B Le Crom

M Assié

Y Blumenfeld

J Guillot

H Sagawa

T Suzuki

M Honma

N L Achouri

B Bastin

R Borcea

W N Catford

E Clément

L Cáceres

M Caamaño

A Corsi

G De France

F Delaunay

N De Séréville

B Fernandez-Dominguez

M Fisichella

S Franchoo

A Georgiadou

J Gibelin

A Gillibert

F Hammache

O Kamalou

A Knapton

V Lapoux

S Leblond

A O Macchiavelli

F M Marqués

A Matta

L Ménager

P Morfouace

N A Orr

J Pancin

X Pereira-Lopez

L Perrot

J Piot

E Pollacco

D Ramos

T Roger

F Rotaru

A M Sánchez-Benítez

M Sénoville

O Sorlin

M Stanoiu

I Stefan

C Stodel

D Suzuki

J-C Thomas

M Vandebrouck

Neutron-proton pairing in the N=Z radioactive

Keywords: radioactive beam, direct reactions, neutron-proton pairing, Cooper pairs

published or not. The documents may come

Pairing correlations are at the origin of superfluidity in strongly interacting quantum many-body systems [START_REF] Brink | Nuclear superfluidity : pairing in finite systems[END_REF][START_REF]Fifty Years of Nuclear BCS[END_REF]. The theoretical description of these correlations is rooted in the microscopic theory of superconductivity developed by Bardeen, Cooper and Schrieffer (BCS) [START_REF] Bardeen | [END_REF] with building blocks made of strongly correlated pairs, the Cooper pairs. For most of the known nuclei, the superfluid states consist of isovector (T=1) neutron and/or proton pairs (nn or pp pairs). In N=Z nuclei, the large overlap between neutron and proton wave functions allows another type of Cooper pairs made of neutron-proton pairs (np pairs) of two different types: either isovector Cooper pairs (L=0, S=0, T=1) as for the nn and pp pairs or isoscalar Cooper pairs (L=0, S=1, T=0), the "deuteron-like" pairing. The latter is a very unique manifestation in nature of cross-species pairing [4] .

Cooper pair transfer is an efficient probe to unravel the properties of the pairing force. Systematic neutron pair transfer (adding or removing a pair), performed within a major shell, reveals the occurence of a paired superfluid phase. For openshell nuclei, like stable singly-magic Sn isotopes, the pair transfer amplitude is enhanced at mid-shell and is driven by the ratio of the pairing gap to the strength of the pairing force, which reflects the number of Cooper pairs contributing coherently [START_REF] Brink | Nuclear superfluidity : pairing in finite systems[END_REF][START_REF]Fifty Years of Nuclear BCS[END_REF]5]. For closed-shell nuclei, like 208 Pb, pair vibrations sign the onset of the superfluid phase and the pair transfer cross-section is governed by the number of phonons involved [START_REF] Brink | Nuclear superfluidity : pairing in finite systems[END_REF][START_REF]Fifty Years of Nuclear BCS[END_REF][START_REF] Mottelson | Nobel prize lecture[END_REF].

Similar arguments on the pair transfer cross-sections hold for np pair [START_REF] Fröbrich | [END_REF]8,9]. In this case, two components enter into play: the isoscalar pair and the isovector pair transfer. Their interplay can be probed starting from an even-even self-conjugate nucleus (J=0 + ,T=0) and populating respectively the lower lying (J=1 + ,T=0) and (J=0 + ,T=1) states in the residual odd-odd nucleus. The relative strength of the isoscalar and isovector channel is obtained from the ratio of the cross-sections: σ(0 + ,T=1)/σ(1 + ,T=0).

The np pair stripping and pick-up reactions performed in direct kinematics in stable N=Z nuclei of the sd-shell have been reviewed in [4] and recently remeasured consistently [10]. Isoscalar pairing remains elusive in the sd-shell due, perhaps, to the limited possible occupation and calls for further studies in heavier nuclei. Indeed, high-j orbitals are more favorable as the typical number of pairs entering into play depends on the degeneracy of the specific orbitals involved. Refs. [11,12] have shown that the region of N=Z≈64 is the most promising to observe a competition between the two channels. This region is out of reach today for transfer reaction studies but higher j orbitals and/or higher shells where N=Z nuclei are unstable can be tackled. These are very challenging measurements given the low cross-sections, the radioactive beam intensities available and the high density of states in the residual odd-odd nuclei. Indeed, only one attempt was made so far to measure 44 Ti( 3 He,p) 46 V with a radioactive (long-lived) 44 Ti beam [4,13]. In the present work, we push the measurements up to the end of the f 7/2 shell by studying the np pair transfer reactions 56 Ni(p, 3 He) 54 Co and 52 Fe(p, 3 He) 50 Mn , involving respectively a doubly magic nucleus and a near mid-shell nucleus to investigate the relative strength of the isovector and isoscalar Cooper pairs. For these heavier nuclei, the LS coupling does not hold anymore due to the effect of the spin-orbit force. In the jj-coupling, the isovector pairing concentrates in the J=0 channel whereas for the isoscalar pairing, two main couplings are favored: the anti-aligned (J=1) and the maximum aligned (J=J max =7 in our case) [14]. Thus the cross-section to the (J=0 + ,T=1) state reflects the contribution of np Cooper pairs in the isovector channel whereas the cross-section to the (J=1 + ,T=0) state reflects the isoscalar np pairs with their two main components: the Cooper pairs (J=1) and the aligned pairs (J=7). To discuss the contribution of the latter, we assume a simplified picture with only the f 7/2 orbit. For 56 Ni(p, 3 He), the transferred pair can only be a neutron and a proton holes coupled to J=1, so that there is no aligned np pairs contribution. In the 52 Fe(p, 3 He) case, IBM calculations [15] show that the ground state of 52 Fe can be described as two aligned np pairs coupled to J=0, but the lower lying (J=1 + ,T=0) state of 50 Mn has very little overlap with 3 aligned np pairs coupled to J=1. Therefore, the aligned np pairs contribution plays a minimal role in both reactions. Thus, the present study concentrates on isovector and isoscalar Cooper pairs.

The experiment was performed at GANIL [16]. The beams of 56 Ni and 52 Fe were produced at 30.5A MeV and 31.2A MeV respectively, with an intensity of 10 5 pps, by fragmentation of a 58 Ni primary beam accelerated to 74.5A MeV on a 1 mm-thick Be target. Two different settings of the LISE spectrometer [17] including the Wien filter were used to select each beam. Two low-pressure multi-wire devices, CATS [18], were placed upstream of the target in order to reconstruct event by event the beam trajectory and its position on the target. They also provide beam time reference and precise determination of the number of incoming ions. Beam contaminants, which represented 40% of the total intensity, were removed offline using a timeof-flight measurement between the cyclotron radio-frequency and the first CATS detector. The secondary beams impinged on a 6.8 mg/cm 2 thick CH 2 target, in which (p, 3 He) reactions occurred. A 12 C target was also used to estimate the carbon background.

