Neutron skin and signature of the $N$ = 14 shell gap found from measured proton radii of $^{17-22}$N

A thick neutron skin emerges from the first determination of root mean square radii of the proton distributions for $^{17-22}$N from charge changing cross section measurements around 900$A$ MeV at GSI. Neutron halo effects are signaled for $^{22}$N from an increase in the proton and matter radii. The radii suggest an unconventional shell gap at $N$ = 14 arising from the attractive proton-neutron tensor interaction, in good agreement with shell model calculations. $Ab$ $initio$, in-medium similarity re-normalization group, calculations with a state-of-the-art chiral nucleon-nucleon and three-nucleon interaction reproduce well the data approaching the neutron drip-line isotopes but are challenged in explaining the complete isotopic trend of the radii.

measurements around 900A MeV at GSI. Neutron halo effects are signalled for 22 N from an increase in the proton and matter radii. The radii suggest an unconventional shell gap at N = 14 arising from the attractive proton-neutron tensor interaction, in good agreement with shell model calculations. Ab initio, in-medium similarity re-normalization group, calculations with a state-of-theart chiral nucleon-nucleon and three-nucleon interaction reproduce well the data approaching the neutron drip-line isotopes but are challenged in explaining the complete isotopic trend of the radii.
Keywords: Proton Radii, Matter Radii, Neutron Skin, Shell structure, Magic Number, Shell Model, Ab initio theory, radioactive beams Neutron-rich nuclei are fertile grounds to search for unexpected features.
Exotic nuclear forms are revealed with the formation of neutron skins and halos approaching the neutron drip-line [1][2][3], many of which relate to modifications of conventional shells. The emerging signatures of changes in the shell structure must be identified and their origins understood. The presence of neutron halos in 11 Li and 11 Be relate to the breakdown of the N = 8 shell gap. Evidence has been found for a new shell gap at N = 16 at the drip-line of carbon to fluorine isotopes. Studies of excited states and momentum distributions have discussed a shell gap at N = 14 between the 1d 5/2 and 2s 1/2 orbitals in oxygen isotopes. However, its reduction for the nitrogen isotopes is signalled and its disappearance due to a level inversion is predicted in the carbon isotopes [4][5][6][7][8][9][10][11][12].
Therefore, further experimental investigation is needed for revealing the cause of this shell gap and its evolution. Systematic trends of proton radii along an isotopic chain can reveal the presence of neutron magic numbers [13].
In this work the first determination of root mean square radii of density distribution of protons treated as point particles, referred to henceforth as point proton radii of neutron-rich isotopes 17−22 N together with those for stable nuclei 14,15 N is presented from a measurement of charge changing cross sections. The proton radii decrease from 17 N to 21  The situation changes in 20,21 N where the P || are explained with 83% and 68% probability, respectively of valence neutrons in the l = 2 orbital with the core in its ground state. A shell gap at N = 14 in 22 O was first indicated from the high excitation energy of its first excited state [5,6]. Proton inelastic scattering [7] affirms this, implying a small quadrupole deformation (β = 0.26(4)) and a B(E2) value deduced to be 21 (8)  This shell gap was found to be strongly reduced to 1.41(17) MeV in 22 N and predicted to disappear in the carbon isotopes [11]. Ref. [12] however deduces a moderately large energy gap of 3.02 MeV at N = 14 from the excited states in 21 N. Therefore, more experimental information is needed to understand if N = 14 is a shell gap in the nitrogen nuclei which is at the transition region between the oxygen and carbon nuclei.
Point proton radii of nuclei reflect deformation and shell effects. Shell gaps can be visible as local minima in these radii along an isotopic chain [13] and can bring new insight into shell evolution. The charge radii of 18−28 Ne [14], from isotope shift measurements show local minima at N =8 and 14.
The first determination of point proton radii (R p ) from measurements of charge-changing cross sections (σ cc ) for the neutron-rich isotopes 17−22 N as well as for the stable isotopes 14,15 N are reported here. The experiment was performed at the fragment separator FRS [15] at GSI in Germany. Beams of 14,15 N and 17−22 N were produced by fragmentation of 22 Ne and 40 Ar, respectively, at 1A GeV, on a 6.3 g/cm 2 thick Be target. The isotopes of interest were separated in flight and identified using their magnetic rigidity (Bρ), time-of-flight and the energy-loss measured in a multi-sampling ionization chamber (MUSIC) [16]. In The σ cc is then obtained from the relation σ cc = t −1 ln(R Tout /R Tin ). Here R Tin and R Tout are the ratios of N SameZ /N in with and without the reaction target, respectively and t is the target thickness. The term R Tout accounts for losses due to interactions with non-target materials and for detection efficiencies. nuclei, the admixture of Z = 6 and 8 relative to Z = 7 is < 10 −4 . Fig. 1 shows a particle identification spectrum before the target where the black circled events denote the isotope of interest and the other events are contaminant fragments.
