A Semiclassical Theory of Multistep Nuclear Reaction Processes

The master equation theory of precompound and compound nuclear reactions has been generalized with inclusion of the conservation of angular momentum and parity. This improved semiclassical theory has been extended for application as an evaluation tool of the calculations of nucleon induced reaction cross sections and double differential cross sections. For structural materials at incident neutron energies below 20 MeV, it is demonstrated that the constructed model contains the Hauser-Feshbach, Weisskopf-Ewing as well as the exciton models as limiting cases. The unified treatment ofpre-equilibrium processes includes a number of interesting features, such as the exciton state densities with the exact Pauli exclusion correction which are renormalized to the back-shifted Fermi-gas formula, the introduction of formation factors of composite particle in calculations of pick-up type composite particle emission and the double differential cross sections (DDCS) for all kinds of particles in terms of the leading particle model.     

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.     

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.     

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.     

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.     

广义不确定性原理下费米气体低温热力学性质

在考虑到广义不确定性原理时, 统计物理中的态密度必须做出修正, 这导致对传统统计物理的所有结果都有不同程度的修正. 在高能、高温条件下, 此修正是颠覆传统观念的, 在低温条件下, 也有一定的修正. 研究了低温条件下考虑到广义不确定性原理时, 理想费米气体和具有弱相互作用费米气体的热力学性质, 分别给出理想费米气体和弱相互作用费米气体的化学势、内能和定容热容的解析表达式, 并以铜电子气体为例进行了具体数值计算, 将计算结果与不考虑广义不确定性原理时的费米气体的热力学性质进行了比较, 探讨了广义不确定性原理对系统热力学性质的影响. 考虑到广义不确定性原理后费米气体的化学势、费米能和基态能增大, 热容减少, 内能随温度的增加先增大, 到某一温度(对于铜电子气体, T/T_F0～0.3)时, 增值为零, 温度再增加内能减少. 这些修正的具体数值主要由粒子数密度决定, 粒子数密度越大, 修正越大.     

Comments on the state densities and the transition rates in the pre-equilibrium exciton model

It is proposed to renormalize the exciton-state densities used in models for precompound decay such that the summed state densities agree with the expressions employed in equilibrium statistical models. In this way a close fit can be guaranteed between preequilibrium model calculations and the results of equilibrium statistical models for the evaporative stage of the reaction. The consequences of this proposal for the internal transition rates of the pre-equilibrium exciton model are analyzed. The matrix element for the residual interaction is obtained not from a phenomenological parametrization, but from the nucleon mean free path in nuclear matter. It is demonstrated that the proposed renormalization, from one-component Fermi-gas formulas to two-fermion expressions for the state densities, leads to strongly improved agreement of the effective exciton-model values for the nucleon mean free path in nuclear matter with realistic estimates. It is proved that the particle-hole state densities for a two-component Fermi gas, summed over the allowed exciton-state numbers, agree with the phenomenological state-density expressions used in statistical Hauser-Feshbach models.     

Polar nano-rods under shear: From equilibrium to chaos

The orientational dynamics of rod-like particles with permanent (electric or magnetic) dipole moments in a plane Couette shear flow is investigated using mesoscopic relaxation equations combined with a generalized Landau free energy. The free energy contribution due to the coupling between average alignment and dipole orientation is derived on a microscopic basis. Numerical results of the resulting eight-dimensional dynamical system are presented for the case of longitudinal dipoles and thermodynamic conditions where the equilibrium state is a (polar or non-polar) nematic. Solution diagrams reveal presence of a large variety of periodic, transient chaotic, and chaotic dynamic states of the average alignment and dipole moment, respectively, appearing as a function of Deborah number and tumbling parameter. Compared to rods without dipoles we observe a significant preference of out-of-plane kayaking-tumbling states and, generally, a higher sensitivity to the initial conditions including bistability. We also demonstrate that the average (electric) dipole moment characterizing most of the observed states yields electrodynamic (magnetic) fields of measurable strength.     

