Continuum-discretized Coupled-channels Research Articles

Breakup dynamic polarization potential forHe6+Pb208at27MeV

We study the local dynamic polarization potential, DPP, component of the $^{6}\mathrm{He}+^{208}\mathrm{Pb}$ interaction generated by the coupling to breakup channels as calculated using the continuum-discretized coupled-channels (CDCC) method. The results are for $^{6}\mathrm{He}$ at a laboratory energy of $27\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$. The elastic channel $S$-matrix is inverted and the bare potential of the CDCC calculation is subtracted to yield the DPP. This has a very long attractive and absorptive tail, with the real component extending beyond $40\phantom{\rule{0.3em}{0ex}}\mathrm{fm}$, generated by Coulomb breakup. Although the long tail of the DPP has a major effect on the elastic $S$-matrix, it is shallow enough to have a more modest effect on the root mean square (rms) radius (increased) and the volume integrals. To facilitate comparison with other theories, the potential tails are fitted to analytic forms.

Breakup and fusion ofLi6andHe6withPb208

The effect of $^{6}\mathrm{Li}$ and $^{6}\mathrm{He}$ breakup on the fusion cross section of these nuclei with $^{208}\mathrm{Pb}$ is investigated by means of continuum-discretized coupled-channels (CDCC) calculations. For $^{6}\mathrm{Li}$ the calculations describe reasonably well the experimental data for elastic scattering, $^{6}\mathrm{Li}\ensuremath{\rightarrow}\ensuremath{\alpha}+d$ breakup and the absorption cross section given by the sum of the $^{6}\mathrm{Li}$ fusion and the $\ensuremath{\alpha}$ production cross section not attributed to breakup. The effect of $^{6}\mathrm{Li}$ breakup on the calculated absorption cross section is found to depend strongly on the imaginary part of the diagonal bare potential. A combination of the CDCC technique and the barrier penetration model generates results close to the measured fusion cross section. For $^{6}\mathrm{He}$ the calculated absorption cross section is much larger than the measured $^{6}\mathrm{He}+^{209}\mathrm{Bi}$ complete fusion cross section values. However, it is found to be relatively independent of the form of the imaginary part of the bare potential. The complete fusion cross section is again found to be reasonably well described by the CDCC∕BPM combination.

Subbarrier fusion in the systems 11,10Be+ 209Bi

The subbarrier fusion cross sections with radioactive ion beams in the systems 10,11Be+ 209Bi have been measured and compared also with the stable system 9Be+ 209Bi. All three cross sections are similar within experimental accuracy; this is somehow unexpected since, in particular, the fusion process with 11Be should be influenced by its halo structure and small binding energy and could behave in a different way from the other two Be isotopes, in particular from 10Be, well bound. Extensive calculations were undertaken within the formalism of continuum discretized coupled channel ( CDCC) that considered only the coupling to two-body breakup, into n+core, of the projectile with excitation to continuum up to 10 MeV. The results are in agreement with the 11,10Be experimental data except at low energy where the cross sections could be influenced by target/projectile collective excitations. The CCFULL theoretical approach, compared to CDCC, seems to reproduce better the subbarrier region and worse the above barrier one. The 9Be cross sections are all underestimated by the CDCC approach while are well reproduced by CCFULL calculations. This suggests that, in this last case, the schematization of the breakup process into 2 only fragments maybe is not adequate and/or other excitations, properly handed by the CCFULL code, could be relevant.

How does breakup influence the total fusion ofLi6,7at the Coulomb barrier?

Total (complete + incomplete) fusion excitation functions of $^{6,7}$Li on $^{59}$Co and $^{209}$Bi targets around the Coulomb barrier are obtained using a new continuum discretized coupled channel (CDCC) method of calculating fusion. The relative importance of breakup and bound-state structure effects on total fusion is particularly investigated. The effect of breakup on fusion can be observed in the total fusion excitation function. The breakup enhances the total fusion at energies just around the barrier, whereas it hardly affects the total fusion at energies well above the barrier. The difference between the experimental total fusion cross sections for $^{6,7}$Li on $^{59}$Co is notably caused by breakup, but this is not the case for the $^{209}$Bi target.

