Continuum-discretized coupled-channels calculations for three-body models of deuteron-nucleus reactions

The formalism and results of truncated coupled channels evaluations of three-body models of deutron-induced nuclear reactions are reviewed. Emphasis is placed on breakup, elastic scattering and stripping. The relations of the coupled channels method to the Faddeev method, the adiabatic approximation and the distorted wave Born approximation are discussed extensively. Although the adiabatic approximation is seen to be excellent for the wavefunction in the elastic channel, it significantly underestimates the contributions of breakup states in stripping. Significant effects are associated with coupling to relative l = 2 breakup states.     

Final state interaction effects in electrodisintegration of the deuteron in the Bethe-Salpeter approach

The electrodisintegration of the deuteron is considered in a relativistic model of nucleon-nucleon interaction based on the Bethe-Salpeter approach with a separable interaction kernel. The exclusive cross section is calculated in the impulse approximation under various kinematic conditions. Final state interactions between the outgoing nucleons are taken into account. The comparison of nonrelativistic and relativistic calculations is presented. Partial-wave states of the neutron-proton pair with total angular momentum J = 0, 1 are considered.     

CDCC Approaches and Roles of Closed Channels in d-Nucleus Reactions

The effect of deuteron breakup in d-nucleus reaction is treated with the continuum discretized coupled channels (CDCC) approach, and the effects on the total reaction cross sections and elastic scattering angular distributions are studied by comparing the calculations of CDCC and spherical optical model with our global deuteron optical potential [Phys. Rev. C 73 (2006) 054605] below 200 MeV, for target nuclei ranging from ¹²C to ²⁰⁸Pb. The contributions from the closed channels to the total reaction and breakup cross sections, and angular distributions of elastic scattering are also seriously discussed.     

Coupling effects of resonant and discretized non-resonant continuum states in4He +6Li scattering at 10 MeV/A

Alpha- particle scattering from the resonant (3 ₁ ⁺ ) and non-resonant continuum states of⁶Li is studied at incident energy 10 MeV/A. Theα+d breakup continuum part within the excitation energyE _ex=1.475–2.475 MeV is discretized in two energy bins. Unlike the results at higher incident energies, here the coupled-channel calculations show significant breakup continuum coupling effects on the elastic and inelastic scattering. It is shown that even when the continuum-continuum coupling effects are strong, the experimental data of the ground state and the resonant as well as discretized non-resonant continuum states impose stringent constraint on the coupling strengths of the non-resonant continuum states.     

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 E_d = 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.     

Second-order breakup corrections to elastic deuteron-nickel scattering between 13 and 80 MeV

Breakup corrections to the elastic scattering matrix elements are calculated in the second-order distorted-wave Born approximation at deuteron incident energies of 45 and 85 MeV. The effects of spin are included. The size of the corrections are found to be generally as large as those obtained in a previous study at 13 and 21.6 MeV. The breakup cross section is calculated to first order in the breakup matrix elements by a distorted-wave Born treatment. Comparison with fully coupled calculations shows that the DWBA method overestimates the breakup cross section by a factor of about three.The continuum of breakup states up to a n-p relative momentum 1 fm^?1 is included in the calculations. This continuum is discretized by subdividing it first into two bins, and then into four bins. The finer discretization does not make a large difference in either the elastic cross section or the breakup cross sections. The higher bins give only a small contribution to either quantity.     

Coupling effects of resonant and discretized non-resonant continuum states in 4He + 6Li scattering at 10 MeV/A

Alpha-particle scattering from the resonant (3 ₁ ⁺ ) and non-resonant continuum states of ⁶Li is studied at incident energy 10 MeV/A. The α + d breakup continuum part within the excitation energy E _ex = 1.475–2.475 MeV is discretized in two energy bins. Unlike the results at higher incident energies, here the coupled-channel calculations show significant breakup continuum coupling effects on the elastic and inelastic scattering. It is shown that even when the continuum-continuum coupling effects are strong, the experimental data of the ground state and the resonant as well as discretized non-resonant continuum states impose stringent constraint on the coupling strengths of the non-resonant continuum states.     

Cross Sections for Neutron–Deuteron Elastic Scattering in the Energy Range 135–250 MeV

We report new measurements of the neutron–deuteron elastic scattering cross section at energies from 135 to 250 MeV and center-of-mass angles from 80^? to 130^?. Cross sections for neutron-proton elastic scattering were also measured with the same experimental setup for normalization purposes. Our nd cross section results are compared with predictions based on Faddeev calculations including three-nucleon forces, and with cross sections measured with charged particle and neutron beams at comparable energies.     

