X.-X Sun

J G Li

M Kimura

N Michel

K Ogata

C X Yuan

Z H Yang
email: zhyang@ribf.riken.jp

Y Kubota

A Corsi

K Yoshida

Q Yuan

G Authelet

H Baba

C Caesar

D Calvet

A Delbart

M Dozono

J Feng

F Flavigny

J.-M Gheller

J Gibelin

A Giganon

A Gillibert

K Hasegawa

T Isobe

Y Kanaya

S Kawakami

D Kim

Y Kiyokawa

M Kobayashi

N Kobayashi

T Kobayashi

Y Kondo

Z Korkulu

S Koyama

V Lapoux

Y Maeda

F M Marqués

T Motobayashi

T Miyazaki

T Nakamura

N Nakatsuka

Y Nishio

A Obertelli

A Ohkura

N A Orr

S Ota

H Otsu

T Ozaki

V Panin

S Paschalis

E C Pollacco

S Reichert

J.-Y Roussé

A T Saito

S Sakaguchi

M Sako

C Santamaria

M Sasano

H Sato

M Shikata

Y Shimizu

Y Shindo

L Stuhl

T Sumikama

Y L Sun

M Tabata

Y Togano

J Tsubota

F R Xu

J Yasuda

K Yoneda

J Zenihiro

S.-G Zhou

W Zuo

T Uesaka

Quasifree Neutron Knockout Reaction Reveals

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A kinematically complete quasi-free (p, pn) experiment in inverse kinematics was performed to study the structure of the Borromean nucleus 17 B, which had long been considered to have neutron halo. By analyzing the momentum distributions and exclusive cross sections, we obtained the spectroscopic factors for 1s 1/2 and 0d 5/2 orbitals, and a surprisingly small percentage of 9(2)% was determined for 1s 1/2 . Our finding of such a small 1s 1/2 component and the halo features reported in prior experiments can be explained by the deformed relativistic Hartree-Bogoliubov theory in continuum, revealing a definite but not dominant neutron halo in 17 B. The present work gives the smallest s-or p-orbital component among known nuclei exhibiting halo features, and implies that the dominant occupation of s or p orbitals is not a prerequisite for the occurrence of neutron halo.

Along an isotopic chain, with increasing neutron number the nuclei gradually lose binding as the dripline (the limit of nuclear existence) is approached [1,2]. When occupying the s or p orbital, the least bound neutrons of near-dripline nuclei can tunnel far out into the "classically forbidden" region, and a novel phenomenon-the neutron halo-occurs [3][4][5][6][7][8][9][10]. It is a prime example of the emergent simplicity in nuclear many-body systems on top of the complexity of interactions among the constituent nucleons [11][12][13][14].

In the context of halo, of particular interest are nuclei with a 2n-halo structure, which generally exhibit a arXiv:2102.03699v1 [nucl-ex] 7 Feb 2021

Borromean character without any bound binary subsystems [4][5][6][7][8][9]. Recently, 2n-halo structure was reported in 22 C and 29 F from the large matter radius [15][16][17] and in 19 B from the enhanced electric dipole strength measurements [18]. It has been of great interest to search for new 2n-halo systems [8] and new halo features such as the core-halo shape decoupling [19][20][21] and the Efimov state [22,23]. Meanwhile, it is also important to have comprehensive investigation of the structure of a variety of nuclei with both well-and less-developed halos, particularly using transfer or knockout reactions, to reach a detailed understanding of 2n halo [6][7][8][9]. Such measurements have so far only been made for light systems with a well-developed 2n halo, 6 He [24][25][26], 11 Li [27][28][29][30][31][32][33][34], 14 Be [34][35][36][37][38]. In all these cases, the valence neutrons dominantly occupy p and s orbitals. This naturally raises the question whether the dominant occupation of s or p orbitals is universal for 2n-halo nuclei and should thus be a criterion to identify 2n-halo systems [6,7]. On the other hand, recent theoretical calculations show that a slight s-wave tail should be sufficient for the occurrence of 2n halo in very weakly bound neutron-rich systems [39].

In this Letter, we report the observation of a surprisingly small s-orbital component in the Borromean nucleus 17 B, which has long been considered as a 2nhalo system [8]. Halo features in 17 B have already been reported-the large matter radius [40,41], the narrow momentum distribution of 15 B [42], and the thick neutron surface [43], but the s-orbital percentage has hitherto not been directly measured. Using a 15 B+n+n 3body model with 15 B being an inert and spherical core, a large s-orbital percentage was deduced from the matter radius and the 15 B momentum distribution-36(19)% [41], 69(20)% [42], 50(10)% [44], 53(21)% [45]. However, Estradé et al. found its neutron skin thickness did not fit in with such a 3-body picture bearing out a dominant sorbital component [43]. The large s-orbital percentage in the neighboring isotopes-14 B (64% ∼ 89%) [46][47][48], 15 B (∼63%) [47], 18 B [49], and 19 B (∼35%) [18]-also makes the s-orbital percentage in 17 B an intriguing question.

In the present work, we have achieved the first direct measurement of the s-orbital percentage in 17 B using the quasi-free (p, pn) reaction in inverse kinematics. This study concerns a kinematically complete measurement, which was made possible by combining the high-intensity beams provided by the Radioactive Isotope Beam Factory of RIKEN Nishina Center and the state-of-the-art detector instruments including the vertex-tracking liquid hydrogen target MINOS [50], in-beam γ-ray spectrometer DALI2 [51], and the SAMURAI spectrometer [52][53][54].

Experiment.-Secondary 17 B beams (∼1.4×10 4 pps, ∼277 MeV/nucleon) were produced from the fragmentation of 48 Ca at 345 MeV/nucleon and prepared using the BigRIPS fragment separator [55,56]. They were then tracked onto the 150 mm-thick MINOS target [50] using two multi-wire drift chambers. At the target region, we placed a γ-ray detector array constructed with 68 NaI crystals of DALI2 [51], a recoil-proton spectrometer composted of a multi-wire drift chamber and a plastic scintillator array, and the recoil-neutron detector array WINDS [57]. The charged fragments and decay neutrons were detected by SAMURAI [52,53] and NEBULA [54]. The relative energy E rel of the unbound nucleus 16 B was reconstructed from the momenta of 15 B and the decay neutron, with a resolution (FWHM) of ∼0. 45 √ E rel (in MeV). When 15 B is in an excited state, the energy of 16 B (E d ) with respect to the 15 B(g.s.) + n threshold can be obtained as the sum of E rel and the excitation energy of 15 B. Population of excited 15 B fragments was reported in a prior breakup experiment of 17 B [58]. Details of the setup can be found in [32,33,38].

Quasi-free (p, pn).-We first confirmed the quasi-free (p, pn) process by checking the kinematical correlation between recoil protons and recoil neutrons [38,59,60]. The correlation between the polar angles, θ p and θ n , agrees nicely with the kinematical simulation (Fig. 1(a)).

The kinematically complete measurement allows us to reconstruct the momentum of the recoil neutron from momentum conservation without detecting it, which largely enhances the statistics. The corresponding angular correlation is presented in Fig. 1(b). The correlation locus gets broadened relative to panel (a), but follows well the expected correlation pattern of quasi-free (p, pn).

Gamma-coincidence.-The Doppler-shift-corrected γray spectrum in coincidence with 16 B (E rel ≤ 5 MeV) is shown in Fig. 1(c). It is well fitted using the response functions of two known γ rays, 1327 keV and 1407 keV from 15 B excited states [58,61,62], and a twoexponential background (χ 2 /ndf =1.4). For each γ ray, the response function was obtained from Geant4 simulations considering the realistic setup and the resolution of each crystal. The inset shows the γ-gated E rel spectrum after correcting for the γ efficiency (∼12%), in comparison to the inclusive one. Obviously, the core-excited component is small (within ∼5%), and thus neglected when fitting the E rel spectrum (see below). 16 B states from γ-coincident analysis [START_REF]of the γcoincident analysis, the distorted-wave impulse approximation (DWIA) calculation, and the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) calculation[END_REF] are presented in Table I.

E rel spectrum of 16 B.-For 17 B with N =12, the knocked-out neutron should mainly come from 1s 1/2 and 0d 5/2 orbitals. The small contribution from p-wave orbitals was confirmed by checking the angular correlation [28]. Ignoring higher-lying 0d 3/2 is justified by the theoretical calculations we employed (see below).

