Mean free path of nucleons in a Fermi gas at finite temperature

The mean free path of a nucleon in a nuclear Fermi gas at finite temperature is calculated by utilizing the free nucleon-nucleon cross section modified to suppress final states excluded by the Pauli principle. The results agree with an earlier zero-temperature calculation but yield substantially smaller values than a previous finite-temperature analysis.The Fermi gas mean free paths are some two to four times shorter than those implied by phenomenological imaginary optical potentials, suggesting that the present Fermi gas model fails to adequately describe the physical processes determining the mean free path. Even so, the present results, taken as lower bounds on the mean free path, require temperatures of some 4.5 MeV before the mean free path of bound nucleons becomes as short as the nuclear diameter. It follows that very high excitation energies are prerequisite to any short mean free path assumption in nuclear heavy-ion collisions.     

The energy dependent nucleus-nucleus optical potential in a nuclear matter approach

We calculated the energy dependent real and imaginary part of the nucleus-nucleus optical potential in a lab. energy domain between 50MeV and 225MeV per nucleon for the systems¹²C-¹²C and⁵⁸Ni-⁵⁸Ni. In our model we assume that the energetical behaviour of the colliding nucleons inside of the two overlapping nuclei is locally that of two colliding nuclear matter systems. Therefore the effective nucleon-nucleon interaction is calculated using a Bethe-Goldstone equation that includes the Pauli blocking caused by the two Fermi spheres and takes account of the presence of the other nucleons by their mean field. Neglecting finite range effects one can calculate the real and imaginary part by a folding procedure.     

Surface and volume contributions to real and imaginary parts of the heavy-ion optical potential

Starting from the Feshbach expression for the optical potential, explicit formulae for the real and the imaginary parts of the optical potential between two heavy ions (HI's) are obtained. They are each composed of a volume and a surface term. The contributions to the volume term are calculated in two nuclear Fermi liquids which flow through each other starting from the realistic Reid soft core nucleon-nucleon (NN) interaction. Since the Fermi surface is formed by two spheres one obtains a complex Brueckner reaction matrix which is approximated by a complex, effective local interaction. It is used in a fully antisymmetrized double folding procedure to obtain the volume terms of the optical potential between the two HI's. The surface contributions are directly calculated in the collision of the two finite HI's. The collective surface vibrations (3^? octupole state and 2⁺, 4⁺ (T = 0) giant resonances for the ¹⁶O?¹⁶O collision) are included as intermediate states. This yields especially an imaginary contribution at the surface which reduces the transparency found with the volume terms alone. The method is applied to ¹⁶O?¹⁶O scattering at 83 and 332 MeV laboratory energy. The local approximations to the real and imaginary parts obtained in this way agree well with phenomenological fits. The surface terms improve the agreement of the differential cross section at 80 MeV where experimental data are available.     

The reaction mechanism of heavy ion scattering at intermediate energies

The contribution to the real and imaginary nucleus-nucleus (N-N) optical potential from nucleon-nucleon scattering in the medium is calculated in a local density approximation from a two Fermi sphere nuclear matter picture for the N-N collision. This reaction mechanism is shown to be dominant for ¹²C + ¹²C scattering at all considered energies (160 MeV < E_lab < 2250 MeV) giving a weakly energy dependent reaction cross section of about 900 mb. Inclusion of the collective 2⁺, 3^? excitations in a coupled channel calculations gives good agreement for both the measured elastic and inelastic 2⁺ cross section at E_lab = 1016 MeV. This fully microscopic parameter free calculation indicates that the energy dependence of the reaction cross section for this system is mostly due to the decrease of the collective contribution with increasing energy contrary to current theoretical models.     

