Unbound excited states in 19,17C

The neutron-rich carbon isotopes 19,17C have been investigated via proton inelastic scattering on a liquid hydrogen target at 70 MeV/nucleon. The invariant mass method in inverse kinematics was employed to reconstruct the energy spectrum, in which fast neutrons and charged fragments were detected in coincidence using a neutron hodoscope and a dipole magnet system. A peak has been observed with an excitation energy of 1.46(10) MeV in 19C, while three peaks with energies of 2.20(3), 3.05(3), and 6.13(9) MeV have been observed in 17C. Deduced cross sections are compared with microscopic DWBA calculations based on p-sd shell model wave functions and modern nucleon-nucleus optical potentials. Jpi assignments are made for the four observed states as well as the ground states of both nuclei.

With the advent of new radioactive beam facilities capable of producing intense beams of various nuclear species far from the stability line, even at the drip-line, an increasingly large amount of information on nuclear levels and modes of excitation is being accumulated throughout the nuclear chart.
New phenomena, such as nuclear halos and skins [1], enhanced E1 transition strengths [2,3], and modifications of shell closures [4,5] have been revealed. Of particular interest in recent years are the neutron-rich carbon isotopes, which have attracted attention not only from their own structural interest, such as the anomalously reduced E2 transition strength in 16 C [6,7], but also for their implications for a new magic number at the neutron number N=16 proposed in oxygen isotopes [5,8].
Neutron number dependence of ground state deformations of carbon isotopes has been investigated in a deformed Hartree-Fock (HF) theory [9]. Generally the prolate deformation is expected at the beginning of the shell whereas oblate deformation arises towards the end of the shell. Note that the occurrence of nuclear deformations in the ground state is a consequence of the spontaneous symmetry breaking effect known in many fields of physics, and there is broad interest in elucidating the intriguing deformation-driving mechanism in atomic nuclei [10], which would be a nuclear physics analogue of the Jahn-Teller effect in molecular physics [11]. The theory predicts prolate deformations for carbon isotopes with N=9-11. For 19 C with N=13 two almost degenerate deformed minima are predicted with spins J π =1/2 + (prolate) and 3/2 + (oblate). Since the shape change at a neutron number smaller than the middle of N=8 and 20 indicates a new shell closure at N=16, it is argued that definite information on the structure of 19 C is important to clarify the possible new shell effect in the neutron-rich carbon isotopes [9,12].
The 19 C nucleus is the heaviest odd carbon isotope. It is loosely bound with a neutron separation energy of S n =0.58(9) MeV [13], and exhibits one-neutron halo structure as evidenced by large Coulomb break-up cross sections [14]. The ground state spin and parity were also investigated via the measurements of longitudinal momentum distributions of charged fragments after the removal of one neutron from 19 C [15,16]. From these measurements the spin-parity was assigned to be J π g.s. =1/2 + . Other possibilities, however, have been suggested by authors in Ref. [17] based on their experiment in search of an isomeric transition. Moreover, there remains a controversy over the different widths of longitudinal momentum distributions measured at different energies [18,19], which has led to a conjecture of a possible resonance state just above the particle decay threshold as a clue to solve such an inconsistency [20].
In this situation it it worthwhile accumulating experimental information on the ground as well as excited states of 19 C. This paper reports a new measurement using the (p, p ′ ) inelastic scattering on 19  There exists one (p, p ′ ) work on 19 C reporting two bound states at 0.20 and 0.27 MeV using γ-ray spectroscopy [24]. These states are tentatively assigned as 3/2 + and 5/2 + , respectively, based on the assumption of J π g.s. =1/2 + . Close proximity of levels near the ground state has made it difficult to identify levels from comparisons in excitation energy between theory and experiment. Note that in shell model calculations [16] the triplet of levels are predicted below 0.62 MeV with no ground state configuration favoured. We have chosen to probe states in the unbound region, where in contrast to the bound region a lower level density is predicted up to about 3 MeV from the threshold in a shell model calculation described later. With no states known above the particle decay threshold, the measurement involved a search for resonances in this region, and we report a new state in this paper. For 17 C eleven new states up to 16.3 MeV excitation energy have been recently reported from the threeneutron transfer reaction 14 C( 12 C, 9 C) 17 C, with limited spin assignments [25].