弱磁场中弱相互作用费米气体的有限粒子数效应(英文)

由弱磁场中弱相互作用费米气体的配分函数,导出有限粒子数条件下系统的配分函数G(β,N ).在此基础上,运用统计平均方法求解有限粒子数弱相互作用费米气体热力学量的解析表达式,给出各种温度条件下的热力学性质.研究结果表明,有限粒子数效应使各个热力学量都产生了一个修正项,除温度趋于0外,粒子数对化学势的修正项有直接影响,对内能和热容量的修正项并不产生直接影响.并且有限粒子数效应总是降低化学势,从而使化学势的0点向低温漂移,粒子数增大,会削弱这种效应,粒子间的相互排斥会加强这种效应.     

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 Thomas–Fermi quark model: Non-relativistic aspects

The first numerical investigation of non-relativistic aspects of the Thomas–Fermi (TF) statistical multi-quark model is given. We begin with a review of the traditional TF model without an explicit spin interaction and find that the spin splittings are too small in this approach. An explicit spin interaction is then introduced which entails the definition of a generalized spin “flavor”. We investigate baryonic states in this approach which can be described with two inequivalent wave functions; such states can however apply to multiple degenerate flavors. We find that the model requires a spatial separation of quark flavors, even if completely degenerate. Although the TF model is designed to investigate the possibility of many-quark states, we find surprisingly that it may be used to fit the low energy spectrum of almost all ground state octet and decuplet baryons. The charge radii of such states are determined and compared with lattice calculations and other models. The low energy fit obtained allows us to extrapolate to the six-quark doubly strange H$H$-dibaryon state, flavor symmetric strange states of higher quark content and possible six quark nucleon–nucleon resonances. The emphasis here is on the systematics revealed in this approach. We view our model as a versatile and convenient tool for quickly assessing the characteristics of new, possibly bound, particle states of higher quark number content.     

Green’s function method in the problem of complex configurations in fermi systems with pairing

The Green’s function method is used to derive general equations for describing effects of pairing in Fermi systems where there are two types of interaction, two-particle and quasiparticle-phonon interaction. These equations generalize Bardeen-Cooper-Schrieffertheory to the case of complex configurations involving “strong” phonons. In the approximation of weak coupling to phonons, realistic equations that make it possible to describe excited states of nonmagic even-even nuclei with allowance for a single-particle continuum and complex configurations of the two quasiparticles ? phonon type are formulated for the first time. These equations are solved for an isovector E 1 resonance in the stable isotope ¹²⁰ Sn and in the unstable isotopes ^104,132Sn. It is shown that complex configurations must be taken into account in order to describe E1 excitations—in particular, in a broad energy region around the nucleon binding energy.

     

Influence of an enlargement of the model space on number projected quasiparticle calculations

Results of number projected quasiparticle calculations for Sn isotopes in large and in small model spaces are compared when the force strengths and single-particle energies are determined consistently within each model space. When extending the model space, one observes that the model parameters extracted from the odd nuclei become more satisfactory. For even nuclei the collective states are not lowered in energy although electromagnetic transition rates increase considerably. Spectroscopic factors for one-nucleon transfer reactions change noticeably only for shells close to the Fermi level. Two-nucleon transfer cross-sections are strongly increased for ground state to ground state transitions only. We criticize a usual approximation formula for theL=0 two-nucleon transfer cross-section.     

Influence of an enlargement of the model space on number projected quasiparticle calculations

Results of number projected quasiparticle calculations for Sn isotopes in large and in small model spaces are compared when the force strengths and single-particle energies are determined consistently within each model space. When extending the model space, one observes that the model parameters extracted from the odd nuclei become more satisfactory. For even nuclei the collective states are not lowered in energy although electromagnetic transition rates increase considerably. Spectroscopic factors for one-nucleon transfer reactions change noticeably only for shells close to the Fermi level. Two-nucleon transfer cross-sections are strongly increased for ground state to ground state transitions only. We criticize a usual approximation formula for theL=0 two-nucleon transfer cross-section.     

The nuclear excitations and particle emission following muon capture

The nuclear muon capture process is studied as an excitation of the analogues of various harmonics of the multipole giant resonance. The excitation energy distribution is calculated with the use of the hydrodynamical model of the nucleus. The following particle emission cascade (including precompound states) is considered and the obtained particle spectra as well as yields of various residual nuclei are in a fair agreement with the existing experimental data.     

