We study the effect of isospin degree of freedom on the balance energy (E$_{bal}$) as well as its mass dependence throughout the mass range for two different sets of isobaric systems with N/Z = 1 and 1.4 at different colliding geometries ranging from the central to peripheral ones. Our findings reveal the dominance of Coulomb repulsion in isospin effects on E$_{bal}$ as well as its mass dependence throughout the range of the colliding geometry. Our results also indicate that the effect of symmetry energy on the energy of vanishing flow is uniform throughout the mass range and throughout the colliding geometry.
The effects of an isospin asymmetry and in-medium corrections to the nucleon collision cross section on the balance energy are explored. The BUU model for intermediate energy heavy-ion collisions is used with isospin-dependent mean fields to calculate the balance energies of 58Fe + 58Fe and 58Ni + 58Ni for a range of impact parameters. We find that we are able to reproduce the impact parameter dependence of the balance energy, and the sign (but not the magnitude) of the shift in balance balance energy as a function of isospin asymmetry.
The effect of isospin degree of freedom on balance energy and its mass dependence has been studied for the mass range between 50 and 350. Our results shows the dominance of Coulomb potential in isospin effects.
The effect of different isospin-dependent cross-section on directed flow is studied for variety of systems(for which experimental balance energies are available) using an isospin-dependent Quantum Molecular Dynamic (IQMD) model. We show that balance energies are sensitive towards isospin-dependent cross-sections for light systems, while nearly no effect exist for heavier nuclei. A reduced cross-section $σ= 0.9σ_{NN}$ with stiff equation of state is able to explain experimental balance energies in most of systems. A power law behaviour is also given for the mass dependence of balance energy, wh…
We study the effect of isospin degree of freedom on the balance energy as well as its mass dependence throughout the mass range 48-270 for two sets of isobaric systems with N/Z = 1 and 1.4 using isospin-dependent quantum molecular dynamics (IQMD) model. Our fndings reveal the dominance of Coulomb repulsion in isospin effects on balance energy as well as its mass dependence throughout the range of the colliding geometry.
Based on the isospin-dependent quantum molecular dynamics (IQMD) picture, we attempt to understand the nature of transverse flow in $_{28}Ni^{58}+_{28}Ni^{58}$ and $_{26}Fe^{58}+_{26}Fe^{58}$ systems at wide range of energies and impact parameters. The isospin dependence of balance energy in transverse flow is clearly visible. The results are compared with the experimental data available.
We study the effect of isospin degree of freedom on balance energy throughout the mass range between 50 and 350 for two sets of isobaric systems with N/A = 0.5 and 0.58. Our fndings indicate that different values of balance energy for two isobaric systems may be mainly due to the Coulomb repulsion. We also demonstrate clearly the dominance of Coulomb repulsion over symmetry energy.
Using the isospin-dependent quantum molecular dynamics model we study the isospin effects on the disappearance of flow for the reactions of 58Ni+58Ni and 58Fe+58Fe as a function of impact parameter. We found good agreement between our calculations and experimentally measured energy of vanishing flow at all colliding geometries. Our calculations reproduce the experimental data within 5%(10%) at central (peripheral) colliding geometries.
We study the effect of isospin degree of freedom on balance energy throughout the mass range between 50 and 350 for two sets of isotopic systems with N/Z = 1.16 and 1.33 as well as isobaric systems with N/Z = 1.0 and 1.4. Our findings indicate that different values of balance energy for two isobaric systems may be mainly due to the Coulomb repulsion. We also demonstrate clearly the dominance of Coulomb repulsion over symmetry energy.
We classify $A$--nucleon forces according to their isospin dependence and discuss the most general isospin structure of the three--nucleon force. We derive the leading and subleading isospin--breaking corrections to the three--nucleon force using the framework of chiral effective field theory.
For this study we have simulated the 28Ni58+28Ni58 and 26Fe58+26Fe58 systems at incident energies ranging from 45 MeV/nucleon to 105 MeV/nucleon by using soft equation of state. The geometry of reactions chosen are = b/bmax = 0.2, 0.4, 0.5, 0.6, and 0.7.The Px/A is becoming more positive for neutron - rich system 26Fi58 as compared to neutron-poor system 28Ni58. This point towards the higher value of balance energy for 26Fi58 system. Further study in this direction is ongoing.
