_{1}

^{*}

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
Δ*t*_{r}, 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”
*T*
_{r }is derived from the Green’s function for unbiased single-particle diffusion and is shown to be consistent with Δ
*t*
_{r}. The finite statistical lifetime of the randomly moving radioactive atom plays an essential part. For stable particles,
*T*
_{r }is of infinite duration, whereas for an unstable particle (such as
^{222}Rn), with diffusivity
*D* and decay rate
λ,
*T*
_{r } 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.

Of the many sources and causes of airborne radioactive contamination [^{238}U), uranium-235 (^{235}U), and thorium-242 (^{242}Th) dispersed widely in rocks and soils throughout the Earth. The radon isotope with the longest half-life, ^{222}Rn, is of particular concern. The World Health Organization (WHO) reports ^{222}Rn to be the leading cause of lung cancer among people who have never smoked, and to significantly increase the risk of lung cancer over that due to smoking alone in those who do smoke or have smoked in the past [^{3} to minimize health hazards (Ref [

In a recent paper [^{222}Rn) by passive diffusion without the requirement, common to virtually all other methods of radon measurement, of a fixed instrumental control volume. This new measurement protocol enables the user to determine radon concentration immediately after a short-term (e.g. 24 hour) sampling time, thereby eliminating the cost, delay, and sample loss of sending samples to an external testing laboratory for analysis. Application of the GM method yielded results of equivalent accuracy and higher precision, particularly in repeated short-term measurements, in comparison with results obtained concurrently with state-of-the-art certified commercial radon detectors also utilizing passive diffusion.

Briefly summarized, the reported GM method employed one GM detector to record counts of^{222}Rn and its polonium progeny ^{218}Po, ^{214}Po in the room air―i.e. not contaminated by emissions from radioactive elements in the construction materials of surrounding surfaces. Theoretical analysis based on the physics of atomic diffusion in air and alpha particle interactions in matter provided the means to convert the alpha count rate (e.g. cpm = counts per minute) into a radon activity concentration (Bq/m^{3} or pCi/L).

The methodology and accompanying analysis in [^{222}Rn to ^{218}Po is about 4 cm in air (at 1 atm and 20˚C). Moreover, the energy loss of an alpha particle per collisional interaction is approximately a constant 35 eV for each ionizing encounter [

whose variation with distance r from the alpha point of origin is shown in the solid red plot of

The measurement procedure described in [

Although the integrations are theoretically over the volume and cross section of the entire room (or, as a practical matter, over an infinite range for a room size^{222}Rn in air yielded

For measurements of radioactivity executed over a period of time in an open volume, it is necessary to determine the rate at which radioactive atoms pass through the region of detectability. This was accomplished in [

in which

is the flow velocity determined by the ^{222}Rn diffusion coefficient [

and decay rate [

In a conventional flow measurement, the sample consists of a stream that passes through a flow cell aligned perpendicular to the detector surfaces. The detectors, which are photomultiplier tubes in the case of flow scintillation counting, therefore sample the radioactive material only for the time that the atoms reside in the flow cell. This residence time is the ratio of the flow velocity and sample flow rate (in units of volume per time). In liquid chromatography the flow velocity is a well-defined, experimentally adjustable quantity, but in the diffusion of a radioactive gas it is a characteristic parameter of a stochastic process. Expression (5) was derived in [

Although not reported in [

in which

is the polylogarithm function. For index

Use of the preceding expressions (8) and (9) together with relations (4) and (5) leads to the following closed form expression for the residence time

Evaluation of relation (12) for passive diffusion of ^{222}Rn in air yields a residence time

Although residence time defined in terms of the diffusive flow of a macroscopic current of radioactive gas resulted in values of radon concentration in accord with measurements by a calibrated standard, it is nevertheless of interest, both conceptually and practically, to examine the concept of residence time from a micro-statistical perspective of particle Brownian motion. For one, this latter approach avoids the adoption of a phenomenological range function with adjustable parameter

