Journal of Environmental Radioactivity 140 (2015) 50e58
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Journal of Environmental Radioactivity journal homepage: www.elsevier.com/locate/jenvrad
The accuracy of radon and thoron progeny concentrations measured through air filtration J.M. Stajic, D. Nikezic* Faculty of Science, University of Kragujevac, R. Domanovica 12, Kragujevac 34000, Serbia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 September 2014 Received in revised form 28 October 2014 Accepted 1 November 2014 Available online
The accuracy and the optimization of determining radon and thoron progeny concentrations in air using air filtration followed by alpha activity measurements were investigated in details. The effects of radon and thoron concentrations, filtering duration and the choice of measuring intervals on relative standard deviations were analyzed. Obtaining satisfactory results by this method should be expected only in the case of high radon and thoron progeny concentrations in air. The optimization process also showed up to be dependent on the progeny concentration. Determinant of the system matrix and its effect on the sensitivity of the results were investigated. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Air filtration Alpha activity Relative standard deviation System determinant
1. Introduction Inhaling high concentrations of radon, thoron and their decay products can cause harmful effects to human lung tissues. Exposure to radon and its progeny has been ranked as the second leading cause of lung cancer after tobacco smoking. This fact justifies the great efforts of scientists all around the world to develop and improve the methods for measuring concentrations of these naturally occurring radionuclides in air. The method based on air filtration, followed by measurements of alpha, beta or gamma activity from filter has been investigated by many authors (EL€ggeler et al., 1995; Iimoto et al., 2001; Hussein et al., 2001; Ga } , 2006; Pogorski and Phillips, 1985; Ruzer and Papp and Dezso Sextro, 1997; Singh et al., 2006). Detecting numbers of counts in several different time intervals, during or after filtering process, should allow solving the system of equations for radon and thoron
progeny concentrations in air. However, those who try to bring this good theoretical concept into practical realization can face the possibility of obtaining incorrect or physically unacceptable results. In our previous work (Stajic and Nikezic, 2014), we investigated the reliability of this method, illustrating the effect of counting statistics on the accuracy of the final results with some concrete and realistic examples. The purpose of this paper was to present a detailed and more general consideration of the accuracy and the optimization of the measuring method involving air filtration followed by alpha counting. The dependences of standard deviations and the system determinant on radon and thoron concentrations, filtering duration and the choice of measuring intervals were investigated.
2. Material and methods
* Corresponding author. Tel.: þ381 34 336 223; fax: þ381 34 335 040. E-mail address: [email protected] (D. Nikezic). http://dx.doi.org/10.1016/j.jenvrad.2014.11.002 0265-931X/© 2014 Elsevier Ltd. All rights reserved.
Radon (222Rn) and thoron (220Rn) decay schemes are given below, with neglecting the decay branches with small probabilities:
J.M. Stajic, D. Nikezic / Journal of Environmental Radioactivity 140 (2015) 50e58
222
a
Rn/
3:824 d
b b a a 214 214 218 214 210 84 Po / 82 Pb / 83 Bi / 84 Po/ 82 Pb 164 ms 26:8min 19:9min 3:05min
Some of the short-lived radon and thoron decay products (218Po, Po, 216Po, 212Bi and 212Po) are alpha emitters. Due to the very short half life, 214Po might be considered in equilibrium with its precursor, 214Bi. Furthermore, 212Pb is assumed to be formed directly from 220Rn due to the short half-life of 216Po. Measuring total numbers of alpha counts during five different time intervals, subsequent to filtering process, would allow determination of individual concentrations of 218Po, 214Pb, 214Bi, 212Pb and 212Bi in air. System of equations used for calculating the numbers of radon and thoron progeny atoms on filter at the moment of filtering termination (N10, N20, N30, N40, N50) can be expressed through the matrix equation: 214
R ¼ G$N0
(1)
where the following notations have been used:
3 2 R1 a1 6 R2 7 6 a2 6 7 6 7 6 R¼6 6 R3 7G ¼ 6 a3 4 R4 5 4 a4 R5 a5 2
51
b1 b2 b3 b4 b5
c1 c2 c3 c4 c5
d1 d2 d3 d4 d5
3 3 2 N10 e1 7 7 6 e2 7 6 N20 7 7 6 N ¼ N e3 7 7 0 6 30 7 4 N40 5 e4 5 N50 e5
(2)
l2 ai ¼ ε expð l1 tPi Þ expð l1 tKi Þ þ εl3 ðl1 l2 Þðl1 l3 Þ l1 ðexpð l1 tPi Þ expð l1 tKi ÞÞ þ ðl2 l1 Þðl2 l3 Þ l1 l2 ðexpð l2 tPi Þ expð l2 tKi ÞÞ þ l3 ðl1 l3 Þðl2 l3 Þ ðexpð l3 tPi Þ expð l3 tKi ÞÞ
(6)
(7)
where ε presents the efficiency of alpha counting, l1, l2, l3, l4 and l5 are decay constants of 218Po, 214Pb, 214Bi, 212Pb and 212Bi, respectively and tPi, tKi are initial and final moments of i-th measuring interval (i ¼ 1,2, …,5). Multiplying the matrix Equation (1) by inverse matrix of the matrix G, the following expression was obtained:
(8)
This equation can be used for obtaining the numbers of radon and thoron progeny on the filter at the moment of filtering termination. These numbers are related to the numbers of radon and thoron progeny in air through the following equations:
n1 ¼ N10
n2 ¼
l1 hn½1 expð l1 TÞ
(9)
1 N20 l2 l2 n1 1 expð l1 TÞ 1 expð l2 TÞ hn l2 l1 l1 þ expð l2 TÞ l2 l1 (10)
n3 ¼
(3) bi ¼ ε
ci ¼ εðexpð l3 tPi Þ expð l3 tKi ÞÞ
ei ¼ ε½expð l5 tPi Þ expð l5 tKi Þ
N0 ¼ G1 $R
The elements R1, R2, R3, R4 and R5 present the numbers of alpha counts detected during five different time intervals and N10, N20, N30, N40, N50 are the numbers of 218Po, 214Pb, 214Bi, 212Pb and 212Bi atoms on filter, respectively. The elements of the system matrix G are determined as:
l3 l2 expð l3 tPi Þ expð l3 tKi Þ l3 l2 l3 expð l2 tPi Þ expð l2 tKi Þ l2
l4 l5 expð l4 tPi Þ expð l4 tKi Þ l4 l5 l4 expð l5 tPi Þ expð l5 tKi Þ l5
di ¼ ε
1 N30 l3 l2 l3 n1 1 1 expð l3 TÞ hn ðl2 l1 Þðl3 l1 Þ l1 l3 expð l2 TÞ expð l1 TÞ þ ðl2 l1 Þðl3 l2 Þ l1 l2 l3 expð l3 TÞ n2 1 ðl3 l1 Þðl3 l2 Þ l3 l2 l2 expð l2 TÞ þ expð l3 TÞ l3 l2 (11)
(4) (5)
n4 ¼ N40
l4 hn½1 expð l4 TÞ
(12)
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1 N50 l5 l5 n4 1 n5 ¼ expð l4 TÞ hn l5 l4 1 expð l5 TÞ l4 þ expð l5 TÞ l5 l4
Ai ¼
where T presents the filtering duration and n1, n2, n3, n4 and n5 are the numbers of 218Po, 214Pb, 214Bi, 212Pb and 212Bi atoms per unit volume of air. Filter efficiency is denoted by h and n is the filtering rate (filtering efficiency could be taken h ¼ 1 for glass fiber filters). Accordingly, the activity concentrations of 218Po, 214Pb, 214Bi, 212 Pb and 212Bi in air (denoted as A1, A2, A3, A4 and A5, respectively) can be expressed directly through the alpha counts detected during five time intervals:
3 2 z11 A1 6 A2 7 6 z21 6 7 6 6 A3 7 ¼ 6 z31 6 7 6 4 A4 5 4 z41 A5 z51
z12 z22 z32 z42 z52
z13 z23 z33 z43 z53
z14 z24 z34 z44 z54
3 2 3 R1 z15 6 R2 7 z25 7 7 6 7 6 7 z35 7 7$6 R3 7 z45 5 4 R4 5 z55 R5
zij $Rj
ði ¼ 1; 2; …; 5Þ
(15)
j¼1
(13)
2
5 X
(14)
where the elements zij of the system matrix Z can be obtained using Equations (3)e(7) and (9)e(13). This can also be rewritten in a shorter way as:
According to error propagation formula (Knoll, 1999), standard deviations of radon and thoron progeny concentrations in air (Equation (15)) can be calculated as:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 5 uX sðAi Þ ¼ t z2ij $Rj
ði ¼ 1; 2; …; 5Þ
(16)
j¼1
pffiffiffi The previous equation was obtained by adopting R as standard deviation of detecting R counts during some measuring time interval. It was also assumed that there was no other source of error in the system i.e. all the other parameters were treated as constants. Two programs written in Fortran 90 were used in order to analyze the previous equation. One of them calculated numbers of counts that would be detected during the specific time intervals, using the presumed values of radon and thoron progeny concentrations in air. These calculations were based on Equation (1), Equations (3)e(7) and Equations (9)e(13). The second program calculated the elements of the matrix Z and it used the previously obtained count numbers in order to determine relative standard deviations of radon and thoron progeny concentrations. It also
Fig. 1. Relative standard deviations of radon and thoron progeny concentrations as functions of radon concentration in air. Thoron concentration was set to 20 Bq/m3 (a), 100 Bq/m3 (b), 200 Bq/m3 (c) and 400 Bq/m3 (d).
