The dominance of the $\nu(0d_{5/2})^2$ configuration in the $N=8$ shell in $^{12}$Be from the breakup reaction on a proton target at intermediate energy

The momentum distribution of $^{11}$Be fragments produced by the breakup of $^{12}$Be interacting with a proton target at 700.5 MeV/$u$ energy has been measured at GSI Darmstadt. To obtain the structure information on the anomaly of the $N=8$ neutron shell, the momentum distribution of $^{11}$Be fragments from the one-neutron knockout $^{12}$Be(p,pn) reaction, measured in inverse kinematics, has been analysed in the distorted wave impulse approximation (DWIA) based on a quasi-free scattering scenario. The DWIA analysis shows a surprisingly strong contribution of the neutron $0d_{5/2}$ orbital in $^{12}$Be to the transverse momentum distribution of the $^{11}$Be fragments. The single-neutron $0d_{5/2}$ spectroscopic factor deduced from the present knock-out data is 1.39(10), which is significantly larger than that deduced recently from data of $^{12}$Be breakup on a carbon target. This result provides a strong experimental evidence for the dominance of the neutron $\nu(0d_{5/2})^2$ configuration in the ground state of $^{12}$Be.

The anomaly of the neutron N = 8 shell has been known since many years, for example, from the abnormal spin-parity of the 11 Be ground state.
The systematic change from the conventional sp shell for neutrons in carbon isotopes to a mixture of 0p 1/2 , 1s 1/2 and 0d 5/2 shells in the neutron rich Be isotopes was discussed already 20 years ago by Tanihata [1]. The recent studies have been focused on the disappearance of the N = 8 shell closure when the 1s 1/2 orbital is shifted below the 0d 5/2 orbital, giving rise to the formation of neutron halos [2]. While the extended sizes of the halo nuclei can be accurately deduced from the measured interaction (or reaction) cross section [3,4], or from the angular distribution of intermediate energy proton elastic scattering in inverse kinematics [5,6,7,8,9,10,11], the shell structure of the unstable neutron-rich nuclei has been studied mainly based on the analysis of momentum distributions of fragments from breakup reactions [12,13,14].
The ground-state structure of 12 Be with N = 8 is of high interest from the shell-model point of view. It should be noted that the ν(0d 5/2 ) 2 (J π = 0 + , T = 1) intruder state in 12 Be was pointed out as highly possible by Barker [15] some 40 years ago. A more recent microscopic particle-vibration coupling study by Gori et al. [16] shows a quite strong coupling between the valence neutrons and the lowest 2 + and 3 − excited states of the 10 Be core, leading to a dominance of the ν(0d 5/2 ) 2 configuration in the ground state of 12 Be. So far, 12 Be has been studied in several experiments, and quite interesting are the measurements of the 12 Be breakup on a 9 Be target by Navin et al. [17], and on a 12 C target by Pain et al. [18]. In the first experiment, the observation of 12 Be fragmentation followed by the γ-emission from the bound states of 11 Be has shown that the 1s 1/2 neutron shell is mixed with the 0p 1/2 shell in the ground state of 12 Be. In the second experiment, the γ rays following the neutron emission from the (unbound) 1.78 MeV (d 5/2 ) and 2.69 MeV (p 3/2 ) excited states of 11 Be fragments [18] have been observed, which show a strong mixing of the neutron p and sd shells in the ground state of 12 Be.
The present work presents the momentum distributions of 11 Be fragments produced in the one-nucleon knockout 12 Be(p,pn) 11 Be reaction measured at GSI Darmstadt in inverse kinematics at an energy of 700.5 MeV/nucleon.
The transverse momentum distribution of the 11 Be fragments was analyzed in the distorted wave impulse approximation (DWIA), using the quasi-free scattering (QFS) assumption for the single nucleon knockout reaction [19].
Evidence for a strong dominance of the ν(0d 5/2 ) 2 configuration in the ground state of 12 Be was found from the present DWIA analysis. At the entrance of the secondary target IKAR [7,9,21], the secondary 12 Be beam energy was 700.5 MeV/u with 1.1 % FWHM. Its intensity was about  [7,9,21]. One module is zoomed as seen in the bottom inset. Details are explained in the text. 6000 ions/s, and the contamination from other isotopes was approximately 1 %. The time projection ionization chamber IKAR [7,9,21], which was filled with hydrogen gas and operated at 10 bar pressure, served simultaneously as a gas target and a recoil proton detector.
