Probing nuclear forces beyond the nuclear drip line: The cases of ^{16}F and ^{15}F
V. Girard-Alcindor, I. Stefan, F. de Oliveira Santos, O. Sorlin, D. Ackermann, P. Adsley, J.C. Angelique, M. Assie, M. Assuncao, D. Beaumel, E. Berthoumieux, R. Borcea, L. Caceres, I. Celikovic, M. Ciemala, V. Chudoba, G. D Agata, F. de Grancey, G. Dumitru, F. Flavigny, C. Fougeres, S. Franchoo, A. Georgiadou, S. Grevy, J. Guillot, V. Guimaraes, F. Hammache, O. Kamalou, J. Kiener, S. Koyama, L. Lalanne, V. Lapoux, I. Matea, A. Matta, A. Meyer, P. Morfouace, J. Mrazek, F. Negoita, M. Niikura, D. Pantelica, L. Perrot, C. Petrone, J. Piot, C. Portail, T. Roger, F. Rotaru, A.M. Sanchez Benitez, N. de Sereville, M. Stanoiu, C. Stodel, K. Subotic, D. Suzuki, V. Tatischeff, J.C. Thomas, P. Ujic, D. Verney
EEPJ manuscript No. (will be inserted by the editor)
Probing nuclear forces beyond the nuclear drip line: The cases of F and F A Tribute to Mahir Hussein
V. Girard-Alcindor , I. Stefan , F. de Oliveira Santos , O. Sorlin , D. Ackermann , P. Adsley , J.C. Ang´elique ,M. Assi´e , M. Assun¸c˜ao , D. Beaumel , E. Berthoumieux , R. Borcea , L. C´aceres , I. Celikovic , M. Ciemala , V.Chudoba , G. D’Agata , F. de Grancey , G. Dumitru , F. Flavigny , C. Foug`eres , S. Franchoo , A. Georgiadou ,S. Gr´evy , J. Guillot , V. Guimaraes , F. Hammache , O. Kamalou , J. Kiener , S. Koyama , L. Lalanne , V.Lapoux , I. Matea , A. Matta , A. Meyer , P. Morfouace , J. Mrazek , F. Negoita , M. Niikura , D. Pantelica ,L. Perrot , C. Petrone , J. Piot , C. Portail , T. Roger , F. Rotaru , A.M. S´anchez Ben´ıtez , N. de S´er´eville , M.Stanoiu , C. Stodel , K. Subotic , D. Suzuki , V. Tatischeff , J.C. Thomas , P. Ujic , and D. Verney Grand Acc´el´erateur National d’Ions Lourds (GANIL), CEA/DRF-CNRS/IN2P3, Bvd Henri Becquerel, 14076 Caen, France Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France LPC Caen, ENSICAEN, Normandie Universit´e, CNRS/IN2P3 Caen, France Departamento de F´ısica, Universidade Federal de S˜ao Paulo, CEP 09913-030, Diadema, S˜ao Paulo, Brazil CEA Saclay, Irfu, DPhN, Universit´e Paris-Saclay, 91191 Gif-sur-Yvette, France Horia Hulubei National Institute of Physics and Nuclear Engineering, P.O. Box MG6 Bucharest-Margurele, Romania Vinˇca Institute of Nuclear Sciences, University of Belgrade Belgrade, Serbia Institute of Nuclear Physics, PAS, Radzikowskiego 152, PL-31342 Krak´ow, Poland Nuclear Physics Institute of the Czech Academy of Sciences, 250 68 Rez, Czech Republic or shortly Nuclear Physics Instituteof the CAS, 250 68 Rez, Czech Republic. Instituto de F´ısica, Universidade de S˜ao Paulo, CEP 05508-090, S˜ao Paulo, SP, Brazil Department of Physics, University of Tokyo, Hongo 7-3-1, Bunkyo, Tokyo 113-0033, Japan Department of Integrated Sciences, Centro de Estudios Avanzados en F´ısica, Matem´aticas y Computaci´on (CEAFMC),University of Huelva, 21071 Huelva, Spain RIKEN Nishina Cente, JapanReceived: date / Revised version: date
Abstract.