We first checked the E rel spectrum gated by the momentum (P ) of the knocked-out neutron to disentangle states populated by 1s 1/2 and 0d 5/2 knockout. The E rel spectra gated by 0 MeV/c ≤ P ≤ 60 MeV/c (selective for 1s 1/2 ) and 60 MeV/c ≤ P ≤ 160 MeV/c (selective for 0d 5/2 ) are shown in Fig. 2(a), together with the inclusive one for comparison. All the spectra are normalized according to the peak at ∼0, since it is 3 -associated purely with 0d 5/2 (see below). Its d-wave character has also been established in [START_REF] Lecouey | [END_REF]. To enhance the visibility, in Fig. 2(b) we presented the ratio of the P -gated spectrum to the inclusive one. Obviously, the prominent peak at ∼1 MeV in Fig. 2(a) is mainly from 0d 5/2 knockout. Meanwhile, the red histogram in Fig. 2(b) clearly shows two 1s 1/2associated states, one at ∼0.2 MeV and another broad bump at 1∼4 MeV. Hence, the E rel spectrum was fitted using four resonances, after taking into account the experimental acceptance and resolutions. The states at ∼0.04 MeV and ∼1 MeV were described with d-wave Breit-Wigner line shapes and the other two with s-wave line shapes [65]. Since the intrinsic width of the ∼0.04-MeV state is much smaller than the experimental resolution, it was fixed to 1 keV in the fitting, and an upper limit of ∼20 keV was estimated. The results are presented in Fig. 2(c) and Table I. Errors of the resonance parameters include both statistical and systematic errors, the latter being dominated by the effect of the fitting range. The 0.046(3)-MeV state agrees with prior reports [49,[START_REF] Lecouey | [END_REF]66], but the resonant energy is refined; meanwhile, the inset of Fig. 1(c) clearly shows that it is associated with 15 B(g.s.). The 2.38(8)-MeV state observed in the γ-coincident analysis also agrees well with the reported state at 2.40 (7) MeV [66].

Results and Discussions.-In Fig. 3, we compare the level scheme of 16 B to theoretical calculations: (i) Shell model (SM) calculation with YSOX interactions, consid- ering the reduction of sd-shell n-n interactions by a factor of 0.75 [67][68][69][70][71]. (ii) Valence-space in-medium similarity renormalization group (VS-IMSRG) calculation [72] using the optimized chiral effective field theory interaction at next-to-next-to-leading order [73] in Hartree-Fock basis with 15 major harmonic-oscillator shells ( ω = 24 MeV). (iii) Antisymmetrized molecular dynamics plus generator coordinate method (AMD) calculation using the Gogny D1S interaction [43,74]. (iv) Gamow shell model (GSM) calculation with a 10 He core [75][76][77]. All partial waves up to l = 3 were included for the valence neutrons, while p 3/2 and p 1/2 basis states were included for the well-bound valence protons. For the two-body force we adopted the Minnesota force [78], and the onebody force was modeled with a Woods-Saxon potential with the potential parameters adjusted to reproduce the separation energies and low-lying states of 14 B and 15

B.

Given that the spin-parity (J π ) of 17 B(g.s) is 3/2 -, J π of the two 1s 1/2 -associated states (0.183 MeV, 2.8 MeV) should be either 1 -or 2 -, while the other two non-1s 1/2 states should be more likely 3 -or 4 -. Tentative assignments of J π can thus be made, as shown in Fig. 3. Though predicted as the ground state by SM, AMD, and VS-IMSRG, 0 -should be safely excluded for the 0.046 MeV state, because of the small spectroscopic factors (<0.03) predicted by these models. In order to analyze the momentum distribution of the knocked-out neutron and extract the 0d 5/2 and 1s 1/2 spectroscopic factors in 17 B, we carried out the distortedwave impulse approximation (DWIA) calculation [79]. This model has recently been applied in several (p, pn) and (p, 2p) experiments [32,33,[80][81][82][83][84][85][86]. The singleparticle wave function and the nuclear density were obtained using the Bohr-Mottelson potential [START_REF] Bohr | Nuclear Structure[END_REF]. The optical potentials for the distorted waves in the initial and final channels were constructed with the microscopic folding model [START_REF] Toyokawa | [END_REF] using the Melbourne g-matrix interaction [89] with the spin-orbit component disregarded. For the p-n interaction, the Franey-Love effective interaction [90] was adopted. DWIA calculations were performed separately for all reaction channels listed in Table I, taking into account E d of 16 B.

Figure 4 shows the transverse momentum (P x ) distributions, together with DWIA calculated curves for 1s 1/2 and 0d 5/2 knockout after folding in the experimental resolution (FWHM) of 45 MeV/c. For the 0.046-MeV state (0 MeV ≤ E rel ≤ 0.1 MeV) and the 1.08-MeV state (0.9

MeV ≤ E rel ≤ 1.3 MeV), the data can be well reproduced by a pure 0d 5/2 component, in line with the above J π assignment of 3 -and 4 -.

For the 0.183-MeV state (0.25 MeV ≤ E rel ≤ 0.45 MeV) and the 2.8-MeV state (2.5 MeV ≤ E rel ≤ 3.5 MeV), we performed the minimum-χ 2 fitting using a combination of 1s 1/2 and 0d 5/2 . A 1s 1/2 fraction of 14(4)% and 28(3)% was obtained, respectively. The errors include both the statistical and systematic errors, the latter being dominated by DWIA and the E rel gate for the data. For the 2.8-MeV state, the same 1s 1/2 fraction is obtained when gating on the left (29(2)%) or right half (28(2)%) of the E rel peak in the analysis, providing further evidence that it is a singlet rather than a doublet.

For γ-coincident 16 B states, the P x distributions are all in agreement with knockout of 0d 5/2 neutrons from 17 B [63]. This naturally leads to a J π of 3 -or 4 -, as shown in Fig. 3. The contribution of the 14 B+2n channel is very small (∼5 %) and thus not included in Table I, and the P x distribution is consistent with 0d 5/2 knockout.

We then deduced the exclusive cross sections (σ exp ), as tabulated in Table I. For the 0.183-MeV and 2.8-MeV states, the 1s 1/2 fraction determined above has been used. The experimental spectroscopic factor (S exp ) is obtained by dividing σ exp with the theoretical cross section for a unit spectroscopic factor (σ th ) from DWIA. Both σ exp and σ th are the integrated cross sections over the detector coverage (35 • < θ p < 55 • ). We have incorporated the experimental conditions into DWIA to facilitate the direct comparison of σ exp and σ th [START_REF]of the γcoincident analysis, the distorted-wave impulse approximation (DWIA) calculation, and the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) calculation[END_REF]. The errors quoted are the combined statistical and systematic errors. For the 0.046-MeV and 1.08-MeV states, the systematic error of σ exp is dominated by the correction of detector efficiencies (9%). For the 0.183-MeV and 2.8-MeV states, the uncertainty of the 1s 1/2 fraction has also been considered. For S exp , the uncertainty on σ th (within 10%) has further been included, estimated by varying the potential parameters and E d of 16 B in the DWIA calculation. The total spectroscopic factors for 1s 1/2 (S s = 0.24(4)) and 0d 5/2 (S d = 2.53 (21)) are obtained by summing up the S exp for 1s 1/2 and 0d 5/2 respectively in Table I, and a 1s 1/2 spectroscopic factor percentage of 9(2)% is thus deduced from the ratio of S s and S s + S d . As shown in Fig. 3, some 16 B states are not observed in the current quasi-free (p, pn) experiment, indicating that these states should have very small spectroscopic factors in 17 B; their effect on S s and the 1s 1/2 spectroscopic factor percentage is negligible compared to the errors quoted above. Note that the 1.08-MeV state could be a doublet of two closely located states, given its relatively large width. But such possibility will not affect our conclusion-a small 1s 1/2 component in 17 B, since the P x distribution in Fig. 4(b) clearly shows it is populated (almost) purely by 0d 5/2 knockout. Following the conventional picture for "2nhalo nuclei" [6], one may expect a 15 B+2n structure for 17 B [41,42,44,45], and the 1s 1/2 spectroscopic factor percentage of 9(2)% thus leads to a percentage of only 9(2)% for the halo-relevant (1s 1/2 ) 2 configuration. As discussed by Estradé et al. [43], 17 B may be better described in a 13 B+4n model, and our result would then lead to a percentage of ∼18% for the halo-relevant (1s 1/2 ) 2 (0d 5/2 ) 2 configuration, which is also small. However, prior experiments indeed consistently point to the formation of halo in 17 B [40][41][42][43][44], which seems to suggest a predominant s orbital component (∼50%) [40-42, 44, 45] in accordance with the conventional picture of "2n-halo nuclei"-an inert core plus two spatially decoupled valence neutrons [6]. To understand this seeming discrepancy, we compared our result to theoretical pre-dictions obtained by summing up the 1s 1/2 and 0d 5/2 spectroscopic factors of states shown in Fig. 3. The 1s 1/2 percentage is largely overestimated by SM (25%), VS-IMSRG (26%), and GSM (45%). For SM and GSM, we have also checked that the result is not sensitive to the one-body Hamiltonian of 1s 1/2 . This may suggest significant impact of deformation or other many-body effects, which can not be sufficiently considered within the shell-model framework. A small 1s 1/2 percentage of 5% is provided by AMD, which is based on the nucleonic degrees of freedom and also predicts a large prolate deformation in 17 B (β = 0.4). But the 1s 1/2 percentage is significantly underestimated by AMD, mainly due to the use of Gaussian wave functions [91].

