Direct Reaction Theory Research Articles

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.

Plane-Wave Matrix Approach to Direct Reactions

A new method based on plane-wave representations is developed and applied to the theory of direct reactions. The coefficients of the plane-wave expansions of propagators are determined either by the requirement that they fit the scattering matrix for an auxiliary potential both on and off the momentum shell or by the use of an effective unit operator. Such expansions lead to the reduction of typical direct reaction problems to simple matrix algebra, the basic matrices being either plane-wave matrices or auxiliary potential matrices. Particular attention is given here to finite-range distorted-wave matrix elements and coupled-channels theory, where the possible advantages of the present approach can be understood even at a formal level.

Consideration on direct reaction mechanisms in the unified microscopic nuclear theory

In the framework of a unified microscopic nuclear theory the starting equations for the discussion of the different phenomenological direct reaction theories are derived. Subsequently in continuation of a previous investigation the6Li(p,3He)4He reaction as a realistic example is discussed to obtain a better physical insight into the different kinds of reaction mechanisms.

Elastic and Inelastic Scattering of Protons on 14N in the Energy Region of 10 to 15 MeV. I. Experimental

The experimental results of the 14 N( p , p ') 14 N reaction in the energy region of 10∼15 MeV were analyzed from both points of view of direct and compound-nucleus formation processes. Although the direct reaction theory reproduces the experimental results fairly well, it is not satisfactory in some aspects, especially in its poor reproduction of the rapid changes of the cross sections and angular distribution shapes with proton energy. On the other hand the compound nucleus formation theory with reasonable choices of spins and parities of the levels of the compound nucleus 15 O can reproduce well not only this energy dependence but also various aspects of the experimental results. Some theoretical considerations on the symmetry natures of the wave functions of the 14 N and 15 O states are described.

The reactions 14N(p, 3He) 12C, 14N(d, α) 12C, 13C(p, d) 12C and the 4 + level of 12C

In a study of the two-nucleon transfer reaction 14N(p, 3He) 12C at 50 MeV, strong transitions are observed to the ground state, 4.43, 12.71, 14.08, 15.11 and 16.11 MeV levels of 12C. The first four of these levels are also strongly excited in the 14N(d, α) 12C reaction, which is investigated at incident energies of 15 and 20 MeV. A detailed comparison of the data is made with the theory of direct reactions, using intermediate coupling, shell-model wave functions for 12C, and a finite-range non-local formulation of the DWBA. Good agreement between theory and experiment is obtained with realistic optical-model potentials for all reaction channels. The analysis identifies the 14.08 MeV level as the lowest 4 + state of 12C. This assignment is supported by a study of the single-nucleon transfer reaction 13C(p, d) 12C at 50 MeV in which a 4 + state cannot be reached by 1p shell neutron pick-up. In this reaction the 14.08 MeV level is not observed. The angular distributions for the observed levels at 0.0, 4.43, 12.71, 15.11 and 16.11 MeV are satisfactorily explained with a finite-range non-local DWBA theory and with spectroscopic factors derived from intermediate coupling shell-model wave functions.

Measurement of the 10B(n, α) reaction at En = 14.4 MeV leading to stable and unstable levels of 7Li

Spectra of alpha particles produced by neutron bombardment of an enriched 10B target have been measured. The angular distribution for two distinct alpha groups in the spectra belonging to the (n, α 0, 1) and (n, α 2) transitions are presented. The cross sections obtained are 8.2 ± 2.5 mb and 9.5 ± 2.8 mb, respectively. The latter has been measured in the angular range θ c.m. = 0°–90°. The angular distributions have been analysed in the framework of the direct reaction theory.