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53
examines the behavior of the determinant of the matrix Z and its effects on the system stability.
3. Results and discussion 3.1. Variation of radon and thoron progeny concentrations in air Fig. 1 presents relative standard deviations, si, defined in Equation (17) below, as functions of radon and thoron concentrations in air.
si ¼
sðAi Þ Ai
ði ¼ 1; 2; …; 5Þ
(17)
The progeny concentrations were calculated from radon and thoron concentrations in air (ARn, ATn) using the relations:
Ai ¼ Fi $ARn
ði ¼ 1; 2; 3Þ
(18)
Aj ¼ Fj $ATn
j ¼ 4; 5
(19)
The following equilibrium factors, F1, F2, F3, F4 and F5 were applied for 218Po, 214Pb, 214Bi, 212Pb and 212Bi, respectively: F1 (218Po) ¼ 0.7249; F2 (214Pb) ¼ 0.3722; F3 (214Bi) ¼ 0.2719; F4 (212Pb) ¼ 0.0576; F5 (212Bi) ¼ 0.0437.
Fig. 2. Radon (a) and thoron (b) progeny concentrations with error bars corresponding to standard deviations. Thoron concentration of 200 Bq/m3 was assumed.
Fig. 3. Relative standard deviations of radon and thoron progeny concentrations as functions of thoron concentration in air. Radon concentration was set to 200 Bq/m.3.
Fig. 4. Radon (a) and thoron (b) progeny concentrations with error bars corresponding to standard deviations. Radon concentration of 200 Bq/m3 was assumed.
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J.M. Stajic, D. Nikezic / Journal of Environmental Radioactivity 140 (2015) 50e58
Radon concentration in air was varied between 10 and 500 Bq/m3 and thoron concentration was set to 20 Bq/m3 (Fig. 1a), 100 Bq/m3 (Fig. 1b), 200 Bq/m3 (Fig. 1c) and 400 Bq/m3 (Fig. 1d). As suggested by some authors (Chunxiang and Daling, 1983; Islam and Haque, 1994), the following counting intervals were used:
Dt1 ¼ 60s 240s; Dt2 ¼ 300s 1200s; Dt3 ¼ 1260s 2400s; Dt4 ¼ 9000s 15000s Dt5 ¼ 21600s 33600s: Realistic values of experimental parameters were assumed:
h ¼ 1 (filter efficiency), n ¼ 0.2 L/s (filtering rate) and ε ¼ 0.1 (the
Fig. 5. Relative standard deviation of radon and thoron progeny concentrations as functions of filtering duration.
These values were obtained using the best estimation of Jacobi room model parameters (Yu et al., 2005): lv ¼ 0.55 h1 (ventilation rate); la ¼ 50 h1 (attachment rate); ld u ¼ 20 h1 (deposition rate of unattached progeny); ld a ¼ 0.2 h1 (deposition rate of attached progeny).
Fig. 6. Standard deviations of radon (a) and thoron (b) progeny concentrations as functions of filtering duration. Radon and thoron concentrations were ARn ¼ ATn ¼ 200 Bq/m.3.
efficiency of alpha counting). Increasing radon concentration in air leads to decrease of relative standard deviations of all three radon progeny concentrations. Accordingly, in the presence of 20 Bq/m3 of thoron in air, radon concentration has to be higher than 45 Bq/m3 in order to keep relative standard deviation of all three radon daughters below 100%. Increasing thoron concentration in air leads to slight increase in standard deviations of radon progeny. Relative standard deviation of 212Bi increased with increasing radon concentration while it showed the opposite behavior when the increase of thoron concentration in air was considered. However, relative standard deviation of 212Pb was quite low in all cases. Fig. 2 presents the actual values of radon and thoron progeny concentrations obtained from Equations (18) and (19) with error bars corresponding to the standard deviations. Thoron concentration of 200 Bq/m3 was chosen and radon concentration was varied between 10 and 500 Bq/m3. The effect of thoron concentration growth on relative standard deviations of radon and thoron progeny is more obviously presented in Fig. 3. Radon concentration in air was assumed to be 200 Bq/m3 and thoron concentration was varied from 10 to 500 Bq/ m3. Obviously, in the presence of 200 Bq/m3 of radon in air, relative standard deviation of 212Bi is less than 100% for ATn > 100 Bq=m3 and it gets below 50% for ATn > 300 Bq=m3. Radon and thoron progeny concentrations with the corresponding error bars were presented in Fig. 4. Results were obtained for radon concentration of 200 Bq/m3.
Fig. 7. The growth of radon and thoron progeny concentrations on filter during the filtering process.
J.M. Stajic, D. Nikezic / Journal of Environmental Radioactivity 140 (2015) 50e58
55
Table 1 Maximal and minimal relative standard deviations and the corresponding filtering durations.