Mass and isospin dependence of symmetry energy coefficients $a_{sym}$ of finite nuclei is investigated with the measured nuclear masses incorporating the liquid drop mass formula. The enhanced $a_{sym}$ for nearly symmetric nuclei are observed. To describe the mass and isospin dependence of $a_{sym}$, a modified formula based on the conventional surface-symmetry term is proposed and the corresponding rms deviation of nuclear masses is checked.
The isospin dependence of the spin-orbit potential is investigated for an effective Skyrme like energy-functional suitable for density-dependent Hartree-Fock calculations. The magnitude of the isospin dependence is obtained from a fit to experimental data on finite spherical nuclei. It is found to be close to that of relativistic Hartree models. Consequently the anomalous kink in the isotope shifts of Pb nuclei is well reproduced.
We investigate the $^1S_0$ neutron and proton superfluidity in isospin asymmetric nuclear matter. We have concentrated on the isospin dependence of the pairing gaps and the effect of a microscopic three-body force. It is found that as the isospin asymmetry goes higher, the neutron $^1S_0$ superfluid phase shrinks gradually to a smaller density domain, while the proton one extends rapidly to a much wider density domain. The three-body force turns out to weaken the neutron $^1S_0$ superfluidity slightly, but it suppresses strongly the proton $^1S_0$ superfluidity at high densities in nuclear mat…
The liquid-gas phase transition in hot neutron-rich nuclear matter is investigated within a self-consistent thermal model using different interactions with or without isospin and/or momentum dependence. The boundary of the phase-coexistence region is shown to be sensitive to the density dependence of the nuclear symmetry energy as well as the isospin and momentum dependence of the nuclear interaction.
The isospin dependence of the spin-orbit potential is investigated for an effective Skyrme-like energy functional suitable for density dependent Hartree-Fock calculations. The magnitude of the isospin dependence is obtained from a fit to experimental data on finite spherical nuclei. It is found to be close to that of relativistic Hartree models. Consequently, the anomalous kink in the isotope shifts of Pb nuclei is well reproduced.
We study the relative contribution of the symmetry energy and isospin dependence of the nucleon-nucleon cross section to the collective transverse in-plane flow for the reactions of Ca+Ca having N/Z varying from 1.0, 1.6 and 2.0. We see that role of symmetry energy to the flow increases with N/Z of the system.
By the use of the latest experimental data on the spectra of $^{133}$Sb and $^{131}$Sn and on the analysis of properties of other odd nuclei adjacent to doubly magic closed shells the isospin dependence of a mean spin-orbit potential is defined. Such a dependence received the explanation in the framework of different theoretical approaches.
Using isospin dependent quantum molecular dynamical model, the studies of the isospin effect on preequilibrium nucleon emission in heavy ion collisions under different entrance channel conditions show that the ratio of preequilibrium neutron number to proton number depends strongly on symmetry potential, beam energy, and the ratio of neutron to proton of the colliding system, but weakly on isospin dependent in-medium nucleon-nucleon cross sections, impact parameter, Pauli potential, and momentum dependent interaction in the energy region from 45MeV/u up to 150 MeV/u where the dynamics is domin…
We study the effect of isospin degree of freedom on balance energy throughout the mass range between 50 and 350 for two sets of isotopic systems with N/Z = 1.16 and 1.33 as well as isobaric systems with N/A = 1.0 and 1.4. Our findings indicate that different values of balance energy for two isobaric systems may be mainly due to the Coulomb repulsion. We also demonstrate clearly the dominance of Coulomb repulsion over symmetry energy.
We aim to explore the effect of isospin dependence of cross section on symmetric and neutron rich system. We also aim to explore whether the analysis is affacted if one discusses in terms of "$E_{bal}$ as a function of N/Z or N/A" of the system.
Based on the systematic investigation of the data available for $A \geq 40$, a $Z^{1/3}$ dependence for the nuclear charge radii is shown to be superior to the generally accepted $A^{1/3}$ law. A delicate scattering of data around $R_c/Z^{1/3}$ is infered as owing to the isospin effect and a linear dependence of $R_c/Z^{1/3}$ on $N/Z$ (or $(N-Z)/2$) is found. This inference is well supported by the microscopic Relativistic Continuum Hartree-Bogoliubov (RCHB) calculation conducted for the proton magic Ca, Ni, Zr, Sn and Pb isotopes including the exotic nuclei close to the neutron drip line. Wit…
The binding energies, shapes and sizes of even-even beta-stable nuclei with A >= 40 and a few chains of isotopes with Z=50, 56, 82, 94 protons and isotones with N=50, 82, 126 neutrons are analyzed. The average isospin dependence of the radii of protons and neutrons evaluated within the relativistic mean field theory is studied. A simple, phenomenological formula for neutron radii is proposed.