The unbound proton-rich nuclei F and F are investigated experimentally and theoretically.Several experiments using the resonant elastic scattering method were performed at GANIL with radioac-tive beams to determine the properties of the low lying states of these nuclei. Strong asymmetry between F- N and F- C mirror nuclei is observed. The strength of the nucleon − nucleon effective interactioninvolving the loosely bound proton in the s / orbit is significantly modified with respect to their mirrornuclei N and C. The reduction of the effective interaction is estimated by calculating the interactionenergies with a schematic zero-range force. It is found that, after correcting for the effects due to changesin the radial distribution of the single-particle wave functions, the mirror symmetry of the n − p interactionis preserved between F and N, while a difference of 63% is measured between the p − p versus n − n interactions in the second excited state of F and C nuclei. Several explanations are proposed.
PACS. ≤ A ≤
19 – 21.60.Cs Shell Model –21.10.Sf Coulomb energy – 21.10.Dr Binding energies and masses – 25.60.-t Reactions induced by unstablenuclei – 25.70.Ef Resonances – 25.40.Cm Elastic proton scattering
The exploration of the nuclear landscape was made pos-sible by the development of radioactive beams. Nuclearmodels have been developed and tested in nuclei far away a Present address:
Technische Universit¨at Darmstadt, Ger-many from the valley of stability [1,2,3,4]. Approaching the shoresof this landscape, the nuclear drip lines, has allowed theobservation of several new phenomena, such as halo nu-clei, modification of the effective nuclear interactions, re-arrangement of the nuclear shells, and clustering. Beyondthe drip line, unbound nuclei usually disappear in theform of waves in the sea of the continuum, i.e. large res- a r X i v : . [ nu c l - e x ] J a n V. Girard-Alcindor et al.: Probing nuclear forces beyond the nuclear drip line: The cases of F and F onances. Theoretical description of the unbound nuclei,identification and understanding the role of specific partsof the nuclear forces, are still challenges to nuclear re-search. Recently, following Ikeda’s work on α -clusteringnear α -emission threshold [5], Oko(cid:32)lowicz et al [6,7] pro-posed to generalize the Ikeda’s conjecture to all near thresh-olds states, including states involving unstable subsystemslike di-neutron or di-proton.In this article, focus is given to the nucleon − nucleon interaction, represented by the Two-Body Two-Body Ma-trix Elements (TBME) in the nuclear shell model. Theidea is to investigate by how much the nucleon − nucleon interaction changes when nuclei are fully embedded in thecontinuum.Experimentally, the effective nucleon − nucleon inter-action energy, labeled δ V exp , can be extracted from themeasured binding energies (BE) [8,9]. For instance, the n − p interaction energy is obtained from the relation δ V exp = BE(Z , N) + BE(Z − , N − − BE(Z − , N) − BE(Z , N − δ V th has been predicted to depend onthe spatial overlap of wave functions of the last particles [9,10,11,12]. Indeed, as one approaches the drip line, radialwave functions for these particles spread and dilute furtherin space. As a result δ V th is expected to decrease since thespatial overlap between a well bound and the unboundnucleon is smaller.A good way to shed light on the effect of the con-tinuum in the nucleon − nucleon interaction is to com-pare level schemes of mirror nuclei involving a bound andan unbound nucleus. The asymmetries observed betweenthe mirror nuclei can be used to single out the role of nucleon − nucleon interaction. In this article, we combinethe results obtained from previous and new experimentsperformed at GANIL for the unbound nuclei F and F.In all these experiments, the resonant elastic scatteringmethod was used with the thick target inverse kinematicstechnique [13,14,15,16]. F The mirror pair, F- N, can be considered a perfect caseto investigate the effect of the continuum in the nucleon − nucleon force. The structure of the N nucleus (S n =+2488.8(2.3) keV) is well known, and the first low ly-ing states are well described assuming pure single-particleconfigurations. The structure of the mirror F nucleus(S p = -535(5) keV) have been previously investigated andthe results are presented in detail in Ref. [17]. This exper-iment had the advantage of combining excellent energyresolution, high statistics, and precise energy calibration,marking a leap in quality and consistency over the prece-dent results obtained for this nucleus. Experimental andtheoretical aspects of this study are presented succinctlyhereafter. Table 1.
