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.     

Inclusive and exclusive measurements in the projectile breakup of 7Li

The inclusive and exclusive measurements were carried out for ⁷Li projectile breakup on ²⁷Al target at 48 MeV. In the inclusive data we have observed a broad peak around the beam velocity for alphas and tritons. The exclusive data for alpha-triton coincidences show good agreement with the post-form DWBA theory of breakup reactions.     

Prior-form DWBA analysis of (3He,pd) elastic breakup at 90 MeV

Proton-deuteron coincident cross sections in ¹²C,⁵¹V,⁹⁰Zr(³He,pd) elastic breakup at 90 MeV have been calculated within the framework of the prior-form distorted-wave Born approximation (DWBA). Sufficient convergence of the calculations was obtained by including the pd relative angular momenta up to l = 4. The calculations reproduced the general trend of the coincident energy spectra and the angular correlation data, though deficiencies of the calculations were seen at some angles. The peripheral feature of the (³He, pd) elastic breakup is discussed from the angular momentum dependence of the transition amplitude.     

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.     

Microscopic description of diffractive deuteron breakup by <Superscript>3</Superscript>He nuclei

A microscopic formalism for describing observed cross sections for deuteron breakup by threenucleon nuclei was developed on the basis of the diffraction nuclear model. A general formula that describes the amplitude for the reaction ²H(³He, ³Hep)n and which involves only one adjustable parameter was obtained by using expansions of the integrands involved in terms of a Gaussian basis. This formula was used to analyze experimental data on the exclusive cross sections for deuteron breakup by ³He nuclei at the projectile energy of 89.4MeV. The importance of employing, in calculations, a deuteron wave function that has a correct asymptotic behavior at large nucleon–nucleon distances was demonstrated.     

The (p,d) reaction at 65 MeV

The (p, d) reaction on ¹²C, ¹⁴N, ¹⁶O, ⁴⁰Ca and ⁹⁰Zr has been studied with 65 MeV incident protons. DWBA calculations have been carried out to investigate the importance of finite range and non-locality effects. The experimental angular distributions for the strong transitions have been compared to DWBA calculations to study the effect of the deuteron optical potential. This comparison supports the use of the adiabatic deuteron potential which includes the effect of deuteron breakup.     

A new formalism of inclusive breakup reactions and validity of the surface approximation

Austern and Vincent's formalism for the inclusive breakup reactions in DWBA is rewritten in a form in which the elastic and inelastic breakup processes are well discernible. This form is calculable exactly and hence we test the validity of the surface approximation used by Baur et al.     

Fragmentation processes in nuclear reactions

Fragmentation processes in nuclear collisions are reviewed. The main emphasis is put on light ion breakup at nonrelativistic energies. The post- and prior-form DWBA theories are discussed. The post-form DWBA, appropriate for the “spectator breakup” describes elastic as well as inelastic breakup modes. This theory can also account for the stripping to unbound states. The theoretical models are compared to typical experimental results to illustrate the various possible mechanisms. It is discussed, how breakup reactions can be used to study high-lying single particle strength in the continuum; how they can yield information about momentum distributions of fragments in the nucleus.     

Deuteron stripping to the single particle states of 17O

Angular distributions for deuteron-¹⁶O elastic scattering and the ¹⁶O(d, p)¹⁷O reaction leading to levels with E_x = 0.0, 0.87, and 5.08 MeV have been measured at energies of 25.4, 36.0 and 63.2 MeV. The elastic deuteron data have been fit with a standard spin one optical model potential to obtain parameters for use in a DWBA analysis of the (d, p) data. The potential found in the search is shown to be consistent with other data taken in the range from 25 to 82 MeV. In addition to this deuteron optical potential, an adiabatic deuteron potential, which includes the effects of deuteron breakup, was used in the DWBA analysis. The neutron form factor was selected independent of the width of any state. The mean square radius, a single particle property, is used to find the well parameters and it determines the width of the single particle state. The spectroscopic factors obtained for the ground and first excited states are between 0.8 and 1.0 and are consistent with a large single particle parentage for these states and lower energy data. The width extracted from the DWBA analysis of the 5.08 MeV unbound state was 20% less than that obtained from elastic neutron scattering to the same state, possibly pointing up some difficulties with DWBA procedures commonly used. The adiabatic deuteron potential yields spectroscopic factors that are energy independent to 20% and gives satisfactory calculated angular distribution shapes for angles less than 40°. The conventional deuteron potential gives less satisfactory calculated shapes with the consequent introduction of some ambiguity in the derived spectroscopic factors.     

