Coupled-channel Born Approximation Calculations Research Articles

Multistep processes in heavy-ion-induced single-nucleon transfer reactions

It was first noted during the 1970s that finite-range distorted wave Born approximation (FR-DWBA) calculations were unable satisfactorily to describe the shape of the angular distributions of many single-proton (and some single-neutron) transfer reactions induced by heavy ions, with calculations shifted to larger angles by up to ~ 4 degrees compared with the data. These reactions exhibited a significant mismatch, either of the reaction Q value or the grazing angular momentum of the entrance and exit channels, and it was speculated that the inclusion of multi-step transfer paths via excited state(s) of the projectile and/or ejectile could compensate for the effect of this mismatch and yield good descriptions of the data by shifting the calculated peaks to smaller angles. However, to date this has not been explicitly demonstrated for many reactions. In this work we show that inclusion of the two-step transfer path via the 4.44-MeV 2+ excited state of the 12C projectile in coupled channel Born approximation calculations enables a good description of the 208Pb(12C,11B)209Bi single-proton stripping data at four incident energies which could not be described by the FR-DWBA. We also show that inclusion of a similar reaction path for the 208Pb(12C,13C)207Pb single-neutron pickup reaction has a relatively minor influence, slightly improving the already good description obtained with the FR-DWBA.

Analyzing powers and the role of multistep processes in the C12(Li7,t)O16 reaction

The analyzing powers $^{T}T_{10}\phantom{\rule{0.16em}{0ex}},^{T}T_{20}$, and $^{T}T_{30}$ for the $^{12}\mathrm{C}(^{7}\mathrm{Li},t$) reaction were measured and combined with new and previously determined angular distributions to probe the role of multistep processes through coupled channel Born approximation (CCBA) calculations employing new $\ensuremath{\alpha}$-particle spectroscopic amplitudes calculated with the phenomenological shell model in the unrestricted $psd$ space. Good descriptions of the cross-section angular distributions could be obtained except for the larger angles for the 6.05 MeV ${0}^{+}$ and 6.13 MeV ${3}^{\ensuremath{-}}$ states, while the analyzing powers were not generally described. The calculations presented demonstrate the sensitivity of the angular distributions of both the differential cross sections and the analyzing powers to various multistep routes. While the measured analyzing power $^{T}T_{30}$ is roughly zero for all states, the various CCBA calculations produced large and highly oscillatory values for this observable, showing that other multistep processes must be present beyond those taken into account in this work. Standard distorted wave Born approximation (DWBA) calculations were also carried out and these show considerable differences from the CCBA results but the description of the analyzing powers, while slightly improved, is still poor.

Cluster structure of O17

The transfer reaction $^{13}\mathrm{C}(^{11}\mathrm{B},^{7}\mathrm{Li})^{17}\mathrm{O}$ leading to the ground and several excited states of $^{17}\mathrm{O}$ was investigated at an incident boron beam energy of 45 MeV. The experimental data were analyzed by means of coupled-channel Born approximation calculations and $^{17}\mathrm{O}=\ensuremath{\alpha}+^{13}\mathrm{C}$ spectroscopic factors were extracted from a comparison of the data and the calculations at forward angles. The largest spectroscopic factor obtained was that for the subthreshold 6.356 MeV $1/{2}^{+}$ state, important for the production of neutrons in the stellar environment, and the squared Coulomb modified asymptotic normalization coefficient, ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}^{2}$, is consistent with previous determinations. A significant rise in the experimental data at backward scattering angles suggested the possibility of $^{17}\mathrm{O}=^{6}\mathrm{Li}+^{11}\mathrm{B}$ clustering in some $^{17}\mathrm{O}$ states, including the ground state. However, explicit inclusion of $^{6}\mathrm{Li}$ transfer in the calculations could not explain the observed effect. Compound nucleus calculations suggested that while the backward angle rise could be explained by such processes for two of the populated $^{17}\mathrm{O}$ states other multistep direct processes must contribute significantly to the other two.

