Statistical multistep compound reactions — with renormalized exciton state densities

The master equation which takes into account angular momentum considerations is used to obtain the emission spectra for statistical multistep compound reactions. Realistic wave functions are used to evaluate the radial integrals appearing in the escape and spreading widths. The renormalized particle hole state density of Akkermans et al. is used in the calculations. The improvements over the one component formulation are discussed.     

Particle decay of resonant state generated by direct reaction

A framework is presented for studying two step reactions illustrated byA+a→B ^*+b, B^*→C+c. In the first step, a direct reaction excites a resonant state (e.g., giant resonances or isobaric analog resonances), which decays by a particle emission in the second step. Feshbach's projection operator method is used to obtain a reaction amplitude, which consists of a compound part and a direct part. The latter corresponds to a knock-out process when the direct reaction is inelastic scattering. A compound part includes contributions from the direct decay of doorway states as well as the multistep decay. The statistical theory of precompound reactions is applied to this multistep decay and an explicit expression is given for the cross section.     

The Angular Distribution of Light Particle Projectile Based on a Semi-Classical Model

A semi-classical model of multi-step direct and compound nuclear reactions is proposed to describe the angular distributions of the light particle projectiles from reaction processes induced by a nucleon with energies of several tens of MeV. The exact closed solution to the time-dependent master equation of the exciton model is applied. Based on the Fermi gas model, the scattering kernal between two-nucleon collieion includes the influences of the Fermi motion and the Pauli exclusion principle, which give the significant improvement to rise of the backward distributions. The energyangular correlation for the first few steps of the collision process (muli-step direct process) yields the further improvement of the angular distribution. The pick-up mechanism is employed to describe the composite particle emission. Thh reasonable physical picture reproduces the experimental data of the energy spectra of the composite particles satisfactorily. The angular distribution of the emitted composite particle is determined by an angular factor in terms of the momentum conservation of the nucleons forming the composite cluster. The generalized master equation is employed for the multi-step compound process. Thus a classical approach has been establised to calculate the double differential cross sections for all kinds of particles emitted in multi-step nucler reaction processes.     

Radiative proton capture to discrete states in 31N at 98 and 176 MeV

A high resolution magnetic pair spectrometer for medium energy photons has been used for studies of the exclusive (p,γ) reaction on ¹²C at 98 and 176 MeV. We present angular distributions of the differential cross section for transitions to the ground state and the first three excited states of ¹³N. The distributions from these four states are compared with predictions from a microscopic continuum shell model calculation using a realistic finite-ranged effective interaction with tensor components. In general, reasonable agreement is obtained for reactions to states described by p shell single particle wave functions, but poor agreement is obtained for transitions described by sd shell single particle wave functions. Nevertheless we find, however, that the dominant reaction mechanism is a direct capture process. We also compare the (p,γ) data with existing data on the exclusive (p,π) reaction at 185 MeV revealing notable differences between the two reactions.     

Statistical multistep reactions from 1 to 100 MeV

An analytical model for statistical multistep direct and multistep compound reactions is presented. It predicts emission spectra for neutrons, protons, alphas, and photons including equilibrium, preequilibrium, direct (collective and non-collective) as well as multiple particle emission processes. Results are presented for nucleon-induced reactions up to 100 MeV.     

Multistep processes in nuclear reactions

Recent applications of the theory of Feshbach, Kerman and Koonin to analyse multistep processes in nuclear reactions are described, and illustrated by detailed comparisons with a range of experimental data for neutron and proton reactions. The techniques used to distinguish between the multistep direct, multistep compound, compound and collective contributions to the cross-sections are described, and their effectiveness assessed. Particular attention is devoted to recent analyses that take into account the transitions from the multistep direct to the multistep compound chain and also to those that evaluate the collective contributions to the continuum region. The variation of the effective nucleonnucleon interaction with incident energy and target nucleus is studied, and the possibility of a parameter-free calculation is discussed. The extensions of the theory to include multiparticle emission from the direct chain and also to studies of alpha-particle emission are described.     