We then resort to the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) [19,20,[92][93][94][95], which self-consistently considers weak binding, deformation, and pairing-induced continuum coupling. We used the effective interaction PK1 [96] and a densitydependent zero-range pairing force [20]. The neutron and matter radius [43], deformation [62,97], and S 2n [98] of 17 B are well reproduced [START_REF]of the γcoincident analysis, the distorted-wave impulse approximation (DWIA) calculation, and the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) calculation[END_REF]. DRHBc provides a small 1s 1/2 orbital percentage of 14% for the valence neutrons, close to the experimental result of 9(2)%. Meanwhile, the neutron density distribution shows a slight but definite low-density tail extending into large radial distances [START_REF]of the γcoincident analysis, the distorted-wave impulse approximation (DWIA) calculation, and the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) calculation[END_REF], indicating a weak halo component in 17 B. Hence, the neutron halo exists in 17 B as reported in prior experiments [40][41][42][43][44], but not as the dominant structure component.

Summary.-We have measured the 0d 5/2 and 1s 1/2 spectroscopic factors in 17 B using the quasi-free (p, pn) reaction in inverse kinematics. A small spectroscopic factor percentage of 9(2)% was determined for the 1s 1/2 orbital. Our result thus reveals a surprisingly small sorbital component in 17 B whether it is described in a simple 15 B+2n model or more properly in a 13 B+4n model. Our finding of such a small 1s 1/2 component and the halo features in 17 B reported in prior experiments [40][41][42][43][44] can be well explained by DRHBc, revealing a definite but not dominant halo component in 17 B. The present work gives the smallest s-or p-orbital component among known nuclei exhibiting halo features, and implies that the dominant occupation of s or p orbitals is not a prerequisite for the occurrence of neutron halo. In weakly bound neutron-rich nuclei, as long as s or p orbitals around the Fermi surface are occupied by the least bound neutrons with an appreciable strength, the halo naturally occurs and coexists with other non-halo configurations [4,[6][7][8][9]. The halo component, whether or not dominant, results in a distinctive diffused surface, and thus manifests itself in reactions sensitive to the surface properties [6][7][8][9].

We would like to thank the RIBF accelerator staff for the primary beam delivery and the BigRIPS team for their efforts in preparing the secondary beams. Z. H. Y. acknowledges fruitful discussions with C.L. Bai, J. Casal, Y. Sato, T.T. Sun, and S.M. Wang, and the support

FIG. 1 .

 1 FIG. 1. Angular correlation between recoil protons and neutrons for events with recoil neutrons detected by WINDS (a) and reconstructed from momentum conservation (b). The black line indicates the kinematical simulation assuming quasi-free (p, pn) off 17 B. (c) Doppler-corrected γ-ray spectrum, fitted using two γ rays (1327 keV and 1407 keV) and a two-exponential background. The inset presents the γ-gated E rel spectrum, together with the inclusive one for comparison.

  FIG. 2. (a) 16 B E rel spectra gated by 0 MeV/c ≤ P ≤ 60 MeV/c (red) and 60 MeV/c ≤ P ≤ 160 MeV/c (blue) in comparison to the inclusive one (black). (b) Ratios of the momentum-gated spectrum to the inclusive one. The grey dashed line, with a constant value of 1, stands for the scenario that the entire spectrum is associated with knockout of 0d 5/2 neutrons. (c) The fitting with a sum of four resonances. The inset is a zoom-in view of the 0-1 MeV region.

FIG. 3 .

 3 FIG. 3. Observed 16 B states compared to Gamow shell model (GSM), valence-space in-medium similarity renormalization group (VS-IMSRG), antisymmetrized molecular dynamics (AMD), and shell model (SM) calculations. The 1nseparation energies are shown in the parentheses (in MeV).

FIG. 4 .

 4 FIG.4. Transverse momentum (Px) distributions for different16 B states. The vertical error bars stand for the statistical error, while the horizontal for the bin size. In (a) and (b), DWIA calculated curves for knockout of 1s 1/2 (red-dashed) and 0d 5/2 (black-dotted) neutrons are normalized to the peak of the experimental spectrum; in (c) and (d), the blue-solid curves represent the fitting with a combination of 1s 1/2 (reddashed) and 0d 5/2 (black-dotted). All DWIA curves have been convoluted with the experimental resolution.

TABLE I .

 I Summary of16 B states and experimental spectroscopic factors (Sexp). The experimental (σexp) and theoretical single-particle cross sections (σ th ) are the integrated value over the detector coverage. Sexp is defined as Sexp = σexp/σ th .

	Ex( 15 B) [MeV]	E rel [MeV]	Γr [MeV]	E d [MeV] a	J π	n orbital	σexp [mb]	σ th [mb]	Sexp
	0	0.046(3)	< 0.02	0.046(3)	3 -	0d 5/2	0.0639(6)	0.199	0.32(4)
	0	0.183(6)	0.44(7)	0.183(6)	2 -	0d 5/2 1s 1/2	0.041(5) 0.007(2)	0.196 0.329	0.21(3) 0.02(1)
	0	1.08(6)	0.5(2)	1.08(6)	4 -	0d 5/2	0.18(2)	0.181	0.97(14)
	0	2.8(1)	1.8(3)	2.8(1)	1 -/ 2 -	0d 5/2 1s 1/2	0.14(2) 0.054(9)	0.160 0.243	0.87(14) 0.22(4)
	1.33	1.05(8)	0.3(2)	2.38(8)	(3 -)	0d 5/2	0.009(3)	0.164	0.06(2)
	1.33 2.73	2.4(2) 0.9(3)	0.4(3) 0.5(2)	b 3.7(2)	(3 -/ 4 -)	0d 5/2	0.011(4)	0.154	0.07(3)
	2.73	2.1(1)	< 0.5	4.8(1)	(3 -/ 4 -)	0d 5/2	0.003(2)	0.144	0.02(1)
						1s 1/2 spectroscopic factor percentage	9(2) %

a E d =E rel +Ex( 15 B). b Taken as the weighted average of the results of the two decay channels.

from the Foreign Postdoctoral Researcher program of RIKEN. This work has been supported by the European Research Council through the ERC Grant No. MINOS-25856; the JSPS KAKENHI Grants No. JP16K05352, No. JP18H05404, and No. JP16H02179; the National Natural Science Foundation of China under Grants No. 11525524, No. 11835001, No. 11921006, No. 11975282, No. 11435014, No. 11775316, and No. 11961141004; the Strategic Priority Research Program of Chinese Academy of Sciences, Grant No. XDB34000000; the Institute for Basic Science (IBS-R031-D1). The computation of this work was partly supported by the HPC Cluster of ITP-CAS and the Supercomputing Center, CNIC of CAS.


A Revel

O Sorlin

F M Marqués

Y Kondo

J Kahlbow

T Nakamura

N A Orr

F Nowacki

J A Tostevin

C X Yuan

N L Achouri

H Al Falou

L Atar

T Aumann

H Baba

K Boretzky

C Caesar

D Calvet

H Chae

N Chiga

A Corsi

H L Crawford

F Delaunay

A Delbart

Q Deshayes

Z Dombrádi

C A Douma

Z Elekes

P Fallon

I Gašparić

J.-M Gheller

J Gibelin

A Gillibert

M N Harakeh

W He

A Hirayama

C R Hoffman

M Holl

A Horvat

Á Horváth

J W Hwang

T Isobe

N Kalantar-Nayestanaki

S Kawase

S Kim

K Kisamori

T Kobayashi

D Körper

S Koyama

I Kuti

V Lapoux

S Lindberg

S Masuoka

J Mayer

K Miki

T Murakami

M Najafi

K Nakano

N Nakatsuka

T Nilsson

A Obertelli

F De

Oliveira Santos

H Otsu

T Ozaki

V Panin

S Paschalis

D Rossi

A T Saito

T Saito

M Sasano

H Sato

Y Satou

H Scheit

F Schindler

P Schrock

M Shikata

Y Shimizu

H Simon

D Sohler

L Stuhl

S Takeuchi

M Tanaka

M Thoennessen

H Törnqvist

Y Togano

T Tomai

J Tscheuschner

J Tsubota

T Uesaka

Z Yang

M Yasuda

K Yoneda

Extending the Southern Shore of the Island of Inversion to 28 F

Detailed spectroscopy of the neutron-unbound nucleus 28 F has been performed for the first time following proton/neutron removal from 29 Ne/ 29 F beams at energies around 230 MeV/nucleon. The invariant-mass spectra were reconstructed for both the 27 F ( * ) + n and 26 F ( * ) + 2n coincidences and revealed a series of well-defined resonances. A near-threshold state was observed in both reactions and is identified as the 28 F ground state, with Sn( 28 F) = -199(6) keV, while analysis of the 2n decay channel allowed a considerably improved Sn( 27 F) = 1620(60) keV to be deduced. Comparison with shell-model predictions and eikonal-model reaction calculations have allowed spin-parity assignments to be proposed for some of the lower-lying levels of 28 F. Importantly, in the case of the ground state, the reconstructed 27 F+n momentum distribution following neutron removal from 29 F indicates that

it arises mainly from the 1p 3/2 neutron intruder configuration. This demonstrates that the island of inversion around N = 20 includes 28 F, and most probably 29 F, and suggests that 28 O is not doubly magic.

