_{1}

A simple method employing a pair of pancake-style Geiger-Mueller (GM) counters for quantitative measurement of radon activity concentration (activity per unit volume) is described and demonstrated. The use of two GM counters, together with the basic theory derived in this paper, permit the detection of alpha particles from decay of and progeny (
^{218}Po,
^{214}Po) and the conversion of the alpha count rate into a radon concentration. A unique feature of this method, in comparison with standard methodologies to measure radon concentration, is the absence of a fixed control volume. Advantages afforded by the reported GM method include: 1) it provides a direct in-situ value of radon level, thereby eliminating the need to send samples to an external testing laboratory; 2) it can be applied to monitoring radon levels exhibiting wide short-term variability; 3) it can yield short-term measurements of comparable accuracy and equivalent or higher precision than a commercial radon monitor sampling by passive diffusion; 4) it yields long-term measurements statistically equivalent to commercial radon monitors; 5) it uses the most commonly employed, overall least expensive, and most easily operated type of nuclear instrumentation. As such, the method is par-ticularly suitable for use by researchers, public health personnel, and home dwellers who prefer to monitor indoor radon levels themselves. The results of a consecutive 30-day sequence of 24 hour mean radon measurements by the proposed GM method and a commercial state-of-the-art radon monitor certified for radon testing are compared.

Radon is a colorless, odorless, chemically inert radioactive gas produced in the decay series of uranium-238 (

In the decay series leading from

Adverse health effects from

Since the half-life of

In response to increasing concern over radon exposure, numerous investigations have been made during the past 50 years or more regarding methods to measure indoor radon concentration. This is a vast subject, but, in brief, contemporary radon methodologies ordinarily employ one of the following categories of detectors [^{3} or (particularly in the US) in pCi/L, where 1 becquerel (Bq) = 1 disintegration/second, and 1 pico-curie per liter (pCi/L) = 37 Bq/m^{3}.

From the standpoint of the convenience of end users of radon activity information, who in the majority of cases are not nuclear scientists or engineers with sophisticated instrumentation to perform radon activity measurements themselves, each of the preceding numbered methodologies, with perhaps exception of (5), requires that the collected sample bearing a fixed volume of radon be sent away to a testing laboratory for analysis. The delay time of at least a few days is not merely an inconvenience, but can significantly affect the uncertainty of the estimated radon activity as a result of sample decay throughout the transit period and waiting time to measurement. (Recall that the half-life of

Commercially available continuous radon monitors (methodology 5 above), pose other kinds of problems, particularly to researchers who need to monitor the short- term radon concentration in their laboratories. University physics labs are often located in the basements of buildings with concrete floors and cinder-block walls, i.e. materials through which radon, originating in the underlying soil, can diffuse and which, themselves, give rise to radon exhalation [

Another drawback is that continuous radon monitors usually record a signal that is proportional to the absolute radon concentration, but for which the proportionality “constant” is an empirical number not deducible from first principles. It must be determined by calibration with a primary standard, can vary among the units sold, and is built into the device algorithm which, as mentioned, is a proprietary secret. Thus, apart from maintaining their own continuously operating nuclear counting equipment, which is (a) costly, (b) ties up apparatus needed for other purposes, and (c) ordinarily requires expertise in alpha, beta, and/or gamma spectroscopy (for examples of such setups, see [

Last, but by no means insignificant, is the relatively high statistical uncertainty in the values of radon activity concentrations measured by commercially available continuous monitors over the short term. The literature of one such state-of-the-art product, claimed by the manufacturer as certified for official radon testing everywhere (except in the US and Canada for which they sell a more expensive model) assigns a relative uncertainty (ratio of standard deviation to mean) of 20% to the displayed 7-day mean activity concentration [

This paper describes a simple, novel method to determine radon concentration directly by means of alpha particle counting in an open volume (rather than fixed control volume) with the most commonly available, overall least expensive, and most easily operated type of nuclear instrumentation: a standard pancake Geiger-Mueller (GM) radiation counter. Although there have been previous attempts to use GM counters to detect the presence of radon [

・ Measurement results are obtained directly after sampling; no external testing laboratory is required to convert the raw data into an absolute radon activity concentration.