An extension of the generalized exciton model and calculations of (p,p') and (p,α) angular distributions

We present a model for calculating the energy-angle double differential cross section of the nucleon emission from preequilibrium states in nucleon-induced reactions. It is an extension of the generalized exciton model of Mantzouranis et al. for treating the kinematics of the NN collision properly. Its final expression is compact and easy to use. Numerical calculations for various reactions were carried out and some improvements over the previous results were obtained. The angular distribution was well reproduced except at backward angles. The validity of the “fast” particle assumption used in the previous models was also examined.We then applied the present model to composite-particle emission by employing the technique for composite-particle emission spectra we developed previously. Here we assumed that the angular distribution of the composite particle is obtained by folding that of the “leading” particle over the internal motion of the composite particle. The numerical calculation for α-emission was performed and a good fit to the experimental data was obtained for various targets and incident energies without any adjustable parameters.     

Fast particle emission from nuclear reactions

The emission of fast particles from continuum states being populated in energetic nuclear reactions is investigated within the framework of different models. Typical bombarding energies range from 20 to 100 MeV per nucleon.The investigation starts with one-step and multistep quantum-mechanical approaches, followed by semi-classical models, especially the exciton model, which can be derived from the quantum-mechanical approaches. Because of its transparency it allows one to study the effects of different approximations. All models need residual interactions having volume integrals larger than those necessary to describe scattering to bound states.Next to the analysis of different types of data in the framework of an extended exciton model, different approaches for complex particle emission were discussed. It is possible to extend the exciton model to include pickup of target nucleons, pickup among recoiling nucleons (coalescence) and even emission of preformed clusters. In any case additional model parameters become necessary. None of thes extensions has so far been able to reproduce data over a large range of energies and projectile mass numbers without model parameter variation. However, the exciton coalescence model reproduces exclusive data from coincidence measurements.     

Relaxation phenomena in nucleon-induced reactions and preequilibrium angular distributions

The intranuclear relaxation process in nucleon-induced reactions is investigated in a Fermi gas model utilizing a statistical operator for the non-equilibrium state, which contains energy, particle number, and linear momentum as relevant observables. In the hydrodynamic stage the phase space is subdivided into subspaces, assumed to be in a quasi-equilibrium, which is characterized by a Fermi distribution function with time-dependent thermodynamic parameters. A set of coupled non-linear equations for the time development of the thermodynamic parameters is derived. For the case of two sybsystems, a numerical solution of these generalized transport equations is provided, with kinetic coefficients calculated microscopically. The initial conditions are fixed in accordance with the energy and angle distribution of both particles occupying states above the Fermi surface after the first collision event. From the results, a fast and a slow stage are established in the relaxation process, with equal relaxation times for all physical quantities under consideration. The dependence of the relaxation time on particle number and excitation energy is estimated. The particle emission from the precompound stage of the reaction is taken into account by using the principle of detailed balance to find expressions for the mean fluxes from the compound system to the outer space, which are included in the equations for the relaxation. From the time evolution of the occupation number for states above the nucleon binding energy, the precompound double-differential cross section cumulating up to a certain time is calculated. The results for the angular distributions in nucleon-induced precompound reactions are compared with measured data as well as with previous predictions from generalized exciton models. Satisfactory agreement with experimental data is obtained. Following the time development of the compound nucleus the consistency of the present model with the evaporation theory is demonstrated by investigating the mean nucleon decay width.     

Multistep scattering in the 160 MeV208Pb(α, α′) reaction

New experimental data has been obtained for the²⁰⁸Pb(α, α′) reaction induced by 160 MeV alpha particles, for inelastic scattering to forward angles. We use these data to investigate the applicability of the multistep scattering theory of Feshbach, Kerman, and Koonin for describing this reaction. The mechanism we study, following the work of Gadioli et al. [1], is of the incident alpha particle remaining intact throughout the scattering process, exciting nucleon particle-hole pairs through multistep process. We conclude that this mechanism, combined with compound nucleus decay at low emission energies, can account for much of the observed data. However, there are indications that other processes also contribute at energies above the compound nucleus emission regime, and we outline future theoretical analyses that are needed.     

Theoretical description of the towing mode through a time-dependent quantum calculation

The target emission for the reaction of ⁵⁸Ni and ⁵⁸Ni at 44 MeV/A is investigated in a time-dependent quantum model. We show that, besides the well-known mechanisms of particle emission, a new phenomenon which gives rise to a specific angular and energetic distribution of the particle emission, is also expected. Properties of this mechanism are similar to the so-called “towing mode”, recently observed experimentally in a similar reaction (J.A. Scarpaci et al., Phys. Lett. B 428 (1998) 241). Characteristics of this new emission type are described.     

