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.

Ruling out Color Transparency in Quasielastic ^{12}C(e,e^{'}p) up to Q^{2} of 14.2 (GeV/c)^{2}.

Quasielastic ^{12}C(e,e^{'}p) scattering was measured at spacelike 4-momentum transfer squared Q^{2}=8, 9.4, 11.4, and 14.2 (GeV/c)^{2}, the highest ever achieved to date. Nuclear transparency for this reaction was extracted by comparing the measured yield to that expected from a plane-wave impulse approximation calculation without any final state interactions. The measured transparency was consistent with no Q^{2} dependence, up to proton momenta of 8.5 GeV/c, ruling out the quantum chromodynamics effect of color transparency at the measured Q^{2} scales in exclusive (e,e^{'}p) reactions. These results impose strict constraints on models of color transparency for protons.

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.

Comparing proton momentum distributions in A = 2 and 3 nuclei via 2H 3H and 3He (e,e′p) measurements

We report the first measurement of the (e,e′p) reaction cross-section ratios for Helium-3 (He3), Tritium (H3), and Deuterium (d). The measurement covered a missing momentum range of 40≤pmiss≤550MeV/c, at large momentum transfer (〈Q2〉≈1.9 (GeV/c)2) and xB>1, which minimized contributions from non quasi-elastic (QE) reaction mechanisms. The data is compared with plane-wave impulse approximation (PWIA) calculations using realistic spectral functions and momentum distributions. The measured and PWIA-calculated cross-section ratios for He3/d and H3/d extend to just above the typical nucleon Fermi-momentum (kF≈250 MeV/c) and differ from each other by ∼20%, while for He3/H3 they agree within the measurement accuracy of about 3%. At momenta above kF, the measured He3/H3 ratios differ from the calculation by 20%−50%. Final state interaction (FSI) calculations using the generalized Eikonal Approximation indicate that FSI should change the He3/H3 cross-section ratio for this measurement by less than 5%. If these calculations are correct, then the differences at large missing momenta between the He3/H3 experimental and calculated ratios could be due to the underlying NN interaction, and thus could provide new constraints on the previously loosely-constrained short-distance parts of the NN interaction.

Spectral line shape and angular momentum

Data have been obtained for ejection of valence shell 4p and 3d electrons in krypton and the 1t2 and 2a1 orbitals in methane. Measurements, with kinematics both on and off the Bethe ridge, examine how the spectral shapes change for fixed energies of 1015 eV for the incident electron, 880 eV for scattered and 120 eV for ejected electron and angular momentum changes. The studied interactions are activated by excitation and ionization of inner shell (-d) electrons and concern complex electron correlations. The triple differential cross sections for the Bethe ridge conditions within the experimental accuracy are similar to the values predicted by the Distorted Wave Impulse Approximation and Born Approximation calculations but need further consideration of electron correlation effects.

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.

High-resolution electron-momentum spectroscopy of the valence orbitals of the benzene molecule

The binding-energy spectrum and electron-momentum spectra of valence orbitals of benzene were measured via a high-resolution electron-momentum spectrometer at incident energies of 600 and 1200 eV plus the binding energy. The experimental momentum profiles were compared with the results from a plane-wave impulse approximation (PWIA) calculation using Dyson orbitals obtained from symmetry-adapted-cluster configuration-interaction calculations. For the ${1e}_{1\mathrm{g}}, {3e}_{2\mathrm{g}}, {3e}_{1\mathrm{u}}$, and ${2b}_{1\mathrm{u}}$ orbitals, the discrepancy between the experimental distributions and the PWIA calculations was tentatively assigned to the distorted wave effects. The current high-resolution electron-momentum spectroscopy results provide the experimental benchmark for rigorous distorted-wave calculations about benzene in the future.

Proton elastic scattering at 200 A MeV and high momentum transfers of 1.7–2.7 fm−1 as a probe of the nuclear matter density of 6He

Differential cross sections of p–6He elastic scattering were measured in inverse kinematics at an incident energy of 200 A MeV, covering the high momentum transfer region of 1.7–2.7 fm−1. The sensitivity of the elastic scattering at low and high momentum transfers to the density distribution was investigated quantitatively using relativistic impulse approximation calculations. In the high momentum transfer region, where the present data were taken, the differential cross section has an order of magnitude higher sensitivity to the inner part of the 6He density relative to the peripheral part (15:1). This feature makes the obtained data valuable for the deduction of the inner part of the 6He density. The data were compared to a set of calculations assuming different proton and neutron density profiles of 6He. The data are well reproduced by the calculation assuming almost the same profiles of proton and neutron densities around the center of 6He, and a proton profile reproducing the known point-proton radius of 1.94 fm. This finding is consistent with the assumption that the 6He nucleus consists of a rigid α-like core with a two-neutron halo.

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.

Analysing powers and spin correlations in deuteron–proton charge exchange at 726 MeV

The charge exchange of vector polarised deuterons on a polarised hydrogen target has been studied in a high statistics experiment at the COSY-ANKE facility at a deuteron beam energy of Td=726 MeV. By selecting two fast protons at low relative energy Epp, the measured analysing powers and spin correlations are sensitive to interference terms between specific neutron–proton charge-exchange amplitudes at a neutron kinetic energy of Tn≈12Td=363 MeV. An impulse approximation calculation, which takes into account corrections due to the angular distribution in the diproton, describes reasonably the dependence of the data on both Epp and the momentum transfer. This lends broad support to the current neutron–proton partial wave solution that was used in the estimation.

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.

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.

Molecular-frame (e,2e) experiment for N2at large momentum transfer

We report molecular-frame ($e$,2$e$) cross sections for N${}_{2}$ at large momentum transfer, obtained using the electron-electron-fragment ion triple-coincidence technique. The measured angular distribution of the ($e$,2$e$) cross section for the inner-valence 2${\ensuremath{\sigma}}_{\mathrm{g}}$ orbital appears to show that the spatial character of the orbital has been directly observed in momentum space. On the other hand, experimental results for ionization to states above the double-ionization threshold suggested a larger intensity in the direction perpendicular to the molecular axis rather than parallel, an observation that our plane-wave impulse-approximation calculations fail to reproduce.

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.

Dirac oscillators and the relativisticRmatrix

The Dirac oscillators are shown to be an excellent expansion basis for solutions of the Dirac equation by $R$-matrix techniques. The combination of the Dirac oscillator and the $R$-matrix approach provides a convenient formalism for reactions as well as bound-state problems. The utility of the $R$-matrix approach is demonstrated in relativistic impulse approximation calculations where exchange terms can be calculated exactly and scattering waves made orthogonal to bound-state wave functions.