Information about the nuclear matter under the extreme conditions of temperature and density and the role of symmetry energy under these conditions is still a topic of crucial importance in the present day nuclear physics research. The multifragmentation, collective flow and the nuclear stopping is among the various rare phenomenon which can be observed in heavy-ion collisions at intermediate energies. The nuclear stopping, which is sensitive towards the symmetry energy has gained a lot of interest because it provides the possibility to examine the degree of thermalization or equilibration in the matter. Aim of the present study is to pin down the nuclear stopping for the different forms of density dependent symmetry energy
The effect of density dependent symmetry energy on nuclear-stopping is studied using isospin-dependent quantum molecular dynamics model(IQMD). We have used the reduced isospin-dependent cross-section with soft(S) equation of state for the systems having different isostopic content, to explore the various aspects of nuclear stopping. The aim is to pin down the nature of the nuclear stopping with density dependent symmetry energy. Nuclear stopping is found to be sensitive towards the various forms of the density dependent symmetry energy. The nuclear stopping tends to decrease for the stiffer eq…
Nuclear stopping in heavy ion collisions (HIC) has been studied by means of rapidity distribution and asymmetry of nucleon momentum distribution. It is an important quantity in determining the outcome of a reaction. Fen Fu {\it et al.,} \cite{2} calculate both the radial flow and the degree of nuclear stopping in Pb + Pb and Ni + Ni at 0.4, 0.8 and 1.2 GeV/neucleon. They found that the expansion velocity as well as the degree of nuclear stopping are higher in the heavier system at all energies.
The role of momentum dependent interactions is studied in nuclear stopping at intermediate energy using an isospin-dependent quantum molecular dynamics model. Present calculations are performed at incident energy between 50 and 1000 MeV/nucleon. Our findings show that nuclear stopping is sensitive towards the impact parameter, incident energies as well as towards the mass of the colliding system. Further, the degree of stopping is suppressed by the inclusion of momentum dependent interactions, whereas the particle production is not affected by the MDI.
In the present analysis, thousand of events were simulated for the neutron rich reaction of 54Xe131 + 54Xe131, at different impact parameters using hard (HMD) and soft (SMD) momentum dependent equation of state. one can clearly see more stopping in the presence of HMD. Production of LCP's and free particles follows the same trend as stopping parameter R. This simply implies that the fragment production can act as a global indicator for the nuclear stopping.
Using an isospin-dependent quantum molecular dynamics (IQMD) model, nuclear stopping is analyzed in asymmetric colliding channels by keeping the total mass fixed. The calculations have been carried by varying the asymmetry of the colliding pairs with different neutron-proton ratios in center of mass energy 250 MeV/nucleon and by switching off the effect of Coulomb interactions. We find sizable effect of asymmetry of colliding pairs on the stopping and therefore on the equilibrium reached in a reaction.
Using the isospin dependent quantum molecular dynamics model, we study the effect of charge asymmetry and isospin dependent cross-section on nuclear stopping and multiplicity of free nucleons and LMF's. Simulations were carried out for the reactions $^{124}X_{m}+^{124}X_{m}$, where m varies from 47 to 59 and for $^{40}Y_{n}+^{40}Y_{n}$, where n varies from 14 to 23. Our study shows that nuclear stopping as well as the production of LMF's depend strongly on the isospin of the cross-section.
Using an isospin-dependent quantum molecular dynamics, nuclear stopping in intermediate heavy ion collisions has been studied. The calculation has been done for colliding systems with different neutron-proton ratios in beam energy ranging from 15MeV/u to 150MeV/u. It is found that, in the energy region from above Fermi energy to 150MeV/u, nuclear stopping is very sensitive to the isospin dependence of in-medium nucleon-nucleon cross section, but insensitive to symmetry potential. From this investigation, we propose that nuclear stopping can be used as a new probe to extract the information on …
Data on stopping in intermediate-energy central heavy-ion collisions are analyzed following transport theory based on the Boltzmann equation. In consequence, values of nuclear shear viscosity are inferred. The inferred values are significantly larger than obtained for free nucleon dispersion relations and free nucleon-nucleon cross sections.
