Higher-order effects on the incompressibility of isospin asymmetric nuclear matter

Analytical expressions for the saturation density of asymmetric nuclear matter as well as its binding energy and incompressibility at saturation density are given up to fourth order in the isospin asymmetry $\delta =({\rho }_n-{\rho }_p)/\rho$ using 11 characteristic parameters defined by the density derivatives of the binding energy per nucleon of symmetric nuclear matter, the symmetry energy $E_{\mathrm{sym}}\hspace{0.5em}(\rho )$, and the fourth-order symmetry energy $E_{\mathrm{sym},4}(\rho )$ at normal nuclear density ${\rho }_0$. Using an isospin- and momentum-dependent modified Gogny interaction (MDI) and the Skyrme-Hartree-Fock (SHF) approach with 63 popular Skyrme interactions, we have systematically studied the isospin dependence of the saturation properties of asymmetric nuclear matter, particularly the incompressibility $K_{\mathrm{sat}}(\delta )=K_0+K_{\mathrm{sat},2}{\delta }^2+K_{\mathrm{sat},4}{\delta }^4+O({\delta }^6)$ at saturation density. Our results show that the magnitude of the higher order $K_{\mathrm{sat},4}$ parameter is generally small compared to that of the $K_{\mathrm{sat},2}$ parameter. The latter essentially characterizes the isospin dependence of the incompressibility at saturation density and can be expressed as $K_{\mathrm{sat},2}=K_{\mathrm{sym}}-6L-\frac{J_0}{K_0}L$, where $L$ and $K_{\mathrm{sym}}$ represent, respectively, the slope and curvature parameters of the symmetry energy at ${\rho }_0$ and $J_0$ is the third-order derivative parameter of symmetric nuclear matter at ${\rho }_0$. Furthermore, we have constructed a phenomenological modified Skyrme-like (MSL) model that can reasonably describe the general properties of symmetric nuclear matter and the symmetry energy predicted by both the MDI model and the SHF approach. The results indicate that the higher order $J_0$ contribution to $K_{\mathrm{sat},2}$ generally cannot be neglected. In addition, it is found that there exists a nicely linear correlation between $K_{\mathrm{sym}}$ and $L$ as well as between $J_0/K_0$ and $K_0$. These correlations together with the empirical constraints on $K_0,L,E_{\mathrm{sym}}\hspace{0.5em}({\rho }_0)$, and the nucleon effective mass lead to an estimate of $K_{\mathrm{sat},2}=-370\pm 120$ MeV.

Figure 1

Energy per nucleon as a function of density for symmetric nuclear matter in the MDI interaction. Also included are results obtained by using Eq. (4) up to ${\chi }^2,{\chi }^3$, and ${\chi }^4$, respectively.

Figure 2

Density dependence of symmetry energy using the MDI interaction with $x=1$ (left), 0 (middle), and $-1$ (right) together with corresponding results obtained by using Eq. (11) up to $\chi ,{\chi }^2,{\chi }^3$, and ${\chi }^4$, respectively.

Figure 3

Density dependence of the fourth-order symmetry energy $E_{\mathrm{sym},4}(\rho )$ using the MDI interaction together with results obtained by using Eq. (16) up to $\chi ,{\chi }^2,{\chi }^3$, and ${\chi }^4$, respectively.

Figure 4

Density dependence of the symmetry energy from the MDI interaction with $x=1$, 0, and $-1$. The results from the widely used APR prediction [115] and the phenomenological MSL model prediction with $y=-15,-7.5$, and 0.75 are also included for comparison.

Figure 5

The density and isospin asymmetry dependence of the binding energy per nucleon for asymmetric nuclear matter in the MDI interaction with $x=1,0$, and $-1$. The saturation points at different isospin asymmetries are also indicated.

Figure 6

The saturation density ${\rho }_{\mathrm{sat}}$ as a function of ${\delta }^2$ in the MDI interaction with $x=1$, 0, and $-1$. Corresponding results from Eq. (23) including terms up to ${\delta }^2$ and up to ${\delta }^4$, respectively, are also included for comparison. The insets display corresponding results at smaller isospin asymmetries with ${\delta }^2⩽0.1$.

Figure 7

Same as Fig. 6 but for the binding energy at saturation density, $E_{\mathrm{sat}}$.

Figure 8

Same as Fig. 6 but for the incompressibility at saturation density, $K_{\mathrm{sat}}$.

Figure 9

The absolute values of $K_{\mathrm{sat},2}$ and $K_{\mathrm{asy}}$ and the value of $K_{\mathrm{sat},4}$ as functions of $L$ for the MDI interaction with $x=1,0$, and $-1$ and the 63 standard Skyrme interactions considered in the present work.

Figure 10

The third derivative parameter $J_0$ and its ratio $J_0/K_0$ to the incompressibility $K_0$ of symmetric nuclear matter at saturation density as functions of $K_0$ from the MSL model, the MDI interaction, and the SHF prediction with the 63 Skyrme interactions shown in Table .

Figure 11

Correlation between $K_{\mathrm{sym}}$ and $L$ from the MSL model with ${\gamma }_{\mathrm{sym}}=4/3$ and $5/3$, the MDI interaction, and the SHF prediction with the 63 Skyrme interactions shown in Table . The results from two simple one-parameter symmetry energies in Eqs. (81) and (82) are also shown for comparison.

Figure 12

$K_{\mathrm{sat},2}$ as a function of $L$ from the MSL model with (a) ${\gamma }_{\mathrm{sym}}=4/3$ and (b) $5/3$ and $m_{s,0}^{*}=0.8m$ and $m_{v,0}^{*}=0.7m$ for different values of $K_0$ and $E_{\mathrm{sym}}({\rho }_0)$. The shaded region indicates constraints within the MSL model with $K_0=240\pm 20$ MeV, $E_{\mathrm{sym}}({\rho }_0)=30\pm 5$ MeV, and $46⩽L⩽111$ MeV limited by the heavy-ion collision data. The results from the SHF approach with 63 Skyrme interactions are also included for comparison. In addition, the constraint of $K_{\tau }=-550\pm 100$ MeV obtained in Refs. [22, 23] from measurements of the isotopic dependence of the GMR in even-A Sn isotopes is also indicated.