Three-body calculations for $(p,pN)$ reactions: Kinematically inclusive, semi-inclusive, and fully exclusive cross sections

Background: Nucleon knockout reactions have been previously used to extract single particle information from nuclei. The analysis of nucleon knockout from a stable projectile in the collision with a proton target and the comparison with the experimental data is a key test for the reaction and structure models used to evaluate the reaction observables.

Purpose: We analyze $p$ and $n$ knockout from ${}^{12}\mathrm{C}$, assuming that only the heavy fragment or core $C$ (taken as inert), the knockout particle $N$, and the proton target $p$ participate in the collision process with the aim of (i) getting insight to the dominant kinematic conditions of the emitted particles; (ii) clarifying the dynamics of the reaction; (iii) exploring the isospin dependence (here $p$ and $n$ knockout) of the calculated reaction cross sections.

Method: We solve three-body Faddeev/Alt-Grassberger-Sandhas (Faddeev/AGS) equations for transition operators and calculate kinematically fully exclusive, semi-inclusive, and inclusive cross sections.

Results: We show that (i) the dominant final-state kinematic conditions are consistent with the assumption of quasifree scattering reaction mechanism; (ii) the distortions due to higher order multiple scattering terms depend on the final-state kinematics, and the $N-C$ and $p-C$ final state interaction provide significant effects in the calculated observables; (iii) the twofold energy–polar angle and polar angle–polar angle cross sections exhibit distinct $p$- and $n$-knockout behaviors. Finally we also show that the Faddeev/AGS formalism is able to a certain extent to reproduce the available experimental data for the $p$ knockout.

Conclusions: Kinematically fully exclusive measurements of $p$ and $n$ knockout are needed to rigorously assess the role of the distortion, as it cannot be taken into account as an overall reduction factor. This is a prerequisite for a reliable understanding of the structure of the projectile and the reaction mechanism. In addition, realistic interactions inferred from ab initio structure models are needed for analyzing experimental data.

Figure 1

Diagrammatic representation of the multiple scattering expansion of the Faddeev/AGS formalism viewed in terms of the transition amplitude of each interaction pair to third order. The first term of the expansion represents the PWIA.

Figure 2

Diagrammatic representation of the multiple scattering contributing terms (up to third order) to the $p$-core FSI and $N$-core FSI.

Figure 3

QFS diagram for the $N$-knockout reaction $A(p,pN)$.

Figure 4

Shell model scheme for the $p-$ knockout from ${}^{12}\mathrm{C}$ leading to the ground state $(3/2_1^{-})$ and the two excited states $(1/2_1^{-})$ and $(3/2_2^{-})$ of ${}^{11}\mathrm{B}$.

Figure 5

Faddeev/AGS single particle total cross sections for $p$ knockout using different potential parametrizations for the ground state $(3/2_1^{-})$ and the two excited states $(1/2_1^{-})$ and $(3/2_2^{-})$ represented here as a function of the nucleon separation energy. Also shown the EIK-DWIA results for the same final states.

Figure 6

Heavy fragment ground state transverse momentum $p_c$ distributions along the Cartesian $X$ axis for $p$ (top) and $n$ knockout (bottom). The solid and dot-dashed lines represent the FADD and the dashed lines represent the renormalized PWIA cross sections. The ratios $\mathcal{R}$ of the FADD to the PWIA total cross section are given in Table 2.

Figure 7

Nucleon transverse momentum along the Cartesian $X$ axis $p_x$ distributions for $p$ and $n$ knockout. The middle and bottom graphs show the momentum distributions for the target proton and the neutron, respectively. The solid and dot-dashed lines represent the FADD and the dashed line represents the renormalized PWIA cross sections. The ratios $\mathcal{R}$ are given in Table 2.

Figure 8

Angular cross sections for $p$ and $n$ knockout. The middle and the bottom graphs show the cross sections for the target proton and the neutron, respectively. The solid and dashed lines represent the FADD and the renormalized PWIA cross sections, respectively, with $\mathcal{R}$ given in Table 2. The KD-KD optical potential parametrization was used.

