From superconductors to atomic nuclei, strongly-interacting many-body systems are ubiquitous in nature. Understanding the emergent macroscopic properties of such systems in terms of the underlying microscopic particle correlations is an outstanding challenge with wide ranging implications. In the case of nuclear systems, significant experimental and theoretical efforts are being devoted towards understanding the dynamics of protons and neutrons in both stable and radioactive nuclei, and its implications for the dynamics of dense matter in neutron stars, and astrophysical nucleosynthesis processes.

Measurements of high-energy scattering reactions are a time-honored method to study nucleons in nuclei. In such cases high-energy projectiles are shot at a stationary nucleus, the scattered projectile and knocked-out nucleons are detected, and the initial momentum of the struck nucleon in the nucleus is reconstructed. While such measurements are fundamental for mapping the structure of atomic nuclei, their interpretation is often complicated by Initial-State Interactions/Final-State Interactions (ISI/FSI) of the incoming and scattered particles. Such interactions reduce the scattered particle flux (attenuation) and distort their kinematics, complicating the relation between the measured reaction cross-sections and the inferred ground-state nuclear momentum distribution.

Here we report on a recent study by the Baryonic Matter at Nuclotron (BM@N) collaboration at the Joint Institute for Nuclear Research (JINR) that overcame this fundamental limitation and extracted the distributions of nucleons and correlated nucleon pairs in nuclei. The experiment measured high-energy inverse-kinematics scattering, where a relativistic ion beam was scattered from a stationary proton target, see Figure 1. Large-angle quasi-elastic proton-proton scattering was measured in coincidence with a bound residual nuclear fragment. The detection of the residual nucleus was shown to choose the transparent part of the reaction, excluding the otherwise large kinematic distortions due to ISI/FSI that would also break the fragment apart.

While much more complex than traditional “normal-kinematics” measurements, such “inverse-kinematics” measurements also allow us to study radioactive nuclei with large neutron excess that are too short-lived to be used as a fixed target in the laboratory, cf. Figure 1. Understanding the structure and properties of radioactive nuclei is fundamental for our understanding of nucleosynthesis processes in astrophysics.

The experiment took place at the JINR using a 48 GeV/c ¹²C ion beam from the Nuclotron accelerator, a stationary liquid hydrogen target, and a modified BM@N experimental setup as shown in Figure 2. A two-arm spectrometer was placed downstream of the target to detect the two protons from the quasi-elastic (QE) (p,2p) scattering at ~90˚ in the pp center-of-mass (c.m.). The residual nuclear fragments (¹¹B, ¹⁰B, or ¹⁰Be) were identified in coincidence, and their momenta were determined, based on their energy deposition in two thin scintillators and their measured trajectories as they passed through a large-acceptance dipole magnet. The data was first published in [1], here we wish to show the results in the context of previous data obtained by different measurement methods.

The suppression of ISI/FSI was demonstrated by comparing the reconstructed knocked-out proton initial momentum distribution for QE ¹²C(p,2p) events with and without the coincidence detection of a bound ¹¹B fragment. The results are shown in Figure 3 (adapted from [1] ) compared with Plane-Wave Impulse Approximation (PWIA) calculations for knockout of p-shell protons from the ¹²C that assumes no distortion due to ISI/FSI. As can be seen, the ¹¹B fragment detection significantly suppresses ISI/FSI effects, especially above the nuclear Fermi momentum (k_F ~ 250 MeV/c), where the kinematical distortion of the scattered nucleons due to secondary processes becomes significant.

Next we studied short-range correlated (SRC) nucleon pairs by measuring two-nucleon knockout ¹²C(p,2p¹⁰B)n and ¹²C(p,2p¹⁰Be)p reactions. SRCs are fluctuations of strongly interacting nucleon pairs at short distances [2] [3] [4]. Their formation and properties are sensitive to the many-body dynamics of nuclear systems, properties of the short-distance nucleon-nucleon interaction, nucleon structure, and properties of cold dense nuclear matter such as in the outer core of neutron stars [5] [6].

In SRC breakup reactions, ¹⁰B and ¹⁰Be nuclei are produced when a proton-neutron (pn) or proton-proton (pp) pair in the projectile interacts with a proton in the target. We measured 23 ¹²C(p,2p¹⁰B)n and two ¹²C(p,2p¹⁰Be)p events. The other isospin-symmetric nn pairs are not accessible here because neutron knockout was not measured. The large ¹⁰B to ¹⁰Be event-yield ratio is consistent with the previously observed dominance of pn-over pp-SRC pairs [7] [8] [9], and fully agrees with predictions based on ab-initio many-body calculations. Contributions from inelastic reactions and from reactions due to mean-field QE scattering followed by FSI are negligible.

