Multistep Direct Reactions Research Articles

Monte Carlo evaluation of the semiclassical multi-step direct reaction series

We show how the multi-step direct reaction series can be evaluated using Monte Carlo methods. Nucleon-nucleon collisions occur according to the random selection of a nucleon’s attenuation factor along its classical trajectory. The particles and hole excited in a collision are selected from a local Fermi distribution. We assume that the particles continue to propagate and possibly collide again before leaving the nucleus. We assume that holes collide in place, to possibly produce other particles and holes.

Nucleon-induced inelastic scattering with statistical strength functions and the ECIS direct reaction code

Modern theoretical descriptions of inelastic scattering make use of multi-step direct reaction approaches together with transition potentials obtained from sophisticated nuclear structure models. Here we demonstrate how the complexity of such calculations can be reduced to permit simpler ones, also using the ECIS code, but providing an almost equally precise alternative to a much more detailed calculation. We have studied the transition form factors within the random phase approximation (RPA), where these are obtained as linear combinations of particle–hole states. At moderate to high excitation energies, where interference effects tend to disappear, we have proposed an independent particle–hole formalism in which particle–hole states are spread in energy with an appropriate strength function obtained from the RPA. The effects of more complex modes, such as 2p–2h ones, are simulated with widths calculated in a semi-classical context. Here, we verify the validity of our approximations for pre-equilibrium proton-induced reactions on $$^{90}$$ Zr target. Our calculations provide a good description of the reaction data and point toward a simplification of the description of nucleon-induced reactions based on averages of microscopic details of the projectile–target interaction.

Collective Reactions in the Continuum

We show that pre-equlibrium inelastic scattering reactions to the continuum contain substantial collective components in addition to the multistep direct and multistep compound reactions. These collective reactions are investigated for the vibrational nuclei 56Fe, 58Ni, 90Zr, 93Nb, 208Pb and 209Bi , and the strongly-deformed, rotational W nucleus. The collective cross-sections are calculated using the experimental data for low-lying collective excitations supplemented where necessary by the giant multipole resonances evaluated using the energy-weight ed sum rule. The MSC and MSD cross-sections are evaluated by the Feshbach-Kerman-Koonin theory using a consistent set of parameters determined by analyses of (p, xn) reactions, that have practically no collective components. The results are compared with high-resolution neutron inelastic scattering data and prove able to account for the absolute magnitude of the cross-sections and also their detailed structure.

The pre-equilibrium and equilibrium double differential cross sections for the nucleons and light nuclei induce nuclear reactions on 27Al nuclei

The pre - equilibrium and equilibrium double differential crosssections are calculated at different energies using Kalbach Systematicapproach in terms of Exciton model with Feshbach, Kerman andKoonin (FKK) statistical theory. The angular distribution of nucleonsand light nuclei on 27Al target nuclei, at emission energy in the centerof mass system, are considered, using the Multistep Compound(MSC) and Multistep Direct (MSD) reactions. The two-componentexciton model with different corrections have been implemented incalculating the particle-hole state density towards calculating thetransition rates of the possible reactions and follow up the calculationthe differential cross-sections, that include MSC and MSD models.The finite well depth, isospin, shell effects, Pauli effect, chargeeffect, pairing, surface, angular and linear momentum distributionscorrections are considered in this work. The nucleons (n and p) andlight nuclei (2D and 3T) have been employed as projectiles at thetarget 27Al nuclei and at different incident energies (4MeV, 14 MeVand 14.8MeV). The results have been compared with the availableexperimental and theoretical published work. The comparisons showan acceptable agreement with the TALAYS code (Tendel 2014) forthe reactions: 27Al (n, n) 27Al, 27Al (p, n) 63Zn, 27Al (p, D) 62Cu, 27Al(p, p) 63Cu and 27Al (p, 4He)60Ni and at different emission energiesand angles.

