Quasifree ( p , 2 p ) Reactions on Oxygen Isotopes: Observation of Isospin Independence of the Reduced Single-Particle Strength

Quasifree one-proton knockout reactions have been employed in inverse kinematics for a systematic study of the structure of stable and exotic oxygen isotopes at the R 3 B = LAND setup with incident beam energies in the range of 300 – 450 MeV = u. The oxygen isotopic chain offers a large variation of separation energies that allows for a quantitative understanding of single-particle strength with changing isospin asymmetry. Quasifree knockout reactions provide a complementary approach to intermediate-energy one-nucleon removal reactions. Inclusive cross sections for quasifree knockout reactions of the type A O ð p; 2 p Þ A − 1 N have been determined and compared to calculations based on the eikonal reaction theory. The reduction factors for the single-particle strength with respect to the independent-particle model were obtained and compared to state-of-the-art ab initio predictions. The results do not show any significant dependence on proton-neutron asymmetry.


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University of Surrey, GU2 7XH Surrey, United Kingdom 33 Universität zu Köln, Institut für Kernphysik, Zülpicher Straße 77, 50937 Köln, Germany Quasifree one-proton knockout reactions have been employed in inverse kinematics for a systematic study of the structure of stable and exotic oxygen isotopes at the R 3 B=LAND setup with incident beam energies in the range of 300-450 MeV=u. The oxygen isotopic chain offers a large variation of separation energies that allows for a quantitative understanding of single-particle strength with changing isospin asymmetry. Quasifree knockout reactions provide a complementary approach to intermediate-energy onenucleon removal reactions. Inclusive cross sections for quasifree knockout reactions of the type A Oðp; 2pÞ A−1 N have been determined and compared to calculations based on the eikonal reaction theory. The reduction factors for the single-particle strength with respect to the independent-particle model were obtained and compared to state-of-the-art ab initio predictions. The results do not show any significant dependence on proton-neutron asymmetry. DOI: 10.1103/PhysRevLett.120.052501 States near the Fermi surface of closed-shell nuclei display single-particle (SP) behavior [1,2]. This fact underpins the success of the nuclear shell model (SM) [3] and motivates a simplified description of nuclei in terms of an independentparticle model (IPM), in which nucleons move freely in an average potential. Deviations from the simple IPM description have been quantified by (e, e 0 p) measurements on stable nuclei, for instance, at the NIKHEF facility, evidencing that the strength of dominant SP states, the so-called spectroscopic factor (SF), is reduced by about 30%-40% in comparison to predictions based on the IPM [4,5]. This deviation can be understood as a consequence of correlations among nucleons leading to a fragmentation of the SP strength and a partial occupation of states above the Fermi energy.
Correlations among the nucleons are taken into account in the SM, which reproduces the resulting configuration mixing and SP strength distribution close to the Fermi surface reasonably well. Still, an overall reduction of SFs compared to the SM has been reported, which is usually quantified by a reduction factor R, defined as the ratio of the experimental cross section to theoretical predictions (based on either the IPM or SM). These remaining deviations are often attributed to correlations beyond those taken into account in the SM such as short-range correlations (SRC), including those induced by the short-range tensor interaction [6][7][8]. We note that signatures of SRC in momentum distributions [9] and strong proton-neutron correlations [10,11] have been observed in high-energy electron scattering.
The first systematic studies on SFs for unstable isotopes have been undertaken by evaluating one-nucleon removal cross sections at intermediate energies close to 100 MeV=u [12] [One-nucleon removal encompasses any process producing an A-1 nucleus in the final state including different reaction mechanisms such as individual nucleonnucleon collisions or inelastic excitation and decay. Still, this process is sometimes referred to as (heavy-ion induced) knockout in the literature.]. A recent compilation of the existing data by Tostevin and Gade [13] reports reduction factors relative to the SM description for a large number of isotopes. While the residual interactions in SM calculations can account for the spread of the SP strength near the Fermi surface, the data of Ref. [13] suggest a very strong dependance of SFs on the isospin asymmetry of nuclei, quantified by the difference between one-proton and oneneutron separation energies AEðS p − S n Þ. In contrast, more recent results from transfer reactions at lower beam energies suggest a constant quenching of SFs and do not indicate such a pronounced isospin dependance [14][15][16]. Ab initio calculations, such as the self-consistent Green's function (SCGF) [17,18] or coupled-cluster theory [19], suggest indeed a reduction of SFs due to correlations but with a weak asymmetry dependance.