CH 2 ,C targets

EXOGAM clovers CATS MUST2

56 Ni, 52 Fe The energies and angles of the emitted light ejectiles were measured by four MUST2 telescopes [19] in the forward direction covering from 10 to 40 degrees in the laboratory frame (see Fig. 1). Each telescope consists of a 300 µm-thick double-sided stripped Silicon detector (DSSD) with 128 strips on each side backed by 16 4 cm-thick CsI scintillators read out by photodiode which provide energy-loss (∆E) and residual energy (E) measurements, respectively. The DSSD were calibrated with a triple alpha source and the CsI calibration for 3 He and deuterons is deduced from their energy losses in the DSSD. The intrinsic energy resolution is of 40 keV (FWHM) for the former and 0.08 √ E CsI MeV for the latter. Light particle identification was obtained from the ∆E-E measurement. The emission angle of the light ejectiles was deduced from the beam direction and impact position on the target given by the two CATS detectors and from the position information given by the DSSD with a precision better than 1 degree. Its energy was corrected for the energy losses in half of the target and in the dead-layers of the DSSD. The associated gamma spectra with a gate on the excitation energy centered on the first (J=1 + ,T=0) state ±2σ (the gate in E * is given explicitly on the spectra). A simplified level scheme of the residual nuclei is shown with the main transitions and associated intensities taken from [22]. The deduced direct feeding by the transfer reaction (in number of counts) of each excited state is shown in blue with its statistical error bars (see text for details). This information is used to constrain the fit of the excitation energy (shown as a thick red line on spectra a) and c)). The population of the ground state (given in red) is deduced from the fit. The contribution of each state to the fit is identified with the same line code as for the associated gamma-ray lines (dotted for the gs, full for the 1 + , dashed dotted for the 2 + and dashed for the 3 (+) state). For 56 Ni(p, 3 He) 54 Co a contribution at higher energy (around 3 MeV) is also shown and detailed in the text.

The calibration and alignement of the detectors were validated with the one-neutron transfer reaction 56 Ni(p,d) 55 Ni where the ground state excitation energy is reconstructed at -40±50 keV with an energy resolution of σ=420 keV. This reaction was also used as a test bench for simulations of the experimental set-up performed with the nptool package [20]. With realistic conditions for beam spread and beam reconstruction, an energy resolution of σ=390 keV was found, confirming the reliability of the simulations.

The missing mass technique was applied to the two-nucleon transfer reactions 56 Ni(p, 3 He) 54 Co and 52 Fe(p, 3 He) 50 Mn to deduce the excitation energies of 54 Co and 50 Mn, respectively, from the measured energy and angle of the outgoing 3 He . The excitation energy spectra are shown in Fig. 2(a) and (c). The carbon background contamination is evaluated from the 12 C target run with 56 Ni beam. For the 52 Fe beam, the same shape was considered and its amplitude was fitted to the data in the negative excitation energy region where only background contaminations contribute. Background is indicated as a thin dotted line in the excitation energy spectra. In both cases, a broad peak centered around 0 is clearly seen together with another peak at higher energy, around 3 MeV. Simulations of the 56 Ni(p, 3 He) and 52 Fe(p, 3 He) reactions give an excitation energy resolution of σ=600 keV and σ=650 keV respectively. As the 52 Fe beam was run after that of 56 Ni, the CATS resolution was degraded due the high beam intensity already handled, resulting in a slightly degraded excitation energy resolution for 52 Fe beam.

With such an energy resolution, the contributions of the lowlying isomeric states of 54 Co (197 keV, 7 + ) and of 50 Mn (225 keV, 5 + ) cannot be disentangled from the ground-state contribution. However, the angular momentum (L) matching of the reaction favors states with L≤4. Given the large L-transfer (L≥5) involved for the isomeric states, they should be very weakly populated.

To check that the first peak of the 54 Co excitation energy spectrum corresponds to its ground state and thus, to a L=0 transfer, its angular distribution has been extracted. The differential cross-section is obtained from a Gaussian fit of the excitation energy spectrum between -2 and 0.6 MeV (corre-sponding to +1σ) for each angular bin. The fit assumes that only the ground state contributes with a fixed width determined from simulations, although the gate includes about 30% of the first excited state (J=1 + at 937 keV) and 8% of the J=2 + state at 1445 keV.

The angular distribution (see Fig. 3) is compared with calculations using the FRESCO code [23]. The two-nucleon pick-up reaction (p, 3 He) was analyzed by a finite range second-order DWBA calculation in which both sequential and simultaneous transfer are taken into account. In the simultaneous twonucleon transfer, the prior and the post form give the same results while in the sequential (p, 3 He) transfer, the prior and then the post form are used successively in order to eliminate the non-orthogonal terms. The optical potentials were taken from the front-end FR2IN code [24] using the Chapel-Hill 89 [25] for the proton partition while the Bechetti-Greenlees parametrization [26] was used for the 3 He partition. The overlap functions for the heavy nuclei are described by single-particle states in a Woods-Saxon potential with the depth fitted to reproduce a state with the given binding energy. For the direct transfer contribution, the two-particle form factor is built from the two oneparticle form factors and the excitation energy is shared equally between the neutron and proton. In the sequential two-nucleon transfer, only the path 56 Ni(p,d) 55 Ni(d, 3 He) 54 Co is considered due to favorable binding energies in the intermediate 55 Ni nucleus. The intermediate state is defined at half the binding energy difference between the initial and final states. The available energy is shared between the picked-up nucleons. The DWBA calculations were performed for two assumptions for the 54 Co ground state configuration. First, we assumed a pure single particle νf 7/2 x πf 7/2 configuration with no pairing (SP). Second, the two-nucleon amplitudes (TNA) were taken from shell model calculation with the GXPF1 interaction in the fpshell valence space and assuming a 40 Ca inert core [27]. For the isovector channel, only TNA with neutron and proton in the same orbits contribute whereas in the isoscalar, spin-orbit partner cross terms also contribute. Fig. 3 shows the DWBA angular differential cross sections for 56 Ni(p, 3 He) in both cases without any normalization factor. The theoretical simultaneous and total cross-sections for both assumptions (SP and GXPF1) are shown. The major contribution to the cross-section for both cases comes from the sequential transfer and not from the direct transfer of a pair. A complete study of two-neutron pair transfer has shown that transfer, whether it is direct or sequential, is a quantitative probe of pairing correlations [28]. The shape of both calculations is very similar to the experimental data and confirms that the main contribution to the first peak of the excitation energy spectrum comes from an angular momentum transfer L=0, as expected for the ground-state to ground-state transfer. The overall agreement of the GXPF1 total cross-section with the experimental data is better than with the SP total cross-section, suggesting that pairing plays a significant role in the ground-state to ground-state transfer.