Dipole polarizability of6Heand its effect on elastic scattering

The elastic scattering of ${}^{6}\mathrm{Li}$ and ${}^{6}\mathrm{He}$ by ${}^{208}\mathrm{Pb}$ at energies in the vicinity of the Coulomb barrier is studied by means of continuum-discretized coupled-channels calculations. It is shown that the strong reduction of the ${}^{6}\mathrm{He}{+}^{208}\mathrm{Pb}$ elastic scattering cross section at forward angles is caused by long-range dipole Coulomb excitation of the projectile. The role of breakup in the fusion of halo nuclei is also discussed.

Dynamic polarization potential for6He+pdue to breakup

We study the local dynamic polarization potential (DPP) component of the ${}^{6}\mathrm{He}+p$ interaction, determining that part which is due to breakup processes. Results are presented for c.m. energies of 21.57 and 35.06 MeV. The DPP is found by subjecting the elastic scattering S matrix from continuum-discretized coupled-channels calculations to S matri$\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}$potential inversion. The iterative-perturbative inversion method is employed, and the method is shown to give reliable potentials down to projectile-target separation close to zero, a necessary requirement for this case. The contributions of different continuum states are compared, and the way they interact to give the overall DPP is studied. The DPP is found to be very different from that determined using the weighted trivially equivalent approximate method.

Continuum-discretized coupled-channels study of the11Be→10Be+nbreakup effect on11Beelastic scattering

The ${}^{11}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{Be}}}^{10}\mathrm{Be}+n$ breakup effect on the elastic scattering of ${}^{11}\mathrm{Be}$ by ${}^{12}\mathrm{C}$ at $E/A=49.3$ MeV is investigated by the method of continuum-discretized coupled-channels (CDCC) based on the ${}^{10}\mathrm{B}\mathrm{e}+n+\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}$ three-body model. The CDCC calculation well reproduces the experimental data of the elastic scattering, which is consistent with the result of recent studies based on the adiabatic three-body models. The breakup effect is significant and the coupling to the p-wave continuum states and the d-wave resonance state have a dominant contribution to the breakup effect. Dynamical polarization potential which simulates the breakup effect is evaluated and found to have a weakly repulsive real part and a long-range imaginary part of absorptive nature, characteristic of the weakly bound projectile with neutron halo.

Scattering of polarized7Lifrom4He

Complete sets of analyzing powers for elastic and inelastic (to the ${1/2}^{\ensuremath{-}}$ first excited and ${7/2}^{\ensuremath{-}}$ resonant states of ${}^{7}\mathrm{Li})$ scattering of polarized ${}^{7}\mathrm{Li}$ by ${}^{4}\mathrm{He}$ have been obtained. Relations between the different analyzing powers were determined by applying the invariant amplitude method. A comparison of these relations with the measured analyzing powers suggests that the odd rank tensor analyzing powers arise from higher order quadrupole interactions. The quadrupole interaction also gives rise to the inelastic analyzing powers. Continuum-discretized coupled-channels calculations using $\ensuremath{\alpha}+t$ cluster-folded potentials confirm this crucial role of the quadrupole interaction in producing the observed ${}^{7}\mathrm{Li}{+}^{4}\mathrm{He}$ analyzing powers. Cluster-folded spin-orbit and third rank tensor potentials were found to have negligible effects, in contrast to polarized deuteron elastic scattering where the spin-orbit potential plays the dominant role, even in producing the second rank analyzing powers. Coupling between the excited states of ${}^{7}\mathrm{Li},$ in particular that between the 0.478 MeV ${1/2}^{\ensuremath{-}}$ and the 6.68 MeV ${5/2}^{\ensuremath{-}}$ states, was also found to be important.

Gaussian expansion method for few-body systems

We review our method of calculation, Gaussian Expansion Method (GEM), for bound and scattering states of few-body systems. The method was proposed in 1988 and has been applied to a variety of few-body systems. The understanding on the structure and the mechanisms of reactions of those systems obtained from such applications is discussed together with some useful techniques for the calculations. A well-chosen set of Gaussian basis functions forms an approximate complete set in a finite coordinate space so that it can describe accurately short-range corielations and long-range asymptotic behavior as well as highly oscillatory character of wave functions in the bound and the scattering states of the systems. Examples of applications of GEM include i) the latest determination of antiproton mass by the analysis of laser spectroscopic data for antiprotonic helium atoms, ii) predictions and experimental verifications on the structure of hypernuclei and hyperon-nucleon interactions, iii) Coulomb three-body calculations of bound and resonant states of muonic molecules as well as muon transfer reactions in muon catalyzed fusion cycles, iv) a new treatment of CDCC (continuum-discretized coupled channels) method for three- and four-body breakup processes, and v) benchmark test calculations for three- and four-nucleon bound states using realistic interactions.