The isospin dependance is still heavily debated and it is unsettled whether this is an indication of correlation effects missing in SM calculations [20] or deficiencies in the reaction model, which is based on the sudden and eikonal approximations [21]. In particular, an asymmetric momentum distribution with a very large tail towards low momenta was observed in Ref. [21] after removing a tightly bound nucleon, indicating strong deviations from the approximations made. An additional potential issue lies in the fact that the sensitivity of the one-nucleon removal reaction induced by light composite nuclear targets, e.g., Be or C, at intermediate beam energies of around 100 MeV=u is concentrated strongly at the nuclear surface [22,23], probing only the outer part of the projectile wave function, which limits the access to deeply bound states.
In this Letter, we introduce a complementary experimental approach based on quasifree scattering (QFS) reactions in inverse and complete kinematics using a proton target bombarded by a high-energy beam of radioactive and stable nuclei. The oxygen isotopic chain provides thereby a large selection of nuclei with different nucleon separation energies that are suitable for a systematic study of the asymmetry dependance of the SP strength.
The usage of proton targets increases the sensitivity to deeply bound states, which in turn allows for a more complete investigation of the SP wave function [24]. Since the nucleon-nucleon (NN) total cross section has a minimum at around 300 MeV, final-state interactions, such as rescattering and absorption effects, are minimized at beam energies of around 400 MeV=u, where the energies of the outgoing nucleons amount to 200 MeVon average. At these energies, the picture of a localized reaction is supported, which can be described as an elementary QFS process between the struck nucleon and the target proton, where both nucleons are scattered at large angles centered around 45° [25]. Below 100 MeV, the NN cross section rises steeply and causes a strong distortion of the outgoing nucleon wave functions; i.e., the nucleus becomes opaque and the reaction thus probes only the surface at lower beam energies.
The theoretical description of QFS used here is based on the eikonal reaction model where the effect of multiple scattering is treated by use of the distorted wave impulse approximation with a complex optical potential [24]. The internal momentum of the knocked-out nucleon is related directly to the recoil momentum of the residual fragment, which is measured experimentally, and can be interpreted in terms of the angular momentum of the corresponding SP state.
The experiment was performed at the R 3 B=LAND setup at GSI Helmholtzzentrum für Schwerionenforschung in Darmstadt, Germany. A primary 40 Ar beam was accelerated up to 500 MeV=u and directed onto a Be target. The heavy reaction fragments were selected in the fragment separator FRS according to their magnetic rigidity [26] and transported to the experimental hall. The secondary beam was delivered as a cocktail beam containing different isotopes around a certain nominal rigidity. The incoming ions were identified on an event-by-event basis. The solid reaction targets were located at the center of the Crystal Ball detector array (CB) [27] and surrounded by double-sided silicon strip detectors (DSSSD) [28] for energy-loss and position measurements. The CB covers a solid angle of close to 4π and was used for the detection of γ rays and high-energy nucleons from the knockout reactions. The heavy reaction products were deflected by the dipole magnet ALADIN and charges and masses were reconstructed by several tracking detectors. A detailed description of the setup can be found in Refs. [25,[29][30][31][32]. The experiment was performed with CH 2 (458 and 922 mg=cm 2 ) and C (558 and 935 mg=cm 2 ) targets as well as with an empty target frame. The C target was used to estimate and subtract C-induced reactions in the CH 2 target, while measurements without target were made to estimate background contributions.
The angular correlations of the knocked-out projectile nucleon and the recoiled target proton shown in Fig. 1 for the reaction 16 Oðp; 2pÞ 15 N exhibit the characteristics of QFS indicating a nearly coplanar back-to-back scattering. Slight modifications compared to free NN scattering are caused by the binding energy and the internal motion of the nucleons in the nucleus [25]. A coincident measurement of the knocked-out and recoiled nucleons as well as of the residual fragment allows an unambiguous and practically background-free reconstruction of QFS channels.
It is emphasized that all reaction channels were selected requiring the simultaneous detection of two protons and a bound residual N fragment (A-1) in the final state. The inclusive cross sections thus contain the population of the ground and bound excited states of the fragment. In order to extract the exclusive cross sections for the population of excited states below the particle threshold, the measurement of γ rays in coincidence has been analyzed for all reaction channels. In the following paragraphs, the reaction 16 Oðp; 2pÞ 15 N will be presented in detail and the results of the other reaction channels will be summarized later. Additional results including γ spectra and momentum distributions for the other isotopes will be presented together with a more detailed description of the analysis procedure in a forthcoming article.
