Analysis of Residence Time in the Measurement of Radon Activity by Passive Diffusion in an Open Volume: A Micro-Statistical Approach

Residence time in a flow measurement of radioactivity is the time spent by a pre-determined quantity of radioactive sample in the flow cell. In a recent report of the measurement of indoor radon by passive diffusion in an open volume (i.e. no flow cell or control volume), the concept of residence time was generalized to apply to measurement conditions with random, rather than directed, flow. The generalization, leading to a quantity Δtr, involved use of a) a phenomenological alpha-particle range function to calculate the effective detection volume, and b) a phenomenological description of diffusion by Fick’s law to determine the effective flow velocity. This paper examines the residence time in passive diffusion from the micro-statistical perspective of single-particle continuous Brownian motion. The statistical quantity “mean residence time” Tr  is derived from the Green’s function for unbiased single-particle diffusion and is shown to be consistent with Δtr. The finite statistical lifetime of the randomly moving radioactive atom plays an essential part. For stable particles, Tr  is of infinite duration, whereas for an unstable particle (such as 222Rn), with diffusivity D and decay rate λ, Tr  is approximately the effective size of the detection region divided by the characteristic diffusion velocity . Comparison of the mean residence time with the time of first passage (or exit time) in the theory of stochastic processes shows the conditions under which the two measures of time are equivalent and helps elucidate the connection between the phenomenological and statistical descriptions of radon diffusion.