Configuration-Space Faddeev Calculation for Proton�CDeuteron Elastic Scattering Observables

A new computational method for solving the nucleon?Cdeuteron breakup scattering problem has been applied to study the elastic nucleon?Cdeuteron scattering on the basis of the configuration-space Faddeev?CNoyes?CNoble?CMerkuriev equations. The Merkuriev?CGignoux?CLaverne approach has been generalized for arbitrary nucleon?Cnucleon potentials and with an arbitrary number of partial waves. The nucleon?Cdeuteron observables at the incident nucleon energy 3?MeV have been calculated using the charge-independent AV14 nucleon?Cnucleon potential including the Coulomb force for the proton?Cdeuteron scattering. Results have been compared with those of other authors and with experimental proton?Cdeuteron scattering data.     

Quasi-elastic nucleon transfer and single charge exchange in 48Ti + 42Ca collisions

Single nucleon transfer and single charge exchange have been studied for quasi-elastic collisions of ⁴⁸Ti and ⁴²Ca at E_lab = 240, 300 and 385 MeV. Specific features below 12 MeV excitation energy are displayed in all of the measured particle energy spectra for each channel. These have been accounted for in the case of nucleon transfer by a one-step direct transfer mechanism in which all of the available hole and bound particle states in the donor and acceptor nucleus, respectively, have been included. This core excitation model was extended to single charge exchange (SCX) two-step transfers in which a proton and a neutron are exchanged. The absolute magnitude and distribution of strength in the SCX channel was obtained. The contributions from direct charge exchange via the isovector parts of the NN interaction has also been investigated in a microscopic model and is found to contribute of the order of 1–5% to the charge exchange cross section. The distribution of Fermi and Gamow-Teller strength is discussed.     

在中能重离子反应中核子的运动特征

在量子分子动力学模型框架下，研究了在中能重离子反应中单个核子的运动形态是规则的还是混沌的，它与反应系统的整体性质、反应机制是有关联的．在稳定的复合系统中，核子的运动显示出规则的行为．规则行为与混沌行为的并存反映了规则区域被不规则区域所包围的排斥极分岔特性，说明有中等质量碎片的局域稳定源存在．更多的不稳定轨道的出现则标志着这种局域稳定源的减少，有更多的核子及小集团发射．核子的混炖运动比核物质状态方程所标志的力学不稳定区域出现的时间更早．     

Analysis of inclusive nucleon spectra fromp- andα-induced reactions in the frame of the INDEX model,and implications for stopped π−-induced reactions

The previously published INDEX model is tested for nucleon spectra fromp- andα-induced reactions. The results of two alternative versions, the INCLUSIVE INDEX model and the EXCLUSIVE INDEX model, quite well agree with the data. It is found that in the INCLUSIVE INDEX model three preequilibrium stages are sufficient to describe single- and multi-nucleon emission. The model provides an useful first order estimate of the influence of the finite Fermi energy on particle spectra. This effect is very strong for nucleon induced reactions while forα-induced reactions it can be neglected. The deduced mean-free-path multiplier corroborates the long stated discrepancy between models in which excitons interact independently or not. Using preequilibrium parameters similar to those found for nucleon induced reactions the important branching ratio of contributingnp andpp pairs in stoppedπ ^?-absorption can be determined by INDEX model calculations. Deduced values from publishedn- andp-spectra agree reasonably well with those of other experimental analyses but deviate significantly from microscopic model predictions.     

Description of deep inelastic transfer reactions at the Fermi energy within transport models

The possibility of studying the dissipative processes near the Fermi energy using the transport equations arising within the Boltzmann-Nordheim-Vlasov approach is discussed. It is shown that this approach makes it possible to reproduce all the main features of deep inelastic collisions of heavy ions: specifically, the deviation curves for reaction fragments and their Coulomb and nuclear rainbows and the strong correlation between the fragment exit angle and kinetic energy loss. The theoretical results are compared with the experimentally obtained data on the isotopic and velocity distributions of light fragments escaping at small angles in the ¹⁸O + ¹⁸¹Ta and ¹⁸O + ⁹Be reactions at an energy of 35 MeV/nucleon. It is established that the experimental velocity distributions contain two components: dissipative (with a velocity smaller than that of the projectile ion) and direct (having a maximum at the particle beam velocity). It is shown that the transport equations describe well the dissipative part of the reaction, which is about 30% of the total output. The direct component arises during fission of the projectile ion nucleus and is described well within the Goldhaber model.