Of the many sources and causes of airborne radioactive contamination [1] , the naturally occurring contaminant radon is perhaps the most pervasive, since it arises as one of the daughter products, and the only radioactive gas, in the decay series of uranium-238 (²³⁸U), uranium-235 (²³⁵U), and thorium-242 (²⁴²Th) dispersed widely in rocks and soils throughout the Earth. The radon isotope with the longest half-life, ²²²Rn, is of particular concern. The World Health Organization (WHO) reports ²²²Rn 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 [2] . Although no level of indoor radon exposure is regarded as safe, the WHO recommends a reference level of 100 Bq/m³ to minimize health hazards (Ref [2] , p. ix). Similarly, the US Environmental Protection Agency (EPA) recommends a level of 4 pCi/L above which remedial action should be taken [3] . (For comparison, note that 1 picocurie per liter = 37 becquerel per cubic meter.) It is desirable, therefore, in view of the health risks posed by radon, to develop methods to measure indoor radon activity in a rapid, accurate, convenient, and inexpensive way.

In a recent paper [4] Silverman proposed and demonstrated the use of two pancake-style Geiger-Mueller (GM) counters to measure quantitatively the activity concentration, i.e. activity per unit volume, of radon-222 (²²²Rn) 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, , and particles, and the other GM detector, covered with an alpha-blocking layer, to record just, and particles. The detectors were located so that the count-rate difference was exclusively attributable to alpha detection from ²²²Rn and its polonium progeny ²¹⁸Po, ²¹⁴Po 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/m³ or pCi/L).

The methodology and accompanying analysis in [4] relied on the short range of alpha transmissivity in matter. For example, the 5.49 MeV alpha emitted in the decay of ²²²Rn to ²¹⁸Po 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 [5] , leading to a transmission probability that is essentially constant over the greater part of the particle’s range, then decreasing rapidly to zero after a total of about interactions. It is this feature that enabled [4] to represent the alpha transmission through air (or a condensed medium like the thin mica window of the GM detector) by a phenomenological range function of the form

whose variation with distance r from the alpha point of origin is shown in the solid red plot of Figure 1 for the 5.49 MeV alpha in air. The red trace calculated from Equation (1) with alpha range parameter and fall-off parameter is visually indistinguishable from the empirical alpha transmissivity curve [6] . The mean range, marked by the dashed black vertical line is defined by the distance at which alpha transmission is 50%, as indicated by the dashed black horizontal line in the figure. The fluctuation in alpha range values about as the particle rapidly comes to rest is closely represented by a Gaussian distribution (dashed blue curve) with relative uncertainty of ~5% [7] . In the figure, the amplitude of the Gaussian is scaled down by a factor 2 to facilitate visual presentation of both plots.

The measurement procedure described in [4] pertained primarily to the diffusive ingress of radon through a single external subterranean (or partly subterranean) wall, such as commonly occurs in the basements of residences and research labs. For purposes of mathematical modeling, these conditions entail one-dimensional atomic diffusion from a planar source. For atoms diffusing from the source at the wall through the room air and over the GM detectors, positioned so that the detector windows are horizontal (in the plane) and facing upward, the effective detection volume in the half space (centered on each detector window) and effective cross section of flow through a surface normal to the GM window were defined and evaluated in [4] by means of as follows

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) rather than over the finite space and surface of an instrumental control volume, the constraint posed by the alpha range function leads to finite results. For example, evaluation of Equations (2) and (3) for the 5.49 MeV alpha of ²²²Rn in air yielded and.

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 [4] by modeling the detection process as a radioactive flow measurement [8] , such as employed in high-performance radioactive liquid chromatography [9] , by defining an effective residence time

in which

is the flow velocity determined by the ²²²Rn diffusion coefficient [10] [11]

and decay rate [12]

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 [4] by solution of the one-dimensional steady-state diffusion equation with implementation of open-volume boundary conditions and account taken of loss by radioactive decay.

Although not reported in [4] , the integrals in Equations (2) and (3) upon substitution of range function (1) can be expressed in terms of known special functions yielding the following closed forms for the effective detection volume and area

in which

is the polylogarithm function. For index, the function is related to the dilogarithm function

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 ²²²Rn 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. And for another, the examination of radon transport by Brownian motion sheds light on the physical process of diffusion, clarifies theoretical points concerning the calculation of expectation values, and distinguishes the statistical concept of residence time from the concept of time of first passage, which frequently arises in the study of stochastic processes. These two approaches―macro-statistical diffusion and micro-statistical Brownian motion―address two differently expressed, but basically equivalent, practical questions. In the former (diffusion), the question is “On average how many particles per unit time flow through the detection region?”; in the latter (Brownian motion), the question is “On average how long does one particle remain in the detection region?”.

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 [4] . In this regard, Equations (8), (9), and (12) will be useful later in the comparison of the residence time as modeled in [4] with the mean residence time obtained from the micro-statistical treatment developed in this paper.