Quasiparticle nature of excited states in random-phase approximation

Quantum models of nuclear reactions require an effective way of taking into account the complex coupling of the components of different excited states as well as the nuclear residual interactions of the composite system. This is usually achieved with the use of distributions giving nonuniform weights for each of the modes involved. The response function of the quantum model permits the inclusion of the residual interaction and thus a unified description of collective and single-particle excitations. We analyze the particle-hole nature of the RPA modes of the response function in the context of multistep direct (MSD) nuclear reactions. The energy distribution of the particle-hole states that contribute to the response function is studied. Although many states make small contributions to the low-energy collective states, we show that the energy distribution of the higher-energy states is concentrated around the energies of the noninteracting components. The spreading energy of the strength function is determined for different nuclei, and a general fit accounting for both the collective and noncollective parts of the spectra is proposed. In addition we also test the randomness assumption commonly applied in models of MSD reactions.

Evaluations of Reactions Cross Section of Radionuclide by Particles Induced Nuclear Reactions using Exifon Code

In this work the reaction cross section for the production radionuclide of and isotopesfrom bismuth-208 were calculated using EXIFON code in the energy range from 0 MeV to 30 MeV. The code is based on an analytical model for statistical multistep direct and multistep compound reactions (SMD/SMC model).The results obtained shows that production ofTlis not observed at this energy range because the interaction cross section is zero, and also the excitation functions for the production of Tl-205 and Tl-204 were obtained.

Evaluation of Excitation Functions of Reactions Used in Production of Some Medical Radioisotopes

In this work, reaction cross-sections were calculated and Excitation Functions were evaluated for productions of 208Bi, 212,211,210At, 211,210Po isotopes using EXIFON code in the energy range from 0 MeV to 30 MeV. The code is based on an analytical model for statistical multistep direct and multistep compound reactions (SMD/SMC model). This work also investigates the shell structure effect on the reaction cross-section, the results obtained show that the cross-sections of (a, na) reaction for both with shell correction and without shell correction are zeros at energies range considered, this shows that the energy of the incident particle is below the threshold of this reaction due to the present of coulomb repulsive force between the projectile and target nucleus.

Calculation of Reactions Cross Section for Neutron-Induced Reactions on <sup>127</sup>I Isotope

In this work, the reaction cross-section for neutron-induced reactions on 127I isotope was calculated using EXIFON code in the energy range of incident particle from 0 MeV to 30 MeV. The code is based on an analytical model for statistical multistep direct and multistep compound reactions (SMD/SMC model). In order to see the effect of nuclear structure on cross sections, the calculation was done using nuclear shell structure effect and without considering shell structure effect. Obtained results and statistical analysis showed that shell structure effect does not give significant changes to the cross-section at considered energy ranges. This shows that EXIFON code is a good tool for investigation of nuclear reaction cross section and is useful in the production of the radioisotopes of Iodine, Antimony and tellurium of high purity and in an efficient manner using cyclotron or nuclear reactors, these isotopes have potential application for field of medical science especially for diagnostics and therapeutic purposes.

Asymmetries of various P- and T-parities in angular distributions of products of cold-polarized-neutron-induced binary and ternary fission of oriented nuclei and T-invariance

It is shown that a quantum system whose Hamiltonian is independent of time is T -invariant if this Hamiltonian contains only those terms that do not change sign upon time reversal. It is also shown that the coincidence of the amplitudes for multistep direct and statistical nuclear reactions with the timereversed amplitudes for the reactions being studied is a condition that ensures the T -invariance of the amplitudes in question, the transition from the original amplitudes to their time-reversed counterparts being accomplished, first, upon introducing the inverse-reactionmatrices T instead of the original-reaction matrix T and, second, upon replacing the wave functions for the initial, final, and intermediate states of the system by the respective time-reversed functions. It is found that the T -even (T -odd) asymmetries in cross sections for nuclear reactions stem from the interference between the amplitudes characterizing these reactions and having identical (opposite) T -parities. It is shown that the T -invariance condition for the above T -even (T -odd) asymmetries is related to the conservation of (change in) the sign of these asymmetries upon going over from original to inverse nuclear reactions. Mechanisms underlying the appearance of possible T -even and T-odd asymmetries in the cross sections for the cold-polarizedneutron- induced binary and ternary fission of oriented target nuclei are analyzed for the case of employing T -invariant Hamiltonians for the systems under study. It is also shown that the asymmetries in question satisfy the T -invariance condition if the reactions being considered have a sequential multistep statistical character. It is concluded that T -invariance is violated in the limiting case where, in ternary nuclear fission, the emission of a light third particle froma fissile compound nucleus formed upon incident-neutron capture by a target nucleus and its separation to two fission fragments are simultaneous events.

