initial and the residue final states knock . The shape of the ground-state residue parallel momentum distribution in the one-proton removal from 44S to 43P unambiguously revealed the knockout of an $s_{1/2}$ proton, determining the ground-state spin of 43P to $1/2^{+}$ Ril08 , in agreement with shell model.

Using the one-nucleon removal reaction methodology of Ref. Gad08 and shell-model spectroscopic factors, the partial cross sections to all bound, shell-model 42Si final states were calculated. These are confronted with experiment in Fig. 3. A reduction factor $R_{s}=0.3$, appropriate for the effective proton-neutron separation energy asymmetry from 43P, $\Delta S\approx$16 MeV AME2016 ; Tos14 , is applied to the calculated cross sections. The $R_{s}$ and $\Delta S$ deduced from the measured and calculated cross sections (using SDPF-MU) are 0.33(2) and 15.6 MeV.

The measured cross-section distribution reflects the rather simple $\gamma$-ray spectrum, dominated by the $2^{+}_{1}\rightarrow 0^{+}_{1}$ transition, with weak higher-energy transitions. The majority of the cross section feeds the ground state and the $2^{+}_{1}$ level, with modest spectroscopic strength distributed between 2 and 3.5 MeV. The partial cross sections calculated with the SDPF-MU spectroscopic factors describe the measured cross section distribution well, including the values of $\sigma_{inc}$, $\sigma(0^{+}_{1})$ and $\sigma(2^{+}_{1})$ on the absolute scale and the fraction of the strength at higher excitation energy. Use of the SDPF-U-Si wavefunctions predicts a larger inclusive cross section and significantly more strength above 1.5 MeV, in particular, if the predicted $2^{+}_{8}$ state at 3.945 MeV were bound. The cross section distribution based on the SDPF-MU spectroscopy also better matches the measured distribution on a detailed level. The states predicted to be populated strongly are calculated to decay predominantly to the first $2^{+}$ state, consistent with our proposed level scheme. The larger strength at higher excitation energy, predicted using SDPF-U-Si, is not supported by the $\gamma$-ray spectrum (see Fig. 1). For example, the $3^{+}_{2}$ state at 3.034 MeV, predicted to carry significant strength, would decay with a $>90$% branch to the $3^{+}_{1}$ state with a $\sim$600 keV $\gamma$-ray transition that should be visible in the data with $\sim$60 peak counts. Similarly, if the $2^{+}_{8}$ state were bound, the measured inclusive cross section should have been 30% higher and a $\gamma$-ray transition of order five times stronger than the 2.7 MeV peak should have been observed near 3.2 MeV. We conclude that the SDPF-MU interaction provides calculations in better agreement with the data than SDPF-U-Si.

This outcome seems rooted in the vastly different 42Si level densities predicted using SDPF-U-Si and SDPF-MU. The insert to Fig. 3(b) illustrates this point through the number of states per $J^{+}$ value below 4 MeV (and also below 2.5 MeV for $0^{+}$ states). SDPF-U-Si offers five more 2+ and three more 3+ states in this energy window, some of them predicted to carry substantial spectroscopic strengths and thus proton-removal cross section.

Perhaps the most remarkable difference is the number of low-lying $0^{+}$ states generated by the two shell-model interactions, namely 4(3) and 3(1) below 4(2.5) MeV (including the ground state), from SDPF-U-Si and SDPF-MU, respectively. In fact, this abundance of low-lying $0^{+}$ states in the SDPF-U-Si calculation appears to drive the high density of low-lying 42Si levels, as compared to SDPF-MU. This is illustrated in Fig. 4 where, for the first ten calculated states for each $J^{+}$ quantum number, the predicted $B(E2)$ electric quadrupole transition strengths to all other levels are indicated by lines. Here the line thickness scales with the $B(E2)$ values. Both calculations show a pronounced yrast line, formed by the strong intraband $E2$ decays between the first states of each even-$J$ spin. For SDPF-MU, the $0^{+}_{2}$ to $0^{+}_{4}$ states are located beyond 2.5 MeV in excitation energy and are weakly connected with $E2$ transitions to the higher-lying $2^{+}$ states that occur with significant level density above 3-4 MeV. The (isomeric) excited $0^{+}_{2}$ state within SDPF-U-Si, however, appears to be the band-head of an even-$J$ band that carries collectivity comparable to the yrast band, as indicated by the similar $B(E2)$ values. The third and fourth $0^{+}$ states are then predicted to be strongly connected to higher-lying $2^{+}$ states which appear with significant level density starting at 2.5 MeV. The level structure from SDPF-U-Si is more compressed than that from the SDPF-MU calculation, leading to the markedly increased level density at low energies. The low-lying $0^{+}$ states within SDPF-U-Si seem to play a role in this, with the second $0^{+}$ state and the band structure built on top, constituting a remarkable case of predicted shape or configuration coexistence with essentially no connecting $E2$ transitions to the yrast band. Figure 4 also shows the neutron particle-hole content of the three lowest-lying $0^{+}$ states relative to the closed-shell configuration 111In this presentation, 2p-2h contains about 10% of 1p-1h and 3p-3h content.. Clearly, the wavefunctions of the $0^{+}$ states differ significantly between the two calculations. Identifying and characterizing the excited $0^{+}$ states and structures built on top of these will be a challenge for future experiments.

