Distorted-wave Impulse Approximation Calculations Research Articles

Pikoe: A computer program for distorted-wave impulse approximation calculation for proton induced nucleon knockout reactions

pikoe, Proton-Induced KnockOut reaction calculation for Exclusive processes, is a Fortran 90 program that calculates triple- and quadruple-differential cross sections, vector analyzing powers, and momentum distributions of reaction residues, for proton-induced nucleon knockout reactions in normal and inverse kinematics. The distorted-wave impulse approximation with the factorization approximation to the nucleon-nucleon transition matrix is adopted, and the distorted waves of the incoming proton and the outgoing two nucleons are calculated quantum mechanically. Kinematics of the reaction particles are treated in a relativistic manner, which gives the proper asymptotics of the three-body scattering wave in the plane-wave limit. Program summaryProgram Title: pikoeCPC Library link to program files:https://doi.org/10.17632/m594h58kck.1Licensing provisions: MITProgramming language: Fortran 90Supplementary material: See https://www.rcnp.osaka-u.ac.jp/~kazuyuki/pikoe/index.phpNature of problem: Calculation of triple- and quadruple-differential cross sections, vector analyzing powers, and momentum distributions of reaction residues, for nucleon knockout reactions with proton beams/targets in normal/inverse kinematics.Solution method: Numerov method for solving single-channel second-order differential equations; multi dimensional integration with the Gauss-Legendre quadrature rule.Additional comments including restrictions and unusual features: The global optical potential by Koning and Deraloche [1] for nucleons at energies from 1 keV to 200 MeV, for (near-)spherical nuclei with mass number between 24 and 209, is implemented in the program. Tables of nucleon-nucleon (NN) elastic differential cross sections and NN transition amplitudes calculated with the NN effective interaction by Franey and Love [2] are provided as external files for pikoe (included in the package). Users are encouraged to prepare tables for the optical potentials and the cross sections of the elementary process according to the reaction system of interest by themselves. With external files thus prepared, pikoe can be applied to various kinds of knockout processes such as α or deuteron knockout by proton, α-induced knockout reactions, etc. Calculation of spin observables, the vector analyzing power in fact, however, is restricted to nucleon-induced nucleon knockout processes with the coplanar kinematics.

Experimental study of (p, 2p) reactions at 197 MeV on 12C, 16O, 40,48Ca, and 90Zr nuclei leading to low-lying states of residual nuclei

Abstract We have measured the differential cross section and analyzing power for (p, 2p) reactions that lead to discrete states of residual nuclei at an incident energy of 197 MeV for five kinds of target nuclei, 12C, 16O, 40,48Ca, and 90Zr. The data are compared with distorted-wave impulse approximation calculations using two kinds of global optical potentials. The spectroscopic factors obtained from these calculations agree with those determined in (e, e′p) studies within 20%, except for one orbital of the 90Zr nucleus where the distortion effect is much greater than that for the other orbitals of 90Zr and for other target nuclei. We also observed a clear and visibly distinct j-dependence in the analyzing power.

Probing for high-momentum protons in He4 via the He4(e,e′p)X reactions

Experimental cross sections for the He4(e,e′p)X reactions in the missing energy range from 0.017 to 0.022 GeV and up to a missing momentum of 0.632 GeV/c at xB=1.24 and Q2=2 (GeV/c)2 are reported. The data are compared to relativistic distorted-wave impulse approximation calculations for the He4(e,e′p)H3 channel. Significantly more events are observed for pm≥0.45 GeV/c than are predicted by the theoretical model, and striking fluctuations in the ratio of data to the theoretical model around pm=0.3GeV/c are possible signals of initial-state multinucleon correlations.5 MoreReceived 11 May 2019Revised 13 March 2022Accepted 14 April 2022DOI:https://doi.org/10.1103/PhysRevC.105.064003©2022 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBreakup reactionsFew-body systemsNucleon-nucleon interactionsPropertiesA ≤ 5Nuclear Physics

