Continuum-discretized Coupled-channels Method Research Articles

CALCULATION OF BREAKUP CROSS SECTION FOR FOUR-BODY BREAKUP REACTION SYSTEMS

We present a new method of smoothing discrete breakup cross sections calculated by the method of continuum-discretized coupled-channels. In the four-body breakup reaction of 12 C (6 He , nn4 He ) at E in = 229.8 MeV , the continuous breakup cross section is evaluated as a function of the excitation energy of 6 He . Convergence of the cross section with respect to extending the modelspace is also confirmed.

Smoothing Method of Discrete Breakup S-Matrix Elements in the Theory of Continuum-Discretized Coupled Channels

We present a practical way of smoothing discrete breakup S-matrix elements calculated by the continuum-discretized coupled-channel method (CDCC). This method makes the smoothing procedure much easier. The reliability of the smoothing method is confirmed for the three-body breakup reactions, 58Ni(d,pn) at 80 MeV and 12C(6He,4He2n) at 229.8 MeV.

Dynamical Relativistic Effects in Breakup Processes of Halo Nuclei

The continuum-discretized coupled-channels (CDCC) method is used to study the breakup of weakly-bound nuclei at intermediate energies collisions. For large impact parameters, the Eikonal CDCC (E-CDCC) method was applied. The effects of Lorentz contraction on the nuclear and Coulomb potentials have been investigated in details. Such effects tend to increase cross sections appreciably. We also show that, for loosely-bound nuclei, the contribution of the so-called close field is small and can be neglected.

Few-body nuclear reactions

Three- and four-body scattering is described in the framework of exact Faddeev-type integral equations. They are solved numerically using momentum-space partial-wave basis. The Coulomb interaction between charged particles is included using a novel implementation of the screening and renormalization method. The technique is applied to three- and four-nucleon scattering. Furthermore, the method has been extended successfully to elastic, transfer, and breakup reactions in three-body-like nuclear systems. Examples are deuteron scattering on stable nuclei ranging from 4He to 58Ni and proton scattering on weakly bound two-body system such as 11Be, 15C, and 17O. These calculations allow to evaluate the accuracy of traditional approximate nuclear reaction approaches like the continuum-discretized coupled-channels (CDCC) method but also to test novel dynamical models such as nonlocal optical potentials.

New Approach for Evaluating Incomplete and Complete Fusion Cross Sections with Continuum-Discretized Coupled-Channels Method

We propose a new method for evaluating incomplete and complete fusion cross sections separately using the Continuum-Discretized Coupled-Channels method. This method is applied to analysis of the deuteron induced reaction on a 7Li target up to 50 MeV of the deuteron incident energy. Effects of deuteron breakup on this reaction are explicitly taken into account. Results of the method are compared with those of the Glauber model, and the difference between the two is discussed. It is found that the energy dependence of the incomplete fusion cross sections obtained by the present calculation is almost the same as that obtained by the Glauber model, while for the complete fusion cross section, the two models give markedly different energy dependence. We show also that a prescription for evaluating incomplete fusion cross sections proposed in a previous study gives much smaller result than an experimental value.

Properties of Nuclear and Coulomb Breakup of 8B

The dependence of breakup cross sections of 8B at 65 MeV/nucleon on the target mass number A_T is investigated by means of the continuum-discretized coupled-channels method (CDCC) with more reliable distorting potentials than those in the preceding study. The A_T^(1/3) scaling law of the nuclear breakup cross section is found to be satisfied only in the middle A_T region of 40 < A_T < 150. The interference between nuclear and Coulomb breakup amplitudes vanishes in very forward angle scattering, independently of the target nucleus. The truncation of the relative energy between the p and 7Be fragments slightly reduces the contribution of nuclear breakup at very forward angles, while the angular region in which the first-order perturbation theory works well does not change essentially.

Dissociation of Relativistic Projectiles with the Continuum-Discretized Coupled-Channels Method

Relativistic effects in the breakup of weakly-bound nuclei at intermediate energies are studied by means of the continuum-discretized coupled-channels method with eikonal approximation. Nuclear coupling potentials with Lorentz contraction are newly included and those effects on breakup cross sections are investigated. We show that relativistic corrections lead to larger breakup cross sections. Coupled-channel effects on the breakup cross sections are also discussed.

Quantum Three-Body Calculation of the Nonresonant Triple- Reaction Rate at Low Temperatures

The triple-\alpha reaction rate is re-evaluated by directly solving the three-body Schr\"odinger equation. The resonant and nonresonant processes are treated on the same footing using the continuum-discretized coupled-channels method for three-body scattering. Accurate description of the \alpha-\alpha nonresonant states significantly quenches the Coulomb barrier between the two-\alpha's and the third \alpha particle. Consequently, the \alpha-\alpha nonresonant continuum states below the resonance at 92.04 keV, i.e., the ground state of 8Be, give markedly larger contribution at low temperatures than in foregoing studies. We show that Nomoto's method for three-body nonresonant capture processes, which is adopted in the NACRE compilation and many other studies, is a crude approximation of the accurate quantum three-body model calculation. We find about 20 orders-of-magnitude enhancement of the triple-\alpha reaction rate around 10^7 K compared to the rate of NACRE.