Since 43P has a $1/2^{+}$ ground state, only positive-parity states up to and including $J^{\pi}=3^{+}$ can be populated directly by the removal of an $sd$-shell proton (see also Fig. 3). So, if the 1413 keV $\gamma$ ray observed in this work corresponds to that reported in  Tak12 , the tentative $(4^{+}_{1})$ assignment made there for the corresponding state is thus not tenable. In the SDPF-MU picture, which is largely consistent with the present measurements, the possibility that the 1413-keV transition is due to indirect feeding is rather unlikely, since the populated $2^{+}_{2}$ and $3^{+}_{1}$ states are predicted to have only minuscule decay branches, of around 0.6% and 2%, respectively, to the $4^{+}_{1}$. From Fig. 3 it seems, rather, that the state at 2150(13) keV may indeed be the first exited $0^{+}$ state, consistent also with its cross section and excitation energy predicted by SDPF-MU calculations. We note that the assignment of $0^{+}_{2}$ for the 2150(13) keV level was also most consistent with the (SDPF-MU) two-proton removal cross section analysis presented in Tos13 . One-proton removal data from 43P with sufficient statistics to examine the shape of the parallel momentum distribution of 42Si in coincidence with the 1.4 MeV $\gamma$-ray transition would allow confirmation of this assignment if an $\ell=0$ shape was found. At least an order of magnitude more statistics would be needed. A similar analysis is also possible for two-proton removal Tos13 . This challenge may have to await future, high-statistics experiments at a new-generation facility.

In summary, high-resolution in-beam $\gamma$-ray spectroscopy with GRETINA was performed for the neutron-rich nucleus 42Si in a one-proton removal reaction from 43P projectiles. Five $\gamma$-ray transitions are reported, four of which have been observed previously. Coincidence data were used to propose a tentative level scheme, which was then utilized to extract a partial cross section distribution for the direct one-proton removal reaction. The measured partial cross sections are confronted with direct reaction calculations that combine eikonal reaction dynamics with SDPF-MU and SDPF-U-Si shell-model spectroscopic information. These two effective interactions predict markedly different low-lying level densities with the scenario painted by the SDPF-MU calculations more consistent with the new data. This underscores the difficulty in extrapolating configuration-interaction calculations towards the neutron dripline and shows that nuclear models must be tested beyond the energy of the lowest $2^{+}$ states. Our results highlight the SDPF-MU interaction as a starting point for understanding the role of weak binding for the isotone 40Mg, for which both shell-model effective interactions fail to describe the observed, rather compressed, spectrum and where continuum effects are suggested to be at play Cra19 . From the selectivity of the reaction mechanism, and in agreement with similar theoretical work on the two-proton removal reaction leading to 42Si, a level at 2150(13) keV is proposed to be the $(0^{+}_{2})$ state rather than the previously suggested $(4^{+}_{1})$ level. The differences in calculations from the two shell-model effective interactions are discussed and the special role of the low-lying $0^{+}$ states is characterized. More final-state-exclusive experimental data are needed to further interrogate 42Si and to clarify its description within the nuclear shell model. Ultimately, ab-initio-based Hamiltonians that incorporate the effects of the continuum are needed.