Benchmarking intranuclear cascade models for neutrino scattering with relativistic optical potentials

The description of final-state interactions (FSI) in the large phase space probed in neutrino experiments poses a great challenge. In neutrino experiments, which operate under semi-inclusive conditions, cascade models are commonly used for this task, while under exclusive conditions FSI can be treated with relativistic optical potentials (ROP). We formulate conditions under which the ROP approach and cascade model can be directly compared. We feed the NEUT cascade with events from a relativistic distorted-wave impulse approximation calculation that uses the real part of an optical potential. Cuts on the missing energy of the resulting events are applied to define a set of events that can be directly compared to RDWIA calculations with the full optical potential. The NEUT cascade and ROP agree for proton kinetic energies $T_p > 150$ MeV for carbon, oxygen and calcium nuclei when a realistic nuclear density is used to introduce events in the cascade. For $T_p < 100$ MeV the ROP and NEUT cross sections differ in shape and differences in magnitude are larger than 50 \%. Single transverse variables allow to distinguish different approaches to FSI, but due to a large non-QE contribution the comparison to T2K data does not give an unambiguous view of FSI. We discuss electron scattering and argue that with a cut in missing energy FSI can be studied with minimal confounding factors in e.g. $e4\nu$. The agreement of the ROP and NEUT for T2K conditions lends confidence to these models as a tool in oscillation analyses for sufficiently large nucleon kinetic energies. These results urge for caution when a cascade model is applied for small nucleon energies. The assessment of model assumptions relevant to this region are strongly encouraged. This paper provides novel constraints on cascade models from proton-nucleus scattering that can be easily applied to other neutrino event generators.

Experimental study of (p, 2p) reactions at 392 MeV on 12C, 16O, 40Ca and 208Pb nuclei leading to low-lying states of residual nuclei

Abstract We have measured the differential cross-sections and analyzing powers for ($p,2p$) reactions at an incident energy of 392 MeV on $^{12}$C, $^{16}$O, $^{40}$Ca, and $^{208}$Pb nuclei, leading to discrete states of the residual nuclei. The data are compared with two kinds of distorted-wave impulse approximation (DWIA) calculations, a standard calculation using a global optical potential and a calculation using wave functions generated in a relativistic Hartree model. The spectroscopic factors deduced from these two calculations agree with those determined in ($e,e'p$) studies mostly within 15% in the case of the lighter three target nuclei. However, those for the $^{208}$Pb target are overestimated compared with the ($e,e'p$) results. In the heavy target case, the DWIA results are very sensitive to the radius parameter of the bound-state potential and thus a careful treatment is required. Regarding the analyzing powers of the present measurement, we confirmed that the $j$-dependence is sufficient for practical spectroscopic use.

Toward a reliable description of (p,pN) reactions in the distorted-wave impulse approximation

Background: Proton-induced nucleon knockout $(p,pN)$ reactions have been successfully used to study the single-particle nature of stable nuclei in normal kinematics with the distorted-wave impulse approximation (DWIA) framework. Recently, these reactions have been applied to rare-isotope beams at intermediate energies in inverse kinematics to study the quenching of spectroscopic factors.Purpose: Our goal is to investigate the effects of various corrections and uncertainties within the standard DWIA formalism on the $(p,pN)$ cross sections. The consistency of the extracted reduction factors between DWIA and other methods is also evaluated.Method: We analyze the $(p,2p)$ and $(p,pn)$ reaction data measured at the ${\mathrm{R}}^{3}\mathrm{B}$-LAND setup at GSI for carbon, nitrogen, and oxygen isotopes in the incident energy range of 300--450 MeV/u. Cross sections and reduction factors are calculated by using the DWIA method. The transverse momentum distribution of the $^{12}\mathrm{C}(p,2p)^{11}\mathrm{B}$ reaction is also investigated.Results: We have found that including the nonlocality corrections and the M\o{}ller factor affects the cross sections considerably. The proton-neutron asymmetry dependence of reduction factors extracted by the DWIA calculation is very weak and consistent with those given by other reaction methods and ab initio structure calculations.Conclusions: The results found in this work provide a detailed investigation of the DWIA method for $(p,pN)$ reactions at intermediate energies. They also suggest that some higher-order effects, which is essential for an accurate cross-section description at large recoil momentum, is missing in the current DWIA and other reaction models.

