A high-resolution study of the electromagnetic response of $^{206}$Pb below the neutron separation energy is performed using a ($vec{gamma}$,$gamma$) experiment at the HI$vec{gamma}$S facility. Nuclear resonance fluorescence with 100% linearly polarized photon beams is used to measure spins, parities, branching ratios, and decay widths of excited states in $^{206}$Pb from $4.9$ to $8.1$MeV. The extracted $Sigma$$B$(E1)$uparrow$ and $Sigma$$B$(M1)$uparrow$ values for the total electric and magnetic dipole strength below the neutron separation energy are 0.9$pm$0.2e$^2$fm$^2$ and 8.3$pm$2.0$mu_{N}^2$, respectively. These measurements are found to be in very good agreement with the predictions from an energy-density functional (EDF) plus quasiparticle phonon model (QPM). Such a detailed theoretical analysis allows to separate the pygmy dipole resonance from both the tail of the giant dipole resonance and multi-phonon excitations. Combined with earlier photonuclear experiments above the neutron separation energy, one extracts a value for the electric dipole polarizability of $^{206}$Pb of $alpha_{D}!=!122pm10$mb/MeV. When compared to predictions from both the EDF+QPM and accurately calibrated relativistic EDFs, one deduces a range for the neutron-skin thickness of $R_{rm skin}^{206}!=!0.12$-$0.19$fm and a corresponding range for the slope of the symmetry energy of $L!=!48$-$60$MeV. This newly obtained information is also used to estimate the Maxwellian-averaged radiative cross section $^{205}$Pb(n,$gamma$)$^{206}$Pb at 30keV to be $sigma!=!130!pm!25$mb. The astrophysical impact of this measurement--on both the s-process in stellar nucleosynthesis and on the equation of state of neutron-rich matter--is discussed.
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