Random-walk model of precompound decay II: Stochastic uncertainties in the lifetimes and cross-sections

The uncertainties arising from the stochastic nature of precompound decay nuclear reactions are analyzed in the framework of the preequilibrium exciton and random-walk models. It is demonstrated that the standard deviations and the mean values of the exciton-state lifetimes are of the same order of magnitude. Their correlations are weakly positive, except for exciton states near the equilibrium number, where the correlations are significant. The usefulness and the limitations of the never-come-back approximation are discussed. A general proof is presented of the conditions under which the master-equation and random-walk approaches to Markov processes are equivalent. Connections between different preequilibrium models, e.g. the multi-step compound model and the microscopic statistical theory of precompound decay, are pointed out. It is shown that the waiting time between subsequent collisions is governed by a Poisson process, suggesting that the variance associated with the nucleon mean free path in nuclear matter, as estimated from preequilibrium models, is considerable. The stochastic uncertainties in the emission cross-sections correspond to those of a Bernoulli process.     

Description of cross sections for photonuclear reactions in the energy range between 7 and 140 MeV

A combination of the exciton and evaporation models is used to describe photonuclear reactions induced in light, medium-mass, and heavy nuclei by photons of energy in the range 7 ≤ E _γ ≤ 140 MeV. Two mechanisms of the photoexcitation of nuclei are considered. These are the formation of a giant dipole resonance at energies in the range E _γ ? 30 MeV and quasideuteron photoabsorption, which is dominant at energies in the region E _γ ? 40 MeV. The density of particle-hole states, which appears in the exciton model, is calculated on the basis of the Fermi gas model. The emission of two preequilibrium particles is taken into account in describing the quasideuteron reaction channel. The effect of isospin conservation on giant-dipole-resonance decay accompanied by photonucleon emission is examined. The model in question is used to describe cross sections for photon-induced reactions on ²⁶Mg, ⁵⁴Fe, ^{112,118,119,124}Sn, and ¹⁸¹Ta nuclei.     

Incident energy dependence of preequilibrium (p,p′) spectra

Proton energy spectra from (p, xp) reactions on⁹⁸Mo and¹⁰⁶Pd have been measured at incident energies of 12, 14, and 16 MeV to investigate the incident energy dependence of preequilibrium (p, p′) process. The (p, xp) energy spectrum for⁶⁰Ni has also been measured at an incident energy of 18 MeV to confirm the mass number dependence. These spectra were compared with a calculation based on the exciton model in which the square of the average effective matrix element ¦M¦² was assumed to beKA ^?3 E ^?1 and isospin conservation was taken into account. The calculated spectra using a constantK-value (430 MeV³) were in good agreement with all the measured ones. The role of isospin conservation in preequilibrium process was discussed through the present analyses.     

Gamma emission in the pre-equilibrium exciton model

The pre-equilibrium exciton model of nuclear reactions is extended to include gamma emission. Possible types of gamma transitions are considered for which densities of final accessible states are derived. The computed gamma spectra in the reaction (n, γ) with E_n = 14.1 MeV on ⁹³Nb and ¹³⁹La are compared with experiment.     

Combined model of photonucleon reactions

A combination of the semimicroscopic, exciton, and evaporation models is used to describe photonucleon reactions induced in medium-mass and heavy nuclei by photons of energy below the mesonproduction threshold. Two mechanisms of the photoexcitation of nuclei are considered. These are the formation of a giant dipole resonance (GDR) at energies in the range E _γ ≲ 30 MeV and quasideuteron photoabsorption, which is dominant at energies in the region E _γ ≳ 40 MeV. The densities of particle-hole states appearing in the exciton model are calculated on the basis of the Fermi gas model. Our combined model takes into account the multiparticle emission of preequilibrium particles. The influence of isospin conservation and collective phenomena on photonucleon emission by giant dipole resonances is considered. The combined model is used to describe cross sections for photonucleon reactions proceeding on the ^40,48Ca, ⁹⁰Zr, ¹³⁹La, ¹⁴²Nd, and ¹⁸¹Ta nuclei, as well as difference (E _γmax = 85–55 MeV) bremsstrahlung photoneutron spectra for the ⁶³Cu, ¹¹⁵In, ¹¹⁸Sn, ¹⁸¹Ta, ²⁰⁷Pb, ²⁰⁹Bi, and ²³⁵U nuclei and bremsstrahlung photoproton spectra for the ⁹⁰Zr nucleus at the energies of E _γmax = 22, 25, and 34 MeV.     

Nuclear interaction dynamics in the 56Fe (α, xn) reaction

Neutron spectra and angular distributions in the ⁵⁶Fe (α, xn) reaction are measured at α-particle energies of 12.5, 16.3, 18.3, 26.8, and 45.2 MeV. The measurements are performed using time-of-flight fast neutron spectrometers on pulsed accelerators. The measured data are analyzed in the context of equilibrium, preequilibrium, and direct nuclear reaction mechanisms. The contributions from equilibrium, preequilibrium, and direct mechanisms of neutron emission are studied over a wide range of α-particle energies.     