We study the N/Z dependence of participant-spectator matter for Ca+Ca reactions at 50 and 250MeV/nucleon. We also study the role of symmetry energy and momentum-dependent interactions on the N/Z dependence of participant-spectator matter.
We studied the influence of a momentum dependent interaction in the context of isopspin effects on fragmentation and dissipation in intermediate energy heavy ion collisions by using an isospin dependent quantum molecular dynamics model. It is shown that the nuclear stopping, the number of nucleons emitted ,and the multiplicity of intermediate mass fragments are larger with a momentum dependent interaction than without. In particular, the differences for these observables, when using an isospin dependent in-medium nucleon nucleon cross section versus an isospin independent one, are also larger …
The in-medium neutron-proton, proton-proton(neutron-neutron) scattering cross sections($σ_{np}^{*}$, $σ_{pp(nn)}^{*}$) are studied based on QHD-II type Lagrangian within the framework of the microscopic transport theory. The results demonstrate that, for free nucleon-nucleon scattering cross sections, the isospin dependence is dominantly caused by $ρ$ meson field. The medium correction of nucleon-nucleon scattering cross sections is also isospin dependent, $σ_{np}^{*}$ depends on the baryon density weakly and $σ_{pp(nn)}^{*}$ depends on the baryon density significantly, which is due to the dif…
Generalized symmetry energy coefficients of asymmetric nuclear matter are obtained as screening functions. The dependence of the isospin symmetry energy coefficient on the neutron proton (n-p) asymmetry may be determined unless by a constant (exponent) $Z$ which depend on microscopic properties. The dependence of the generalized symmetry energy coefficients with Skyrme forces on the n-p asymmetry and on the density, only from .5 up to 1.5 $ρ_0$, are investigated in the isospin and scalar channels. The use of Skyrme-type effective forces allows us to obtain analytical expressions for these para…
In the present paper, we explore the idea of isospin conservation in new situations and contexts based on the directions provided by our earlier works. We present the results of our calculations for the relative yields of neutron-rich fission fragments emitted in fast neutron-induced fission, 238U (n, fission) reaction by using the concept of the conservation of isospin and compare with the experimental data. Our results successfully reproduced the gross features of partition wise fission fragments distribution of 238U (n, fission). This confirms that in all kinds of fission, isospin remains p…
We study the one-body momentum distribution at different densities in nuclear matter, with special emphasis on its components at high momentum. Explicit calculations for finite neutron-proton asymmetry, based on the ladder self-consistent Green's function approach, allow us to access the isospin dependence of momentum distributions and elucidate their role in neutron-rich systems. Comparisons with the deuteron momentum distribution indicate that a substantial proportion of high-momentum components are dominated by tensor correlations. We identify the density dependence of these tensor correlat…
The isospin dependence of the nucleon effective mass is investigated in the framework of the Dirac Brueckner-Hartree-Fock (DBHF) approach. The definition of nucleon scalar and vector effective masses in the relativistic approach is clarified. Only the vector effective mass is the quantity related to the empirical value extracted from the analysis in the nonrelatiistic shell and optical potentials. In the relativistic mean field theory, where the nucleon scalar and vector potentials are both energy independent, the neutron vector potential is stronger than that of proton in the neutron rich nuc…
A systematic analysis of the ground state and fission properties of actinides and superheavy nuclei important for the $r$ process modeling has been performed within the framework of covariant density functional theory for the first time in Ref. [1]. A brief review of the results related to the heights of primary fission barriers and systematic uncertainties in their prediction is presented. In addition, new results on the potential impact of the isospin dependence of pairing on fission barriers in fission cycling regions is provided for the first time.
