Theory of radon exhalation into accumulators placed at the soil-atmosphere interface

A theoretical analysis is provided for the phenomenon of radon diffusion into cylindrical accumulators (inverted cups), which are commonly used for the measurement of radon exhalation rates from the earth's surface. Analytical solutions to the diffusion equations in the soil and the cup spaces are obtained by a two-dimensional (2-D) analysis that takes into account the perturbation in the horizontal and vertical radon concentration profiles brought about by the presence of the cup. The mixed nature of the boundary conditions at the soil surface and the cup-soil interface is handled by dual integral equation techniques. The treatment includes steady-state and time-dependent situations with and without ventilation. Formulae are derived for the effective time constant of radon build-up in the cup and for the back diffusion correction factors in different experimental situations. It is found that the effective time constant of radon build-up is much larger than the radon decay constant and it increases for smaller cups. The back diffusion correction factors to be employed in one-time measurement methods also increase as the cup dimensions decrease. The present work provides an analytical basis of an earlier numerical treatment for the growth curve analysis of the same problem. Although the results are based on the assumptions of somewhat ideal nature for mathematical tractability, they provide upper bound estimates of the phenomenon of back diffusion. Some practical applications of the results for extracting diffusion length of radon in materials are also suggested.