The development and investigation of such a micro-statistical model is the principal objective of this paper. This is accomplished in the following sections by calculating the exact time-dependent probability density (Green’s function) for Brownian motion of radon from a planar source and showing that a certain integral of this function over time provides the sought-for statistical expectation value corresponding to the residence time in the macro-statistical model developed in [

The equation for the conditional probability

and leads to the solution

The Green’s function expressed in Equation (14) is a three-dimensional Gaussian conditional probability density―i.e. a probability per unit volume―as indicated by the dimensionality

In a coordinate system with vertical z-axis and horizontal x-axis directed normally outward from the particle source plane (

where

The function

Reduction of the problem of 3-dimensional diffusion from a planar source to diffusion from a point source on the horizontal axis is a consequence of the isotropic symmetry of Equation (14), which permits factorization into a product of three independent one-dimensional degrees of freedom. The isotropic symmetry itself comes from neglect of gravity, the effects of which are shown in Appendix 1 to be negligible under the circumstances of measuring radon activity by the method developed in [

From Equation (15), one can calculate the cumulative probability

region) during the time interval t begun when the particle was placed at

The error function in Equation (16) is defined by

and is an anti-symmetric function of its argument.

The time interval

in which

is the characteristic diffusion length. (See Ref. [^{222}Rn, it follows from relations (6) and (7) that

In the limit that

The mean value

in which

A procedure entirely equivalent to (21) for computing expectation values is to use the cumulative distribution function (cdf)

It can then be shown (in Appendix 2) that the expectation value

For a random variable defined on the positive real axis only, the second term on the right side of Equation (23) vanishes. This is the case for

where the second equality in Equation (24) follows from the definition (22) of the cumulative distribution function. Since

For purposes of comparison, the two expressions for residence time are collected below with substitution of the characteristic diffusion length (19)

Phenomenological:

Statistical:

Numerical evaluation of the statistical residence time (27) requires substitution of a specific length ^{222}Rn is about 2.3 m and the range of the alpha particle in air is only about 0.04 m, the exponent in Equation (27) is

with the Taylor series expansion of relation (27) truncated at the lowest order of

For the two expressions (28) and (29) to yield the same residence time interval, the effective length of the detection volume L would be related to the mean alpha range by

For the case of a 5.49 MeV alpha in air with experimental mean range

which are in excellent accord.

From the definition (22) of the cdf, one obtains the corresponding pdf by application of the Leibniz rule [

It then follows from relation (24) that the pdf for the random variable T whose expectation is the residence time

^{222}Rn with

simply by relation (25) or much more onerously from expression (21) with

In the following section the residence time is obtained in a different way by examining stochastic events referred to as “first passage” or “first return”, which occur widely in problems involving Brownian motion with boundaries.

The time of first passage (FPT) is the time for a particle undergoing Brownian motion to first reach a specified site. The problem has wide applicability in fields as diverse as physics, chemistry, biology, economics, and others [

Mathematically, it is required to find a probability density function

The method of images entails placing an “anti-source” of particles at just such a location as to null the net probability at^{222}Rn. The atom is free to wander over the range

The probability

which is seen to be of identical form to the cumulative probability (16) of finding the particle in the region

is identical under the stated circumstances to the residence time

is identical to pdf (33) for

from the unconstrained pdf (15) mathematically and visually (as seen by comparing

identical to probability (16).

A plot of the survival probability

From the analyses of the previous two sections one might expect that the residence time of radon within the detection region could also be modeled as the exit time or time of first passage of a particle to either of two absorbing boundaries representing the entrance and exit planes of the detection region. This inference would, in fact, be incorrect. It is instructive to understand why this latter system gives rise to a residence time of different functional form and entirely different magnitude.

