Free Nucleon-nucleon Cross Sections Research Articles

Theoretical uncertainties on the extraction of in-medium NN cross sections by different Pauli blocking algorithms * *This work was partly inspired by the transport code comparison project, and it was supported by the National Natural Science Foundation of China (11875323, 11705163, 11790320, 11790323, 11961141003), the National Key R&D Program of China (2018YFA0404404), the Continuous Basic Scientific Research Project (WDJC-2019-13, BJ20002501)

Three typical Pauli blocking algorithms in quantum molecular dynamics type models are investigated in the nuclear matter, the nucleus, and heavy ion collisions. In nuclear matter, the blocking ratios obtained with the three algorithms are underestimated by 13%-25% compared to the corresponding analytical values. For a finite nucleus, spurious collisions occur around the surface of the nucleus owing to the defects of the Pauli blocking algorithms. In the simulations of heavy ion collisions, the uncertainty of stopping power arising from the different Pauli blocking algorithms is less than 5%. Furthermore, the in-medium effects of nucleon-nucleon (NN) cross sections on the nuclear stopping power are discussed. Our results show that the transport model calculations with free NN cross sections result in the stopping power decreasing with beam energy when the beam energy is less than 300 MeV/u. To increase or decrease the values of the stopping power, the transport model calculations need enhanced or suppressed model dependent in-medium NN cross sections that are expected to be smaller than the true in-medium NN cross sections.

Analysis of the flow systematics in Au+Au collisions

The new implementation of the Boltzmann-Uhling-Uhlenbeck equation, the VdWBUU simulation (with EoS-dependent in-medium nucleon-nucleon cross sections) appears to reproduce the flow observables in the Au+Au collisions in the energy range from 400 AMeV to 10 AGeV. The range of the feasible stiffness of the EoS can be identified, based on the analysis presented here, as encompassing compressibilities starting from 250-260 MeV and above, and thus consistent with the results of re-analysis of the giant monopole resonance data (250-310 MeV). Using that additional constraint, the range of feasible values of the stiffness of density dependence can be set as γ=1−1.25, with the value γ=1 appearing as as a global value of stiffness of the symmetry energy feasible over the whole range of constrained compressibilities. The implementation of BUU with the free nucleon-nucleon cross sections can not describe correctly the global trends of flow observables.

Effects of In-Medium Nucleon-Nucleon Cross Section and Nuclear Density Distribution on the Proton-Nucleus Total Reaction Cross Section

We present the proton-total reaction cross sections σR by comparing the experimental data with the Glauber model calculation using four sets of nucleon-nucleon cross sections used in literature (free nucleon-nucleon cross sections (NNCS), phenomenological NNCS, Bruckner-type NNCS and Pauli NNCS) and different nuclear densities for symmetric and asymmetric nuclei. For light symmetric target nuclei, the experimental σR data are well reproduced by the former three NNCS with the ground state density distributions of relativistic mean-field theory. On the other hand, for asymmetry heavier target nuclei, the calculations of the σR data depend significantly on the nuclear density distribution at large radius in the low energy region. The experimental σR of these nuclei are well reproduced by the empirical 3 pF density distribution and the in-medium NNCS. The neutron surface distributions are also discussed based on the 3 pF nuclear charge distributions.

Medium effects in direct reactions

We discuss medium corrections of the nucleon-nucleon (NN) cross sections and their influence on direct reactions at intermediate energies 50 MeV/nucleon. The results obtained with free NN cross sections are compared with those obtained with a geometrical treatment of Pauli-blocking and Dirac-Bruecker methods. We show that medium corrections may lead to sizable modifications for collisions at intermediate energies and that they are more pronounced in reactions involving weakly bound nuclei.