PACS numbers:

Introduction.-The study of nuclei located at the neutron dripline, beyond which they are no longer bound with respect to neutron emission, has become possible due to significant technological developments in highintensity neutron-rich beams and high-efficiency detection arrays [1]. Despite these advances, the neutron dripline is only accessible experimentally for light nuclei (Z 10) [2], and even in this region it remains a theoretical challenge to predict it [3]. Models incorporating the effect of three-nucleon forces [4][5][6] have led to a better reproduction of the dripline. However, the effect of the continuum, which can drastically change the shell structure [7,8], is not taken into account except for lighter nuclei [9]. The comparison between the isotopic chains of carbon, nitrogen and oxygen on the one hand, and of fluorine on the other, is particularly interesting: the dripline is located at N = 16 for the former, while the fluorine chain extends to N = 22 ( 31 F [2]). The reason for this, however, is not fully understood.

In the fluorine chain, the odd neutron-number 28,30 F isotopes are unbound, as they lack the extra binding energy provided by pairing. Christian et al. [10] found that 28 F is unbound by 220 (50) keV, and based on the agreement with the predictions of USDA/USDB shell-model calculations 28 F was placed outside the "Island of Inversion" (IoI) [11]. This means that the ground state of 28 F could be described by a particle-hole configuration (π0d 5/2 × ν0d -1 3/2 ) with respect to an unbound core of 28 O, forming a multiplet of J π = 1 + -4 + states as in 26 F [12,13]. On the other hand, the relatively low energies of the first excited states in 27,29 F suggest the presence of intruder neutron pf -shell contributions [14]. If 28 F contains such contributions, negative-parity states, like the J π = 1 --4 -multiplet resulting from the π0d 5/2 ⊗ν1p 3/2 coupling, should appear at low energy.

The location of the dripline in fluorine at N = 22 suggests a profound change in shell structure around doublymagic 24 O [15][16][17]. A direct experimental signature of these structural changes can be found in the evolution of the energies of the 3/2 + , 7/2 -and 3/2 -states, arising from the neutron 0d 3/2 , 0f 7/2 and 1p 3/2 orbits, in the N = 17 isotones from Z = 14 to 10 (Fig. 3 of [18]). In 31 14 Si, the spacing between the ground 3/2 + and the 7/2 - states, which is linked to the size of the N = 20 gap, is 3.2 MeV, and the 3/2 -state lies 0.5 MeV above the 7/2 -. In 27 10 Ne, the 3/2 + -7/2 -gap is reduced to 2 MeV, and the 3/2 -level moves below the 7/2 -state, at only 0.8 MeV above the 3/2 + g.s. [19][20][21]. In 29 Ne, the 3/2 -intruder state becomes the ground state [22]. This migration of levels has been suggested to be due to the hierarchy of the p-n forces present above 24 O [18], and in particular to the central and tensor components [23][24][25].

This Letter reports on the first detailed spectroscopic study of 28 F, which has been carried out using proton and neutron removal from high-energy 29 Ne and 29 F beams, respectively. In the former reaction, the 29 Ne neutron configuration will remain unchanged and negative-parity states are expected to be populated at low energy in 28 F through the removal of a 0d 5/2 proton. Neutron removal, however, can lead to both positive-and negative-parity levels in 28 F depending on the degree to which intruder (2p2h and beyond) configurations are present in 29 F. This study was possible owing to the high luminosity provided by a thick liquid H 2 target and the relatively intense secondary beams, coupled with state-of-the-art arrays for the detection of the high-energy neutrons and charged fragments and of the de-excitation γ-rays. The results indicate that 28 F, and most probably 29 F, lie within the IoI, and also suggest that 28 O is not doubly magic.

Experimental setup.-The experiment was performed at the Radioactive Isotope Beam Factory (RIBF) of the RIKEN Nishina Center. Secondary beams of 29 Ne (∼ 8.1 kHz, 228 MeV/nucleon) and 29 F (∼ 90 Hz, 235 MeV/nucleon) were produced by fragmentation of a 345 MeV/nucleon 48 Ca beam (∼ 500 pnA) on a 15mmthick Be target, and prepared using the BigRIPS fragment separator [26,27]. Secondary beam particles were identified via their energy loss and time of flight as measured using thin plastic scintillators, and tracked on to the object point of the SAMURAI spectrometer [28] using two sets of multi-wire drift chambers, where the MINOS target [29] was located. The latter consisted of a 15cm-thick liquid-hydrogen cell surrounded by a timeprojection chamber, that allowed the reconstruction of the reaction vertex with a precision of 3 mm (sigma) in the beam direction using the intersection between the trajectory of the incoming beam and the measured track(s) of the outgoing proton(s) for the (p, pn) and (p, 2p) reactions. The DALI2 NaI array [30] surrounded the target for the detection of the in-flight de-excitation of fragments (with an efficiency of ε γ ∼ 15% at 1 MeV).

The beam-velocity reaction products were detected in the forward direction using the SAMURAI setup, including the NEBULA [31] and NeuLAND demonstrator [32] neutron arrays, placed respectively some 14 and 11 m downstream of the target. The SAMURAI superconducting dipole magnet [33], with a central field of 2.9 T and a vacuum chamber equipped with thin exit windows [34], provided for the momentum analysis of the charged fragments. Their trajectories and magnetic rigidity were de- Right: same for the 26 F+2n system populated from (e) 29 Ne(-1p) and (f) 29 F(-1n). The fit in red corresponds to a sum of resonances (blue, with the resonance energy in keV) plus a non-resonant distribution (dashed black). Center: same as left, obtained in coincidence with the 915 keV excited state of 27 F (after background subtraction) populated from (c) 29 Ne(-1p) and (d) 29 F(-1n). The energy axis E on the top right is given with respect to Sn( 28 F), and orange dots mark resonances in coincidence with the corresponding fragment γ-rays (see Fig. 2).

termined using drift chambers at the entrance and exit of the magnet [28]. This information, combined with the energy loss and time of flight measured using a 24-element plastic scintillator hodoscope, provided the identification of the projectile-like fragments. The neutron momenta were derived from the time of flight, with respect to a thin plastic start detector positioned just upstream of the target, and the hit position as measured using the NEBULA and NeuLAND arrays [35], with efficiencies of ε n ∼ 50% and ε nn ∼ 10% for decay energies of 0-3 MeV. Energy spectra.-The relative energy (E rel ) of the unbound 28 F system was reconstructed from the momenta of the 26,27 F fragments and neutron(s) [35]. The 27 F+n spectra for both reactions are shown on the left of Fig. 1. The resolution is considerably improved compared to previous studies of neutron-unbound systems [10,36], owing to the high-granularity NeuLAND array as well as the MINOS target. The resolution of E rel ( 27 F+n) varied as fwhm ∼ 0.18 E 0.63 rel MeV. In order to deduce the character of resonances in 28 F, the spectra were described using single-level R-matrix line-shapes [37], which were used as the input for a complete simulation of the setup (including the beam characteristics, the reaction, and the detector resolutions and acceptances), combined with a nonresonant component obtained from event-mixing [38,39] and from the simulation of independent fragments and neutrons, respectively for the two-and three-body spectra. The results of the fit are listed on the figure and summarized in Ref. [40].