A systematic study of the in-medium correction factor ($F$) on nucleon-nucleon elastic cross section is performed within the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model. The effects of the beam energy dependence of $F$ on the directed, elliptic flow and nuclear stopping in $^{197}$Au+$^{197}$Au collisions with energy ranging from $0.09$ to $0.8A$ GeV are explored. It is found that the directed, elliptic flow and nuclear stopping at relatively low energies are very sensitive to $F$, and the sensitivity gradually weakens with increasing beam energy. The beam energy dependent in-m…
The transverse momentum distribution of produced charged particles is investigated for gold-gold collisions at $\sqrt{s_{NN}}=200$ GeV. A simple parameterization is suggested for the particle distribution based on the nuclear stopping effect. The model can fit very well both the transverse momentum distributions at different pseudo-rapidities and the pseudo-rapidity distributions at different centralities. The ratio of rapidity distributions for peripheral and central collisions is calculated and compared with the data.
We present transport calculations for heavy ion reactions in which the mean field and the in-medium nucleon-nucleon cross section are consistently based on the same effective interaction, i.e. the in-medium T-matrix from microscopic Dirac-Brueckner calculations. Doing so, the stopping in central reactions in terms of the recently proposed $var_{\rm tl}$ observable and the correlation to the behavior of the directed flow is investigated. The relation to the nuclear shear viscosity is discussed.
By using the Isospin Dependent Quantum Molecular Dynamics Model (IQMD), we study the dependence of nuclear stopping Q_{ZZ}/A and R in intermediate energy heavy ion collisions on system size, initial N/Z, isospin symmetry potential and the medium correction of two-body cross sections. We find the effect of initial N/Z ratio, isospin symmetry potential on stopping is weak. The excitation function of Q_{ZZ}/A and R depends on the form of medium correction of two-body cross sections, the equation of state of nuclear matter (EOS). Our results show the behavior of the excitation function of Q_{ZZ}/A…
Nuclear stopping in the heavy ion collisions over a beam energy range from SIS, AGS up to SPS is studied in the framework of the modified UrQMD transport model, in which mean field potentials of both formed and "pre-formed" hadrons (from string fragmentation) and medium modified nucleon-nucleon elastic cross sections are considered. It is found that the nuclear stopping is influenced by both the stiffness of the equation of state and the medium modifications of nucleon-nucleon cross sections at SIS energies. At the high SPS energies, the two-bump structure is shown in the experimental rapidity…
We present a complete systematics (excitation function, impact parameter, system size, isospin asymmetry, and equations of state dependences) of global stopping and fragment production for heavy-ion reactions in the energy range between 50 and 1000 MeV/nucleon in the presence of symmetry energy and an isospin-dependent cross section. It is observed that the degree of stopping depends weakly on the symmetry energy and strongly on the isospin-dependent cross section. However, the symmetry energy and isospin-dependent cross section has an effect of the order of more than 10% on the emission of li…
We study the nuclear stopping in high energy nuclear collisions using the constituent quark model. It is assumed that wounded nucleons with different number of interacted quarks hadronize in different ways. The probabilities of having such wounded nucleons are evaluated for proton-proton, proton-nucleus and nucleus-nucleus collisions. After examining our model in proton-proton and proton-nucleus collisions and fixing the hadronization functions, it is extended to nucleus-nucleus collisions. It is used to calculate the rapidity distribution and the rapidity shift of final state protons in nucle…
The correlation between the nuclear stopping and the scale invariant nucleon sideward flow at energies ranging from those available at the GSI heavy ion synchrotron (SIS) to those at the CERN Super Proton Synchrotron (SPS) is studied within ultrarelativistic quantum molecular dynamics (UrQMD). The universal behavior of the two experimental observables for various colliding systems and scale impact parameters are found to be highly correlated with each other. As there is no phase transition mechanism involved in the UrQMD, the correlation may be broken down by the sudden change of the bulk prop…
Within a Boltzmann transport model, we demonstrate correlation between stopping observables and shear viscosity in central nuclear collisions at intermediate energies (on the order of 10 to 1000 MeV/nucleon). The correlation allows us to assess the viscosity of nuclear matter, by tuning the in-medium nucleon-nucleon cross section in our transport model to agree with nuclear stopping data. We also calculate the ratio of shear viscosity to entropy density to determine how close the system is to the universal quantum lower limit proposed in the context of ultrarelativistic heavy ion collisions.