The equation for the conditional probability of diffusion of an unstable particle from some source point at time to a field point at time t with diffusion coefficient D and decay rate takes the form [13] [14]

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 of the right side of the equation.

In a coordinate system with vertical z-axis and horizontal x-axis directed normally outward from the particle source plane (), the probability density for arrival of a particle at irrespective of the initial location of the particle in the source plane is obtained by integrating (14) over the coordinates, as follows:

where is the prior probability (in a Bayesian sense), assumed to be uniform, of the particle distribution in the source plane. Since is a constant, it can be assigned whatever numerical value makes a function whose integral over the entire x-axis is unity. Taking leads to the second equality in (15), which is equivalent to the Green’s function for one-dimensional diffusion with decay from a delta function source.

The function in Equation (15), plotted in Figure 2, is the conditional probability density for finding a particle within the vertical plane located at horizontal coordinate x at time t, given that the particle was in the plane at. To keep symbolic notation as simple as possible where there is no confusion, the initial (0,0) condition is not explicitly shown in the argument. Nevertheless, it bears emphasizing that the coordinates expressed in Equation (15) are actually spatial and temporal intervals. Moreover, radioactive decay is a Markov or “memoryless” process [15] ―that is, an unstable nucleus does not age, but has the same probability of decay within a specified time interval irrespective of when the interval was begun. Therefore, Equation (15) applies whether the source plane is taken to be the wall through which radon enters the room or a plane (to be adopted shortly) bisecting the detection volume above the detector window.

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 [4] and elaborated further in this paper.

From Equation (15), one can calculate the cumulative probability of the particle being somewhere between the plane (entrance plane of the detection region) and (exit plane of the detection

region) during the time interval t begun when the particle was placed at (the vertical plane bisecting the detection region centered above the detector window)

The error function in Equation (16) is defined by

and is an anti-symmetric function of its argument.

The time interval (mean statistical residence time) of a particle within the detection region is then obtained by averaging over all values of the interval t in Equation (16) by the following integral

in which

is the characteristic diffusion length. (See Ref. [4] , Appendix 1) For ²²²Rn, it follows from relations (6) and (7) that

In the limit that in Equation (18), the residence time, the statistical lifetime of the unstable nucleus. (Note: the lifetime and half-life are related by.)

The mean value expressed in Equation (18) calls for a brief explanation because its justification comes from a statistical relation that may not be widely familiar. Ordinarily, the expectation value of a distributed quantity (i.e. random variable) T is performed by an integration weighted by a probability density function (pdf) in an expression like

in which is the probability that a realization t of random variable T falls within the interval. However, in Equation (16) is not the probability density for residence time. Moreover, it is dimensionless, so that to multiply by t and integrate over a range of t would lead to a quantity that has the incorrect dimension of time squared.

A procedure entirely equivalent to (21) for computing expectation values is to use the cumulative distribution function (cdf) defined as the probability given by

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

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, which can be related to the cdf of the residence time by the following argument. The probability that the particle lies in a region between and after time interval t means that the actual residence time T must be greater than t, otherwise the particle would have left the region. Thus, one can write the equivalence

where the second equality in Equation (24) follows from the definition (22) of the cumulative distribution function. Since for, it then follows from relations (23) and (18) that

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

Phenomenological: (26)

Statistical:. (27)

Numerical evaluation of the statistical residence time (27) requires substitution of a specific length of the detection volume. Since a fixed length implies a sharply defined volume, such a substitution corresponds to a phenomenological residence time (26) in the limit, as illustrated by plot (f) in Figure 3. Moreover, since the characteristic diffusion length (20) for ²²²Rn 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. It is therefore appropriate to compare

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, relation (30) leads to a detection volume of horizontal length, which, as shown in Figure 1, corresponds very closely to the distance through air at which the alpha range function is perceived to drop to 0. Exact numerical evaluation of residence times (26) and (27) for L given by relation (30) yields

which are in excellent accord.

From the definition (22) of the cdf, one obtains the corresponding pdf by application of the Leibniz rule [16] for differentiating an integral

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

Figure 4 shows a plot of pdf (33) as a function of time for parameters characteristic of ²²²Rn with given by expressions (29) and (30). The sharp rise (to a maximum) with long “fat tail” of the pdf resembles an inverse Gaussian or Levy distribution. The first term () on the right hand side of pdf (33), known as the Smirnov density [17] , derives exclusively from continuous Brownian motion; the second term depends on radioactive decay and would be absent if the diffusing particle were stable (). It is to be noted that the mean residence time for a diffusing stable particle, whether calculated

simply by relation (25) or much more onerously from expression (21) with replaced by pdf (33), is infinite. One sees this explicitly by taking the limit of Equation (27) as. The interpretation of this result for a stable particle undergoing one-dimensional Brownian motion, as is demonstrated in books on stochastic processes [18] , is that the particle will eventually return to a specified location with a probability of 100%, but on average only after a random walk of infinite duration. However, an unstable particle, which does not survive long enough to take an infinite walk, has a finite mean residence time―a result that one would of course expect on physical grounds.