As the main goal of the experiment is to measure the crosssection not only to the ground-state (J=0 + , T=1) but also to the first excited (J=1 + ,T=0) state (located at 937 and 651 keV for 56 Ni and 52 Fe respectively) and to deduce their ratio, the 3 Hereconstructed excitation energy spectra alone are not sufficient. To improve the resolving power by almost a factor of 20, the detection set-up was supplemented by four Germanium clovers of the EXOGAM array [21] placed at 55 mm around the target with front faces angles spanning from 45 to 135 degrees. The add-back photopeak efficiency of the array was measured with radioactive sources to be 8.6(1)% at 1 MeV. After Doppler correction, the energy resolution (FWHM) for gamma-rays emitted during the decay of the populated excited states in the heavy residues is 80 keV at 1 MeV according to simulations. Particlegamma coincidence measurements give unambiguous identification of the states populated provided that the level schemes of the residual nuclei are available.

For 54 Co and 50 Mn, the low-lying excited states are wellknown [22]. The gamma spectra shown in Fig. 2(b) and(d) are gated by a 4σ-wide gate on the excitation energy centered on the J=1 + ,T=0 state of interest (thus including 95% of the contribution of this state). The corresponding gate is reported on each spectrum. The gamma-ray lines observed are shown by vertical arrows on the gamma spectra with the same line style as in the level scheme. Within the applied gate, only the 1 + , 2 + and 3 (+) states are observed, as expected from the L-matching condition of the transfer reaction. The contributions between 100 and 300 keV correspond to the random coincidences with the gamma-rays from the decay of the implanted beams of 56 Ni and 52 Fe.

The gated gamma spectra of Fig. 2(b) and (d) are used to determine the cross-section for the direct feeding by the transfer reaction of the low-lying excited states in 54 Co and 50 Mn. In order to extract the number of counts for each gamma-ray line, the spectra were efficiency corrected. An L-dependent Lorentz boost efficiency was further applied. Then, each line was fitted by a gaussian combined with a linear background. The number of counts expected in the excitation energy spectrum for each state within the gate (reported in blue on the level schemes of Fig. 2) was deduced after subtracting the feeding from higher lying states.

In the case of 54 Co, the 3 + state feeding by the transfer reaction (within the excitation energy gate) is given by its main line at 375 keV. The total population of the 2 + state is deduced from its main line at 508 keV, relying on the tabulated gamma intensities. The deduced number of 1445 keV transitions is consistent with the measured one. Then, its direct feeding by the transfer reaction is deduced by subtracting the contribution of the 375 keV transition and is shown in blue in the level scheme. The same procedure is applied for the 1 + state where the feeding from the 508 keV transition is subtracted.

For 50 Mn, the total population of the 2 + state is inferred from its line at 800 keV. The feeding from the (3) state at 2477 keV is subtracted to deduce its direct feeding by the transfer reaction. The same method is applied for the 1 + state where the feeding from the 150 keV transition is accounted for. The error bars on the direct feeding of the excited states by the transfer reaction are shown in parenthesis. They take into account only the statistical error. For both measurement, the top feeding contribution to the 1 + low-lying state is important and contributes to the large error bars. As a result, the transfer cross-section to the 1 + state of 50 Mn could be compatible with zero.

With the information on the direct feeding of the excited levels, a multi-gaussian fit of the excitation energy spectra can be performed to deduce precisely the population of the ground state. The fit is applied between -2 MeV (to include fully the ground state) and the energy of the 1 + state +2σ (shown by the vertical red lines in Fig. 2a) and c)). The positions of the excited states are taken from the database but a global shift of the excitation energy is allowed. The width of the state is fixed as deduced from simulations. The amplitudes of the gaussians for the excited states are determined by the direct population of the states deduced from the gamma spectra and are allowed to vary within the error bars. The amplitudes are corrected from the percentage of the gaussian that is included in the fit range. The total fit is shown as a full red line in Fig. 2 with each individual contribution shown with a line code corresponding to the one in the level scheme. The p-value of both fits is higher than 0.8, showing that the fit, with a limited number of parameters, reproduces well the data. The deduced ground state populations are shown in red in the level schemes together with the fit error bar. Given the limited excitation energy resolution, the tail of the peak located between 2 and 3 MeV contributes below 2 MeV. In the case of 52 Fe(p, 3 He), the contribution of the (3) state at 2477 keV is included in the fit as it is observed in the γ-ray spectrum. Its effect is included in the fit error bars. For 56 Ni, no γ-ray from this peak is identified within our gate. If we extend the gate up to 4 MeV, we observe a transition at 1650 keV that could correspond to the 3094 keV level. The contribution of this state is shown on Fig. 2(a) with a thin line. The possible overestimation of the ground-state population due to the tail of this peak is lower than 2%.

The experimental cross-sections obtained for 56 Ni(p, 3 He) and 52 Fe(p, 3 He) transfer to the lower lying (J=1 + ,T=0) and (J=0+,T=1) states are reported in Tab.1 with their associated statistical and systematic error bars (which is dominated by the uncertainty on the target thickness). The associated theoretical predictions relying on second-order DWBA calculations using either the SP approximation ("no pairing") or the TNA from the Shell Model calculations with the GXPF1 interaction ("pairing") are also shown. Both for 54 Co and 50 Mn, the contribution of ∆T=1 transfer to the ground-state (J=0 + ,T=1) largely dominates over the contribution of ∆T=0 transfer to the excited state (J=1 + ,T=0). This is reproduced by the theoretical calculations. For the deuteron-like transfer, the SP or GXPF1 calculations give similar results, showing no enhancement related to isoscalar pairing. The weak population of the (J=1 + ,T=0) state has also been noticed in the measurement of 44 Ti( 3 He,p) 46 V [4] but not in the two-nucleon transfer measurements performed in the sd-shell. This observed lower cross-sections in the fp-shell may be explained by the fact that isoscalar np pairing is more hindered by the spin-orbit effect than in the sd-shell [11,30,31,32]. Indeed, the np pair configurations for the isoscalar channel are not only built with the same orbitals (as for the isovector channel) but also with the spin-orbit partner orbitals. In this case, the f 7/2 and f 5/2 orbitals have large spin-orbit splitting which disfavors isoscalar deuteron-like pairing. As discussed in [32], it reflects in the ground states of the odd-odd fp-shell nuclei that have the configuration (J=0 + ,T=1) except for 58 Cu which has (J=1 + ,T=0) configuration because the odd neutron and the odd proton occupy the 2p-orbit which has smaller spin-orbit splitting.