Studying the ejection of light charged particles induced by 50 MeV <Superscript>3</Superscript>He ions upon interacting with a <Superscript>112</Superscript>Sn nucleus

Experimental double-differential and integral spectra of (³He, xp), (³He, xd), (³He, xt), (³He, x³He) and (³He, xα) reactions on ¹¹²Sn nuclei induced by 50 MeV 3He ions are presented. Theoretical calculations of the experimental inclusive spectra of the reactions are performed using the exciton model of preequilibrium decay. The corresponding mechanisms of reactions are determined. The experimental results can be used to develop new approaches in the theory of nuclear reactions, and to design safe and wasteless hybrid nuclear power plants.     

Analysis of excitation functions from light ion induced reactions in the EXCLUSIVE INDEX model

The previously introduced EXCLUSIVE INDEX model allows to predict the population of 6 residual nuclei including the primary compound nucleus through two stages of the preequilibrium phase. The present version is limited to maximum two-nucleon emission. The preequilibrium ejectiles may reduce the brought-in rotational energy by a model of maximum angular momentum decoupling. Subsequent evaporation of protons, neutrons andα-particles is treated in the frame of the Weisskopf-Ewing ands-wave approximation considering pairing effects only in compound nucleus state densities. The sensitivity of essential preequilibrium parameters on the shape of calculated excitation functions is tested. The model predictions well compare to excitation functions fromp, d,³He and⁴He induced reactions including the large set from the reaction⁹³Nb(⁴He,xn yp) up to 170 MeV bombarding energy. The general importance of two-nucleon preequilibrium emission is accentuated in several examples. The deduced preequilibrium parameters corroborate the results from the INDEX model analysis of nucleon spectra.     

Light charged particle emission in40Ar induced reactions on68Zn at 14.6, 19.6 and 35 MeV/nucleon

The light charged particles emitted in the⁴⁰Ar+⁶⁸Zn reaction performed at 14.6, 19.6 and 35 MeV/nucleon have been studied inclusively. The energy spectra have been analysed in terms of preequilibrium emission, moving source and coalescence models. A complete coherence between the present data and those obtained separately from heavy fragment studies is achieved. These data are consistent with the onset of projectile fragmentation between 20 and 35 MeV/nucleon correlated with the formation of a highly excited region of the nuclear system inducing light particle emission. The other sources of light particles can be interpreted as statistical and sequential decay of the quasi projectile.     

High energy proton emission in reactions induced by 315 MeV 16O ions

Inclusive proton spectra have been measured for the reaction ¹⁹⁷Au(¹⁶O, p)X at 315 MeV. The data, which are consistent with emission from a moving source, are compared with the fireball model and with models preequilibrium emission.     

Precompound emission in the reaction 65Cu(p,xn) and the nucleon-nucleon scattering mechanism

The reaction ⁶⁵Cu(p, xn) has been studied with E_p = 26.7 MeV for angles 3° ? θ ? 177°0. Whereas the angle-integrated spectrum can be explained by a semiclassical preequilibrium model, the nucleon-nucleon scattering approach fails for the preequilibrium neutron emission at far backward angles. The failure is shown to be due to principal shortcomings. In contrast, the quantummechanical statistical multistep calculations give good agreement with the main contribution for residual energies U < 9 MeV coming from the SMDE component.     

Continuous spectra of protons emitted in α-particle induced reactions

Proton spectra resulting from (α, p) reactions on ⁵⁴Fe, ⁵⁶Fe, ⁵⁹Co, ⁵⁸Ni, ⁶¹Ni and ⁶³Cu were measured at E_α = 23.0 MeV. Energy and angular distributions of protons present two main components due to evaporative and preequilibrium processes. The hybrid model was chosen for analysing the data: no free parameters were allowed in the computations. The analysis is consistent with the one performed for (α, n) reactions on the same target nuclei. Hybrid model calculations succeed in explaining simultaneously both neutron and proton emission in α-particle induced reactions.     

Angular distribution calculation based on the exciton model taking account of the influence of the Fermi motion and the Pauli principle

An exact closed form solution to the time-integrated master equation of the exciton model is applied to the calculation of the angular distribution for both the preequilibrium and equilibrium decays of the neutron-induced reaction. The distribution probability of two-nucleon collision fromΩ toΩ′ based on the Fermi gas model and the influence of the Fermi motion and the Pauli principle on the shape of the angular distribution are studied in detail. We have concluded that the influence of these effects on the shape of the angular distribution is rather significant for reactions in the energy range of several tens of MeV. As an example to compare with the experiments we have calculated the neutron-induced reaction⁹³Nb(n,n′) atE _n =15 MeV. It seems that the most significant improvement of the present approach is the rise of the backward direction of the double differential cross section for the higher energy emitted neutrons.     

