Two-step Contributions Research Articles

Coupled channels analysis of positive pion inelastic scattering from 28Si at 50 MeV.

The 2/sub 1//sup +/, 4/sub 1//sup +/, and 0/sub 2//sup +/ states in /sup 28/Si have been observed with ..pi../sup +/ scattering at 50 MeV. These data, along with existing data for /sup 28/Si are compared to coupled channel calculations. This analysis produces a larger ..beta../sub 2/ than previously found in low energy distorted wave calculations and is consistent with results from resonance energy studies. The interference of the one- and two-step contributions make the calculated 4/sup +/ cross section quite sensitive to the sign of ..beta../sub 4/. The sensitivity to the ..pi..A optical model is discussed.

One- and two-step contributions to the 90Zr(p,n)90Nb cross section at 0 degrees.

Within a multiple collision theory based on the eikonal approximation we study the $^{90}\mathrm{Zr}$(p,n${)}^{90}$Nb 0\ifmmode^\circ\else\textdegree\fi{} cross section at 200 MeV incident proton energy. Both the one- and two-step contributions are calculated and compared to the experimental inclusive cross section. Good agreement between theory and experiment is found. Possible implications for the ``missing Gamow-Teller strength'' are discussed.

Semiclassical analysis of two-particle elastic transfer

Two-particle elastic transfer reactions between heavy ions are studied using the semiclassical approach including both one- and two-step contributions to the two-particle transfer process. Good agreement with the data is obtained with form factors and optical potential calculated from nuclear structure. Because the phases of one- and two-step processes turn out to be very similar it seems, however, difficult to extract, from the interference pattern, clear-cut information on the relative importance of the simultaneous and the successive transfer processes.

36-MeV-triton-induced charge exchange: Mass measurements and energy levels of neutron-rich nuclei and the charge exchange reaction mechanism.

Energy spectra and differential cross sections have been obtained for the charge exchange reaction ${(}^{3}$H${,}^{3}$He) on targets of $^{30}$,28Si and $^{26}\mathrm{Mg}$ at an incident energy of 36 MeV. Previously unobserved energy levels of $^{30}\mathrm{Al}$ and $^{26}\mathrm{Na}$ are reported and compared to shell model predictions. Microscopic form factors based on the M3Y effective nucleon-nucleon interaction are used in distorted-wave Born approximation codes to fit the data. The sensitivity of the model to input parameters is discussed and some spin assignments made. Coupled channels calculations are performed to fit the two-step contributions to the data via sequential one-nucleon transfers.

CCBA description of one-nucleon pickup by polarized helions and polarized tritons on 30Si

CCBA and DWBA calculations are presented for the ( 3He, 4He) and ( 3H, 4He) reactions on 30Si. The differential cross sections and analysing powers for 7 levels in 29Si and 3 levels in 29Al show strong evidence for sequential or two-step contributions via the 2 + state in 30Si. These contributions, and the inelastic couplings in the exit channels are included in the CCBA calculations, leading to an improved description of the data. The analysing powers are shown to be sensitive to the strength and character of the sequential amplitudes. The 3 2 + states require large s 1 2 sequential amplitudes, in agreement with previous work. The 7 2 + states are populated by second order processes, and the direct amplitudes are found to be zero. The extracted spectroscopic factors are much smaller for the CCBA analyses than those found in the DWBA analyses. The reactions provide no additional information about the 3He spin-orbit potential.

Double excitation in high energy inelastic scattering.

The contribution of two-step processes in quasielastic nucleon-nucleus collisions is estimated. We use a simple model for the single-step inelastic excitation based on the random-phase-approximation response of semi-infinite nuclear matter. In the 200 MeV (p,n) reaction on /sup 90/Zr, the two-step background is found to be significant, contributing about 15% at 13/sup 0/ scattering angle in the quasielastic peak region. The relative magnitude of the two-step contribution depends strongly on energy, becoming less important in the 800 MeV (p,p') reaction on /sup 116/Sn.