218
Po Pb Bi 212 Pb 212 Bi 214 214
Activity concentration in air [Bq/m3]
Maximal relative standard deviation smax [%]
Filtering duration corresponding to smax [min]
Minimal relative standard deviation smin [%]
Filtering duration corresponding to smin [min]
144.9 74.4 54.4 11.5 8.7
75.77 64.68 61.54 10.57 265.27
60 1 1 1 1
39.15 11.77 21.38 1.38 44.23
7 36 22 60 60
3.2. Variation of filtering duration 3.2.1. Relative standard deviation The previous results were obtained by setting the filtering duration to 20 min, as suggested by some authors. Fig. 5 presents relative standard deviations of radon and thoron progeny concentrations as functions of filtering duration varied from 1 to 60 min. Radon and thoron concentrations in air were both assumed to be 200 Bq/m3. The effect of filtering duration was also illustrated in terms of radon and thoron progeny concentrations and their standard deviations (presented by error bars) in Fig. 6. Increasing filtering duration causes the decrease of relative standard deviations of both thoron progeny concentrations (Figs. 5 and 6b). However, relative standard deviation of 218Po decreases during the first several minutes and after that it has an obvious tendency of increase with increasing filtering duration (Figs. 5 and 6a). This behavior can be related to the growth of radon and thoron progeny activities on filter during the filtering process (presented in Fig. 7). As previously demonstrated, relative standard deviations of radon and thoron progeny concentrations depend on their concentrations in air i.e. on their activities “collected” on filter during the filtration. Due to the simultaneous determination, relative standard deviation of each individual progeny concentration depends on the concentrations of all five progeny. The growth of one single progeny concentration leads to decrease of its relative standard deviation, but it increases standard deviations of the others. Accordingly, the behavior of relative standard deviation of 218 Po is related to saturation of its activity on filter, achieved after only several minutes of filtering (Fig. 7).
Standard deviations of 214Pb and 214Bi also show slight increase after reaching the minimum values, although it is less pronounced and not that obvious. Table 1 presents minimal and maximal relative standard deviations of radon and thoron progeny and their corresponding filtering durations obtained for the current conditions. In order to determine the most appropriate filtering duration, a threshold value of relative standard deviation was chosen and it was successively reduced until the minimal value was found, satisfying the condition that relative standard deviation of each progeny concentration was less or equal to that value. For the current measuring conditions and radon and thoron concentrations of 200 Bq/m3, the minimal threshold value was found to be 56.46% and the corresponding filtering duration was 28 min. The same optimal filtering duration was also obtained by minimizing the total sum of all five relative standard deviations. This filtering duration shows up to be the most appropriate from this point of view and it should provide the most accurate results of simultaneous measuring of all five progeny concentrations in a single experiment. The previous consideration was based on the assumption that radon and thoron concentrations in air were both 200 Bq/m3. Simultaneous variations of radon and thoron concentrations lead to changes of relative standard deviations, but the filtering duration corresponding to minimal sum of all five relative standard deviations remains the same for as long as the ratio between these two concentrations does not change (ARn =ATn ¼ 1). However, if the ratio between radon and thoron concentrations is disrupted, the filtering duration corresponding to minimal value of total sum of relative standard deviations will change. Fig. 8 illustrates this, presenting the dependence of the sum of all five relative standard deviations on filtering duration for different thoron activity concentrations in air, assuming ARn ¼ 200 Bq/m3. Obviously, decreasing thoron concentration (in comparison to radon concentration) increases the optimal filtering duration. In contrast, decreasing the ratio ARn =ATn moves the optimal filtering duration to lower values due to the domination of relative standard deviations of radon progeny. The minimum values of the functions presented in Fig. 8 and the corresponding filtering durations were also given in Table 2.
Table 2 Minimal values of total sum of relative standard deviations and the corresponding optimal filtering durations for different values of thoron activity concentration in air. Radon concentration was assumed to be 200 Bq/m3.
Fig. 8. Total sum of all five relative standard deviations vs. filtering duration (for different ratios of radon and thoron concentrations in air).
ARn/ATn
Minimal value of total sum of relative standard deviations [%]
The corresponding filtering duration [min]
4 2 1 0.5 0.25
223.81 176.69 148.6 135.07 133.98
60 38 28 23 20
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J.M. Stajic, D. Nikezic / Journal of Environmental Radioactivity 140 (2015) 50e58 Table 4 Time intervals that provided the smallest values of total sum of relative standard deviations. Filtering time was set to 28 min. Radon and thoron progeny concentrations were obtained assuming 200 Bq/m3 of radon and 200 Bq/m3 of thoron in air.
Fig. 9. System determinant vs. filtering duration.
3.2.2. System determinant In the course of investigating the reliability of the method, the system presented in Equation (14) was also analyzed. The determinant of the matrix Z can be used for assessing the sensitivity of the obtained progeny concentrations to the changes of counts detected during five time intervals. A lower determinant provides higher stability and better accuracy of the results, as will be demonstrated later. Fig. 9 presents the effect of increasing filtering duration on the determinant of the matrix Z (presented in Equation (14)), assuming the previously mentioned measuring intervals andARn ¼ ATn ¼ 200 Bq=m3 . Increasing filtering duration from 1 to 60 min, system determinant decreases from 10 to 106. Considering the fact that higher system determinant provides higher sensitivity of the results, long filtering is more convenient from this point of view. The previously obtained filtering duration of 28 min has the corresponding value of 8.3$106 of the system determinant. System determinant does not depend on radon and thoron activity concentrations in air but only on the choice of counting intervals and filtering parameters e filter efficiency and filtering rate (decreasing filter efficiency and filtering rate both lead to increase in system determinant)
1. Interval Starteend [min]
2. Interval Starteend [min]
3. Interval Starteend [min]
4. Interval Starteend [min]
5. Interval Starteend [min]
Total sum System of relative determinant standard [106] deviations [%]
1e3 1e5 1e5 1e5
4e34 6e34 6e36 6e36
35e65 35e65 37e67 37e67
66e306 66e306 68e308 68e308
307e547 307e547 309e539 309e549
138.386 138.013 137.930 137.383
1.443 0.986 1.012 0.97
first three measuring intervals were varied between 1 and 30 min, with a step of 2 min, while the durations of the last two intervals were increased from 10 min to 4 h, with a step of 10 min. The time delay between each two successive measurements was always 1 min. Totally, 1.944.000 different combinations of measuring intervals were tested. Standard deviations of radon and thoron progeny concentrations were calculated for each combination. Table 4 shows the combinations that provided the lowest values of the total sum of relative standard deviations of all five progeny concentrations. The activity concentration of 200 Bq/m3 was assumed for both - radon and thoron in air. The lowest value of the total sum of relative standard deviations and the corresponding value of the system determinant were written in bold. The determinant of the matrix Z was also calculated for each of these 1.944.000 combinations and wide range of values was obtained. Fig. 10 shows these values for the combinations made according to Table 3. The first four intervals were set to the shortest values and the fifth interval was varied over the whole range of values. After that, the fourth interval was increased step by step and the fifth interval was varied all over again. Further, the third interval was varied and so on. This is the reason of the periodic behavior of the function presented in Fig. 10. The highest peaks correspond to the variation of the first measuring interval, assuming the minimal durations of the second, third, fourth and fifth measurements. Two combinations of counting intervals corresponding to the highest and the lowest value of the system determinant are presented in Table 5. Obviously, the highest value of the system determinant corresponds to the shortest measuring intervals.
3.3. Variation of measuring time intervals Taking 28 min as optimal filtering duration, measuring time intervals were changed as presented in Table 3. Measuring of alpha activity from the filter started 1 min after filtering termination. The
Table 3 Variation of alpha counting intervals. First interval Second interval Third interval Forth interval Fifth interval
Start1 Duration1 Start2 Duration2 Start3 Duration3 Start4 Duration4 Start5 Duration5
1 min 2 mine30 min Start1 þ Duration1 2 mine30 min Start2 þ Duration2 2 mine30 min Start3 þ Duration3 10 mine240 min Start4 þ Duration4 10 mine240 min
þ 1 min þ 1 min þ 1 min þ 1 min
Fig. 10. The values of the determinant obtained for different combinations of counting intervals.