Dominance of Tensor Correlations in High-Momentum Nucleon Pairs Studied by (p,pd) Reaction.

The isospin character of p-n pairs at large relative momentum has been observed for the first time in the ^{16}O ground state. A strong population of the J,T=1,0 state and a very weak population of the J,T=0,1 state were observed in the neutron pickup domain of ^{16}O(p,pd) at 392MeV. This strong isospin dependence at large momentum transfer is not reproduced by the distorted-wave impulse approximation calculations with known spectroscopic amplitudes. The results indicate the presence of high-momentum protons and neutrons induced by the tensor interactions in the ground state of ^{16}O.

Validity of the distorted-wave impulse-approximation description of Ca40(e,e′p)K39 data using only ingredients from a nonlocal dispersive optical model

The nonlocal implementation of the dispersive optical model (DOM) provides all the ingredients for distorted-wave impulse-approximation (DWIA) calculations of the $(e,e'p)$ reaction. It provides both the overlap function, including its normalization, and the outgoing proton distorted wave. This framework is applied to describe the knockout of a proton from the $0\textrm{d}\frac{3}{2}$ and $1\textrm{s}\frac{1}{2}$ orbitals in ${}^{40}$Ca with fixed normalizations of 0.71 and 0.60, respectively. Data were obtained in parallel kinematics for three outgoing proton energies: 70, 100, and 135 MeV. Agreement with the data is as good as, or better than, previous descriptions employing local optical potentials and overlap functions from Woods-Saxon potentials - both with standard nonlocality corrections - whose normalization (spectroscopic factor) and radius were fitted to the data. The present analysis suggests that slightly larger spectroscopic factors are obtained when nonlocal optical potentials are employed than those generated with local potentials. The results further suggest that the chosen kinematical window around 100 MeV proton energy provides the best and cleanest method to employ the DWIA for the analysis of this reaction. The conclusion that substantial ground-state correlations cannot be ignored when describing a closed-shell atomic nucleus is therefore confirmed in detail. To reach these conclusions, it is essential to have a complete description of the nucleon single-particle propagator that accounts for all elastic nucleon-scattering observables in a wide energy domain up to 200 MeV. The current nonlocal implementation of the DOM fulfills this requirement.

Cross sections and analyzing powers for ( p⃗,np ) reactions of H2, Li6 , and C12 at 296 MeV

We report measurements of cross sections and analyzing powers for nucleon-knockout $(\stackrel{P\vec}{p},np)$ reactions of $^{2}\mathrm{H}, ^{6}\mathrm{Li}$, and $^{12}\mathrm{C}$ at 296 MeV using the RCNP NPOL3 neutron detector and large acceptance spectrometer. The $1s$ and $1p$ knockout reactions were successfully separated with a separation-energy resolution of approximately 6 MeV. The resulting data were compared with plane-wave and distorted-wave impulse approximation calculations employing the nucleon-nucleon $t$ matrix in free space. No significant differences were identified in the case of the present analyzing power data, in contrast to the significant reductions observed in $(\stackrel{P\vec}{p},2p)$ values. The conventional Pauli blocking and nuclear binding effects with the $g$ matrix and the relativistic effect of nucleon mass reduction in the nuclei are also discussed.