Proton-neutron correlation in the deuteron breakup at 56 MeV and prior-form DWBA analysis

Proton-neutron angular correlations in the ¹²C, ⁵¹V and ¹¹⁸Sn(d, pn) reactions have been measured at 56 MeV to investigate the deuteron breakup process. The elastic breakup which leaves the target nucleus in its ground state dominates the coincident spectra. The elastic breakup cross sections are estimated to be 36–48% of the inclusive breakup yields at 15° or 17.5°. In the angular correlations the protons are emitted predominantly on the side of the beam opposite to the neutrons. The experimental data have been analyzed using the prior-form DWBA. For both nuclear and Coulomb breakup, sufficient convergence of the calculations is obtained by including the pn angular momenta up to l = 2. For the nuclear breakup calculations, the l = 0 and 2 contributions dominate the cross sections. For the Coulomb breakup the l = 1 contribution is predominant. In the calculations the effect of the Coulomb breakup is seen at forward angles of the angular correlation. The DWBA calculations reproduce fairly well the coincident energy spectra and the angular correlations in the angular region where the protons are emitted on the side of the beam opposite to the neutrons. On the other hand the calculations overestimate the break-up cross sections by a factor of 2 to 10 in the angular region where the protons are emitted on the same side of the beam as the neutrons. The distributions of deuteron c.m. angular momenta that contribute to the breakup amplitude are examined to obtain information on the region of space in which the breakup reactions takes place.     

Some effects of spin-orbit interaction on rotational levels and rotational line intensities in vibrationally unexcited 2A, 2E, and 2F electronic states of open-shell XY4 molecules

Rotational energy levels in vibronic ground states of ²A, ²E, and ²F electronic states of open-shell XY₄ molecules, as well as rotational line intensities for allowed transitions between such states, are discussed, including the effects of spin-orbit interaction and tetrahedral splittings. Jahn-Teller effects are assumed to be small, and are only taken into account implicitly, through their contributions to various parameters in the effective Hamiltonian. Qualitative information is obtained by considering several limiting-case coupling schemes among the electron spin angular momentum S, the electron orbital angular momentum L, and the pure rotational angular momentum R. These limiting cases are similar in spirit to Hund's coupling cases in diatomic molecules, but differ sufficiently from the latter to make detailed correspondences unhelpful. Quantitative information on rotational energy levels and line intensities is obtained numerically by diagonalizing a Hamiltonian matrix set up in a basis set characterized by uncoupled moleculefixed projections of S, L, and the total angular momentum J, and symmetrized so that all basis set functions belong to a definite species in the subgroup D_2d of the true point group T_d. Hamiltonian matrix elements are determined by ladder operator techniques. Three sample calculated spectra, corresponding to $\text{p(}{}^2\text{F}{}_2\text{)-s(}{}^2\text{A}{}_1\text{)}$, $\text{d(}{}^2\text{E)-p(}{}^2\text{F}{}_2\text{)}$, and $\text{d(}{}^2\text{F}{}_2\text{)-p(}{}^2\text{F}{}_2\text{)}$ are presented. As one might expect, when the spin-orbit constant A is set equal to zero, then both qualitative and quantitative aspects of the rotational-electronic problem in open-shell XY₄ molecules can be mapped easily onto discussions of the rotation-vibration problem from the CH₄ literature.     

Large-angle quasi-free scattering in 6Li(p,pd)4He at 670 MeV

The ⁶Li(p, pd)⁴He reaction was investigated at 670 MeV by a coincidence experiment with a large-angle scattering geometry. The energy sharing and angular correlation of the reaction products were measured and the momentum distribution of the recoil nucleus was determined for transitions leading to residual nuclei in the ground and excited states. Results were analysed in terms of a simplified distorted-wave impulse approximation using cluster-model and three-body wave functions. The observed momentum distribution of the p-n pair in the p-shell of ⁶Li is in agreement with three-body calculations, while the spectroscopic factor is larger than predicted by theory. Transitions to breakup states of the α-particle also have the characteristics of quasi-free scattering on deuteron clusters.     