Pauli blocking and medium effects in nucleon knockout reactions

We study medium modifications of the nucleon-nucleon (NN) cross sections and their influence on the nucleon knockout reactions. Using the eikonal approximation, we compare the results obtained with free NN cross sections with those obtained with a purely geometrical treatment of Pauli-blocking and with NN obtained with more elaborated Dirac-Bruecker methods. The medium effects are parametrized in terms of the baryon density. We focus on symmetric nuclear matter, although the geometrical Pauli-blocking allows for the treatment of asymmetric nuclear matter. It is shown that medium effects can change the nucleon knockout cross sections and momentum distributions up to 10% in the energy range E=50-300 MeV/nucleon. The effect is more evident in reactions involving halo nuclei.

Effect of finite range of theNNforce andNNcross section on reaction cross section for neutron rich nuclei

The reaction cross section ({sigma}{sub R}) is calculated using the optical limit of the Glauber theory. A density-dependent effective nucleon-nucleon (NN) cross section {sigma}{sub NN} is considered. Finite and zero range NN interactions are studied. The effect of finite range and an appropriate local density can increase {sigma}{sub R} up to 20% compared to the zero range at constant density (0.16 fm{sup -3}), while a zero range calculation with free NN cross section increases {sigma}{sub R} up to 13%. These factors affect the values of the rms radii for neutron rich nuclei extracted from {sigma}{sub R}.

Glauber model and the heavy ion reaction cross section

We reexamine the Glauber model and calculate the total reaction cross section as a function of energy in the low and intermediate energy range, where many of the corrections in the model are effective. The most significant effect in this energy range is by the modification of the trajectory due to the Coulomb field. The modification in the trajectory due to nuclear field is also taken into account in a self-consistent way. The energy ranges in which particular corrections are effective are quantified, and it is found that when the center of mass energy of the system becomes 30 times the Coulomb barrier, none of the trajectory modification to the Glauber model is really required. The reaction cross sections for light and heavy systems, from the Coulomb barrier to intermediate energies, have been calculated. The exact nuclear densities and free nucleon-nucleon (NN) cross sections have been used in the calculations. The center of mass correction, which is important for light systems, has also been taken into account. There is an excellent agreement between the calculations with the modified Glauber model and the experimental data. This suggests that the heavy ion reactions in this energy range can be explained by the Glauber model in terms of free NN cross sections without incorporating any medium modification.

Effects of the in-mediumNNinteraction on total reaction and neutron removal cross sections

We discuss calculations of total reaction cross sections between complex nuclei, ${\ensuremath{\sigma}}_{R},$ using optical limit and few-body Glauber models in which the free nucleon-nucleon cross sections ${\ensuremath{\sigma}}_{\mathrm{NN}}$ are replaced by their (reduced) values in the nuclear medium. This replacement lowers ${\ensuremath{\sigma}}_{R}$ by at most a few percent when ${\ensuremath{\sigma}}_{\mathrm{NN}}$ is determined from the local matter density in each overlapping volume element of the significant projectile-target trajectories. This relatively small effect contrasts with reductions of about 10% in ${\ensuremath{\sigma}}_{R}$ reported by Xiangzhou et al., who assume a global value for the matter density throughout the interaction region. For two-neutron halo nuclei, we investigate the significance of these in-medium effects for the neutron-removal cross sections, ${\ensuremath{\sigma}}_{\ensuremath{-}2n}.$ We show that use of an in-medium ${\ensuremath{\sigma}}_{\mathrm{NN}}$ raises ${\ensuremath{\sigma}}_{\ensuremath{-}2n}$ for ${}^{6}\mathrm{He}$ but lowers it for ${}^{11}\mathrm{Li},$ because of their different halo sizes.

Elliptic Flow in Heavy-Ion Collisions near the Balance Energy

The proton elliptic flow in collisions of Ca on Ca at energies from 30 to 100 MeV/nucleon is studied in an isospin-dependent transport model. With increasing incident energy, the elliptic flow shows a transition from positive to negative flow. Its magnitude depends on both the nuclear equation of state (EOS) and the nucleon-nucleon scattering cross section. Different elliptic flows are obtained for a stiff EOS with free nucleon-nucleon cross sections and a soft EOS with reduced nucleon-nucleon cross sections, although both lead to vanishing in-plane transverse flow at the same balance energy. The study of both in-plane and elliptic flows at intermediate energies thus provides a means to extract simultaneously the information on the nuclear equation of state and the nucleon-nucleon scattering cross section in medium.