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 [19] . Consider the first-passage of an unstable particle with decay rate and diffusion coefficient D undergoing a random walk from an initial position to the boundary. Since there is no further interest in the particle once it has reached, the particle is considered absorbed at the boundary. One might therefore interpret the residence time as the mean time for a particle to diffuse from the initial location to the exit of the detection region. The particle is then absorbed at the exit and cannot return.

Mathematically, it is required to find a probability density function for one-dimensional diffusion of an unstable particle such that. Since the diffusion equation is linear, the sought-for function is readily found from solution (15) by the method of images [20] to be

The method of images entails placing an “anti-source” of particles at just such a location as to null the net probability at. From the form of expression (34), one sees that vanishes identically for. Figure 5 shows plots of pdf (34) as a function of x for different values of t with and (marked by a vertical dashed line) for the diffusion and decay parameters of ²²²Rn. The atom is free to wander over the range with vanishing probability at the barrier. The spatial range of physical significance does not extend beyond.

The probability that the particle has not reached the absorbing barrier in time t, referred to as the survival probability, is then given by

which is seen to be of identical form to the cumulative probability (16) of finding the particle in the region with no absorbing boundaries. Thus, the mean time of first passage to the boundary

is identical under the stated circumstances to the residence time in (18). Likewise, the probability density function of the random variable whose mean is, which is given by

is identical to pdf (33) for. Although the constrained pdf (34) differs

from the unconstrained pdf (15) mathematically and visually (as seen by comparing Figure 2 and Figure 5), the equivalence of the spatial integrals (16) and (35) follows by making the transformation of variable in the integral (35), leading to the expression

identical to probability (16).

A plot of the survival probability as a function of time is shown in Figure 6. Over a short time interval, as shown in the insert, starts flat and then decreases approximately linearly in time. Over a long time interval, exhibits a heavy tail due to the asymptotic behavior of the error function.

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 such that a particle diffusing from initial position to either boundary is absorbed, i.e.. The solution is

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 () of image functions placed at and with must be employed to null the probability density at both boundaries. The complete solution takes the form

in which alone enforces only the boundary at as shown in Figure 5.

Figure 7 shows plots of pdf (39), truncated at, for and different values of time. Clearly, the particle is confined between the two boundaries, indicated by vertical dashed lines. However, as time increases, the Gaussian functions corresponding to spread, and each image function eventually crosses the opposite boundary. To maintain the two boundary conditions, therefore, additional image functions are required. Figure 8 shows spatial plots of pdf (39) at the fixed time interval for increasing numbers of image pairs. A minimum of 7 image pairs (in addition to the Gaussian) is required to maintain absorption of the particle at the two boundaries. As time increases further, additional image pairs must be added. One sees, therefore, why the exact solution for arbitrary time comprises an infinite number of image functions.

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 given by expression (19). Evaluating relation (43) for the diffusion and decay parameters of ²²²Rn with leads to an exit time, which is very much shorter than the previously obtained residence time (27).

The difference between and is not just a matter of magnitude, but also of analytical structure. Series expansion of to the first non-vanishing term in (which is for ²²²Rn) and leads to

independent of the decay rate, in contrast to

The discrepancy between (44) and (45), which one might have thought described the same physical system―namely, a decaying particle initially at in Brownian motion between two locations at―arises because the two physical systems are not at all equivalent.

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 overlying the detector. The particle in this system can exit the region of detectability or diffuse back into it with a probability of 50%. The space open to the radon atom is effectively infinite, whereas the space open to the particle between two absorbing barriers is finite. In the system with one absorbing barrier at, whose time of first passage is equal to the residence time, the region of space available to the particle is still infinite.

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 is maximum. Since the statistical lifetime is very long () compared with the mean residence time (), one can, for the purposes of finding the maximum, set equal to 0, i.e. drop the second term on the right side of the second equality in (33). The location of the peak of the first term (i.e. Smirnov density) is then obtained by setting the time derivative of its logarithm to zero, and leads to

which, to within a numerical factor, has the same form as. Evaluated for radon diffusivity and length, one obtains, in close agreement with the maximum of the plot in Figure 4.

It is worth noting here, although the derivation is given elsewhere [21] , that the mean exit time (referred to as time to capture) of a stable particle diffusing from some initial position to either of two absorbing boundaries can be calculated directly from solution of a Poisson equation

with implementation of appropriate boundary conditions. For a stable particle undergoing a one-dimensional random walk between absorbing boundaries at and the solution to (47) is

which reduces to for the chosen initial condition. As expected for consistency, this is also the result obtained by taking the limit of in relation (43) as approaches 0

In the case of an unstable particle with decay rate diffusing between two absorbing boundaries, it can be shown [22] that the corresponding time to capture follows from an equation of the form

and leads to a solution that for reduces exactly to in relation (43).