Weak charge form factor and radius of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mrow/><mml:mn>208</mml:mn></mml:msup></mml:math>Pb through parity violation in electron scattering

We use distorted wave electron scattering calculations to extract the weak charge form factor ${F}_{W}(\overline{q})$, the weak charge radius ${R}_{W}$, and the point neutron radius ${R}_{n}$ of ${}^{208}$Pb from the Lead Radius Experiment (PREX) parity-violating asymmetry measurement. The form factor is the Fourier transform of the weak charge density at the average momentum transfer $\overline{q}=0.475$ fm${}^{\ensuremath{-}1}$. We find ${F}_{W}(\overline{q})=0.204\ifmmode\pm\else\textpm\fi{}0.028\phantom{\rule{0.28em}{0ex}}(\mathrm{exp})\ifmmode\pm\else\textpm\fi{}0.001\phantom{\rule{0.28em}{0ex}}(\mathrm{model})$. We use the Helm model to infer the weak radius from ${F}_{W}(\overline{q})$. We find ${R}_{W}=5.826\ifmmode\pm\else\textpm\fi{}0.181\phantom{\rule{0.28em}{0ex}}(\mathrm{exp})\ifmmode\pm\else\textpm\fi{}0.027\phantom{\rule{0.28em}{0ex}}(\mathrm{model})\phantom{\rule{4pt}{0ex}}\mathrm{fm}$. Here the experimental error includes PREX statistical and systematic errors, while the model error describes the uncertainty in ${R}_{W}$ from uncertainties in the surface thickness $\ensuremath{\sigma}$ of the weak charge density. The weak radius is larger than the charge radius, implying a ``weak charge skin'' where the surface region is relatively enriched in weak charges compared to (electromagnetic) charges. We extract the point neutron radius ${R}_{n}=5.751\ifmmode\pm\else\textpm\fi{}0.175\phantom{\rule{4pt}{0ex}}(\mathrm{exp})\ifmmode\pm\else\textpm\fi{}0.026\phantom{\rule{0.28em}{0ex}}(\mathrm{model})\ifmmode\pm\else\textpm\fi{}0.005\phantom{\rule{0.28em}{0ex}}(\mathrm{strange})\phantom{\rule{4pt}{0ex}}\mathrm{fm}$ from ${R}_{W}$. Here there is only a very small error (strange) from possible strange quark contributions. We find ${R}_{n}$ to be slightly smaller than ${R}_{W}$ because of the nucleon's size. Finally, we find a neutron skin thickness of ${R}_{n}\ensuremath{-}{R}_{p}=0.302\ifmmode\pm\else\textpm\fi{}0.175\phantom{\rule{4pt}{0ex}}(\mathrm{exp})\ifmmode\pm\else\textpm\fi{}0.026$ (model) $\ifmmode\pm\else\textpm\fi{}$ 0.005 (strange) fm, where ${R}_{p}$ is the point proton radius.