Nearside-farside analyses of heavy-ion direct reactions to the continuum

Decompositions into nearside and farside (NF) components are made of heavy-ion continuum cross sections calculated within the multi-step direct reaction (MSDR) method including one- and two-step processes. The approach is applied to the 16O + 48Ti system at E lab around 100 MeV and to the 16O + 208Pb system at E lab = 192 MeV, and both cross sections and polarizations of the residual nucleus are considered. It is seen that at higher energy losses the two-step contributions become dominant, and that for scattering angles beyond the grazing angle these two-step contributions are mainly farsided. The relative importance of the NF components depends sensitively on the balance between Coulomb repulsion and nuclear attraction. The magnitude of the farside contribution also depends critically on the use of a polarization potential in the excited states. This potential is treated empirically. The decompositions are usefully interpreted in terms of a simple model for multi-step deflection functions. The results of the present quantum-mechanical approach to continuum heavy-ion reactions exhibit many of the features of classical calculations of deepinelastic processes.

Pion-induced monopole transitions

The importance of imposing the orthogonality condition between the initial and final state wave functions for a correct description of the direct contribution to the monopole cross section is demonstrated in the context of a study of $^{28}\mathrm{Si}(\ensuremath{\pi},{\ensuremath{\pi}}^{\ensuremath{'}})$. Two step effects, previously found important for the $^{12}\mathrm{C}$ monopole excitation, are less important in this reaction. Very general estimates of two-step contributions are derived. These indicate that multistep effects will be small in medium energy monopole excitations away from forward angles.

An eikonal treatment including two-step mechanisms for the (p,n) reaction on light nuclei

The reactions 7Li(p, n) 7Be(g.s.), 7Li(p, n) 7Be(0.43 MeV), and 13C(p, n) 13N(g.s.) are treated in an eikonalized distorted-wave impulse approximation (DWIA), including the contribution of a two-step mechanism involving isovector correlations for the last reaction. Our calculation uses the free on-shell nucleon-nucleon t-matrix, with spin and isospin dependence, and incorporates antisymmetrization of it. We find agreement at roughly the 20% level with the forward experimental charge-exchange results in the energy range 120 to 200 MeV. The two-step contribution for the 13C case is rather small, being of the order of 10% at 0°; it interferes constructively, thus improving somewhat the agreement with experiment.

Two-step contributions in thePb206(t→, p)Pb208two-neutron transfer reaction

The /sup 206/Pb(t,p) reaction has been studied over the angular range from 10/sup 0/ to 60/sup 0/ at a triton energy of 17 MeV. Cross sections and analyzing powers were measured for 11 states between 2.6 and 5.0 MeV excitation, including the 4/sup -/ state at 3.475 MeV. The two-neutron transfer reaction to this state was represented by a sequential transfer calculation. Data for three natural parity states, in particular the analyzing powers, were not fitted well by one-step distorted-wave Born approximation calculations. Numerous sequential transfer calculations with different transfer configurations were tried for each of these states. While most of the calculations produced results similar to the distorted-wave Born approximation results, at least one calculation was found for each state which gave an improved representation over that of the distorted-wave Born approximation. The failure of the distorted-wave Born approximation documents the need to apply sequential transfer reaction processes in general descriptions of two-nucleon transfer reactions.

One-step and two-step contributions to two-nucleon transfer reactions

Qualitative arguments are given to show that although first-order distorted-wave Born approximation calculations are adequate for the treatment of single-nucleon transfer reactions, there is little reason to trust them for two-nucleon or multinucleon transfer reactions. A first-plus-second Born approximation calculation is presented for the $^{208}\mathrm{Pb}$($^{16}\mathrm{O}$, $^{18}\mathrm{O}$)$^{206}\mathrm{Pb}$ ground state transition, at bombarding energies below and slightly above the Coulomb barrier. It is found that successive transfer (two-step) processes account for almost all the observed cross section.NUCLEAR REACTIONS Theory of one- and two-step contributions to $^{208}\mathrm{Pb}$($^{16}\mathrm{O}$, $^{18}\mathrm{O}$)$^{206}\mathrm{Pb}$.