Probing surface distributions ofαclusters inNe20viaα-transferreaction

Direct evidence of the $\alpha$-cluster manifestation in bound states has not been obtained yet, although a number of experimental studies were carried out to extract the information of the clustering. In particular in conventional analyses of $\alpha$-transfer reactions, there exist a few significant problems on reaction models, which are insufficient to qualitatively discuss the cluster structure. We aim to verify the development of the $\alpha$-cluster structure from observables. As the first application, we plan to extract the spatial information of the cluster structure of the $^{20}$Ne nucleus in its ground state through the cross section of the $\alpha$-transfer reaction $^{16}$O($^6$Li,~$d$)$^{20}$Ne. For the analysis of the transfer reaction, we work with the coupled-channel Born approximation (CCBA) approach, in which the breakup effect of $^6$Li is explicitly taken into account by means of the continuum-discretized coupled-channel method based on the three-body $\alpha + d + {}^{16}$O model. The two methods are adopted to calculate the overlap function between $^{20}$Ne and $\alpha + {}^{16}$O; one is the microscopic cluster model (MCM) with the generator coordinate method, and the other is the phenomenological two-body potential model (PM). We show that the CCBA calculation with the MCM wave function gives a significant improvement of the theoretical result on the angular distribution of the transfer cross section, which is consistent with the experimental data. Employing the PM, it is discussed which region of the cluster wave function is probed on the transfer cross section. It is found that the surface region of the cluster wave function is sensitive to the cross section. The present work is situated as the first step in obtaining important information to systematically investigate the cluster structure.

T20Tanalyzing powers fromC12(Li7,α)N15

The transverse analyzing powers $^{T}T_{20}$ were obtained from a polarized beam study of the reaction $^{12}\mathrm{C}(^{7}\mathrm{Li},\ensuremath{\alpha})^{15}\mathrm{N}$ at ${E}_{\text{lab}}({}^{7}\text{Li})=34$ MeV for several well known, strongly populated and isolated states in $^{15}\mathrm{N}$ to determine if they contained a sufficiently distinctive signature to be used to provide a test against which selectively populated and strong, but not well understood states, such as the one at $13.17$ MeV could be used to determine their spins and parities. Unfortunately, current finite range distorted wave Born approximation and coupled channels Born approximation calculations were not able to describe the data to the well-known states. The current work presents these experimental analyzing powers with the expectation that they can be used as a test of future detailed reaction model calculations.

Evidence for transfer followed by breakup in 7Li + 65Cu

The observation of a large cross-section for the α+d channel compared to breakup into the α+t channel from an exclusive measurement for the 7Li+65Cu system at 25 MeV is presented. A detailed analysis of the angular distribution using coupled channels Born approximation calculations has provided clear evidence that the observed α+d events arise from a two step process, i.e., direct transfer to the 2.186 MeV (3+) resonance in the α+d continuum of 6Li followed by breakup, and are not due to final state interaction effects.

Asymptotic normalization coefficients for14N+p15Oand the astrophysicalSfactor for14N(p,γ)15O

The ${}^{14}\mathrm{N}(p,\ensuremath{\gamma}{)}^{15}\mathrm{O}$ reaction, which controls energy production in the CNO cycle, has contributions from both resonance and direct captures to the ground and excited states. The overall normalization of the direct captures is defined by the corresponding asymptotic normalization coefficients (ANCs). Especially important is the ANC for the subthreshold state in ${}^{15}\mathrm{O}$ at $\ensuremath{-}0.504\mathrm{keV}$ since direct capture through this state dominates the reaction rate at stellar energies. In order to determine the ANCs for ${}^{14}\mathrm{N}+{p}^{15}\mathrm{O},$ the ${}^{14}\mathrm{N}{(}^{3}\mathrm{He}{,d)}^{15}\mathrm{O}$ proton transfer reaction has been measured at an incident energy of 26.3 MeV. Angular distributions for proton transfer to the ground and five excited states were obtained. ANCs were then extracted from comparison to both distorted-wave Born approximation and coupled-channels Born approximation calculations. Using these ANCs, we calculated the astrophysical factor and reaction rates for ${}^{14}\mathrm{N}(p,\ensuremath{\gamma}{)}^{15}\mathrm{O}.$ Our analysis favors a low value for the astrophysical factor.

12C+12C→8Beg.s.+16Og.s.resonance reaction aroundEc.m.=32.5MeV

A resonance observed in the excitation function of the ${}^{12}\mathrm{C}{+}^{12}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{C}}}^{8}{\mathrm{Be}}_{\mathrm{g}.\mathrm{s}.}{+}^{16}{\mathrm{O}}_{\mathrm{g}.\mathrm{s}.}\ensuremath{\alpha}$-transfer reaction around ${E}_{\mathrm{c}.\mathrm{m}.}=32.5$ MeV is studied by the method of coupled-channel Born approximation (CCBA), in which the distorted waves are given by the microscopic coupled-channel (CC) calculations and the $\ensuremath{\alpha}$-transfer reaction processes are treated as the one-step transitions. The CCBA calculation reproduces well the characteristic behavior of the excitation function as well as the angular distribution at the on-resonance energy. The result of the analysis suggests that the $(\ensuremath{\alpha}+\ensuremath{\alpha})+(\ensuremath{\alpha}{+}^{12}\mathrm{C})$ type four-cluster states are populated in this reaction.

