Diffusion of $^{210}\text{Pb}$ and $^{210}\text{Po}$ in Nylon

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, $^{222}$Rn and its long-lived daughter $^{210}$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 $^{210}$Pb and its daughter $^{210}$Po under varying humidity conditions. Our results show that both $^{210}$Pb and $^{210}$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 $^{210}$Pb was found to be lower than 1.14 $\times$ 10$^{-15}$ cm$^2$/s at 40$\%$ relative humidity (RH), and to be (4.03 $\pm$ 1.01) $\times$ 10$^{-13}$ cm$^2$/s at 95$\%$ RH. The diffusivity of $^{210}$Po at 95$\%$ RH was measured to be (3.94 $\pm$ 0.98) $\times$ 10$^{-13}$ cm$^2$/s. These findings underscore the importance of controlling environmental humidity and material exposure to radon in the design of ultra-low background experiments.