COUPLED-CHANNELS ANALYSES OF 6He BREAKUP REACTIONS

We present analyses of breakup effects of 6 He on the elastic scattering by the continuum-discretized coupled-channels method, in which the reaction system is described as a four-body model, n+n+4 He +target. In this analysis, three-body breakup continuum of 6 He is discretized by daiagonalizing the internal Hamiltonian of 6 He in a space spanned by the Gaussian basis functions. The calculated elastic cross sections are in good agreement with the experimental data, which shows that nuclear and Coulomb breakup effects are significant.

Description of Four-Body Breakup Reaction with the Method of Continuum-Discretized Coupled-Channels

We present a method of smoothing discrete breakup S-matrix elements calculated by the method of continuum-discretized coupled-channels (CDCC). This smoothing method makes it possible to apply CDCC to four-body breakup reactions. The reliability of the smoothing method is confirmed for two cases, 58 Ni(d, pn) at 80 MeV and the E1 transition of 6 He. We apply CDCC with the smoothing method to a 6 He breakup reaction at 22.5 MeV. Multistep breakup processes are found to be important. Subject Index: 221, 223, 227

One-neutron removal reactions ofC18andC19on a proton target

We have studied the one-neutron removal reactions of $^{18}\mathrm{C}$ at 81 MeV/nucleon and $^{19}\mathrm{C}$ at 68 MeV/nucleon on a liquid hydrogen target. Transverse-momentum distributions of the core fragments $^{17}\mathrm{C}$ and $^{18}\mathrm{C}$, and their partial cross sections, are measured using coincidences with the $\ensuremath{\gamma}$ rays. These are compared to theoretical calculations based on the continuum-discretized coupled-channels (CDCC) method. We have found that the transverse-momentum distributions of $^{17}\mathrm{C}$ are well described by the CDCC calculations, and the agreement provides evidence for the spin-parity assignments of ${J}^{\ensuremath{\pi}}=1/{2}^{+}$ for the 0.21-MeV state and ${J}^{\ensuremath{\pi}}=5/{2}^{+}$ for the 0.33-MeV state in $^{17}\mathrm{C}$. The ${J}^{\ensuremath{\pi}}=(2,3){}^{+}$ assignment for the 4.0-MeV state in $^{18}\mathrm{C}$ is obtained for the first time based on the momentum distribution of $^{18}\mathrm{C}$. The measured partial cross sections are consistent with the CDCC calculations combined with shell-model spectroscopic factors. The present study demonstrates that the transverse-momentum distribution obtained in a one-neutron removal reaction on a proton target is as useful as the conventional longitudinal-momentum distribution measured with the heavier Be or C targets for determining the orbital angular momentum of a removed neutron.

THREE AND FOUR-BODY BREAKUP REACTIONS

Breakup reactions provide spectroscopic information on exotic nuclei. Coulomb breakup indirectly measures the astrophysical S factor for radiative-capture reactions. The validity of first-order perturbation theory is limited for extended systems such as halo nuclei. More elaborate reaction models are necessary: semi-classical time-dependent Schrödinger equation, eikonal and dynamical eikonal approximations, continuum-discretized coupled-channel method. Breakup experiments do not provide much information on the structure of a two-cluster halo nucleus but accurate exclusive experiments should be more interesting for three-cluster nuclei.

Continuum discretization methods in a composite particle scattering off a nucleus: Benchmark calculations

The direct comparison of two different continuum discretization methods toward the solution of a composite particle scattering off a nucleus is presented. The first approach---the continuum-discretized coupled-channel method---is based on the differential equation formalism, while the second one---the wave-packet continuum discretization method---uses the integral equation formulation for the composite-particle scattering problem. As benchmark calculations, we have chosen the deuteron off $^{58}\mathrm{Ni}$ target scattering (as a realistic illustrative example) at three different incident energies: high, middle, and low. Clear nonvanishing effects of closed inelastic channels at small and intermediate energies are established. The elastic cross sections found in both approaches are very close to each other for all three considered energies.

Full coupled-channel description of three-body and four-body breakup reactions

Recent development of the continuum-discretized coupled-channels method (CDCC) is reported with introducing the eikonal CDCC method (E-CDCC) and four-body CDCC. The former is applied to analysis of 8B breakup by 208Pb at 52 A MeV and the astrophysical factor S 17 ( 0 ) for the 7Be( p , γ ) 8B reaction is extracted with the asymptotic normalization coefficient method. We obtain S 17 ( 0 ) = 20.9 −1.9 +2.0 eV b , which is significantly larger than the previous one, S 17 ( 0 ) = 18.9 ± 1.8 eV b , determined from an analysis with the virtual photon theory. The difference between the two values is found to be due to the contributions from nuclear breakup and higher-order processes. The latter, four-body CDCC, is applied to 6He+ 209Bi scattering near the Coulomb barrier energy and found to reproduce the elastic cross sections well, in contrast to the standard three-body CDCC. Accurate treatment of the three-body continuum states of 6He turns out to be essential to correctly describe the 6He+ 209Bi scattering.

