Angular distribution in exciton and hybrid models for preequilibrium reactions

Simple expressions for the calculation of angular distributions of nucleons emitted in nucleon-induced preequilibrium reactions are derived within the framework of both the exciton and hybrid model.     

Angular distribution of nucleons in nucleon-induced preequilibrium reactions

We present a model, leading to a generalised master equation, from which both angular distributions and spectra of preequilibrium nucleons can be successfully calculated. Implications and limitations of this model are discussed.     

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.     

Angular distributions in quasi-fission reactions

Angular distributions for fission-like fragments were measured in the systems ⁵⁰Ti, ⁵⁶Fe + ²⁰⁸Pb by applying an off-line K X-ray activation technique. The distributions d²σ/dθdZ exhibit forward-backward asymmetries that are strongly Z-dependent. They result from a process (quasi-fission) which yields nearly symmetric masses in times comparable to the rotational period of the composite system. A method for obtaining the variance of the tilting angular momentum, K₀², from these skewed, differential angular distributions is described. The results indicate that the tilting mode is not fully excited in quasi-fission reactions. The results are compared to the sum of the variances of all statistical spin components, measured via γ-multiplicities. Integration of the angular distributions d²σ/dθdZ over all values of Z yields the integral angular distributions dσ/dθ and dσ/dΩ symmetric around 90°. The associated unusually large anisotropies do not at all provide an adequate basis for tests or modifications of the transition-state theory. A deconvolution of d²σ/dθdZ is performed with gaussian distributions depending on rotational angles Δθ extending over a range of up to 540°. From the mean values 〈Δθ〉 a time scale for the evolution of K₀ is calculated.     

Angular distributions in pre-equilibrium reactions

A new model is proposed for calculating angular distributions in preequilibrium reactions. In this model, as in the model of Feshbach et al. the system consisting of target plus projectile initially branches into two sets of states with either no particle in the continuum (multistep compound states) or with at least one particle in the continuum (multistep direct states). The multistep compound emission is assumed to be isotropic while the angular distribution of the multistep direct emission is described using the fast particle model of Mantzouranis et al. A similar master equation is used for both chains of states differing only in the angular dependence of the emission rates. The two chains of states are treated independantly neglecting inter-branch transitions. The angular distributions for 14.6 MeV neutrons calculated using this model are found to be in better agreement with the data than the fast particle model.     

On the calculation of mean lifetimes in the preequilibrium exciton model

An explicit and exact calculation method of mean lifetimes in the nuclear exciton model describing preequilibrium decay is presented. A simple algorithm is suggested. Relations of existing approaches with the analytical solution of the time-integrated master equation are discussed.     

The finite depth of the nuclear potential well in the exciton model of preequilibrium decay

The density of states and the transition rates are derived for the equidistant spacing model with the limitation on the hole maximum energy. The previous results of Blann are generalized. The influence of the finite depth of nuclear potential well on the pre-equilibrium decay within the exciton model is investigated. The calculations of proton and deuteron spectra from the reaction¹²⁰Sn+p are presented.     

Angular distributions in photonucleon and nucleon radiative capture reactions

Angular distributions in photonucleon and nucleon radiative capture reactions are investigated by means of the direct-semidirect model. It is shown that the introduction of higher-order multipoles and the choice of an appropriate coupling interaction allow reproduction of experimental data without recourse to free parameters.     

Fast and precise exciton model calculations of nuclear reactions

Conversion of a set of differential master equations of a preequilibrium decay exciton model into a system of algebraic equations for mean lifetimes is discussed. The connection of the master equations with the closed-form expression is shown. A simple and computationally efficient method for obtaining the mean lifetimes of exciton states is presented.     

Two-component equilibration in the exciton model of nuclear reactions

A generalized two-components master-equation approach for a nuclear system towards equilibration is described, two-components being the proton and the neutron components. An approximate closed form solution of the two-component equation is discussed. Further an effective one-component master equation is derived from it. Explicit expressions for effective transition and emission rates are derived under a binomial model. The mean free path of an exciton has been found to be reduced in accounting for the two-component equilibration.     

Dual analytic model of generalized parton distributions

A model for generalized parton distributions (GPDs) in the form of ∼(x/g ₀)^{(1−x)ᾶ(t)}, where ᾶ(t) = α(t) − α(0) is the nonlinear part of the Regge trajectory and g ₀ is a parameter, g ₀ > 1, is presented. For linear trajectories, it reduces to earlier proposals. We compare the calculated moments of these GPDs with the experimental data on form factors and find that the effects from the nonlinearity of Regge trajectories are large. By Fourier transforming the obtained GPDs, we access the spatial distribution of protons in the transverse plane. The relation between dual amplitudes with Mandelstam analyticity and composite models in the infinite-momentum frame is discussed, the integration variable in dual models being associated with the quark longitudinal-momentum fraction x in the nucleon. The text was submitted by the authors in English.     

