Correlation between symmetry energy and effective $k$-mass splitting with an improved isospin- and momentum-dependent interaction

The isospin- and momentum-dependent interaction using the isospin-dependent quantum molecular dynamics model is improved. The momentum dependence of the interaction is fitted to the optical potentials extracted with proton-nucleus scattering data, while the isospin dependence is adjusted based on the current constraints on the symmetry energy and its density slope at normal density. The resulting parameters of the interaction are used to calculate the effective masses characterized by the nonlocality of the nuclear potential in the spatial coordinates (the so-called effective $k$ masses). It is found that this parametrization can reproduce the effective $k$ masses calculated by the Dirac-Brueckner-Hartree-Fock model. The current constraints on the total symmetry energy can accommodate a wide-range adjustment of the local symmetry energy and effective $k$-mass splitting. Four groups of parameters, which provide different density dependences of symmetry energy and effective $k$-mass splitting, are applied to investigate the isospin diffusion and neutron to proton double ratios in collisions involving ${}^{124}\mathrm{Sn}$ and ${}^{112}\mathrm{Sn}$ nuclei at 50 and 120 MeV/nucleon. Calculations confirm the sensitivity of symmetry energy to the isospin diffusion, but indicate the synchronous effects of symmetry energy and effective $k$-mass splitting on the double ratios.

Figure 1

Energy per nucleon as a function of density in symmetric nuclear matter at zero temperature.

Figure 2

Energy dependence of single particle potentials in symmetric nuclear matter at (a) $0.5{\rho }_0$, (b) ${\rho }_0$, and (c) $1.5{\rho }_0$. For comparison, the figures also show the real parts of the optical potential calculated using the nonlinear derivative model by Gaitanos et al. (solid curves) [45], calculated using the relativistic mean field model by Typel et al. (dash-dot-dotted curves) [46], phenomenologically fitted from the proton-nucleus scattering data by Hama et al. (dash-dotted curve for one fitting and dotted curve for another fitting) [19, 20], calculated with the Dirac-Brueckner approach by ter Haar and Malfliet (open circles) [47], and phenomenologically fitted the proton-nucleus scattering data by Arnold et al. (open stars) [18].

Figure 3

Effective masses as a function of momentum in symmetric nuclear matter at normal density. The solid line shows our calculations. The dotted line displays the calculation by DBHF model considering both nonlocality in space and nonlocality in time [12]. The dashed line displays the calculation by DBHF model without considering nonlocality in time [11].

Figure 4

Effective $k$ masses of neutron and proton as a function of momentum in asymmetric nuclear matter at normal density for (a) $x=-0.4$ and (b) $x=0.4$.

Figure 5

Momentum-dependent part of symmetry energies as a function of density.

Figure 6

Symmetry energies as a function of density within (a) Pos-S and Pos-H parameters which provide $m_n^{*}>m_p^{*}$ in neutron-rich nuclear matter, (b) Neg-S and Neg-H parameters which provide $m_n^{*}<m_p^{*}$ in neutron-rich nuclear matter.

Figure 7

Isospin transport ratios obtained from the yield ratios of $A=7$ isotopes in collisions involving ${}^{124}\mathrm{Sn}$ and ${}^{112}\mathrm{Sn}$ nuclei at impact parameters of b = 6 fm and at 50 MeV/nucleon. Data shown as solid circles are taken from Ref. [34]. The inside panel shows the total square $X^2$ computed from the difference between data and calculations.

Figure 8

Double ratios of the coalescence invariant neutron and proton spectra in central ${}^{124}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ and ${}^{112}\mathrm{Sn}+{}^{112}\mathrm{Sn}$ collisions at (a) 50 and (b) 120 MeV/nucleon. Calculations within different parameters are shown as lines. Data shown as solid circles are taken from Ref. [33].