Skyrme interaction and nuclear matter constraints

This paper presents a detailed assessment of the ability of the 240 Skyrme interaction parameter sets in the literature to satisfy a series of criteria derived from macroscopic properties of nuclear matter in the vicinity of nuclear saturation density at zero temperature and their density dependence, derived by the liquid-drop model, in experiments with giant resonances and heavy-ion collisions. The objective is to identify those parametrizations which best satisfy the current understanding of the physics of nuclear matter over a wide range of applications. Out of the 240 models, only 16 are shown to satisfy all these constraints. Additional, more microscopic, constraints on the density dependence of the neutron and proton effective mass $\beta$-equilibrium matter, Landau parameters of symmetric and pure neutron nuclear matter, and observational data on high- and low-mass cold neutron stars further reduce this number to 5, a very small group of recommended Skyrme parametrizations to be used in future applications of the Skyrme interaction of nuclear-matter-related observables. Full information on partial fulfillment of individual constraints by all Skyrme models considered is given. The results are discussed in terms of the physical interpretation of the Skyrme interaction and the validity of its use in mean-field models. Future work on application of the Skyrme forces, selected on the basis of variables of nuclear matter, in the Hartree-Fock calculation of properties of finite nuclei, is outlined.

Figure 1

Constraint SM3: Pressure in SNM as a function of density up to 3$\frac{\rho }{{\rho }_{\mathrm{o}}}$ as predicted by consistent Skyrme parametrizations. The shaded area in the region 2 $<\frac{\rho }{{\rho }_{\mathrm{o}}}<$ 4.6 is taken from Ref. [78].

Figure 2

Constraint SM4: Pressure in symmetric nuclear matter as a function of density in the region 1.2 $<\frac{\rho }{{\rho }_{\mathrm{o}}}<$ 2.2. The shaded area represents an educated guess discussed in Ref. [80] around the pressure-density relation available from kaon production data [81]. The dashed line extrapolates the pressures consistent with GMR data to higher densities in the region 1.2 $<\frac{\rho }{{\rho }_{\mathrm{o}}}<$ 1.7.

Figure 3

Constraint PNM1: Energy per particle in PNM as a function of density. The gray bands were based on Ref. [40] for two different parametrizations (dashed red and dashed-dotted green lines). The region inside the solid blue line illustrates the universal limit constraint given by Eq. (30) [39]. For more explanation see text.

Figure 4

EoS of low-density pure neutron matter. (a) Band defined by results of theoretical calculations summarized in Ref. [42]. See the reference for explanation of the legend. (b) The same as in Fig. 3, but with the additional band (a) included.

Figure 5

Constraint PNM2: Pressure in the PNM as a function of density as calculated by consistent Skyrme parametrizations up to 3$\frac{\rho }{{\rho }_{\mathrm{o}}}$. The bands are in the region 2 $<\frac{\rho }{{\rho }_{\mathrm{o}}}<$ 4.6. For detailed explanation see Ref. [110].

Figure 6

Constraints on symmetry energy ${\mathcal{S}}_{\mathrm{o}}$, its first derivative $L$, and $P_{\mathrm{o}}=P_{\mathrm{PNM}}\left({\rho }_{\mathrm{o}}\right)$, all at saturation density, as derived from HIC [90], PDR [91, 93], IAS [95], and FRDM [89]. Predictions of the consistent Skyrme parametrizations all lie within the blue dashed rectangle. For full explanation see text and [97].

Figure 7

Density dependence of neutron (left panel) and proton (right panel) effective mass in $\beta$-equilibrium matter as calculated by Skyrme interactions consistent with the macroscopic constraints. For more detail see text.

Figure 8

Landau parameters calculated by CSkP sets, which passed microscopic constraints derived from the effective mass considerations, in SNM. See text for more explanation.

Figure 9

The same as Fig. 8, but for PNM.

Figure 10

Density dependence of the symmetry energy $\mathcal{S}$ as a function of $\frac{\rho }{{\rho }_{\mathrm{o}}}$ as calculated by Skyrme interactions consistent with macroscopic constraints.

Figure 11

Energy per particle in PNM and SNM as a function of particle number density $\rho$ for three selected Skyrme parametrizations, Skxs20, SQMC700, and GSkII.

Figure 12

Gravitational mass vs radius for cold nonrotational neutron stars as calculated using a Skyrme EoS augmented by BPS EoS at low density [146] (left panel) and matched by a FQMC EoS at high densities [149] and by BPS EoS at low densities (right panel). The dashed lines indicate the limits on the maximum mass of the most massive neutron star observed up to date [147]. Only Skyrme parametrizations which are consistent with both macroscopic and microscopic constraints are used. For more explanation see text.

Figure 13

Relation between the gravitational mass, $M_{\mathrm{g}}$, for the selected Skyrme models, and the corresponding baryonic mass, $M_{\mathrm{b}}.$ The boxes represent constraints derived by Podsiadlowski et al. [150] (full line box) and more recently by Kitaura et al. [151] (dashed line box) based on the proposed properties of system J0737-3039, as discussed in the text. Results are shown for a Skyrme interaction consistent with both macroscopic and microscopic constraints.

Figure 14

Density dependence of the energy per particle, resulting only from the contributions of the nonstandard terms in Eq. (1) for all nonstandard Skyrme parametrizations used in this work.