The energy spectra of Fig. 1(a,b), from the 29 Ne(-1p) and 29 F(-1n) reactions, exhibit a lowest-lying resonance with a width of Γ = 180(40) keV at respectively 204( 16) and 198 (6) keV above threshold, without any coincident γ-ray. The weighted mean, 199(6) keV, provides therefore a determination of the g.s. energy of 28 F (-S n ). This is compatible with the less precise value of 220(50) keV from Ref. [10] using the 29 Ne(-1p) reaction. As shown in Fig. 1(a), we observe a second peak in the (-1p) channel at 363 (17) keV, which is in coincidence with the 915 (12) keV transition (inset of Fig. 1c) from the decay of the excited state of 27 F [14]. As such, the resonance lies at the sum energy of 1278 (21) keV above threshold. As this value matches the energy of the fourth peak at 1280 (30) keV, we propose that the 1280 keV state, populated in 29 Ne(-1p), decays both to the ground and firstexcited states of 27 F, with corresponding branching ratios of 60% and 40%. The 2810 keV resonance is also observed in coincidence with the 915 keV γ-ray. It is thus placed at an energy of 3725 keV (Fig. 2). Three other resonances identified in Fig. 1(a) at 940, 1840 and 3660 keV are also placed in Fig. 2. The spectrum of Fig. 1(b), obtained from 29 F(-1n), displays three clear resonances, including the g.s. (see above). The resonance at 996 (13) keV does not fully match the 940 (20) keV observed in the (-1p) reaction. We thus propose that they correspond to two different states, as shown in Fig. 2. Given the uncertainties, the 1880(80) and 1840 (30) keV resonances observed in both reactions can correspond to the same state. If we require a coincidence with the 915 keV γ-ray of 27 F, one can see in Fig. 1(d) the two resonances at 406 (28) and 3180(260) keV plus an additional structure at 1200(80) keV, corresponding therefore respectively to levels at 1321, 4095 and 2115 keV (Fig. 2). Note that the 406(28) keV resonance overlaps with that at 363 (17) keV, which was proposed to decay in competition with the 1280 (30) keV transition in the (-1p) channel with similar intensities. However, as the fit of the (-1n) data does not allow the placement of a 1280 (30) keV resonance with the required intensity, we propose that the 363 (17) and 406 (28) keV transitions come from the decay of different states located respectively at 1280 and 1321 keV. Finally a resonance is placed at 3980 keV.

Resonances in 28 F decaying by 2n emission have been identified after applying cross-talk rejection conditions to the 26 F+2n events [41]. As can be seen in Fig. 1(e,f), the lowest-lying peak produced in both the (-1p) and (-1n) reactions has compatible energies of respectively E rel = 245(32) and 227(88) keV. The states observed in the 2n decay correspond to excitation energies of E rel +S n ( 27 F), when referenced to the 28 F g.s., or to an excitation energy of E rel + S 2n ( 28 F). According to AME2016 [42], the uncertainty on S n ( 27 F) = 1270(410) keV is large, which also influences the present determination of S 2n ( 28 F), making the placement of the resonances very uncertain.

However, we first note that the two low-energy resonances are, as for the 1840 and 1880 keV resonances in the 27 F+n decay, produced in both reactions. Second, they have compatible intrinsic widths [40], independent of the decay mode. Third, the ratios between the 245 and 1840 keV resonances in (-1p), and the 227 and 1880 keV resonances in (-1n), are the same (∼ 10%). This suggests that they all originate from a single state at ∼ 1860 keV, that decays both by 1n and 2n emission. Excellent agreement between the 1n and 2n decay spectra is obtained using S n ( 27 F) = 1620(60) keV and S 2n ( 28 F) = 1420(60) keV, the latter being deduced from the present determination of S n ( 28 F). A summary of all the levels identified is reported in Fig. 2.

Momentum distributions.-In the (-1n) reaction, the reconstructed momentum distribution of the 27 F+n system allows the orbital angular momentum of the removed neutron to be deduced [START_REF] Hansen | Direct Reactions With Exotic Nuclei[END_REF]. The transverse-momentum distribution corresponding to the feeding of the 28 F g.s. is fitted in Fig. 3(a) with eikonal-model calculations [45,46] using a combination of ℓ = 1, 3 components. This choice of negative-parity ℓ values is guided by the fact that the g.s. is also produced in the 29 Ne(-1p) reaction, which, as discussed earlier, is expected to lead to negative-parity states at low E rel . The fit, which gives a spectroscopic factor of C 2 S = 0.40 (6), is dominated by the ℓ = 1 component (79%), meaning that the g.s. of 28 F is mainly composed of an intruder p-wave component.

The momentum distribution of the resonance at 406 keV, Fig. 3(b), is obtained after gating on the 915 keV γ-ray transition. It is well reproduced by a pure ℓ = 2

4 - 3 + 2 - 2 + 4 + 1 - 1 + 3 - 2 + 4 + 1 + 2 - 6 - 3 + 5 - 1 - 0 - 6 - 3 - 4 - 0 + 5 - 0 - 0 + 4 - 2 - 3 + 1 - 2 + 3 - 4 + 2 - 6 - 1 + 1 - 1 + 2 + 0 + 2 - 4 - 3 - 3 + 4 + 2 +
29 Ne(-1p) component, meaning that the parity of the 1321 keV state is positive, with C 2 S = 0.012 (4). In order to account for its highly favored 1n decay through the (1/2 + ) excited state of 27 F, rather than to the (5/2 + ) g.s. despite the higher energy available, we propose that it has J π = 1 + . Indeed, this would result in an ℓ = 0 neutron decay to the excited state, as opposed to an ℓ = 2 decay to the ground state. Other (higher) spin values would not account for such a unique behavior. For the resonance at 996 keV, the momentum distribution, Fig. 3(c), is very well reproduced by an admixture of ℓ = 2 (72%) and ℓ = 0 (28%), making it another candidate for a positive-parity state, with C 2 S = 0.30 (4).

As for the (-1p) reaction, the four most populated states, with energies 204, 940, 1280 and 1840 keV, all display momentum distributions compatible with ℓ = 2 proton removal from the d 5/2 orbital, with C 2 S of respectively 0.20(3), 0.46 (7), 0.50(8) and 0.22(4), summing up to about 1.4, as compared to the maximal expected occupancy of 2 for the d 5/2 orbital in 29 Ne.

Shell-model calculations.-These have been performed using the sdpf-u-mix interaction [54] in order to predict the energy, J π (Fig. 2, right) and C 2 S values of negativeand positive-parity states in 28 F. In order to assess the sensitivity to the level scheme, the sdpf-mu interaction [55] has also been used. The sdpf-u-mix interaction has been refined in order to reproduce the observed 3/2 - and 7/2 -level crossing and location of the pf intruder orbitals in 27 Ne, 29 Mg and 31 Si, and the dripline at 31 F.

Both calculations predict about 15 negative-and positive-parity states below 2 MeV, demonstrating that the normal and intruder configurations in 28 F are very close in energy. The first 10 states have relatively pure configurations (60-80%) mostly originating from the proton 0d 5/2 and neutron 0d 3/2 and 1p 3/2 orbits, with the exception of the 5 -and 6 -levels that arise from a neutron in the 0f 7/2 orbit. The π0d 5/2 ⊗ ν1p 3/2 and π0d 5/2 ⊗ ν0d 3/2 couplings lead to a multiplet of J = 1-4 states with negative and positive parity, respectively.

The calculations predict that four negative-parity states J π = (4, 2, 1, 3) -are mainly populated in the (-1p) reaction with dominant ℓ = 1 components and C 2 S values of 0.75, 0.44, 0.35 and 0.19, in rather good agreement with experiment. We thus think we have populated this multiplet of states. Among them, a J π = 4 - g.s. is predicted by both calculations, with Γ of about 180 keV, in agreement with experiment. Using similar arguments, the 940 keV state is proposed to be J π = 2 -. The 1 -level is predicted to decay both to the ground (5/2 + ) and first-excited (1/2 + ) states of 27 F with ℓ = 1, and could correspond to the state identified at 1280 keV.

As it has the highest energy in both calculations, the 1840-keV resonance is tentatively assigned as J π = 3 -.

In the (-1n) reaction, the 4 -g.s. is calculated to be the most populated among other negative-parity states with C 2 S = 0.36, coming mostly (90%) from an ℓ = 1 removal, to be compared with C 2 S = 0.40 (6), with 79% of ℓ = 1 fraction. As for the positive-parity states, produced only in the (-1n) reaction, both the sdpf-u-mix interaction predicts the lowest state as J π = 3 + with C 2 S = 0.54, in reasonable agreement with the 996 keV state with C 2 S = 0.30(4). The 1 + state is predicted to decay principally to the first excited state of 27 F with ℓ = 0, making the 1321 keV state a good J π = 1 + candidate. The calculated C 2 S value of the 1 + state, 0.31, is however much larger than experiment.

The first positive-parity states are predicted too low in energy, which could be explained by effects of the continuum (not taken into account explicitly in the present calculations) that change the effective two-body matrix elements [13,56] and induce lingering of the ℓ = 1 states compared to ℓ = 2 [8]. Another feature that could be related to the effects of the continuum, discussed in Ref. [7] as an apparent reduction of pairing, is the damping of the |S n (N ) -S n (N + 1)| amplitude when approaching the dripline. While these amplitudes are correctly reproduced in lighter (N 16) fluorine isotopes by the present calculations, our experimental S n ( 27 F) -S n ( 28 F) value of 1.82 (6) MeV is significantly smaller than the predicted 2.8 MeV.