J.M. Stajic, D. Nikezic / Journal of Environmental Radioactivity 140 (2015) 50e58
57
Table 5 The effect of system determinant on system stability. Sys. det.
Measuring time intervals [min]
3.574 103
1e3
336.23
4e6
307.30
7e9
291.18
8.563 10
7
Exact number of counts
10e20
1355.16
21e31
1225.92
1e9
1243.39
10e40
3705.85
41e71
2564.46
72e312
6859.41
313e553
3794.348
Changing number of counts
218
Po [Bq/m3]
214
þ1 1 þ1 1 þ1 1 þ1 1 þ1 1 þ1 1 þ1 1 þ1 1 þ1 1 þ1 1
173.2 117.2 58.3 232. 221.8 68.5 141. 149.3 145.8 144.6 146.2 143.7 144.0 145.9 146.1 145.1 144.9 145.2 145.1 144.9
452.9 303.6 1401.2 1550.3 1660.6 1511.4 46.3 195.5 97.8 51.2 74.3 74.5 74.5 74.4 74.5 74.5 74.4 74.5 74.5 74.4
The assumed radon and thoron concentrations of 200 Bq/m3 correspond to the following radon and thoron progeny concentrations in air:
. AA 218 Po ¼ 144:9 Bq m3 ;
. AB 214 Pb ¼ 74:4 Bq m3 ;
. AC 214 Bi ¼ 54:4 Bq m3 ;
. A4 212 Pb ¼ 11:5 Bq m3 ;
. A5 212 Bi ¼ 8:7 Bq m3 Two programs written in Fortran 90 were used in order to investigate the behavior of the system of equations presented in matrix form in Equation (14). One of them used the previous progeny concentrations and the specific measuring time intervals (given in Table 5) in order to calculate the numbers of alpha counts detected during these time intervals. The exact count numbers were also presented in Table 5 and they were given as decimal numbers, just as they were obtained in the output of the program. The other program had the opposite purpose - it used these count numbers and the system of equations corresponding to the matrix Equation (14) in order to calculate radon and thoron progeny concentrations in air. The solution was in a good agreement with the previously assumed radon and thoron progeny concentrations and it confirmed that the programs worked correctly and the method was functional. However, changing the exact numbers of counts (in the input of the same program) for one single count gives the progeny concentrations presented in Table 5. Obviously, the system of equations with the lowest system determinant is relatively stable to these small changes and it gives the solutions which are similar to the correct values. On the contrary, the system of equations with the largest determinant of the matrix Z is extremely sensitive to the changes of count numbers in any time interval and it provides the solutions which are totally different from the previous values. It actually appears that variations of count numbers even in the second decimal place causes significant changes of the obtained progeny concentrations. Since the system determinant does not depend on radon and thoron concentrations in air, the same conclusion could be made for any other combination of progeny concentrations as well.
Pb [Bq/m3]
214
Bi [Bq/m3]
642.6 533.9 2255.4 2363.9 2561.2 2452.7 141.7 250.2 93.1 15.3 54.4 54.3 54.5 54.3 54.2 54.4 54.4 54.4 54.4 54.4
212
Pb [Bq/m3]
231.1 207.6 854.6 878.1 959.9 936.4 64.2 87.7 27.5 4.1 11.5 11.5 11.5 11.5 11.5 11.5 11.5 11.5 11.5 11.5
212
Bi [Bq/m3]
596.6 613.6 2379.5 2362.3 2557.7 2574.9 208.2 190.9 30.8 48.1 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.8
4. Conclusion The method for determining radon and thoron progeny concentrations in air based on air filtration is rather unreliable and it can ensure obtaining satisfactory results only in the case of high radon and thoron concentrations in air. Increasing an individual progeny concentration leads to decrease of its relative standard deviation, but on the other hand, it causes increase in relative standard deviations of the others. Relative standard deviation of 218Po is generally higher than that of 214Pb and 214Bi (assuming equilibrium concentrations). The accuracy of measuring concentration of 212Bi is highly dependent on radon and thoron concentrations in air. However, the method can be considered rather convenient for measuring concentration of 212Pb since its relative standard deviation is quite low in all cases. Optimization of the method also depends on radon and thoron progeny concentrations in air, so it is quite difficult to determine the best measuring parameters without previous prediction of the concentration levels. Generally, filtering intervals of 20e30 min are rather convenient for determining radon progeny concentrations, while the accuracy of measuring thoron progeny concentrations increases with increasing filtering duration. Short measuring intervals are not recommended due to the counting statistics, but they are also not convenient from the point of view of the stability of the results. The most perfect optimization would be the one involving measurement of each individual progeny concentration in a separate experiment. Acknowledgment The present work was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, under the Project No. 171021. References Chunxiang, Z., Daling, L., 1983. Measurement of mixed radon and thoron daughter concentrations in air. Nucl. Instr. Meth. 215, 481e488. EL-Hussein, A., Mohamemed, A., Abd EL-Hady, M., Ahmed, A.A., Ali, A.E., Barakat, A., 2001. Diurnal and seasonal variation of short-lived radon progeny concentration and atmospheric temporal variations of 210Pb and 7Be in Egypt. Atmos. Environ. 35, 4305e4313.
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€ggeler, H.W., Jost, D.T., Baltensperger, U., Schwikowski, M., 1995. Radon and Ga thoron decay product and 210Pb measurements at Jungfraujoch, Switzerland. Atmos. Environ. 29 (5), 607e616. Iimoto, T., Kosako, T., Sugiura, N., 2001. Measurements of summer radon and its progeny concentrations along with environmental gamma dose rates in Taiwan. J. Environ. Radioact. 57, 57e66. Islam, G.S., Haque, A.K.F., 1994. Measurement of mixed radon and thoron daughter concentrations using alpha and beta activities filtered from air. Radiat. Meas. 23 (4), 737e742. Knoll, G.F., 1999. Radiation Detection and Measurement, third ed. John Wiley & Sons, Chapter 3, New York. }, Z., 2006. Measuring radon progeny and thoron progeny in air by Papp, Z., Dezso absolute beta counting subsequent to grab sampling. Radiat. Meas. 41 (5), 617e626.
Pogorski, S.R., Phillips, C.R., 1985. An optimised four-count total-alpha measurement scheme for the detection of radon progeny and lead-212 in air. Radiat. Prot. Dosim. 12 (1), 45e53. Ruzer, L., Sextro, R., 1997. Measurement of radon decay products in air by alpha and beta spectrometry. Radiat. Prot. Dosim. 72 (1), 43e48. Singh, K., Singh, S., Mehra, R., Singh, M., Sahota, H.S., Papp, Z., 2006. Measurement of radon and thoron progeny outdoors in Malout, India, using grab aerosol sampling and beta counting. Radiat. Meas. 41, 108e111. Stajic, J.M., Nikezic, D., 2014. Analysis of radon and thoron progeny measurements based on air filtration. Radiat. Prot. Dosim. http://dx.doi.org/10.1093/rpd/ncu183 (in press). http://rpd.oxfordjournals.org/content/early/2014/06/10/rpd.ncu183.full.pdf. Yu, K.N., Nikezic, D., Ng, F.M.F., Leung, J.K.C., 2005. Long-term measurements of radon progeny concentrations with solid state nuclear track detectors. Radiat. Meas. 40, 560e568.
