Isospin dependence of incompressibility in relativistic and nonrelativistic mean field calculations

The isospin dependence of incompressibility is investigated in 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 using the Thomas-Fermi approximation. The Coulomb term is also studied by using various different Skyrme Hamiltonians and RMF Lagrangians. Microscopic $\mathrm{HF}+\mathrm{random}$ phase approximation (RPA) calculations are performed with Skyrme interactions for ${}^{208}\mathrm{Pb}$ and Sn isotopes to study the strength distributions of isoscalar giant monopole resonances. The symmetry term of nuclear incompressibility is extracted to be $K_{\tau }=-(500\pm 50)$ MeV from the recent experimental data of isoscalar giant monopole resonances (ISGMR) in Sn isotopes.

Figure 1

Correlation between the nuclear matter incompressibility $K_{\infty }$ and the symmetry term $K_{\tau }$ in the finite nucleus incompressibility calculated by using 14 parameter sets of the SHF model (open circles) and 7 parameter sets of the RMF model (solid circles). The solid line is drawn for the RMF model with the correlation coefficient $r=0.759$, while the dashed line is for the SHF model with $r=0.827$. The numbers denote the different parameter sets: 1 for SI, 2 for SIII, 3 for SIV, 4 for SVI, 5 for Skya, 6 for SkM, 7 for SkM${}^{*}$, 8 for SLy4, 9 for MSkA, 10 for SkI3, 11 for SkI4, 12 for SkX, 13 for SGII, 14 for SGI, 15 for NLSH, 16 for NL3, 17 for NLC, 18 for TM1, 19 for TM2, 20 for DD-ME1, and 21 for DD-ME2.

Figure 2

Correlation between the nuclear matter incompressibility $K_{\infty }$ and the Coulomb contribution $K_{\mathrm{Coul}}$ in the finite nucleus incompressibility calculated by using 14 parameter sets of the SHF model (open circles) and 7 parameter sets of the RMF model (solid circles). See the legend to Fig. 1 and the text for details.

Figure 3

The difference of incompressibility $\Delta K=K_A-K_{A=112}$ as a function of $\delta =\frac{N-Z}{A}$. Experimental data are determined by using the excitation energies of ISGMR in Ref. [25] and the HF mass radii. See the text and the caption to Table for details.

Figure 4

Continuum $\mathrm{HF}+\mathrm{RPA}$ response functions of Skyrme interactions SIII, SGI, and SkM${}^{*}$ with the different incompressibilities. The experimental data are taken from Ref. [30].

Figure 5

$\mathrm{HF}+\mathrm{RPA}$ response functions of ${}^{112}\mathrm{Sn}$, ${}^{116}\mathrm{Sn}$, ${}^{120}\mathrm{Sn}$, and ${}^{124}\mathrm{Sn}$ nuclei with Skyrme interaction SkM${}^{*}$. The experimental data are taken from Ref. [30] (Texas A&M) and Ref. [25] (RCNP).