Conclusions.-In summary, detailed spectroscopy of 28 F has been undertaken using nucleon removal from secondary beams of 29 F and 29 Ne, with statistics orders of magnitude higher than the previous study and unprecedented energy resolution. This was made possible through the unique combination of a thick liquid target and state-of-the-art arrays for the detection of highenergy neutrons and charged fragments, as well as deexcitation γ-rays. They proved essential to cope with the high density of states in 28 F and allowed the identification of the 1n and 2n decay modes, including transitions to bound excited states of 26,27 F. In addition to making comparisons with shell-model calculations, the 28 F transverse-momentum distributions following neutron removal, combined with eikonal-model calculations, allowed the ℓ configuration of the removed neutron to be deduced.

The 28 F g.s. resonance was unambiguously identified, with S n ( 28 F) = -199 (6) keV. It has a negative parity with an ℓ = 1 content of about 80%, which places 28 F inside the IoI. Based on the comparison to shell-model calculations of the decay patterns, resonance widths and C 2 S values, we propose that the multiplet of J π = (1-4) -states originating from the π0d 5/2 ⊗ ν1p 3/2 configuration has been identified. The first positive-parity resonance (3 + ) is proposed at 996 keV, about 560 keV higher than shell-model predictions. A candidate for a J π = 1 + resonance is proposed at 1321 keV. As opposed to 26 F, that has well-identified positive-parity states from p-n configurations above a doubly-magic 24 O core, 28 F displays mixed negative-and positive-parity states, with the negative-parity states being more bound. These features strongly suggest that N = 20 magicity is not restored at 28 O. Moreover, the single-neutron removal, including the strong ℓ = 1 feeding of the negative-parity 28 F g.s., supports the suggestion, based on mass measurements, that 29 F also lies within the IoI [57].

Finally, we propose a very precise value of S n ( 27 F) = 1620(60) keV, as compared to the tabulated value of 1270(410) keV, which combined with S n ( 28 F) = -199(6) keV leads to a reduced oscillation in the S n values of about 35% at the dripline, as compared to shell-model calculations. This damping in the oscillations has also been recently observed in the boron isotopic chain [39], suggesting that a reduced pairing force may be a generic feature of dripline nuclei.

29 FIG. 1 :

 291 FIG.1: Left: relative-energy spectra of the27 F+n system populated from the reactions (a)29 Ne(-1p) and (b) 29 F(-1n). Right: same for the26 F+2n system populated from (e)29 Ne(-1p) and (f) 29 F(-1n). The fit in red corresponds to a sum of resonances (blue, with the resonance energy in keV) plus a non-resonant distribution (dashed black). Center: same as left, obtained in coincidence with the 915 keV excited state of 27 F (after background subtraction) populated from (c)29 Ne(-1p) and (d) 29 F(-1n). The energy axis E on the top right is given with respect to Sn( 28 F), and orange dots mark resonances in coincidence with the corresponding fragment γ-rays (see Fig.2).

FIG. 3 :

 3 FIG.3: Experimental transverse-momentum distributions of the27 F+n system following the 29 F(-1n) reaction compared to eikonal-model calculations for: (a) removal of a neutron with ℓ = 1, 3 (respectively blue, brown lines) for the g.s. at E rel = 198 keV; (b) a pure ℓ = 2 distribution for the resonance at 406 keV (corresponding to the state at 1321 keV); (c) a mixture of ℓ = 0, 2 (respectively purple, green lines) for the state at E rel = 996 keV. In (a,c) the red line is the total fit.

We thank M. Ploszajczak for fruitful discussions, and the accelerator staff of the RIKEN Nishina Center for their efforts in delivering the intense 48 Ca beam. N.L.A., F.D., J.G., F.M.M. and N.A.O. acknowledge partial support from the Franco-Japanese LIA-International Associated Laboratory for Nuclear Structure Problems as well as the French ANR-14-CE33-0022-02 EXPAND. J.A.T. acknowledges support from the Science and Technology Facilities Council (U.K.) grant No. ST/L005743/1. I.G. was supported by HIC for FAIR and Croatian Science Foundation under projects No. 1257 and 7194. Z.D., Z.E. and D.S. were supported by projects No. GINOP-2.3.3-15-2016-00034 and K128947, and I.K. by project No. PD 124717. J.K. acknowledges support from RIKEN as short-term International Program Associate. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract No. DE-AC02-06CH11357 (ANL). This project was funded in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project-ID 279384907, SFB 1245, and the GSI-TU Darmstadt cooperation agreement.


Li Y Kubota

A Corsi

G Authelet

H Baba

C Caesar

D Calvet

A Delbart

M Dozono

J Feng

F Flavigny

J.-M Gheller

J Gibelin

A Giganon

A Gillibert

K Hasegawa

T Isobe

Y Kanaya

S Kawakami

D Kim

Y Kikuchi

Y Kiyokawa

M Kobayashi

N Kobayashi

T Kobayashi

Y Kondo

Z Korkulu

S Koyama

V Lapoux

Y Maeda

F M Marqués

T Motobayashi

T Miyazaki

T Nakamura

N Nakatsuka

Y Nishio

A Obertelli

§ K Ogata

A Ohkura

N A Orr

S Ota

H Otsu

T Ozaki

V Panin

S Paschalis

E C Pollacco

S Reichert

J.-Y Roussé

A T Saito

S Sakaguchi

M Sako

C Santamaria

M Sasano

H Sato

M Shikata

Y Shimizu

Y Shindo

L Stuhl

T Sumikama

Y L Sun

M Tabata

Y Togano

J Tsubota

Z H Yang

J Yasuda

K Yoneda

J Zenihiro

T Uesaka

Surface localization of the dineutron in 11

The formation of a dineutron in the nucleus 11 Li is found to be localized to the surface region.

The experiment measured the intrinsic momentum of the struck neutron in 11 Li via the (p, pn) knockout reaction at 246 MeV/nucleon. The correlation angle between the two neutrons is, for the first time, measured as a function of the intrinsic neutron momentum. A comparison with reaction calculations reveals the localization of the dineutron at r ∼ 3.6 fm. The results also support the density dependence of dineutron formation as deduced from Hartree-Fock-Bogoliubov calculations for nuclear matter.

The structures of fermionic many-body systems are often characterized by two-particle correlations [1,2]. In nuclei, Bardeen-Cooper-Schrieffer-(BCS)-like pairings between two nucleons are known to stabilize nuclei similar to the behavior in superconducting materials [3].

On the other hand, a completely different type of two-neutron correlation, dineutron correlation, has attracted much attention in nuclear physics and other relevant fields [4]. In the 1970's, A. B. Migdal postulated the dineutron as a spatially compact two-neutron pair [5].

Behind the formation of a dineutron, quantum interference between odd and even parity orbits plays a crucial role [6]. The appearance of the dineutron correlation is presumed to be a key ingredient to elucidate the stabilities and exotic structures of nuclei near and beyond the neutron drip-line nuclei [7].

The dineutron correlation in a finite nuclear system may provide insight into dilute neutron-rich matter found in the inner crust of neutron stars. Recently, Hartree-Fock-Bogoliubov calculations predicted that the correlation length of two neutron pairs drastically changes with the matter density. Consequently, the dineutron correlation should appear below the saturation density ρ 0 of 10 -4 ρ/ρ 0 0.5 [8]. Hence, laboratories studies on finite nuclei should elucidate information about the existence and properties of dineutrons.

Studies on the formation and the density dependence of 11 Li as well as 6 He [9][10][11] are crucial because 11 Li has a peculiar structure, which is known as the halo. That is, the matter density gradually varies from the saturated core to the very low-density tail where only valence neutrons exist. If the spatial distribution of the dineutron can be experimentally determined, the density-dependent properties of dineutron may be revealed.

Many studies have examined the dineutron correlation in 11 Li using the transfer [12] or the Coulomb breakup reactions [13][14][15]. One pioneering study measured the electric dipole (E1) response of 11 Li using the Coulomb breakup reaction [16][17][18]. The E1 cluster sum rule value is directly related to the opening angle θ 12 of the two valence neutrons with respect to the core [4]. In the absence of a correlation, the θ 12 value is 90 • and it becomes smaller as the dineutron correlation becomes stronger. Hence, θ 12 , which can be determined by the E1 cluster sum rule value, provides a good measure of the dineutron correlation strength.

A latest result from RIKEN [18] showed that the E1 strength for 11 Li integrated over the relative energy region of E rel ≤ 3 MeV is consistent with θ 12 = 48 +14 -18 degrees, indicating that 11 Li has a strong dineutron correlation. Recent theoretical works have reevaluated the opening angle from the same data, and extracted larger values by including the excitation of the 9 Li core [19], the distance between the 9 Li core and a halo neutron [20], and the effect of the final state interaction (FSI) on the B(E1) distribution [21].