Measured energy, spin, width and spectroscopic fac-tor of the low-lying states in F. The properties of the 3 − state were adopted from Ref. [20].E x (keV) J π Γ p (keV) S0 0 − ± ± − ± ± − ± ±
16 3 − ± . A radioactive beam of O ions was produced at the SPI-RAL facility at GANIL through the fragmentation of a95 AMeV O primary beam impinging on a thick carbongraphite production target. The ions were post-acceleratedby means of the CIME cyclotron up to the energy of 1.2AMeV. It was possible to obtain an O beam with anintensity of 1.0(2)x10 pps and 97(1) % purity. The ionswere implanted onto a thick polypropylene (CH ) n tar-get. Some ions underwent proton elastic scattering and thescattered protons were detected promptly to the reactionin a 300 µ m thick silicon detector that covered an angularacceptance of ± ◦ downstream of the target ( θ cm = 180 ◦ ).The measured excitation function is shown in Fig. 1.An energy resolution of 23.5(3) keV FWHM was obtainedin lab, which corresponds to (cid:39) F( O + p ) were studied through an R-matrix analysis ofthe spectrum. Three resonances can be observed corre-sponding to the 0 − ground state and the 1 − and 2 − firstexcited states of the unbound F nucleus. The measured
Fig. 1.
Excitation function of the reaction O( p , p ) O mea-sured in inverse kinematics. The black line corresponds to theR-matrix fit. The presence of the 0 − ground state, and twoexcited states 1 − and 2 − is clearly observed. The resolution ofthe experiment was (cid:39) properties are presented in Table 1. The deduced spectro-scopic factors of the low-lying states in F are all closeto 1. Similar values were found for the mirror nucleus N[18,19]. . Girard-Alcindor et al.: Probing nuclear forces beyond the nuclear drip line: The cases of F and F 3 F
0- 0 1- 193 2- 424 3- 722 N
2- 0 0- 120 3- 298 1- 397
Fig. 2.
Level schemes of the two mirror nuclei F and N.The states corresponding to the 1p / ⊗ / and 1p / ⊗ / single-particle configurations are shown in blue and red respec-tively. n − p interaction energies The level schemes of the two mirror nuclei N and Fare compared in Fig. 2. Large differences can be observed:the ground state of F has J π = 0 − while that of Nhas J π = 2 − [20]. In F, compared to N, it lookslike both 0 − and 1 − states are down shifted in energyby about 500 keV relatively to 2 − and 3 − states. The low-lying states of the N nucleus can be well described usinga single-particle configuration with a closed core of Cand a deeply bound proton in the 1p / orbital (S p ( N)=+10.2 MeV) coupled either to a neutron in the 2s / or-bital (S n ( C)= +1.22 MeV), leading to J π =0 − , − states,or to a 1d / neutron (S n ( C*)= +0.48 MeV), formingthe J π =2 − , − states. In the same way, (0,1) − and (2,3) − states in F can be described as a O core and a neu-tron in the 1p / orbital (S n ( O)= +13.22 MeV) cou-pled either to a proton in the 2s / orbital (S p ( F gs )=-1.27 MeV) or to a proton in the 1d / orbital (S p ( F*)=-2.79 MeV). None of the states have the same spin, so,they don’t mix, and if we neglect interaction with higherenergy states of the same spin value, the spacing betweenthe two members of a given multiplet is only due to theresidual interaction. For example, in the case of the J π =0 − state in N ( C+ n + p ), originating from the n − p cou-pling π / ⊗ ν / , the effective n − p interaction iscalculated with δ V exp ( N) − = BE( N) − − BE( N) / − +BE( C) + − BE( C) / + The same method was also applied to the J π =1 − state,and to the J π =2 − , − states originating from the n − p coupling π / ⊗ ν / .The obtained experimental n − p interaction energiesare given in Table 2. It is observed that the effective n − p interaction changes by 40% for the (0 − ,1 − ) multiplet ascompared with the mirror nucleus N, and by 10% for the(2 − ,3 − ). In all cases it is weaker in F, in agreement withexpectations that δ V will decrease as the nucleus goes inthe direction of the drip line.