Spin-flip probablility in the inelastic scattering of 15—40 MeV protons on light nuclei

Spin-flip probability angular distributions for proton scattering from the ${2}_{1}^{+}$ level of $^{24}\mathrm{Mg}$ have been determined between 25\ifmmode^\circ\else\textdegree\fi{} and 165\ifmmode^\circ\else\textdegree\fi{} (in steps of 10\ifmmode^\circ\else\textdegree\fi{}) at 15 incident energies between 14.5 and 35.5 MeV and, to obtain a more detailed energy dependence at backward angles, spin-flip probabilites have also been measured between 115\ifmmode^\circ\else\textdegree\fi{} and 165\ifmmode^\circ\else\textdegree\fi{} at ten additional energies. In order to extend the study of spin-flip probabilities also to other nuclei and to complement previous angular distributions, further backward angle data have been taken at 13 energies for $^{32}\mathrm{S}$ and near 40 MeV for $^{12}\mathrm{C}$ and $^{28}\mathrm{Si}$. The experimental results have been compared with coupled-channel calculations based on the rotational model and, for $^{24}\mathrm{Mg}$, also with the predictions of microscopic antisymmetrized distorted wave calculations including two-step contributions. The collective analysis proves that spin-flip probabilities are particularly sensitive to the sign of the quadrupole deformation.

An investigation of the importance of multi-step processes in the three-nucleon (p, α) transfer reactions

DWBA calculations were carried out using one-step and two-step processes that included both sequential transfer and inelastic transfer processes in the three-nucleon (p, α) transfer reaction. A comparison of the theoretical calculations with our previously presented data of the 56Fe(p, α) 53Mn and 55Mn(p, α) 52Cr reactions at 35 MeV indicates the importance of the two-step (p, α')(α', α) and (p, t)(t, α) process in some weak direct (p, α) transitions. However, for the strong (p, α) transitions in the reaction 92 Zr( p, α) 89 Y at 35 MeV the two-step contributions are very small.

The 48Ca( 3He, t) 48Sc reaction at 66 and 70 MeV : Reaction mechanism and Gamow-Teller strength

The 48Ca( 3He, t) 48Sc reaction has been studied at E 3 He = 66 MeV and 70 MeV. Angular distributions are given from θ = 8° to 35°. The 0 +–7 + multiplet in 48Sc is strongly excited at these energies and the spectra are further characterized by some broad structures on a continuous background. Broad peaks at E ex = 7.8, 10.6 and 13.3 MeV have an angular distribution similar to the low-lying 1 + state. The broad peaks are interpreted as envelopes of groups of 1 + states carrying a significant part of the Gamow-Teller strength. The strength distribution is consistent with a shell-model calculation. The reaction mechanism has been examined at 66 MeV. The cross section is calculated for the 0 +–7 + multiplet in 48Sc as a coherent sum of one-step and two-step processes. The two-step contribution is calculated in a full finite-range 2nd-order DWBA. The result is similar contributions from ( 3He, α, t) and ( 3He, d, t). The calculations account reasonably well for the data with the exception of the transition to 0 +, IAS.

(He3,t) reaction mechanism via theAr38(He3,t)K38reaction at 40 MeV

The $^{38}\mathrm{Ar}$($^{3}\mathrm{He}$,$t$)$^{38}\mathrm{K}$ reaction has been studied using a 40 MeV $^{3}\mathrm{He}$ beam. Angular distributions for 26 levels are presented and theoretical analysis of the results has been carried out. Firstly, the conclusions of a macroscopic study of the $^{38}\mathrm{K}$ level spectrum are presented. Secondly, the difficulties encountered using a onestep microscopic model to describe the five lowest states of $^{38}\mathrm{K}$ are discussed. Thirdly, the improvements obtained by including two-step contributions in our microscopic model are shown and the importance of the transitions occuring via the [$(^{3}\mathrm{He},\ensuremath{\alpha})+(\ensuremath{\alpha},t)$] channel is emphasized.NUCLEAR REACTIONS $^{38}\mathrm{Ar}$($^{3}\mathrm{He}$,$t$), $E=40$ MeV; measured $\ensuremath{\sigma}(\ensuremath{\theta})$; resolution 75 keV. Microscopic DWBA analysis; coupled-channel analysis; deduced $L$ transfers; deduced strengths of microscopic ($^{3}\mathrm{He}$,$t$) effective force.

