Theory Of Direct Nuclear Reactions Research Articles

An Approximation of Cross Sections of Channels of the 9Be(α, n) Reaction on the Basis of the Gamow Interaction Radius

Radiation transport of α-particles requires cross sections of inelastic interaction of α-particles with nuclei. At low energies the inelastic interactions are limited to the 9Be(α, n) reaction, whose channel cross sections are assessed only in the specialized evaluated nuclear data library JENDL/AN-2005. The new TPT (Toolkit for Particle Transport) approximation of the 9Be(α, n) reaction cross sections for different channels takes into account the lowest resonance and, at high energies, relies on the exponential decrease in the S-factor, which is a consequence of the theory of direct nuclear reactions.

TheJ-dependent distorting optical-model potential in heavy-ion transfer reactions

The theory of direct nuclear reactions is considered. Different heavy-ion reactions with different particle transfer are calculated in the framework of the exact finite-range distorted-wave Born-approximation formalism, in which the distorting potentials depend explicitly on the total angular momentumJ of the system. The bound states of the transferred particle with the core nucleus forming the projectile or target nucleus in the initial channel and forming the residual nucleus in the final channel are represented by double-Gaussian potentials. Theoretical expressions are evaluated for the form factors and the differential cross-sections. Numerical calculations are carried out for the angular distributions for different heavy-ion reactions with particle transfer at different energies. The results of the present calculations of the differential cross-sections are in good agreement with the experimental measurements, which investigate the important contributions of theJ-dependent distorting potentials in reproducing the oscillatory structure in the shapes of the angular distributions. The obtained values of the extracted spectroscopic factors are reasonable.

Direct reaction mechanism for heavy ion reactions with particle transfer

The theory of direct nuclear reactions is considered. Heavy ion reactions with particle transfer are treated using a direct reaction mechanism. The exact finite-range distorted wave Born approximation (EFR-DWBA) is used in investigating these direct heavy ion reactions. The particle-nucleus interactions for the bound states in the target and residual nuclei and represented by Yukawa plus exponential potentials. Theoretical expressions are evaluated for the differential cross sections of the particle transfer heavy ion reaction process. Numerical calculations are carried out for the form factors and angular distributions for different heavy ion reactions with different particle transfer. The present theoretically calculated differential cross sections are in good agreement with the experimental measurements. The obtained values of the extracted spectroscopic factors are reasonable.

Bare Potential DWBA for (d, p) Reactions

Applicability of potential (BPDWBA), recently applied for heavy ion inelastic reactions with success, to light ion induced transfer reactions is investigated. The reaction 160(d, p) 17 0 and 4°Ca(d, p) 41 Ca at Ed=10.5MeV are studied, for which coupled channel (CC) calculations exist. Search is made of the potential which fit the CC results of (d, d) and (p, p) scatterings. The potentials and/ or the potentials (the distorting potentials in the CC equations) are used in DWBA calculations of (d, p) cross sections. Results of the various types of DWBA distinguished by the choice of the distorting potentials are compared with the CC results. It is found that none of such DWBA's reproduces the CC cross sections. Physical implications of these types of DWBA are discussed in detail. Justification of DWBA has been a long standing problem in the theory of direct nuclear reactions. In a pioneering work on the subject, Perey and Satchlerl) analyzed a (p, p') process with one-phonon excitation. These authors compared DWBA with the method of coupled channels (CC) and concluded that first order perturbation theory in the channel coupling potential was unwarranted when the coupling was strong, but DWBA was still valid even in such a case if the optical potentials, UoPt, were used as the distorting potentials. They argued that the higher order effect of the coupling was taken into account effectively by the Uopt since they correctly described the elastic scattering in the initial and final chan­ nels. Hereafter we shall call this type of DWBA DWBA. Later, however, Satchler 2 ) questioned the theoretical basis of conventional DWBA. As a theoretically more transparent alternative to it he introduced asymmetric DWBA (ADWA) in which the distorting potential was the optical potential for the entrance channel while for the exit channel it was the bare potential, the distorting potential, U, of the CC calculation. The two methods were compared with CC for the case of 56Fe(p, p') with a 2+-phonon excitation. It was found that ADWA agreed with CC better, though only slightly, than conventional DWBA. Also other cases of inelastic scattering were mentioned for which ADW A gave better agreement with experimental data. DWBA with the potential both for the entrance and exit channels was also tested for 56Fe(P, p') and found to be inadequate.

