The break-up of the deuteron and stripping to unbound states

We treat the break-up of the deuteron in the field of a heavy nucleaus. Our theory is based on the post formalism of the DWBA, which is discussed in detail. In particular we consider the case where the energy of the incoming deuteron is below the Coulomb barrier. This sub-Coulomb break-up permits a test of the DWBA by comparing experimental data with the theory. Small effects are investigated in this context, and other treatments of this process are reviewed critically. To give an intuitive picture of the break-up process we introduce a semiclassical model.In the second part the theory is extended to stripping reactions to unbound (resonant) states. We show that this reaction which interferes with the direct break-up process, can again be well described in the post-form of the DWBA. Using simple models we discuss the most important features of such reactions. Particular attention is paid to the “l-enhancement”, “parallelism” and the influence of the specific wave functions on resonance line shapes. A survey and comparison of various theories of stripping reactions to unbound states is given and finally we discuss possible future developments.     

On the reversibility of stripping reactions

The basis for the equivalence of prior and post forms of the T-matrix in both the Born and the distorted wave Born approximations is here studied in a one-dimensional model that facilitates clarification of some of the underlying mechanism. One sees how important it is that changes in the wave functions bring it about that averages of different perturbing potentials over different regions can still be equal. The post form commonly employed for a (d, p) reaction makes it seem at first sight that the potential stripping the deuteron apart is the potential that holds it together. The prior form DWBA has as a perturbation the intuitively appealing interaction responsible for the classical break-up of the deuteron and contributing to the “surface reaction” nature of stripping.     

An approximate three-body theory of deuteron stripping

A treatment of deuteron stripping is developed in which the three-body effects associated with deuteron break-up in the nuclear field are included explicitly. The essence of the method is the choice of a convenient discrete set of n-p eigenfunctions as a representation of the three-body continuum effects. This approach leads to a distorted wave stripping matrix element similar to that of the DWBA, except that the elastic deuteron wave is replaced by a three-body wave function given as the solution of a set of coupled two-body Schrödinger equations. The adiabatic theory of Johnson and Soper appears as the solution in a suitable first approximation. This new formalism should prove useful in the evaluation of corrections to three-body models of the deuteron-nucleus system, in particular those models in which the nucleon-target interaction is represented by a complex local optical potential.     

Qualitative features of Pauli break-up from the configuration space viewpoint

A dimensionless measure of the effect of Pauli exclusion upon the neutron-proton interaction in a projectile deuteron is calculated in the zero range approximation for a Hulthen deuteron. One learns thereby that the Pauli break-up matrix element is distributed differently in space from that for nuclear break-up and that the effect with decreasing energy to a large value at the lowest deuteron energies.     

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.     

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.     

On the validity of dwba for deuteron stripping: (I). The Mitra three-body model

Previous work showing that there exists an exact formulation of the DWBA for stripping in the S-wave, separable potential, three-body model of Mitra is discussed and extended. The one-body equation obeyed by the c.m. wave function used in the reformulated DWBA is derived and compared with the equation obeyed by the wave function used in the standard formulation of DWBA, viz., the deuteron elastic scattering wave function. Results obtained by other workers on application of three-body methods to direct reactions are discussed in light of the fact that an exact DWBA exists for the separable potential model.     

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.     

Polarization transfer in the deuteron break-up reaction at 56 MeV

The polarization transfer coefficients K^yy′ of the break-up spectra in the reactions ¹²C, ⁵⁸Ni, ²⁰⁹Bi($\text{d}$, $\text{p}$X) were measured at 56 MeV. In the forwardd angles (13.5° and 20.0°), the K_y^y′ data at the break-up peak are nearly equal to $\text{2}\text{3}$ (the spectator model prediction) and they decrease at both sides of the peak. The deviation due to the deuteron D-state is estimated, using PWBA and DWBA.     