The experiment was performed at the RIKEN Accelerator Research Facility (RARF). The radioactive beams of 19 C and 17 C at 70 MeV/nucleon were produced using the projectile-fragment separator, RIPS [26], from a 22 Ne primary beam at 110 MeV/nucleon. Typical beam intensities were 260 cps for 19 C and 10.4 kcps for 17 C with momentum spreads ∆P/P of 3.0% and 0.1%, respectively. The secondary target was a cryogenic hydrogen target [27] having a cylindrical shape, 3 cm in diameter and 120±2 mg/cm 2 in thickness.
The target was surrounded by forty-eight NaI(Tl) scintillators used to detect de-excitation γ-rays from the charged fragments. Each crystal had a dimension of 6.1×6.1×12.2 cm 3 . The charged fragments were bent by a dipole magnet behind the target and were detected by a plastic counter hodoscope.
Multi-wire drift chambers placed before and after the magnet were used to extract trajectory information of the charged particles. Neutrons were detected by a neutron hodoscope consisting of two walls of a plastic scintillator array placed 4.6 and 5.8 m behind the target. Each wall had a dimension of 2.14 W ×0.72 H (or 0.90 H )×0.12 T m 3 . The total efficiency of the hodoscope was 24.1±0.8% for a threshold setting of 4 MeVee. This was deduced by measuring the 7 Li(p, n) 7 Be(g.s.+0.43 MeV) reaction at E p =70 MeV and using existing cross section data [28]. The invariant mass of the final system was calculated event-by-event from the momentum vectors of the charged particle and the neutron. A series of studies probing unbound resonance states in beryllium isotopes has been successfully performed using a detector setup similar to the present one [29,30,31]. The experimental spectra were analyzed to extract the resonance energy E r and the width Γ r in the following way. Firstly, a single Breit-Wigner shape function was generated for certain values of E r and Γ r : The shift function ∆E l (E rel ) and the level width Γ l (E rel ), which depend on the relative energy E rel , were calculated by using the penetration P l and shift S l factors [34] by the relations: where l refers to the decay angular momentum. Decay neutrons were supposed to be in the l=2 orbit. The channel radius was taken to be MeV), and 3/2 + 2 (3.08 MeV) are predicted above the decay threshold and below 3.5 MeV.
In order to further clarify the nature of states involved in transitions shown in Table 1, microscopic DWBA calculations were performed using the code DW81 [39]. The optical potential was taken from the global parameterization KD02 [40]. A microscopic optical potential [41] based on the approach of Jeukenne, Lejeune, and Mahaux (JLM) [42] was also tested. The projectilenucleon effective interaction was the M3Y interaction [43]. The transition density was calculated with the shell model using the WBT interaction [37]. The single-particle wave function was generated in a harmonic oscillator well. The oscillator parameter was chosen so that the rms radius corresponding to the wave function reproduces the experimental value [44]: b=2.07 fm for 19 C and 1.83 fm for 17 C. The effect of core polarization on quadrupole transition amplitudes was taken into account in the isoscaler channel, by introducing isospin dependent polarization charges obtained in the FH+RPA particle-vibration model and parameterized in Ref. [10]: δ T =0 =0.17 for 19 C and 0.22 for 17 C. Integrated DWBA cross sections are given in Table 1 for each transition listed.
The J π values are discussed below.
In Fig. 2, the DWBA predictions of the differential cross section leading to the 1.46 MeV state in 19 C are shown for three possible ground-state configurations. The solid line was obtained by using the KD02 optical potential for the supposed 1/2 + 1 →5/2 + 2 transition, while the dot-dashed line by using the JLM potential for the same transition. The latter was specifically derived for the p+ 19 C system at E p =70 MeV. These curves agree closely with each other, justifying an extrapolated use in mass number of the KD02 potential (the nominal range is A=27-209) down to the A=19 region even for nuclei with large neutron/proton ratios. We therefore adopt the KD02 potential below.
Dashed and dotted curves respectively assume J π g.s. =3/2 + 1 and 5/2 + 1 , and the same final state. These curves fail to reproduce the magnitude of the cross section. Clearly, the data are much better described by the solid and dot-  [14,15,16], and that of the 1.46 MeV state is 5/2 + .