The measured cross sections were subject to various corrections such as the 2p detection efficiency, which was crucial since its uncertainty dominates the systematic uncertainty of the deduced cross sections. This efficiency has been obtained from simulations of (p, 2p) events according to the QFS kinematics at the various beam energies listed in Table I. The simulation of the experiment was performed within the R3BRoot framework [33,34] based on the GEANT4 toolkit [35] and using different physics models [36][37][38] for the treatment of reactions in the detector material. The observed 6% variation of the deduced detection efficiency of 63% with the different model inputs was treated as a systemic uncertainty. For the reaction 16 Oðp; 2pÞ 15 N, for instance, an inclusive cross section of 26.8(9)[1.7] mb was deduced, where the systematic uncertainty is given in square brackets (see Table I). This cross section includes proton knockout from the 0p 1=2 orbit to the ground state (g. s.) of 15 N and from the 0p 3=2 PHYSICAL REVIEW LETTERS 120, 052501 (2018) orbit to bound excited states (see discussion below). The removal of a proton from the 0s 1=2 orbit can only populate unbound states of 15 N and is thus not considered. Figure 2 shows the projection of the transverse momentum distribution of 15 N on the y axis (symbols). Since this includes proton knockout from the 0p 1=2 and 0p 3=2 orbits, it is compared to the sum of the theoretical distributions for both orbits. The theoretical cross sections were calculated with the eikonal theory of Ref. [24] and amount to 13.2 and 25.1 mb assuming knockout from completely filled 0p 1=2 and 0p 3=2 orbits, respectively. The reduction factor R amounts to R ¼ 0.70ð5Þ and agrees well with the result R ¼ 0.65ð5Þ from (e, e 0 p) data [5]. The dash-dotted curve in Fig. 2 shows the distribution of the total spectrum (solid) scaled by R. The scaled distribution describes the experimental data well, confirming our assumption that the data is dominated by proton knockout from orbits of l ¼ 1.
Exclusive cross sections were extracted from a fit to the coincident γ spectrum as shown in Fig. 3 for the 16 Oðp; 2pÞ 15 N reaction. Besides the simulated two transitions from the excited 3=2 − states at 6.63 and 9.93 MeV, a background contribution arising from (p, 2p) reactions without γ-ray emission was included in the fit. The population of the g. s. was obtained by subtracting the contribution of the excited states from the total cross section resulting in SF values of 1.60(39), 2.01(23), and 0.58 (13) for populating the g. s. and the 3=2 − states at 6.63 and 9.93 MeV, respectively. Note that the measured SF for the 1=2 − g: s: amounts to 80% of the IPM, while the 0p 3=2 strength adds up to 65%, whereas the SCGF calculation discussed below predicts 78% and 80%, respectively. However, theory does not reproduce the observed TABLE I. Measured and calculated (p, 2p) cross sections for the reactions given in the first column. The second and third columns give neutron and proton separation energies of the residual A−1 N, respectively [39,40]. In the fourth column, the mean beam energy in the middle of the CH 2 target is given. In the fifth column, inclusive cross sections for all bound states are listed along with statistical (round brackets) and systematic uncertainties (square brackets). The predictions from eikonal theory (sixth column) are shown for the knockout of 0p 1=2 protons except for 16 O, where the sum of 0p 1=2 and 0p 3=2 contributions is given. The last column gives the resulting reduction factor R relative to the IPM with its total uncertainty.   fragmentation of 3=2 − strength, which is collected in one single state. The experimental SF values for the states discussed above are consistent with the results from (e, e 0 p) data [41,42]. The measured inclusive cross sections for proton knockout are listed in Table I. Since only bound states of the residual A−1 N are detected, the results fluctuate with changes of the separation energies along the isotopic chain as a consequence of the very different nucleon separation energies of the daughter nuclei. 16 Oðp; 2pÞ 15 N has the largest cross section since both knockout from 0p 1=2 and 0p 3=2 populate bound states in 15 N. For the 15 Oðp; 2pÞ 14 N and 18 Oðp; 2pÞ 17 N reactions, the 0p 1=2 protons contribute fully, but only part of the (fragmented) 0p 3=2 strength is below the continuum threshold. The case is similar for the 22 O projectile, albeit with a larger contribution of the 0p 3=2 proton strength due to the relatively large neutron separation energy of 4.59 MeV of the daughter nucleus 21 N [39]. The case of 13 Oðp; 2pÞ 12 N is at the other extreme, since the knockout from the 0p 1=2 orbit contributes only partially to the cross section due to the very weakly bound protons in 12 N (S p ¼ 0.6 MeV [39]). The rest of the reaction channels can be safely considered as arising from the full 0p 1=2 proton knockout alone. Table I also gives the corresponding theoretical cross sections, assuming the IPM occupation.