As for the ∆T=1 transfer to the ground state, its cross-section is almost an order of magnitude larger than the deuteron-like transfer. As expected, the transfer cross-section increases at the middle of the shell where the number of contributing pairs is larger. However the measured cross-section for the groundstate to ground-state transfer 52 Fe(p, 3 He) 50 Mn is lower than the predicted one. It might reflect the fact that T=1 pairing is affected by deformation [33].

Ratios of cross-sections give valuable information on the relative strength of the isoscalar and isovector pairing. Given the large statistical errors on the cross-section for the (J=1 + ,T=0) state, the usual error propagation formula based on the gaussian approximation for small errors are not valid. Following Ref. [29] for large errors, we have used the log normal distribution which results into asymmetric error bars. The ratios of cross-sections with their error bars are reported in Tab. 1 and on Fig. 4 where they are compared with theoretical predictions. For 52 Fe(p, 3 He), the cross-section to the 1 + state of 50 Mn is compatible with zero, our sensitivity being 9 µb (corresponding to one count in the gamma-ray line of this state). Thus the associated cross-section ratio σ(0 + ,T=1)/σ(1 + ,T=0) gives only a lower limit. Our measurement extends the experimental measurements reviewed in ref. [4] towards the fp-shell. The experimental ratios are compared to the ratios deduced from second-order DWBA calculations performed with GXPF1 TNA for 56 Ni and 52 Fe as well as 48 Cr(p, 3 He) in Fig. 4. As a reference the ratios for the SP case are also reported. The trend of the experimental points is well reproduced by the DWBA calculations with GXPF1. Even with the large error bars, the SP, which gives an almost flat behavior, can be dismissed. When the experimental ratio is higher (lower) than the SP ratio, isovector pairing (isoscalar) dominates over the isoscalar (isovector) pairing. Both for 54 Co and 50 Mn, the observed ratios point to a dominating isovector pairing.

On Fig. 4, the ratio for 40 Ca(p, 3 He) measured in Ref. [10] is also reported to complete the f -shell and to give a reference of closed shell nucleus for comparison with the 56 Ni(p, 3 He) case. For the former, the measured ratio is 1.75, i.e. the cross-section to the (J=0 + , T=1) ground-state is less than twice the crosssection to the first (J=1 + ,T=0) excited state. The experimental ratio is below the SP ratio so we can infer that, in this case, isoscalar pairing is dominant over isovector pairing. This is the opposite for 56 Ni(p, 3 He) , where the experimental ratio lies higher than the SP limit, indicating that isovector pairing dominates over isocalar pairing.

In summary, the (p, 3 He) two-nucleon transfer reactions on 52 Fe (open-shell) and on 56 Ni (closed-shell) from the fp-shell clearly show the important role of pairing correlations and, at the same time, do not provide evidence for an isoscalar deuteron-like (J=1) pairing condensate. Either the T=0 strength is mainly concentrated in the aligned (J=J max ) configuration or, more likely, the reduced effective degeneracy to form J=1 pairs due to the spin-orbit splitting fragments the strength to higher energies with individual components not strong enough to be observed above background. Similar experiments on heavier N=Z nuclei in the g-shell appear as the next logical step for further studies of the competition between isovector vs. isoscalar pairing but this will have to wait for the new generation of radioactive beam facilities.

Figure 1 :

 1 Figure 1: Experimental set-up (see text for details).

Figure 2 :

 2 Figure 2: (Color online) (Left) Excitation energy spectra for 56 Ni(p,3 54 Co (a) and for 52 Fe(p,3 He) 50 Mn (c). The background contribution deduced from the measurement on a 12 C target is shown as a light dotted line (see text for details). (Right) The associated gamma spectra with a gate on the excitation energy centered on the first (J=1 + ,T=0) state ±2σ (the gate in E * is given explicitly on the spectra). A simplified level scheme of the residual nuclei is shown with the main transitions and associated intensities taken from[22]. The deduced direct feeding by the transfer reaction (in number of counts) of each excited state is shown in blue with its statistical error bars (see text for details). This information is used to constrain the fit of the excitation energy (shown as a thick red line on spectra a) and c)). The population of the ground state (given in red) is deduced from the fit. The contribution of each state to the fit is identified with the same line code as for the associated gamma-ray lines (dotted for the gs, full for the 1 + , dashed dotted for the 2 + and dashed for the 3 (+) state). For 56 Ni(p, 3 He)54 Co a contribution at higher energy (around 3 MeV) is also shown and detailed in the text.

Figure 3 :

 3 Figure 3: (Color online) Angular distribution for 56 Ni(p, 3 He) 54 Co ground state obtained in this experiment (full dots) compared with second-order DWBA calculations with GXPF1 in black (dotted line for direct transfer and full line for direct+sequential transfer) and with SP configuration in red. The error bars correspond to the statistical ones.

Figure 4 :

 4 Figure 4: (Color online) Ratio σ(0 + , T = 1)/σ(1 + , T = 0) obtained in this experiment (black dots) and for second order DWBA calculations with GXPF1(red triangle) and SP picture (blue squares). 40 Ca results are taken from ref.[10].

Table 1 :

 1 Theoretical (based on second-order DWBA calculations) and experimental cross-sections for 56 Ni(p,3 He) and 52 Fe(p,3 He). For cross-sections, the first error bar given corresponds to the statistical one and the second one to the systematics errors. For the ratios, the error bar is only the statistical one (see text for details).

		σ(0+,T=1) (µb) σ(1+,T=0) (µb) Ratio
			56 Ni(p, 3 He) 54 Co
	this work	109	stat ± 5	sys ± 10	17	stat ± 7	sys ± 2	6.3 +3.1 -2.1
	SP		73			19			3.8
	GXPF1		136			21			6.4
			52 Fe(p, 3 He) 50 Mn
	this work	156	stat ± 12	sys ± 15	16 +29 -16	sys ± 2	9.8 +∞ -4.0
	SP		69			16			4.3
	GXPF1		257			17			15.1

Acknowledgments

The authors are very grateful to GANIL staff, particularly Vincent Morel for assistance with the set-up. The research leading to these results has received funding from the European Union under Seventh Framework Programme FP7 Infrastructures project ENSAR, grant agreement No. 262010. 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-AC02-05CH11231 (LBNL). A.O. Macchiavelli thanks the Université Paris-Sud for support as an invited professor.


