- Email: [email protected]
Comprehensive software for the assessment of 222Rn and 220Rn decay products based on air sampling measurements
ARTICLE IN PRESS Applied Radiation and Isotopes 67 (2009) 867–871
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Comprehensive software for the assessment of 222Rn and products based on air sampling measurements
220
Rn decay
Octavian Sima Physics Department, University of Bucharest, Bucharest-Magurele, P.O. Box MG-11, RO-077125, Romania
a r t i c l e in fo
abstract
Keywords: Radon decay products Analysis software Uncertainty analysis Measurement optimization
A computational tool dedicated to the measurement of 222Rn and 220Rn decay products by air sampling is presented. a- or g-spectrometry measurements, gross a or b counting, as well as a combination of them are considered. Special attention is given to the evaluation of the uncertainty budget of the results. Besides typical applications in the analysis of experimental data, the software can be used for assessing the expected quality of a measurement protocol and for optimizing it, by generating and analyzing sets of realistic synthetic data. & 2009 Elsevier Ltd. All rights reserved.
1. Introduction Radon and radon decay products represent subjects of interest in various fields. Indeed, radon and its decay products are of major concern for the occupational exposure e.g. of uranium miners, but also for the irradiation of the public, delivering approximately half of the natural exposure dose. They have also applications as tracers in various meteorology and hydrology studies, and the correlation with the seismic activity is a subject of current interest. Radon reduction is a main issue in low-level or ultra-lowlevel laboratories. In this work we consider active methods for the measurement of the radon (222Rn and 220Rn) decay products. The early methods of this kind, pioneered by the work of Tsivoglou (Tsivoglou et al., 1953), were specifically developed for a particular type of measurement (gross a or a-spectrometry) and for a particular measurement protocol. Later the advantage of simultaneous a- and g-spectrometry was clearly demonstrated and the ag-spectrometry method (Paul et al., 1999) became the standard metrological procedure for radon progeny measurement in radon reference atmospheres. In spite of the long history of the active methods, it seems that a general and friendly procedure for analyzing simultaneously various types of measurements, for an arbitrary measurement protocol, is still lacking. Also, certain features of the statistical processes involved in the methods seem to have been disregarded. Active methods for the assessment of the activity concentration of 222Rn and 220Rn decay products were applied since long by our group in the University of Bucharest and in the Environmental
Fax: +40 214574521.
E-mail address: [email protected] 0969-8043/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2009.01.022
Radioactivity Laboratory, Bucharest (Sima, 1978; Dovlete, 1983; Sima et al., 1986; Sonoc, 1989; Sonoc and Sima, 1992; Baciu, 2005). The computational methods developed in these early studies were recently extended and implemented in a dedicated software called AMERATHOR (Active Methods for Radon and Thoron decay products). The adopted theoretical model, the software and its applications will be presented in this paper.
2. Theoretical model In the active methods for the measurement of 222Rn progenies (218Po, 214Pb, 214Bi) and of 220Rn progenies (216Po, 212Pb, 212Bi, 208 Tl) the radon decay products are collected from the air by retention on a filter through which the air is sampled with a controlled flow rate. The loaded filter is measured and the activity concentrations of the decay products in the air are evaluated. The time evolution of the numbers Ni of nuclei of type i (i ¼ 1, 2 and 3 for 218Po, 214Pb and 214Bi, respectively, in the 222Rn decay series) collected on the filter is obtained by solving the set of Bateman equations. For example, N1 ðtÞ ¼
N1 ðtÞ ¼
C1 Z D
l1 C1 Z D
l1
½1 expðl1 tÞ
for toT
½1 expðl1 TÞ exp½l1 ðt TÞ
(1)
for t4T
(2)
where Ci and li represent the concentration in the air and the decay constant for type i nuclide, T the air sampling time, D the flow rate and Z the filter collection efficiency (for simplicity assumed equal for all the nuclides). The equations for N2(t) and N3(t) contain similar terms describing the contribution of the nuclides of type 2 or 3 directly sampled from the air and more complex terms representing cases when the nuclides of type 2 or
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3 are produced by the decay of the parents collected on the filter; clearly N2(t) depends linearly on C2 and C1, N3(t) on C3, C2 and C1. Consider the counts mk(s1,s2) registered in the time interval (s1,s2) due to the detection (with efficiency ek) of the radiations of type k emitted by nuclide i with emission probability pk mk ðs1 ; s2 Þ ¼ k pk li
Z
s2 s1
Ni ðtÞ dt
(3)
As Ni(t) are linear combinations of the concentrations, the number of counts will also be linear combinations of concentrations. The case of 220Rn decay products is described by a similar formalism. By making a set of count measurements and fitting the data, e.g. by the least squares method, it is possible to obtain the concentrations. The uncertainties of the concentrations depend on the uncertainties of the number of counts, of the emission probabilities, of the efficiencies, of the flow rate and filter efficiency. In the conventional approach the uncertainty of the number of counts is obtained by assuming Poisson distribution. The effect of the uncertainty of the flow rate and of the other quantities is obtained by using the error propagation formula. The above approach for obtaining the uncertainty is not completely satisfactory. Generally speaking, the Eqs. (1)–(3) are valid for the mean values of the corresponding quantities, i.e. for the first moment of the distributions of all the random variables included in the equations, e.g. the number of collected nuclei or the number of counts, the flow rate. The error propagation formula applied to these equations gives indeed the contribution of the uncertainty of the mean value of the flow rate to the uncertainty of the mean value of the number of counts, but it does not describe the variance of the actual number of counts due to flow rate fluctuations during sampling. In fact, the equations for the first moment of the distributions by themselves do not include any information on the second moment of the distributions, e.g. the variance of the number of counts; this is obtained conventionally by assuming Poisson distribution. However, deviations from Poisson distribution may occur, because the distribution of the number of counts depends on the statistical properties of the process of generating the nuclei by collection on the filter, dependent in turn on the statistical properties of the flow rate, and by the serial decay. Indeed, if the flow rate has big fluctuations it will induce big fluctuations in the number of collected nuclei, which should be reflected in an increased variance of the number of counts. Also, the number of nuclei collected around time t and the number collected around time t0 can be correlated if the flow rate at t is correlated with the flow rate at t0 . We developed and implemented in AMERATHOR a more realistic approach based on a stochastic model describing the collection on the filter, the disintegration and the measurement processes. The model (Appendix A) takes into account the flow rate fluctuations during sampling. Of course, the mean values of the quantities are given by the same equations as in the traditional approach. The main differences with respect to the traditional approach are (a) the variance of the number of counts is higher than the Poisson term, including additional contributions, depending on the statistical properties of the flow rate and of the distribution of the particles in air; (b) the number of counts produced by the radiations emitted by different nuclides are correlated to a certain degree, due to the fact that flow rate fluctuations affect simultaneously the collection of all types of nuclides and due to the correlation between genetically related particles.