Breakup reactions of the halo nuclei 11Be and 8B.

We calculate the nuclear induced breakup of $^{11}\mathrm{Be}$ and $^{8}\mathrm{B}$ using a more realistic treatment of the diffraction and stripping processes than in previous work. The breakup is treated in the eikonal approximation with a profile function calculated from a realistic optical potential at low energies and from free nucleon-nucleon cross sections at high energies. This treatment gives a good description of measured breakup cross sections, as well as the longitudinal momentum distribution of the corelike fragments, which is narrower than predicted in the transparent limit. The real part of the potential is found to be significant and enhances the diffractive breakup at low energies. \textcopyright{} 1996 The American Physical Society.

Compressibility probed by linear momentum transfer.

The linear momentum transfer occurring in central heavy ion collisions is studied within the Landau Vlasov model. A sensitivity to the equation of state is found for this observable. Assuming the free nucleon-nucleon cross section, a soft compressibility modulus (around 210 MeV) is required to reproduce the large amount of data available for linear momentum transfer. This result stands for local and nonlocal effective potentials. \textcopyright{} 1996 The American Physical Society.

Probing the EOS and nn cross section by analyzing the collective flow at high and intermediate energy in the RVUU approach

We have studied the collective flow at high and intermediate energy in a relativistic Vlasov-Uehling-Uhlenbeck (RVUU) approach based on Walecka's QHD-I model, with the aim to probe the nuclear-matter equation of state (EOS) and the in-medium nucleon-nucleon cross section σ. At high energy (1.2 GeV/u), the out-of-plane azimuthal correlation function C( Ψ) is only sensitive to the effective mass m ∗ and insensitive to the nuclear compressibility K and the effective nucleon-nucleon cross section σ within a reasonable range. We have found that the preferred value of m ∗ is about 0.85 m. With this value of m ∗ , from the in-plane mean transverse momentum 〈 P x ( Y)〉 which is sensitive to both m ∗ and σ we have drawn an effective nn cross section σ, namely σ ∼ 0.8 σ f where σ f is the free nucleon-nucleon cross section in Cugnon's parametrization. Taking advantage of the fact that the energy of vanishing flow (EVF) at intermediate energy (around 100 MeV/u) is only sensitive to the nucleon-nucleon cross section σ, we have drawn some information on the nucleon-nucleon cross section σ, namely σ = (1.4±0.2) σ f.

Analysis of flow disappearance in the RVUU approach

A theoretical analysis of flow dissappearance in heavy-ion collisions is performed in the relativistic Vlasov-Uehling-Uhlenbeck (RVUU) approach. We find that the energy of vanishing flow (EVF) is insensitive to the impact parameter b and the momentum dependent mean field. By comparing the numerical results to the experimental data, a nucleon-nucleon cross section σ=(1.4±0.2) σ f is inferred where σ f is the energy-dependent free nucleon-nucleon cross section.

Heavy-ion excitation of giant resonances at intermediate energies

We develop a Glauber-type formalism to study the excitation of giant resonances in heavy-ion collisions at intermediate energies. Contributions from the nuclear and the Coulomb interaction as well as their interference are easily accomplished within this approach. Application is done for the excitation of isovector giant dipole and isoscalar giant quadrupole resonances in 208Pb by means of 17O projectiles at 84 MeV nucleon laboratory energy. A good agreement with the experimental data is found. The formalism is parameter-free, having as inputs the free nucleonnucleon cross sections, the ground-state densities and the transition densities of the nuclei.

Kinetic approach to the damping of giant quadrupole resonances.