Nuclear matter approach to the heavy-ion optical potential

An energy-dependent local potential for heavy-ion (HI) scattering is derived from Reid's softcore interaction using the Brueckner theory. The Bethe-Goldstone equation in momentum space is first solved with the outgoing boundary condition for two colliding systems of nuclear matter with the relative momentum K_r per nucleon. The Fermi distribution is assumed to consist of two spheres without double counting of their intersection separated by the relative momentum K_r. The angle-averaged Pauli projection function is given in the form of a one-dimensional integral. Secondly the optical potential for HI scattering is evaluated using the energy-density formalism. The energy density is calculated for two limiting cases: (i) the sudden approximation in which the spatial distribution of the two HI is described by an antisymmetrized cluster wave function, and (ii) the adiabatic limit represented by an antisymmetrized two-centre wave function. The complex HI potential is defined in terms of the energy density from nuclear matter so that both volume elements in the finite and the infinite systems have the same nucleon and kinetic energy density. This method is applied to the ¹⁶O + ¹⁶O, ⁴⁰Ca + ¹⁶O, and ⁴⁰Ca + ⁴⁰Ca potentials. The calculated results are compared with phenomenological potentials. Though in principle our approach can generate an imaginary part for the HI potential, the magnitude is too small. Reasons and possible improvements of this point are discussed.     

Microscopic optical model analysis of elastic scattering cross sections for nucleon-40Ca systems at low energy

Neutron and proton-⁴⁰Ca elastic scattering angular distributions and total reaction cross sections are calculated between 10 and 40 MeV using the microscopic optical potential derived by Bouyssy et al. within the nuclear structure approach. Direct comparison with experiment confirms that our calculation reproduces the imaginary potential at low energy for protons, but it gives insufficient absorption above the deuteron pick-up threshold. A renormalization of both the real and the imaginary parts of the potential leads to good agreement with the data. For both parts of the potential this renormalization is compatible with the renormalization factors obtained by Bouyssy et al. from a comparison of calculated and phenomenological volume integrals.     

A phenomenological model of deep-inelastic collisions between complex nuclei

A simple model of heavy-ion collisions is proposed. Classical equations of motion with inclusion of a phenomenological two-body friction force are integrated numerically along trajectories. The nucleus-nucleus interaction potential which is used in the calculations includes deformation degrees of freedom in the exit channel. Both entrance and exit channel potentials are based on the boundary conditions following the liquid-drop model. The energy-angle distributions of deep-ineIastic reactions are very well reproduced by the model. The existing data on fusion cross sections are compared with the model predictions. This comparison indicates that in the nucleus-nucleus potential a repulsive core is present.     

Modelling heavy-ion energy deposition in extended media

We present recent developments of the Monte Carlo model for heavy-ion therapy (MCHIT), which is currently based on the Geant4 toolkit of version 9.2. The major advancement of the model concerns the modelling of violent fragmentation reactions by means of the Fermi break-up model, which is used to simulate decays of hot fragments created after the first stage of nucleus-nucleus collisions. By means of MCHIT we study the dose distributions from therapeutic beams of carbon nuclei in tissue-like materials, like water and PMMA. The contributions to the total dose from primary beam nuclei and from charged secondary fragments produced in nuclear fragmentation reactions are calculated. The build-up of secondary fragments along the beam axis is calculated and compared with available experimental data. Finally, we demonstrate the impact of violent multifragment decays on energy distributions of secondary neutrons produced by carbon nuclei in water.     

Optical potential in the gas approximation

Using the gas approximation the optical potential is expressed as a function of the scattering matrix of nucleon-nucleon scattering. The formalism of Green functions is used. The imaginary part of the optical potential in nuclear matter is calculated for high energies using the experimental phase shifts. In addition, a calculation of the imaginary part is made using theS-wave part of the Tabakin potential.     

A criterion for the existence of spin waves in polarized gases in a wide temperature range

We derive a kinetic equation for a polarized paramagnetic gas that is exact in the Boltzmann (binary) approximation. The equation is written in a compact form and applies to both Fermi and Bose gases in a wide temperature range as long as the Boltzmann approximation remains applicable. The derived equation is used to analyze the conditions for the propagation of spin waves in polarized Fermi and Bose gases. We deduce a universal criterion for the propagation of weakly damped spin waves in a wide temperature range. The criterion is reduced to the condition that the real parts of the particle zero-angle scattering amplitudes (or T matrices) be much larger than their imaginary parts. We derive dispersion equations for spin waves at high and low gas temperatures and show that spin waves can propagate in both these limiting cases.     