3. The software The basic application of AMERATHOR (Section 4) is to evaluate the concentrations of the decay products by solving a system of linear equations relating the measured numbers of counts to the concentrations. This is done by a least squares technique, using the variances of the measured counts as weighting factors. The application of this technique to the measured data automatically provides both the values of the concentrations and their covariance matrix. The procedures applied have a controlled numerical accuracy; this is important because in specific cases the matrix that should be inverted to solve the problem has a very small determinant, being close to a singular matrix. The second application of AMERATHOR (Section 5) is to provide computersimulated values of the number of counts (including fluctuations) expected in specific conditions, useful for testing the quality of a measurement protocol and of the analysis procedures. The input data completely defining the measurement protocol comprise the flow rate D, the air sampling time T and the type and parameters of the measurements: efficiency, background and time schedule. In AMERATHOR it is supposed that for each loaded filter one or several mutually independent measurement channels Mk (1pkp20) are used for activity measurement. Each a or g peak independently measured represents a channel; a gross a counting corresponds also to a measurement channel. Every channel is characterized by the detection efficiency of each of the radon decay products in that channel. Concerning the efficiency, the best situation is achieved when in a given channel the efficiency is maximal for a certain decay product and vanishes for all the others (typically the case of a- and g-spectroscopy channels); the worst—when several decay products have similar detection efficiencies in the same channel (e.g. gross a or gross b channels). In the case of closely lying, low statistics peaks it can be advantageous to consider the entire multiplet as a measurement channel instead of resolving it into components; this channel is characterized by the efficiency values for each of the decay products contributing to the multiplet. In each measurement channel the counting can be carried out in one or in several time intervals. In AMERATHOR measurements carried out after the end of sampling (which is the usual procedure) as well as during the air sampling can be accommodated. The number of the measurement intervals and the time schedule of each complete the characterization of the measurement channel. The data pertaining to each measurement protocol are saved in an ASCII file associated to that protocol. The measured numbers of counts, the computer-simulated numbers of counts as well as the results are also written in specific ASCII files. Note: AMERATHOR is available free of charge from the author.
4. Assessment of the activity concentration of decay products
222
Rn and
220
Rn
Active methods for the measurement of 222Rn and 220Rn decay products are applied for various purposes, requiring a more or less accurate and complete description of the concentrations of the decay products. The most ambitious task is the assessment of the concentrations of each of the nuclides: 218Po, 214Pb, 214Bi, 216Po, 212Pb, 212Bi and 208Tl; 214Po and 212Po are not considered separately, because due to the very short half-life they are always in equilibrium with 214 Bi and 212Bi. Commonly it is very difficult to solve this ambitious task, even if a complex measurement protocol is adopted. For example, due to its short T1/2 (0.15 s), 216Po decays practically before starting the measurements if they are not performed during air sampling; in AMERATHOR 216Po is always
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assumed in equilibrium with 212Pb. Furthermore, if only a measurement channels are used, the experimental data do not include any information on the activity concentration of 208Tl. The difficulties are higher in the case of low levels of 222Rn and 220Rn, because generally the relative uncertainties of the results are higher in this case and may even increase to the limit when the results become meaningless. In this case it is recommended to simplify the problem, by adopting specific approximations. In some cases the focus of the measurement is on 222Rn decay products, neglecting the contribution of 220Rn decay products. A simplified measurement protocol and analysis method can be adopted in this case. In the case when the measurement is carried out with the purpose of dose assessment the measurement of the potential alpha energy concentration cp of 222Rn and of 220Rn decay products is sufficient instead of the assessment of the activity concentration of each decay product. In order to cover all the cases of interest in AMERATHOR the same experimental data are automatically analyzed using various approximations. The detailed output of the program for a given set of experimental data includes the results of the analysis for each approximation implemented. In the case when the number of experimental data available is at least five, the following cases are implemented: (1) Equilibrium is assumed between 208Tl and 212Bi. The concentrations of 218Po, 214Pb, 214Bi, 212Pb, and 212Bi are independently evaluated (five unknowns). (2) Equilibrium is assumed between each 220Rn decay product. This equilibrium concentration represents one unknown, the others are the concentrations of 218Po, 214Pb and 214Bi (four unknowns). (3) Besides equilibrium between each 220Rn decay product, equilibrium between 218Po and 214Pb is also assumed (three unknowns). (4) 220Rn decay products are neglected. The concentrations of 222 Rn decay products are independently evaluated (three unknowns). (5) Equilibrium is assumed between 220Rn decay products and also between 222Rn decay products (two unknowns). (6) 220Rn decay products are neglected, equilibrium is assumed between each 222Rn decay products (one unknown). Besides the activity concentrations of the decay products, the output for each of the above cases includes also the values of the potential alpha energy concentration cp of the 222Rn decay products and of the 220Rn decay products. The values computed for cp are more robust than the values of the individual concentrations, as they are less sensitive to the uncertainties of the input data. If the number of experimental data available is smaller than five, the program automatically selects the cases which are appropriate from the above list of six cases. If the efficiency for the 220Rn decay products in each measurement channel is negligible, then the cases in which 220Rn decay products are evaluated are automatically excluded. Finally, the user can provide input data for the degree of equilibrium between each parent and decay product. Then the data are analyzed assuming fixed ratios of the decay product to parent activity concentrations. The number of unknowns is correspondingly reduced. The magnitude of the contribution of various sources of uncertainty to the uncertainty of the results depends on the details of the measurement protocol and on the activity concentration of radon decay products (e.g. Sonoc and Sima,
869
1992). In the case of typical outdoor concentrations the final uncertainty is dominated by the statistical distribution of the number of counts; this contribution depends roughly inversely proportional on the square root of the concentrations. We will illustrate the uncertainty budget by an example. We consider activity concentrations around 2–4 Bq m3 (222Rn decay products) and around 0.2–0.4 Bq m3 (220Rn decay products). The measurement protocol comprises 15 min sampling followed by measurements in three a-spectrometry channels: around 6 MeV (contribution from 218Po and 212Bi), 7.68 MeV (214Po) and 8.78 MeV (212Po). In every channel three measurements of 10 min each are conducted immediately after the end of sampling, followed by two delayed measurements in the intervals (60–90 min) and (300–330 min). In the above conditions the relative uncertainties of the concentrations resulting from counting statistics are around 10% (218Po and 212Bi), 8% (214Bi) and 6% (214Pb and 212Pb); the contribution of the flow rate fluctuations during sampling to the standard deviation of the number of counts is around 1% (but this contribution increases proportionally to the concentrations). The contribution of the uncertainty of the mean value of flow rate is 2%; that of the filter collection efficiency is similar. The contribution of the decay constants is between 0.1% and 3%, that of the detection efficiency around 2%. Other sources of uncertainty (time intervals, transition probabilities) have smaller contributions.