・ A relatively short measurement period of 24 hours suffices to provide adequate precision (relative uncertainty of about 10% at 100 Bq/m^{3}) for most personal health- related purposes, such as to ascertain whether a local radon concentration exceeds the published standards set by the US Environmental Protection Agency [^{3} set by the World Health Organization [

・ The operational relations derived in this paper, based on fundamental physical principles and known properties of alpha particles and detector materials, are easy to implement and straightforward to interpret.

The simplicity of the measurement method together with the transparency of the analysis makes it possible for a broad demographic such as researchers, public health officials, and home residents, particularly in developing countries, to estimate local radon activity concentration in real time without the need of specialized and expensive nuclear equipment and without having to rely on public or private testing laboratories or on the readings of commercial devices whose operation and programming are shrouded in secrecy.

The method described in this paper calls for use of two pancake-style GM radiation counters: one with a detector window unobstructed so as to record incident

As the only naturally occurring radioactive gas to emanate from common building materials or percolate into a room from the underlying soil, radon is the only alpha emitter and progenitor of alpha emitters (isotopes of polonium) that can contribute to the GM alpha particle count rate under the conditions described above. Alpha particles issuing from radioactive materials in the walls, floor, and ceiling are not detected because of the intrinsically short range of alpha particles in matter. As a relatively heavy ion (in comparison to the much lighter beta particle), an alpha particle emitted in radioactive decay loses energy almost entirely by creating electron-ion pairs in its passage through matter [

Apart from slight variations due to fluctuations in air density, an alpha track is effectively straight with a range closely correlated with initial energy E according to the relation [

where range

For passage through condensed matter, the range

where

with weight

For a mixture of molecules (such as air) of effective mass M,

For purposes of this paper, the molar (or volume) composition of air is 78% N_{2}, 21% O_{2}, and 1% Ar

leading to the effective atomic mass number of air

Given the density of air

As applied to a GM window, which is here taken to be Muscovite mica (KAl_{3}Si_{3}O_{10}(OH)_{2}) of density

The above examples serve to illustrate the very short range of alpha particles produced by radon and its progeny and provide a basis for the effective alpha range function and alpha transmission function to be constructed shortly.

It is assumed in this paper that all radionuclides decay exponentially in time in accordance with standard nuclear physics, although this assumption has been challenged in the literature and is a matter of current research. For example, see [

where

from which follows the relation between half-life

If, as in the case of

where

A general method to solve coupled linear differential equations like (11) is given in [

which is particularly useful for numerical solution by computer. Of relevance to the present work is the limiting case known as secular equilibrium, a steady-state condition of equal activities between a long-lived parent and much shorter-lived daughter radionuclides. Mathematically, the criteria for secular equilibrium are

In light of the preceding background, consider a GM detector with circular window of radius

in which the various factors are defined as follows:

Explicit expressions for the functions

Effective detection volume (in the half space

Effective cross section of radon flow through surface

The upper limit to the radial integrals (15) and (16) is given as

The overall efficiency

Thus, the volume average of Equation (14) takes the much simplified form

The number

where the radon current density

is the product of the diffusion velocity

and a characteristic diffusion length

are obtained from solution of the diffusion equation based on Fick’s law [

The diffusion of radon over the detector results in an effective residence time in the detection volume

analogous to the residence time in radioactive flow measurements [

Substitution of Equations ((21) and (24)) into Equation (18) leads to the compact expression

where

Inversion of Equation (26) therefore yields the sought-for radon activity concentration (e.g. in Bq/m^{3})

The bracketed product in the numerator of Equation (25) yields a dimensionless quantity, the number of radon disintegrations that occur in a time interval given by the residence time in the denominator. In other words, under conditions of 100% detection efficiency

Derivation of the activity concentration relation (27) assumed (a) effectively equivalent ranges for the three alphas produced by radon and polonium decays and (b) secular equilibrium among radon and its progeny. Examination of these two assumptions will lead to an extension of relation (27).