Inclusive neutron spectra from the break-up of3He in peripheral collisions with27Al,63Cu,93Nb,197Au

The break-up contribution to the continuous neutron spectra measured with³He projectiles of 34.4MeV for targets with masses 27≦A≦197 shows a broad peak centering the more belowE _3He/3 the heavier the target is. The DWBA theory of elastic and inelastic break-up underestimates the inclusive spectra, but gives the correct shapes. Both, calculated and experimental peak positions, are indicative of a peripheral collision.     

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.     

Final-state interaction between nucleons in elastic deuteron breakup within the three-body potential model

The diffraction deuteron breakup has been investigated within the three-body potential model, where the final-state interaction between nucleons is described nonperturbatively with respect to the pn potential. The triple differential cross section of the (d, pn) reaction on ⁴⁰Ca at deuteron energies of 140, 200, and 270 MeV and small emission angles of nucleons have been calculated for two schemes of the kinematically complete experiment. Noneikonal and nonadiabatic corrections to the cross section are small and do not change the shape of the peak in the proton energy spectra.     

The quasi-free breakup of He

A DWBA formalism is presented for the quasi-free breakup of ³He into elastic, inelastic and particle-transfer channels. The model assumes that only a subset of the projectile interacts strongly with the target while the remaining fragment acts as a spectator. The full DWBA is reduced to a rather transparent quasi-free reaction mechanism. Distortion of the spectator wave function due to the Coulomb field of the target is included. Calculations of angular correlations and projected energy spectra resulting from processes involving the binary breakup of ³He on ²⁸Si at E(³He) = 52 MeV are, apart from a normalization factor, in good agreement with the data.     

Finite-range DWBA analyses of (dt) and (d3He) reactions on 64Zn at Ed = 13 MeV

Exact finite-range DWBA analyses of ($\text{d}\text{, t}\text{)}$ and $\text{(}\text{d}\text{d}$, ³He) reactions have been performed for an ⁶⁴Zr target at an incident deuteron energy of 13.0 MeV, leading to the ground states of the residual nuclei. The microscopic overlap functions for (d, t) and (d, ³He) systems obtained by using the Phillips triton wavefunctions have been used as projectile form factors into the calculation. The results are compared with zero-range DWBA calculations taking the finite-range effects into account by means of a local energy approximation. The results obtained by finite-range and zero-range calculations for differential cross sections as well as vector and tensor analysing powers are compared with the experimental data. The range of validity of the local energy approximation is discussed.     

Observation of deuteron D-state and compound nucleus effects in (d,p) reactions on 46Ti and 52Cr with a polarized deuteron beam

The cross section and the vector and tensor analyzing powers have been measured for ⁴⁶Ti(d, p)⁴⁷Ti at deuteron energies of 6 and 10 MeV and ⁵²Cr(d, p)⁵³Cr at 6 MeV. Transitions were observed to the states at E_x=0.159, 1.549 and 1.793 MeV in ⁴⁷Ti and the states at E_x=0.0, 0.564, 1.006 and 2.321 MeV in ⁵³Cr. In addition, the cross sections and vector analyzing powers for deuteron elastic scattering were measured for the same targets and deuteron energies and compared to optical model calculations. The choice of optical parameters for the DWBA analysis of the (d, p) reactions is discussed. Calculations made with the DWBA method show that the deuteron D-state must be included to reproduce even qualitatively the (d, p) tensor analyzing power measurements. The j-dependence of the tensor analyzing power T₂₂ is discussed. The validity of the local energy approximation (which was used to incorporate the deuteron D-state into the DWBA calculations) is evaluated by comparison to finite range calculations. The contribution of compound nucleus reactions to the measured cross sections and analyzing powers was investigated. In order to determine the compound cross section, the Ericson fluctuations in excitation functions of cross section and vector analyzing power were measured from 5 to 7 MeV on each target. The formulas used to calculate the polarization observables from the Hauser-Feshbach theory are presented.     