Neutron occupancy of the [formula omitted] orbital and the [formula omitted] shell closure in 24O

One-neutron knockout from 24O leading to the first excited state in 23O has been measured for a proton target at a beam energy of 62 MeV/nucleon. The decay energy spectrum of the neutron unbound state of 23O was reconstructed from the measured four momenta of the 22O fragment and emitted neutron. A sharp peak was found at Edecay=50±3 keV, corresponding to an excited state in 23O at 2.78±0.11 MeV, as observed in previous measurements. The longitudinal momentum distribution for this state was consistent with d-wave neutron knockout, providing support for a Jπ assignment of 5/2+. The associated spectroscopic factor was deduced to be C2S(0d5/2)=4.1±0.4 by comparing the measured cross section (σ−1nexp=61±6 mb) with a distorted wave impulse approximation calculation. Such a large occupancy for the neutron 0d5/2 orbital is in line with the N=16 shell closure in 24O.

Cluster knockout reactions

Cluster knockout reactions are expected to reveal the amount of clustering (such as that of α, d and even of heavier clusters such as12C, 16O etc.) in the target nucleus. In simple terms, incident medium high-energy nuclear projectile interacts strongly with the cluster (present in the target nucleus) as if it were existing as a free entity. Theoretically, the relatively softer interactions of the two outgoing particles with the residual nucleus lead to optical distortions and are treated in terms of distorted wave (DW) formalism. The long-range projectile–cluster interaction is accounted for, in terms of the finite range (FR) direct reaction formalism, as against the more commonly adopted zero-range (ZR) distorted wave impulse approximation (DWIA) formalism. Comparison of the DWIA calculations with the observed data provide information about the momentum distribution and the clustering spectroscopic factor of the target nucleus. Interesting results and some recent advancements in the area of (α, 2 α) reactions and heavy cluster knockout reactions are discussed. Importance of the finite-range vertex and the final-state interactions are brought out.

Alpha-cluster structure in the ground state of 40Ca displayed in a (p,pα) knockout reaction

The analyzing power is very sensitive to details of the reaction mechanism of (p, pα) knockout reactions in the incident energy range of approximately 100 MeV and higher. Whereas distorted wave impulse approximation calculations in the past proved to give an excellent reproduction of analyzing power angular distributions for quasifree (p, pα) reactions on light targets such as 6Li, 9Be and 12C, the situation for 40Ca was not as simple. It is now shown that the theory also offers good agreement with the experimental distribution of the heaviest target nucleus if care is taken to use proper distorted waves which treat α−36Ar properly as a system for which α–elastic scattering is anomalous. Thus it is shown that 40Ca reveals its ground state α–cluster structure in an unambiguous way similar to the light target nuclei.

Complete set of polarization transfer observables for the16O(p⃗,n⃗)16F reaction at 296 MeV and 0 degrees

We report measurements of the cross section and a complete set of polarization transfer observables for the ${}^{16}{\rm O}(\vec{p},\vec{n}){}^{16}{\rm F}$ reaction at a bombarding energy of $T_p$ = 296 MeV and a reaction angle of $\theta_{\rm lab}$ = $0^{\circ}$. The data are compared with distorted-wave impulse approximation calculations employing the large configuration-space shell-model (SM) wave functions. The well-known Gamow-Teller and spin-dipole (SD) states at excitation energies of $E_x$ $\lesssim$ 8 MeV have been reasonably reproduced by the calculations except for the spin--parity $J^{\pi}$ = $2^-$ state at $E_x$ = 5.86 MeV. The SD resonance at $E_x$ $\simeq$ 9.5 MeV appears to have more $J^{\pi}$ = $2^-$ strength than $J^{\pi}$ = $1^-$ strength, consistent with the calculations. The data show significant strength in the spin-longitudinal polarized cross section $ID_L(0^{\circ})$ at $E_x$ $\simeq$ 15 MeV, which indicates existence of the $J^{\pi}$ = $0^-$ SD resonance as predicted in the SM calculations.