5. Performance test and optimization of the measurement protocol Before starting extensive measurements of 222Rn and 220Rn decay products it is useful to choose the measurement protocol that is best suited for the specific purpose of the measurements. In this way the balance between the quality of the results and the required experimental effort can be optimized. In order to assist the user to choose the best measurement protocol AMERATHOR incorporates a module for providing computer-simulated data. Starting with user-defined values of the concentrations of each 222 Rn and 220Rn decay product in the air, the program simulates the build-up and decay of the activity on the filter. Then the program prepares computer-simulated values of the numbers of counts corresponding to the selected measurement protocol. Two types of output files are produced. The first contains the values of the mean number of counts expected in the conditions stipulated in the measurement protocol. The second contains values statistically distributed around the mean values, with the variance realistically computed using the model presented in Section 2 and Appendix A. The number of equivalent files containing randomly distributed data is at the option of the user. The computer-simulated data are then processed as the actual measurement data by AMERATHOR. The activity concentrations, the potential alpha energy concentrations and their uncertainties are evaluated as presented in Section 4. The uncertainties reported by the program reflect the statistical nature of the sampling, decay and measurement processes. The bias of the computed results with respect to the initial values which were used for generating the data gives information on the uncertainties of type B associated with the given measurement protocol and analysis scheme. The values of the uncertainties of type A and of type B estimated as above can be used to assess the quality of the measurement protocol. The optimization of the protocol can be achieved by repeating the computations with modified values of the parameters and by observing the dependence of the uncertainties on these values (e.g. Sima et al., 1986; Sonoc and Sima, 1992). It is important to
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mention that the computed values of the uncertainties of type A decrease if specific approximations (e.g. cases 1–6 in Section 4) are applied with the purpose of decreasing the number of unknowns. On the other hand, the uncertainties of type B increase if the approximations are not very good. Consequently the balance between the statistical and the systematic uncertainties should be taken into account for defining the optimal measurement protocol. Finally, as the dependence of the statistical uncertainties on the values of the concentrations of 222Rn and 220Rn decay products in the air is different from the dependence of the systematic uncertainties, the optimization is appropriate to a specific range of concentrations of the 222Rn and 220Rn decay products.
The statistical properties of the number of counts m(s1, s2) registered by the detector in the interval (s1, s2), can be directly computed using P(k), f(t1, t2,.., tk|k) and q(t, s). In particular the mean value and the variance are given by (Sima, 1993) Z T Z s2 mðs1 ; s2 Þ ¼ dt ds qðt; sÞ f ðtÞ (A.2)
varðmðs1 ; s2 ÞÞ ¼ mðs1 ; s2 Þ þ
Appendix A For a realistic evaluation of the uncertainties of the results provided by the active methods for the measurement of radon decay products a stochastic birth and death process that describes the collection on the filter, the decay and the measurement, is considered. The stochastic model is similar to that applied previously to characterize the uncertainties in neutron activation analysis resulting from the fluctuations of the neutron flux (Sima, 1993). For simplicity we will consider explicitly only the case of 218Po nuclei. Let P(k), k ¼ 0, 1, 2,ybe the probability of collecting exactly k nuclei during the sampling interval (0,T). Define by f(t1, t2,y, tk|k) the probability density function that if k nuclei were collected then the first was collected at time t1, the second at t2 and so on. Let q(t, s) be the probability density function (as a function of s) of detecting at time s the decay of the nucleus collected on the filter at time t with a detector that has an efficiency e for the decay radiation, emitted with probability p per decay qðt; sÞ ¼ p l1 exp½l1 ðs tÞ qðt; sÞ ¼ 0
if sot
if sXt, (A.1)
T
dt
Z
T
dt 0
Z
s2
ds s1
0
Z
s2
ds0 qðt; sÞ
s1
qðt 0 ; s0 Þ ½f ðt; t 0 Þ f ðtÞ f ðt 0 Þ
(A.3)
0
where the functions f(t) and f ðt; t Þ are defined by 1 X
PðkÞ
f ðt; t 0 Þ ¼
1 X
k X
ðiÞ
f ðtjkÞ
(A.4)
i¼1
k¼1
In this paper the structure and the applications of the AMERATHOR program were presented. In the program the experimental results obtained using different types of measurements can be specifically processed. The user should only input the proper efficiency values in order to switch from the measurement of an a peak to the measurement of a g peak or to a gross a or b counting. Also, in any measurement channel various measurement intervals can be specified (including the possibility of counting during air sampling). Using AMERATHOR, the complete information available about the activity of a filter, obtained by making various measurements on the same filter, is used for the evaluation. The output of the program is represented by the values and the uncertainties of the concentrations of the 222Rn and 220Rn decay products and of the potential alpha energy concentrations. The program has a flexible output, presenting specific results obtained when various approximations are made. Finally, the program can be used to generate synthetic data simulating the number of counts expected in a given measurement. The analysis of the synthetic data reveals the performance of the measurement and evaluation method in the conditions applied in the simulation. This is useful in order to optimize the measurement protocol before starting extensive measurements using that protocol.