With the desired nitrogen isotope events (N in ) selected for the incident beam, the events having Z = 7 after the reaction target were counted using the MUSIC detector placed downstream of the target. This yielded N SameZ .
The TPC and the plastic scintillator detectors placed downstream of the second MUSIC provided additional Z information and their correlation with the MUSIC detector ensured proper Z identification and MUSIC efficiency determination. The 1σ Z resolution for nitrogen in the MUSIC was ∆Z∼0.11. To obtain N SameZ , a selection window of width ∼4σ was put around the Z = 7 peak ( Fig. 1(c)) in the low-Z side of the spectrum. On the high-Z side, the selection window includes the peak at Z = 8 ( Fig. 1(c)). This is because an increase in Z does not arise from interactions between protons in the incident N isotope and target nucleons. Therefore, they need to be included in the unchanged charge events (N SameZ ) used to measure σ cc for extracting R p .
For the stable nucleus 14 N the measured σ cc is 828±5 mb. However, oneneutron removal here can lead to 13 N, in proton-unbound excited states since the one-proton separation energy is very low (S p = 1.9 MeV). In such a situation decay by proton emission changes Z, although this process does not involve any interaction with the protons. The corrected σ cc for 14 N is therefore obtained by subtracting the one-neutron removal cross section that leads to proton unbound states in 13 N. This cross section is estimated to be 35±7 mb, using the Glauber model and spectroscopic factors based on pickup reaction measurements [18] as well as from shell model calculations using the WBP Hamiltonian in the p-shell [19]. For 15 N, the one-neutron removal cross section to 14 N states above the proton threshold is estimated to be 12±5 mb using spectroscopic factors from Ref. [20]. The measured cross section of 828±20 mb for 15 N is therefore corrected to eliminate this neutron removal effect. The corrected σ cc for 14,15 N are listed in Table 1. For neutron-rich nuclei 17−22 N the proton separation energy gradually increases thereby greatly decreasing the neutron removal cross section to proton unbound states. Hence, no correction of σ cc is necessary for these nuclei.
The measured cross sections with one standard deviation total uncertainties are listed in Table 1. The uncertainties contain a ∼0.1% contribution from target thickness, uncertainties from contaminants in the beam events and fluctuations within the selected phase space. The σ cc values decrease with lowering of the event rejection threshold of the veto detector. Table 1 shows the σ cc values with complete rejection of events hitting the veto detector and no rejection of veto hit events. It may be mentioned that the σ cc reported in Ref. [21] have large uncertainties thereby making them unsuitable for discussing nuclear structure evolution and neutron skin thickness. No radii were determined in Ref. [21].
The cross sections are analyzed within the Glauber model framework described in Ref. [22] and used in Refs. [23][24][25] to derive the point proton radii.
A harmonic oscillator point proton density distribution is considered without any recoil effect from the neutron distribution. The reaction cross section (σ R ) is calculated with the nucleon-target formalism in Glauber theory [26] which effectively includes the multiple-scattering effect missing in the optical-limit approximation [26,27]. At the high beam energies of this experiment, the interaction cross section (σ int ) is approximately equal to σ R . Both the σ cc and σ R are evaluated with the finite-range profile function parametrized in Ref. [28].
Once the input densities for the projectile and target, and the profile function parameters are fixed, the theory has no adjustable parameters. A study of σ R and σ cc for 12 C+ 12 C in Ref. [22] shows that the uncertainty from profile function parameters at energies around 900A MeV is less than 1% from the consistency of finite range calculations with known density of 12 C. This is also seen from the comparison of proton radii for 12,13 C from σ cc and e − scattering [25]. The systematic uncertainty in σ R from different projectile densities with the same radius is around 5% [29] mainly due to differences in the neutron density tail.
Ref. [30] shows that a 2% uncertainty in σ R leads to 5% uncertainty in the matter radius with different densities. This uncertainty is smaller for the proton radius since the density of the deeply bound protons does not have an extended tail and the σ cc measured have ∼0.6-1% uncertainty. The harmonic oscillator density profile used here is well justified and therefore no significant systematic uncertainty can be foreseen in the radii reported here. Systematic uncertainties however, do not change the relative isotopic variation of radii.