Consider assumption (a) first. The energy of an alpha particle determines its range and therefore the effective range function

where indices

of the activities

The ranges in air and mica, calculated from Equations ((2) and (8)) respectively, increase approximately as the 3/2 power of alpha energy.

It is important to bear in mind that the three residence times

still depend on the diffusion velocity

Consider next assumption (b) regarding secular equilibrium, expressed by relation (13). Solution (12) of the Bateman equation (11) for a closed system of radon and its

Nuclide | Energy (MeV) | Air (cm) | Mica (μm) |
---|---|---|---|

5.5 | 4.0 | 20.4 | |

6.0 | 4.6 | 23.5 | |

7.7 | 6.9 | 35.1 |

Nuclide | ^{−1}) | ^{2}/s) | ||
---|---|---|---|---|

2.1 (-6) | 1.1 (-5) | 4.8 (-6) | 2.3 | |

3.8 (-3) | 5.3 (-6) | 1.4 (-4) | 3.7 (-2) | |

4.2 (3) | 5.3 (-6) | 1.5 (-1) | 1.3 (-9) |

progeny leads to the results displayed graphically in

Depending on environmental conditions (e.g. relative humidity, temperature, aerosol content, and other factors), the radon progeny, whether initially charged or neutral, are subject shortly after generation to physical and chemical interactions with the surfaces of a room as well as with particles and molecules within the room’s atmosphere. Because of these interactions, the daughter nuclides―in particular

In place of the Bateman equations, the new equilibrium conditions can be determined from solution of the Jacobi equations [

where

is the loss rate,

In the experimental section of this paper indoor radon activity concentrations are reported for measurements taken in an unventilated basement room with cement block walls and very low dust and aerosol content in the air. Progeny resulting from radon decay are more likely to become part of air molecular clusters than to plate out. Experimental values for the deposition velocity of attached radon progeny have been found experimentally to range from about 0.03 to 0.20 m/h [

which is in excellent accord with the value 0.0117 m/h obtained from a recent statistical fit to data taken in indoor dwellings [

Substitution of

from which follow the equilibrium proportions and number of alpha particles per radon decay

Inversion of Equation (28) then yields the general operational relation for radon activity concentration measured with two GM counters in sampling time T

With the values given in Equations ((31) and (37)), the constant ratio

To good approximation, the standard deviation of activity (38) is then

in which the uncertainty

It is to be noted explicitly that relation (39) does not depend on a non-calculable empirical proportionality constant that must be determined independently by calibration against a primary standard. Rather, the physical quantities whose numerical values enter Equation (38) are measurable properties of particles (e.g. alphas), materials (e.g. mica), and local environment (e.g. air). As such, the more accurately these physical quantities are known for the conditions under which measurements are made, the better will be the resulting estimate of radon activity concentration. Nevertheless, the numerical values in relation (39) show that, for moderate levels of radon in an unventilated room with clean air, the analysis based on Equation (28) with equilibrium values (37) gives results comparable to the analysis which assumed secular equilibrium values (31).

where

The solid angle subtended by a detector in an experiment to measure nuclear radiations is ordinarily defined only for configurations with a point-like or planar source symmetrically located on or about the symmetry axis normal to the detecting surface [

where, from

The problem, then, is to express angles

The first step is to obtain the lengths

from the law of cosines. The second step is to express

After some algebraic manipulation, it then follows that

As a quick plausibility check of Equation (46), set

Upon arrival at the surface of the GM mica window, an alpha particle will fail to pass through the mica and ionize the interior fill gas if its path length exceeds the range in mica, as given in

where d is the window thickness, as shown in

Examination of

The area of intersection of two circles of respective radii

where, in the context of the geometry of the GM window,

The operation Re in Equation (49) to “take the real part” of the bracketed expression is necessary to include the case of non-overlapping circles. In that case, expression (49) returns 0; otherwise it can return an imaginary number if

which peaks at about 15% at a source separation of 1.13 cm.