Polarization transfer in deuteron stripping reactions

Polarization phenomena involving the spins of a and b in the A(a, b)B reaction are discussed using a complete set of irreducible tensors carrying definite spin transfer. The linear model independent equations relating the cross section and the polarization observables with these tensors are shown to be particularly appropriate for the study of spin dependent interactions, preferentially associated with particular values of spin transfer. The DWBA theory of polarization transfer in deuteron stripping reactions is thoroughly discussed and among the 17 polarization observables we distinguish those likely to be more sensitive to spin dependent distortion, to have stronger deuteron D-state effects, to exhibit the sign-rule j-dependence and other forms of j-dependence. For certain deuteron polarizations, when the spin transfer is pure $\text{s =}\text{1}\text{2}$, it is shown that deuteron stripping reactions are transparent to vector polarization transfer and the outgoing nucléon polarization independent of scattering angle and deuteron energy. DWBA calculations including contributions from spin transfer $\text{3}\text{2}$ through the deuteron D-state and spin-orbit distortion show that polarization transfer in such deuteron polarizations can be explored as a method of producing fast polarized neutrons with known polarization.     

Coupled channels approach to the deuteron break-up in the field of a nucleus

The elastic scattering and break-up of the deuteron in the field of a nucleus is formulated in terms of coupled differential equations together with correct boundary conditions. Cross sections for the two processes are given in terms of the scattering coefficients. An approximate treatment of the Coulomb distortion in the break-up channel is suggested.     

The reactiond+d→p+t and elastic deuteron-deuteron scattering at 51.5 Mev

The elastic deuteron-deuteron scattering and the reactionsd(d,p)t andd(d,³He)n have been investigated at an incident deuteron energy of 51.5 MeV. Time-of-flight technique was used for the particle identification. The observed angular distributions for the (d, p) and (d, n) reaction are identical within the accuracy of the experimental data. The experimental results for the reactiond(d, p)t are compared with a theoretical prediction based on the application of a generalized separable potential model to the four nucleon systems. Information on the single deuteron break-up reactiond+d→d+p+n and on the double deuteron break-up reactiond+d→p+p+n+n is contained in the observed continuous proton and deuteron spectra. The double break-up seems to be contribute only with a rather small amount to the whole break-up cross section.     

Spin-orbit coupling effects in deuteron stripping angular correlations

Spin-orbit contributions in the DWBA analysis of particle-gamma angular correlations following deuteron stripping reactions have been investigated for several l=2 transitions. It is shown that in general these effects are quite large.     

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.     

Experimental and theoretical study of the elastic break-up of the alpha particle on58Ni nuclei atE α=172.5 MeV

The triton-proton angular correlation function has been measured for the⁵⁸Ni(α, t p)⁵⁸Ni reaction leading to the ground state of the target nucleus. The obtained experimental data have been interpreted in terms of the direct break-up (fragmentation) of the alpha particle on the target nucleus potential. It is shown that the main features of the measured double differential cross section are well predicted by the DWBA theory of the break-up. The dependence of the elastic and inclusive break-up cross sections upon the fragment-target interaction potentials is discussed.     

DWBA Analysis of Polarization of Protons from the 9Be(d,p1)10Bex Reaction at Ed = 5.5, 15.0, and 20.5 MeV

The polarization of protons from the ⁹Be(d, p₁)¹⁰Be^x reaction has been analyzed in terms of the DWBA theory at deuteron energies 5.5, 15.0 and 20.5 MeV. Good fits were obtained which show the usefulness of the DWBA theory for describing polarization by using the optimum optical model parameters.     

Deuteron break-up in the field of the nucleus

The break-up of deuterons in Coulomb and nuclear fields is calculated in the framework of a distorted-wave Born approximation (DWBA). The matrix element in the post-interaction formulation is evaluated using a partial-wave decomposition. The results are in very good agreement with recent experiments, but only if nuclear forces are included. In addition, other theoretical models for the disintegration of deuterons are reviewed critically.     

Vector analysing power and cross section for 90Zr(d, p)91Zr at 11 and 12 MeV

Angular distributions of the vector analysing power and the cross section were measured for ⁹⁰Zr($\text{d}$, p)⁹¹Zr. Measurements were made on two transitions at a deuteron energy of 11 MeV and on 20 transitions at 12 MeV. The observed j-dependence of the vector analysing power provided unambiguous spin assignments for most final states. Measurements of the cross section and vector analysing power were also made for deuteron elastic scattering at 11 MeV in order to determine the potential parameters for DWBA calculations. The DWBA predictions are in good agreement with the measured (d, p) cross sections and in qualitative agreement with the analysing powers. Enriched targets.