Precise Extraction of the Induced Polarization in theHe4(e,e′p→)H3Reaction

We measured with unprecedented precision the induced polarization P(y) in (4)He(e,e'p)(3)H at Q(2)=0.8 and 1.3 (GeV/c)(2). The induced polarization is indicative of reaction-mechanism effects beyond the impulse approximation. Our results are in agreement with a relativistic distorted-wave impulse approximation calculation but are overestimated by a calculation with strong charge-exchange effects. Our data are used to constrain the strength of the spin-independent charge-exchange term in the latter calculation.

Polarization transfer measurements forC12(p→,n→)N12(g.s.,1+)at 296 MeV and nuclear correlation effects

Differential cross sections and complete sets of polarization observables are presented for the Gamow-Teller {sup 12}C(p-vector,n-vector){sup 12}N(g.s.,1{sup +}) reaction at a bombarding energy of 296 MeV with momentum transfers q of 0.1 to 2.2 fm{sup -1}. The polarization transfer observables are used to deduce the spin-longitudinal cross section, ID{sub q}, and spin-transverse cross sections, ID{sub p} and ID{sub n}. The data are compared with calculations based on the distorted-wave impulse approximation (DWIA) using shell-model wave functions. Significant differences between the experimental and theoretical results are observed for all three spin-dependent ID{sub i} at momentum transfers of q > or approx. 0.5 fm{sup -1}, suggesting the existence of nuclear correlations beyond the shell model. We also performed DWIA calculations employing random-phase approximation (RPA) response functions and found that the observed discrepancy is partly resolved by the pionic and {rho}-mesonic correlation effects.

Analyzing power and cross section distributions of theC12(p,pα)Be8cluster knockout reaction at an incident energy of 100 MeV

The ($p,p\ensuremath{\alpha}$) reaction on $^{12}\mathrm{C}$ was investigated experimentally using polarized incident protons of 100 MeV. The scattered proton and $\ensuremath{\alpha}$ particle, from the knockout reaction, were detected in coincidence. Coincident data, which were obtained at ten quasifree angle pairs for proton angles ranging from ${25}^{\ifmmode^\circ\else\textdegree\fi{}}$ to ${110}^{\ifmmode^\circ\else\textdegree\fi{}}$, were analyzed in terms of the distorted-wave impulse approximation (DWIA). Calculated energy-sharing cross section and analyzing power distributions reproduce the data reasonably well, indicating that a quasifree knockout mechanism dominates the reaction. Since measurements of analyzing powers were made, spin-orbit distortions were included in the DWIA calculations. The effects of this were found to be very small near zero recoil momentum and did not destroy the validity of the factorization approximation where the two-body $p\text{\ensuremath{-}}\ensuremath{\alpha}$ cross section enters as a multiplicative factor in the three-body ($p,p\ensuremath{\alpha}$) cross section expression. Spectroscopic factors derived from the data are consistent with theoretical predictions. Analyzing power data also follow the trend of free $p\text{\ensuremath{-}}^{4}\mathrm{He}$ scattering data, and comparisons with DWIA predictions are in reasonable agreement. Because the two-body interaction response between the projectile and the $\ensuremath{\alpha}$ cluster was found to resemble the scattering of protons from a free $\ensuremath{\alpha}$ particle to a remarkable degree, the present results would strongly imply the existence of preformed $\ensuremath{\alpha}$ clusters in $^{12}\mathrm{C}$.