Z 0
f ðtÞ ¼ 6. Summary
s1
0
PðkÞ
k¼2
k X
f
ði;jÞ
ðt; t 0 jkÞ
(A.5)
i;j¼1;iaj ðiÞ
ði;jÞ
In the above equations f ðtjkÞ and f ðt; t 0 jkÞ give the probability density functions of collecting the i-th nucleus at moment t, respectively, the i-th nucleus at t and the j-th nucleus at t0 if k nuclei were collected on the filter in (0,T); these functions are easily obtained by integrating f ðt 1 ; t 2 ; . . . ; t k jkÞ over all the variables except the i-th or the i-th and the j-th, respectively, over the interval (0,T). In Eq. (A.2) we considered for simplicity that s1 4T (counting after the end of sampling). In the case of 218Po build-up on the filter the function f(t) is given simply by f ðtÞ ¼ C 1 Z D
(A.6) 218
Po in the air where C 1 is the mean value of the concentration of and D is the mean value of the flow rate. Then Eq. (A.2) reduces to the equation obtained in the traditional approach. The first term on the right side of Eq. (A.3) is the usual Poisson term. The term in the square brackets reflects the correlation of the numbers of particles collected on the filter at t and t0 ; in the absence of correlations that term would be zero (except for t ¼ t0 ), because f(t,t0 ) would be equal to f ðtÞ f ðt 0 Þ. The second term in Eq. (A.3) means that the correlations present in the collection of the particles on the filter give rise to correlations, exponentially attenuated, in the disintegration and counting of the particles. Correlations between the particles collected on the filter can result also from correlations present in the spatial distribution of the particles in the air. Due to the lack of more detailed information, we consider that the numbers of particles distributed in air in disjoint spatial domains are uncorrelated and that the distribution function of the number of particles in a given domain depends only on the volume of that domain. Evidently the statistical properties, e.g. the variance, of the number of particles in a given domain, depend on the volume of that domain; the same is true for the ratio between the number of particles and the volume, i.e. for the concentration. Indeed, even if the ratio between the number n of particles from 1 m3 and the volume V ¼ 1 m3 has the same mean value as the ratio between the number n0 of particles from 1 cm3 and the volume V0 ¼ 1 cm3, the variances of the two ratios are different: the variance of each ratio is inversely proportional with the corresponding volume and varðn=VÞ V ¼ varðn0 =V 0 Þ V 0 . Therefore in order to describe the statistical properties of the concentration as a random variable it is necessary to specify the volume; for concreteness we choose a volume V0 and define the concentration as the ratio between the number of particles from a domain of volume V0 and V0. Similarly, we define the flow rate as the volume sampled in the time interval T0. Then, assuming that the flow rate correlations have a very
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short range in time (e.g. compared with the half life of the decay products), it can be shown (Sima, 1987) that 2
f ðt; t 0 Þ f ðtÞ f ðt 0 Þ ¼ Z2 ½D varðC 1 Þ V 0 þ C 1 varðDÞ 0
T 0 C 1 D dðt t Þ
(A.7)
As stated above varðC 1 Þ V 0 is independent of V0; also varðDÞ T 0 is independent of T0. Note that the contribution of var(D) increases with the square of the concentration. Note also that if the number of particles in a volume V obeys the Poisson distribution, then the first term in the square brackets becomes equal to D C 1 and then this term is exactly cancelled by the last term; this approximation is adopted in AMERATHOR. The final formula for the variance of number of counts corresponding to the detection of 218Po nuclei is obtained by replacing Eq. (A.7) in Eq. (A.3). The model sketched above, extended for the case of nuclides belonging to radioactive series, is implemented in AMERATHOR. References Baciu, A.C., 2005. Radon ant thoron progeny concentration variability in relation to meteorological conditions at Bucharest (Romania). Journal of Environmental Radioactivity 83 (1), 171–189.
871
Dovlete, C., 1983. Determination of the gamma emitting daughters of the atmospheric 222Rn and 220Rn. Studii si Cercetari de Fizica 35 (4), 333–338 (in Romanian). Paul, A., Ro¨ttger, S., Honig, A., et al., 1999. Measurement of short-lived radon progenies by simultaneous ag-spectrometry at the German radon reference chamber. Nuclear Instruments and Methods in Physics Research A 434 (2), 303–312. Sima, O., 1978. Natural radioactivity of aerosols. In: Proceedings of the Annual Scientific Conference of the Institute for Meteorology and Hydrology, Part I. Bucharest, pp. 361–380 (in Romanian). Sima, O., Sonoc, S., Dovlete, C., 1986. Critical analysis of the least squares method applied to the determination of environmental radon and thoron daughters. Studii si Cercetari de Fizica 38 (5), 464–495 (in Romanian). Sima, O., 1987. Improved precision estimation in neutron activation analysis and other related methods. Studii si Cercetari de Fizica 39 (3), 177–189 (in Romanian). Sima, O., 1993. Uncertainties in neutron activation analysis resulting from reactor noise. Journal of Radioanalytical and Nuclear Chemistry 174 (1), 65–72 Articles. Sonoc, S., 1989. On the determination of 218Po concentration in aerosol samples by gamma spectrometry. Studii si Cercetari de Fizica 41 (9), 851–857 (in Romanian). Sonoc, S., Sima, O., 1992. Optimal method for environmental radon and thoron daughters determination by alpha spectrometry. Radiation Protection Dosimetry 45 (1/4), 51–52. Tsivoglou, E.C., Ayer, H.E., Holaday, D.A., 1953. Occurrence of nonequilibrium atmospheric mixtures of radon and its daughters. Nucleonics 11 (9), 40–45.