Table 2.
Measured effective n − p interaction energies. Theuncertainties are lower than 10 keV on all values. Calculatedfrom Ref. [17,21]State (J π ) δ V exp ( N) δ V exp ( F)(MeV) (MeV)0 − -1.151 -0.7351 − -0.874 -0.5422 − -2.011 -1.8343 − -1.713 -1.536 r(fm) - - u (r) N F Fig. 3.
Comparison of the calculated single-particle wave func-tions for the 2 s / orbital in F and N. The difference is dueto the fact that the neutron in N is bound whereas the protonin F is unbound. The wave functions have been calculatedusing a Woods-Saxon potential.
This apparent breaking of the symmetry of the nuclearforce between N- F mirror nuclei is explained by thelarge coupling to the continuum. The single-particle wavefunctions have been calculated considering a Woods-Saxon,Coulomb and spin-orbit potentials, using a standard set ofparameters ( V W S ≈ a = 0 . r =1.26 fm, V SO =6 MeV). As can be seen in Fig. 3, the wave functionscalculated for the 2s / orbit are different since the neu-tron is bound in N and the proton is unbound in F.This difference modifies the overlap between the 2s / andthe 1p / wave functions, and so, the effective n − p in-teraction. This effect has been investigated in details byOgawa et al [11] using a Woods-Saxon plus M3Y force,and by Yuan et al [12] using a monopole-based-universalinteraction (VMU) in the Woods-Saxon basis. Here, weuse a schematic zero-range interaction v = − a δ ( r p − r n )to calculate the effective nucleon − nucleon interaction. Inthe definition, a is the positive strength of the interactiongiven in units of MeV.fm − , and the minus sign servesto emphasize its attractive nature. Despite its seemingsimplicity, the δ interaction reproduces fairly well manyproperties of nuclei [10]. One gets δV th = − a J π (cid:90) ∞ r [ u p ( r ) u n ( r )] dr V. Girard-Alcindor et al.: Probing nuclear forces beyond the nuclear drip line: The cases of F and F Table 3.
Calculated reduction factor for the effective nucleon − nucleon interaction. δV th Reduction Factor < p / s / | v pn | p / s / > < p / d / | v pn | p / d / > < (2 s / ) | v pp | (2 s / ) > - F 0
16 - - - - F 1/2 F A sy mm e t r y F a c t o r ) ) Fig. 4.
Asymmetry factors F (see text) are shown for severalmirror states. For the 4 states (0 − , 1 − , 2 − , 3 − ) studied in F- N, the mirror symmetry is well preserved since the F is closeto 1. In the case of the 1/2 − narrow resonance in F, F canbe more or less close to 1.0 depending on the its structure.The case of a pure N+ π (1d / ) structure is indicated by ablack star, for a pure N+ π (2s / ) by the blue star, and for He cluster+ N core by the red star. The associated errorsare much smaller than the size of the stars. where a J contains the strength of the n − p nuclear inter-action in the state with spin J , and u n ( r ) and u p ( r ) arethe radial wave functions calculated with a Woods-Saxonpotential. By virtue of the charge symmetry of nuclearforces, the same a J coefficients have been used to calcu-late the interaction energies in the mirror nuclei F and N.The calculated Reduction Factors δV th ( F) δV th ( N) are pre-sented in Table 3. These factors are of the same order asthose calculated in Ref. [12], even if they were not cal-culated with the same core nucleus ( O instead of O).The Asymmetry Factor F = δV exp ( F) δV th ( F) × δV th ( N) δV exp ( N) can be used to determine the degree of mirror asymmetryremaining after correction for wave functions differences inthe mirror states. If the observed asymmetry is due solelyto this effect, F must be equal to 1.0. The calculated valueof F for all states studied in this article are presented inFig. 4. As can be observed in the figure, these factors areclose to the expected value of 1.0 for the 4 states. Smalldeviations of about 10% are observed for the 2 − and 3 − states, but the model used here is too simplistic to give realcredit to these small deviations. Despite the fact that theexperimental n − p interaction energies δV exp are reduced,the mirror symmetry is preserved. In agreement with theconclusion of Ref. [11] and [12], we confirm that the largedifferences in effective n − p interaction energies between Table 4.