The orthogonal projection method in scattering theory

This work gives a detailed account of the orthogonal projection method in the theory of two- and three-body scattering, which is based on the employment of orthogonal projecting pseudopotentials. The method is applied to a number of physical problems, of which the following the most important: the improvement of convergence and the rearragement of Born series to make them convergent at low energies in the presence of bound states in a system, as well as the consideration of the Pauli exclusion principle in the scattering of composite particles and in the integral theory of direct nuclear reactions. The properties of eigenvalues of kernels of the equations obtained are investigated and the conditions for the convergence of their iterations are derived. For the three-body problem, the general case of three different particles is considered, as well as two particular cases, namely, two particles in the field of a heavy core and three identical particles. The proven theorems are illustrated by numerical examples. Other useful applications of the proposed approach are listed and discussed, in particular, those in solid-state physics and in the theory of electromagnetic transitions. The approach suggested is compared with those of the other authors and the prospects of using the developed formalism are discussed.

Outline of the development of the theory of direct nuclear reactions

A brief review is given of the development and current status of the theoretical descriptions of direct nuclear reactions.

Multi-Interaction, Finite-Range, Two-Mode, DWBA Analyses of Direct Nuclear Reactions

The finite-range distorted-wave Born approximation (DWBA) theory of direct nuclear reactions is modified to facilitate inclusion of both direct and exchange modes of transfer reactions and all effective interaction potentials for single-cluster transfers. While spin-orbit terms are neglected many other of the usual approximations are not made. Application of the formalism using the computer code FANLU2 is demonstrated in several examples designed to demonstrate useful features of such analyses. Contributions from all reaction mechanisms and both modes are shown to be necessary for an adequate description of the reaction $d(\ensuremath{\alpha},t)^{3}\mathrm{He}$ at 82 Mev with no parameter adjustment. The reaction $^{9}\mathrm{Be}(p,\ensuremath{\alpha})^{6}\mathrm{Li}$ at 6 MeV illustrates angular momentum dependence of the reduced widths in a more complex reaction. Finite-range effects are demonstrated in a simplified analysis of the $^{6}\mathrm{Li}(^{3}\mathrm{He},p)^{8}\mathrm{Be}$ reaction at 17 MeV. Various aspects of such analyses with the code FANLU2 are discussed.

DWBA analysis of the reaction 12C(α, n) 15O

Angular distributions of neutrons from the reaction 12C(α, n) 15O(g.s.) have been measured at lab energies from 18.4 to 23.1 MeV and angles ranging from 0° to 130°, using a time-of-flight technique. The experimental curves generally show a forward peaking and a strong dependence on the incident energy. The data were compared with the angular distributions predicted by the distorted-wave theory of direct Nuclear reactions, and no agreement could be obtained when only a stripping mechanism was taken into account.

Study of the spectra of fast deuterons produced in the interaction of 670 MeV protons with nuclei

The spectra of deuterons emitted at angles of 6.5°, 9.5°, 13.5° and 16° (lab system) in the interaction of protons with the nuclei 1H, 2H, 6Li, 7Li, C, Al, Cu, Rh and Pb have been measured in the momentum range 700–1700 MeV/ c. The secondary particles were analyzed with the help of magnetic and time-of-flight spectrometers. The differential cross sections for deuteron production in the reactions p + 〈2N〉 → N + d and p + 〈N〉 → d + π have been measured for selected nuclei and angles. The effective numbers of nucleons and of two-nucleon clusters for the investigated nuclei were determined. The results obtained are discussed in terms of the dispersion theory of direct nuclear reactions.