Figure 9

$pp$ (top) and $pn$ (bottom) elastic cross sections at 400 MeV beam energy. The dashed line for $pp$ represents the angular distribution without the Coulomb interaction.

Figure 10

Contour plot of the energy-polar angle correlation cross section $\frac{d^2\sigma }{d{E}_{p}d{\theta }_p}$ for $p$ knockout and $\frac{d^2\sigma }{d{E}_{N}d{\theta }_N}$ for $n$ knockout. The cross sections are in units of mb/(MeV rad). The KD-KD optical potential parametrization was used.

Figure 11

Calculated $\frac{d^2\sigma }{d{E}_{p}d{\theta }_p}$ (solid line) and $\frac{d^2\sigma }{d{E}_{N}d{\theta }_N}$ (dashed line) at fixed angles $\theta$ of the measured particle, for $p$ and $n$ knockout. The KD-KD optical potential parametrization was used.

Figure 12

Contour plot and 3D graphical representations of the angular correlation cross sections $\frac{d^2\sigma }{d{\theta }_{p}d{\theta }_N}$ for $p$ (top) and $n$ knockout (bottom). The cross sections are in units of ${\mathrm{mb}/\mathrm{rad}}^2$. The KD-KD optical potential parametrization was used.

Figure 13

Angular correlation cross sections $d^3\sigma /\left(d{\theta }_{p}d{\theta }_{N}d\varphi \right)$ with $\varphi ={180}^{\circ }$ and fixed target proton angles for $p$ (top) and $n$ knockout (bottom). The solid and dashed lines represent the FADD and renormalized PWIA result, respectively, with $\mathcal{R}$ given in Table 2. The KD-KD optical potential parametrization was used.

Figure 14

Contour plot of the PWIA azimuthal angle correlations $\frac{d^2\sigma }{d\mathrm{\Phi }d{\theta }_p}$ and $\frac{d^2\sigma }{d\mathrm{\Phi }d{\theta }_N}$, with $\mathrm{\Delta }\mathrm{\Phi }={\mathrm{\Phi }}_N-{\mathrm{\Phi }}_p$ for $p$ and $n$ knockout, where ${\mathrm{\Phi }}_N({\mathrm{\Phi }}_p$) is the azimuthal angle for the knockout nucleon (proton). The cross sections are in units of mb/sr. The KD-KD optical potential parametrization was used.

Figure 15

$S$ parameter for different kinematic configurations as discussed in the text.

Figure 16

The top and middle parts of the figure represent the kinematically fully exclusive cross sections as a function of the $S$ parameter, $\frac{d^5\sigma }{dSd{\mathrm{\Omega }}_{p}d{\mathrm{\Omega }}_N}$, as defined in the text, for fixed polar angles and coplanar geometry, using two different optical potential parametrizations for $p$ (left) and $n$ (right) knockout. Also shown is the ratio between the FADD result and the PWIA approximation. The cross sections are in units of mb/(MeV ${\mathrm{sr}}^2$). The bottom part of the figure shows the relative energies between each interacting pair.

Figure 17

The top and middle parts of the figure represents the kinematically fully exclusive cross sections as a function of the $S$ parameter, $\frac{d^5\sigma }{dSd{\mathrm{\Omega }}_{p}d{\mathrm{\Omega }}_N}$, as defined in the text, for fixed polar angles and coplanar geometry, using two different optical potential parametrizations for $p$ (left) and $n$ (right) knockout. Also shown is the ratio between the FADD result and the PWIA approximation. The cross sections are in units of mb/(MeV ${\mathrm{sr}}^2$). The bottom part of the figure shows the relative energies between each interacting pair.

Figure 18

Calculated heavy fragment (core) transverse momentum distribution $\sqrt{{{\left(p_x\right)}^2+{\left(p_y\right)}^2}^{\phantom{0}}}$ for the $p{(}^{12}\mathrm{C},2p){}^{11}\mathrm{B}$ reaction leading to the ground state $(3/2_1^{-})$ and the two excited states $(1/2_1^{-})$ and $(3/2_2^{-})$. The experimental data are taken from Ref. [14].