Measured energy, spin, width and spectroscopic fac-tors of the low-lying states in F. The numbers in bracketscorrespond to statistical and systematic uncertainties.E R (keV) J π Γ p (keV) S1270(10)(10) 1 / + +2000 ) 0.82763(9)(10) 5 / + ≈ / − ≈ the mirror states are mainly explained by the differentoverlaps between the wave functions. F The isotope F is located two neutrons away from theproton drip line. The study of this nucleus is briefly pre-sented below. More details about this study can be foundin Ref. [22,23,24,25]. In addition, new results have beenobtained recently and are presented here for the first time.
Three experiments were performed at GANIL to study theunbound F nucleus. The third experiment was focusedon the study of the two-proton decay. The obtained results[25] will be published soon in a refereed journal [24]. Theobjective of the first two experiments was the study of thestructure of the low lying states. The excitation function ofthe elastic scattering reaction O(p,p) O was measuredat 180 ◦ (c.m.), as can be seen in Fig. 5. It was obtainedin inverse kinematics using a thick polyethylene (proton)target. This spectrum has high statistics, good energy res-olution with 16.5(5) keV FWHM ( ≈
95 keV FWHM forthe second experiment) and large energy covering from0.4 MeV to 5.6 MeV (4 MeV to 5.6 MeV). The anal-ysis of the excitation function was performed using theR-matrix method with the code AZURE2 [26]. The fit isshown by the continuous blue line. The measured spectro-scopic properties are presented in Table 4, and they wereused to produce the level schemes shown in Fig. 6.The obtained values for the two first states are inagreement with those from the literature [22]. In addition,the second excited state was clearly observed as a narrowdip in the excitation function, see right side of Fig. 5. Inthe mirror nucleus C, the second excited state has spinJ π = 1 / − . The shape of the dip observed in the excita-tion function presented in Fig. 5 could be reproduced byR-matrix calculation assuming J π = 1 / − . The R-matrixfit of the excitation function was performed taking intoaccount the experimental resolution.A second experiment was performed at GANIL in or-der to confirm the existence of this 1/2 − state and tosearch for new states at higher energy. The same tech-nique of resonant elastic scattering was used, although themeasurement was made with a completely different exper-imental setup. A much thinner ( ≈ µ m) polyethylenetarget was used and the protons were detected with the . Girard-Alcindor et al.: Probing nuclear forces beyond the nuclear drip line: The cases of F and F 5 (MeV) r E ( b / s r) C M W / d s d x 0.1 + + - Fig. 5.
Measured excitation function of the reaction O(p,p) O. Left: Data and calculation renormalized by a factor 0.1. Theexperimental points in black are from Ref. [22]. The red points are new data confirming the existence of the dip and the 1 / − resonance. The continuous blue line corresponds to the R-Matrix fit. Three unbound states are identified: The 1 / + groundstate, the 5 / + first excited state, and the 1 / − very narrow resonance. N+2p O+p F C C+2n C+n Fig. 6.