The overall detection efficiencies of the individual 5.5, 6.0 and 7.7 MeV alpha particles arriving at the detector from anywhere within the half-space above the GM window are obtained by substitution in Equation (17) of the respective range functions to yield

The mean alpha detection efficiencies under conditions of secular equilibrium (31) and Jacobi equilibrium (37) are

which agree closely with the empirically determined alpha efficiency 4.2% provided by the manufacturer of the author’s GM counters [

A comparison of the GM method with a commercial radon monitor to measure short- term and long-term indoor radon concentrations was made by means of two Inspector Radiation Alert detectors (to be referred to as GM1 and GM2) manufactured by S.E. International Inc (SEI), each with a halogen-quenched, pancake-style GM tube with mica window of areal density ~2.0 mg/cm^{2} and effective window diameter 4.5 cm. The manufacturer-specified accuracy (in counts per minute) is

The two GM counters were placed, as described in Section 2.1, on a raised (0.5 m) platform in a basement room with concrete walls, at a distance of 1.5 m from the one wall that was lower than the outside ground level due to the exterior sloping terrain. From data of radon penetration and mobility [

On the same platform and at the same distance from the subterranean wall as the GM counters, were placed two Corentium Home radon monitors [^{3} after 7 days. As pointed out in Section 1, the relative uncertainty (ratio of standard deviation to mean) of a 24-hour measurement would then be about 53%. The monitor displays three readings, the mean radon activity concentration of the preceding (a) 24 hours, (b) 7 days, and (c) period up to 1 year. The measurement durations of relevance to the experiment reported here are 24 hours and 30 days.

^{3} (or 2.7 pCi/L), GM1 recorded about 62,300

As ^{3}, in comparison with

the mean reading of COR1 and COR2.

The method to measure radon concentration with GM counters, although especially useful for short-term (24 h) sample periods, also provides accurate long-term radon levels with an uncertainty that decreases with the square root of sampling time. From the data in

The results (54) show that, with respect to the Corentium monitors which are marketed as a standard for radon testing, the proposed GM-based method yielded a statistically equivalent mean long-term value of radon concentration. Assuming that the two sets of measurements are samples from normal distributions, the statistic z for testing the equivalence of two means [

is itself normally distributed as

The conventional statistical threshold for judging whether an event could have occurred through pure chance is 5%. Since probability (56) well exceeds the 5% threshold, the difference between the two means in (54) is regarded as statistically insignificant. It

No. | GM1 | GM2 | COR1 pCi/L | COR2 pCi/L | ACOR pCi/L | |||
---|---|---|---|---|---|---|---|---|