8Bbreakup in elastic and transfer reactions

We have studied the transfer reaction $^{14}$N($^7$Be,$^8$B)$^{13}$C at (\mathrm{E}_{\mathrm{lab}}=84) MeV, paying special attention to the effects of the coupling to the continuum in the exit channel. Using the Continuum Discretized Coupled Channels (CDCC) formalism, we find that these effects are important for the description of the elastic scattering observables. However, for the transfer process, differences between the predictions of the differential cross section within the Distorted Wave Born Approximation (DWBA) and the CDCC-Born approximation (CDCC-BA) are found to be negligible. This result supports the use of the DWBA method as a reliable tool to extract the $S_{17}(0)$ factor in this case.

A CDCC Study of11Be →10Be +nBreakup Effect on the11Be Elastic Scattering

The 1 1 Be→ 1 0 Be+n breakup effect on 1 1 Be+ 1 2 C elastic scattering at E/A=49.3 MeV is studied by continuum-discretized coupled-channels (CDCC) method based on the 1 0 Be+n+ 1 2 C three-body model. The CDCC calculation well reproduces the experimental data of the elastic scattering, and the breakup effect of the 1 1 Be nucleus is found to be significant. A dynamical polarization potential (DPP) evaluated shows a characteristic of the halo nucleus.

Fusion calculations for the6,7Li+16Osystems

Recently reported measurements of fusion for the ${}^{6,7}\mathrm{Li}{+}^{16}\mathrm{O},{}^{13}\mathrm{C},$ and ${}^{12}\mathrm{C}$ systems using the characteristic $\ensuremath{\gamma}$-ray yield method were significantly larger than previous measurements for the same systems using the evaporation residue technique. The question of which set of measurements more accurately represents fusion and, indeed, the precise definition of what constitutes fusion is an important one, considering the recent interest in the question of whether fusion is inhibited or enhanced for weakly bound exotic nuclei such as ${}^{6}\mathrm{He}$ and ${}^{11}\mathrm{Li}.$ We present here fusion cross sections derived from continuum discretized coupled channels (CDCC) calculations using the barrier penetration model (BPM) that agree very well with the characteristic $\ensuremath{\gamma}$-ray method results.

Uncorrelated scattering approximation for the scattering and breakup of weakly bound nuclei on heavy targets

The scattering of a weakly bound (halo) projectile nucleus by a heavy target nucleus is investigated. A new approach, called the Uncorrelated Scattering Approximation, is proposed. The main approximation involved is to neglect the correlation between the fragments of the projectile in the region where the interaction with the target is important. The formalism makes use of hyperspherical harmonics, Raynal–Revay coefficients and momentum-localized wave functions to expand projectile channel wave functions in terms of products of the channel wave function of the individual fragments. Within this approach, the kinetic energy and angular momentum of each fragment is conserved during the scattering process. The elastic, inelastic and breakup S-matrices are obtained as an analytic combination involving the bound wave function of the projectile and the product of the S-matrices of the fragments. The approach is applied to describe the scattering of deuteron on 58Ni at several energies. The results are compared with experimental data and continuum-discretized coupled-channels calculations.

αbreakup of6Liand7Linear the Coulomb barrier

Angular distributions of the $\ensuremath{\alpha}$-particle production differential cross section from the breakup of ${}^{6}\mathrm{Li}$ and ${}^{7}\mathrm{Li}$ projectiles incident on a ${}^{208}\mathrm{Pb}$ target have been measured at seven projectile energies between 29 and 52 MeV. The $\ensuremath{\alpha}$-breakup cross section of ${}^{6}\mathrm{Li}$ was found to be systematically greater than that of ${}^{7}\mathrm{Li}$ across the entire energy range. These data have been compared with previously reported results and with the predictions of continuum-discretized coupled channels (CDCC) calculations including resonant and nonresonant projectile breakup. The present data compare well with previous measurements, while the CDCC calculations provide a reasonable prediction of the relative $\ensuremath{\alpha}$-breakup cross sections but underpredict their absolute values. The calculations confirm that a major factor in the enhancement of the ${}^{6}\mathrm{Li}$ to ${}^{7}\mathrm{Li}$ $\ensuremath{\alpha}$-breakup cross section is the difference between the $\ensuremath{\alpha}$-breakup thresholds of the two isotopes. These results have implications for structural studies of light exotic nuclei based on elastic scattering.