V Girard-Alcindor

I Stefan

F De

Oliveira Santos

O Sorlin

D Ackermann

D Assunção

Beaumel

Mahir Hussein

F De Oliveira

Santos

P Adsley

J C Angélique

M Assié

M Assunção

D Beaumel

E Berthoumieux

R Borcea

L Cáceres

I Celikovic

M Ciemala

V Chudoba

G D'agata

F De Grancey

G Dumitru

F Flavigny

C Fougères

S Franchoo

A Georgiadou

S Grévy

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 Sánchez Benítez

N De Séréville

M Stanoiu

C Stodel

K Subotic

D Suzuki

V Tatischeff

J C Thomas

P Ujic

D Verney

Probing nuclear forces beyond the nuclear drip line: The cases of 16 F and 15 F A Tribute to

Keywords: 21, 10, -k Properties of nuclei; nuclear energy levels -27, 20, +n 6 ≤ 25, 70, Ef Resonances -25, 40, Cm Elastic proton scattering

    Probing nuclear forces beyond the nuclear drip line: the cases of 16 F and 15 F: A Tribute to Mahir Hussein

Introduction

The exploration of the nuclear landscape was made possible by the development of radioactive beams. Nuclear models have been developed and tested in nuclei far away a Present address: Technische Universität Darmstadt, Germany from the valley of stability [1,2,3,4]. Approaching the shores of this landscape, the nuclear drip lines, has allowed the observation of several new phenomena, such as halo nuclei, modification of the effective nuclear interactions, rearrangement of the nuclear shells, and clustering. Beyond the drip line, unbound nuclei usually disappear in the form of waves in the sea of the continuum, i.e. large res-onances. Theoretical description of the unbound nuclei, identification and understanding the role of specific parts of the nuclear forces, are still challenges to nuclear research. Recently, following Ikeda's work on α-clustering near α-emission threshold [5], Oko lowicz et al [6,7] proposed to generalize the Ikeda's conjecture to all near thresholds states, including states involving unstable subsystems like di-neutron or di-proton.

In this article, focus is given to the nucleon -nucleon interaction, represented by the Two-Body Two-Body Matrix Elements (TBME) in the nuclear shell model. The idea is to investigate by how much the nucleon -nucleon interaction changes when nuclei are fully embedded in the continuum.

Experimentally, the effective nucleon -nucleon interaction energy, labeled δV exp , can be extracted from the measured binding energies (BE) [8,9]. For instance, the n -p interaction energy is obtained from the relation δV exp = BE(Z, N) + BE(Z -1, N -1) -BE(Z -1, N) -BE(Z, N -1) Theoretically, δV th has been predicted to depend on the spatial overlap of wave functions of the last particles [9,[START_REF] Heyde | The Nuclear Shell Model[END_REF][START_REF] Ogawa | [END_REF]12]. Indeed, as one approaches the drip line, radial wave functions for these particles spread and dilute further in space. As a result δV th is expected to decrease since the spatial overlap between a well bound and the unbound nucleon is smaller.

A good way to shed light on the effect of the continuum in the nucleon -nucleon interaction is to compare level schemes of mirror nuclei involving a bound and an unbound nucleus. The asymmetries observed between the mirror nuclei can be used to single out the role of nucleon -nucleon interaction. In this article, we combine the results obtained from previous and new experiments performed at GANIL for the unbound nuclei 16 F and 15 F. In all these experiments, the resonant elastic scattering method was used with the thick target inverse kinematics technique [13,14,15,16].

The case of 16 F

The mirror pair, 16 F-16 N, can be considered a perfect case to investigate the effect of the continuum in the nucleonnucleon force. The structure of the 16 N nucleus (S n = +2488.8(2.3) keV) is well known, and the first low lying states are well described assuming pure single-particle configurations. The structure of the mirror 16 F nucleus (S p = -535(5) keV) have been previously investigated and the results are presented in detail in Ref. [17]. This experiment had the advantage of combining excellent energy resolution, high statistics, and precise energy calibration, marking a leap in quality and consistency over the precedent results obtained for this nucleus. Experimental and theoretical aspects of this study are presented succinctly hereafter.

Table 1. Measured energy, spin, width and spectroscopic factor of the low-lying states in 16 F. The properties of the 3 - state were adopted from Ref. [20].

Ex (keV) J π Γp (keV) S 0 0 - 25 ± 5 1.1(2) 193 ± 6 1 - 70 ± 5 0.91(8) 424 ± 5 2 - 6 ± 3 1.2(5) 722 ± 16 3 -15.1 ± 6.7 1.3(6)

Experiment

A radioactive beam of 15 O ions was produced at the SPI-RAL facility at GANIL through the fragmentation of a 95 AMeV 16 O primary beam impinging on a thick carbon graphite production target. The ions were post-accelerated by means of the CIME cyclotron up to the energy of 1.2 AMeV. It was possible to obtain an 15 O 6+ beam with an intensity of 1.0(2)x10 6 pps and 97(1) % purity. The ions were implanted onto a thick polypropylene (CH 2 ) n target. Some ions underwent proton elastic scattering and the scattered protons were detected promptly to the reaction in a 300 µm thick silicon detector that covered an angular acceptance of ±1 • 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 obtained in lab, which corresponds to 7 keV FWHM in the center of mass. The resonances in the compound nucleus 16 F ( 15 O +p) were studied through an R-matrix analysis of the spectrum. Three resonances can be observed corresponding to the 0 -ground state and the 1 -and 2 -first excited states of the unbound 16 F nucleus. The measured 15 O measured in inverse kinematics. The black line corresponds to the R-matrix fit. The presence of the 0 -ground state, and two excited states 1 -and 2 -is clearly observed. The resolution of the experiment was 7 keV FWHM in cm. Adapted from Ref. [17].

properties are presented in Table 1. The deduced spectroscopic factors of the low-lying states in 16 F are all close to 1. Similar values were found for the mirror nucleus 16 N [18,19]. Fig. 2. Level schemes of the two mirror nuclei 16 F and 16 N.

The states corresponding to the 1p 1/2 ⊗ 1d 5/2 and 1p 1/2 ⊗ 2s 1/2 single-particle configurations are shown in blue and red respectively.