The rapidity dependence of the single- and double- neutron to proton ratios of nucleon emission from isospin-asymmetric but mass-symmetric reactions Zr+Ru and Ru+Zr at energy range $100 \sim 800$ A MeV and impact parameter range $0\sim 8$ fm is investigated. The reaction system with isospin-asymmetry and mass-symmetry has the advantage of simultaneously showing up the dependence on the symmetry energy and the degree of the isospin equilibrium. We find that the beam energy- and the impact parameter dependence of the slope parameter of the double neutron to proton ratio ($F_D$) as function of ra…
The isospin dependence of incompressibility is investigated in the Skyrme Hartree-Fock (SHF) and relativistic mean field (RMF) models. The correlations between the nuclear matter incompressibility and the isospin dependent term of the finite nucleus incompressibility is elucidated by using the Thomas-Fermi approximation. The Coulomb term is also studied by using various different Skyrme Hamiltonians and RMF Lagrangians. The symmetry energy coefficient of incompressibility is extracted to be K_τ=-(500\pm50) MeV from the recent experimental data of isoscalar giant monopole resonances (ISGMR) in …
Correlated Basis Function theory and Fermi Hypernetted Chain technique are extended to study medium-heavy, doubly closed shell nuclei in j-j coupling scheme, with different single particle wave functions for protons and neutrons and isospin dependent two-body correlations. Central semirealistic interactions are used. Ground state energies, one-body densities, distribution functions and momentum distributions are calculated for 12C, 16O, 40Ca, 48Ca and 208Pb nuclei. The values of the ground state energies provided by isospin dependent correlations are lower than those obtained with isospin inde…
By using the Furnstahl, Serot and Tang's model, the effect of density dependence of the effective nucleon-nucleon-rho-meson (NN-rho) coupling on the liquid-gas phase transition in hot asymmetric nuclear matter is investigated. A limit pressure p_lim has been found. We found that the liquid-gas phase transition cannot take place if p>p_lim. The binodal surface for density dependent NN-rho coupling situation is addressed.
Properties of symmetric and asymmetric nuclear matter have been investigated in the relativistic Dirac-Brueckner-Hartree-Fock approach based on projection techniques using the Bonn A potential. The momentum, density, and isospin dependence of the optical potentials and nucleon effective masses are studied. It turns out that the isovector optical potential depends sensitively on density and momentum, but is almost insensitive to the isospin asymmetry. Furthermore, the Dirac mass $m^*_D$ and the nonrelativistic mass $m^*_{NR}$ which parametrizes the energy dependence of the single particle spect…
We report quantum Monte Carlo calculations of single-$Λ$ hypernuclei for $A<50$ based on phenomenological two- and three-body hyperon-nucleon forces. We present results for the $Λ$ separation energy in different hyperon orbits, showing that the accuracy of theoretical predictions exceeds that of currently available experimental data, especially for medium-mass hypernuclei. We show the results of a sensitivity study that indicates the possibility to investigate the nucleon-isospin dependence of the three-body hyperon-nucleon-nucleon force in the medium-mass region of the hypernuclear chart, whe…
The relativistic mean field theory in combination with the analytic continuation in the coupling constant method is used to determine the energies and widths of single-particle resonant states in Sn isotopes. It is shown that there exists clear shell structure in the resonant levels as appearing in the bound levels. In particular, the isospin dependence of pseudospin symmetry is clearly shown in the resonant states, is consistent with that in the bound states, where the splittings of energies and widths between pseudospin doublets are found in correlation with the quantum numbers of single-par…
Thermal properties of asymmetric nuclear matter, including the temperature dependence of the symmetry energy, single-particle properties, and differential isospin fractionation, are investigated with different neutron-proton effective mass splittings using an improved isospin- and momentum-dependent interaction. In this improved interaction, the momentum-dependence of the isoscalar single-particle potential at saturation density is well fitted to that extracted from optical model analyses of proton-nucleus scattering data up to nucleon kinetic energy of 1 GeV, and the isovector properties, i.e…
The discrepancy between experimental data and theoretical calculations in one-nucleon removal reactions at intermediate energies (quantified by the so-called "quenching factors") and its dependence on the isospin asymmetry of the nuclei has been an open problem in nuclear physics for the last fifteen years. In this work, we propose an explanation for this long-standing problem, which relies on the inclusion of the process of core destruction due to its interaction with the removed nucleon. To include this effect, we extend the commonly used eikonal formalism via an effective nucleon density, a…
We have developed an efficient isospin projection method in the shell model Monte Carlo approach for isospin-conserving Hamiltonians. For isoscalar observables this projection method has the advantage of being exact sample by sample. The isospin projection method allows us to take into account the proper isospin dependence of the nuclear interaction, thus avoiding a sign problem that such an interaction introduces in unprojected calculations. We apply our method in the calculation of the isospin dependence of level densities in the complete $pf+g_{9/2}$ shell. We find that isospin-dependent co…
In the present work we take the non relativistic limit of relativistic models and compare the obtained functionals with the usual Skyrme parametrization. Relativistic models with both constant couplings and with density dependent couplings are considered. While some models present very good results already at the lowest order in the density, models with non-linear terms only reproduce the energy functional if higher order terms are taken into account in the expansion.
