Radon and its progeny constitute a major source of background in rare-event physics experiments, such as those searching for dark matter, neutrinos, and neutrinoless double $\beta$ decay, due to their origin as unavoidable decay products of natural uranium. In particular, ${}^{222}\mathrm{Rn}$ and its long-lived daughter ${}^{210}\mathrm{Pb}$ can diffuse from detector material surfaces, resulting in sustained background contributions. To investigate this process, a system was developed using a controlled radon source, a vacuum chamber with a high electric field, and a thin nylon-6 film to enable deposition of radon progeny onto the film surface. Nylon-6 was selected for the initial measurement given its history in low-background experiments. We intend to systematically study diffusion in various polymers in the future. Our setup allowed for controlled study of the diffusion behavior of ${}^{210}\mathrm{Pb}$ and its daughter ${}^{210}\mathrm{Po}$ under varying humidity conditions. Our results show that both ${}^{210}\mathrm{Pb}$ and ${}^{210}\mathrm{Po}$ diffuse significantly in nylon under high relative humidity, which can potentially lead to internal contamination and increased background in low-background detectors. The diffusivity of ${}^{210}\mathrm{Pb}$ was found to be lower than $1.14\times {10}^{-15}{\mathrm{cm}}^2/\mathrm{s}$ at 40% relative humidity (RH) and to be $(4.03\pm 1.01)\times {10}^{-13}{\mathrm{cm}}^2/\mathrm{s}$ at 95% RH. The diffusivity of ${}^{210}\mathrm{Po}$ at 95% RH was measured to be $(3.94\pm 0.98)\times {10}^{-13}{\mathrm{cm}}^2/\mathrm{s}$. These findings underscore the importance of controlling environmental humidity and material exposure to radon in the design of ultralow-background experiments.

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• Dark matter
• Nuclear astrophysics
• Nuclear structure & decays
• Particle dark matter