Distorted-wave Impulse Approximation Calculations Research Articles

C12(p,p′)C12reaction at 155 and 200 MeV and precritical phenomena

Cross-section and analyzing-power data are presented for the $^{12}\mathrm{C}(p,{p}^{\ensuremath{'}})^{12}\mathrm{C}$ reaction at bombarding energies of 155 and 200 MeV. Comparisons are made with the systematics of data from other energies. Distorted-wave impulse-approximation calculations for the "pion-like" ${1}^{+}$, $T=1$ transition to the 15.11-MeV state are successful for $q<280$ MeV/c, but are substantially smaller than the data at larger momentum transfers. The implications of the data regarding precritical behavior associated with the pion-condensation threshold are discussed.NUCLEAR REACTIONS $^{12}\mathrm{C}(p,{p}^{\ensuremath{'}})$, $E=155$ MeV, measured $\ensuremath{\sigma}(E,\ensuremath{\theta})$; $E=200$ MeV, measured $\ensuremath{\sigma}(E,\ensuremath{\theta})$, ${A}_{y}(\ensuremath{\theta})$; $\ensuremath{\theta}=6\ensuremath{-}50\ifmmode^\circ\else\textdegree\fi{}$, $\ensuremath{\Delta}\ensuremath{\theta}=2\ensuremath{-}3\ifmmode^\circ\else\textdegree\fi{}$, ${E}_{x}=11\ensuremath{-}17$ MeV. DWIA analysis. Precritical phenomena near pion-condensation threshhold.

Inelasticπ+scattering fromC12and Si at low energies

Angular distributions for elastic and inelastic scattering of ${\ensuremath{\pi}}^{+}$ projectiles from natural carbon and silicon targets at energies of \ensuremath{\sim}35 and 68 MeV are reported. The elastic data, along with other reported low energy pion scattering data for $^{12}\mathrm{C}$, is fitted using the zero-range four-parameter (${b}_{0},{b}_{1}$) Kisslinger optical potential. The best-fit potential is found to be nearly independent of energy between 30 and 68 MeV, the largest energy dependence being in $\mathrm{Re}({b}_{0})$ which changes by 30%. Simple distorted-wave impulse approximation calculations which succeed in describing the ${2}^{+}$ 4.4 MeV cross section at 48.5 MeV do much worse at 67.5 MeV for angles where the ${2}^{+}$ cross section is comparable to or larger than the elastic cross section. This suggests that a coupled channels calculation is needed. The poor agreement between data and theory for the ${0}^{+}$ (7.6 MeV) and ${3}^{\ensuremath{-}}$ (9.6 MeV) $^{12}\mathrm{C}$ transitions found earlier at 48.5 MeV persists at 67.5 MeV. The shape of the pion energy spectra and the energy integrated inelastic angular distribution at 67.5 MeV is poorly described by a quasifree pion-nucleon scattering process.NUCLEAR REACTIONS Elastic and inelastic scattering of 35 and 68 MeV ${\ensuremath{\pi}}^{+}$ from $^{12}\mathrm{C}$ and $^{28}\mathrm{Si}$. Angular distributions: $25\ifmmode^\circ\else\textdegree\fi{}<~\ensuremath{\theta}<~150\ifmmode^\circ\else\textdegree\fi{}$. Optical model and DWIA analysis of cross-section data.

A survey of selected pion-nucleus reactions using phase-shift-equivalent potentials

Since pion-nucleus elastic scattering data can only constrain the asymptotic behavior of the scattered wave, it remains for inelastic scattering and reaction data to test the wave function in the nuclear interior. To this end, distorted-wave impulse approximation calculations have been performed for a variety of low-energy pion reactions using phase-shift-equivalent pion-nucleus potentials: inelastic excitation, photoproduction, and the ( π, p) reaction. Sensitivity of the reaction cross sections to variations in the equivalent potentials is examined, with an eye toward understanding what can be learned about pion-nucleus dynamics from a given set of reaction data.

Differential Cross Sections forO16(γ,π+)N16andB10(γ,π+)Be10in theΔ(1236)Region

Differential cross sections for $^{10}\mathrm{B}$($\ensuremath{\gamma}$, ${\ensuremath{\pi}}^{+}$)$^{10}\mathrm{Be}$ to the ground and first excited states of $^{10}\mathrm{Be}$ separately and $^{16}\mathrm{O}$($\ensuremath{\gamma}$ ${\ensuremath{\pi}}^{+}$,)$^{16}\mathrm{N}$ to the sum of the four lowest-lying states in $^{16}\mathrm{N}$ have been measured at laboratory angles of 45\ifmmode^\circ\else\textdegree\fi{} and 90\ifmmode^\circ\else\textdegree\fi{} for pions with kinetic energies from 80 and 210 MeV. The results, which are the first to discrete nuclear final states in the $\ensuremath{\Delta}(1236)$ region, are in qualitative agreement with several distorted-wave impulse-approximation calculations.