We investigate the equation of state of asymmetric nuclear matter and its isospin dependence in various spin-isospin $ST$ channels within the framework of the Brueckner-Hartree-Fock approach extended to include a microscopic three-body force (TBF). It is shown that the potential energy per nucleon in the isospin-singlet T=0 channel is mainly determined by the contribution from the tensor SD coupled channel. At high densities, the TBF effect on the isospin-triplet T=1 channel contribution turns out to be much larger than that on the T=0 channel contribution. At low densities around and below th…
We study the sensitivity of isospin dependence of nucleon-nucleon cross section and symmetry energy on the collective transverse in-plane flow. We find that collective transverse flow in Ca+Ca reactions shows great sensitivity to isospin dependence of nn cross section and symmetry energy.
The Lagrangian density of relativistic mean-field (RMF) theory with non-linear derivative (NLD) interactions is applied to isospin asymmetric nuclear matter. We study the symmetry energy and the density and energy dependences of nucleon selfenergies. At high baryon densities a soft symmetry energy is obtained. The energy dependence of the isovector selfenergy suppresses the Lane-type optical potential with increasing energy and predicts a $ρ$-meson induced mass splitting between protons and neutrons in isospin asymmetric matter.
A microscopic approach to the proton-neutron nuclear response is formulated in the finite-temperature relativistic nuclear field theory framework. The approach is based on the meson-nucleon Lagrangian of quantum hadrodynamics and advances the relativistic field theory for spin-isospin response beyond the finite-temperature random phase approximation. The dynamical contribution to the in-medium proton-neutron interaction amplitude is described in a parameter-free way by the coupling between the single nucleons and strongly-correlated particle-hole excitations (phonons) within the newly develope…
In this article we investigate the structure of the non-local part of the symmetry term, that leads to a splitting of the effective masses of protons and neutrons in asymmetric matter. Based on microscopic transport simulations we suggest some rather sensitive observables in collisions of neutron-rich (unstable) ions at intermediate ($RIA$) energies. In particular we focus the attention on pre-equilibrium nucleon emissions. We discuss interesting correlations between the N/Z content of the fast emitted particles and their rapidity or transverse momentum, that show a nice dependence on the pres…
The $π^-/π^+$ ratio is examined as a function of isospin asymmetry and beam energy for central collisions induced by neutron-rich nuclei within an isospin-dependent hadronic transport model. The $π^-/π^+$ ratio is found to increase with the isospin asymmetry, but decrease with the beam energy of the reaction. For neutron-rich systems, such as, $^{124}Sn+^{124}Sn$ and $^{132}Sn+^{124}Sn$, the sensitivity of $π^-/π^+$ ratio to nuclear symmetry energy is almost beam energy independent.
We introduce a new microscopic approach to calculate the dependence of fusion barriers and cross-sections on isospin dynamics. The method is based on the time-dependent Hartree-Fock theory and the isoscalar and isovector properties of the energy density functional (EDF). The contribution to the fusion barriers originating from the isoscalar and isovector parts of the EDF is calculated. It is shown that for non-symmetric systems the isovector dynamics influence the sub-barrier fusion cross-sections. For most systems this results in an enhancement of the sub-barrier cross-sections, while for oth…
The short-range properties of the KN interaction are studied within the meson-exchange model of the Juelich group. Specifically, dynamical explanations for the phenomenological short-range repulsion, required in this model for achieving agreement with the empirical KN data, are explored. Evidence is found that contributions from the exchange of a heavy scalar-isovector meson (a_0(980)) as well as from genuine quark-gluon exchange processes are needed. Taking both mechanisms into account a satisfactory description of the KN phase shifts can be obtained without resorting to phenomenological piec…
The constituent quark model based on a hypercentral approach takes into account three-body force effects and standard two-body potential contributions. The quark potential contains a hypercentral interaction, to which a hyperfine term is added. While the hypercentral potential supplies good values for the centroid energies of the resonance multiplets and a realistic set of quark wave functions, the hyperfine splittings are sometimes not sufficient to account for the observed masses. In this work we have introduced an improved form of the hyperfine interaction and an isospin dependent quark pot…
Within a transport model we search for potential probes of the isospin dependence of the in-medium nucleon-nucleon (NN) cross sections. Traditional measures of the nuclear stopping power are found sensitive to the magnitude but they are ambiguous for determining the isospin dependence of the in-medium NN cross sections. It is shown that isospin tracers, such as the neutron/proton ratio of free nucleons, at backward rapidities/angles in nuclear reactions induced by radioactive beams in inverse kinematics is a sensitive probe of the isospin dependence of the in-medium NN cross sections. At forwa…
The computation of the thermodynamic properties of nuclear matter is a central task of theoretical nuclear physics. The nuclear equation of state is an essential quantity in nuclear astrophysics and governs the properties of neutron stars and core-collapse supernovæ. The framework of chiral effective field theory provides the basis for the description of nuclear interactions in terms of a systematic low-energy expansion. In this thesis, we apply chiral two- and three-nucleon interactions in perturbative many-body calculations of the thermodynamic equation of state of infinite homogeneous nucle…
The squeeze-out flow in reactions of $^{124}$Sn + $^{124}$Sn and $^{124}$Ba + $^{124}$Ba at different incident energies for different impact parameters is investigated by means of an isospin-dependent quantum molecular dynamics model. For the first time, it is found that the more neutron-rich system ($% ^{124}$Sn + $^{124}$Sn) exhibits weaker squeeze-out flow. This isospin dependence of the squeeze-out flow is shown to mainly result from the isospin dependence of nucleon-nucleon cross section and the symmetry energy.
