Single-Nucleon Knockout Cross Section

Updated 4 January 2026

Single-nucleon knockout cross section is a measure of the probability of removing a single nucleon from a nucleus via external probes, providing insights into nuclear structure and correlations.
It is computed using reaction models like the eikonal/Glauber approach, DWIA, and impulse approximations that integrate structural inputs such as spectroscopic factors and optical-model potentials.
The observable helps benchmark nuclear mean-field theories, informs on spectroscopic quenching, and tests reaction mechanisms, thereby refining nuclear structure and reaction models.

A single-nucleon knockout cross section quantifies the probability for the removal (knockout) of a single nucleon—either a proton or a neutron—from a nucleus as a result of an external probe (hadron, electron, or neutrino). This observable underpins spectroscopic studies of the single-particle structure of nuclei, tests nuclear mean-field and correlation effects, and serves as a benchmark for reaction and nuclear-structure models. Knockout reactions are interpreted, depending on the probe and energy, via eikonal/Glauber theory, distorted-wave impulse approximation (DWIA), or relativistic impulse approximation (RIA) frameworks, with crucial input from nuclear-structure calculations and nucleon-nucleus optical-model potentials.

1. Theoretical Frameworks for Single-Nucleon Knockout

Single-nucleon knockout cross sections are computed within reaction-theoretical approaches that accommodate the relevant quantum dynamics and final-state effects:

Eikonal-Glauber Model: At energies above ~50–80 MeV/nucleon for hadronic probes, the eikonal (sudden) and Glauber approximations are justified. The inclusive removal cross section for population of all bound residue states factorizes:

$\sigma_{-1N} = \sum_{n\ell j} C^2S_{n\ell j}\; \sigma_{\text{sp}}(n\ell j, S^*_\alpha)$

where $C^2S_{n\ell j}$ is the spectroscopic factor, and $\sigma_{\text{sp}}$ is the "single-particle" cross section for the specific orbital and separation energy, computed via eikonal S-matrices for core–target and nucleon–target interactions (Tostevin et al., 2014, Hebborn et al., 2019).

DWIA (Distorted-Wave Impulse Approximation): In electron and neutrino-induced knockout, cross sections are commonly described by DWIA, where the knocked-out nucleon's final state is distorted by a complex optical potential. The differential cross section reads

$\frac{d^3\sigma}{dE' d\Omega_{e'} d\Omega_N} \propto \sum_{j} S_j \left|\langle \chi^{(-)}_{\mathbf{k}_N} | \hat{J}^\mu(q) | \phi_j \rangle \right|^2$

with $S_j$ the spectroscopic factor, and $\chi^{(-)}_{\mathbf{k}_N}$ the distorted outgoing nucleon wave function (González-Jiménez et al., 2019, Shim et al., 2023).

Impulse Approximation: In neutrino-nucleus scattering, quasielastic single-nucleon knockout is modelled in the impulse approximation, where the weak probe acts on an individual nucleon embedded in a mean field, with nuclear binding and correlations encoded via spectral functions and nuclear response functions (Ankowski et al., 2015, Casale et al., 27 Jul 2025).

2. Structure of the Reaction Cross Section

In both eikonal/Glauber and DWIA approaches, the inclusive single-nucleon knockout cross section decomposes naturally into contributions from the spectroscopic occupancy of the initial orbital (the "structural factor") and a reaction-specific transmission and absorption component (the "dynamical factor"):

Framework	Structural Factor	Reaction Component
Eikonal/Glauber	$C^2S_{n\ell j}$	$\sigma_{\text{sp}}$
DWIA (hadronic, (e,e'p), (p,pN))	$S_j$	$\|\langle \chi^{(-)} \| J \| \phi_j \rangle\|^2$
Relativistic IA (CCQE $\nu$ - $A$ )	SF, shell occupancy	Nuclear response tensors

Structural factor: $C^2S$ or $S_j$ accounts for the shell-model probability to find the nucleon in the relevant quantum state, and is subject to quenching by correlations (Kay et al., 2013, Crespo et al., 2018).
Reaction component: Eikonal S-matrix or DWIA matrix elements model multiple scattering, absorption, and the detailed dynamics of the outgoing nucleon (Shim et al., 2023).

3. Reaction Mechanism: Stripping, Diffraction, and Absorption

The detailed reaction mechanism involves:

Stripping ("inelastic breakup"): The nucleon is inelastically absorbed by the target, residue survives. Dominant in proton and heavy-ion targets.
Diffraction ("elastic breakup"): Projectile breaks up into residue + nucleon, both survive, driven by the coherent action of elastic S-matrices.
DWIA Absorption: Nuclear absorption and loss to inelastic channels are mediated by the imaginary part of the optical potential in the distorted waves. The absorption factor $R = \sigma^{\rm DW}/\sigma^{\rm PW}$ parameterizes the reduction from the unattenuated plane-wave impulse approximation, with $R$ decreasing with increasing target mass and for more localized (low- $n$ , low- $l$ ) orbitals (Shim et al., 2023).

In both eikonal and DWIA the S-matrix or optical potential is constructed using empirical or microscopic parametrizations, typically Woods–Saxon forms, and adjusted to nucleon–nucleus systematics (Hebborn et al., 2022).