This paper reviews our recent findings on the dynamics of transport of baryon number in proton-induced reactions at the CERN SPS. These are put in a more general context of the present understanding of baryon stopping phenomena up to LHC and cosmic ray energies. The implications of our studies, advantages to be provided by modern and high quality experimental data, and perspectives of new measurements are shortly discussed.
The detailed analysis of wide excitation function of nuclear stopping has been studied within a transport model, Isospin-dependent Quantum Molecular Dynamics model (IQMD) and an overall good agreement with the INDRA and FOPI experimental data has been achieved. It is found that mean value of isotropy in central Heavy-Ion Collision (HIC) reaches a minimum near Fermi energy and approaches a maximum around 300 - 400$A$ MeV. This suggests that, in statistical average, the equilibration is far from being reached even in central HIC especially near Fermi energy. A hierarchy in stopping of fragments,…
Using the isospin-dependent quantum molecular dynamical (IQMD) model, we systematically study the role of momentum dependent interactions in global stopping and analyze the effect of symmetry energy in the presence of momentum dependent interactions. For this, we simulate the reactions by varying the total mass of the system from 80 to 394 at different beam energies from 30 to 1000 MeV/nucleon over central and semi-central geometries. The study is carried in the presence of momentum dependent interactions and symmetry energy by taking into account hard equation of state. The nuclear stopping i…
We propose a numerical definition for baryon stopping in relativistic heavy ion collisions that is obtainable from final hadron rapidity distributions as well as from bremsstrahlung measurements. Thus a new channel of communication is opened between the two methods.
We address the stopping in heavy-ion induced reactions around the Fermi energy using central collisions recorded with \emph{INDRA} $4π$ array. The stopping is minimal around the Fermi energy and corresponds to the disappearance of the Mean-Field effects and the appearance of nucleon-nucleon elastic collisions. This phenomenon is attributed to a change in the dissipation regime going from 1-body (Mean-Field) dissipation at low incident energy to 2- body (nucleonic collisions) above the Fermi energy. A connection to the in-medium transport properties of nuclear matter is proposed and comprehensi…
We formulate a lattice Hamiltonian model for intermediate energy heavy ion collisions. Our approach incorporates a momentum-dependent nuclear mean field that yields an optical potential that agrees with proton-nucleus experiments and that also receives support from measurements involving heavy ions. We emphasize the precision of our numerical results in connection with energy and momentum conservation. We first outline the static properties of our solutions, after which we consider recent stopping power results for heavy ion collisions. The results obtained with different types of nuclear mean…
With the newly updated version of the ultrarelativistic quantum molecular dynamics (UrQMD) model, a systematic investigation of the effects of in-medium nucleon-nucleon ($NN$) elastic cross section on the collective flow and the stopping observables in $^{197}\text{Au}+^{197}\text{Au}$ collisions at beam energies from 40 to 150 MeV/nucleon is performed. Simulations with the medium correction factor $\mathcal{F}=σ^{\text{in-medium}}_{NN}/σ^{\text{free}}_{NN}=0.2,~0.3,~0.5$, and the one obtained with the FU3FP1 parametrization which depends on both the density and the momentum are compared to th…
By considering three different Nucleon-Nucleon (NN) elastic differential cross sections: the Cugnon \emph{et al.} parameterized differential cross section [Nucl. Instrum. Methods Phys. Res., Sect. \textbf{B111}, 215 (1996)], and the differential cross section derived from the collision term of the self-consistent relativistic Boltzmann-Uehling-Uhlenbeck equation proposed by Mao \emph{et al.} [Z.\ Phys.\ A {\bf 347}, 173 (1994)], as well as the isotropic differential cross section, within the newly updated version of the ultrarelativistic quantum molecular dynamics (UrQMD) model, the influence …