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Comprehensive software for the assessment of 222Rn and products based on air sampling measurements
220
Rn decay
Octavian Sima Physics Department, University of Bucharest, Bucharest-Magurele, P.O. Box MG-11, RO-077125, Romania
a r t i c l e in fo
abstract
Keywords: Radon decay products Analysis software Uncertainty analysis Measurement optimization
A computational tool dedicated to the measurement of 222Rn and 220Rn decay products by air sampling is presented. a- or g-spectrometry measurements, gross a or b counting, as well as a combination of them are considered. Special attention is given to the evaluation of the uncertainty budget of the results. Besides typical applications in the analysis of experimental data, the software can be used for assessing the expected quality of a measurement protocol and for optimizing it, by generating and analyzing sets of realistic synthetic data. & 2009 Elsevier Ltd. All rights reserved.
1. Introduction Radon and radon decay products represent subjects of interest in various fields. Indeed, radon and its decay products are of major concern for the occupational exposure e.g. of uranium miners, but also for the irradiation of the public, delivering approximately half of the natural exposure dose. They have also applications as tracers in various meteorology and hydrology studies, and the correlation with the seismic activity is a subject of current interest. Radon reduction is a main issue in low-level or ultra-lowlevel laboratories. In this work we consider active methods for the measurement of the radon (222Rn and 220Rn) decay products. The early methods of this kind, pioneered by the work of Tsivoglou (Tsivoglou et al., 1953), were specifically developed for a particular type of measurement (gross a or a-spectrometry) and for a particular measurement protocol. Later the advantage of simultaneous a- and g-spectrometry was clearly demonstrated and the ag-spectrometry method (Paul et al., 1999) became the standard metrological procedure for radon progeny measurement in radon reference atmospheres. In spite of the long history of the active methods, it seems that a general and friendly procedure for analyzing simultaneously various types of measurements, for an arbitrary measurement protocol, is still lacking. Also, certain features of the statistical processes involved in the methods seem to have been disregarded. Active methods for the assessment of the activity concentration of 222Rn and 220Rn decay products were applied since long by our group in the University of Bucharest and in the Environmental
Fax: +40 214574521.
E-mail address: [email protected] 0969-8043/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2009.01.022
Radioactivity Laboratory, Bucharest (Sima, 1978; Dovlete, 1983; Sima et al., 1986; Sonoc, 1989; Sonoc and Sima, 1992; Baciu, 2005). The computational methods developed in these early studies were recently extended and implemented in a dedicated software called AMERATHOR (Active Methods for Radon and Thoron decay products). The adopted theoretical model, the software and its applications will be presented in this paper.
2. Theoretical model In the active methods for the measurement of 222Rn progenies (218Po, 214Pb, 214Bi) and of 220Rn progenies (216Po, 212Pb, 212Bi, 208 Tl) the radon decay products are collected from the air by retention on a filter through which the air is sampled with a controlled flow rate. The loaded filter is measured and the activity concentrations of the decay products in the air are evaluated. The time evolution of the numbers Ni of nuclei of type i (i ¼ 1, 2 and 3 for 218Po, 214Pb and 214Bi, respectively, in the 222Rn decay series) collected on the filter is obtained by solving the set of Bateman equations. For example, N1 ðtÞ ¼
N1 ðtÞ ¼
C1 Z D
l1 C1 Z D
l1
½1 expðl1 tÞ
for toT
½1 expðl1 TÞ exp½l1 ðt TÞ
(1)
for t4T
(2)
where Ci and li represent the concentration in the air and the decay constant for type i nuclide, T the air sampling time, D the flow rate and Z the filter collection efficiency (for simplicity assumed equal for all the nuclides). The equations for N2(t) and N3(t) contain similar terms describing the contribution of the nuclides of type 2 or 3 directly sampled from the air and more complex terms representing cases when the nuclides of type 2 or
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3 are produced by the decay of the parents collected on the filter; clearly N2(t) depends linearly on C2 and C1, N3(t) on C3, C2 and C1. Consider the counts mk(s1,s2) registered in the time interval (s1,s2) due to the detection (with efficiency ek) of the radiations of type k emitted by nuclide i with emission probability pk mk ðs1 ; s2 Þ ¼ k pk li
Z
s2 s1
Ni ðtÞ dt
(3)
As Ni(t) are linear combinations of the concentrations, the number of counts will also be linear combinations of concentrations. The case of 220Rn decay products is described by a similar formalism. By making a set of count measurements and fitting the data, e.g. by the least squares method, it is possible to obtain the concentrations. The uncertainties of the concentrations depend on the uncertainties of the number of counts, of the emission probabilities, of the efficiencies, of the flow rate and filter efficiency. In the conventional approach the uncertainty of the number of counts is obtained by assuming Poisson distribution. The effect of the uncertainty of the flow rate and of the other quantities is obtained by using the error propagation formula. The above approach for obtaining the uncertainty is not completely satisfactory. Generally speaking, the Eqs. (1)–(3) are valid for the mean values of the corresponding quantities, i.e. for the first moment of the distributions of all the random variables included in the equations, e.g. the number of collected nuclei or the number of counts, the flow rate. The error propagation formula applied to these equations gives indeed the contribution of the uncertainty of the mean value of the flow rate to the uncertainty of the mean value of the number of counts, but it does not describe the variance of the actual number of counts due to flow rate fluctuations during sampling. In fact, the equations for the first moment of the distributions by themselves do not include any information on the second moment of the distributions, e.g. the variance of the number of counts; this is obtained conventionally by assuming Poisson distribution. However, deviations from Poisson distribution may occur, because the distribution of the number of counts depends on the statistical properties of the process of generating the nuclei by collection on the filter, dependent in turn on the statistical properties of the flow rate, and by the serial decay. Indeed, if the flow rate has big fluctuations it will induce big fluctuations in the number of collected nuclei, which should be reflected in an increased variance of the number of counts. Also, the number of nuclei collected around time t and the number collected around time t0 can be correlated if the flow rate at t is correlated with the flow rate at t0 . We developed and implemented in AMERATHOR a more realistic approach based on a stochastic model describing the collection on the filter, the disintegration and the measurement processes. The model (Appendix A) takes into account the flow rate fluctuations during sampling. Of course, the mean values of the quantities are given by the same equations as in the traditional approach. The main differences with respect to the traditional approach are (a) the variance of the number of counts is higher than the Poisson term, including additional contributions, depending on the statistical properties of the flow rate and of the distribution of the particles in air; (b) the number of counts produced by the radiations emitted by different nuclides are correlated to a certain degree, due to the fact that flow rate fluctuations affect simultaneously the collection of all types of nuclides and due to the correlation between genetically related particles.