Recoil Damping in Heavy-Ion Transfer Reactions

An earlier suggestion by the authors that the unexpected features of angular distributions in heavy-ion transfer reactions at energies above the Coulomb barrier may be explained by including recoil and finite-range effects in a direct-reaction theory, is examined in detail. It is shown that the finite mass of the transferred particle may be taken into account approximately by the inclusion of a recoil phase factor in the transfer function of the usual distorted-wave Born amplitude. The implications of modifying the transfer function are worked out with the help of a sharp-cutoff diffraction model for the scattering of the strongly absorbing nuclear cores. Simple, closed expressions for the transfer differential cross sections are obtained. Unlike the earlier work, these expressions are valid for arbitrary angular momentum transfers, and intrinsic spins are included. When the zero-range limit is used or the mass of the transferred particle is neglected, the model predicts extreme diffraction oscillations in the angular distributions. However, if finite-range and recoil terms are retained, then, at sufficiently high energies and large angular momentum transfers, the theory gives strong damping of the diffraction oscillations. The resulting structureless angular distributions fall off with a $\frac{1}{{q}^{3}}$ dependence on the linear momentum transfer $q$, in excellent agreement with experiment. The theory is applied to the recent experimental results of Birnbaum, Overley, and Bromley for the ${\mathrm{C}}^{12}$(${\mathrm{N}}^{14}$, ${\mathrm{N}}^{13}$)${\mathrm{C}}^{13}$ reaction. Substantial damping of the angular distributions is predicted.

A study of the 6Li(p, n) 6Be reaction

Angular distributions for the ground state neutrons have been measured at 8.3, 9.0, 9.6, 10.0, 10.5, 15.0, 16.55 and 18.0 MeV using a time-of-flight spectrometer. The excitation and width of the first excitated state of 6Be has been measured to be 1.67±0.05 MeV and 1.18±0.07 MeV, respectively. The angular distributions of the neutrons from the 1.67 MeV state has also been measured. Below 11 MeV, the angular distributions are simple and indicate a compound nucleus process. Above 11 MeV, the results cannot easily be interpreted in terms of the compound nucleus, but on the other hand a direct-reaction theory has also failed to give agreement with experiment. Evidence has been found for a possible second excited state in 6Be at about 3.0 MeV.

The theory of direct nuclear knock-out reactions with polarized beams

A previous statement of the theory of direct nuclear knockout reactions is generalized to include reactions involving the exchange of two-fermion clusters and reactions involving polarized incident beams. The formalism is developed in terms of the distorted-wave Born approximation. Spin-dependent distortions and a finite-range direct interaction are included, but kinematical considerations are neglected assuming the infinitely heavy core for the target and residual nuclei. Coulomb effects, neglected in the previous paper, are taken into account in the present treatment. For cases of the incident beam being unpolarized and polarized, explicit expressions are obtained for the angular distribution, the total cross-section, and the polarization of the emitted particles.

Angular distributions of the recoil nuclei from the reactions 12C( 3He, α) 11C and 12C( 3He, d) 13N

Angular distributions of the 11C and 13N recoil nuclei from the reactions of 24.8 MeV 3He particles with 12C have been determined by assay of the radioactive nucleides after their collection in catcher foils. Data for laboratory angles from 7° to 170° are presented. Structure in the 11C angular distribution is readily apparent. A calculated 11C angular distribution, based upon the distorted-wave theory of direct reactions, can reproduce essentially all of the details of the experimental data. It is concluded from these data that the 12C( 3He, α) 11C reaction proceeds predominantly by direct pick-up. Contrastingly, the 13N recoil data, when compared with the data of Wegner and Hall for the 12C( 3He, d 0) 13N reaction which can be fitted with the results of a direct stripping calculation, indicate that the ( 3He, pn) reaction is a significant contributor (> 60 %) to the 12 C → 13 N cross section. Neither the shape nor the magnitude of the 13N angular distribution can be explained on the basis of ( 3He, d) stripping alone.

Effects of the DeuteronDState andJDependence in (d, p) and (p, d) Reactions

It is shown that the $D$ component of the deuteron internal wave function gives rise to corrections to the direct reaction theory of ($d, p$) and ($p, d$) reactions which depend strongly on the angular momentum of the transferred neutrons. For an orbital angular momentum transfer of 3 the corrections are large and give an important contribution to the $J$ dependence observed in the reactions ${\mathrm{Fe}}^{56}(p, d){\mathrm{Fe}}^{55}$ and ${\mathrm{Ni}}^{57}(p, d){\mathrm{Ni}}^{56}$.

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.