Effect of theα-nucleus potential on the28Si(α,p)31Preaction

The differential cross section of the ${}^{28}\mathrm{Si}(\ensuremath{\alpha}{,p)}^{31}\mathrm{P}$ reaction for 26 MeV incident energy has been analyzed in the distorted wave Born approximation (DWBA) with zero and full-finite range using a deep and shallow optical, Michel and molecular potentials in the incident channel, and a usual optical model potential for proton in the final channel. The parameters of potential in the entrance channel are determined from the elastic scattering data. The calculations done with the deep optical and Michel potentials reproduce the structure of the angular distributions reasonably well, but fail to account for the absolute magnitudes by a few orders. The shallow optical one is satisfactory up to about ${\ensuremath{\theta}}_{\mathrm{c}.\mathrm{m}.}=100\ifmmode^\circ\else\textdegree\fi{}.$ The molecular potential, on the other hand, reproduces both the absolute cross sections and the pattern of the angular distributions. Coupled-channels Born approximation calculations improve fits to the data over the DWBA predictions.

Heavy-Ion Resonance and Multi-Cluster Structure of Nuclei

New kind of heavy-ion resonances recently discovered in nuclear collisions of C+C and O+O systems leading to various multi-cluster exit channels, such as C+C →3α+12C, 3α+3α, 2α+O and O+O → (α+C)+O, (α+C)+(α+C), are studied theoretically by the coupled-channel (CC) and coupled-channel Born approximation (CCBA) calculations using microscopic cluster-model wave functions of the colliding nuclei, C and O, and the nucleus-nucleus interactions based on the realistic G-matrix interaction. These resonance states correspond to highly excited states of the compound systems, Mg and S nuclei, at excitation energies more than 40∼55 MeV, having the associated multi-cluster structures and large decay widths to these reaction channels. In this talk, we show that characteristic features of these newly-found resonances as well as those of the classical resonances of di-nucleus molecular nature are well reproduced by the microscopic CC and CCBA calculations. The reaction mechanism leading to the formation of these exotic resonances is also discussed in some detail for the O+O → (α+C)+O, (α+C)+(α+C) and C+C → Be+O reactions.

Three-particle cluster structure aboveEx=11 MeVin15N

${}^{12}\mathrm{C}{(}^{7}\mathrm{Li},\ensuremath{\alpha})$ angular distributions have been obtained for 16 states in ${}^{15}\mathrm{N}$ at ${E(}^{7}\mathrm{Li})=52.5 \mathrm{MeV}$. Also, ${}^{12}\mathrm{C}{(}^{6}\mathrm{Li}\ensuremath{\rightarrow}{,}^{3}\mathrm{He})$ angular distributions and vector analyzing powers have been measured at ${E(}^{6}\mathrm{Li}\ensuremath{\rightarrow})=50 \mathrm{MeV}$. Finite-range distorted-wave Born-approximation (FRDWBA) triton cluster transfer calculations are able to describe the ${(}^{7}\mathrm{Li},\ensuremath{\alpha})$ data for transfer to states with known ${J}^{\ensuremath{\pi}}$ values. Comparison with known levels in ${}^{19}\mathrm{F}$ suggests that the ${}^{12}\mathrm{C}{(}^{7}\mathrm{Li},\ensuremath{\alpha})$ reaction selectively populates negative parity states. FRDWBA calculations support this idea. Several previously suggested ${J}^{\ensuremath{\pi}}$ levels are confirmed and new ${J}^{\ensuremath{\pi}}$ values are proposed for six other levels. The previously observed ${(}^{6}\mathrm{Li}\ensuremath{\rightarrow}{,}^{3}\mathrm{He}) J$ dependence has been used along with coupled-channel Born-approximation calculations to confirm the ${J}^{\ensuremath{\pi}}$ values for states in ${}^{15}\mathrm{N}$.