Three-body description of direct nuclear reactions: Comparison with the continuum discretized coupled channels method

The continuum discretized coupled channels (CDCC) method is compared with the exact solution of the three-body Faddeev equations in momentum space. We present results for (i) elastic and breakup observables of $d+$$^{12}\mathrm{C}$ at ${E}_{d}=56$ MeV, (ii) elastic scattering of $d+$$^{58}\mathrm{Ni}$ at ${E}_{d}=80$ MeV, and (iii) elastic, breakup, and transfer observables for $^{11}\mathrm{Be}$$+p$ at ${E}_{{}^{11}\mathrm{Be}}/A=38.4$ MeV. Our comparative studies show that in the first two cases, the CDCC method is a good approximation of the full three-body Faddeev solution, but for the $^{11}\mathrm{Be}$ exotic nucleus, depending on the observable or the kinematic regime, it may miss some of the dynamic three-body effects that appear through the explicit coupling to the transfer channel.

Coupled-channel effects in elastic scattering and near-barrier fusion induced by weakly bound nuclei and exotic halo nuclei

The influence on fusion of coupling to the breakup process is investigated for reactions where at least one of the colliding nuclei has a sufficiently low binding energy for breakup to become an important process. Elastic scattering, excitation functions for sub- and near-barrier fusion cross sections, and breakup yields are analyzed for $^{6,7}\mathrm{Li}+^{59}\mathrm{Co}$. Continuum-discretized coupled-channels (CDCC) calculations describe well the data at and above the barrier. Elastic scattering with $^{6}\mathrm{Li}$ (as compared to $^{7}\mathrm{Li}$) indicates the significant role of breakup for weakly bound projectiles. A study of $^{4,6}\mathrm{He}$ induced fusion reactions with a three-body CDCC method for the $^{6}\mathrm{He}$ halo nucleus is presented. The relative importance of breakup and bound-state structure effects on total fusion is discussed.

Coulomb breakup effects on the elastic cross section ofHe6+Bi209scattering near Coulomb barrier energies

We accurately analyze the $^6$He+$^{209}$Bi scattering at 19 and 22.5 MeV near the Coulomb barrier energy, using the continuum-discretized coupled-channels method (CDCC) based on the $n$+$n$+$^4$He+$^{209}$Bi four-body model. The three-body breakup continuum of $^6$He is discretized by diagonalizing the internal Hamiltonian of $^6$He in a space spanned by the Gaussian basis functions. The calculated elastic and total reaction cross sections are in good agreement with the experimental data, while the CDCC calculation based on the di-neutron model of $^6$He, i.e., the $^2n$+$^{4}$He+$^{209}$Bi three-body model, does not reproduce the data.

Three-body continuum-discretized coupled-channel calculations forHe6scattering from heavy nuclei

Data for scattering of $^{6}\mathrm{He}$ from $^{197}\mathrm{Au}$, $^{208}\mathrm{Pb}$, and $^{209}\mathrm{Bi}$ targets at low energies were consistently analyzed by use of the continuum-discretized coupled-channels method and the dineutron model of the projectile. A very good description of the experimental data was obtained with the strength of the dipole couplings reduced by 50%. We find that the dipole couplings are responsible for the suppression of the Coulomb rainbow and that the quadrupole couplings must be included in the calculations in order to obtain good agreement with the elastic-scattering data at more backward angles.

Effect of halo structure on 11Be + 12C elastic scattering

The 11Be → 10Be + n breakup effect on 11Be + 12C elastic scattering at E/A = 49:3 MeV is studied by continuum-discretized coupled-channels (CDCC) method based on the 10Be + n + 12C three-body model. The CDCC calculation well reproduces the experimental data of the elastic scattering, and the breakup effect of the 11Be nucleus is found to be significant. Furthermore, the reaction dynamics of the 11Be → 10Be + n breakup process is investigated and a dynamical polarization potential (DPP) is evaluated.

Continuum-discretized coupled-channels method for four-body nuclear breakup inHe6+C12scattering

We propose a fully quantum-mechanical method of treating four-body nuclear breakup processes in scattering of a projectile consisting of three constituents, by extending the continuum-discretized coupled-channels method. The three-body continuum states of the projectile are discretized by diagonalizing the internal Hamiltonian of the projectile with the Gaussian basis functions. For $^{6}\mathrm{He}+^{12}\mathrm{C}$ scattering at 18 and $229.8\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$, the validity of the method is tested by convergence of the elastic and breakup cross sections with respect to increasing the number of the basis functions. Effects of the four-body breakup and the Borromean structure of $^{6}\mathrm{He}$ on the elastic and total reaction cross sections are discussed.