Applied Modeling and Computations in Nuclear Science - American

Chapter 18

Downloaded by NATL UNIV OF SINGAPORE on May 5, 2018 | https://pubs.acs.org Publication Date: November 16, 2006 | doi: 10.1021/bk-2007-0945.ch018

Corrections for Overdispersion Due to Correlated Counts in Radon Measurements Using Grab Scintillation Cells, Activated Charcoal Devices, and Liquid Scintillation Charcoal Devices 1

2

Phillip H . Jenkins , James F. Burkhart , and Carl J. Kershner

3

1

Bowser-Morner, Inc., 4518 Taylorsville Road, Dayton, O H 45424 University of Colorado, 1420 Austin Bluffs Parkway, Colorado Springs, C O 80918 Mound Laser and Photonics Center, Inc., 720 Mound Road, Miamisburg, O H 45342 2

3

Lucas and Woodward (1) published values for a factor, " J , " for correcting estimates of counting error that were based on Poisson statistics. Their work considered radon measurements using the "Lucas cell." We expanded on their work by considering four different types of scintillation cell currently in use. Also, we investigated the relationships of the individual counting efficiencies for the alpha particles emitted by R n , P o and P o in these types of scintillation cells and showed that they were not equal as assumed by Lucas and Woodward (1) but varied with cell geometry. Further, we measured the proportions among the individual counting efficiencies at two locations of different elevations, Dayton, Ohio and Colorado Springs, Colorado, and showed that the variation of the individual counting efficiencies was affected by air density. Using values of individual counting efficiencies for the three radionuclides, we calculated values of J for the four types of scintillation cell at both locations. We calculated values of J 222

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© 2007 American Chemical Society

Semkow et al.; Applied Modeling and Computations in Nuclear Science ACS Symposium Series; American Chemical Society: Washington, DC, 2006.

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250


for one type of activated charcoal device and showed that these values were much smaller than those for scintillation cells because of smaller counting efficiencies. We also calculated values of J for liquid scintillation charcoal devices and showed that these values were larger than those for scintillation cells because of the potential inclusion of increased correlated counts and because of larger counting efficiencies. A n example calculation of how the J might be used to correct the Poisson statistics for measurements from two device types was included.

1. Introduction It is common for Poisson, or "counting," statistics to be applied to measurements of radioactive materials to determine an estimate of the statistical uncertainty due to the random nature of radioactive decay and measurement of radioactive emissions. Unfortunately, it is also common for limitations of this approach to be ignored. The criteria that must be met in order for Poisson statistics to be valid are: (1) the number of atoms present in the sample must be very much greater than 1, (2) the counting time must be very small compared with the half-life of the radionuclide being detected, and (3) the observed counts must not be correlated. The first criterion is usually not an issue with radon measurements. The effect of violating the second criterion is a true variance that is smaller than the estimate obtained from "counting" statistics (underdispersion), but this criterion is also usually not an issue with radon measurements. The effect of violating the third criterion is a true variance that is larger than the estimate obtained from "counting" statistics (overdispersion). It is the third criterion that often is not met with radon measurements. There are potentially several sources of overdispersion, such as instability of the measurement system, as thoroughly discussed by Semkow (2); however, this chapter addresses only overdispersion due to the inclusion of correlated counts in radon measurements. Inkret et al. (3) also thoroughly discussed sources of overdispersion and underdispersion in radon and radon progeny measurements. A l l radon measurement techniques are based on detecting emissions from radon itself and/or emissions from one or more of its short-lived decay products. In cases where emissions from more than one of these radionuclides are observed, correlated counts may result, potentially making Poisson statistics invalid. Simply stated, for Poisson statistics to be valid only one observed count from each atom of radon and its subsequent progeny is allowed.


251 Lucas and Woodward (/) described a factor, '\7," for correcting Poisson statistics for measurements of radon using the specific type of alpha scintillation cell known as the "Lucas cell." The factor V is defined as the ratio of the expected variance to the expected mean of the count, or 2


J=a /p.

(1)

This factor is frequently called the "coefficient of dispersion." (2) Since μ is the value of the variance predicted by Poisson statistics, J is the ratio of the true variance to the Poisson variance. In this chapter, we expanded on the work of Lucas and Woodward (1) by calculating values of J for four types of grab scintillation cell currently used by the authors, for devices that adsorb radon and are analyzed by gamma-ray spectroscopy and for liquid scintillation charcoal devices. Note that in all three of these techniques the radon is sealed in a container and later analyzed by some type of counting method. Poisson statistics are commonly, and incorrectly, applied to measurements of radon and radon progeny, leading to unfair comparisons among measurement methods and misleading statements in the advertisement of products. The authors hope that this chapter will help in correcting this situation.

2. Calculation of Values of J Two assumptions are made that simplify the calculations. First, because of the small half-life of P o (164 μ5 (4)\ it is assumed that the decay of B i can be described as the emission of a beta particle immediately followed by the emission of an alpha particle. Second, because of the long half-life of P b (22.3 y (4)) compared with those of the short-lived radon decay products, it is assumed that P b is stable and therefore its decay constant is zero. H is defined as the maximum number of particles that can be detected by a specific method from the decay of a single radon atom and its subsequent short lived decay products. Because we expanded on the work of Lucas and Woodward (7) to consider radon measurement techniques other than scintillation cells, we used the maximum possible value of H, i.e., H =5. The values of σ and μ can be calculated i f the counting efficiencies of all detected radiation emissions are known. The approach is to track a single radon atom and calculate the probabilities of all possible outcomes from the decay of that radon atom until it becomes an atom of P b . These probabilities then also 2 l 4

2 1 4

210

210

2

210


252 apply to all the other radon atoms initially present. Consider that the radon nucleus exists in five states as it progresses through the decay series: 222

State 0 - it is still R n , State 1 - it has become P o , State 2 - it has become P b , State 3 - it has become B i / P o , and State 4 - it has become P b . 218

214

2 l 4

2 l 4


210

The mean of a count tracking the decay of one radon atom is calculated from the following equation: p=Zhq{h),

(A = 0, 1, . . . , / / )

(2)

where q(h) is the probability of observing h counts during the counting period. The variance of a count is calculated from the following equation: 2

2

2

a = [Eh q(h)]-p .