1 | 63,140 | 70,570 | 3.53 | 3.84 | 4.24 | 3.02 | 2.81 | 2.92 |

2 | 62,360 | 68,940 | 2.94 | 3.20 | 3.53 | 3.48 | 2.56 | 3.02 |

3 | 62,260 | 68,020 | 2.37 | 2.58 | 2.85 | 2.54 | 2.02 | 2.28 |

4 | 62,410 | 68,350 | 2.49 | 2.71 | 3.00 | 2.70 | 3.16 | 2.93 |

5 | 61,480 | 67,200 | 2.34 | 2.55 | 2.81 | 2.78 | 2.56 | 2.67 |

6 | 61,870 | 67,260 | 2.11 | 2.30 | 2.54 | 3.24 | 2.32 | 2.78 |

7 | 61,480 | 67,600 | 2.62 | 2.85 | 3.15 | 3.37 | 3.37 | 3.37 |

8 | 62,260 | 67,930 | 2.30 | 2.51 | 2.77 | 3.18 | 3.24 | 3.21 |

9 | 62,250 | 67,830 | 2.24 | 2.44 | 2.70 | 3.32 | 3.08 | 3.20 |

10 | 61,690 | 67,830 | 2.63 | 2.86 | 3.17 | 3.78 | 2.24 | 3.01 |

11 | 62,450 | 67,570 | 1.92 | 2.09 | 2.31 | 3.24 | 2.89 | 3.07 |

12 | 61,860 | 68,070 | 2.68 | 2.92 | 3.22 | 3.67 | 2.37 | 3.02 |

13 | 61,460 | 67,400 | 2.49 | 2.71 | 3.00 | 2.56 | 3.18 | 2.87 |

14 | 61,940 | 68,430 | 2.87 | 3.13 | 3.46 | 3.64 | 4.05 | 3.85 |

15 | 61,590 | 68,170 | 2.94 | 3.20 | 3.53 | 3.51 | 3.51 | 3.51 |

16 | 62,790 | 67,640 | 1.73 | 1.89 | 2.09 | 2.62 | 2.08 | 2.35 |

17 | 61,500 | 66,820 | 2.06 | 2.24 | 2.48 | 3.02 | 2.81 | 2.92 |

18 | 61,520 | 67,930 | 2.82 | 3.07 | 3.39 | 2.51 | 3.02 | 2.77 |

19 | 62,470 | 67,540 | 1.89 | 2.06 | 2.27 | 3.37 | 2.27 | 2.82 |

20 | 61,780 | 67,240 | 2.16 | 2.35 | 2.60 | 3.37 | 2.59 | 2.98 |

21 | 62,000 | 68,120 | 2.62 | 2.85 | 3.15 | 2.24 | 2.83 | 2.54 |

22 | 62,680 | 67,620 | 1.80 | 1.96 | 2.16 | 2.43 | 2.45 | 2.44 |

23 | 62,290 | 67,440 | 1.94 | 2.12 | 2.34 | 2.51 | 3.00 | 2.76 |

24 | 61,810 | 68,160 | 2.78 | 3.02 | 3.34 | 2.56 | 2.27 | 2.42 |

25 | 62,060 | 68,670 | 2.96 | 3.22 | 3.56 | 3.40 | 2.83 | 3.12 |

26 | 62,020 | 68,060 | 2.56 | 2.79 | 3.08 | 3.67 | 3.40 | 3.54 |

27 | 61,320 | 68,120 | 3.09 | 3.36 | 3.72 | 3.02 | 3.67 | 3.35 |

28 | 61,500 | 69,000 | 3.57 | 3.89 | 4.30 | 3.83 | 2.75 | 3.29 |

29 | 62,100 | 68,510 | 2.82 | 3.07 | 3.39 | 2.37 | 2.43 | 2.40 |

30 | 62,490 | 68,770 | 2.73 | 2.97 | 3.28 | 2.91 | 3.89 | 3.40 |

SE = secular equilibrium; JE = Jacobi equilibrium

is worth emphasizing, however, that the difference of the two mean long-term measurements in (54) would be statistically significant if those two concentrations were both measured by the GM method. In that case, the relevant statistic would be

with associated p-value

which is considerably below the 5% threshold of significance. The reason for the difference in outcomes between (56) and (58) is due to the much lower standard error of the GM measurements in comparison to measurements by the commercial monitor. This seminal point is elucidated further in

be distributed with a Gaussian density centered on the intersecting theoretical red or blue line (depending on equilibrium conditions) given by Equation (38). One can infer from the figure that many such points would be required before the ML line of regression to the radon concentration estimated by commercial monitor would be statistically equivalent to theoretical relation (38) of the GM method.