Effective n -p interaction energies

The level schemes of the two mirror nuclei 16 N and 16 F are compared in Fig. 2. Large differences can be observed: the ground state of 16 F has J π = 0 -while that of 16 N has J π = 2 - [20]. In 16 F, compared to 16 N, it looks like both 0 -and 1 -states are down shifted in energy by about 500 keV relatively to 2 -and 3 -states. The lowlying states of the 16 N nucleus can be well described using a single-particle configuration with a closed core of 14 C and a deeply bound proton in the 1p 1/2 orbital (S p ( 15 N)= +10.2 MeV) coupled either to a neutron in the 2s 1/2 orbital (S n ( 15 C)= +1.22 MeV), leading to J π =0 -, 1 -states, or to a 1d 5/2 neutron (S n ( 15 C*)= +0.48 MeV), forming the J π =2 -, 3 -states. In the same way, (0,1) -and (2,3) - states in 16 F can be described as a 14 O core and a neutron in the 1p 1/2 orbital (S n ( 15 O)= +13.22 MeV) coupled either to a proton in the 2s 1/2 orbital (S p ( 15 F gs )= -1.27 MeV) or to a proton in the 1d 5/2 orbital (S p ( 15 F*)= -2.79 MeV). None of the states have the same spin, so, they don't mix, and if we neglect interaction with higher energy states of the same spin value, the spacing between the two members of a given multiplet is only due to the residual interaction. For example, in the case of the J π =0 - state in 16 N ( 14 C+n+p), originating from the n -p coupling π1p 1/2 ⊗ ν2s 1/2 , the effective n -p interaction is calculated with

δV exp ( 16 N) 0 -= BE( 16 N) 0 --BE( 15 N) 1/2 - +BE( 14 C) 0 + -BE( 15 C) 1/2 +
The same method was also applied to the J π =1 -state, and to the J π =2 -, 3 -states originating from the n -p coupling π1p 1/2 ⊗ ν1d 5/2 . The obtained experimental n -p interaction energies are given in Table 2. It is observed that the effective n -p interaction changes by 40% for the (0 -,1 -) multiplet as compared with the mirror nucleus 16 N, and by 10% for the (2 -,3 -). In all cases it is weaker in 16 F, in agreement with expectations that δV will decrease as the nucleus goes in the direction of the drip line. Comparison of the calculated single-particle wave functions for the 2s 1/2 orbital in 16 F and 16 N. The difference is due to the fact that the neutron in 16 N is bound whereas the proton in 16 F is unbound. The wave functions have been calculated using a Woods-Saxon potential.

Interpretation

This apparent breaking of the symmetry of the nuclear force between 16 N-16 F mirror nuclei is explained by the large coupling to the continuum. The single-particle wave functions have been calculated considering a Woods-Saxon, Coulomb and spin-orbit potentials, using a standard set of parameters (V W S ≈50-60 MeV, a = 0.6 fm, r 0 =1.26 fm, V SO =6 MeV). As can be seen in Fig. 3, the wave functions calculated for the 2s 1/2 orbit are different since the neutron is bound in 16 N and the proton is unbound in 16 F. This difference modifies the overlap between the 2s 1/2 and the 1p 1/2 wave functions, and so, the effective n -p interaction. This effect has been investigated in details by Ogawa et al [START_REF] Ogawa | [END_REF] using a Woods-Saxon plus M3Y force, and by Yuan et al [12] using a monopole-based-universal interaction (VMU) in the Woods-Saxon basis. Here, we use a schematic zero-range interaction v = -a δ(r p -r n ) to calculate the effective nucleon-nucleon interaction. In the definition, a is the positive strength of the interaction given in units of MeV.fm -3 , and the minus sign serves to emphasize its attractive nature. Despite its seeming simplicity, the δ interaction reproduces fairly well many properties of nuclei [START_REF] Heyde | The Nuclear Shell Model[END_REF]. One gets For the 4 states (0 -, 1 -, 2 -, 3 -) studied in 16 F- 16 N, the mirror symmetry is well preserved since the F is close to 1. In the case of the 1/2 -narrow resonance in 15 F, F can be more or less close to 1.0 depending on the its structure.

δV th = - a J 4π ∞ 0 1 r 2 [u p (r )u n (r )] 2 dr
The case of a pure 13 N+π(1d 5/2 ) 2 structure is indicated by a black star, for a pure 13 N+π(2s 1/2 ) 2 by the blue star, and for 2 He cluster+ 13 N core by the red star. The associated errors are much smaller than the size of the stars.

where a J contains the strength of the n -p nuclear interaction in the state with spin J, and u n (r) and u p (r) are the radial wave functions calculated with a Woods-Saxon potential. By virtue of the charge symmetry of nuclear forces, the same a J coefficients have been used to calculate the interaction energies in the mirror nuclei 16 F and 16 N.

The calculated Reduction Factors δV th ( 16 F) δV th ( 16 N) are presented in Table 3. These factors are of the same order as those calculated in Ref. [12], even if they were not calculated with the same core nucleus ( 16 O instead of 14 O).

The Asymmetry Factor

F = δV exp ( 16 F) δV th ( 16 F) × δV th ( 16 N) δV exp ( 16 N)
can be used to determine the degree of mirror asymmetry remaining after correction for wave functions differences in the mirror states. If the observed asymmetry is due solely to this effect, F must be equal to 1.0. The calculated value of F for all states studied in this article are presented in Fig. 4. As can be observed in the figure, these factors are close to the expected value of 1.0 for the 4 states. Small deviations of about 10% are observed for the 2 -and 3 - states, but the model used here is too simplistic to give real credit to these small deviations. Despite the fact that the experimental n-p interaction energies δV exp are reduced, the mirror symmetry is preserved. In agreement with the conclusion of Ref. [START_REF] Ogawa | [END_REF] and [12], we confirm that the large differences in effective n -p interaction energies between the mirror states are mainly explained by the different overlaps between the wave functions.

3 The case of 15 F

The isotope 15 F is located two neutrons away from the proton drip line. The study of this nucleus is briefly presented below. More details about this study can be found in Ref. [22,23,24,25]. In addition, new results have been obtained recently and are presented here for the first time.