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The accuracy of radon and thoron progeny concentrations measured through air filtration J.M. Stajic, D. Nikezic* Faculty of Science, University of Kragujevac, R. Domanovica 12, Kragujevac 34000, Serbia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 September 2014 Received in revised form 28 October 2014 Accepted 1 November 2014 Available online
The accuracy and the optimization of determining radon and thoron progeny concentrations in air using air filtration followed by alpha activity measurements were investigated in details. The effects of radon and thoron concentrations, filtering duration and the choice of measuring intervals on relative standard deviations were analyzed. Obtaining satisfactory results by this method should be expected only in the case of high radon and thoron progeny concentrations in air. The optimization process also showed up to be dependent on the progeny concentration. Determinant of the system matrix and its effect on the sensitivity of the results were investigated. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Air filtration Alpha activity Relative standard deviation System determinant
1. Introduction Inhaling high concentrations of radon, thoron and their decay products can cause harmful effects to human lung tissues. Exposure to radon and its progeny has been ranked as the second leading cause of lung cancer after tobacco smoking. This fact justifies the great efforts of scientists all around the world to develop and improve the methods for measuring concentrations of these naturally occurring radionuclides in air. The method based on air filtration, followed by measurements of alpha, beta or gamma activity from filter has been investigated by many authors (EL€ggeler et al., 1995; Iimoto et al., 2001; Hussein et al., 2001; Ga } , 2006; Pogorski and Phillips, 1985; Ruzer and Papp and Dezso Sextro, 1997; Singh et al., 2006). Detecting numbers of counts in several different time intervals, during or after filtering process, should allow solving the system of equations for radon and thoron
progeny concentrations in air. However, those who try to bring this good theoretical concept into practical realization can face the possibility of obtaining incorrect or physically unacceptable results. In our previous work (Stajic and Nikezic, 2014), we investigated the reliability of this method, illustrating the effect of counting statistics on the accuracy of the final results with some concrete and realistic examples. The purpose of this paper was to present a detailed and more general consideration of the accuracy and the optimization of the measuring method involving air filtration followed by alpha counting. The dependences of standard deviations and the system determinant on radon and thoron concentrations, filtering duration and the choice of measuring intervals were investigated.
2. Material and methods
* Corresponding author. Tel.: þ381 34 336 223; fax: þ381 34 335 040. E-mail address: [email protected] (D. Nikezic). http://dx.doi.org/10.1016/j.jenvrad.2014.11.002 0265-931X/© 2014 Elsevier Ltd. All rights reserved.
Radon (222Rn) and thoron (220Rn) decay schemes are given below, with neglecting the decay branches with small probabilities:
J.M. Stajic, D. Nikezic / Journal of Environmental Radioactivity 140 (2015) 50e58
222
a
Rn/
3:824 d
b b a a 214 214 218 214 210 84 Po / 82 Pb / 83 Bi / 84 Po/ 82 Pb 164 ms 26:8min 19:9min 3:05min
Some of the short-lived radon and thoron decay products (218Po, Po, 216Po, 212Bi and 212Po) are alpha emitters. Due to the very short half life, 214Po might be considered in equilibrium with its precursor, 214Bi. Furthermore, 212Pb is assumed to be formed directly from 220Rn due to the short half-life of 216Po. Measuring total numbers of alpha counts during five different time intervals, subsequent to filtering process, would allow determination of individual concentrations of 218Po, 214Pb, 214Bi, 212Pb and 212Bi in air. System of equations used for calculating the numbers of radon and thoron progeny atoms on filter at the moment of filtering termination (N10, N20, N30, N40, N50) can be expressed through the matrix equation: 214
R ¼ G$N0
(1)
where the following notations have been used:
3 2 R1 a1 6 R2 7 6 a2 6 7 6 7 6 R¼6 6 R3 7G ¼ 6 a3 4 R4 5 4 a4 R5 a5 2
51
b1 b2 b3 b4 b5
c1 c2 c3 c4 c5
d1 d2 d3 d4 d5
3 3 2 N10 e1 7 7 6 e2 7 6 N20 7 7 6 N ¼ N e3 7 7 0 6 30 7 4 N40 5 e4 5 N50 e5
(2)
l2 ai ¼ ε expð l1 tPi Þ expð l1 tKi Þ þ εl3 ðl1 l2 Þðl1 l3 Þ l1 ðexpð l1 tPi Þ expð l1 tKi ÞÞ þ ðl2 l1 Þðl2 l3 Þ l1 l2 ðexpð l2 tPi Þ expð l2 tKi ÞÞ þ l3 ðl1 l3 Þðl2 l3 Þ ðexpð l3 tPi Þ expð l3 tKi ÞÞ
(6)
(7)
where ε presents the efficiency of alpha counting, l1, l2, l3, l4 and l5 are decay constants of 218Po, 214Pb, 214Bi, 212Pb and 212Bi, respectively and tPi, tKi are initial and final moments of i-th measuring interval (i ¼ 1,2, …,5). Multiplying the matrix Equation (1) by inverse matrix of the matrix G, the following expression was obtained:
(8)
This equation can be used for obtaining the numbers of radon and thoron progeny on the filter at the moment of filtering termination. These numbers are related to the numbers of radon and thoron progeny in air through the following equations:
n1 ¼ N10
n2 ¼
l1 hn½1 expð l1 TÞ
(9)
1 N20 l2 l2 n1 1 expð l1 TÞ 1 expð l2 TÞ hn l2 l1 l1 þ expð l2 TÞ l2 l1 (10)
n3 ¼
(3) bi ¼ ε
ci ¼ εðexpð l3 tPi Þ expð l3 tKi ÞÞ
ei ¼ ε½expð l5 tPi Þ expð l5 tKi Þ
N0 ¼ G1 $R
The elements R1, R2, R3, R4 and R5 present the numbers of alpha counts detected during five different time intervals and N10, N20, N30, N40, N50 are the numbers of 218Po, 214Pb, 214Bi, 212Pb and 212Bi atoms on filter, respectively. The elements of the system matrix G are determined as:
l3 l2 expð l3 tPi Þ expð l3 tKi Þ l3 l2 l3 expð l2 tPi Þ expð l2 tKi Þ l2
l4 l5 expð l4 tPi Þ expð l4 tKi Þ l4 l5 l4 expð l5 tPi Þ expð l5 tKi Þ l5
di ¼ ε
1 N30 l3 l2 l3 n1 1 1 expð l3 TÞ hn ðl2 l1 Þðl3 l1 Þ l1 l3 expð l2 TÞ expð l1 TÞ þ ðl2 l1 Þðl3 l2 Þ l1 l2 l3 expð l3 TÞ n2 1 ðl3 l1 Þðl3 l2 Þ l3 l2 l2 expð l2 TÞ þ expð l3 TÞ l3 l2 (11)
(4) (5)
n4 ¼ N40
l4 hn½1 expð l4 TÞ
(12)
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J.M. Stajic, D. Nikezic / Journal of Environmental Radioactivity 140 (2015) 50e58
1 N50 l5 l5 n4 1 n5 ¼ expð l4 TÞ hn l5 l4 1 expð l5 TÞ l4 þ expð l5 TÞ l5 l4
Ai ¼