A study of the 58Ni, 90Zr and 208Pb(p, d) reactions at 121 MeV

Abstract The (p, d) reaction has been studied on 58Ni, 90Zr and 208Pb at 121 MeV in order to test the applicability of the usual DWBA methods to higher energy data. The calculations describe the angular distribution for the strongly excited low-lying states reasonably well when adiabatic-deuteron optical potentials are used. Some discrepancies in shape persist, however, and some values of the spectroscopic factors differ from lower energy data in spite of many variations in the calculations. By use of exact finite-range calculations a value of D02 = 1.23 × 104 MeV2 · fm3 was found for use at 121 MeV. Deuteron D-state contributions were negligible at forward angles and two-step contributions do not appear more significant than for data at lower energy.

Forbidden transitions in theCa40(N15,O16)K39reaction and theL=1problem

Cross sections of the $^{40}\mathrm{Ca}$($^{15}\mathrm{N}$, $^{16}\mathrm{O}$)$^{39}\mathrm{K}$ reaction were measured and analyzed. The transition of the $\frac{9}{{2}^{\ensuremath{-}}}$ state, which is of weak-coupling nature, is shown to provide an excellent test of coupled-channel Born-approximation theory and the coupled-channel Born-approximation theory analysis for the $L=1$ problem in the $^{40}\mathrm{Ca}$($^{13}\mathrm{C}$, $^{14}\mathrm{N}$) reaction. Calculated two-step contributions to the $L=1$ transitions in the ($^{15}\mathrm{N}$, $^{16}\mathrm{O}$) reaction are large enough to affect the phases of their angular distributions.NUCLEAR REACTIONS $^{40}\mathrm{Ca}$($^{15}\mathrm{N}$, $^{16}\mathrm{O}$), $E=60$ MeV: measured $\frac{d\ensuremath{\sigma}}{d\ensuremath{\Omega}}(\ensuremath{\theta})$. CCBA and DWBA analyses.

Strong Two-Step Contributions in Direct Allowed Two-Nucleon Transfers

The h/sub 9/2/g/sub 9/2/ multiplet in /sup 210/Bi was populated by /sup 208/Pb(..cap alpha.., d) in experiments at 33 and 48 MeV. The strong favoring of the five unnatural-parity states of the multiplet disagrees with one-step transfer distorted-wave Born-approximation predictions. Coupled-reaction-channels calculations explain this effect as interference of direct np transfer and successive stripping amplitudes of comparable magnitude. This holds for weak (low-spin) and strong (high-spin) members of the multiplet.

A study of the ( 3He, t) reaction on 60Ni

The 60Ni( 3He, t) 60Cu reaction was studied at 38 MeV. Angular distributions were obtained. from 2.5° to 65° for transitions to 40 levels in 60Cu. The data were compared to calculations done using a coupled-reaction-channels formalism. It was found that when two-step contributions arising from inelastic excitation of low-lying 2 + and 4 + levels in 60Ni followed by charge exchange were added to the direct charge-exchange contributions, and exchange contributions were considered as well, the J-dependence of the interaction strength V τ was removed. Attempts to describe the shapes of several transitions to 1 + levels in 60Cu with the inclusion of a tensor force in the interaction were unsuccessful as the angular distributions preferred an l = 1 shape rather than a mixture of l = 0 and l = 2 transition amplitudes as predicted by calculations. In order to provide spectroscopic factors for the pickup-stripping calculations data were also obtained for the 60Ni( 3He, α) 59Ni reaction at 38 MeV. Spectroscopic factors were obtained from the eighteen measured angular distributions and found to be in fair agreement with theoretical predictions using the MSDI wave functions of Koops and Glaudemans.

Isovector correlations and pion single charge exchange reactions in light nuclei

The reactions 7Li(π +, π 0) 7Be(g.s.), 7Li(π +, π 0) 7Be∗(429keV) and 13C(π +, π 0) 13N(g.s.) are treated in DWIA, including corrections arising from two-step processes in which an intermediate state is reached in the course of the charge exchange. The distortions are introduced by means of a first-order optical potential; no special assumptions of extreme high energy limits are made. The closure sum over intermediate states in the two-step charge exchange mechanism leads to an isovector two-particle correlation function whose properties for p-shell nuclei are discussed in detail. Consequences of the introduction of effective transition operators for the single-step mechanism, with and without spin-flip, and for the two-step contribution are discussed, as are the allowable modifications in nuclear shell radii consistent with present knowledge of Coulomb energies. Comparison is made with experiment — in the case of the separated 7Be states, for the first time — and with other theoretical approaches.