C12(p, n)N12reaction at 99 MeV

Cross sections were measured for /sup 12/C( p,n)/sup 12/N reaction at 99.1 MeV for nine laboratory angles between 0/sup 0/ and 48.5/sup 0/. Two detector stations were used: one at 68.0 m was for angles between 0/sup 0/ and 24.5/sup 0/, the other at 76.3 m for angles between 24/sup 0/ and 48.5/sup 0/. Overall energy resolutions of 260 and 230 keV for neutrons of about 80 MeV were achieved at the first and second detector stations, respectively, with large-volume mean-timed counter assemblies. Cross sections for the /sup 12/N ground state (1/sup +/) and 0.96 MeV state (2/sup +/) are compared with a distorted-wave impulse approximation calculation using an effective nucleon-nucleon interaction derived from nucleon-nucleon scattering data. For the ground state, the magnitudes of the calculated cross sections agree with the measured cross sections at forward angles but significantly underestimate the measured values beyond about 30/sup 0/; for the 0.96 MeV state, the calculated cross sections must be reduced by 30% to obtain agreement with the cross sections measured at forward angles.

Excitation of Unnatural-Parity States inC12by 800-MeV Polarized Protons

Cross sections and analyzing powers have been measured for 800-MeV proton inelastic transitions to unnatural-parity states in $^{12}\mathrm{C}$. Data for the 15.11-MeV ${1}^{+}$, $T=1$ state are well explained by a distorted-wave impulse-approximation calculation based on proton-neutron charge-exchange cross sections. Negative analyzing powers were observed for the first time at 800 MeV, for the 12.71-MeV ${1}^{+}$, $T=0$ state. Values of ${A}_{y}$ appear to be characteristic of the isospin transfer, and support isospin assignments for states at 18.3 and 19.4 MeV.

Quasi-free scattering of polarized protons

Cross section and analyzing power, energy-sharing spectra have been measured for the 16O( p , 2p) reaction at an incident energy of 200 MeV for twenty-four different pairs of angles of the detected final-state protons. This general survey is intended to test the validity of the distorted wave impulse approximation (DWIA), particularly with respect to the predicted j-dependent analyzing power caused by the distorting optical potentials and nuclear spin-orbit coupling. Although the data in general confirm this j-dependence, agreement in detail between DWIA calculations and the analyzing power data becomes worse as the momentum of the recoiling nucleus increases. Predictably, the cross-section calculations show more sensitivity to the optical-model parameters than do the analyzing powers. Within the present limits imposed by scanty elastic scattering data, there seems to remain a choice between predicted ( p , 2p) cross sections which either are unreasonably large or which have the incorrect energy-sharing shapes. More comprehensive elastic scattering and total reaction cross section data are clearly needed.

Quasi-free (α,2α) reaction induced by 140 MeV alpha particles onBe9,C12,O16, andNe20

Measurements of the ($\ensuremath{\alpha}$,$2\ensuremath{\alpha}$) reaction on $^{9}\mathrm{Be}$, $^{12}\mathrm{C}$, $^{16}\mathrm{O}$, and $^{20}\mathrm{Ne}$ targets at ${E}_{\ensuremath{\alpha}}=140$ MeV were made at 20 angle pairs. The quasi-free knockout mechanism appears to dominate the reaction. The experimental data were analyzed with distorted-wave impulse approximation calculations. The factorization approximation employed in the calculations was tested explicitly and found to be valid. The shapes of the calculated energy sharing spectra are in satisfactory agreement with the data. The predicted absolute cross sections were found to be very sensitive to the cluster-core bound state radius parameter, and a bound state radius $R=2.52{{A}_{c}}^{\frac{1}{3}}$ fm is necessary to obtain absolute spectroscopic factors consistent with existing theoretical values. Several possible explanations for this excessive value are suggested. Comparisons are made with other alpha knockout and transfer reactions.NUCLEAR REACTIONS $^{9}\mathrm{Be}$, $^{12}\mathrm{C}$, $^{16}\mathrm{O}$, $^{20}\mathrm{Ne}$ ($\ensuremath{\alpha}$,$2\ensuremath{\alpha}$), ${E}_{\ensuremath{\alpha}}=140$ MeV; measured $\ensuremath{\sigma}({E}_{\ensuremath{\alpha}1},{E}_{\ensuremath{\alpha}2},{\ensuremath{\theta}}_{\ensuremath{\alpha}1},{\ensuremath{\theta}}_{\ensuremath{\alpha}2})$ DWIA analysis; deduced absolute spectroscopic factors.