59Co($\vec p$,α) reaction at 100 MeV incident energy – statistical multistep direct reaction into the continuum of outgoing energies

The pre-equilibrium proton induced emission of light complex nuclei with energies in the continuum has been studied comprehensively for many years. Double–differential cross sections and especially analyzing power distributions are typical of an intranuclear nucleon-nucleon multistep statistical reaction mechanism. The final stage of the reaction may be a result of a direct pickup or knockout of the ejectile. The discussion on this subject continues to be a hot topic for both theoretical and experimental investigations. In this paper the results from the latest studies of the inclusive (, α) reactions on 59 Co to the continuum will be reported. The formalism based on the statistical multistep direct emission formulation of Feshbach, Kerman and Koonin is found to give a reasonably good reproduction of cross section and analyzing power distributions at various emission energies. Special attention will be paid to the details of the calculations.

Investigation of the Shell Effect on Neutron Induced Cross Section of Actinides

Investigations has been made concerning the development of the effect of shell structure on neutron induced cross section, evaluations for Actinides elements in the energy range of 0-30 MeV. The EXIFON code, which is based on the analytical model for statistical multistep direct and multistep compound reactions, was used for the calculation of the cross section for (n-2n), (n-p) and (n-α) reaction channels. Results are compared with data from the experimental database, EXFOR from the IAEA nuclear data bank to deduce the shell effect. An empirical relation for the reaction cross section has been established for magic numbers nucleus in the (n-2n) reaction channel in the energy range. Results shows that the odd-even effect has also been observed as the cross section for odd-even nuclei are higher than their neighbouring even-even nuclei and with comparison in term of Shell correction, the cross section is higher when the Shell correction is not considered.

A multistep direct reaction approach for neutron-induced reactions at intermediate energy

The Tamura–Udagawa–Lenske (TUL) formulation of the multistep direct reaction (MSDR) approach to the preequilibrium processes is reviewed. It takes into account consistently the coherent and the statistical aspects of such processes. We present calculations of neutron-induced charge exchange reactions at incident energies of 30–100 MeV, an energy region of great interest in applications to accelerator-driven systems.

Cross Sections for Multistep Direct Reactions

Inelastic scattering and charge-exchange reactions have been analysed at energies ranging from 14 to 27 MeV using the modified multistep direct reaction theory (MSD) of Feshbach, Kerman and Koonin.1) The modified theory considers the non-DWBA matrix elements in the MSD cross section formulae and includes both incoherent particle- hole excitations and coherent collective excitations in the continuum, according to the prescription of. 2,3) The results show important contributions from multistep processes at all energies considered.

NWY Model Calculations for Nucleon-Induced Multistep Direct Reactions

Calculations of the preequilibrium process are performed with the theory of Nishioka, Weidenmüller, and Yoshida (NWY), which is a quantum mechanical approach to the preequilibrium nuclear reactions. For a one-step process, cross sections given by the NWY theory are sum of DWBA cross sections which excite various lp-lh states, and those are averaged over the residual state weighted by the final states density. A true level density which is based on a random matrix model is used for this. A comparison of the calculated NWY one-step process with the experimental data is shown. For a two-step process, the NWY theory adopts a “sudden approximation.” An example of the two-step cross section calculation is made for 208Pb(p, p’) reaction.