The GM counters and commercial radon monitors used in this experimental comparison both sampled environmental radon through a process of passive diffusion. One may inquire, therefore, why it is that the GM method can make consistent short-term measurements with much lower statistical uncertainty, whereas short-term measurements of the commercial monitor show wide scatter and insensitivity to the total alpha count rate. An explanation, at least in part (since the programming of the commercial monitor is a proprietary secret), is that the built-in control volume according to Corentium is 24 cm^{3}, whereas the effective detection volumes of the 5.5, 6.0, and 7.7 MeV alpha particles in open air are, according to Equation (15), about 139, 210, and 700 cm^{3} respectively.

In the comprehensive description of methodologies in the World Health Organization Handbook [

While the preceding statements are not altogether incorrect, they do not constitute valid arguments against the use of GM counters as an effective methodology to measure―and not merely detect―indoor concentration of radon. The statements presuppose the use of only a single GM counter and assume the necessity of an initially determined volume of radon. As shown in this paper, neither supposition is justified.

In this paper I describe and demonstrate a method to measure indoor radon using two GM counters, which, together with the theoretical analysis derived here for converting an alpha count rate into a radon concentration, provides a number of advantages over alternative methodologies:

・ The method is especially suitable for daily monitoring of radon since it provides an activity concentration after just 24 h in situ sampling that eliminates the need to send a prepared sample to a testing laboratory. For continuous monitoring, this saves users much time and considerable expense. Other methodologies (e.g. alpha-track counting) may have higher accuracy, but are not a convenient or economical option for researchers or home dwellers who require frequent or continuous radon monitoring.

・ The GM method yields more precise mean short-term and long-term radon concentrations than do commercial radon monitors employing alpha detection in a passive diffusion chamber.

・ The GM method yields radon concentrations of comparable accuracy to that of commercial monitors employing sampling by passive diffusion.

・ The method employs the simplest, most versatile, and overall least expensive radiation detection instrumentation (simply two pancake GM counters) for non-nuclear researchers and home dwellers who are not trained in nuclear spectrometry or do not have their own nuclear spectrometry instrumentation. As such, the GM method can be especially useful in geographical areas where access to specialized radon testing equipment or testing laboratories is inadequate. Or, in fact, useful in any milieu for people who want to monitor indoor radon levels themselves.

The principles behind the experimental method and theoretical analysis described in this paper are physically verifiable and fully transparent (in contrast to the operation and programming of commercial monitors which are patent-protected intellectual property). The use of two monitors, so placed as to receive alpha particles only from radon and its polonium progeny, makes it possible to separate the alpha signal from the beta and gamma background. Whereas a commercial radon monitor may use energy selection to count alphas from radon only, the method described here counts alphas from both sources and uses the laws of physics and properties of materials to calculate the mean number of alphas detected per radon decay.

An essentially novel feature to the method described here is the means by which the alpha count rate is converted into a radon concentration even though there is no built-in control volume. This is accomplished by calculating an effective detection volume (15), detection cross section (16), and residence time (24), which are determined by the physical laws governing alpha particle interactions in matter and the diffusion of atoms in air. Moreover, the theoretical analysis presented here can be applied to different environmental conditions of indoor radon measurement, once the effects of these conditions on the diffusion of radon gas and the equilibrium of radon and its progeny have been ascertained.

Silverman, M.P. (2016) Method to Measure Indoor Radon Concentration in an Open Volume with Gei- ger-Mueller Counters: Analysis from First Principles. World Journal of Nuclear Sci- ence and Technology, 6, 232-260. http://dx.doi.org/10.4236/wjnst.2016.64024

Mass conservation of nuclei in a process of one-dimensional diffusive flow with nuclear decay is expressed by the differential equation

in which

with diffusion constant D. In the case of a steady-state flow, the derivative

with characteristic diffusion length

The solution to Equation (61) and the associated current density take the general form

with diffusion velocity

Coefficients

For radon measured in an open volume, the condition that

and from Equation (63) that

Define the following measurable quantities:

The alpha count rate (cpm) and corresponding standard deviation is then determined from the expressions

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