3. The software The basic application of AMERATHOR (Section 4) is to evaluate the concentrations of the decay products by solving a system of linear equations relating the measured numbers of counts to the concentrations. This is done by a least squares technique, using the variances of the measured counts as weighting factors. The application of this technique to the measured data automatically provides both the values of the concentrations and their covariance matrix. The procedures applied have a controlled numerical accuracy; this is important because in specific cases the matrix that should be inverted to solve the problem has a very small determinant, being close to a singular matrix. The second application of AMERATHOR (Section 5) is to provide computersimulated values of the number of counts (including fluctuations) expected in specific conditions, useful for testing the quality of a measurement protocol and of the analysis procedures. The input data completely defining the measurement protocol comprise the flow rate D, the air sampling time T and the type and parameters of the measurements: efficiency, background and time schedule. In AMERATHOR it is supposed that for each loaded filter one or several mutually independent measurement channels Mk (1pkp20) are used for activity measurement. Each a or g peak independently measured represents a channel; a gross a counting corresponds also to a measurement channel. Every channel is characterized by the detection efficiency of each of the radon decay products in that channel. Concerning the efficiency, the best situation is achieved when in a given channel the efficiency is maximal for a certain decay product and vanishes for all the others (typically the case of a- and g-spectroscopy channels); the worst—when several decay products have similar detection efficiencies in the same channel (e.g. gross a or gross b channels). In the case of closely lying, low statistics peaks it can be advantageous to consider the entire multiplet as a measurement channel instead of resolving it into components; this channel is characterized by the efficiency values for each of the decay products contributing to the multiplet. In each measurement channel the counting can be carried out in one or in several time intervals. In AMERATHOR measurements carried out after the end of sampling (which is the usual procedure) as well as during the air sampling can be accommodated. The number of the measurement intervals and the time schedule of each complete the characterization of the measurement channel. The data pertaining to each measurement protocol are saved in an ASCII file associated to that protocol. The measured numbers of counts, the computer-simulated numbers of counts as well as the results are also written in specific ASCII files. Note: AMERATHOR is available free of charge from the author.
4. Assessment of the activity concentration of decay products
222
Rn and
220
Rn
Active methods for the measurement of 222Rn and 220Rn decay products are applied for various purposes, requiring a more or less accurate and complete description of the concentrations of the decay products. The most ambitious task is the assessment of the concentrations of each of the nuclides: 218Po, 214Pb, 214Bi, 216Po, 212Pb, 212Bi and 208Tl; 214Po and 212Po are not considered separately, because due to the very short half-life they are always in equilibrium with 214 Bi and 212Bi. Commonly it is very difficult to solve this ambitious task, even if a complex measurement protocol is adopted. For example, due to its short T1/2 (0.15 s), 216Po decays practically before starting the measurements if they are not performed during air sampling; in AMERATHOR 216Po is always
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assumed in equilibrium with 212Pb. Furthermore, if only a measurement channels are used, the experimental data do not include any information on the activity concentration of 208Tl. The difficulties are higher in the case of low levels of 222Rn and 220Rn, because generally the relative uncertainties of the results are higher in this case and may even increase to the limit when the results become meaningless. In this case it is recommended to simplify the problem, by adopting specific approximations. In some cases the focus of the measurement is on 222Rn decay products, neglecting the contribution of 220Rn decay products. A simplified measurement protocol and analysis method can be adopted in this case. In the case when the measurement is carried out with the purpose of dose assessment the measurement of the potential alpha energy concentration cp of 222Rn and of 220Rn decay products is sufficient instead of the assessment of the activity concentration of each decay product. In order to cover all the cases of interest in AMERATHOR the same experimental data are automatically analyzed using various approximations. The detailed output of the program for a given set of experimental data includes the results of the analysis for each approximation implemented. In the case when the number of experimental data available is at least five, the following cases are implemented: (1) Equilibrium is assumed between 208Tl and 212Bi. The concentrations of 218Po, 214Pb, 214Bi, 212Pb, and 212Bi are independently evaluated (five unknowns). (2) Equilibrium is assumed between each 220Rn decay product. This equilibrium concentration represents one unknown, the others are the concentrations of 218Po, 214Pb and 214Bi (four unknowns). (3) Besides equilibrium between each 220Rn decay product, equilibrium between 218Po and 214Pb is also assumed (three unknowns). (4) 220Rn decay products are neglected. The concentrations of 222 Rn decay products are independently evaluated (three unknowns). (5) Equilibrium is assumed between 220Rn decay products and also between 222Rn decay products (two unknowns). (6) 220Rn decay products are neglected, equilibrium is assumed between each 222Rn decay products (one unknown). Besides the activity concentrations of the decay products, the output for each of the above cases includes also the values of the potential alpha energy concentration cp of the 222Rn decay products and of the 220Rn decay products. The values computed for cp are more robust than the values of the individual concentrations, as they are less sensitive to the uncertainties of the input data. If the number of experimental data available is smaller than five, the program automatically selects the cases which are appropriate from the above list of six cases. If the efficiency for the 220Rn decay products in each measurement channel is negligible, then the cases in which 220Rn decay products are evaluated are automatically excluded. Finally, the user can provide input data for the degree of equilibrium between each parent and decay product. Then the data are analyzed assuming fixed ratios of the decay product to parent activity concentrations. The number of unknowns is correspondingly reduced. The magnitude of the contribution of various sources of uncertainty to the uncertainty of the results depends on the details of the measurement protocol and on the activity concentration of radon decay products (e.g. Sonoc and Sima,
869
1992). In the case of typical outdoor concentrations the final uncertainty is dominated by the statistical distribution of the number of counts; this contribution depends roughly inversely proportional on the square root of the concentrations. We will illustrate the uncertainty budget by an example. We consider activity concentrations around 2–4 Bq m3 (222Rn decay products) and around 0.2–0.4 Bq m3 (220Rn decay products). The measurement protocol comprises 15 min sampling followed by measurements in three a-spectrometry channels: around 6 MeV (contribution from 218Po and 212Bi), 7.68 MeV (214Po) and 8.78 MeV (212Po). In every channel three measurements of 10 min each are conducted immediately after the end of sampling, followed by two delayed measurements in the intervals (60–90 min) and (300–330 min). In the above conditions the relative uncertainties of the concentrations resulting from counting statistics are around 10% (218Po and 212Bi), 8% (214Bi) and 6% (214Pb and 212Pb); the contribution of the flow rate fluctuations during sampling to the standard deviation of the number of counts is around 1% (but this contribution increases proportionally to the concentrations). The contribution of the uncertainty of the mean value of flow rate is 2%; that of the filter collection efficiency is similar. The contribution of the decay constants is between 0.1% and 3%, that of the detection efficiency around 2%. Other sources of uncertainty (time intervals, transition probabilities) have smaller contributions.
