The low-energy spectrum of ¹⁴C was determined by proton inelastic scattering using a¹⁴C beam of about 12.4 A MeV and an intensity of about 2,000 pps for about 25 h of beam time. The AT-TPC was filled with 300 torr of pure hydrogen gas H 2 under static pressure. The beam energy after the AT-TPC window (also 12 μ m of PPTA) was about 12.4 A MeV. The magnetic field was set to 2.85 T. The trajectory of the reaction products was determined on a event-by-event basis, enabling the inference of the angle and the magnetic rigidity through the track point cloud. The magnetic rigidity and the energy loss are used to identify the reaction products, as shown in the left panel of Figure 3. The most intense band on this plot corresponds to scattered protons.

The excitation energy spectrum of ¹⁴C has been obtained after selecting the protons in the identification matrix and correcting for the energy loss of the beam in the detector. The characteristic kinematic lines of ¹⁴C excited states are shown in the right panel of Figure 3. The dashed lines refer to the calculated kinematics at the center of the detector for the ground state and the first excited state. The magnetic rigidity vastly increases the dynamic range of the detector as can be seen in the proton energy range covered in this reaction. It is important to highlight that at high proton energies there is a systematic deviation of the data with respect to the calculated kinematics. This discrepancy is likely caused by the electric field edge effects at the outer radius of the detector volume, which impact the reconstruction of high-rigidity particles.

The excitation energy spectrum of ¹⁴C is shown in the upper left panel of Figure 4. Besides the ground state, we are able to resolve the first excited state (6.091 MeV, 1 1 − ) and the 2 2 + at 8.317 MeV. The group of states at around 7 MeV has been identified as 6.728 MeV 3 1 − , 7.012 2 1 + and 7.341 MeV 2 1 − , in agreement with Ref. Lozowski [36]. The values of the energy levels were extracted from the Nuclear Structure and Decay Data (NuDat) database [37]. The experimental resolution in this case, determined from a gaussian fit to the ground state peak, is 150 keV (standard deviation), with an accuracy of 30 keV [19]. The apparent peak at about 9 MeV is attributed to an excited state in ¹⁴N above the proton emission threshold, which is populated through the (p,n) charge-exchange reaction. Such events are identified by momentum conservation since the efficiency for the detection of neutrons in the AT-TPC is very low, although not negligible working as a proton target. The angular distribution associated to the 1 1 − state, shown in the upper right panel of Figure 4, was directly deduced from 20 ° to 100 ° in CM. In this angular domain, the peak is well isolated from the neighboring states. The angular distribution was corrected for acceptance and reconstruction efficiency effects. This correction utilized a comprehensive simulation that accounted for both the geometry and response of the AT-TPC. The simulated angular distribution, shown in the lower panel of Figure 4, was obtained generating events by sampling from a flat distribution between 0 ° and 180 ° in center of mass (CM). The gradual loss of efficiency between 0 ° and 40 ° can be attributed to the limited acceptance imposed by the pad plane’s hole for particles emitted at forward angles. At angles above 110 ° , the energy loss of the protons becomes insufficient to ensure 100 % of trigger efficiency. It is evident from the region of the distribution where the efficiency exceeds one that a fraction of misreconstructed events is not rejected but instead assigned incorrect angles or energies.

Among the intricate structures of neutron-rich beryllium isotopes, ¹²Be stands as a candidate to observe a halo-like structure built on an excited state of a nucleus [38]. Its structure can be understood as the coupling between a valence neutron and a¹¹Be core. Therefore, one could expect the possibility of observing an excited state on ¹²Be with a strong overlap to the ¹¹Be ground state, a paradigmatic neutron halo nucleus. The bound structure of ¹²Be favors this hypothesis because the 1 1 − state is located around 400 keV below the neutron emission threshold ( S n = 3.170 MeV), a common feature of weakly-bound systems with a large spatial distribution. We designed an experiment to investigate an enhanced transition l = 1 from the 0 2 + isomeric state in ¹²Be as a possible signature of a halo structure in an excited state via inelastic scattering as primary probe. To validate the detection method, the setup was commissioned to detect the scattered proton in coincidence with the beam-like ¹²Be isomer. Concurrently, we measured cross sections for the ¹²Be (p,d) transfer reaction, which provides valuable information on the ¹²Be-¹⁰Be ⊗ n overlap. In this work, we present results on the latter reaction.

The experiment was conducted using a low-intensity ¹²Be beam of about 150 pps at 12 A MeV. The AT-TPC was filled with 600 torr of pure hydrogen gas. The data analysis was performed in the same fashion as discussed for the proton inelastic scattering on ¹⁴C.

The kinematics for the ¹²Be (p,d)¹¹Be reaction and the ¹¹Be excitation energy spectrum are shown in the left and right panels or Figure 5, respectively. Within our experimental resolution of 200 keV (standard deviation) and an accuracy of about 20 keV, we observe the population of several states of ¹¹Be with established J π : ground state, 0.320 MeV (1/ 2 1 − ), 1.78 MeV (5/ 2 1 + ), 2.65 MeV (3/ 2 1 − ) and a doublet consisting of the 3.89 (3/ 2 − or 5/ 2 − ) and 3.96 MeV (3/ 2 3 − ). Although the ground and first excited states are unresolved, we can infer quantitative information on the population strength by considering the corresponding angular distributions. Extracting spectroscopic information with such a low-intensity beam clearly demonstrates the outstanding capabilities of Active Targets for experiments with radioactive beams. A detailed analysis of the angular distributions will be addressed in a separate publication to allow for a more thorough exploration.