The stopping power of protons and deuterons in low energy collisions with helium gas targets is investigated with the numerical solution of the time-dependent Schrödinger coupled-channels equations using molecular orbital wavefunctions. It is shown that at low projectile energies the energy loss is mainly due to nuclear stopping, charge exchange, and the excitation of low energy levels in the target. The second and third mechanisms, called electronic stopping, dominate for E>200 eV. At lower energies it is also shown that a threshold effect is responsible for a quick drop of the energy loss. T…
Three typical algorithms of Pauli blocking in the quantum molecular dynamics type models are investigated in the nuclear matter, the nucleus and the heavy ion collisions. The calculations in nuclear matter show that the blocking ratios obtained with the three algorithms are underestimated 13-25\% compared to the analytical values of blocking ratios. For the finite nucleus, the spurious collisions occur around the surface of the nucleus owing to the defects of Pauli blocking algorithms. In the simulations of heavy ion collisions, the uncertainty of stopping power from different Pauli blocking a…
he velocity dependence of the stopping power of swift protons and deuterons in low energy collisions with hydrogen and helium gas targets is investigated with the numerical solution of the time-dependent Schroedinger coupled-channels equations using molecular orbital wavefunctions. At low projectile energies the stopping is mainly due to nuclear stopping, charge exchange of the electron, and excitation of the lowest levels in the target. The second and third mechanisms dominate at E < 200 eV. At lower energies it is also shown that a threshold effect is responsible for a quick drop of the ener…
We investigate the density behaviour of the symmetry energy with respect to isospin equilibration in the combined systems $Ru(Zr)+Zr(Ru)$ at relativistic energies of 0.4 and $1.528 AGeV$. The study is performed within a relativistic framework and the contribution of the iso-vector, scalar $δ$ field to the symmetry energy and the isospin dynamics is particularly explored. We find that the isospin mixing depends on the symmetry energy and a stiff behaviour leads to more transparency. The results are also nicely sensitive to the "fine structure" of the symmetry energy, i.e. to the covariant prope…
A parton-hadron cascade model which is the extension of hadronic cascade model incorporating hard partonic scattering based on HIJING is presented to describe the space-time evolution of parton/hadron system produced by ultra-relativistic nuclear collisions. Hadron yield, baryon stopping and transverse momentum distribution are calculated and compared with HIJING and VNI.
With the semiclassical Landau-Vlasov transport model we studied the stopping observable $R_E$, the energy-based isotropy ratio, for the $^{129}$Xe\,+\,$^{120}$Sn reaction at beam energies spanning 12$A$ to 100$A$ MeV. We investigated the impacts of the nonlocality of the nuclear mean field, of the in-medium modified nucleon-nucleon ($NN$) cross section and of the reaction centrality. A fixed set of model parameters yields $R_E$ values that favorably compare with the experimental ones, but only for energies below the Fermi energy $E_F$. Above $E_F$ agreement is readily possible, but by a smooth…
We present a phenomenological model for the initial conditions needed in a one-fluid hydrodynamical description of ultrarelativistic nuclear collisions at CERN--SPS. The basic ingredient is the parametrization of the baryon stopping, i.e. the rapidity distribution, as a function of the thickness of the nuclei. We apply the model to S + S and Pb + Pb collisions and find after hydrodynamical evolution reasonable agreement with the data.
We study nuclear stopping in central collisions for heavy-ion induced reactions in the Fermi energy domain, between $15$ and $100$ A\,\textrm{MeV}. Using the large dataset of exclusive measurements provided by the $4π$ array \emph{INDRA}, we determine the relative degree of stopping as a function of system mass and bombarding energy. We show that the stopping can be directly related to the transport properties in the nuclear medium. By looking specifically at free nucleons (here protons), we present for the first time a comprehensive body of experimental results concerning the mean free path, …
We study the different nucleus-nucleus collisions, O+Au, S+S, S+Ag, S+Au and Pb+Pb, at the CERN-SPS energy in a one-fluid hydrodynamical approach using a parametrization based on baryon stopping in terms of the thickness of colliding nuclei. Good agreement with measured particle spectra is achieved. We deduce the mass number scaling behaviour of the initial energy density. We find that the equilibration time is nearly independent of the size of the colliding nuclei.