Coulomb breakup and polarizability of the deuteron

After a brief review and optical model analysis of electric polarizability of the deuteron, continuum discretized coupled channels (CDCC) analysis is presented for 208 Pb(d, d) elastic scattering around 3–7 MeV. Effective deuteron polarizability is extracted from the CDCC analysis. Importance of coupling of 3P states with 3S and 3D scattering states is reported.

Convergence of the solution of the continuum discretized coupled channels method

The validity of the method of continuum discretized coupled channels (CDCC) is numerically tested and confirmed for a realistic example of ${d+}^{58}\mathrm{Ni}$ scattering at 80 MeV. Elastic and breakup S-matrix elements, calculated with the CDCC method, numerically converge as the model space is extended, viz., the internal angular momentum of the proton-neutron pair and the matching radius are increased. Two methods of discretization, the average method and midpoint method, of the continuum of the internal linear momentum of the $p\ensuremath{-}n$ pair are tested and found to yield the same results.

Deuteron elastic scattering and (d, p) reactions on 208Pb at Ed = 22 MeV and j-dependence of T20 in (d,p) reaction

An experimental study of deuteron elastic scattering and (d,p) reactions at Ed = 22 MeV was made for 208Pb target. A new j-dependence of T20 for (d,p) reaction at backward angles was observed. A “model independent” optical potential method was applied to analyze deuteron elastic scattering in the 10–22 MeV energy range. A continuum discretized coupled channels (CDCC) analysis was performed for deuteron elastic scattering at 22 MeV and the deuteron breakup effect was elucidated. DWBA and CDCC (d,p) analyses were made for (d,p) reactions.

Energy dependence of the 6Li - nucleus interactions at intermediate energies

Elastic scattering of by , , , and nuclei at various incident energies up to 100 MeV per nucleon is investigated by the continuum-discretized coupled-channel (CDCC) method based on a double-folding model of the - nucleus interaction with a realistic energy- and density-dependent nucleon - nucleon interaction, called DDM3Y. The projectile breakup effect is found to diminish with increasing incident energies. A comparison of the present result with an earlier CDCC calculation, based on the cluster-folding-model interaction, shows a clear difference between the two for large angle scattering. Possible implications of this discrepancy are discussed.

Continuum-discretized coupled-channels analysis of6Li→+4Hescattering atEc.m.=11.1MeV

Data for the elastic scattering of 27.8 MeV polarized {sup 6}Li from {sup 4}He and new data for the {sup 6}Li{r_arrow}{alpha}+d sequential breakup have been analyzed using the coupled-channels (CC) method with potentials derived via the cluster-folding (CF) procedure. The three low-lying T=0 resonant excited states of {sup 6}Li as well as nonresonant continuum states were taken into account in the calculations by means of the continuum-discretized coupled-channels (CDCC) method. No adjustable parameters were used. The analysis shows a dominant role of the projectile breakup into {alpha}+d in the {sup 6}Li+{sup 4}He elastic scattering. The deuteron transfer process is found to be of less importance. The calculations reproduce the values of the differential cross section for the sequential breakup channels at forward scattering angles. Predictions for the direct breakup near the {sup 6}Li{r_arrow}{alpha}+d breakup threshold are made. {copyright} {ital 1997} {ital The American Physical Society}

Three-body reaction theory in a model space.

A distorted waves Faddeev method for three-body systems is introduced, following Bencze and Redish and Birse and Redish, by using a multipole expansion to select parts of the interactions in rearrangement channels to serve as three-body distorting potentials in the first Faddeev equation. The consequence is a greatly reduced role for the second and third Faddeev equations. Truncating the multipole expansion and discarding the second and third equations reduces us back to the well-known continuum discretized coupled channels (CDCC) method. The relations among these methods are discussed. Despite the difficulties of the CDCC method, it is much easier than a full Faddeev calculation. \textcopyright{} 1996 The American Physical Society.