The low-energy spectrum obtained for ¹⁴C via proton inelastic scattering can be compared to ab initio calculations employing nuclear interactions from chiral effective field theory. To solve the quantum many-body problem, we employ the coupled-cluster (CC) approach, where one starts from a mean-field solution | Φ 0 〉 and parametrizes the nuclear ground state wavefunction as (Equation 1) | Ψ 0 〉 = e T | Φ 0 〉 (1) Here, T is the so-called cluster operator, which can be expanded as a sum of n -particle- n -hole excitations: T = T 1 + T 2 + T 3 + …   . In this framework, excited states can be accessed employing the equation-of-motion coupled-cluster (EOM-CC) method [39]. In EOM-CC, the target state | Ψ f 〉 is computed via the ansatz (Equation 2) | Ψ f 〉 = R e T | Φ 0 〉 (2) where also the EOM excitation operator R can be written in terms of a particle-hole expansion. In CC theory, both the cluster operator T and the EOM operator R are truncated due to computational limitations. Coupled-cluster singles and doubles (CCSD), where T and R are truncated at the 2p-2h level, is the most frequently used approximation. Adding linear 3p-3h excitations in the so-called CCSDT-1 approximation [40] leads to increased precision.

As an example, we focus here on the first 1 − state in the spectrum of ¹⁴C and compare it to the experimental spectrum obtained with the AT-TPC. To this aim, we employ the chiral Δ NNLO GO (394) and Δ NNLO GO (450) interactions [41]. These nuclear force models, given at next-to-next-to-next-to-leading order in the chiral expansion, contain the Δ isobar as an explicit degree of freedom and they have been successfully employed in several applications [42, 43]. We performed CC calculations starting from a Hartree-Fock Slater determinant including up to 15 major harmonic oscillator shells, and we studied convergence by varying the harmonic oscillator frequency ℏ Ω between 12 and 16 MeV.

Our results for the excitation energy of the first 1 − state are shown in Table 1. Theoretical uncertainties account for the residual dependence on the model space parameters, and for the truncation of the coupled-cluster expansion according to the strategy employed in Simonis et al. [44]; Acharya et al. [45]. We observe that our predictions lie higher than the experimental determination at around 6.1 MeV. However, it is worth pointing out that the addition of linear 3p-3h excitation reduces the excitation energy of an amount varying between 15 % and 18 % on the basis of the interaction, moving theory in the direction of the experimental result. A complete analysis of model uncertainties, including the effect of the chiral EFT truncation and of different optimization protocols for the low-energy constants, is left for future work.

The experimental results on the first excited state of ¹⁴C can also be compared to available shell-model calculations. In Ref. Yuan et al. [46], the first 1 − excited state of ¹⁴C is calculated with three different shell-model interactions (YSOX [46], SFO [47], WBP [48]) optimized for the p s d -shell region. The latter predict excitation energies ranging from 5.5 to 6 MeV, in close proximity to the experimental data.

Future experimental campaigns will exploit the AT-TPC to study electromagnetic responses up to the giant dipole resonance region. Electromagnetic strength data could be compared to calculations combining CC theory with the Lorentz Integral Transform technique [49] in the so-called LIT-CC method [50, 51]. This approach allows for an ab initio desciption of electromagnetic reaction observables in nuclei at and in the vicinity of closed-shells [44, 52, 53]. It is based on the calculation of an integral transform with Lorentzian kernel of the response. Considering small values of the Lorentzian width, we can construct a discretized strength function, where continuum excited states of the nucleus are represented by bound pseudo-states. As an example, let us focus on the E1 strength function of ¹⁴C, shown in Figure 6. At low energy, below 8 MeV, we distinguish the first 1 − excited state under analysis in this work. Its transition strength amounts to around 5 % of the one observed for states at excitation energies above 15 MeV.

We have studied the structure of ¹¹Be from a qualitative point of view from the spectrum obtained in the transfer measurement. We have applied shell model calculations, with the YSOX interaction [46] to calculate the spectroscopic factors of the ¹²Be ( p , d ) 11 Be reaction. This interaction works in a full p − s d model space, including ( 0 − 3 ) ℏ Ω excitations, and it can give good description of the energy, quadrupole and spin properties of the psd-shell nuclei. The calculated spectroscopic factors are compared to the experimental results shown in Figure 7. The spectroscopic factors represent the neutron occupancy of the 0 p 1 / 2 , 0 p 3 / 2 , 1 s 1 / 2 and 0 d 5 / 2 orbitals. It can be seen that the ground state and the first two excited states are populated, showing that the shell model is predicting a very strong configuration mixing in the ground state of ¹²Be due to the breakdown of N = 8 magic number. This is in agreement with the experimental spectrum, except that the ground and the first excited states are not well isolated. However, it is expected that their individual contributions can be determined by the angular distribution, owing to their very different shapes. The higher 3 / 2 − excited states are populated due to the removal strength from the 0 p 3 / 2 orbital. Significantly, the energy of the 1 / 2 − state deviates from the experimental results because the effective energy of the interaction is not optimized for nuclei far from the stability, and the continuum coupling effect was not accounted for. The experimental results presented in this work show strong agreement with previous findings from knock-out [54] and transfer [55] experiments, although a comprehensive discussion will be provided in a forthcoming publication.