The symmetry energy coefficients for nuclei with mass number A=20~250 are extracted from more than 2000 measured nuclear masses. With the semi-empirical connection between the symmetry energy coefficients of finite nuclei and the nuclear symmetry energy at reference densities, we investigate the density dependence of symmetry energy of nuclear matter at subnormal densities. The obtained results are compared with those extracted from other methods.
The velocity dependence of the stopping power of swift protons and deuterons in low energy collisions is investigated. At low projectile energies the stopping is mainly due to nuclear stopping and charge exchange of the electron. The second mechanism dominates after Ep >=200 eV. A dynamical treatment of the charge exchange mechanism based on two-center electronic wavefunctions yields very transparent results for the exchange probability. We predict that the stopping cross sections vary approximately as v to the 1.35 power for projectile protons on hydrogen targets in the 1 keV energy region.
By using the functional path integral method, we obtain a model-independent formula for nuclear symmetry energy, which explicitly shows the relation between nuclear symmetry energy and isospin susceptibility. The latter one is found to be a probe to the QCD chiral phase transition. We further found that, the nuclear symmetry energy has an abrupt change at the critical nuclear density where the chiral symmetry restores partially, which could be detected from the experiments.
We formulate a Lattice Hamiltonian approach for the modeling of intermediate energy heavy ion collisions. After verifying stationary ground state solutions, we implement this in a calculation of nuclear stopping power and compare our results with experimental data. Our findings support a relatively soft nuclear equation of state, with a momentum-dependent self-consistent mean field.
Extrapolations from nuclei to neutron stars hinge on the symmetry term in nuclear binding formulas. The term describes reduction in the binding associated with neutron-proton (np) imbalance. Regrettably, binding formulas in the literature commonly lack an intrinsic consistency with regard to the symmetry term. Our elementary considerations determine the universal macroscopic limit for the term and predict its weakening in light nuclei due to emergence of the nuclear asymmetry-skin. Experimental systematic of isobaric-analogue states allows for a determination of the volume and surface coeffici…
The liquid-gas phase transition in hot asymmetric nuclear matter is investigated within relativistic mean-field model using the density dependence of nuclear symmetry energy constrained from the measured neutron skin thickness of finite nuclei. We find symmetry energy has a significant influence on several features of liquid-gas phase transition. The boundary and area of the liquid-gas coexistence region, the maximal isospin asymmetry and the critical values of pressure and isospin asymmetry all of which systematically increase with increasing softness in the density dependence of symmetry ene…
Based on the general analysis of the grand canonical partition function in the S-matrix framework, the calculated results on symmetry energy, free energy and entropy of dilute warm nuclear matter are presented. At a given temperature and density, the symmetry energy or symmetry free energy of the clusterized nuclear matter in the S-matrix formulation deviates, particularly at low temperature and relatively higher density, in a subtle way, from the linear dependence on the square of the isospin asymmetry parameter $X=(ρ_n-ρ_p)/(ρ_n+ρ_p)$, contrary to those obtained for homogeneous nucleonic mat…
Using the relativistic impulse approximation with empirical NN scattering amplitude and the nuclear scalar and vector densities from the relativistic mean-field theory, we evaluate the Dirac optical potential for neutrons and protons in asymmetric nuclear matter. From the resulting Schrö% dinger-equivalent potential, the high energy behavior of the nuclear symmetry potential is studied. We find that the symmetry potential at fixed baryon density is essentially constant once the nucleon kinetic energy is greater than about 500 MeV. Moreover, for such high energy nucleon, the symmetry potential …
We examine how nuclear masses are related to the density dependence of the symmetry energy. Using a macroscopic nuclear model we calculate nuclear masses in a way dependent on the equation of state of asymmetric nuclear matter. We find by comparison with empirical two-proton separation energies that a smaller symmetry energy at subnuclear densities, corresponding to a larger density symmetry coefficient L, is favored. This tendency, which is clearly seen for nuclei that are neutron-rich, nondeformed, and light, can be understood from the property of the surface symmetry energy in a compressibl…
These are personal recollections of how Gerry Brown's work and thinking influenced the development of the chiral effective Lagrangian based theory of nuclear forces.