A schematic view of the experimental layout is presented in FIG. 1. The proton recoil signal, obtained from the IKAR detector, was coincident with that of the scattered Be particle. The scattering angle θ s and the vertex point were determined from the coordinates measured by a tracking system consisting of 2 pairs of 2-dimensional multi-wire proportional chambers (MWPC1-MWPC2 and MWPC3-MWPC4), arranged upstream and downstream with respect to IKAR. The scintillators S1 and S2 were used for triggering and beam identification. The beam was identified via the time-of-flight (ToF) between scintillators S1 and S8 (located at the FRS, not shown in FIG 1) and energy loss measurements in the scintillators S1 and S2, while a circular-aperture scintillator VETO with a 2 cm diameter hole at its center selected the projectiles which entered IKAR within an area of 2 cm in diameter around the central axis. The helium bags were used to reduce the multiple Coulomb scattering of the incoming and outgoing particles. The analysis for the elastic 12 Be(p,p) channel was presented in details in Ref. [10]. For the present one-neutron knockout 12 Be(p,pn) channel, the difference is the selec- At the high incident energy considered in the present work, the oneneutron knockout from the loosely bound 12 Be projectile occurs promptly, inducing almost no perturbation on the other nucleons (adiabatic approximation). The non participating nucleons in 12 Be can be considered as "spectators" [22] which scatter elastically off the p target. Following this idea, the kinematic of the elastic 11 Be + p scattering was used to determine the momentum of the outgoing 11 Be fragment (p out ), with the scattering angle θ s determined as discussed above, and the velocity of the incoming 11 Be core (with momentum p in ) assumed to be equal to that of the 12 Be projectile. At very low momentum transfer, the x projection of the fragment's transverse momentum can be calculated as where, θ in x (θ x ) are angle of the incoming (outgoing) particle in the laboratory frame. As the result, the x−component distribution of the 11  The motivation of the present study is focused on the structure information concerning the anomaly of the N = 8 neutron shell in the Be isotopes. Therefore, the momentum distribution of the 11 Be fragments from the oneneutron knockout 12 Be(p,pn) reaction (in inverse kinematics) was carefully studied in the DWIA analysis of the knockout reaction. The DWIA has been well proven as a reliable approach to describe the single-nucleon knockout reaction [19,23,24]. There are three particles emerging in the exit channel of the 12 Be(p,pn) reaction, whose exact kinematics cannot be determined by the present experimental setup. Therefore, as discussed above, the 11 Be core of the 12 Be projectile was assumed to scatter on the proton target elastically during the breakup reaction, and the elastic 11 Be + p kinematics was used to determine the momenta of the 11 Be fragments. This is in fact the quasi-free scattering (QFS) approximation usually adopted in the DWIA studies of the quasi-elastic scattering at high energies [19].
In such a QFS scenario, the DWIA scattering amplitude for the A(p,pn)B reaction can be determined [19,23] as where χ in k and χ out k are the incoming and outgoing distorted waves of the proton and knockout neutron, S(lj) is the spectroscopic factor of the lj component in the wave function of the valence neutron in A, which is described by the ψ jlm function. τ pn is the proton-neutron scattering matrix that depends on the energy and the relative proton-neutron momenta in the entrance (k pn ) and exit (k pn ) channels.
At high energies, the distorted waves are determined in the eikonal approximation as in is the pA scattering matrix in the entrance channel, and S (p) out and S (n) out are the pB and nB scattering matrices in the exit channel. They are obtained from the (real) nucleon optical potentials given by the folding model, and the corresponding imaginary parts given by the tρρ approach [25]. A recoil correction α = (A − 1)/A due to the center of mass (c.m.) motion is also introduced [19,23]. The single-neutron wave function ψ jlm is generated by the standard method using a Woods-Saxon potential supplemented with a spin-orbit term, whose parameters were adjusted to reproduce the observed neutron separation energy of 12 Be. Adopting the free proton-neutron scattering amplitude for τ pn , the DWIA transition amplitude (2) at a given impact parameter b can be written as T p,pn = S(lj)τ pn (k pn , k pn ; E) d 3 r S(b, θ p , θ n ) exp(−iq · r)ψ jlm (r), (5) where θ p and θ n are the c.m. angles of the outgoing proton and neutron, respectively, and the momentum transfer q = k p + k n − αk p . The total scattering matrix is a product of the nucleon scattering matrices Expressing q explicitly in terms of the transverse (q t ) and longitudinal (q z ) momentum transfers, the transverse momentum distribution of the neutrons knocked out from the single-particle state ψ jlm is obtained (after integrating over q z ) as [19] dσ lj where dσ pn dΩ and S(b) are averaged over the energies of the outgoing proton and neutron (at the given transverse momentum q t ), the later also averaged over the scattering angles of the protons. u lj (r) is the radial part of the single-particle wave function ψ jlm (r); J m (q t b) and P lm (b, z) are the cylindrical Bessel function and Legendre polynomials, respectively, and To compare with the experimental momentum distribution, we need to reduce the distribution (7) to that along the x-component of q t . Inserting q t = p 2 x + p 2 y , we obtain [26] the inclusive momentum distribution of the one-neutron knockout 12 Be(p,pn) 11 Be reaction as The measured transverse momentum distribution of 11 Be fragments ejected from the one-neutron knockout 12 Be(p,pn) 11 Be reaction was subjected to the DWIA analysis (7)- (8), with the spectroscopic factors S(lj) of the singleparticle configurations ν1s 1/2 , ν0d 5/2 and ν1p 3/2 of the valence neutron in The present result provides an important evidence for the dominance of the (ν0d 5/2 ) 2 configuration in the ground state of 12 Be, with the spectroscopic factor S(νd 5/2 ) ≈ 1.39 which is significantly larger than that reported in Ref. [18] (S(νd 5/2 ) ≈ 0.48). The dominance of the d-wave in the ground state of 12 Be was also shown by a recent microscopic particle-vibration coupling study by Gori et al. [16] as due to a strong coupling between the valence neutron and the 2 + and 3 − excitations of the 10 Be core. At the relatively high energy of 700.5 MeV/u, the 11 Be fragment produced in the present 12 Be(p,pn) 11 Be knockout reaction could well populate the known 1/2 − , 5/2 + and 3/2 − excited states seen in the 12 Be breakup reaction on a carbon target [18]. With the small spectroscopic factors found for the ν1s 1/2 and ν1p 3/2 configurations, such a strong preference of the neutron knockout channel to the 5/2 + excitation of 11 Be found in the present work poses a challenge for the future experimental and theoretical structure studies of the neutron rich Be isotopes.
In conclusion, the transverse momentum distribution of 11 Be fragments ejected from the one-neutron knockout 12 Be(p,pn) 11 Be reaction has been measured at an energy around 700 MeV/nucleon. A dominance of the dwave in the ground state of 12 Be was found from the DWIA analysis of the present neutron knockout data. This result provides a clear evidence for the intruder ν0d 5/2 level in the single-particle scheme that breaks the magicity of the N = 8 shell in the Be isotopes.