where T presents the filtering duration and n1, n2, n3, n4 and n5 are the numbers of 218Po, 214Pb, 214Bi, 212Pb and 212Bi atoms per unit volume of air. Filter efficiency is denoted by h and n is the filtering rate (filtering efficiency could be taken h ¼ 1 for glass fiber filters). Accordingly, the activity concentrations of 218Po, 214Pb, 214Bi, 212 Pb and 212Bi in air (denoted as A1, A2, A3, A4 and A5, respectively) can be expressed directly through the alpha counts detected during five time intervals:
3 2 z11 A1 6 A2 7 6 z21 6 7 6 6 A3 7 ¼ 6 z31 6 7 6 4 A4 5 4 z41 A5 z51
z12 z22 z32 z42 z52
z13 z23 z33 z43 z53
z14 z24 z34 z44 z54
3 2 3 R1 z15 6 R2 7 z25 7 7 6 7 6 7 z35 7 7$6 R3 7 z45 5 4 R4 5 z55 R5
zij $Rj
ði ¼ 1; 2; …; 5Þ
(15)
j¼1
(13)
2
5 X
(14)
where the elements zij of the system matrix Z can be obtained using Equations (3)e(7) and (9)e(13). This can also be rewritten in a shorter way as:
According to error propagation formula (Knoll, 1999), standard deviations of radon and thoron progeny concentrations in air (Equation (15)) can be calculated as:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 5 uX sðAi Þ ¼ t z2ij $Rj
ði ¼ 1; 2; …; 5Þ
(16)
j¼1
pffiffiffi The previous equation was obtained by adopting R as standard deviation of detecting R counts during some measuring time interval. It was also assumed that there was no other source of error in the system i.e. all the other parameters were treated as constants. Two programs written in Fortran 90 were used in order to analyze the previous equation. One of them calculated numbers of counts that would be detected during the specific time intervals, using the presumed values of radon and thoron progeny concentrations in air. These calculations were based on Equation (1), Equations (3)e(7) and Equations (9)e(13). The second program calculated the elements of the matrix Z and it used the previously obtained count numbers in order to determine relative standard deviations of radon and thoron progeny concentrations. It also
Fig. 1. Relative standard deviations of radon and thoron progeny concentrations as functions of radon concentration in air. Thoron concentration was set to 20 Bq/m3 (a), 100 Bq/m3 (b), 200 Bq/m3 (c) and 400 Bq/m3 (d).
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53
examines the behavior of the determinant of the matrix Z and its effects on the system stability.
3. Results and discussion 3.1. Variation of radon and thoron progeny concentrations in air Fig. 1 presents relative standard deviations, si, defined in Equation (17) below, as functions of radon and thoron concentrations in air.
si ¼
sðAi Þ Ai
ði ¼ 1; 2; …; 5Þ
(17)
The progeny concentrations were calculated from radon and thoron concentrations in air (ARn, ATn) using the relations:
Ai ¼ Fi $ARn
ði ¼ 1; 2; 3Þ
(18)
Aj ¼ Fj $ATn
j ¼ 4; 5
(19)
The following equilibrium factors, F1, F2, F3, F4 and F5 were applied for 218Po, 214Pb, 214Bi, 212Pb and 212Bi, respectively: F1 (218Po) ¼ 0.7249; F2 (214Pb) ¼ 0.3722; F3 (214Bi) ¼ 0.2719; F4 (212Pb) ¼ 0.0576; F5 (212Bi) ¼ 0.0437.
Fig. 2. Radon (a) and thoron (b) progeny concentrations with error bars corresponding to standard deviations. Thoron concentration of 200 Bq/m3 was assumed.
Fig. 3. Relative standard deviations of radon and thoron progeny concentrations as functions of thoron concentration in air. Radon concentration was set to 200 Bq/m.3.
Fig. 4. Radon (a) and thoron (b) progeny concentrations with error bars corresponding to standard deviations. Radon concentration of 200 Bq/m3 was assumed.
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J.M. Stajic, D. Nikezic / Journal of Environmental Radioactivity 140 (2015) 50e58
Radon concentration in air was varied between 10 and 500 Bq/m3 and thoron concentration was set to 20 Bq/m3 (Fig. 1a), 100 Bq/m3 (Fig. 1b), 200 Bq/m3 (Fig. 1c) and 400 Bq/m3 (Fig. 1d). As suggested by some authors (Chunxiang and Daling, 1983; Islam and Haque, 1994), the following counting intervals were used:
Dt1 ¼ 60s 240s; Dt2 ¼ 300s 1200s; Dt3 ¼ 1260s 2400s; Dt4 ¼ 9000s 15000s Dt5 ¼ 21600s 33600s: Realistic values of experimental parameters were assumed:
h ¼ 1 (filter efficiency), n ¼ 0.2 L/s (filtering rate) and ε ¼ 0.1 (the
Fig. 5. Relative standard deviation of radon and thoron progeny concentrations as functions of filtering duration.
These values were obtained using the best estimation of Jacobi room model parameters (Yu et al., 2005): lv ¼ 0.55 h1 (ventilation rate); la ¼ 50 h1 (attachment rate); ld u ¼ 20 h1 (deposition rate of unattached progeny); ld a ¼ 0.2 h1 (deposition rate of attached progeny).
Fig. 6. Standard deviations of radon (a) and thoron (b) progeny concentrations as functions of filtering duration. Radon and thoron concentrations were ARn ¼ ATn ¼ 200 Bq/m.3.
efficiency of alpha counting). Increasing radon concentration in air leads to decrease of relative standard deviations of all three radon progeny concentrations. Accordingly, in the presence of 20 Bq/m3 of thoron in air, radon concentration has to be higher than 45 Bq/m3 in order to keep relative standard deviation of all three radon daughters below 100%. Increasing thoron concentration in air leads to slight increase in standard deviations of radon progeny. Relative standard deviation of 212Bi increased with increasing radon concentration while it showed the opposite behavior when the increase of thoron concentration in air was considered. However, relative standard deviation of 212Pb was quite low in all cases. Fig. 2 presents the actual values of radon and thoron progeny concentrations obtained from Equations (18) and (19) with error bars corresponding to the standard deviations. Thoron concentration of 200 Bq/m3 was chosen and radon concentration was varied between 10 and 500 Bq/m3. The effect of thoron concentration growth on relative standard deviations of radon and thoron progeny is more obviously presented in Fig. 3. Radon concentration in air was assumed to be 200 Bq/m3 and thoron concentration was varied from 10 to 500 Bq/ m3. Obviously, in the presence of 200 Bq/m3 of radon in air, relative standard deviation of 212Bi is less than 100% for ATn > 100 Bq=m3 and it gets below 50% for ATn > 300 Bq=m3. Radon and thoron progeny concentrations with the corresponding error bars were presented in Fig. 4. Results were obtained for radon concentration of 200 Bq/m3.
Fig. 7. The growth of radon and thoron progeny concentrations on filter during the filtering process.
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55
Table 1 Maximal and minimal relative standard deviations and the corresponding filtering durations.