In the study of the 14 C( 12 C, 9 C) 17 C reaction [25], a state was observed at 2.06 MeV in 17 C, which was assigned as either 3/2 + 2 or 7/2 + 1 using the results of  Fig. 3 for three initial-state configurations: solid line assumes J π g.s. =3/2 + 1 , dashed one 5/2 + 1 , and dotted one 1/2 + 1 . The 1/2 + 1 →7/2 + 1 assumption gives much lower cross sections than the data. Of the remaining two we see that the 3/2 + 1 assumption for the ground state better describes the data by reproducing the slope of the angular distribution, although the 5/2 + 1 assumption also gives a reasonable description of the data.
In Fig. 3, differential cross sections leading to the 3.05 MeV state are also compared with DWBA predictions obtained by assuming different configurations for the ground state, and the 9/2 + 1 state for the excited state. This state was populated only weakly. It had a sizeable cross section only at backward angles, where transition amplitudes with large angular momentum transfers are involved. It could be identified as the 9/2 + state reported at 3.10 MeV in the three-neutron transfer work [25], and predicted at 3.01 MeV in the present shell model calculations. The cross sections leading to this state are in good agreement with the solid curve calculated for the 3/2 + 1 →9/2 + 1 transition, excluding the possibility of J π g.s. =5/2 + 1 for 17 C. This further corroborates the earlier 3/2 + assignments for the ground state of this nucleus [16,21,22,23], giving us confidence in the current procedure employing the shell model wave functions and DWBA calculations.
The 6.13 MeV state in 17 C appears to correspond to the 6.20 MeV state reported close in energy in the three-neutron transfer work [25]. In that study it was tentatively assigned as either 5/2 + 4 or 5/2 + 5 , with a preference for the latter assignment due to larger occupancies predicted for the 0d shells. The 4%, 4%, and 6% for the ground, 7/2 + 1 , and 9/2 + 1 states, respectively. This indicates that promoting one neutron from lower-lying shells to the 0d3/2 shell is the dominant excitation process of this state. The deduced narrow width is consistent with the reported value of Γ r =0.35 (15) MeV in Ref. [25]. ] orbit, one would obtain a value for the quadrupole deformation parameter of β 2 ≈ 0.4, by referring to neutron single-particle levels in a deformed Woods-Saxon potential [45], and by equating the excitation energy of 1.46 MeV with the energy difference of the two Nilsson orbits. Interestingly, this value is consistent with the result of more recent deformed Skyrme HF calculations predicting a local prolate minimum with J π =1/2 + at β 2 =0.39 [10]. In the HF calculation, however, the ground state was predicted to be oblate with β 2 =−0.36 and J π =3/2 + , and to be more bound by 2.05 MeV than the prolate minimum. Moreover a local oblate minimum having J π =1/2 + with β 2 =−0.35 was also predicted to be almost degenerate with the J π =3/2 + ground state.
The present cross section for the 1/2 + →5/2 + transition observed in 19 C will be useful in distinguishing between the two 1/2 + states with different signs of β 2 , and provide a clue to further investigate the persistence of the new magic number N=16 proposed in oxygen isotopes [5,8] down to the carbon isotopes.
The HF model [10] gives a good account of the ground state spin of 3/2 + for 17 C. It is of interest to see if it also accounts for other states observed in this experiment.
In summary, we have demonstrated that the (p, p ′ ) reaction using the invariant mass method in inverse kinematics leading to unbound resonance states in the residual nucleus is feasible for structure studies even for nuclei far from stability. The measurements were made on 19,17 C at 70 MeV/nucleon.
One resonance in 19 C and three in 17 C were observed above the particle decay threshold. A DWBA analysis employing shell model wave functions and modern nucleon-nucleus optical potentials was used to identify the transitions observed. The spin-parity of the ground state of 19 C was found to be consistent with 1/2 + , and that of the strongly excited 1.46 MeV state was assigned to be 5/2 + 2 . For 17 C the observed states corresponded well with those reported in a three-neutron transfer study [25]. By adding information from this experiment spin-parity assignments of 7/2 + 1 for the 2.20 MeV state and 5/2 + 4 for the 6.13 MeV state were made. The spectroscopic information from this study will impose stringent constraints on further theoretical investigations of light neutron-rich nuclei in this region.  J π =5/2 + 2 is assumed for the excited state.