Dynamics of nuclear interaction in the reaction <Superscript>56</Superscript>Fe(<Emphasis Type="Italic">α</Emphasis>, <Emphasis Type="Italic">xn</Emphasis>)

The spectra and angular distributions of neutrons from the reaction ⁵⁶Fe(α, xn) were measured at the alpha-particle energies of 12.3, 16.3, 18.3, 26.8, and 45.2 MeV. The measurements were performed with time-of-flight fast-neutron spectrometers at pulsed accelerators of charged particles. The measured data were analyzed within models of equilibrium, preequilibrium, and direct reaction mechanisms. The calculations were performed by using an exact formalism of Hauser-Feshbach statistical theory. The required nuclear-level densities in nickel isotopes excited in this reaction were determined on the basis of new experimental data on low-lying levels, neutron resonances, and spectra of evaporated neutrons. The contributions of equilibrium, preequilibrium, and direct neutron emission were studied over a broad range of alpha-particle energies.     

Preequilibrium model of photonucleon reactions that is based on Fermi gas densities

A combination of the exciton and evaporation models is used to describe photonucleon reactions induced in heavy and medium-mass nuclei by photons of energy in the range 7 ≤ E _γ ≤ 140 MeV. The formation of a giant dipole resonance and quasideuteron absorption are considered as two mechanisms that are responsible for the photoexcitation of a nucleus in the energy regions E _γ ? 20 MeV and E _γ ? 40 MeV, respectively. As is well known, the densities of particle-hole states are employed in the exciton model, and these quantities are calculated on the basis of the Fermi gas model. This makes it possible to take into account the effect of the energy dependence of single-particle and single-hole densities of states on the rate of emission and intranuclear processes. The model in question is applied to describing partial photonucleon cross sections for ¹¹⁹Sn, ¹⁴⁰Ce, ¹⁸¹Ta, and ²⁰⁸Pb nuclei.     

Preequilibrium GDR excitation and entrance channel angular momentum effects

The energy spectra of the γ-rays emitted in the ³⁵Cl + ⁹²Mo reaction at incident energy E = 260 MeV were measured in coincidence with the ejectiles produced in dissipative reaction events. The cumulative energy spectrum of the γ-rays coming from the decay of the ejectiles was calculated within the statistical model and its comparison to the experimental spectrum evidences an excess in the data for E _γ = 8 to 12 MeV. Such an excess, fitted with a Lorentz curve, is attributed to the preequilibrium GDR γ-decay of the intermediate dinuclear system. The centroid energy of the Lorentz curve corresponds to a dipole oscillation along the symmetry axis of the system and its width is found to be comparable to that of the ground state GDR low energy component of the deformed dinucleus. The small quantal dispersion Δ? = (10.3 ± 0.1)? of the entrance channel angular momentum, determined by analysing the dissipative fragment angular distribution in the framework of the Strutinsky model, is suggested to limit the broadening of the preequilibrium GDR width.     

Fast emission ofα-particles in deep-inelastic36Ar+197Au collisions

Angular and energy correlations betweenα-particles and deep-inelastic projectile-like fragments were measured for the system³⁶Ar+¹⁹⁷Au atE _Lab=380 MeV. At most half of the coincident events can be attributed to statistical emission ofα-particles from the fully accelerated projectile-like fragments. The remainder of the events may be due to direct emission during the first 10^?22 s of the scattering process and to preequilibrium processes taking place within some 10^?21 s.     

On the mechanism of the (n, 2n) reaction with 14MeV projectiles on127I and209Bi

The angular and spectral distributions of coincident neutrons from the reactions¹²⁷I(n, 2n)¹²⁶I and²⁰⁹Bi(n, 2n)²⁰⁸Bi have been measured with two time-of-flight detectors at the incident energy 14.1 MeV. Neutron emission has been studied for reaction anglesθ _lab between 10° and 150° (relative angles from 60° to 270°). Energy spectra, angular distributions and coincidence yields are compared with statistical model calculations including preequilibrium decay modes, and nuclear level density and spin cut off parameters are derived.     

High spin selectivity in α-particle induced reactions

Distinct maxima have been observed in the spectra of protons emitted from the ²⁴Mg(α, p)²⁷Al reaction in the region of the excitation energies of ²⁷Al between 5 and 15 MeV, where continuous spectra are expected according to the statistical model. To investigate the reaction mechanism responsible for the selective excitation of several states of ²⁷Al, angular distributions of protons in the angular range from 20° to 170° in the lab system were measured at the α-particles energies E_lab = 25.4 25.8 and 26.2 MeV. Thirteen out of thirty levels observed in proton spectra, exhibit symmetry of their angular distributions around 90° the c.m. system, when averaged over energy. The contribution of preequilibrium emission was calculated and found to be negligible. A Hauser-Feshbach analysis was carried out to determine spins of states of ²⁷Al whose angular distributions were symmetric around 90° in the c.m. system. Results of this analysis indicated high spin selectivity in the ²⁴Mg(α, p)²⁷Al reaction.