4. Nuclear Structure Inputs: Spectroscopic Factors and Correlations

Nuclear-structure information is encoded via:

Shell-Model Spectroscopic Factors ( $C^2S$ ): Determined by diagonalization in appropriate truncated spaces with effective interactions ( $sd$ , $pf$ shells etc.), with a normalization (center-of-mass) correction factor applied (Tostevin et al., 2014).
Spectroscopic Quenching: Empirical analyses of (e,e'p), (p,2p), and transfer reactions show a uniform reduction of measured strength relative to mean-field theory: $F_q \simeq 0.55 \pm 0.10$ , attributed to short-range correlations (SRC) and tensor forces not captured by the shell model (Kay et al., 2013, Crespo et al., 2018). Ab initio QMC and GFMC calculations provide overlaps and cross-section ratios $\mathcal{R}_\sigma \approx \mathcal{R}_\Sigma$ in close agreement with experiment for light nuclei (Crespo et al., 2018).
Asymptotic Normalization Coefficient (ANC): For weakly bound ("halo") systems, the cross section and the width of the parallel-momentum distribution of the residue are governed by the ANC, with the knockout being purely peripheral and insensitive to the interior wave function or the continuum structure (Hebborn et al., 2019, Hebborn et al., 2019).

5. Model and Medium-Uncertainty, Core-Destruction, Isospin Effects

Interpretation of measured single-nucleon knockout cross sections requires quantification of theoretical uncertainties and possible corrections:

Optical-Model Uncertainty: Bayesian analyses show theoretical uncertainties of at least 20% for halo nuclei and at least 40% for tightly-bound nuclei in knockout cross section predictions, critical for evaluation of extracted spectroscopic factors or ANCs (Hebborn et al., 2022). Ambiguities in the geometry and depth of the imaginary optical potential, especially for deeply bound nucleons, dominate the error budgets.
Core Destruction Effects: Inclusion of the process wherein the removed nucleon continues to interact with and potentially destroy the residue (core) after being absorbed reduces the isospin asymmetry of the "quenching factor" $R_s(\Delta S)$ , aligning knockout results with transfer and (p,pN) systematics and explaining observed flattening of $R_s$ for large separation energies (Gomez-Ramos et al., 2023).
Medium Modifications and Pauli Blocking: Eikonal S-matrices must include in-medium nucleon–nucleon cross-section effects, notably Pauli blocking, especially for halo nuclei, modifying cross sections by 5–10% (Bertulani et al., 2010).

6. Systematics, Quenching, and Physical Interpretation

Extensive experimental systematics have established:

Quenching Systematics: The ratio $R_s = \sigma_{\text{exp}}/\sigma_{\text{th}}$ decreases approximately linearly with removal-nucleon separation-energy asymmetry $\Delta S$ (Tostevin et al., 2021):

$R_s(\Delta S) \approx 0.61 - 0.016\,\Delta S \qquad (\Delta S\;\text{in MeV})$

Ranging from $R_s \simeq 1$ for weakly bound nucleons ( $\Delta S \sim -20$ MeV) to $R_s \simeq 0.3$ for deeply bound nucleons ( $\Delta S \sim +20$ MeV).

Isospin and Deficient-Species Trends: Knockout of the more correlated, deficient nucleon species (protons in neutron-rich, neutrons in proton-rich systems) displays stronger quenching, consistent with observed enhancements in high-momentum tails and the occupation of single-particle orbits (Crespo et al., 2018, Hebborn et al., 2019).
Role of Meson-Exchange Currents (MEC) and SRCs in QE Knockout: In charged-current (CC) neutrino scattering, one-body–two-body current interference suppresses the dominant transverse 1p1h response functions and thus the CCQE single-nucleon knockout cross section by 10–20%, with similar effects observed across RFG, RMF, and superscaling (SuSAM*) approaches (Casale et al., 27 Jul 2025, Cuyck et al., 2017). Including these corrections is necessary for accurate neutrino-nucleus event generator implementations.

7. Practical Calculation and Experimental Comparison

A typical calculation of a single-nucleon knockout cross section involves:

Projectile and Target Specification: Choose the $(Z,N)$ and target nucleus, select the beam energy.
Structure Inputs: Shell-model diagonalization to obtain final states, excitation energies, and $C^2S(\alpha,j)$ values with c.m. correction.
Single-Particle Orbitals: Woods–Saxon parameters fitted to reproduce separation energies for each final state.
Densities and NN Profile: Obtain densities from self-consistent models (e.g., SkX Hartree–Fock), select free or in-medium NN parameters.
Eikonal/DWIA Calculation: Compute S-matrices or optical potentials, evaluate $\sigma_{\text{sp}}$ for each orbital and final state, sum to obtain $\sigma_{\text{th}}$ .
Comparison to Data and $R_s$ Extraction: Measure $\sigma_{\text{exp}}$ , form $R_s$ for physical interpretation and for benchmarking nuclear structure models.

Uncertainties are typically $\sim20\%$ for nucleon-removal cross section predictions dominated by shell-model truncations, optical-potential ambiguities, and NN parameters (Hebborn et al., 2022, Tostevin et al., 2014).

In conclusion, the single-nucleon knockout cross section is a primary probe of the single-particle structure and quantum correlations in nuclei. Its calculation and interpretation draw on advanced reaction and structure theory, careful quantification of uncertainties, and a growing systematics of high-precision experimental data. Robust connections to transfer and (p,pN) reaction systematics, as well as precise formal expressions for core-momentum observables, have positioned this observable at the core of contemporary nuclear-structure research (Tostevin et al., 2014, Tostevin et al., 2021, Kay et al., 2013, Crespo et al., 2018, Hebborn et al., 2019, Hebborn et al., 2019, Hebborn et al., 2022, Gomez-Ramos et al., 2023).