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.     

Elastic <Superscript>16</Superscript>O<Superscript>16</Superscript>O scattering at <Emphasis Type="Italic">E</Emphasis>(<Superscript>16</Superscript>O) = 350 MeV and new method for a model-independent determination of the scattering matrix

A new approach that can be used to extract the scattering matrix S(l) as a complex-valued function of the orbital angular momentum l directly from experimental data on elastic nucleus-nucleus scattering at intermediate energies without resort to additional model assumptions is developed on the basis of the evolutionary algorithm. Owing to the fact that the behavior of the derivatives of S(l) is automatically monitored within this approach, the scattering matrix obtained for ¹⁶O¹⁶O interaction at an energy of 350 MeV is determined by its absolute value and the nuclear phase shift, which are smooth monotonic functions of the orbital angular momentum. Moreover, the result involves no distortions, the form of the quantum deflection function being typical of the nuclear-rainbow pattern. It is shown that the ultimate form of S(l) is independent of input representations of the scattering matrix, which were usually taken on the basis of various phenomenological models.     

Antisymmetry and channel coupling contributions to the absorption for p+αd+3He

To understand recently established empirical p+α potentials, RGM calculations followed by inversion are made to study contributions of the d+³He reaction channels and deuteron distortion effects to the p+α potential. An equivalent study of the d+³He potential is also presented. The contributions of exchange non-locality to the absorption are simulated by including an phenomenological imaginary potential in the RGM. These effects alone strongly influence the shape of the imaginary potentials for both p+α and d+³He. The potentials local-equivalent to the fully antisymmetrised-coupled channels calculations have a significant parity-dependence in both real and imaginary components, which for p+α is qualitatively similar to that found empirically. The effects on the potentials of the further inclusion of deuteron distortion are also presented. The inclusion of a spin-orbit term in the RGM, adds additional terms to the phase-equivalent potential, most notably the comparatively large imaginary spin-orbit term found empirically.     

Double folding analysis of ~(3)He elastic and inelastic scattering to low lying states on ~(90)Zr,~(116)Sn and ~(208)Pb at 270 MeV

The experimental data on elastic and inelastic scattering of 270 MeV~3 He particles to several low lying states in~(90)Zr,~(116)Sn and~(208)Pb are analyzed within the double folding model(DFM). Fermi density distribution(FDD) of target nuclei is used to obtain real potentials with different powers. DF results are introduced into a modified DWUCK4 code to calculate the elastic and inelastic scattering cross sections. Two choices of potentials form factors are used; Woods Saxon(WS) and Woods Saxon Squared(WS~2) for real potential, while the imaginary part is taken as phenomenological Woods Saxon(PWS) and phenomenological Woods Saxon Squared(PWS~2). This comparison provides information about the similarities and differences of the models used in calculations.     

超冷费米气体的膨胀动力学研究新进展

多体系统的非平衡动力学演化是当前物理学中最具挑战性的问题之一.超冷量子费米原子气体具有较强的可控性,是研究多体非平衡动力学的理想系统,可以用来模拟和理解大爆炸后的早期宇宙、重离子碰撞中产生的夸克-胶子以及核物理等动力学.一般多体系统演化是非常复杂的,往往需要利用对称性来研究.利用Feshbach共振可以制备标度不变的费米原子气体:无相互作用和幺正费米量子气体.当远离平衡态时,可利用普适的指数和函数来刻画,其动力学可以通过对系统的时空演化进行标度变换来识别.本文主要介绍近年来强相互作用超冷费米气体的膨胀动力学研究进展,包括原子气体的各向异性展开、标度动力学和Efimovian膨胀动力学.