On the theory of direct reactions (II)

Distorted wave theories for a direct reaction amplitude M 12 connecting channels 1 and 2 are usually obtained from the formal theory of scattering by means of a long chain of approximations. Here we consider a version of distorted wave theory which involves just one basic assumption and which can be further approximated to yield all the currently used expressions. We show that this basic assumption, which underlies all the usual theories, leads plausibly to the relation Im M 12=M 12f 11 †+M 12 †f 22, where and f 11 are the elastic scattering amplitudes in the initial and final channels. The result is discussed by means of a simple model of nuclear reactions and also on the basis of a more general theory. The implications of this “truncated unitarity relation” were brought out in an earlier paper, where it was shown to give very stringent conditions for the validity of the distorted wave treatment.

Theory of direct transfer nuclear reactions

When in direct reactions a group of transferred nucleons in the projectile or in the residual nucleus is described by the shell model, the expansion of the shell-model wave functions into cluster wave functions leads to mixture of the orbital and total momenta of the transferred cluster. The cluster reduced widths for such reactions differ from those for resonance reactions, as the group may not be transferred in its ground state. Reduced widths, calculated by the plane wave theory of direct transfer reactions, permit a straightforward comparison with experimental data.

An interaction radius independent theory of resonance and direct nuclear reactions

A general theory of nuclear reactions is formulated which is valid for arbitrary short-range interactions between identical nucleons and which is independent of any assumptions about the existence of a nuclear radius. Assumptions are made, instead, about the analytic character of the reduced transition matrix. Both resonance and direct reactions, as well as the relation between them, follow directly from the formalism.

Direct reaction analysis of the [formula omitted] reaction

Thirty-nine angular distributions of the 15 N( p, α 0) 12 C reaction over a 9 MeV range were analysed using a plane wave direct reaction theory incorporating pick-up and exchange pick-up (or heavy-particle stripping) coherently mixed (PWBAE theory). The results of this analysis indicate that the reaction proceeds predominantly by a direct reaction mechanism.

A study of the reactions O 17(He 3, α0), O 18(He 3, α0) and O 18(He 3, α1)

The angular distribution of the alpha particles from each of the reactions O 17( He 3, α 0) O 16, O 17( He 3,α 0), O 18( He 3,α 0) and O 18( He 3,α 1) O 17∗ has been measured over an angular range of 0–160° at thirteen values of incident energy between 2.68 and 6.47 MeV. The experimental results and a comparison of these results with the theoretical prediction of a plane-wave direct-reaction theory with two modes, pick-up and heavy particle stripping, are presented.

Remarks on the theory of direct reactions

The effect of coupling to channels other than those representing scattering in the initial and final states is investigated. Necessary and sufficient conditions are obtained for such couplings to be small and the extent to which these conditions are valid is examined for (d, p), (t, p) and (d, α) reactions. It is concluded that only (d, p) reactions are likely to be well described by the usual DWBA theory which ignores these couplings. Plausible arguments are given to account for the empirical agreement between experiment and the cut off DWBA theory.

Direct reaction analysis of the F 19 (p, α0)O 16 reaction

Thirty-two angular distributions of the F 19(p, α 0)O 16 reaction over a 23 MeV range were analysed using a plane-wave direct reaction theory incorporating pick-up and exchange pick-up (or heavy-particle stripping) coherently mixed (PWBAE theory). The results of this analysis indicate that the reaction proceeds predominantly by a direct reaction mechanism, but that compound nucleus contributions cannot be ignored at certain energies. Only partial success in obtaining a fit to the extreme backward peak at 11 000 MeV proton bombarding energy, plus the fact that the poorest fits are obtained for other angular distributions with large backward peaking, indicate that other mechanisms than first-order exchange effects are possibly contributing to the backward peaking observed.

Excited states in 29P by the reaction 32S(p, α) 29P

The reaction 32S(p, α) 29P was investigated by use of 14.5 MeV protons. Alpha particles were analysed with a broad range magnetic spectrograph and were detected with nuclear plates. Seven alpha groups from the reaction 32S(p, α) 29P were found. These alphas to to the ground state and the excited states in 29P at 1385±20, 1959±20, 2410±20, 3108±20, 3446±20 and 4076±20 keV. A Q value of this reaction was measured as Q = 4171±20 keV. The angular distribution for each alpha group was measured and analysed in terms of direct reaction theory. A brief discussion is presented on the properties of the excited states in 29P and the (p, α) reaction.