Analyzing powers for mirror states in12C(6Li,7Li/7Be)atE6Li=50MeV: Evaluation of channel coupling

A complete set ${(iT}_{11}{,T}_{20}{,T}_{21}{,T}_{22})$ of analyzing powers (AP's) has been obtained for the single nucleon transfer reaction ${}^{12}\mathrm{C}{(}^{6}\mathrm{Li}{,}^{7}\mathrm{Li}{)}^{11}$C. In addition, ${\mathrm{iT}}_{11}$ and ${T}_{21}$ AP data have been obtained for ${}^{12}\mathrm{C}{(}^{6}\mathrm{Li}{,}^{7}\mathrm{Be}{)}^{11}$B. A finite range distorted wave Born approximation calculation provides a qualitatively good description of the ground-state transfer data ${\mathrm{iT}}_{11},$ but is out of phase at small angles. It describes the magnitude but not the detail of the rank-2 AP over the measured angular range. A coupled-channels Born approximation (CCBA) analysis describes the elastic, inelastic, and ground-state transfer reaction cross section and AP data well. The excited-state transfer cross section description is also good, but the AP data are less well described. CCBA calculations indicate that coupling to states in ${}^{6}$Li and ${}^{12}$C is not crucial to the description, but that coupling in the final state between the ground and first excited states of ${}^{7}$Li or ${}^{7}$Be is needed to describe the small-angle oscillations in the ground-state transfer ${\mathrm{iT}}_{11}$ data.

26Mg(6Li

Angular distributions of differential cross section and vector analyzing power have been measured for the $^{26}\mathrm{Mg}$${(}^{6}$Li\ensuremath{\rightarrow}${,}^{7}$Li${)}^{25}$Mg reaction at 60 MeV bombarding energy. Finite-range distorted-wave Born approximation calculations were found to reproduce much of the structure of the data, although the predictions were approximately 3\ifmmode^\circ\else\textdegree\fi{} out of phase with the data. Finite-range coupled-channels Born approximation calculations including inelastic excitations in $^{6}\mathrm{Li}$ and $^{7}\mathrm{Li}$, using coupling schemes derived from elastic scattering analyses, modified the predictions only slightly and did not remove the phase discrepancy.

Effects of 6Li excitation in 64Zn(d

Cross sections and the analyzing powers ${\mathit{A}}_{\mathit{y}}$, ${\mathit{A}}_{\mathit{y}\mathit{y}}$, and ${\mathit{A}}_{\mathit{x}\mathit{x}}$ for the $^{64}\mathrm{Zn}$(d${,}^{6}$Li${)}^{60}$Ni reaction forming the ground and first-excited states of $^{60}\mathrm{Ni}$ are measured in 5\ifmmode^\circ\else\textdegree\fi{} steps from ${\mathrm{\ensuremath{\theta}}}_{\mathrm{lab}}$=25\ifmmode^\circ\else\textdegree\fi{} to 80\ifmmode^\circ\else\textdegree\fi{} using a 16.4-MeV vector- and tensor-polarized deuteron beam. The $^{6}\mathrm{Li}$ ground state and seven other states, each representing $^{6}\mathrm{Li}$ continuum states of a given spin and internal orbital angular momentum, are included in finite-range, coupled-channel Born approximation (CCBA) calculations for $^{64}\mathrm{Zn}$(d${,}^{6}$Li${)}^{60}$Ni. The CCBA calculations demonstrate that \ensuremath{\alpha} transfers forming these continuum states, particularly those forming the first ${3}^{+}$ state in $^{6}\mathrm{Li}$, affect the $^{64}\mathrm{Zn}$(d${,}^{6}$Li${)}^{60}$Ni cross sections and analyzing powers strongly.

70Zn(p,t)68Zn reaction at Ep=35 MeV.

We have investigated the $^{70}\mathrm{Zn}$(p,t${)}^{68}$Zn reaction using a beam energy of 35 MeV. Angular distributions were extracted for 18 levels up to 4.25 MeV excitation in $^{68}\mathrm{Zn}$. Distorted-wave Born-approximation calculations have been performed and compared to each angular distribution. Because the distorted-wave Born-approximation angular distributions to the lowest ${2}^{+}$, ${3}^{\mathrm{\ensuremath{-}}}$, and ${4}^{+}$ states in $^{68}\mathrm{Zn}$ are found to be a poor representation of the data, we have also performed coupled-channels Born-approximation calculations for the $^{70}\mathrm{Zn}$(p,t) reaction leading to these states.

A macroscopic description of two nucleon transfer reactions with deformed nuclei

We present a formalism, based on macroscopic approximations, for treating heavy-ion pair transfer reactions with well-deformed targets. Using the relative motion wave functions provided by the angle dependent phase shift method and macroscopic form factors, an approximate expressions for the coupled channel Born approximation (CCBA) transition amplitude can be obtained. The treatment also provides a simple method for studying the influence of different reaction routes and coupling potentials. Numerical calculations for the 154Sm( 12C, 14C) 152Sm ( E 1ab = 65 MeV) and 186W( 12C, 14C) 184W ( E 1ab = 70 MeV) reactions leading to the 0 + and 2 + states are reported and a comparison with experimental data and conventional CCBA calculations is made.