Using the σ cc with full veto rejection, the point proton radius R ex,veto p of 14 N derived from σ cc is consistent with that (R including pulse height with ADC overflow) after scaling is listed in Table 1 (3) 3.08 (12) that 1245±49 mb [32]. Considering an admixture of the 1d 5/2 neutron orbital with the 21 N core in the 3/2 − , 1.16 MeV excited state the data are consistent with a valence neutron probability of 50% -100% in the 2s 1/2 orbital. This agrees within the upper end of the uncertainty band with the momentum distribution measurement [4]. Therefore the evolution of R ex p sheds new light on the halo-like structure in 22 N. The R ex p of 15,18−22 N are consistent with the relativistic mean field (RMF) predictions of Ref. [33,35] (Fig.2b) while the radii predicted in Ref. [34] are larger than the data.  Fig.2b shown for comparison). However, as N increases, the depth of the one-body potential for proton must increase due to more attractive neutron-proton interaction. This effect is therefore, taken into account for neutron-rich isotopes by adopting a WS potential, whose strengths Ab initio calculations were performed with the chiral nucleon-nucleon and three-nucleon potential N 2 LO sat [38], which has been shown to describe binding energies and radii of light and medium-mass nuclei accurately [38][39][40][41]. We employ two different theoretical frameworks, namely the coupled-cluster [42,43] and valence-space in-medium similarity renormalization group (VS-IMSRG) methods [44][45][46][47]. Coupled-cluster calculations use a Hartree-Fock basis of 15 major oscillator shells withhω = 16 MeV, while VS-IMSRG use a basis of 11 major oscillator shells withhω = 22 MeV. In our coupled-cluster calculations we apply the normal-ordered two-body approximation [48][49][50] for the N N N interaction, with an additional energy cut on three-body matrix elements e 1 + e 2 + e 3 ≤ E 3max = 16hω. The VS-IMSRG calculations use a Hartree-Fock like reference which is constructed with respect to ensemble states above 4 He for 14−15 N, and states above 10 He for 16−23 N following Ref. [47]. This allows for a similar normal-ordered two-body approximation for the N N N interaction, with E 3max = 14hω. The coupled-cluster calculations of 13−17 N and 21−23 N are done by employing generalized equation-of-motion states obtained by charge-exchange, particle-removed, and particle-attached from closed shell calculations of carbon and oxygen isotopes [43,[51][52][53][54][55]. Charge-exchange, particle-removed, and particle-attached calculations are approximated at the coupled-cluster singles and doubles, 2p − 1h, and 1p − 2h excitation level, respectively. In addition, we augment the particle-attached (removed) calculations with the newly developed 3p − 2h (2p − 3h) perturbative corrections [56]. These are in essence based on the completely renormalized coupled-cluster formalism [57][58][59]. On the other hand, the VS-IMSRG approach generates an effective shell-model interaction which can be diagonalized by conventional means such that all nitrogen isotopes within the valence space can be calculated.  (Fig. 2b), and are in agreement with the measured radii for 19−22 N and just slightly lower than the data for 14 N. The decrease in R ex p observed experimentally for 17−21 N is not successfully predicted by either of the ab initio frameworks. An indication of a flattening in R p for N = 13-15 is seen in VS-IMSRG calculations, which predict a prominent dip in radius for 14 N. We note that ab initio calculations of neutron-rich calcium isotopes yielded a similar discrepancy with observed radii [40].
The matter radii (R ex m ) of 14,15,17−22 N are extracted in this work using the Glauber model where the proton radius of R ex,avg p is used and the neutron radius from harmonic oscillator density is varied to get the different matter radii that reproduce the measured σ int [32]. For 17 N, the σ int data at 710A MeV [32] are adopted for R ex m and neutron skin. The extracted R ex m , listed in Table 1 and shown in Fig. 3 neutron-neutron interaction between the 1d 5/2 -2s 1/2 (1d 5/2 -1d 5/2 ) orbitals as the neutron 1d 5/2 orbital is filled while the proton-neutron interaction is attractive for p(1d 5/2 )-n(1d 5/2 ) leading to less configuration mixing, correspondingly smaller deformation, and a smaller point proton radius. For nitrogen isotopes the proton-neutron tensor interaction is more attractive for p(1p 1/2 )-n(1d 5/2 ) orbitals, thereby reducing the gap between proton 1p 1/2 and 1p 3/2 orbitals when more neutrons are added in the d 5/2 orbital, hence resulting in a small point proton radius because of the lowering of the 1p 1/2 orbital. This attractive interaction also lowers the neutron 1d 5/2 orbital leading to the N = 14 shell gap.
It is therefore, reflected also in a dip in the matter radius and neutron skin thickness at N = 14.
In summary, the point proton radii for neutron-rich 17