Inelastic pion scattering fromO18

The inelastic scattering of 164 MeV pions from /sup 18/O is studied using the distorted-wave impulse approximation in momentum space. The sensitivity of the predictions to various shell model descriptions of the yrast 2/sup +/ state is examined in detail. The excitation of the triplet of states (4/sup +//sub 1/, 0/sup +//sub 2/, 2/sup +//sub 2/) near 3.7 MeV and the (3/sup -//sub 1/, 0/sup +//sub 3/, 2/sup +//sub 3/) triplet at about 5.2 MeV are also investigated within the context of the shell model. In order to explain the scattering to natural parity states (those with angular momentum J and parity (-1)/sup J/), one must enhance the theoretical results by about the same factor as needed to explain the B (EJ; 0 ..-->.. J) transition rates obtained from electromagnetic excitation. The sensitivity of the results to the isoscalar or isovector character of this enhancement is examined. Agreement between distorted-wave impulse approximation calculations and experiment is found to be satisfactory.

Comparison of the ReactionHe4(p, 2p)H3at Intermediate Energies with the Distorted-Wave Impulse Approximation

The /sup 4/He( p,2p)/sup 3/H reaction has been measured at 250, 350, and 500 MeV using symmetric geometries. The energy dependence of the cross section at zero recoil momentum and angular distributions are presented. The data are compared to distorted-wave impulse-approximation (DWIA) calculations. Unlike at lower energies, the zero-recoil-momentum data are well described by the DWIA at these energies. For the angular distributions, the DWIA increasingly underestimates the data as the recoil momentum is increased.

Total Cross Section for (γ,π−) Reactions in theΔRegion

Total cross sections for $^{12}\mathrm{C}$($\ensuremath{\gamma}$,${\ensuremath{\pi}}^{\ensuremath{-}}$)$^{12}\mathrm{N}$ and $^{7}\mathrm{Li}$($\ensuremath{\gamma}$,${\ensuremath{\pi}}^{\ensuremath{-}}$)$^{7}\mathrm{Be}$ have been measured from threshold to 360 MeV photon energy by detecting the radioactivity of the residual nuclei, thereby singling out the ground state of $^{12}\mathrm{N}$ and the ground and first excited states of $^{7}\mathrm{Be}$. The cross sections are found to peak at about 40 MeV pion energy and then to fall gradually. In contrast to pion charge exchange and other photopion experiments, these results are well reproduced both in shape and in magnitude by distorted-wave impulse-approximation calculations.

Positive-pion photoproduction onC12near threshold

The ..pi../sup +/ photoproduction on /sup 12/C has been measured relative to the proton from 0--20 MeV above threshold. Total cross sections from 0--12 MeV, summed over states in /sup 12/B have been extracted with accuracies between 5% and 8% relative to the proton photoproduction cross section. Distorted wave impulse approximation calculations have been performed and show good agreement with the data. The yield from 0--3 MeV above threshold is well described by a one parameter fit for the ground state cross section using only the sigma-arrow-right x epsilon-arrow-right term of the Hamiltonian.

The 12C(p,2p)11B Reaction at 100 MeV

The 12C(p,2p)l1B reaction has been measured at 100 MeV in both symmetric and asymmetric geometries. Distorted wave impulse approximation calculations were used to obtain spectroscopic factors for all states of l1B below 7 MeV excitation. From these spectroscopic factors and the shapes of the measured angular correlations, it is concluded that (i) no large 1f components are present in the wavefunction of the ground state of 12C and (ii) this experiment gives no evidence for the formation of giant resonances as the intermediate step in multistep reaction processes to the 4�44 MeV (5/2-) and 6�74 MeV (7/2-) states of 11B.

Experimental Test of the Factorization Approximation in the ReactionCa40(p, 2p)K39at 148.2 MeV

We present data for the reaction $^{40}\mathrm{Ca}(p, 2p)^{39}\mathrm{K}$ at 148.2 MeV. Energy sharing cross sections for the ${\frac{3}{2}}^{+}$ ground state and ${\mathrm{\textonehalf{}}}^{+}$ excited state are well described by distorted-wave impulse-approximation calculations. An explicit comparison of the results of the factorization approximation with these experimental results provides considerable support for the usual distorted-wave impulse-approximation treatment.

Nucleon Knockout by Kaons

The reaction (K/sup +/,K/sup +/p) is considered as a probe of the deep hole states in nuclei. Distorted-wave impulse-approximation calculations are presented for the knockout of a 1s/sub 1/2/ proton from /sup 40/Ca. We conclude that the experiment should be valuable.