Feasibility of extracting aΣ−admixture probability in the neutron-richΛ10Lihypernucleus

We theoretically examine production of the neutron-rich ${}_{\ensuremath{\Lambda}}^{10}\mathrm{Li}$ hypernucleus by a double-charge exchange (${\ensuremath{\pi}}^{\ensuremath{-}}$, ${K}^{+}$) reaction on a $^{10}\mathrm{B}$ target with distorted-wave impulse approximation calculations. We calculate the inclusive spectrum at the incident momentum 1.20 GeV/$c$ by a one-step mechanism ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\Sigma}}^{\ensuremath{-}}$ via ${\ensuremath{\Sigma}}^{\ensuremath{-}}$ doorways caused by a ${\ensuremath{\Sigma}}^{\ensuremath{-}}p\ensuremath{\leftrightarrow}\ensuremath{\Lambda}n$ coupling. The resultant spectrum can explain the magnitude of the recent experimental data, so that the ${\ensuremath{\Sigma}}^{\ensuremath{-}}$ admixture probability in ${}_{\ensuremath{\Lambda}}^{10}\mathrm{Li}$ is found to be of the order of ${10}^{\ensuremath{-}1}%$. The (${\ensuremath{\pi}}^{\ensuremath{-}}$, ${K}^{+}$) reaction provides the ability to extract properties of wave functions with $\ensuremath{\Lambda}\text{\ensuremath{-}}\ensuremath{\Sigma}$ coupling effects in neutron-rich nuclei, as well as the reaction mechanism.

Study of nuclear correlation effects via [formula omitted] at 296 MeV

We report measurements of the cross section and a complete set of polarization observables for the Gamow–Teller C12(p→,n→)N12(g.s.,1+) reaction at a bombarding energy of 296 MeV. The data are compared with distorted wave impulse approximation calculations employing transition form factors normalized to reproduce the observed beta-decay ft value. The cross section is significantly under-predicted by the calculations at momentum transfers q≳0.5 fm−1. The discrepancy is partly resolved by considering the non-locality of the nuclear mean field. However, the calculations still under-predict the cross section at large momentum transfers of q≃1.6 fm−1. We also performed calculations employing random phase approximation response functions and found that the observed enhancement can be attributed in part to pionic correlations in nuclei.

Limitations of the distorted-wave impulse approximation in describing the energy dependence of theB10(n,p)Be10(g.s.) reaction

We report differential cross section measurements for the stretched transition from the $^{10}\mathrm{B}$ ground state (${J}^{\ensuremath{\pi}}={3}^{+}$) to the $^{10}\mathrm{Be}$ ground state (${J}^{\ensuremath{\pi}}={0}^{+}$) in the $^{10}\mathrm{B}$($n,p$)$^{10}\mathrm{Be}$ (g.s.) reaction. These data were obtained over the energy range from 70 to 240 MeV, covering momentum transfer values from 0.6 to 2.5 fm${}^{\ensuremath{-}1}$. In this momentum transfer range, the isovector tensor effective interaction dominates the transition. Cross sections are compared to zero- and finite-range distorted wave impulse approximation calculations using modern free and density-dependent effective interactions and a transition density consistent with ($e,{e}^{'}$) data. Good agreement is observed at energies above 120 MeV, but below this energy the cross sections are larger than the calculated values by more than 40%. The implications for DWIA calculations are discussed.

Isovector effective NN interaction in Si28(p→,n→)P28(6−) at 198 MeV

Abstract We report measurements of the cross section and a complete set of polarization observables for the Si 28 ( p → , n → ) P 28 ( 6 − ) reaction at a bombarding energy of 198 MeV. The data are compared with distorted wave impulse approximation calculations employing response functions normalized to inelastic electron scattering. The spin-longitudinal polarized cross section I D q is slightly over-predicted by the calculations, while the normal spin-transverse polarized cross section I D n is significantly under-predicted. The calculated in-plane spin-transverse I D p and spin-scalar I D 0 polarized cross sections agree well with the experimental data. These results are consistent with those for Si 28 ( p → , p → ′ ) Si 28 ( 6 − , T = 1 ) scattering at the same energy, and thus it is concluded that isospin-mixing effects are not responsible for the discrepancy between theory and experiment in the ( p → , p → ′ ) case. Energy half-off-shell effects as medium effects on the effective nucleon–nucleon interaction are also investigated and found to be too small to be responsible for the discrepancy.