Interdependence of different symmetry energy elements

Relations between the nuclear symmetry energy coefficient and its density derivatives are derived. The relations hold for a class of interactions with quadratic momentum dependence and a power-law density dependence. The structural connection between the different symmetry energy elements as obtained seems to be followed by almost all reasonable nuclear energy density functionals, both relativistic and nonrelativistic, suggesting a universality in the correlation structure. This, coupled with known values of some well-accepted constants related to nuclear matter, helps in constraining values of different density derivatives of the nuclear symmetry energy, shedding light on the isovector part of the nuclear interaction.

Figure 1

The correlation between $K_{\mathrm{sym}}^0$ and $[3C_2^0-L_0]$ as obtained from 500 EDFs [9, 10]. The black circles correspond to the Skyrme-inspired EDFs, and the red squares refer to those obtained from RMF models. The models consistent with all the constraints demanded by Dutra et al. are highlighted by orange circles for Skyrme EDFs [9] and blue squares for RMF EDFs [10]. The inner (outer) colored regions around the best-fit straight line through these points depict the loci of $95\%$ confidence (prediction) bands of the regression analysis. The inset shows the correlation line obtained from the Skyrme-RMF models, and the magenta triangles are the results obtained from EDFs with realistic interactions, MDI(0), MDI(1), MDI(-1) [15], APR [29], BHF [30], BCPM [31], and SBM [32], respectively. The green diamonds represent results from a few Gogny interactions [14].

Figure 2

Contours of constant $L_0, K_{\mathrm{sym}}^0, Q_{\mathrm{sym}}^0$, and $\mathrm{\Delta }{m}_0^{*}$ in color shades (as indicated on the right side of each panel) as functions of the input nuclear constants $C_2^0, C_2^1$, and ${\rho }_0$ depicting the interdependence between various symmetry energy elements. The values of $L_0, K_{\mathrm{sym}}^0$, and $Q_{\mathrm{sym}}^0$ are in units of MeV and those for $\mathrm{\Delta }{m}_0^{*}$ are in units of the free nucleon mass. For details, see text.