Mathematically, it is now required to find a probability density function

again obtained by the method of images, where, in marked contrast to the simple solution (34) for the case of one absorbing boundary, an infinite number (

in which

The survival probability that the particle has not reached either boundary within time t is given by

in which

The corresponding first-passage or exit time is

Performing the integrations and taking the limit in Equation (42) leads to an exit time

with diffusion length ^{222}Rn with

The difference between ^{222}Rn) and

independent of the decay rate

The discrepancy between (44) and (45), which one might have thought described the same physical system―namely, a decaying particle initially at

When a randomly moving particle reaches one or the other of the two absorbing boundaries it is removed from the system. This is not the case for radon diffusing through a region of width

To understand the physical significance of the time interval (44) in the context of radon diffusion in an open volume (no boundaries) consider the temporal value at which pdf (33) for the residence time

which, to within a numerical factor, has the same form as

It is worth noting here, although the derivation is given elsewhere [

with implementation of appropriate boundary conditions. For a stable particle undergoing a one-dimensional random walk between absorbing boundaries at

which reduces to

In the case of an unstable particle with decay rate

and leads to a solution that for

^{1}From a thermodynamic perspective, the potential responsible for diffusion is the chemical potential

The preceding sections have demonstrated the equivalence of residence times calculated from models of a) free diffusion of a macroscopic sample of radon gas under a concentration gradient, and b) unconstrained Brownian motion of individual radon atoms moving forward or backward independently with equal probability―i.e. oblivious to the concentration gradient. Since approach a) appears to involve directed motion attributable to a kind of force (i.e. gradient of a potential^{1}) and approach b) clearly involves random motion in the absence of any force, it may not be immediately apparent why the two approaches actually describe the same physical process. Here, in brief, is an explanation, as illustrated schematically in

For equal probabilities

that captures the physical content of Fick’s law. The greater transport of particles to the left does not arise because there is an asymmetric force acting on individual particles, but because of the greater number of particles to enter the region dx from the right (closer to the source) than from the left. For each particle, however, the two directions of motion are equally probable. The applicability of Fick’s law to radon diffusion has, in fact, been tested and confirmed experimentally [

Implementation of the newly introduced procedure [

The analyses in this paper have established that one must exercise caution in adopting a stochastic model to describe the time spent by a radioactive atom in the detection region. For example, although it may seem reasonable to define (and presumably measure) the residence time as the mean time for an atom within the detection region to first reach either the entrance or exit plane, this choice is physically unsatisfactory. It ignores the possibility that the atom, having reached a boundary, can proceed away from or back into the detection region with equal probability. As a consequence, this first-passage or exit time, given approximately by

is too short and would lead to a measurement of radon concentration that is significantly lower than actual (see Equation (27) of Ref. [

Given the system parameters D, ^{222}Rn is ~3.8 days.

Silverman, M.P. (2017) Analysis of Residence Time in the Measurement of Radon Activity by Passive Diffusion in an Open Volume: A Micro-Statistical Approach. World Journal of Nuclear Science and Technology, 7, 252-273. https://doi.org/10.4236/wjnst.2017.74020

The point at issue is whether the vertical density profile of radon gas is affected by gravity over a dimension of order of the alpha-particle range

The barometric law for equilibrium gas pressure

where ^{-2} close to the Earth’s surface. The gas is assumed to behave ideally with equation of state [

where

Elimination of pressure by substitution of Equation (56) into (55) leads to a logarithmic differential equation for density whose solution is

with scale parameter

In the second equality of (58) M is the radon molar mass and

For ^{222}Rn (M = 0.222 kg) at room temperature (

(B) Adiabatic Atmosphere (Constant Entropy)

Although a room (e.g. a laboratory) or a building may be kept under isothermal conditions, the lowest section of the Earth’s atmosphere (the troposphere) is not isothermal, but more accurately described by the adiabatic constitutive relation [

where

with the scale parameter

It is to be concluded, therefore, that gravity has a negligible effect on the equilibrium density profile of radon within the detection volume, and that use of the isotropic Green’s function for calculating the radon residence time is justified.

If

Integration by parts of the two integrals on the second line of (61) yields

where use was made of the properties

It then follows from (61) that

For an intrinsically positive variable X the second term on the right side of (63) vanishes. In that case, the same procedure can be used to show that the variance of X is given by