Experiments

Three experiments were performed at GANIL to study the unbound 15 F nucleus. The third experiment was focused on the study of the two-proton decay. The obtained results [25] will be published soon in a refereed journal [24]. The objective of the first two experiments was the study of the structure of the low lying states. The excitation function of the elastic scattering reaction 14 O(p,p) 14 O was measured at 180 • (c.m.), as can be seen in Fig. 5. It was obtained in inverse kinematics using a thick polyethylene (proton) target. This spectrum has high statistics, good energy resolution with 16.5 (5) keV FWHM (≈ 95 keV FWHM for the second experiment) and large energy covering from 0.4 MeV to 5.6 MeV (4 MeV to 5.6 MeV). The analysis of the excitation function was performed using the R-matrix method with the code AZURE2 [26]. The fit is shown by the continuous blue line. The measured spectroscopic properties are presented in Table 4, and they were used to produce the level schemes shown in Fig. 6. The obtained values for the two first states are in agreement with those from the literature [22]. In addition, the second excited state was clearly observed as a narrow dip in the excitation function, see right side of Fig. 5. In the mirror nucleus 15 C, the second excited state has spin J π = 1/2 -. The shape of the dip observed in the excitation function presented in Fig. 5 could be reproduced by R-matrix calculation assuming J π = 1/2 -. The R-matrix fit of the excitation function was performed taking into account the experimental resolution.

A second experiment was performed at GANIL in order to confirm the existence of this 1/2 -state and to search for new states at higher energy. The same technique of resonant elastic scattering was used, although the measurement was made with a completely different experimental setup. A much thinner (≈ 100 µm) polyethylene target was used and the protons were detected with the MUST2 [START_REF] Pollacco | MUST2: A new generation array for direct reaction studies[END_REF] ensemble of silicon detectors. Details and results of this experiment are presented in Ref. [24,25]. As shown with the red dots in the right side of Fig. 5, the new measurement perfectly confirms the presence of a dip in the excitation function. The resonance is measured at E R = 4.880(140) MeV with Γ = 23 [START_REF] Heyde | The Nuclear Shell Model[END_REF] keV. It is in good agreement with the results of the first experiment, confirming the existence and properties of this narrow resonance.

The two protons emission from this narrow resonance is energetically possible, see Fig. 6. Since there is no intermediate state accessible to 14 O, it should be a direct twoproton emission to the g.s. of 13 N. Considering that the available energy is only Q 2p = 130 keV [21], the branching ratio for the two proton emission is negligible (Γ 2p < 10 -10 eV), and the measured width corresponds only to the emission of a single proton towards the g.s. of 14 O.

Effective p -p and n -n interaction energies

The observation of this narrow resonance in 15 F is quite surprising since it is located 1.6 MeV above the Coulomb plus the =1 centrifugal barrier for proton emission. There is no barrier to retain the proton inside the nucleus. Moreover, it is difficult to calculate the single-particle width for such a loosely bound state. The Wigner width is Γ W ≈7.4 MeV. Experimentally, the spectroscopic factor for the oneproton emission is S gs =0.005 (calculated with the Wigner width). It is clear that the structure of this state is far from being a proton coupled to a 14 O gs core. It is interesting to look for the 99.5% missing component.

The situation is not completely symmetrical in the mirror 15 C nucleus. The 1/2 -state in the mirror nucleus is at the excitation energy E x =3.10 MeV, it is unbound for the emission of one neutron by 1.89 MeV, and bound for the emission of two neutrons by 6.29 MeV, see Fig. 6. This state has also been measured as a narrow resonance, with a width of 38 keV [START_REF] Fortune | [END_REF]. It is thought to be primarily an (sd) 2 excitation, i.e. 13 C(gs)×(sd) 2 0 + [START_REF] Fortune | [END_REF]29]. Theoretically, shell model calculations were performed with the code NUSHELLX [30] in the psd space and with the psdmk interaction [31]. It is calculated that the spectroscopic factor to the ground state is

S gs = | < 14 O gs + p| 15 F 1/2 -> | 2 =0
.025, confirming the weak overlap with 14 O gs .

Canton et al [32] used the multichannel algebraic scattering (MCAS) theory with Pauli-hindered method to predict the properties of the low-lying states in 15 F. The calculations was recently updated and several unusual narrow resonances are predicted at high energies [33]. A very narrow width Γ =5 keV was predicted for the 1/2 -state [32], later updated to Γ =107 keV [33].

Fortune and Sherr [29], using a potential model, determined the single-particle widths for 15 C. These values were scaled down to reproduce the measured widths in 15 C, and the extracted dimensionless reduced widths θ 2 were used to calculate widths in the mirror nucleus 15 F. The results of these calculations confirmed that narrow resonances are to be expected in 14 O+p, with a prediction of Γ =55 keV for the 1/2 -state. Refined values were later published by Fortune [START_REF] Fortune | [END_REF], with Γ =38 keV. Canton et al [34] objected that θ 2 do not necessary scale with the single-particle widths, especially when θ 2 is small [35]. Moreover, the presence of a particle continuum can significantly impact spectroscopic properties of weakly bound nuclei and excited nuclear states close to, and above, the particle emission threshold [36], which is the case here. Gamow Shell Model in the coupled channel representation [37] was also used to determine the properties of this state. The calculations were performed considering a 12 C core and three valence protons. It was confirmed that the overlap with 14 O g.s. +p is very weak, with S gs =0.0035. It was found that the 1/2 -state is 97% composed of two quasi-bound protons located in the 2s 1/2 shell [38].

The formalism presented in the previous section to deduce the n -p effective interaction energies was applied to 15 F(1/2 -) to deduce the p -p effective interaction energy. We suppose that the 1/2 -state can be described by an inert core of 13 N gs (1/2 -) coupled to two protons in the 2s 1/2 orbit. Shell model calculations show that the first excited state of 14 O 1 -(E x =5.173 MeV) is well described by a 13 N gs plus one proton in the 2s 1/2 shell, the spectroscopic factor being S=0.70. In this case, we can write for 15 F

δV exp pp = BE( 15 F) 1/2 -+ BE( 13 N) 1/2 --2 × BE( 14 O) 1 -
The n-n interaction energy is determined in the same way with 15 C. The obtained interaction energies are listed in Table 5. However, both valence protons in 15 F are charged and the Coulomb repulsion energy must be corrected for. This was calculated using the formula [39]

W = 1 2 ρ(r)φ(r)dv
where ρ(r) is the density of charge, φ(r) the electrical potential and dv a volume element. The calculated correction is W = 179 keV, which gives δV exp pp =-1.142 MeV. This value indicates that the interaction energy is weaker in 15 F by a factor 1.9, once again in agreement with the expectation that the interaction decreases as the nucleus approaches, or goes further, the drip line.