Stripping Distortions inB11(d,pγ)B120.95Angular Correlations

$p\ensuremath{-}\ensuremath{\gamma}$ angular correlations were measured in the ${\mathrm{B}}^{11}{{(d,p\ensuremath{\gamma})}^{12*}}_{0.95}$ reaction in order to determine how its distortions depend on deuteron energy and proton angle. Various proton angles were used, with ${E}_{d}=1.0,1.8,3.0,\mathrm{and} 5.5$ MeV. These parameters were chosen to systematically test predictions based on an application of the dispersion theory of direct nuclear reactions. The observed failure of the correlation anisotropy to increase as the stripping denominator $D$ is reduced is most easily interpreted as a negative result for the application of this concept to the ${\mathrm{B}}^{11}(d,p){{\mathrm{B}}^{12}}_{0.95}$ reaction. Trends in the behavior of the data are discussed. In addition, spectroscopic information about the captured neutron is deduced, and the excited state of ${\mathrm{B}}^{12}$ is compared with shell-model theory.

The distorted-waves theory of direct nuclear reactions with spin-orbit effects

A general and flexible formulation is given of the distorted-waves theory of direct nuclear reactions which takes account of the effects of including spin-orbit coupling in the distortion, and which does not make the zero-range approximation. The evaluation of the transition amplitude is shown to involve two distinct steps, one the construction of a form factor which characterizes the particular reaction mode and the particular nuclei, and the other the treatment of the dynamics of the distorted waves, which is similar for all reactions. Expressions are given for various differential cross-sections and polarizations. As examples, procedures for treating inelastic scattering and stripping and knock-on reactions are described. The symmetry properties of the reaction amplitudes are also discussed.

Inelastic effects in stripping and other direct reactions

The distorted-wave theory of direct nuclear reactions, such as deuteron stripping, is generalized to include in the distorted waves the effects of the strong inelastic scattering which may be present when the nuclei show collective behaviour. The possible changes in parity and angular momentum selection rules are discussed in detail. Formulae are given for the general case, and also for a simple model in which only one level of the target nucleus (assumed even) can be excited.

On the dispersion theory of direct nuclear reactions

The Feynman graphs with singularities closest to the physical region of variables are assumed to be responsible for the main contribution to the amplitude of a direct nuclear process. Using the dispersion relations with respect to energy a singular integral equation is obtained for calculating interactions in the initial and final states. This equation admits an accurate solution and simple iteration, the first-order iteration corresponding to the method of distorted waves. It is found that a substantial contribution to the direct process mechanism may come not only from pole graphs corresponding to the Butler mechanism, exchange stripping and heavy pick-up reactions, but from more complex graphs as well. The reactions Be 9(d, n)B 10, Be 9(α, t)B 10 and C 12(d, p)C 13 are considered to illustrate this point. Some reactions of the type (x, yz) and in particular, the “knock out” of nuclear clusters, are analyzed from the same angle.

On the dispersion theory of direct nuclear reactions

The Feynman graphs with singularities closest to the physical region of variables are assumed to be responsible for the main contribution to the amplitude of a direct nuclear process. Using the dispersion relations with respect to energy a singular integral equation is obtained for calculating interactions in the initial and final states. This equation admits an accurate solution and simple iteration, the first-order iteration corresponding to the method of distorted waves. It is found that a substantial contribution to the direct process mechanism may come not only from pole graphs corresponding to the Butler mechanism, exchange stripping and heavy pick-up reactions, but from more complex graphs as well. The reactions Be 9(d, n)B 10, Be 9(α, t)B 10 and C 12(d, p)C 18 are considered to illustrate this point. Some reactions of the type (x, yz) and in particular, the “knock out” of nuclear clusters, are analyzed from the same angle.

Symmetry properties of the distorted wave theory of direct nuclear reactions

It is shown that under certain conditions often nearly realized experimentally the transition amplitudes in the distorted wave Born approximation theory of direct nuclear reactions have a simple symmetry under interchange of the distorted waves for initial and final states. For observations on the reaction the symmetry is equivalent to rotating coordinate axes by π about the recoil direction. This property is used to show in detail how the initial and final distortions have opposing effects for some phenomena (such as polarization of emitted particles or change of symmetry axis for the distribution of subsequent γ-rays) but are cooperative for others (such as the differential cross-section). Some conclusions are drawn about distortion effects in various types of reaction.