The level scheme of F compared to C. MUST2 [27] ensemble of silicon detectors. Details and re-sults of this experiment are presented in Ref. [24,25]. Asshown with the red dots in the right side of Fig. 5, thenew measurement perfectly confirms the presence of a dipin the excitation function. The resonance is measured atE R = 4.880(140) MeV with Γ = 23(10) keV. It is in goodagreement with the results of the first experiment, con-firming the existence and properties of this narrow reso-nance.The two protons emission from this narrow resonanceis energetically possible, see Fig. 6. Since there is no inter-mediate state accessible to O, it should be a direct two-proton emission to the g.s. of N. Considering that theavailable energy is only Q p = 130 keV [21], the branchingratio for the two proton emission is negligible ( Γ p < − eV), and the measured width corresponds only to theemission of a single proton towards the g.s. of O. p − p and n − n interaction energies The observation of this narrow resonance in F is quitesurprising since it is located 1.6 MeV above the Coulombplus the (cid:96) =1 centrifugal barrier for proton emission. Thereis no barrier to retain the proton inside the nucleus. More-over, it is difficult to calculate the single-particle width forsuch a loosely bound state. The Wigner width is Γ W ≈ S gs =0.005 (calculated with the Wignerwidth). It is clear that the structure of this state is far frombeing a proton coupled to a O gs core. It is interestingto look for the 99.5% missing component.The situation is not completely symmetrical in the mir-ror C nucleus. The 1 / − state in the mirror nucleus isat the excitation energy E x =3.10 MeV, it is unbound forthe emission of one neutron by 1.89 MeV, and bound forthe emission of two neutrons by 6.29 MeV, see Fig. 6. Thisstate has also been measured as a narrow resonance, witha width of 38 keV [28]. It is thought to be primarily an( sd ) excitation, i.e. C(gs) × ( sd ) + [28,29].Theoretically, shell model calculations were performedwith the code NUSHELLX [30] in the psd space and withthe psdmk interaction [31]. It is calculated that the spec-troscopic factor to the ground state is S gs = | < O gs + p | F / − > | =0.025, confirming the weak overlap with O gs .Canton et al [32] used the multichannel algebraic scat-tering (MCAS) theory with Pauli-hindered method to pre-dict the properties of the low-lying states in F. The cal-culations was recently updated and several unusual nar-row resonances are predicted at high energies [33]. A verynarrow width Γ =5 keV was predicted for the 1 / − state[32], later updated to Γ =107 keV [33].Fortune and Sherr [29], using a potential model, de-termined the single-particle widths for C. These valueswere scaled down to reproduce the measured widths in C, and the extracted dimensionless reduced widths θ were used to calculate widths in the mirror nucleus F.The results of these calculations confirmed that narrow
V. Girard-Alcindor et al.: Probing nuclear forces beyond the nuclear drip line: The cases of F and F Table 5.
Measured effective interaction energies in F and C State (J π ) δV exp nn δV exp pp (MeV) (MeV)1/2 − -2.127 -0.963Coulomb corrected - -1.142 resonances are to be expected in O+p, with a predic-tion of Γ =55 keV for the 1 / − state. Refined values werelater published by Fortune [28], with Γ =38 keV. Canton et al [34] objected that θ do not necessary scale withthe single-particle widths, especially when θ is small [35].Moreover, the presence of a particle continuum can signif-icantly impact spectroscopic properties of weakly boundnuclei and excited nuclear states close to, and above, theparticle emission threshold [36], which is the case here.Gamow Shell Model in the coupled channel represen-tation [37] was also used to determine the properties ofthis state. The calculations were performed considering a C core and three valence protons. It was confirmed thatthe overlap with O g.s. + p is very weak, with S gs =0.0035.It was found that the 1 / − state is 97% composed of twoquasi-bound protons located in the 2 s / shell [38].The formalism presented in the previous section to de-duce the n − p effective interaction energies was applied to F(1/2 − ) to deduce the p − p effective interaction energy.We suppose that the 1 / − state can be described by aninert core of N gs (1 / − ) coupled to two protons in the2 s / orbit. Shell model calculations show that the firstexcited state of O − (E x =5.173 MeV) is well describedby a N gs plus one proton in the 2 s / shell, the spectro-scopic factor being S=0.70. In this case, we can write for F δ V exppp = BE( F) / − + BE( N) / − − × BE( O) − The n − n interaction energy is determined in the same waywith C. The obtained interaction energies are listed inTable 5. However, both valence protons in F are chargedand the Coulomb repulsion energy must be corrected for.This was calculated using the formula [39] W = 12 (cid:90) ρ (r) φ (r) dv where ρ ( r ) is the density of charge, φ ( r ) the electricalpotential and dv a volume element. The calculated cor-rection is W = 179 keV, which gives δV exp pp =-1.142 MeV.This value indicates that the interaction energy is weakerin F by a factor 1.9, once again in agreement with theexpectation that the interaction decreases as the nucleusapproaches, or goes further, the drip line.