Non-equilibrium dynamics in high-energy nucleus-nucleus collisions: (II). Calculation of inclusive particle spectra

We present theoretical results for proton, deuteron, triton, ³He and ⁴He inclusive particle spectra measured in 250 MeV/amu and 400 MeV/amu nucleus-nucleus reactions. The calculations are based on a multiple collision model which is derived from the relativistic Boltzmann equation and which includes composites formation through thermodynamics. The agreement with the data is reasonable to good and the remaining discrepancies are discussed.     

The deuteron breakup reaction induced by protons of 65, 85 and 100 MeV

Coplanar energy sharing spectra for p + d breakup at 65, 85 and 100 MeV proton bombarding energies were measured using the University of Maryland sectored isochronous cyclotron, by measuring the energies of either two protons or one proton and one neutron in coincidence. The detector angles were chosen to enhance either the p-p or p-n quasifree scattering, or the p-n final state interaction. The energy dependence of the peak cross section at equal symmetric quasifree scattering angle pairs was extracted for the ²H(p, 2p)n and ²H(p, pn)p reactions. Quasifree angular distributions were obtained for the reaction ²H(p, 2p)n at 65 MeV and for the reaction ²H(p, pn)p at 65, 85 and 100 MeV. The plane wave impulse approximation theory can only qualitatively reproduce the shape of the quasifree scattering peak in the energy sharing spectra and the shape of the p-p quasifree angular distribution. The discrepancies observed between the plane wave impulse approximation theory and the experimental data imply that the presence of the spectator particle (i.e., the multiple scattering effects) has a strong influence on the magnitude and the shape of the experimental results. Multiple scattering calculations were carried out in the three-body model of Aaron, Amado and Yam except that the S-wave separable two-body amplitudes were modified to fit two-nucleon elastic scattering data at higher energies. Comparisons of the results of these multiple scattering calculations to the experimental data show excellent quantitative agreement throughout the energy range and the angular region of this experiment, except for a few cases in which this model is inherently insufficient; namely, regions in which the Coulomb interaction is important, or regions for which a Hulthén wave function is inaccurate and the off-shell effects are not properly taken into account.     

Three-body model of deuteron breakup and stripping

For a three-body model Hamiltonian, the scattering eigenfunction that corresponds to an incident deuteron is expanded in terms of eigenfunctions of the neutron-proton relative Hamiltonian, as suggested by Johnson and Soper. In this expansion, breakup is represented by an integral over the continuum of neutron-proton scattering states. Only states of zero relative angular momentum are included; the validity and advantages of this approximation are discussed. The continuum is divided into five discrete channels, whose coupling to each other and to the deuteron channel is treated by solving coupled differential equations with appropriate boundary conditions. It is found necessary to use a simple WKB method to take account of the long-range coupling among breakup channels; this method introduces potential matrices W and S that describe local and derivative coupling of the channels. The reaction of breakup on the elastic channel is neglected.The properties of W and S and the breakup wavefunction are examined for the case of 22.9 MeV deuterons incident on a target of mass number A ≈ 40. The Coulomb interaction is ignored, and a local Gaussian shape is used for both the real and imaginary parts of the nucleon-nucleus optical potential.It is found that a rather broad spectrum of n-p continuum states is excited, especially for low center-of-mass angular momentum. This result weakens the justification for the Johnson-Soper adiabatic theory, which emphasizes breakup into states of low relative energy.The breakup part of the wavefunction at zero n-p separation is comparable with the elastic part, but is important only over a surprisingly short range in the center-of-mass coordinate, with the result that breakup cross sections are quite small. Nevertheless, breakup produces major modifications of (d, p) cross sections. These modifications can to some extent be simulated by the Johnson-Soper method. The breakup wavefunctions show several interesting effects in their dependence on angular momentum and radius.