(A = 0, 1,...,//)

(3)

2

Once μ and σ are calculated, J is calculated as the ratio of the two, as shown in Eq. 1. The bulk of the problem is calculating the probabilities q(h).

A . Calculation of/7/, First, the probabilities, p of the nuclide we are tracking going from State / to State j during any time period, are calculated. Lucas and Woodward (7) used the traditional equations of Bateman (5) in their calculations of these factors. However, a much more efficient approach is to use the recurrence formula described by Hamawi (6) for linear serial transformations. This formula has been discussed by Scherpelz and DesRosiers (7). Jenkins (8) presented an example of the use of this formula, comparing it with the use of the Bateman equations, for modeling grab samples of radon decay products in air. The first step in the process is to define a set of intermediate factors, fy. First, the "first-tier" factors are calculated, which simply describe the exponential decay of each of the five radionuclides. Note that the exponentials are evaluated only in the first-tier factors. ij9

First-tier factors:

f = exp(-/l 0 /u-expH!/) /22 e x p R 0 f = exp(-X t) 00

0

=

2

33

3


(4) (5) (6) (7)

253

/ 4 = expH 0. 4

W

4

Once the "first-tier" factors are calculated, they are used to calculate the "second-tier" factors. f \ = (f -fχ) I (λ\ - λ^)

(9)

/i2 = Wi-/22)/02-*i)

(10)

/34 = (^33 - / 4 4 ) / Λ - Α ) .

(12)

Second-tier factors:

0

00


3

Likewise, the "second-tier" factors are used to calculate the "third-tier" factors. Third-tier factors:

f

02

= (f \ -fn) I (λ - λ ) 0

2

(13)

0

/i3 = ( / i - / 3 ) / a 3 - A i )

(14)

/24 = (^23 - / 3 4 ) / α 4 - 4 ) .

(15)

2

2

The pattern of the recurrence formula should be obvious at this point. The "third-tier" factors are used to calculate the "fourth-tier" factors. Fourth-tier factors:

f

03

= (foi -fn) I

- λ)

(16)

0

/i4 = ( / i 3 - / 4 ) / ( A - A ) . 2

4

(17)

1

And finally, the "fourth-tier" factors are used to calculate the "fifth-tier" factor. Fifth-tier factor:

/

0 4

= 2 >Ml^2 P03 ~ λ λ\ À fo3

(21) (22)

Po4 = h λ\ λ X f)4-

(23)

0

2

2

3

218

The next set, p are the probabilities that if the nucleus is in State 1 ( Po) at the beginning of the time period t, that it will be in State j > 0 at the end of the time period: lj9


254 (24) (25) (26) (27)

Pu =/n Pu

P\3

=λ\/ΐ2 =

Pl4

-

^1 M

^ / l 3 ^2

^3/l4-

The next set, p are the probabilities that i f the nucleus is in State 2 ( Pb) at the beginning of the time period /, that it will be in State j > 1 at the end of the time period:


2ji

P22

=

P23

=

(28) (29) (30)

fl2 Λ2./23 =

P24 h ^3/24-

The next set, /? , are the probabilities that i f the nucleus is in State 3 ( B i / P o ) at the beginning of the time period t, that it will be in State j > 2 at the end of the time period: 3>

214

214

(31) (32) And finally, p is the probability that i f the nucleus is in State 4 ( Pb) at the beginning of the time period t, that it will still be in State 4 at the end of the time period. Because it is assumed that P b is stable, /? always equals 1, but the equation is included here for completeness: 44

210

44

(33)

B. States at the Ends of Delay Time and Counting Time There is always a delay time, t , between the capture of radon in a sampling device and the counting of the sample in the analysis laboratory. The next step in the calculation process is to determine the state in which the radon nucleus exists at the end of the delay time. Equations 19 through 33 are typically used in a computer subroutine for the calculation of the p factors. The time, t, is set equal to the delay time, / and the ρ factors are calculated using that subroutine. The needed probabilities d are merely obtained from the p factors: d

i}

A

ϋ

0J

0J

(34) (35) (36) (37)


255 (38)

- P04-

4)4

The next step is to calculate the probabilities that the nucleus will go from State ι to State j during the counting time. This is done by setting the time, t, equal to the counting time, / , and calculating the ρ factors again. c

ϋ

C . Calculation of q(h) Once the d and py factors are calculated, then it is possible to calculate the values of q(h). ε , ε and ε are the counting efficiencies in counts per disintegration (c d" ) for the alpha particles emitted from R n , P o and P o , respectively. ε and e are the counting efficiencies (c d" ) for the beta particles or gamma rays (depending on measurement method) emitted from P b and B i , respectively. The calculation of q{h) is done in steps. First, the probabilities, q , of detecting / counts from k emissions are calculated:


0J

0

{

3σ

1

222

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1

2

3b

2I4

2 I 4

ik

=

Applied Modeling and Computations in Nuclear Science - American

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