In addition, the dineutron correlation has been investigated by measuring the charge radius [22,23]. 11 Li has a larger charge radius than the 9 Li, indicating the two halo neutrons in 11 Li are distributed on one side forming the compact dineutron and not distributed symmetrically.

The above two methods average dineutron information over the whole volume of the nucleus. Therefore, neither the location nor the density dependence of the dineutron formation can be determined. However, the spatial distribution of the dineutron can be investigated by the neutron knock-out reaction. The missing momentum, which is the momentum of the knock-out neutron in the 11 Li nucleus, can be determined experimentally assuming the impulse approximation. Then the spatial distribution of the dineutron can be interpreted through the Fourier transformation.

The neutron knock-out reaction was pioneered in experiments at GSI Helmholtz Centre for Heavy Ion Research (GSI) at a high energy (300 MeV/nucleon) with a carbon target [24,25]. The correlation angle between the valence neutrons θ nf is defined for momentum space in the so-called Y-type Jacobi coordinates [START_REF] Betounes | Differential Equations: Theory and Applications[END_REF]. Schematic diagram of θ nf , which is comparable with 180 • -θ 12 , is shown later in the inset of Fig. 4. Thus, the spatially compact dineutron should show an angle greater than 90 • . Indeed, the GSI experiment exhibited an enhancement at θ nf > 90 • , providing evidence for the existence of the dineutron in 11 Li. Another study constructed an analytical wave function to reproduce these data, and from the wave function, the averaged opening angle θ 12 was deduced to be 61.7 • for the J = 0 pair [START_REF] Shulgina | [END_REF]. This result is consistent with the reevaluated results [20,28,29] of the E1 measurement [18]. Although the GSI experiment also showed that the slope of the cos θ nf distribution depends on the relative energy of 10 Li ( 9 Li-n) [25], the dependence was not discussed in detail.

Here, we report the kinematically complete measurement of 11 Li(p, pn) 10 Li in inverse kinematics with an unprecedentedly high statistics at 246 MeV/nucleon. The nucleon target allows the whole volume of 11 Li to be probed with the least effect of the absorption [30]. In addition, the flexibility in kinematics can realize the quasi-free condition where the knockout neutron is free from the FSI from 10 Li. Thus, the initial momentum of the neutron before the knock-out can be derived from the measured momenta of the recoil proton and knock-out neutron [31]. This measurement was made possible due to the unique combination of a highly-intense 11 Li beam at the RI Beam Factory (RIBF) at RIKEN and a thick liquid hydrogen target MINOS device [32,33].

The experiment was performed at RIBF, which is operated by RIKEN Nishina Center and the Center for Nuclear Study, The University of Tokyo. A 11 Li secondary beam was produced as a cocktail beam, which included 14 Be and 17 B, through a projectile fragmentation reaction from a 48 Ca beam at 345 MeV/nucleon bombarding 9 Be as the primary target.

The averaged beam intensity on the target was 400 pnA. The secondary beam was selected and purified using the BigRIPS fragment separator [34]. The average 11 Li beam energy was 246 MeV/nucleon and the typical intensity was 1 × 10 5 particles per second with a purity of 70%. 11 Li beam particles were identified on an event-by-event basis.

Figure 1 shows the main elements of the experimental setup around the secondary target.

The same setup was used in Ref. [35]. To achieve a high luminosity without degrading the resolution, the MINOS device, which was composed of a 15-cm-thick liquid hydrogen target coupled to a cylindrical time projection chamber (TPC) to determine the reaction vertex [36], was installed. The reaction vertex in the target was determined with an uncertainty of 6 mm at the full width at half maximum (FWHM) using a combination of the trajectory information of the incident 11 Li beam and the recoil proton.

After a (p, pn) reaction, the target proton (recoil proton) and one of the valence neutrons of 11 Li (knock-out neutron) were scattered at large polar angles centered at approximately 45 • . Then a recoil particle detector (RPD) was installed on the right-hand side of the beam line to detect the recoil proton scattered to θ p = 30-65 • in the laboratory frame. A large momentum transfer of > 1.5 fm -1 , which is much larger than the typical two-neutron momentum in 11 Li of 0.1 fm -1 [12], was selected to ensure a clean quasi-free knock-out reaction condition. The RPD was composed of a multi-wire drift chamber and a plastic scintillator hodoscope, which measured the scattering angle and the time of flight, respectively. The momentum vector of the recoil proton at the reaction vertex was reconstructed by considering the energy loss. The neutron detector array WINDS [37] was installed on the left-hand side of the beam line (-60 • < θ n < -25 • ). WINDS measured the scattering angle and the time of flight of the knock-out neutron. An unbound reaction residue 10 Li was emitted in the very forward direction but immediately decayed into the heavy fragment 9 Li and another neutron. These momenta were analyzed by a SAMURAI spectrometer [START_REF] Kobayashi | {XVIth} International Conference on ElectroMagnetic Isotope Separators and Techniques Related to their Applications[END_REF][START_REF] Shimizu | {XVIth} International Conference on ElectroMagnetic Isotope Separators and Techniques Related to their Applications[END_REF] and a neutron detector array NEBULA [START_REF] Nakamura | proceedings of the {XVIIth} International Conference on Electromagnetic Isotope Separators and Related Topics (EMIS2015)[END_REF], respectively.

The reaction channel of 11 Li(p, pn) 10 Li * → 9 Li+n was identified by detecting all particles in the final state. The missing momentum k, which is the initial momentum vector of the knock-out neutron in the beam rest frame, is derived as

k := k n1 = k ′ n1 + k ′ p -k p , (1)
where k i and k ′ i represent the momentum vectors of particle i [n1: knock-out neutron, p: target (recoil) proton] in the initial and final states, respectively. Following Refs. [24,25], the correlation angle θ nf is defined as

cos θ nf = K ′ • k |K ′ ||k| , (2)
K ′ = k ′ n2 -k ′ f , (3)
where k ′ n2 and k ′ f represent the momentum vectors of the decay neutron and the heavy fragment 9 Li, respectively.

The relative energy E rel was obtained by subtracting the sum of the decay neutron and the heavy fragment 9 Li masses from the invariant mass of 10 Li.

The acceptance and efficiency of the setup were evaluated by Geant4 [START_REF] Agostinelli | [END_REF]42] simulations.

The overall acceptance is 0.6%, which is limited mainly by the azimuthal angle coverage of RPD and WINDS. The average resolutions on the missing momentum k and the correlation angle θ nf are 0.17 fm -1 (FWHM) and 12 • (FWHM), respectively.

Figure 2 shows the cross section as a function of k for three different ranges of the relative energy E rel . The theoretical curves show that the calculated distribution shapes for different angular momentum components are quite distinct. Hence, the calculations according to the distorted-wave impulse approximation (DWIA) [30] can be fitted to the measured k distribution to determine each multipole component. Table I summarizes the two-neutron configurations of (1s) 2 , (0p) 2 , and (0d) 2 considered for the fitting and their integrated fractions. Including the contributions from f -and higher orbits did not change the fitting result.

The fitting was performed for each 9 Li+n relative energy bin. Figures 2(a The red, green, and blue curves in Fig. 2(c) are the fitted results for the 1s, 0p, and 0d components, respectively. Assuming the J = 0 pair of two valence neutrons, the integrated fractions of the two-neutron configurations (1s) 2 , (0p) 2 , and (0d) 2 are 35 ± 4%, 59 ± 1%, and 6 ± 4%, respectively. The error is attributed to the limited information on the 9 Li-n optical potential at energies above 100 MeV. Our results confirm comparable contributions from s 2 and p 2 , which are accompanied with a small d 2 contribution. Table I compares the values in this study to those from previous experimental and theoretical results.

To simplify the argument, here, we concentrate on the ratio of the s 2 and p 2 components. The s/p ratio is 0.59, which is consistent with the analysis using the multi-step transfer calculation of the 11 Li(p, t) measurement at 3 MeV/nucleon [45]. a larger s/p value of ≥ 1.3 has been reported for 11 Li(p, d) measured at 6 MeV/nucleon [44]. However, the s-wave fraction was not directly determined from the experimental data, and the estimate may have a large uncertainty. The knock-out measurement with a carbon target at 260 MeV/nucleon shows s/p ∼ 1 [25] but with the error is on the order 40%. This is marginally consistent with the present result of 0.59 within a 1σ variation. However, it should be noted that the strong peripherality of the heavy-ion-induced knock-out reaction employed in Ref. [25] may underestimate the p-wave, which has a larger fraction in the inner part of 11 s-wave. This may overestimate the s/p ratio. The effects of the different peripheralities in the (p, pn) and the heavy-ion-induced knock-out reactions will be discussed later.

The experimental s/p ratio was compared to the theoretical calculations. A three-body model calculation with a density dependent force predicts an s/p ratio of less than 0.7 [46].