Multistep direct reactions at low energies

The cross sections for neutron inelastic scattering by niobium at 20 MeV and for the charge-exchange (p,n) reaction on copper at 27 MeV have been calculated using the multistep direct reaction theory of Feshbach, Kerman and Koonin [1]. The theory was modified to include the non-DWBA matrix elements and the collective vibrations. The results show enhanced contributions from two-, three- and four-step direct reactions in agreement with experiment.

Multistep processes in charge-exchange reactions

Cross sections for the charge-exchange 65Cu(p, n) 65Zn reaction at the incident energy of 27 MeV and the 100Mo(p, n) 100Tc reaction at the incident energy of 26 MeV have been calculated using the multistep direct reaction theory of Feshbach, Kerman and Koonin. The theory was modified to include the non-DWBA matrix elements and the isovector collective vibrations according to the prescription of Marcinkowski and Mariański. The results show enhanced contributions from two-, three- and four-step direct reactions in agreement with experiment.

Multistep direct reactions at low energies

It is shown that in order to obtain realistic cross sections for the multistep direct reactions, that fit experiment and are in accord with the gradual absorption of incoming flux, the enhanced non-DWBA matrix elements involving the biorthogonally conjugated wave functions have to be used.

Nucleon scattering to the continuum in terms of the two-fermion theory of multistep direct reactions

Cross sections of the (p, xp) reaction on 93Nb were measured at an incident energy of 26.5 MeV and analysed together with the existing data for nucleon scattering between 18 and 26 MeV. The (p,p′) and (n,n′) emission spectra and angular distributions have been described consistently in the framework of the two-fermion multistep direct theory of Feshbach, Kerman and Koonin. We have confirmed the predictions following our earlier analyses that the strength of the effective nucleon-nucleon interaction does not depend on the reaction channel provided that all pre-equilibrium reactions are included. The latter requires allowing for the giant resonances in continuum from the energy-weighted sum rules. The method applied avoids double-counting of collective and onestep direct excitations.

Average strength of the effective interaction in multistep direct reactions

The effective NN-interaction strength for the first NN collision in multistep reactions, averaged along the trajectory of the incident nucleon with respect to both the nuclear density and the first NN-collision probability, is obtained. Good agreement between the calculated values of this quantity and its phenomenological values is found. The account for the finite-range corrections to the involved local density approximation has a small effect on the average strength of the NN interaction. It is concluded that the nuclear-density dependence of the effective NN interaction may account for the phenomenological V{sub 0} values which have been underestimated at low energies when the trend of the optical-model potential only has been followed. {copyright} {ital 1997} {ital The American Physical Society}

Microscopic two-component multistep direct theory for continuum nuclear reactions

We present a multistep direct reaction theory for analyzing nucleon-induced reactions to the continuum for incident energies up to 200 MeV. Two principal advances in multistep direct theory are studied. (1) A microscopical approach is given for calculating distorted-wave Born approximation (DWBA) transitions to the continuum, where transitions to all accessible 1$p$1$h$ shell model states are explicitly determined. These states, obtained from a simple noninteracting Nilsson model, are assumed to be spread according to Gaussian distributions. In this approach, therefore, state densities are not used. We also provide a link with more conventional methods that utilize particle-hole state densities, and present a more accurate technique for sampling the DWBA strength. (2) A two-component formulation of multistep direct reactions is given, where neutron and proton excitations are explicitly accounted for in the evolution of the reaction, for all orders of scattering. We show that the attractive convolution structure for multistep processes persists within a two-component formalism, and conveniently automatically generates the many reaction pathways that can occur in the Feshbach-Kerman-Koonin expansion of the multistep cross section when neutron and proton excitations are followed. This formalism is particularly important for the simultaneous analyses of neutron and proton emission spectra. The multistep direct theory is applied, along with theories for multistep compound, compound, and collective reactions, to analyze experimental emission spectra for a range of targets and energies. Particular attention is paid to a complete and comprehensive analysis of all important decay channels and reaction mechanisms. We show that the theory correctly accounts for measured neutron and proton emission angle-integrated spectra, as well as angular distributions. Additionally, we note that these microscopic and two-component developments facilitate more fundamental studies into effective nucleon-nucleon interactions in multistep calculations.