218
Po Pb Bi 212 Pb 212 Bi 214 214
Activity concentration in air [Bq/m3]
Maximal relative standard deviation smax [%]
Filtering duration corresponding to smax [min]
Minimal relative standard deviation smin [%]
Filtering duration corresponding to smin [min]
144.9 74.4 54.4 11.5 8.7
75.77 64.68 61.54 10.57 265.27
60 1 1 1 1
39.15 11.77 21.38 1.38 44.23
7 36 22 60 60
3.2. Variation of filtering duration 3.2.1. Relative standard deviation The previous results were obtained by setting the filtering duration to 20 min, as suggested by some authors. Fig. 5 presents relative standard deviations of radon and thoron progeny concentrations as functions of filtering duration varied from 1 to 60 min. Radon and thoron concentrations in air were both assumed to be 200 Bq/m3. The effect of filtering duration was also illustrated in terms of radon and thoron progeny concentrations and their standard deviations (presented by error bars) in Fig. 6. Increasing filtering duration causes the decrease of relative standard deviations of both thoron progeny concentrations (Figs. 5 and 6b). However, relative standard deviation of 218Po decreases during the first several minutes and after that it has an obvious tendency of increase with increasing filtering duration (Figs. 5 and 6a). This behavior can be related to the growth of radon and thoron progeny activities on filter during the filtering process (presented in Fig. 7). As previously demonstrated, relative standard deviations of radon and thoron progeny concentrations depend on their concentrations in air i.e. on their activities “collected” on filter during the filtration. Due to the simultaneous determination, relative standard deviation of each individual progeny concentration depends on the concentrations of all five progeny. The growth of one single progeny concentration leads to decrease of its relative standard deviation, but it increases standard deviations of the others. Accordingly, the behavior of relative standard deviation of 218 Po is related to saturation of its activity on filter, achieved after only several minutes of filtering (Fig. 7).
Standard deviations of 214Pb and 214Bi also show slight increase after reaching the minimum values, although it is less pronounced and not that obvious. Table 1 presents minimal and maximal relative standard deviations of radon and thoron progeny and their corresponding filtering durations obtained for the current conditions. In order to determine the most appropriate filtering duration, a threshold value of relative standard deviation was chosen and it was successively reduced until the minimal value was found, satisfying the condition that relative standard deviation of each progeny concentration was less or equal to that value. For the current measuring conditions and radon and thoron concentrations of 200 Bq/m3, the minimal threshold value was found to be 56.46% and the corresponding filtering duration was 28 min. The same optimal filtering duration was also obtained by minimizing the total sum of all five relative standard deviations. This filtering duration shows up to be the most appropriate from this point of view and it should provide the most accurate results of simultaneous measuring of all five progeny concentrations in a single experiment. The previous consideration was based on the assumption that radon and thoron concentrations in air were both 200 Bq/m3. Simultaneous variations of radon and thoron concentrations lead to changes of relative standard deviations, but the filtering duration corresponding to minimal sum of all five relative standard deviations remains the same for as long as the ratio between these two concentrations does not change (ARn =ATn ¼ 1). However, if the ratio between radon and thoron concentrations is disrupted, the filtering duration corresponding to minimal value of total sum of relative standard deviations will change. Fig. 8 illustrates this, presenting the dependence of the sum of all five relative standard deviations on filtering duration for different thoron activity concentrations in air, assuming ARn ¼ 200 Bq/m3. Obviously, decreasing thoron concentration (in comparison to radon concentration) increases the optimal filtering duration. In contrast, decreasing the ratio ARn =ATn moves the optimal filtering duration to lower values due to the domination of relative standard deviations of radon progeny. The minimum values of the functions presented in Fig. 8 and the corresponding filtering durations were also given in Table 2.
Table 2 Minimal values of total sum of relative standard deviations and the corresponding optimal filtering durations for different values of thoron activity concentration in air. Radon concentration was assumed to be 200 Bq/m3.
Fig. 8. Total sum of all five relative standard deviations vs. filtering duration (for different ratios of radon and thoron concentrations in air).
ARn/ATn
Minimal value of total sum of relative standard deviations [%]
The corresponding filtering duration [min]
4 2 1 0.5 0.25
223.81 176.69 148.6 135.07 133.98
60 38 28 23 20
56
J.M. Stajic, D. Nikezic / Journal of Environmental Radioactivity 140 (2015) 50e58 Table 4 Time intervals that provided the smallest values of total sum of relative standard deviations. Filtering time was set to 28 min. Radon and thoron progeny concentrations were obtained assuming 200 Bq/m3 of radon and 200 Bq/m3 of thoron in air.
Fig. 9. System determinant vs. filtering duration.
3.2.2. System determinant In the course of investigating the reliability of the method, the system presented in Equation (14) was also analyzed. The determinant of the matrix Z can be used for assessing the sensitivity of the obtained progeny concentrations to the changes of counts detected during five time intervals. A lower determinant provides higher stability and better accuracy of the results, as will be demonstrated later. Fig. 9 presents the effect of increasing filtering duration on the determinant of the matrix Z (presented in Equation (14)), assuming the previously mentioned measuring intervals andARn ¼ ATn ¼ 200 Bq=m3 . Increasing filtering duration from 1 to 60 min, system determinant decreases from 10 to 106. Considering the fact that higher system determinant provides higher sensitivity of the results, long filtering is more convenient from this point of view. The previously obtained filtering duration of 28 min has the corresponding value of 8.3$106 of the system determinant. System determinant does not depend on radon and thoron activity concentrations in air but only on the choice of counting intervals and filtering parameters e filter efficiency and filtering rate (decreasing filter efficiency and filtering rate both lead to increase in system determinant)
1. Interval Starteend [min]
2. Interval Starteend [min]
3. Interval Starteend [min]
4. Interval Starteend [min]
5. Interval Starteend [min]
Total sum System of relative determinant standard [106] deviations [%]
1e3 1e5 1e5 1e5
4e34 6e34 6e36 6e36
35e65 35e65 37e67 37e67
66e306 66e306 68e308 68e308
307e547 307e547 309e539 309e549
138.386 138.013 137.930 137.383
1.443 0.986 1.012 0.97
first three measuring intervals were varied between 1 and 30 min, with a step of 2 min, while the durations of the last two intervals were increased from 10 min to 4 h, with a step of 10 min. The time delay between each two successive measurements was always 1 min. Totally, 1.944.000 different combinations of measuring intervals were tested. Standard deviations of radon and thoron progeny concentrations were calculated for each combination. Table 4 shows the combinations that provided the lowest values of the total sum of relative standard deviations of all five progeny concentrations. The activity concentration of 200 Bq/m3 was assumed for both - radon and thoron in air. The lowest value of the total sum of relative standard deviations and the corresponding value of the system determinant were written in bold. The determinant of the matrix Z was also calculated for each of these 1.944.000 combinations and wide range of values was obtained. Fig. 10 shows these values for the combinations made according to Table 3. The first four intervals were set to the shortest values and the fifth interval was varied over the whole range of values. After that, the fourth interval was increased step by step and the fifth interval was varied all over again. Further, the third interval was varied and so on. This is the reason of the periodic behavior of the function presented in Fig. 10. The highest peaks correspond to the variation of the first measuring interval, assuming the minimal durations of the second, third, fourth and fifth measurements. Two combinations of counting intervals corresponding to the highest and the lowest value of the system determinant are presented in Table 5. Obviously, the highest value of the system determinant corresponds to the shortest measuring intervals.
3.3. Variation of measuring time intervals Taking 28 min as optimal filtering duration, measuring time intervals were changed as presented in Table 3. Measuring of alpha activity from the filter started 1 min after filtering termination. The
Table 3 Variation of alpha counting intervals. First interval Second interval Third interval Forth interval Fifth interval
Start1 Duration1 Start2 Duration2 Start3 Duration3 Start4 Duration4 Start5 Duration5
1 min 2 mine30 min Start1 þ Duration1 2 mine30 min Start2 þ Duration2 2 mine30 min Start3 þ Duration3 10 mine240 min Start4 þ Duration4 10 mine240 min
þ 1 min þ 1 min þ 1 min þ 1 min
Fig. 10. The values of the determinant obtained for different combinations of counting intervals.
