Proton-Induced Quasifree Knockout Reactions

• Proton-induced quasifree knockout reactions are high-energy processes where a proton collides with a nucleus, ejecting nucleons via the impulse approximation.
• The eikonal and distorted-wave frameworks enable precise mapping of momentum distributions and extraction of spectroscopic factors in these reactions.
• Experimental implementations using inverse-kinematics rare-isotope beams offer critical insights into short-range correlations and nuclear structure evolution.

Proton-induced quasifree knockout reactions are high-energy processes in which an incident proton collides with a target nucleus and scatters from a single nucleon or correlated pair, ejecting one or more nucleons nearly instantaneously, while the rest of the nucleus acts as a spectator. These reactions, denoted as (p,2p), (p,pn), (p,3p), etc., underpin modern ab initio and mean-field explorations of single-particle and correlated nuclear structure, short-range nucleon-nucleon correlations, and the evolution of nuclear properties far from stability. Their theoretical description relies on the impulse approximation, eikonal or optical-potential distorted waves, and advanced many-body nuclear wave functions. They are a core tool for spectroscopic studies on exotic and stable nuclei at facilities employing inverse-kinematics rare-isotope beams.

1. Fundamental Reaction Mechanism and Impulse Approximation

In the quasifree limit, the incident proton imparts a large momentum transfer to a single bound nucleon (or nucleon pair) in the target. The hard collision leads to the ejection of this nucleon, typically with minimal energy or momentum transfer to the residual core. The impulse approximation posits the reaction proceeds through a single, instantaneous hard scattering, with all other nucleons acting as non-interacting spectators (Aumann et al., 2013, Yoshida et al., 2024).

The factorized cross section under the impulse and spectator assumptions is:

$$\frac{d^3\sigma}{d\Omega_1 d\Omega_2 dE_2} = K\,S(p_m)\,|T_{fi}|^2$$

where $K$ is a kinematic factor (including Jacobians and phase space), $S(p_m)$ is the nuclear spectral function (i.e., momentum distribution), and $T_{fi}$ is the distorted-wave transition amplitude. The missing momentum $$p_m = \mathbf{p}_1 + \mathbf{p}_2 - \mathbf{p}_{\text{proj}}$$ corresponds to the negative of the initial bound-nucleon momentum in the target frame (Nakada et al., 19 Dec 2025, Patsyuk et al., 2021).

In the (p,2p) reaction, only a single bound proton participates in the elementary p–p collision. Sequential or correlated two-nucleon ejection processes are handled within generalized frameworks [(p,3p), (p,pn)], employing extensions described below.

2. Theoretical Formalism: Distorted-Wave and Glauber Approaches

The distorted-wave impulse approximation (DWIA) is the standard theoretical tool, in which distorted waves for projectile and ejectiles are generated with complex optical potentials. For high-energy beams, the eikonal (Glauber) approximation further simplifies the multiple-scattering problem, yielding tractable and physically transparent cross-section formulae.

For (p,2p), the high-energy Glauber cross section is:

$$\sigma_{(p,2p)}(E_p) = \int d^2b\, e^{-2\,\mathrm{Im}\chi(b)}\,|S_{pN}(b)|^2$$

where $\chi(b)$ is the cumulative eikonal phase, $S_{pN}(b)$ is the profile function of the elementary p–N collision, and $b$ is the impact parameter (Lehr et al., 2021). Nuclear attenuation (initial- and final-state interactions) is encoded in $e^{-2\,\mathrm{Im}\chi}$, and leads to a strong suppression of central collisions.

In the factorized DWIA, the transition matrix element for one-nucleon knockout is:

$$T_p = \langle\chi^{(-)}_1\chi^{(-)}_2|t_{pp}| \chi^{(+)}_0\phi_p\rangle$$

where $\chi_i$ are distorted waves for the protons, $t_{pp}$ is the proton–proton t-matrix, and $\phi_p$ is the bound-state wave function of the removed nucleon. In appropriate kinematics, the triple-differential cross section maps the distorted overlap—reflecting the bound-state momentum distribution—onto the measured cross sections (Nakada et al., 19 Dec 2025, Yoshida et al., 2024).

For (p,3p), the eikonal sequential-quasifree formalism involves two independent p–p collisions with the core assumed inert, and the amplitude is constructed as a double integral over the two hard scatterings and eikonal propagation between them (Gómez-Ramos, 2024). The two-nucleon removal cross section probes the two-nucleon amplitude (TNA) and spectroscopic strength for pair removal.

3. Nuclear Structure Inputs and Spectroscopic Factors

Quantitative interpretation of knockout data requires realistic nuclear structure inputs. Contemporary approaches employ:

• Hartree-Fock or Hartree-Fock-Bogoliubov single-particle wave functions and occupation amplitudes (Bertulani et al., 3 Jan 2026).
• Ab initio quantum Monte Carlo wave functions for light nuclei, used for generating overlaps and spectroscopic factors (Crespo et al., 2018).
• Empirical dispersive optical models (DOM) providing fully consistent self-energies, bound and scattering wave functions, and spectroscopic factors through a single nonlocal, energy- and density-dependent potential (Atkinson et al., 2024).

Measured exclusive cross sections for specific final states are related to the theoretical single-particle cross section $\sigma_{sp}$ via the spectroscopic factor $C^2S$:

$\sigma_{\text{exp}} = C^2S\,\sigma_{sp}$

This enables direct extraction of $C^2S$ and empirical tests of theoretical models for nucleon correlations and occupation probabilities. Notably, DOM-based analyses reveal reduced spectroscopic factors in (p,2p) compared to (e,e′p) reactions, which are attributed to inadequacies in the in-medium p–p interaction or to missing dynamical correlations (Atkinson et al., 2024).