A unified model of preequilibrium decay

A microscopic exciton model has been developed which describes the time evolution of a highly excited nucleus through a series of binary collisions creating additional particle-hole pairs. Starting from an arbitrary exciton density distribution, the spectral shapes of the succeeding stages are generated through recursion relations which keep track of the “exciton flux” explicitely. The computer code BEEFALO provides particle spectra and average life times of the successive stages. The results are compared with the predictions of other preequilibrium models using “equal probability” density distributions. The controversy about the use of single particle orn-exciton state life times has been resolved by showing that both approaches lead to the same particle spectra if proper book-keeping is made through appropriate recursion relations.     

Multiplicity distributions in the generalized Feynman gas model

Multiplicity distributions Ψ_n^(k) in the generalized Feynman gas model of order k (defined by saying that all integrated correlation functions f_n except f₁,…,f_k are zero) are derived and expressed in terms of Poisson distributions with different ”average multiplicities”, which are related to the integrated correlation functions. The relations between Ψ_n^(k) and Ψ_n^(j) for arbitrary positive integers k,j are found. An intuitive picture to gain a better feeling for these relations is developed.On the basis of our formulae we show that the experimentally observed multiplicity distributions (between 50 GeV/c and 303 GeV/c incoming momentum) can be well reproduced by those of a Feynman gas model of order two. Other applications of our formulae are suggested.     

Angular distributions of penning electrons

Angular distributions of electrons ejected in thermal collisions of He(2¹ S, 2³ S)-metastables with Ar, Kr, Xe, Hg, CO, N₂ are measured with respect to the vector of relative velocity of the colliding particles. Except for preliminary results on the He(2₁S, 2³ S)-/Ar-system, these are the first such distributions reported. They are found to be strongly anisotropic and asymmetric in some cases, and approximately isotropic in those cases where the interaction potential between the metastable- and the target particle is strong. For the cases of weak interaction potential, the connection between the “internal” angular distribution in the frame fixed to the internuclear axis and the laboratory angular distribution, has been derived. Calculations of laboratory angular distributions involving an assumed “internal” distribution and the kinematics of a hard core collision show good qualitative agreement with the experimental results.     

Partial breakdown of the evaporation model and possible preequilibrium α-particle emission in heavy-ion fusion reactions

Relative cross sections for residual nuclei following fusion reactions were measured by γ-ray spectroscopy in the reactions ⁵²Cr+ ¹²C at 56.00 MeV and ⁴⁸Ti+ ¹⁶O at 57.74 MeV, in which the common “compound nucleus” ⁶⁴Zn was excited at the same energies and the relative distributions of the entrance spins were nearly identical. It is shown that ratios of the measured cross sections in both entrance channels can be precisely determined experimentally and are insensitive to small changes of the parameters in calculations based on the evaporation model. Using these particular quantities, we have shown that the evaporation process is not the sole mechanism producing the observed residues, especially those resulting from α-particle emission. The assumption of preequilibrium α-particle emission at high channel spins is shown to reproduce the present data fairly well.     

Angular distributions in quasi-fission reactions: Evidence for incomplete relaxation of the tilting mode

The integral β^?-spectra of²³⁵U and²³⁹Pu fission products have been measured with a plastic scintillator telescope at an external neutron guide tube at the high flux reactor of the ILL in Grenoble. The highly enriched targets (150 – 800 γg/cm²) were placed in a fission chamber at a distance of approximately 110 m from the reactor core. From the measured beta-spectra absolute counting rates per MeV and fission have been calculated, which are compared with the results of earlier experiments of other authors and with recent theoretical calculations.     

Inclusive and exclusive preequilibrium spectra in a model of independently interacting excitons

Starting from a previous ansatz ofindependently interacting excitons the more refined model INCLUSIVE INDEX is developed yieldinginclusive preequilibrium proton and neutron spectra. It is shown that three stages are sufficient to describe single- and multi-nucleon emission to all orders of practical importance. By use of recurrence relations the loss of particles and energy due to emission from previous stages is fully accounted for. In order to predict the population of specific residual nuclei the precompound process is reformulated in terms ofexclusive emission spectra. This intricate statistical problem is solved in the EXCLUSIVE INDEX model by algorithms representing the average probability for a specified number of particles not to be emitted but to reach the next stage of nuclear relaxation. Differential excitation probabilities can be calculated for those six nuclei which are populated by the emission of none, one, or two preequilibrium nucleons in two stages via alltogether 15 principal paths. Typical features of the INCLUSIVE and EXCLUSIVE INDEX model are discussed. The strength of one- and two-nucleon emission is compared with HYBRID model predictions.     

Properties of multi-Skyrmions and their quantization within a generalized chiral soliton model

A semiclassical quantization of the Skyrme model featuring a sixth-order term in the derivatives of the chiral field in the Lagrangian is performed. The orbital, isotopic, interference, and flavor tensors of inertia are calculated. For this version of the model, numerical calculations are performed for the excitation energies of flavors in baryon systems.