The effect of one- and two-body dissipation on the damping of giant quadrupole resonances is studied in a semiclassic approach solving a Vlasov equation with a collisional relaxation time. The latter is microscopically evaluated from the equilibration of a distorted momentum distribution in a kinetic approach. Important effects from energy and angle dependent nucleon-nucleon (NN) cross sections and from the time variation of Pauli blocking are stressed. Once these points are suitably treated, a good agreement with the experimental systematics is obtained from the use of a free NN cross section.

Proton propagation in nuclei studied in the A dependence of the (e,e'p) reaction in the quasifree region.

The {ital A} dependence of the ({ital e},{ital e}{prime}{ital p}) reaction in the quasifree region has been measured at an average {ital Q}{sup 2} of 0.33 (GeV/{ital c}){sup 2} for targets of {sup 12}C, {sup 27}Al, {sup 58}Ni, and {sup 181}Ta. The outgoing proton kinetic energy was 180{plus minus}30 MeV. By comparing the ratio of ({ital e},{ital e}{prime}{ital p}) coincidence to ({ital e},{ital e}{prime}) singles yields, average proton transmissions are obtained for each target. The resulting mean free path'' or, more precisely, the attenuation length for protons in the nucleus is significantly longer than expectations based on the free nucleon-nucleon cross section.

Description of heavy ion reactions with medium dependent forces

The collision between two nuclei is calculated in the Quantum Molecular Dynamics (QMD) approach for C-C and Nb-Nb reactions using a collision term for the nucleon-nucleon cross section which is determined in the nuclear medium using the Brueckner G-matrix with the appropriate Pauli operator and the self consistent single particle energies for the densities of the projectile and the target at the collision point of the two nucleons. The QMD approach is very similar to the Vlasov-Uehling-Uhlenbeck or Boltzmann-Uehling-Uhlenbeck approach, but taking into account the finite extension of the wave packet representing a nucleon in position and momentum space. Usually one uses in the collision term a constant (40 mb) or the free nucleon-nucleon cross section. Using a local density approximation the Bethe-Goldstone equation is solved for two colliding nuclear matters of which the densities are determined by the local densities obtained in the QMD approach at the collision point of the two nucleons inside the nuclear medium. Comparing the momentum transfer in transverse and longitudinal direction and the number of emitted particles calculated with the G-matrix cross section and the isotropic constant cross section (40 mb) we find for this inclusive data only minor differences between the results in the C-C collision, but larger ones for the Nb-Nb case. The double differencial cross section for charge particle emission in the C-C reaction at E lab = 84 MeV/n is described in good agreement with the data.

Covariant Boltzmann-Uehling-Uhlenbeck approach for heavy-ion collisions.

We present a covariant transport theory for heavy-ion collisions based on the sigma-..omega.. model. The two-body collision term is introduced in line with relativistic classical kinetic theory assuming the free nucleon-nucleon cross section in the collision integral. Within this approach, we study collisions of /sup 16/O+/sup 16/O at 600 MeV/nucleon for two different parameter sets of the underlying Lagrangian density, i.e., different equations of state. The most striking result is the strong sensitivity of the transverse momentum distribution on the momentum dependence of the mean field which is self-consistently included in the relativistic approach. We find that residual two-particle collisions might even slightly reduce the transverse momentum p/sub t/ due to an increase of stopping power obtained from the collisions. An increase of the cross section by a factor of 2 leads to a much smaller enhancement of p/sub t/ than observed in nonrelativistic calculations with a momentum independent mean field. We furthermore present a first application to high-energy photon production and study the influence of the equation of state on the differential photon yield.

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.

High momentum nucleons from relativistic nuclear collisions

A hard-scattering mechanism is proposed to describe proton and pion spectra from high-energy nuclear collisions. Results of a parameter-free model using free nucleon-nucleon cross sections and the empirical momentum distribution for nucleons within the nuclei are compared with data from C + C, Ne + NaF, and C + Pb collisions at 800 MeV/nucleon.