Effect of breakup on the deuteron nucleus spin orbit and tensor potentials

A previous treatment of breakup in terms of a coupled channel formulation is generalized so as to include the spin of the nucleus. The break up spectrum is described in terms of the eigenstates of the free neutron proton Hamiltonian, based on the Reid soft core potential. Nearly spherically symmetric as well as nearly quadrupole deformed break up states are included. The latter give large contributions to the correction to the spin orbit and tensor parts of the deuteron-nucleus elastic optical model. A numerical application for 13 MeV and 21.6 MeV deuterons incident on Ni is presented, in which virtual breakup is included to second order in the breakup transition matrix element. The breakup correction to the deuteron potential is strongly dependent on the orbital angular momentum and the energy of the deuteron-nucleus motion. The spin orbit potential is rendered less diffuse and acquires an imaginary part. The tensor potential correction is comparable in radial shape to the Watanabe static folding model result, and tends to increase its magnitude by nearly 100%.     

Relativistic calculation for the reactions\bar p d → 5πp and\bar p d → 3πp at rest

We present a fully relativistic calculation for ¯pd → 5πp and ¯pd → 3πp that includes angular momentum and spin dynamics. We calculate the inclusive proton distributions from two diagrams: the leading “tree” diagram, and the diagram for pion rescattering. Pion-nucleon rescattering proceeds through the Δ, in the RaritaSchwinger formalism, thus preserving the correct angular dependence. We use realistic parameterizations of the deuteron, keeping both theS andD states. The loop integrations for the rescattering amplitude and the phase space integrations have been done numerically. We find that the combined effects of a correct treatment of the rescattered pion and relativity are small in comparison with a simple non-relativistic calculation.     

HCNO as a semirigid bender: The degenerate ν4 state

The semirigid bender Hamiltonian for fulminic acid HCNO (Bunker, Landsberg, and Winnewisser, J. Mol. Spectrosc.74, 9–25 (1979)) is extended. The extended Hamiltonian describes the manifold of large amplitude vibrational states (due to the ν₅ HCN bending mode) superimposed on a high frequency vibrational state involving excited quanta of the ν₄ CNO bending mode. Such high frequency vibrational states may be degenerate when the large amplitude coordinate is zero, and the semirigid bender Hamiltonian is modified to account for the ν₄ vibrational angular momentum around the molecular axis in the linear limit, and for l-doubling effects. The extended Hamiltonian is used to fit HCN bending and rotation energy level separations for HCNO superimposed in the ν₄ fundamental level. It is found that the effective HCN bending potential in the ν₄ state is very similar to that in the high frequency vibrational ground state. The results obtained confirm the conclusion reached by Bunker, Landsberg, and Winnewisser: HCNO is linear at equilibrium.     

The break-up reactionp+d→p+p+n

The three-particle reactionp+d→p+p+n has been investigated at a deuteron bombarding energy of 52.0 MeV. Single counter experiments as well as coincidence measurements have been carried out. The laboratory angular distribution of the break-up reaction and the total break-up cross-section have been determined by means of the single counter experiments. The angular distribution for final-state interacting neutron-proton pairs was obtained by means of coincidence measurements. The coincidence experiments provide a strong evidence for very different shapes of the angular distributions for neutron-proton singlet and triplet final-state interaction.     

基于CDCC理论对氘核破裂效应进行初步研究(英文)

为研究氘核的破裂效应对弹性散射角分布和反应截面的影响, 基于连续离散化耦合道 (CDCC) 理论编制了程序CDCCOM。 从中心点波函数的初始值出发, 利用P3C5算法求解耦合道方程组, 进而通过边界点上内、 外区的波函数相匹配求得S矩阵元。 P3C5算法提高了计算精度, 同时验证了程序CDCCOM的有效性。 通过与其他工作的计算结果及实验数据进行比较, 认为在氘核入射能量低于200 MeV的情况下, 对于大多数靶核, 通过CDCCOM都能够得出合理的结果, 表明该程序可用于进一步研究氘核诱发的非弹性核反应。     

New treatment of breakup in few-body systems

A new approach to determining breakup amplitudes in few-body systems in the context of a Faddeev formalism based on lattice discretization of a continuum is described. Due to such discretization and use of finite-dimensional representations for all operators in the kernels of integral equations, breakup in few-body systems is interpreted as a partial case of multi-channel scattering and corresponds to transitions between the states of the discretized continuum of an asymptotic channel Hamiltonian. The case study is based on amplitudes of three-nucleon breakup n + d → n + n + p with semi-realistic NN interaction potentials.