Multi-step compound model of heavy-ion fusion

Heavy-ion fusion reactions have been analyzed within a multi-step compound model composed of a dinucleus configuration, coupled to particle and breakup channels as well as to an equilibrated compound nucleus configuration. The resulting fusion cross sections, defined as the summed particle emission cross sections from the equilibrated compound nucleus, are in reasonable agreement with the data for several systems. The resulting angular distributions as well as the time evolution of the system are also discussed.     

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.     

Polarization bremsstrahlung at collisions of fast ions with multiatomic targets

The processes of polarization bremsstrahlung at collisions of fast ions with linear chains consisting of isolated atoms have been considered. The intensities and angular distributions of radiation spectra have been obtained for an arbitrary number of atoms in a chain. It has been shown that the interference of the photon emission amplitudes leads to a noticeable change in the spectral angular distributions of polarization bremsstrahlung as compared to the distributions at collisions with an isolated atom. The results allow standard generalization to the cases of polarization bremsstrahlung at channeling of fast ions over surfaces and in solid lattices.     

THE WIDTH FLUCTUATION CORRECTION IN STATISTICAL THEORY OF NUCLEAR REACTIONS

In the width fluctuation correction calculation of compound nuclear reaction,we have made a unified treatment of the reaction processes of particle emission,gamma emission and fission processes,in which the residual nuclei can be in discrete states or in the continuum.The calculated results of inelastic cross sections and the cross sec-tions for particle emission,gamma emission and fission processes are self-consistent.     

Multistep direct emission from nucleon-induced reactions

Unbound 1p1h states are excited together with the bound ones in one-step direct processes induced by nucleons of energy greater than the particle binding energy. The cross-sections of the one-step direct reactions to bound final states are folded into a convolution integral to obtain the multistep cross-sections. This is done in the framework of the theory of Feshbach, Kerman and Koonin (FKK) that describes the emission of one particle. The processes to unbound 1p1h final states give rise to more complicated direct reactions, that are evaluated by an approximate method. The comparison of the theory with inclusive proton inelastic scattering by iron and with the charge exchange (p,n) reaction on zirconium is revised, at energies from 25 MeV to 120 MeV. It is shown that in the case of the (p,n) reaction the theory of FKK does not account for approximately 35% of the direct processes.Received: 11 November 2003, Published online: 10 August 2004PACS: 25.40.Kv Charge-exchange reactions - 25.40.Ep Inelastic proton scattering - 24.60.Gv Statistical multistep direct reactions     

Fusion-fission angular distributions: A new probe of fast fission fractionation in nucleus-nucleus collisions

Fragment angular distributions in heavy ion-induced fission reactions have been analysed in terms of a two component model—fission following compound nucleus formation and fast fission events. It is seen that, contrary to the general assumption, fast fission competes with compound nucleus fission even when the composite system is formed with a spin less than the rotating liquid drop model limit for vanishing fission barrier.     

Quantum Barrier Penetration Studies with Oriented Nuclei: Proton and Neutron Emission from Exotic Isotopes

We report a programme of novel experiments involving direct proton emission and β-delayed proton and neutron emission for exotic nuclei oriented at low temperatures. Full modelling of the quantum tunneling and angular distribution has been made for the first time for these processes. Their study has application in many fields of natural science. The study of barrier effects involving angular momentum and non-spherical deformation can be approached more directly via single-particle emission than via alpha decay, which suffers from complications relating to preformation of the alpha particle. Using the new formalism, predictions are presented for angular distributions of β-delayed neutrons from oriented ¹³⁷I and of direct proton emission from oriented ¹⁴⁷Tm. This revised version was published online in September 2006 with corrections to the Cover Date.     

Direct surface-transfer reaction to the continuum states induced by a 1760 MeV 40Ar beam on 27Al and natTi target nuclei

The inclusive energy spectra of few-nucleon transfer reactions obtained by bombarding ²⁷Al and ^natTi target nuclei with the GANIL 1760 MeV ⁴⁰Ar beam have been analysed in terms of direct surface-transfer reactions populating the continuum states. The shape at forward angles of the various energy spectra and the corresponding angular distributions are well reproduced by a diffractional model calculation based on mutual excitation of the two residual partners. It turns out that one of the essential ingredients of the calculations is the Williams partial level density either of the target or of the projectile constructed only with the transferred nucleons or residual holes coupled to the ground state.