Interpretation

If the 1/2 -state in 15 F is described as a 13 N core+ two proton in the 2s 1/2 orbital, a huge reduction factor of 0.33 is calculated, see Table 3. This is in agreement with the results of Ref. [12] where the greatest reduction factor has been observed for this TBME. Actually, their calculated value is 0.68, which is twice the value of the present work. However, it should be emphasize that their values has been calculated with a different interaction. Also, the asymmetry factor F , shown in Fig. 4, is 63% larger than the expected value of 1. Such a large difference is significant and needs an explanation. One of them is the fact that core polarization effects, which should be included, has not been taken into account in our simple model. Also, it is possible that the a J strength coefficients are different between the two nuclei. This may be due to the fact that the 1/2 -resonance in 15 F is located only 130 keV above the two-proton emission threshold, whereas in the mirror nucleus 15 C, the 1/2 -state is bound by 6.3 MeV for two-neutron emission. This asymmetry in 15 F/ 15 C would be an indication of the generalized Ikeda's conjecture [6,7]. The p -p correlation might be enhanced in 15 F compared to the n -n correlation in the mirror nucleus 15 C. To estimate the impact of this correlation enhancement, we treated 15 C(1/2 -) as 13 C + 2 n where the 2 n cluster is bound in a Woods Saxon potential well. We then applied the same potential to calculate the resonance energy of the mirror system 15 F as 13 N+ 2 He, correcting for the Coulomb energy in 2 He (≈ 200 keV). The calculated asymmetry factor is F =1.16, as shown in Fig. 4 by a blue star. Thus, to explain the 15 F/ 15 C difference it would be necessary to describe the 1/2 -state of 15 F as being 100% composed of 13 N+ 2 He cluster.

Another much simpler solution would be to add to the (2s 1/2 ) 2 wave function a part of (1d 5/2 ) 2 component, which would have the effect of increasing the overlap of the wave functions, reducing the asymmetry. To get the asymmetry factor F = 1, the states in 15 F and 15 C should be described as composed with 58% of (1d 5/2 ) 2 and 42% of (2s 1/2 ) 2 . Fortune et al [START_REF] Fortune | [END_REF] predicted 46% and 54% respectively, using the Lawson-Serduke-Fortune two-body matrix elements. Using these values, they predicted the energy of the 15 F 1/2 -state at E R =4.63 MeV [29], which is only 127 keV below our measured energy.

Conclusion

In the present investigation we have explored the nucleonnucleon force in nuclei beyond the proton drip line. Strong asymmetry between 16 F-16 N and 15 F-15 C mirror nuclei was observed. For the 16 F and 16 N pair, the apparent breaking of the symmetry can be explained by the difference in the overlaps of the wave functions due to the coupling with the continuum. The recently measured and confirmed 1/2 -state in 15 F, on the other hand, shows asymmetry that cannot be explained by the difference in the overlaps of pure (2s 1/2 ) 2 wave functions. This may be an indication of an increase in correlation between the two least bound protons due to the proximity with the twoproton emission threshold. It may also be an indication that the states are described by an almost equal mixture of (2s 1/2 ) 2 and (1d 5/2 ) 2 components.

Fig. 1 .

 1 Fig. 1. Excitation function of the reaction 15 O(p,p) 15 O measured in inverse kinematics. The black line corresponds to the R-matrix fit. The presence of the 0 -ground state, and two excited states 1 -and 2 -is clearly observed. The resolution of the experiment was 7 keV FWHM in cm. Adapted from Ref. [17].

  Fig.3. Comparison of the calculated single-particle wave functions for the 2s 1/2 orbital in16 F and16 N. The difference is due to the fact that the neutron in16 N is bound whereas the proton in16 F is unbound. The wave functions have been calculated using a Woods-Saxon potential.

Fig. 4 .

 4 Fig.4. Asymmetry factors F (see text) are shown for several mirror states. For the 4 states (0 -, 1 -, 2 -, 3 -) studied in16 F-16 N, the mirror symmetry is well preserved since the F is close to 1. In the case of the 1/2 -narrow resonance in15 F, F can be more or less close to 1.0 depending on the its structure. The case of a pure 13 N+π(1d 5/2 ) 2 structure is indicated by a black star, for a pure13 N+π(2s 1/2 ) 2 by the blue star, and for 2 He cluster+13 N core by the red star. The associated errors are much smaller than the size of the stars.

Fig. 5 . 14 Fig. 6 .

 5146 Fig.5. Measured excitation function of the reaction 14 O(p,p)14 O. Left: Data and calculation renormalized by a factor 0.1. The experimental points in black are from Ref.[22]. The red points are new data confirming the existence of the dip and the 1/2 - resonance. The continuous blue line corresponds to the R-Matrix fit. Three unbound states are identified: The 1/2 + ground state, the 5/2 + first excited state, and the 1/2 -very narrow resonance.

Table 2 .

 2 Measured effective n -p interaction energies. The uncertainties are lower than 10 keV on all values. Calculated from Ref.[17,21] State (J π ) δV exp ( 16 N) δV exp ( 16 F)

		(MeV)	(MeV)
	0 -	-1.151	-0.735
	1 -	-0.874	-0.542
	2 -	-2.011	-1.834
	3 -	-1.713	-1.536

Table 3 .

 3 Calculated reduction factor for the effective nucleon -nucleon interaction.

			δV th		Reduction Factor
		< 1p 1/2 2s 1/2 |v pn |1p 1/2 2s 1/2 >		0.66
		< 1p 1/2 1d 5/2 |v pn |1p 1/2 1d 5/2 >		0.80
		< (2s 1/2 ) 2 |v pp |(2s 1/2 ) 2 >		0.33
		2
	F Asymmetry Factor	1 1.2 1.4 1.6 1.8				2 ) He cluster 1/2 100% (2s 2
		0.8
		0.6				5/2 100% (1d	2 )
		-F 0 16	-1	-2	-3	-F 1/2 15

Table 4 .

 4 Measured energy, spin, width and spectroscopic factors of the low-lying states in15 F. The numbers in brackets correspond to statistical and systematic uncertainties.

	ER (keV)	J π	Γp (keV)		S
	1270(10)(10) 1/2 + 376(70)( +200 0 2763(9)(10) 5/2 + 305(9)(10)	)	0.8 ≈1.2
	4757(6)(10)	1/2 -	36(5)(14)		≈0.005

Table 5 .

 5 Measured effective interaction energies in15 F and

	15 C
	State (J π )	δV exp nn	δV exp pp
		(MeV) (MeV)
	1/2 -	-2.127	-0.963
	Coulomb corrected	-	-1.142

Acknowledgments

We would like to thank the GANIL accelerator staff for their support. We also received support by OP RDE, MEYS Czech Republic under the project EF16-013/0001679, by LIA NuAG and COSMA, and by IRP France-Brésil.