Beyond the suppression of ISI/FSI, the fragment momentum distribution is equal to the SRC pair c.m. momentum distribution. This distribution is consistent with that of a Gaussian.

In Figure 4 we compare the Gaussian width measured directly by BM@N with previous indirect extractions from electron scattering measurements at the Thomas Jefferson National Accelerator Facility (JLab) done in “normal kinematics” and with several theoretical predictions [10]. All measurements agree with each other and with the theoretical calculations, showcasing the probe-independence of SRC measurements and the success of its theoretical description.

Lastly, detecting the residual nucleus allows, also for the first time, measuring the interaction between the SRC pair and the residual nuclear system. From a theoretical standpoint, the strong two-body interaction between the nucleons in SRC pairs is expected to be significantly stronger than the average mean-field nuclear interaction [11] [12]. Therefore, the pair interaction should be scale separated from that of the residual nuclear system, allowing us to model the distributions of SRC pairs by independent functions of the pair c.m. and relative momenta that do not depend on the angle between the two.

This is shown in Figure 5 (adapted from [1] ). A clear back-to-back correlation between the momenta of the nucleons in the SRC pair is observed, as expected for strongly-correlated nucleons. The width of the distribution is driven by the pair c.m. motion and agrees with the Generalized Contact Formalism (GCF) simulation [11] [12]. In contrast, no correlation is observed between the angles of SRC c.m. momentum (i.e. ¹⁰B) and the relative momentum between the nucleons in the pair, providing the first direct experimental evidence for the factorization of SRC pairs from the many-body nuclear medium.

To conclude, we demonstrated the feasibility of directly accessing properties of single nucleons and SRC nucleon pairs in nuclei using a 48 GeV/c ion beam from the JINR Nuclotron accelerator and a stationary liquid hydrogen target to measure high-energy inverse kinematics scattering. We identified SRC pairs and performed the first direct measurement of their c.m. momentum distribution. We used this to experimentally verify the SRC factorization assumption which leads to a universal description of the high-momentum tail in all nuclei. This measurement opens the way for SRC studies in radioactive nuclei at the forthcoming FAIR and FRIB facilities, focusing on the dynamics of high-density nucleon-pair fluctuations in very neutron-rich systems. These studies will be pivotal for developing a microscopic understanding of the structure and properties of nuclei far from stability and the formation of visible matter in the universe.

The study of the structure of radioactive nuclei and its relation to nucleon correlations is a growing frontier of nuclear physics, motivated by the importance of neutron-rich nuclei for modeling astrophysical processes. Measurements have shown that nuclear shell occupations (magic numbers) evolve with nuclear asymmetry where for neutron-rich nuclei “known” shells disappear and new shells appear. This evolution of shells has dramatic astrophysical implications as “magic number nuclei” serve as a waiting point for element formation in the r-process. Extending the study to radioactive nuclei using the new experimental approach reported here, based on measurements of hard nucleon-knockout reactions in inverse kinematics, is paramount for a ground-breaking experimental and theoretical program to understand very asymmetric cold dense nuclear systems from unstable nuclei to neutron stars.

From an ab-initio theory perspective, understanding the manner by which two-body correlations impact the emergence of collective phenomena such as shell closure in neutron-rich nuclei is a significant challenge, undertaken by nuclear EFT calculations using many-body numerical techniques such as lattice, coupled-cluster, and Quantum Monte-Carlo calculations. Measuring the ground-state distributions of particles and particle-pairs in such exotic systems is crucial to improving our understanding of such correlation effects in other strongly interacting many-body systems.

We acknowledge the efforts of the BM@N Collaboration, staff of the Nuclotron-NICA Accelerator Division, and Veksler and Baldin High Energy Physics Laboratory at JINR that made this experiment possible. The research was supported by the Israel Science Foundation, the Pazy Foundation, and the BMBF via project no. 05P15RDFN1, through the GSI/TU Darmstadt cooperation agreement, by the US DoE under grant no. DE-FG02-08ER41533 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project ID 279384907, SFB 1245, and the RFBR under grant numbers 18-02-40046 and 18-02-40084/19.

The authors declare no conflicts of interest regarding the publication of this paper.

Kahlbow, J., Patsyuk, M., Lenivenko, V., Segarra, E.P., Johansson, G., Klimanskiy, D.I. and Maksymchuk, A. (2022) Probing Nuclei with High-Energy Hadronic Probes at Inverse Kinematics. Journal of Modern Physics, 13, 761-767. https://doi.org/10.4236/jmp.2022.135044