If the 1 / − state in F is described as a N core+ twoproton in the 2s / orbital, a huge reduction factor of 0.33is calculated, see Table 3. This is in agreement with the results of Ref. [12] where the greatest reduction factor hasbeen observed for this TBME. Actually, their calculatedvalue is 0.68, which is twice the value of the present work.However, it should be emphasize that their values has beencalculated with a different interaction. Also, the asymme-try factor F , shown in Fig. 4, is 63% larger than the ex-pected value of 1. Such a large difference is significant andneeds an explanation. One of them is the fact that core po-larization effects, which should be included, has not beentaken into account in our simple model.Also, it is possible that the a J strength coefficients aredifferent between the two nuclei. This may be due to thefact that the 1 / − resonance in F is located only 130 keVabove the two-proton emission threshold, whereas in themirror nucleus C, the 1 / − state is bound by 6.3 MeV fortwo-neutron emission. This asymmetry in F/ C wouldbe an indication of the generalized Ikeda’s conjecture [6,7]. The p − p correlation might be enhanced in F com-pared to the n − n correlation in the mirror nucleus C.To estimate the impact of this correlation enhancement,we treated C(1/2 − ) as C + n where the n clusteris bound in a Woods Saxon potential well. We then ap-plied the same potential to calculate the resonance energyof the mirror system F as N+ He, correcting for theCoulomb energy in He ( ≈
200 keV). The calculated asym-metry factor is F =1.16, as shown in Fig. 4 by a blue star.Thus, to explain the F/ C difference it would be nec-essary to describe the 1 / − state of F as being 100%composed of N+ He cluster.Another much simpler solution would be to add tothe (2s / ) wave function a part of (1d / ) component,which would have the effect of increasing the overlap ofthe wave functions, reducing the asymmetry. To get theasymmetry factor F = 1, the states in F and C shouldbe described as composed with 58% of (1d / ) and 42%of (2s / ) . Fortune et al [40] predicted 46% and 54% re-spectively, using the Lawson-Serduke-Fortune two-bodymatrix elements. Using these values, they predicted theenergy of the F 1 / − state at E R =4.63 MeV [29], whichis only 127 keV below our measured energy. In the present investigation we have explored the nucleon − nucleon force in nuclei beyond the proton drip line. Strongasymmetry between F- N and F- C mirror nucleiwas observed. For the F and N pair, the apparentbreaking of the symmetry can be explained by the dif-ference in the overlaps of the wave functions due to thecoupling with the continuum. The recently measured andconfirmed 1/2 − state in F, on the other hand, showsasymmetry that cannot be explained by the difference inthe overlaps of pure (2s / ) wave functions. This may bean indication of an increase in correlation between the twoleast bound protons due to the proximity with the two-proton emission threshold. It may also be an indicationthat the states are described by an almost equal mixtureof (2s / ) and (1d / ) components. . Girard-Alcindor et al.: Probing nuclear forces beyond the nuclear drip line: The cases of F and F 7
We would like to thank the GANIL accelerator staff fortheir support. We also received support by OP RDE, MEYSCzech Republic under the project EF16-013/0001679, byLIA NuAG and COSMA, and by IRP France-Br´esil.
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