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.     

Three-body formalism for deuteron stripping reactions

A three-body formalism for deuteron stripping reactions has been developed. The equations of Altet al (1967) (AGS) for the three particle system (target A, n, p) are reduced to a set of coupled one-dimensional integral equations with the use of (i) angular momentum basis for representation and (ii) separable approximation for the two bodyt-matrices (which delineate the interactions between the particle pairs). The on-shell solutions of this set of integral equations are then related to the cross sections of the rearrangement processes. The inputs in this calculation, viz., the separable interactions between the particle pairs in the respective channels are simply constructed from the respective two body bound state in accordance with the bound state approximation (BSA) conforming to the ‘unitarity’ requirement. Using this formalism preliminary calculations for the (d, p) and (d, n) reaction cross sections on¹⁶O have been carried out and they seem to have considerable semblance with the observed cross sections.     

Transparency to vector polarization transfer in deuteron stripping reactions

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 nucleon polarization independent of scattering angle and deuteron energy. With $\text{s =}\text{3}\text{2}$ contributions there is still considerable transparency at the stripping main peak in DWBA calculations.     

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.     

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.     

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.     

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.     

Theory of deuteron stripping on doubly even nuclei

Using the method of quantum Green functions we derive a rigorous formula for the deuteron stripping on a doubly even nucleus. Starting from this formula we discuss the distorted-wave Born approximation, some additional assumptions usually made in numerical calculations and the high-energy case. In addition to this we show that the formfactor can be calculated using a model for the nucleon self energy.     

Nucleon and deuteron induced spallation reactions

The intranuclear cascade (INC)+evaporation model is used to display the gross features of proton, neutron and deuteron induced spallation reactions between 200 MeV and 2 GeV incident energy per nucleon. Particular attention is focused on average particle multiplicities and their dependencies upon target mass number and incident energy. Simple formulae are derived to allow an easy use of these results. They are expected to provide a rapid survey of the gross features of spallation reactions. The decomposition of particle multiplicities and of cross-sections into cascade and evaporation components is performed and discussed. The impact parameter dependence is investigated. The excitation energy left in the target at the end of the cascade step is also studied, as well as the gross properties of the subsequent evolution of the target remnant. It is shown that neutron differential cross-sections can be split into components with reasonably identified physical origin, which exhibit simple power laws as functions of the target mass number.     

Electron-induced reactions on the deuteron

We report on theoretical investigations of elastic and inelastic scattering of electrons on deuterons. Special emphasis is laid on the role of the underlying nucleon-nucleon interaction. In particular, the role of the short range behavior of the interaction, of the strength of the tensor force, and of the pion-nucleon coupling is examined with regard to electron-deuteron scattering observables.Lectures held at the Indian Summer School on Electromagnetic and Weak Interactions of Particles with Nuclei, Sázava, Czechoslovakia, 6–11 September 1992Most of this work was done in collaboration with colleagues from the universities of Adelaide, Florence, Graz, and Hannover, and I want to thank my coworkers J. Adam, I. Allmer, J. Haidenbauer, H. Henning, B. Mosconi, J. Pauschenwein, W. Plessas, P. Ricci, and K. Schwarz for the fruitful cooperation.     

The reduction of deuteron stripping cross sections by a change of deformation

The effect of a change in deformation on stripping cross sections is discussed nearN=88–90, where the nuclear deformations strongly depend on the neutron number. For a series of initial and final deformations a reduction factor is calculated by which the stripping cross section as given bySatchler and others has to be multiplied. For instance, differences in deformations ofΔσ=0.10 which could be expected in this region should give a reduction factor of about 0.08.     

Does deuteron stripping in nuclear collisions test the wave function?

In a simple, parameter-free model for deuteron stripping we calculate the inclusive proton spectrum in the deuteron fragmentation region. The spectrum does not reflect in any simple manner the behavior of the deuteron wave function. We find that in the deuteron rest frame the transverse distribution is substantially wider than the longitudinal one. The physical reasons for this result are discussed. We obtain good agreement with existing data and make predictions for expected ones.     

9.38 GeV/c deuteron stripping on photoemulsion nuclei

The interaction of relativistic deuterons with photoemulsion nuclei has been investigated. The photoemulsion method makes it possible to precisely measure small angles between relativistic particles in reactions and to select unambiguously the events of inelastic deuteron absorption. Proton stripping in the interaction of 9.38 GeV/c deuterons has been investigated.     

On three particle stripping reactions

The formalism for the coupled-channel analysis of stripping reactions of complex nuclear projectiles by nuclei is presented. The general adiabatic approximation is developed. The indirect transitions that are considered are those which arise via intermediate rotational excitations of the target and product nuclei, i.e. the adiabatic approximation for rotational bands is used. At the same time the generalized DWBA procedure is considered for the intrinsic states. Also the antisymmetrization problem is solved. A new method for the calculation of the matrix elements of the (α,n) stripping reaction with finite range effective forces is developed. The method is based on an expansion of the radial functions which describe the relative motion in terms of harmonic oscillator wave functions multiplied by a Gauss one. Effective forces of Gauss type are assumed between the outgoing neutron and each captured nucleon     

Reaction rates of proton and deuteron induced reactions on lithium

Reaction rates are calculated for recently available low-energy data on⁷Li(p,α)⁴He,⁶Li(p,³He)⁴He and⁶Li(d,α)⁴He and compared to literature values. While the new rates are considered to be more accurate, their absolute magnitude agrees within about 10% with the most recent compilation.     

Three-body model for stripping nuclear reactions

A three-body model for the deuteron stripping nuclear reactions is presented. A set of three integral equations is obtained for the wave functions of the three-body problem by introducing a decomposition into angular momentum states into the Lippmann-Schwinger equation. Simple two-particle interactions with separable potentials are used. These separable potentials reduce the three-body problem to the solution of coupled sets of one-dimensional Fredholm integral equations. The angular distributions for²⁸Si(d,p)²⁹Si and⁴⁰Ca(d, p)⁴¹Ca stripping reactions are calculated. From the extracted spectroscopic factors, good agreement with the experimental measurements is obtained.     

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.     

Analyzing power in two-neutron stripping reactions

Analyzing powers have been measured for six states from the ${}^{208}\text{Pb(}\text{t}\text{, p)}{}^{210}\text{Pb}$ reaction and seven states from the ${}^{90}\text{Zr(}\text{t}\text{, p)}{}^{92}\text{Zr}$ reaction. These studies were performed with a beam from a new polarized triton source, and the reaction protons were detected in a Q3D spectrometer. The analyzing powers and differential cross sections are adequately described by DWBA calculations for both the medium and heavy targets up through L-transfers of 4. Above this value, some disagreement is seen.