Although the s-wave fraction was not explicitly shown, the result is consistent with the present result. The tensor-optimized shell model, which considers the tensor and pairing correlations in the 9 Li core, predicts an s/p ratio of 0.9-1.1 [19,[START_REF] Ikeda | Clusters in nuclei[END_REF]. Applying the transferto-the-continuum reaction framework to the 11 Li(p, pn) data [49] gives an s/p ratio as large as 2.2 [START_REF] Gómez-Ramos | [END_REF]. The reason that the s/p ratio differs among theoretical models is a topic for future studies. at k ∼ 0.3 fm -1 . The correlation angle of 103.4 ± 2.1 • obtained with the knock-out reaction by a carbon target [24] was averaged over the whole momentum range. However, it is even larger than the maximum θ nf value in the present work. This discrepancy may be due to the different peripheralities of the probes. The carbon target used in Ref. [24] selectively probed the surface of 11 Li, where the dineutron correlation is favored. Theoretical models of the knock-out process, including the probe transparency, should realize a quantitative comparison of these results.

The mean correlation angle θ nf decreases at smaller and larger k values. It crosses 90 • at k ∼ 0.6 fm -1 , suggesting that the dineutron is not present in the tail of the halo or in the inner part of 11 Li. The increase of the θ nf value in the limited region may be a signature of radial localization of the dineutron in 11 Li around the 9 Li core. With the help of the quasi-free model described below, the peak structure of θ nf at k ∼ 0.3 fm -1 can be interpreted as the dineutron correlation, which is maximized at r ∼ 3.6 fm from the center of the 9 Li core. This is consistent with the three-body model calculation that includes many-body correlations [50] where the root-mean-square radius between two neutron in 11 Li takes a minimum at r ∼ 3.2 fm.

The quasi-free model, which consists of a combination of the 9 Li+n+n three-body model and the knock-out reaction model in Ref. [30], well reproduces the obtained k-dependence of the correlation angle. The three-body model considered the contributions from the excited 9 Li core via the coupled-channel calculation [19]. The folding potential of the effective N N potential with the 9 Li core density was used for the 9 Li-n interaction and its parameter was modified to reproduce the observed two-neutron separation energy. For the n-n interaction, the realistic Argonne v8 ′ force [51] was used. To describe the knock-out process, the final scattering states were approximated with the products of the plane wave of the knockout neutron and the remaining 9 Li + n system. The FSI in the 10 Li resonance was also taken into account [30]. The absorption effect by the proton target was considered in a manner similar to that in Ref. [30] except that the same function form of the damping factor and its parameters were modified to reproduce the observed value of θ nf in this experiment since the optical potential between 9 Li and the proton target is unknown. The successful reproduction of the experimental data with the reaction model highlights the simple mechanism of the quasi-free (p, pn) reaction, which enables the radial-dependent properties of dineutron in nuclei to be extracted.

It is interesting to compare the results with the theoretical predictions on the dineutron correlation in infinite nuclear matter. The present analysis implies that the dineutron correlation is prominent only around the 9 Li core surface where the density is 10 -3 ρ/ρ 0 10 -2 , and it becomes weaker at the tail of the halo where the density is extremely low. Hartree-Fock-Bogoliubov calculations [8] predict a similar density dependence as the dineutron develops in a density region of 10 -4 ρ/ρ 0 0.5 and vanishes at lower and higher densities.

If this is a universal characteristic of the dineutron correlation, it should exist at the lowdensity surface of any neutron-rich nuclei. Actually, the three-body model calculation [50] predicts that the dineutron is formed in a limited region around the surface of 6 He, 16 C, and 24 O. Future (p, pn) experiments should reveal the nature of the dineutron correlation in these nuclei.

In summary, herein the dineutron correlation in 11 Li was investigated via the quasi-free (p, pn) knock-out reaction. High-statistics data were acquired using the high-intensity 11 Li beam at the RIBF and the MINOS device, enabling a detailed analysis of the 11 Li structure.

The fraction of each two-neutron configurations is determined as 35±4% (1s) 2 , 59±1% (0p) 2 , and 6 ± 4% (0d) 2 . The correlation angle cos θ nf distribution has an asymmetric shape and a missing momentum k dependence, indicating that the dineutron correlation is localized radially on the 11 Li surface. The k-dependence of the θ nf is well reproduced by the quasifree model calculations. The dineutron in 11 Li is localized at k ∼ 0.3 fm -1 , corresponding to the nuclear surface r ∼ 3.6 fm. This behavior is consistent with the Hartree-Fock-Bogoliubov calculation for infinite nuclear matter.

We extend our appreciation to the RIKEN Nishina Center, the Center for Nuclear Study,

The University of Tokyo, and the accelerator staff for their efforts in delivering an intense beam. We also would like to thank K.

FIG. 1 .

 1 FIG. 1. Schematic of the setup. Arrows denote particle trajectories.

  ) and (b) show the results of the fitting for 0 ≤ E rel < 0.5 MeV and 2.0 ≤ E rel < 3.0 MeV where the 1sand 0p-components dominate, respectively.

FIG. 2 .

 2 FIG. 2. Missing momentum k distribution for (a) 0 ≤ E rel < 0.5 MeV, (b) 2.0 ≤ E rel < 3.0 MeV, and (c) all data. Horizontal and vertical axes show the missing momentum of the knock-out neutron and the differential cross section, respectively. Black dotted curve represents the experimental acceptance. Red, green, and blue solid curves represent DWIA calculations for s-, p-, and d-waves, respectively. Black thick solid lines represent the fitting result. Four orange colored areas (i)-(iv) represent regions with characteristic angular momentum distributions. See text for details.

Figure 3 FIG. 3 .

 33 Figure3shows the correlation-angle distributions at different k. There is an apparent kdependence. Herein we picked up four k regions (i)-(iv). At (ii) k = 0.25-0.35 fm -1 , where the s-and p-wave fractions obtained by the aforementioned multipole decomposition are similar, the distribution shows a strong enhancement in cos θ nf < 0 (θ nf > 90 • ), providing evidence of a dineutron correlation. The trend is less prominent at the lower (i: 0.0-0.1 fm -1 )

Figure 4

 4 Figure 4 illustrates a clear k dependence of θ nf . θ nf takes a maximum value of ∼ 100 •

2 FIG. 4 .

 24 FIG. 4. Mean values of the correlation angle θ nf in momentum space. Red points show the data in this study. Error bars and green line show the statistical and systematic uncertainty, respectively. Blue curve shows the quasi-free model calculation. Black hatched area shows the average correlation angle obtained in a previous study [24]. Black dashed line shows the expected θ nf value for the two uncorrelated neutrons. Inset shows a schematic diagram of the correlation angle θ nf in 11 Li. Black dot on the line between 9 Li and n 1 represents the center of mass of 11 Li.

  Hagino for the helpful discussion. The development of MINOS and the core MINOS team were supported by the European Research Council through the ERC (Grant No. MINOS-258567). This work was supported in part by KAKENHI (Grant No. JP16K05352, JP16H02179, and JP18H05404). D. K. acknowledges support from the Rare Isotope Science Project of Institute for Basic Science funded by the Ministry of Science and ICT and NRF of Korea (numbers NRF-2013M7A1A1075764 and NRF-2019R1I1A1A01058354). L. S. acknowledges support from the Institute for Basic Science (IBS-R031-D1). J. G., F. M. M., and N. A. O. acknowledge partial support from the French-Japanese LIA-International Associated Laboratory for Nuclear Structure Problems as well as the French ANR14-CE33-0022-02 EXPAND. * kubota@ribf.riken.jp; Present address: Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany † Present address: LPC Caen, ENSICAEN, Université de Caen Normandie, CNRS/IN2P3, F-14050 Caen Cedex, France ‡ Present address: Center for Exotic Nuclear Studies, Institute for Basic Science (IBS), Daejeon

TABLE I .

 I Comparison of the integrated fraction for each multipole in percentage (%) of experimental and theoretical studies.(1s 1/2 ) 2 (0p 3/2 ) 2 (0p 1/2 ) 2 (0d 5/2 ) 2 (0d 3/2 ) 2

		This work; quasi-free (p, pn)		35 ± 4	59 ± 1	6 ± 4
		C-induced knock-out	[25] 45 ± 10 3-5 45 ± 10	10 ± 8
		Detailed analysis of Ref. [25] [27]	36.8	9.9	46.8
	Exp.	(p, pn)	[43]				11 ± 2
		(p, d)	[44] ≥ 44		33 ± 12
		(p, t)	[45] 31-45		51-64
		Few-body	[46]			59.1
	Theor.	Coupled-channel Tensor-optimized shell model [47] * 46.9 [19] 44.0	2.5 2.5	46.9 42.7	3.1 4.1	1.7 1.9
		Transfer to the continuum	[48]	67		31	1

* 0.6% and 0.5% for (f 7/2 ) 2 and (f 5/2 ) 2 , respectively.