The12C(t,α)11B reaction at 33 MeV

Differential cross sections for the 12C(t, alpha )11B reaction at 33 MeV have been measured and analysed for the transition to the 0.0, 2.125, 4.445, 5.021, 6.743, 7.286, 7.978 and 8.559 MeV states in 11B. The results of CRC (coupled reaction channels) calculations are compared with DWBA (distorted wave Born approximation) and CCBA (coupled channels Born approximation) calculations for the single-step and two-step transitions respectively. The CRC calculations resulted in improved agreement with the experimental data for the strong transitions.

Elastic and inelastic scattering of 100 MeV protons from the even-even titanium isotopes

The differential cross sections for elastic and inelastic scattering to the first excited (${2}^{+}$) states of 100 MeV protons from $^{46,48,50}\mathrm{Ti}$ were measured. The angular ranges covered were 9\ifmmode^\circ\else\textdegree\fi{} to 110\ifmmode^\circ\else\textdegree\fi{} for $^{46,50}\mathrm{Ti}$ and 9\ifmmode^\circ\else\textdegree\fi{} to 168\ifmmode^\circ\else\textdegree\fi{} for $^{48}\mathrm{Ti}$. The elastic scattering data were analyzed with a local optical model potential. The quality of fits to the data covering the smaller angular range ( 110\ifmmode^\circ\else\textdegree\fi{}) is good for all three isotopes, with $\frac{{\ensuremath{\chi}}^{2}}{N}\ensuremath{\sim}1.5$; whereas the fit to the large angle data of $^{48}\mathrm{Ti} (\ensuremath{\lesssim}{168}^{\ensuremath{\circ}})$ is not as satisfactory ($\frac{{\ensuremath{\chi}}^{2}}{N}\ensuremath{\sim}3.75$). Various modifications to the Woods-Saxon shape of the real central potential showed no real improvement. Nevertheless, the large angle data appear to favor a slightly sharper falloff in the real central potential near the surface, such as the Woods-Saxon-squared potential. Systematics of the phenomenological optical model potential of 100 MeV protons were studied. The derived volume integrals and rms radii seem to follow the usual $A$ dependence. The isospin dependence of the real potential was also extracted. The mean free paths computed directly from the imaginary potential strength have large uncertainties due to the insensitivity of the data to the central region of the imaginary potential. Distorted-wave Born approximation and coupled-channels Born approximation calculations for the collective excitation of the target nuclei to the first ${2}^{+}$ states provided deformation parameters, ${B}_{2}$, in agreement with those from other proton and alpha scattering data and electromagnetic measurements.

Mg24(O18,O16)Mg26reaction atEO18=50MeV including scattering in entrance and exit channel

Cross section angular distributions of $^{24}\mathrm{Mg}$($^{18}\mathrm{O}$, $^{16}\mathrm{O}$)$^{26}\mathrm{Mg}$ reaction at 50 MeV incident energy leading to the first four excited states in $^{26}\mathrm{Mg}$ and the elastic and inelastic scattering for $^{18}\mathrm{O}$ + $^{24}\mathrm{Mg}$ at 50 MeV and $^{16}\mathrm{O}$ + $^{26}\mathrm{Mg}$ at 56 MeV have been measured in order to investigate the reaction mechanism of the two-nucleon transfer reaction including the analysis of incident and exit channels. The measured angular distributions were analyzed in the framework of the exact finite-range distorted-wave Born approximation. Two neutron transfer amplitudes were determined using the $2s\ensuremath{-}1d$ shell model amplitudes calculated by Chung and Wildenthal. The large experimental cross section in contrast to the smallness of the direct transfer amplitude and the peculiar shape of the measured angular distribution of the lowest ${2}_{1}^{+}$ state suggests that this level is mainly excited by two-step processes. It is shown that a coupled-channel Born approximation calculation describes successfully the angular distributions of the ground state and the ${2}_{1}^{+}$ state in $^{26}\mathrm{Mg}$.

A coupled-channels born-approximation analysis of rare-earth (d, t) transitions

A CCBA (coupled-channels Born approximation) analysis has been performed on a very large set of rare-earth (d, t) transitions. Calculations were performed for states in 159Gd, 161, 163Dy and 165, 167Er which have been identified as members of rotational bands built on the Nilsson single- hole states 11 2 −[505], 1 2 −[521], 3 2 −[521], 5 2 −[523], 5 2 +[642], 5 2 +[633] and 7 2 +[404]. In the cases where the DWBA works well, the CCBA gives results similar to those of DWBA. For the anomalous transitions the CCBA calculations tend to fit better than do the DWBA calculations — but the overall agreement is still rather poor.