For the discussion of the reduction factor R, we concentrate on the aforementioned isotopes, where it is reasonable to assume that the full 0p 1=2 strength is collected in bound states, while the 0p 3=2 strength is exclusively located in the continuum. We also include the one exception for 16 O, where also the 0p 3=2 hole states are bound. We exclude cases where the 0p 3=2 strength is located close to the particle separation threshold and is fragmented. Such a selection is possible since the structure of the produced nuclei is known and, in addition, the γ spectra of the final states were analyzed. For the selected cases, we can then compare the measured cross sections directly to the theoretical ones based on the IPM without the need for additional theoretical structure input, which would complicate the discussion on the asymmetry dependence.
The resulting R values are summarized in the last column of Table I and are displayed in Fig. 4 as a function of the difference of g. s. separation energies ðS p − S n Þ as filled circles and as a square for 16 O, where the sum of 0p 1=2 and 0p 3=2 contributions is shown as discussed above. The error bars represent the statistical uncertainty while the horizontal square brackets indicate the total uncertainty including the systematic errors. This allows a direct comparison of R relative to each other without identical systematic uncertainties. The data from this work show a fluctuation of R around 0.66. The solid and dotted lines display fits with a linear function and with a constant value resulting in a reduced χ 2 of 1.29 and 1.91, respectively. We conclude that the data are consistent with weak or even no dependance of the SP strength on the neutron-proton asymmetry. This trend differs drastically from the result of one-nucleon removal reactions at intermediate energies as compiled in Ref. [13]. Note that R is the ratio of the experimental cross section to the theoretical one based on the IPM, while the R values of Ref. [13] are given relative to a particular SM calculation. For the cases selected here, however, the fragmentation is small and the sum of the SM SF values reflects the sum-rule value given by the IPM. We estimated the uncertainties of the calculated cross sections related to possible variations of the input parameters within a reasonable range (NN cross sections, densities, and SP wave functions) to be less than 5%, i.e., significantly smaller than the experimental uncertainties. Our conclusion agrees with Ref. [16], where transfer data on 14 O have been analyzed. We note that our deduced reduction factor of 0.68 (7) is in very good agreement with the one of 0.73(10)(10), derived from the 14 Oðd; 3 HeÞ transfer [16]. Furthermore, we have performed state-of-the-art ab initio calculations of the proton-hole strength in 14;16;22 O based on the SCGF theory, using the third-order algebraic diagramatic construction approach [ADC(3)] [18,43]. This is the method of choice for calculating the nuclear spectral function and yields the most accurate SF results near subshell closures. The theoretical SF can be sensitive to particle-hole gaps and the density of states at the Fermi surface [44]. Hence, we based our calculations on the saturating chiral interaction NNLO-sat [45], which guarantees the best possible predictions of radii and gaps in this region of the nuclear chart [46]. The resulting SF values shown as blue triangles in Fig. 4 for proton removal to the ground states of 13 N and 21 N and for summed p-shell states in 15 N are in reasonable agreement with the present measurements, although they seem to overestimate the 3=2 − strength in 15 N, where theory does not reproduce the correct fragmentation as explained above. These results are also compatible with earlier microscopic studies [47] as well as (e, e 0 p) data [5]. As was seen for other nuclear interactions [17,18], the SF from NNLO-sat depend little on isospin asymmetry. Note that continuum effects can further affect the quenching of SP strength in 22 O but not to the extent of altering this trend [19]. Thus, ab initio results do not support a significant dependence on isospin asymmetry, in agreement with the experimental results presented in this Letter.
In summary, we have measured inclusive (p, 2p) cross sections for stable and unstable oxygen isotopes using the quasifree scattering technique in inverse kinematics and extracted the single-particle reduction factor R from the comparison with eikonal theory. The reduction obtained from the reaction 16 Oðp; 2pÞ 15 N shows good agreement with the results obtained from (e, e 0 p) measurements. The results for stable and exotic nuclei indicate a weak or even no dependence on the proton-neutron asymmetry. This finding is compatible with the ab initio Green's function and coupled cluster calculations but contradicts the trend derived from intermediate-energy one-nucleon removal cross section measurements. This disagreement calls for further investigations of the reaction mechanism of nucleon removal from deeply bound states at intermediate energies.
In the future, quasifree knockout reactions in inverse kinematics will allow for a systematic investigation of proton and neutron knockout from exotic nuclei covering a wide range of neutron-to-proton asymmetry, which will be important to corroborate the observed trend and to improve our understanding on the evolution of the single-particle structure as a function of neutron-to-proton asymmetry.