We simulate the central reactions of nearly symmetric, and asymmetric systems, for the energies at which the maximum production of IMFs occurs (E$_{c.m.}^{peak}$).This study is carried out by using hard EOS along with cugnon cross section and employing MSTB method for clusterization. We study the various properties of fragments. The stability of fragments is checked through persistence coefficient and gain term. The information about the thermalization and stopping in heavy-ion collisions is obtained via relative momentum, anisotropy ratio, and rapidity distribution. We find that for a complet…
A study of the correlation between nuclear temperatures and symmetry energy is presented for heavy-ion collisions at intermediate energies via the isospin-dependent quantum molecular-dynamics model. It is found that different symmetry energy parameters will change the density and kinetic energy distribution of the hot nuclei. More importantly, nuclear temperatures that are based on kinetic energy properties can be used to study symmetry energy information.
Interplay between the dependence of symmetry energy on density and the variation of nucleonic densities across nuclear surface is discussed. That interplay gives rise to the mass dependence of the symmetry coefficient in an energy formula. Charge symmetry of the nuclear interactions allows to introduce isoscalar and isovector densities that are approximately independent of the magnitude of neutron-proton asymmetry.
We review the main achievements of the research programme for the study of nuclear forces in the framework of chiral symmetry and discuss some problems which are still open.
The pervasive role of the nuclear symmetry energy in establishing some nuclear static and dynamical properties, and in governing some attributes of neutron star properties is highlighted.
Two recent experiments have demonstrated that the effective $ρ$-mass in nuclear medium, as extracted from the $^3He(γ, π^+ π^-)$ reaction, is substantially reduced. This has been advocated as an indication of partial restoration of chiral symmetry in nuclear matter. We show that even in the absence of chiral symmetry, effective mean field nuclear matter models can explain these findings quantitatively.
We recall the main features of the Td (tetrahedral) symmetry in atomic nuclei and present realistic mean-field calculations supporting the existence such a symmetry all over the nuclear chart. A few potential candidate-nuclei are investigated and the possible experimental signatures of the tetrahedral symmetry are also briefly discussed.
Using the relativistic mean-field theory, we adopt two different methods, namely, the coexisting phase method and the self-consistent Thomas-Fermi approximation, to study the impacts of the nuclear symmetry energy on properties of neutron star crusts within a wide range of densities. It is found that the nuclear symmetry energy and its density slope play an important role in determining the pasta phases and the crust-core transition.
With the global color symmetry model being extended to finite chemical potential, we study the density dependence of the local and nonlocal scalar quark condensates in nuclear matter. The calculated results indicate that the quark condensates increase smoothly with the increasing of nuclear matter density before the critical value (about 12$ρ_0$) is reached. It also manifests that the chiral symmetry is restored suddenly as the density of nuclear matter reaches its critical value. Meanwhile, the nonlocal quark condensate in nuclear matter changes nonmonotonously against the space-time distance…
The Interacting Boson Model of nuclear structure is introduced to the unitarity limit. Non relativistic conformal symmetry creates a scale invariant state from which the stationary states of atomic nuclei are obtained. A brief discussion of the nature of the conformal field theory is included along with the BCS-BEC crossover and hydrodynamic properties.
Within a dynamical description of nuclear fragmentation, based on the liquid-gas phase transition scenario, we explore the relation between neutron-proton density fluctuations and nuclear symmetry energy. We show that, along the fragmentation path, isovector fluctuations follow the evolution of the local density and approach an equilibrium value connected to the local symmetry energy. Higher density regions are characterized by smaller average asymmetry and narrower isotopic distributions. This dynamical analysis points out that fragment final state isospin fluctuations can probe the symmetry …
