Origin of the neutron skin thickness of ${}^{208}\mathrm{Pb}$ in nuclear mean-field models

We study whether the neutron skin thickness $\Delta r_{\mathit{np}}$ of ${}^{208}\mathrm{Pb}$ originates from the bulk or from the surface of the nucleon density distributions, according to the mean-field models of nuclear structure, and find that it depends on the stiffness of the nuclear symmetry energy. The bulk contribution to $\Delta r_{\mathit{np}}$ arises from an extended sharp radius of neutrons, whereas the surface contribution arises from different widths of the neutron and proton surfaces. Nuclear models where the symmetry energy is stiff, as typical of relativistic models, predict a bulk contribution in $\Delta r_{\mathit{np}}$ of ${}^{208}\mathrm{Pb}$ about twice as large as the surface contribution. In contrast, models with a soft symmetry energy like common nonrelativistic models predict that $\Delta r_{\mathit{np}}$ of ${}^{208}\mathrm{Pb}$ is divided similarly into bulk and surface parts. Indeed, if the symmetry energy is supersoft, the surface contribution becomes dominant. We note that the linear correlation of $\Delta r_{\mathit{np}}$ of ${}^{208}\mathrm{Pb}$ with the density derivative of the nuclear symmetry energy arises from the bulk part of $\Delta r_{\mathit{np}}$. We also note that most models predict a mixed-type (between halo and skin) neutron distribution for ${}^{208}\mathrm{Pb}$. Although the halo-type limit is actually found in the models with a supersoft symmetry energy, the skin-type limit is not supported by any mean-field model. Finally, we compute parity-violating electron scattering in the conditions of the ${}^{208}\mathrm{Pb}$ parity radius experiment (PREX) and obtain a pocket formula for the parity-violating asymmetry in terms of the parameters that characterize the shape of the ${}^{208}\mathrm{Pb}$ nucleon densities.

Figure 1

Comparison of sharp density profiles having radii $C$, $R$, and $Q$ with the mean-field and 2pF density distributions for the neutron density of ${}^{208}\mathrm{Pb}$. The RMF interaction NL3 has been used in the mean-field calculation.

Figure 2

Correlation of the neutron skin thickness of ${}^{208}\mathrm{Pb}$ and of its bulk and surface parts with the density derivative $L$ of the nuclear symmetry energy.

Figure 3

Comparison of the nucleon densities predicted in ${}^{208}\mathrm{Pb}$ by the mean-field models MSk7 ($C_n-C_p\approx 0$ fm) and NL3 ($C_n-C_p\approx 0.2$ fm).

Figure 4

Parity-violating asymmetry for 1 GeV electrons at $5^{°}$ scattering angle calculated from the 2pF neutron and proton density distributions of ${}^{208}\mathrm{Pb}$ in nuclear mean-field models.