The isospin-asymmetry dependence of the nuclear matter equation of state obtained from microscopic chiral two- and three-body interactions in second-order many-body perturbation theory is examined in detail. The quadratic, quartic and sextic coefficients in the Maclaurin expansion of the free energy per particle of infinite homogeneous nuclear matter with respect to the isospin asymmetry are extracted numerically using finite differences, and the resulting polynomial isospin-asymmetry parametrizations are compared to the full isospin-asymmetry dependence of the free energy. It is found that in…
The in-medium $NN\rightarrow NΔ$ cross section and its differential cross section in isospin asymmetric nuclear medium are investigated in the framework of the one-boson exchange model by including the isovector mesons, i.e., $δ$ and $ρ$ mesons. Our results show that the in-medium $NN\rightarrow NΔ$ cross sections are suppressed with density increasing, and the differential cross sections become isotropic with the density increasing at the beam energy around the $Δ$ threshold energy. The isospin splitting on the medium correction factor, $R=σ_{ NN\rightarrow NΔ}^*/σ_{NN\rightarrow NΔ}^{\text{f…
The neutron-proton differential flow is shown to be a very useful probe of the isospin-dependence of the nuclear equation of state (EOS). This novel approach utilizes constructively both the isospin fractionation and the nuclear collective flow as well as their sensitivities to the isospin-dependence of the nuclear EOS. It also avoids effectively uncertainties associated with other dynamical ingredients of heavy-ion reactions at intermediate energies.
The understanding of the interaction of nucleons in nuclear and neutron-rich matter at non-zero temperature is important for a variety of applications ranging from heavy-ion collisions to nuclear astrophysics. In this papre we apply the Dirac-Brueckner-Hartree-Fock method along with the Bonn B potential to predict single-particle properties in symmetric nuclear matter and neutron-rich matter at finite temperature. It is found that temperature effects are generally small but can be significant at low density and momentum.
We study the effect of isospin degree of freedom on the balance energy (E$_{bal}$) as well as its mass dependence throughout the mass range 48-270 for two sets of isobaric systems with N/Z = 1 and 1.4 at different colliding geometries ranging from central to peripheral ones. Our findings reveal the dominance of Coulomb repulsion in isospin effects on E$_{bal}$ as well as its mass dependence throughout the range of the colliding geometry.
T. Li {\it et al.}[Phys. Rev. C {\bf 81}, 034309 (2010)] have analyzed their measured breathing-mode energies of some tin isotopes in terms of a first-order leptodermous expansion, and find for the symmetry-incompressibility coefficient $K_τ$ the value of -550 $\pm$ 100 MeV. Removing an approximation that they made, we find that the first-order estimate of $K_τ$ shifts to -661 $\pm$ 144 MeV. However, taking into account higher-order terms in the leptodermous expansion shows that the data are compatible with the significantly lower magnitudes indicated by both another experiment and some theore…
Temperature dependence of the nucleon-nucleon inelastic cross section in an isospin-asymmetric nuclear medium
School of Science, Huzhou University（湖州学院理学院）；Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所）；School of Nuclear Science and Technology, University of Chinese Academy of Sciences（中国科学院大学核科学与技术学院）；School of Nuclear Science and Engineering, Shenyang Institute of Engineering（沈阳工程学院核科学与工程学院）