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57
Table 5 The effect of system determinant on system stability. Sys. det.
Measuring time intervals [min]
3.574 103
1e3
336.23
4e6
307.30
7e9
291.18
8.563 10
7
Exact number of counts
10e20
1355.16
21e31
1225.92
1e9
1243.39
10e40
3705.85
41e71
2564.46
72e312
6859.41
313e553
3794.348
Changing number of counts
218
Po [Bq/m3]
214
þ1 1 þ1 1 þ1 1 þ1 1 þ1 1 þ1 1 þ1 1 þ1 1 þ1 1 þ1 1
173.2 117.2 58.3 232. 221.8 68.5 141. 149.3 145.8 144.6 146.2 143.7 144.0 145.9 146.1 145.1 144.9 145.2 145.1 144.9
452.9 303.6 1401.2 1550.3 1660.6 1511.4 46.3 195.5 97.8 51.2 74.3 74.5 74.5 74.4 74.5 74.5 74.4 74.5 74.5 74.4
The assumed radon and thoron concentrations of 200 Bq/m3 correspond to the following radon and thoron progeny concentrations in air:
. AA 218 Po ¼ 144:9 Bq m3 ;
. AB 214 Pb ¼ 74:4 Bq m3 ;
. AC 214 Bi ¼ 54:4 Bq m3 ;
. A4 212 Pb ¼ 11:5 Bq m3 ;
. A5 212 Bi ¼ 8:7 Bq m3 Two programs written in Fortran 90 were used in order to investigate the behavior of the system of equations presented in matrix form in Equation (14). One of them used the previous progeny concentrations and the specific measuring time intervals (given in Table 5) in order to calculate the numbers of alpha counts detected during these time intervals. The exact count numbers were also presented in Table 5 and they were given as decimal numbers, just as they were obtained in the output of the program. The other program had the opposite purpose - it used these count numbers and the system of equations corresponding to the matrix Equation (14) in order to calculate radon and thoron progeny concentrations in air. The solution was in a good agreement with the previously assumed radon and thoron progeny concentrations and it confirmed that the programs worked correctly and the method was functional. However, changing the exact numbers of counts (in the input of the same program) for one single count gives the progeny concentrations presented in Table 5. Obviously, the system of equations with the lowest system determinant is relatively stable to these small changes and it gives the solutions which are similar to the correct values. On the contrary, the system of equations with the largest determinant of the matrix Z is extremely sensitive to the changes of count numbers in any time interval and it provides the solutions which are totally different from the previous values. It actually appears that variations of count numbers even in the second decimal place causes significant changes of the obtained progeny concentrations. Since the system determinant does not depend on radon and thoron concentrations in air, the same conclusion could be made for any other combination of progeny concentrations as well.
Pb [Bq/m3]
214
Bi [Bq/m3]
642.6 533.9 2255.4 2363.9 2561.2 2452.7 141.7 250.2 93.1 15.3 54.4 54.3 54.5 54.3 54.2 54.4 54.4 54.4 54.4 54.4
212
Pb [Bq/m3]
231.1 207.6 854.6 878.1 959.9 936.4 64.2 87.7 27.5 4.1 11.5 11.5 11.5 11.5 11.5 11.5 11.5 11.5 11.5 11.5
212
Bi [Bq/m3]
596.6 613.6 2379.5 2362.3 2557.7 2574.9 208.2 190.9 30.8 48.1 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.8
4. Conclusion The method for determining radon and thoron progeny concentrations in air based on air filtration is rather unreliable and it can ensure obtaining satisfactory results only in the case of high radon and thoron concentrations in air. Increasing an individual progeny concentration leads to decrease of its relative standard deviation, but on the other hand, it causes increase in relative standard deviations of the others. Relative standard deviation of 218Po is generally higher than that of 214Pb and 214Bi (assuming equilibrium concentrations). The accuracy of measuring concentration of 212Bi is highly dependent on radon and thoron concentrations in air. However, the method can be considered rather convenient for measuring concentration of 212Pb since its relative standard deviation is quite low in all cases. Optimization of the method also depends on radon and thoron progeny concentrations in air, so it is quite difficult to determine the best measuring parameters without previous prediction of the concentration levels. Generally, filtering intervals of 20e30 min are rather convenient for determining radon progeny concentrations, while the accuracy of measuring thoron progeny concentrations increases with increasing filtering duration. Short measuring intervals are not recommended due to the counting statistics, but they are also not convenient from the point of view of the stability of the results. The most perfect optimization would be the one involving measurement of each individual progeny concentration in a separate experiment. Acknowledgment The present work was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, under the Project No. 171021. References Chunxiang, Z., Daling, L., 1983. Measurement of mixed radon and thoron daughter concentrations in air. Nucl. Instr. Meth. 215, 481e488. EL-Hussein, A., Mohamemed, A., Abd EL-Hady, M., Ahmed, A.A., Ali, A.E., Barakat, A., 2001. Diurnal and seasonal variation of short-lived radon progeny concentration and atmospheric temporal variations of 210Pb and 7Be in Egypt. Atmos. Environ. 35, 4305e4313.
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€ggeler, H.W., Jost, D.T., Baltensperger, U., Schwikowski, M., 1995. Radon and Ga thoron decay product and 210Pb measurements at Jungfraujoch, Switzerland. Atmos. Environ. 29 (5), 607e616. Iimoto, T., Kosako, T., Sugiura, N., 2001. Measurements of summer radon and its progeny concentrations along with environmental gamma dose rates in Taiwan. J. Environ. Radioact. 57, 57e66. Islam, G.S., Haque, A.K.F., 1994. Measurement of mixed radon and thoron daughter concentrations using alpha and beta activities filtered from air. Radiat. Meas. 23 (4), 737e742. Knoll, G.F., 1999. Radiation Detection and Measurement, third ed. John Wiley & Sons, Chapter 3, New York. }, Z., 2006. Measuring radon progeny and thoron progeny in air by Papp, Z., Dezso absolute beta counting subsequent to grab sampling. Radiat. Meas. 41 (5), 617e626.
Pogorski, S.R., Phillips, C.R., 1985. An optimised four-count total-alpha measurement scheme for the detection of radon progeny and lead-212 in air. Radiat. Prot. Dosim. 12 (1), 45e53. Ruzer, L., Sextro, R., 1997. Measurement of radon decay products in air by alpha and beta spectrometry. Radiat. Prot. Dosim. 72 (1), 43e48. Singh, K., Singh, S., Mehra, R., Singh, M., Sahota, H.S., Papp, Z., 2006. Measurement of radon and thoron progeny outdoors in Malout, India, using grab aerosol sampling and beta counting. Radiat. Meas. 41, 108e111. Stajic, J.M., Nikezic, D., 2014. Analysis of radon and thoron progeny measurements based on air filtration. Radiat. Prot. Dosim. http://dx.doi.org/10.1093/rpd/ncu183 (in press). http://rpd.oxfordjournals.org/content/early/2014/06/10/rpd.ncu183.full.pdf. Yu, K.N., Nikezic, D., Ng, F.M.F., Leung, J.K.C., 2005. Long-term measurements of radon progeny concentrations with solid state nuclear track detectors. Radiat. Meas. 40, 560e568.