5. Performance test and optimization of the measurement protocol Before starting extensive measurements of 222Rn and 220Rn decay products it is useful to choose the measurement protocol that is best suited for the specific purpose of the measurements. In this way the balance between the quality of the results and the required experimental effort can be optimized. In order to assist the user to choose the best measurement protocol AMERATHOR incorporates a module for providing computer-simulated data. Starting with user-defined values of the concentrations of each 222 Rn and 220Rn decay product in the air, the program simulates the build-up and decay of the activity on the filter. Then the program prepares computer-simulated values of the numbers of counts corresponding to the selected measurement protocol. Two types of output files are produced. The first contains the values of the mean number of counts expected in the conditions stipulated in the measurement protocol. The second contains values statistically distributed around the mean values, with the variance realistically computed using the model presented in Section 2 and Appendix A. The number of equivalent files containing randomly distributed data is at the option of the user. The computer-simulated data are then processed as the actual measurement data by AMERATHOR. The activity concentrations, the potential alpha energy concentrations and their uncertainties are evaluated as presented in Section 4. The uncertainties reported by the program reflect the statistical nature of the sampling, decay and measurement processes. The bias of the computed results with respect to the initial values which were used for generating the data gives information on the uncertainties of type B associated with the given measurement protocol and analysis scheme. The values of the uncertainties of type A and of type B estimated as above can be used to assess the quality of the measurement protocol. The optimization of the protocol can be achieved by repeating the computations with modified values of the parameters and by observing the dependence of the uncertainties on these values (e.g. Sima et al., 1986; Sonoc and Sima, 1992). It is important to
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mention that the computed values of the uncertainties of type A decrease if specific approximations (e.g. cases 1–6 in Section 4) are applied with the purpose of decreasing the number of unknowns. On the other hand, the uncertainties of type B increase if the approximations are not very good. Consequently the balance between the statistical and the systematic uncertainties should be taken into account for defining the optimal measurement protocol. Finally, as the dependence of the statistical uncertainties on the values of the concentrations of 222Rn and 220Rn decay products in the air is different from the dependence of the systematic uncertainties, the optimization is appropriate to a specific range of concentrations of the 222Rn and 220Rn decay products.
The statistical properties of the number of counts m(s1, s2) registered by the detector in the interval (s1, s2), can be directly computed using P(k), f(t1, t2,.., tk|k) and q(t, s). In particular the mean value and the variance are given by (Sima, 1993) Z T Z s2 mðs1 ; s2 Þ ¼ dt ds qðt; sÞ f ðtÞ (A.2)
varðmðs1 ; s2 ÞÞ ¼ mðs1 ; s2 Þ þ
Appendix A For a realistic evaluation of the uncertainties of the results provided by the active methods for the measurement of radon decay products a stochastic birth and death process that describes the collection on the filter, the decay and the measurement, is considered. The stochastic model is similar to that applied previously to characterize the uncertainties in neutron activation analysis resulting from the fluctuations of the neutron flux (Sima, 1993). For simplicity we will consider explicitly only the case of 218Po nuclei. Let P(k), k ¼ 0, 1, 2,ybe the probability of collecting exactly k nuclei during the sampling interval (0,T). Define by f(t1, t2,y, tk|k) the probability density function that if k nuclei were collected then the first was collected at time t1, the second at t2 and so on. Let q(t, s) be the probability density function (as a function of s) of detecting at time s the decay of the nucleus collected on the filter at time t with a detector that has an efficiency e for the decay radiation, emitted with probability p per decay qðt; sÞ ¼ p l1 exp½l1 ðs tÞ qðt; sÞ ¼ 0
if sot
if sXt, (A.1)
T
dt
Z
T
dt 0
Z
s2
ds s1
0
Z
s2
ds0 qðt; sÞ
s1
qðt 0 ; s0 Þ ½f ðt; t 0 Þ f ðtÞ f ðt 0 Þ
(A.3)
0
where the functions f(t) and f ðt; t Þ are defined by 1 X
PðkÞ
f ðt; t 0 Þ ¼
1 X
k X
ðiÞ
f ðtjkÞ
(A.4)
i¼1
k¼1
In this paper the structure and the applications of the AMERATHOR program were presented. In the program the experimental results obtained using different types of measurements can be specifically processed. The user should only input the proper efficiency values in order to switch from the measurement of an a peak to the measurement of a g peak or to a gross a or b counting. Also, in any measurement channel various measurement intervals can be specified (including the possibility of counting during air sampling). Using AMERATHOR, the complete information available about the activity of a filter, obtained by making various measurements on the same filter, is used for the evaluation. The output of the program is represented by the values and the uncertainties of the concentrations of the 222Rn and 220Rn decay products and of the potential alpha energy concentrations. The program has a flexible output, presenting specific results obtained when various approximations are made. Finally, the program can be used to generate synthetic data simulating the number of counts expected in a given measurement. The analysis of the synthetic data reveals the performance of the measurement and evaluation method in the conditions applied in the simulation. This is useful in order to optimize the measurement protocol before starting extensive measurements using that protocol.