(p, pd), (p, pt) and (p, pHe3) reactions onC12andO16at 75 MeV

($p, \mathrm{pd}$), ($p, \mathrm{pt}$), and ($p, p^{3}\mathrm{He}$) reactions have been studied at 75 MeV on $^{12}\mathrm{C}$ and $^{16}\mathrm{O}$. A good energy resolution (\ensuremath{\sim}400 keV) permitted the observation of several levels of the different residual nuclei. The results are analyzed with distorted-wave impulse approximation calculations and spectroscopic factors are extracted and compared with theoretical predictions. A comparison of our results with experimental data from transfer reactions on the same nuclei is made. Spectroscopic and mechanism information are discussed; in particular, the excitation of $T=1$ levels in $^{10}\mathrm{B}$ and $^{14}\mathrm{N}$ allowed the comparison with $p\ensuremath{-}{d}^{*}$ ($S=0$, $T=1$) scattering obtained from other reactions.NUCLEAR REACTIONS $^{12}\mathrm{C}$, $^{16}\mathrm{O}(p, px)$, $x=d$, $t$, $^{3}\mathrm{He}$, ${T}_{0}=75$ MeV measured $\ensuremath{\sigma}({T}_{p}, {\ensuremath{\theta}}_{p}, {T}_{x}, {\ensuremath{\theta}}_{x})$; deduced $S$ from DWIA interpretation; comparison with transfer reactions.

Distorted-wave impulse-approximation calculations for quasifree cluster knockout reactions

A formalism for distorted-wave impulse-approximation calculations of quasifree cluster knockout reactions is described. Results are presented for ($p, p\ensuremath{\alpha}$) and ($\ensuremath{\alpha}, 2\ensuremath{\alpha}$) reactions. In both cases distortion effects are large. It is concluded that incident energies between \ensuremath{\sim} 100 and \ensuremath{\sim} 200 MeV are most suitable for ($p, p\ensuremath{\alpha}$) studies of nuclear structure, whereas somewhat higher energies are more appropriate for ($\ensuremath{\alpha}, 2\ensuremath{\alpha}$) studies.NUCLEAR REACTIONS DWIA reaction theory of cluster knockout.

Absolute spectroscopic factors from the (p, pα) reaction at 100 MeV on1p-shell nuclei

The ($p, p\ensuremath{\alpha}$) reaction on $^{6}\mathrm{Li}$, $^{7}\mathrm{Li}$, $^{9}\mathrm{Be}$, and $^{12}\mathrm{C}$ has been investigated experimentally at a bombarding energy of 100 MeV. Data were obtained at a large number of angle pairs in order to study the reaction mechanism. The data are in good agreement with distorted-wave impulse-approximation calculations; unlike most analyses of transfer reactions we obtain absolute $\ensuremath{\alpha}$-particle spectroscopic factors. These are in good agreement with $1p$ shell model predictions. Thus we find no evidence for additional $\ensuremath{\alpha}$-particle-like correlations.NUCLEAR REACTIONS $^{6}\mathrm{Li}$, $^{7}\mathrm{Li}$, $^{9}\mathrm{Be}$, $^{12}\mathrm{C}(p, p\ensuremath{\alpha})$, ${E}_{0}=100$ MeV; measured $\frac{{d}^{3}\ensuremath{\sigma}}{d{\ensuremath{\Omega}}_{p}d{\ensuremath{\Omega}}_{\ensuremath{\alpha}}d{E}_{p}}({\ensuremath{\theta}}_{p}, {\ensuremath{\theta}}_{\ensuremath{\alpha}}, {E}_{p}, {E}_{\ensuremath{\alpha}})$; coplanar geometry; DWIA analysis; deduced $\ensuremath{\alpha}$-particle spectroscopic factors.

Observation ofj-Dependent Effects in Quasifree Scattering of Polarized Protons

Jacob, Maris, Schneider, and Teodoro have suggested that spin-orbit coupling in nuclei, combined with absorption in ($p,2p$) reactions, will select nuclear protons with a given polarization resulting in $j$-dependent ($\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}},2p$) cross sections. Using a 200-MeV polarized proton beam at TRIUMF we have observed an asymmetry which has the expected $j$ dependence and agrees with the predictions of a distorted-wave impulse-approximation calculation.

The quasi-elastic knockout of alpha particles from 16O and 28Si by 0.65 and 0.85 GeV alpha particles

Measurements of the (α, 2α) reaction in the GeV region on 16O and 28Si are described. The results are compared with a calculation employing the plane wave impulse approximation (PWIA) and with a calculation employing the distorted wave impulse approximation (DWIA). The PWIA calculation reproduces the shape of the angular distribution of the observed quasi-elastic events. The main effect of the distortion introduced in the DWIA calculation is to attenuate the quasi-elastic cross section by a factor of five. As a consequence, the (α, 2α) reaction in the GeV region is expected to be more useful to measure the spectroscopic factor N eff than in the region below 100 MeV since the attenuation is considerably larger (≈ 10 3) in the latter case.