4. Momentum Distributions, Angular Momentum Selectivity, and Density Sensitivity

The longitudinal and transverse momentum distributions of the knockout residue are sensitive to the orbital angular momentum (ℓ) of the removed nucleon—the width of the distribution narrows for s-waves relative to d-waves:

$$\frac{d\sigma}{dp_z}\propto \sum_{l,j} C^2S_{lj}|\widetilde\phi_{lj}(p_z)|^2$$

The nuclear interior can be probed even in the presence of strong absorption; for example, high-energy (p,2p) from s1/2 orbits in ^12C accesses mean effective densities ~0.3ρ_0 (where ρ_0 is nuclear saturation density) (0904.0914). This capability distinguishes hadronic knockout from electromagnetic probes (e, e′p), which are less attenuated and access higher densities.

Angular-momentum decomposition by fitting momentum distributions provides nearly model-independent probes of orbital occupancy; e.g., the l=0 (s-wave) and l=2 (d-wave) contributions in the halo candidate ^17Ne were determined to ~3% accuracy by comparing measured and calculated momentum spectra (Lehr et al., 2021).

5. Multi-Nucleon Knockout, Short-Range Correlations, and Sequential Processes

Proton-induced quasifree (p,3p) and (p,pn) reactions enable the study of two-nucleon amplitudes and short-range correlations (SRC). In the (p,3p) formalism, two sequential p–p collisions dominate at intermediate energies (E/A > 200 MeV) (Gómez-Ramos, 2024). The cross section factorizes into a product of the free p–p cross section, eikonal S-matrices for the three outgoing protons, and the two-proton overlap from shell model or ab initio structure calculations.

SRC-driven two-nucleon knockout exhibits back-to-back emission and a striking enhancement of pn over pp pairs, reflecting the dominance of tensor-correlation-induced pn pairs at high momenta. The integrated fraction of SRC pn pairs over the typical k-range is found to be ≳90% in ^12C and exhibits weak asymmetry dependence, confirming previous observations with electromagnetic probes (Stevens et al., 2017).

For Borromean two-neutron halo systems, the opening angle and momentum correlations among the emitted neutrons in (p,pn) knockout provide direct access to dineutron spatial correlations, which can be optimally explored by gating away from sequential resonant decay regions (e.g., from the ^5He resonance in ^6He) and employing transparent proton targets (Kikuchi et al., 2016, Casal et al., 2021).

6. Systematics: Neutron-Skin Effects, Multi-Mechanism Contributions, and Spectroscopic Quenching

Modern theoretical developments include unified Glauber-type frameworks combining probabilistic many-body multiple-scattering with Skyrme-HFB densities, allowing for precision calculations of (p,2p) and (p,3p) cross sections and momentum dispersions over isotopic chains. Both cross section and longitudinal width decrease with increasing neutron-skin thickness, particularly for (p,3p), yielding a sensitive probe of isovector structure and the symmetry-energy slope parameter L (Bertulani et al., 3 Jan 2026).

At intermediate energies (∼100 MeV/u), non-quasifree mechanisms such as inelastic excitation and direct nucleon transfer contribute significantly—sometimes as much as 50% to inclusive cross sections. Multi-step and multi-mechanism models must be employed to avoid erroneous extraction of spectroscopic strengths and to properly account for the observed quenching of cross sections and the weak ∆S dependence in exclusive residue yields (Pohl et al., 2023).

In neutron-rich systems, consistent ∼3× quenching of two-proton removal cross sections observed in both (p,3p) and heavy-target-induced two-proton knockout is interpreted as a signature of structural correlation effects rather than methodological deficiencies in the reaction mechanism (Gómez-Ramos, 2024).

7. Experimental Implementations, Observables, and Future Prospects

Exclusive and kinematically complete measurements—detecting all outgoing nucleons and heavy fragments—are essential for separating quasifree single-particle removal from core and multifragmentation processes, directly accessing momentum distributions, and quantifying spectroscopic factors with minimal model dependence (Lehr et al., 2021, Patsyuk et al., 2021).

Advanced setups such as MINOS (liquid-H₂ target with high-granularity TPC) and high-resolution spectrometers allow measurement of momentum distributions with Δp/p ≈ 10⁻³. Techniques such as fragment tagging (detecting heavy residues like ^11B in ^12C(p,2p)^11B) suppress ISI/FSI-induced distortions and isolate “transparent” single-step events, enabling precise mapping of proton momentum distributions and short-range correlated pair structure (Patsyuk et al., 2021).

Future directions emphasize:

• Systematic two-nucleon knockout measurements across isotopic chains to constrain the density dependence of symmetry energy (Bertulani et al., 3 Jan 2026).
• Enhanced reaction theory: CDCCIA for correlated clusters and pairs (Yoshida et al., 2024), fully in-medium p–p interactions constructed with realistic DOM propagators (Atkinson et al., 2024).
• Coincidence measurement strategies to unambiguously separate sequential and simultaneous removal and to experimentally validate theoretical spectroscopic and correlation predictions.

Such studies are pivotal for advancing quantitative nuclear spectroscopy with rare-isotope beams, constraining ab initio nuclear many-body theory, and unraveling the interplay of mean-field, shell evolution, short-range correlations, and collective phenomena in nuclei far from stability.