Z 0
f ðtÞ ¼ 6. Summary
s1
0
PðkÞ
k¼2
k X
f
ði;jÞ
ðt; t 0 jkÞ
(A.5)
i;j¼1;iaj ðiÞ
ði;jÞ
In the above equations f ðtjkÞ and f ðt; t 0 jkÞ give the probability density functions of collecting the i-th nucleus at moment t, respectively, the i-th nucleus at t and the j-th nucleus at t0 if k nuclei were collected on the filter in (0,T); these functions are easily obtained by integrating f ðt 1 ; t 2 ; . . . ; t k jkÞ over all the variables except the i-th or the i-th and the j-th, respectively, over the interval (0,T). In Eq. (A.2) we considered for simplicity that s1 4T (counting after the end of sampling). In the case of 218Po build-up on the filter the function f(t) is given simply by f ðtÞ ¼ C 1 Z D
(A.6) 218
Po in the air where C 1 is the mean value of the concentration of and D is the mean value of the flow rate. Then Eq. (A.2) reduces to the equation obtained in the traditional approach. The first term on the right side of Eq. (A.3) is the usual Poisson term. The term in the square brackets reflects the correlation of the numbers of particles collected on the filter at t and t0 ; in the absence of correlations that term would be zero (except for t ¼ t0 ), because f(t,t0 ) would be equal to f ðtÞ f ðt 0 Þ. The second term in Eq. (A.3) means that the correlations present in the collection of the particles on the filter give rise to correlations, exponentially attenuated, in the disintegration and counting of the particles. Correlations between the particles collected on the filter can result also from correlations present in the spatial distribution of the particles in the air. Due to the lack of more detailed information, we consider that the numbers of particles distributed in air in disjoint spatial domains are uncorrelated and that the distribution function of the number of particles in a given domain depends only on the volume of that domain. Evidently the statistical properties, e.g. the variance, of the number of particles in a given domain, depend on the volume of that domain; the same is true for the ratio between the number of particles and the volume, i.e. for the concentration. Indeed, even if the ratio between the number n of particles from 1 m3 and the volume V ¼ 1 m3 has the same mean value as the ratio between the number n0 of particles from 1 cm3 and the volume V0 ¼ 1 cm3, the variances of the two ratios are different: the variance of each ratio is inversely proportional with the corresponding volume and varðn=VÞ V ¼ varðn0 =V 0 Þ V 0 . Therefore in order to describe the statistical properties of the concentration as a random variable it is necessary to specify the volume; for concreteness we choose a volume V0 and define the concentration as the ratio between the number of particles from a domain of volume V0 and V0. Similarly, we define the flow rate as the volume sampled in the time interval T0. Then, assuming that the flow rate correlations have a very
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short range in time (e.g. compared with the half life of the decay products), it can be shown (Sima, 1987) that 2
f ðt; t 0 Þ f ðtÞ f ðt 0 Þ ¼ Z2 ½D varðC 1 Þ V 0 þ C 1 varðDÞ 0
T 0 C 1 D dðt t Þ
(A.7)
As stated above varðC 1 Þ V 0 is independent of V0; also varðDÞ T 0 is independent of T0. Note that the contribution of var(D) increases with the square of the concentration. Note also that if the number of particles in a volume V obeys the Poisson distribution, then the first term in the square brackets becomes equal to D C 1 and then this term is exactly cancelled by the last term; this approximation is adopted in AMERATHOR. The final formula for the variance of number of counts corresponding to the detection of 218Po nuclei is obtained by replacing Eq. (A.7) in Eq. (A.3). The model sketched above, extended for the case of nuclides belonging to radioactive series, is implemented in AMERATHOR. References Baciu, A.C., 2005. Radon ant thoron progeny concentration variability in relation to meteorological conditions at Bucharest (Romania). Journal of Environmental Radioactivity 83 (1), 171–189.
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Dovlete, C., 1983. Determination of the gamma emitting daughters of the atmospheric 222Rn and 220Rn. Studii si Cercetari de Fizica 35 (4), 333–338 (in Romanian). Paul, A., Ro¨ttger, S., Honig, A., et al., 1999. Measurement of short-lived radon progenies by simultaneous ag-spectrometry at the German radon reference chamber. Nuclear Instruments and Methods in Physics Research A 434 (2), 303–312. Sima, O., 1978. Natural radioactivity of aerosols. In: Proceedings of the Annual Scientific Conference of the Institute for Meteorology and Hydrology, Part I. Bucharest, pp. 361–380 (in Romanian). Sima, O., Sonoc, S., Dovlete, C., 1986. Critical analysis of the least squares method applied to the determination of environmental radon and thoron daughters. Studii si Cercetari de Fizica 38 (5), 464–495 (in Romanian). Sima, O., 1987. Improved precision estimation in neutron activation analysis and other related methods. Studii si Cercetari de Fizica 39 (3), 177–189 (in Romanian). Sima, O., 1993. Uncertainties in neutron activation analysis resulting from reactor noise. Journal of Radioanalytical and Nuclear Chemistry 174 (1), 65–72 Articles. Sonoc, S., 1989. On the determination of 218Po concentration in aerosol samples by gamma spectrometry. Studii si Cercetari de Fizica 41 (9), 851–857 (in Romanian). Sonoc, S., Sima, O., 1992. Optimal method for environmental radon and thoron daughters determination by alpha spectrometry. Radiation Protection Dosimetry 45 (1/4), 51–52. Tsivoglou, E.C., Ayer, H.E., Holaday, D.A., 1953. Occurrence of nonequilibrium atmospheric mixtures of radon and its daughters. Nucleonics 11 (9), 40–45.