- Email: [email protected]
The recoil factor of 214Pb
Aerosol Science 35 (2004) 1041 – 1050 www.elsevier.com/locate/jaerosci
The recoil factor of
214
Pb
N. Stevanovi"ca , D. Nikezi"ca;∗ , A. Djordjevichb a
Physics Faculty of Science, University of Kragujevac, R. Domanovic 12, 34000 Kragujevac Serbia and Montenegro b City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong Received 19 November 2003; received in revised form 16 February 2004; accepted 17 February 2004
Abstract The majority of the 218 Po atoms (the 2rst progeny of radon) are attached to natural aerosols in the air. After their alpha decay, the newly formed nuclei of 214 Pb may detach from the carrier aerosol due to recoil. The average fraction of the nuclei detached in this manner is called the “recoil factor”. It is an important parameter in the calculation of the radon-progeny concentration and the related radiation dose in lungs. We determine this parameter for eight common aerosol types and show that it cannot be treated as a 2xed constant since it varies strongly with the aerosol particle size, shape, material, and surface/volume type of contamination. The range of values that the recoil factor can take broadens with the hygroscopic growth of the aerosol particles, to the maximum range between approximately 0.1 and 0.8. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Aerosol; Radon progeny;
214
Pb; Recoil
1. Introduction Radium (226 Ra) is present in the materials used for the construction of most buildings and the terrain underneath. A progeny of its radioactive decay is radon (222 Rn). This chemically inert gas may migrate gradually to the buildings’ interior. Its accumulation there represents a signi2cant radiation hazard. A short-lived progeny of its decay is polonium (218 Po). Some of the 218 Po may remain as a free or unattached fraction that is in the atomic form or bound to small species of molecular dimensions. However, most of it is likely to attach to natural aerosols normally present in the air, giving rise to the “attached fraction” of radon progeny. The rather complex behavior of both fractions contributes to the uncertainty of measurement of the total indoor radon-progeny concentration ∗
Corresponding author. Tel.: +381-34-336223; fax: +381-34-335040. E-mail address: [email protected] (D. Nikezi"c).
0021-8502/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jaerosci.2004.02.006
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D (Ili"c & Sutej, 1997). This concentration depends on several factors including the radon source strength, the ventilation rate in the environment, the aerosol concentration and its size distribution, the progeny’s rate of attachment to the aerosol particles, and these particles’ deposition rates (PorstendEorfer, 1984; Fleischer, 1997; Jacobi, 1972). A model that accounts for these factors and calculates the concentration of the indoor radon progeny has been reported (Jacobi, 1972). The 2ne-tuning of a parameter in this model known as the “recoil factor” is the objective of this paper. Attached to an aerosol particle, the 218 Po radon progeny decays by the emission of an alpha particle. The remaining nucleus of lead (214 Pb) undergoes recoil with energy of 117 keV. This energy is suHcient to detach the 214 Po nucleus from the aerosol particle on which 218 Po had been attached. The average fraction of detached nuclei is called the recoil factor, denoted by p. Mercer (1976) has estimated it at p ≈ 0:8. This value, assumed to be a constant between 0.8 and 0.83, has since been used in virtually all reported calculations. Research reported subsequent to Mercer’s work has provided new information about the stopping power of nuclei in diKerent media (Zeigler, 2003) and their size distribution (Reineking et al., 1994). Based on such additional insight, we re-evaluate the recoil factor in this paper. Our motivation to do that stems from this factor’s signi2cance in the calculation of the indoor radon-progeny concentration (Jacobi, 1972). The behavior and characteristics of aerosols depend strongly on their particle size. The diameter d is used as a measure of that size when a spherical shape may be assumed. An equivalent diameter de = (6V=)1=3 is evaluated for irregularly shaped aerosol particles with a volume of V (Hinds, 1998). Typically, the particle size varies from 0.001 to 100 m according to the log-normal distribution. It was thought previously that the aerosol distribution had only one maximum. Such single-mode distribution was assumed by Mercer (1976) when calculating the recoil factor. However, a second maximum has been detected in the meantime (PorstendEorfer & Reineking, 2000). The corresponding distribution would therefore have to be more than just a single-mode distribution. Detailed measurements have actually identi2ed three distribution modes. They were referred to as the nucleation, accumulation and course modes (Reineking et al., 1994). Each had its own diameter AMTDi (median of aerosol thermodynamic diameter) and geometric standard deviation ( gi ) (i = 1; 2; 3). The notion of there being three modal distributions of attached radon progeny has been accepted widely. For example, the dose conversion factor has been calculated on that basis (Marsh & Birchall, 2000; Marsh et al., 2002). We too adopt the three-mode distribution concept in our calculations in this work. The three-mode distribution of the attached progeny is described by the following equation (PorstendEorfer & Reineking, 2000): f(d) =
3 i=1
2
e−(ln d−ln AMTDi ) =2 ln √ fi 2 ln( gi ) d
2
( gi )
;
(1)
where fi is a fraction of the mode i; i = 1 for the nucleation mode, i = 2 for the accumulation mode and i = 3 for the coarse mode. Numerical values in Eq. (1) recommended by Marsh and Birchall (2000) in their “sensitivity analysis” are: AMTD1 = 0:05 m, g1 = 2:0 f1 = 0:28; AMTD2 = 0:25 m g2 = 2:0 f2 = 0:70; AMTD3 = 1:50 m, g3 = 1:5 f3 = 0:02. The corresponding three modes and the cumulative distribution curve are shown in Fig. 1.
N. Stevanovi-c et al. / Aerosol Science 35 (2004) 1041 – 1050
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10 AMTD 1 =0.05 σg1 =2.0 f1 =0.28 AMTD 2 =0.25 σg2 =2.0 f2 =0.70 AMTD 3 =1.50 σg3 =1.5 f3 =0.02
DISTRIBUTIONS
1
0.1
0.01
0.001 0.001
0.01
0.1
1
10
Equivalent diameter (µm)
Fig. 1. Radioactive aerosol distribution: solid line is the total distribution; dotted line is the nucleation mode (subscript i = 1); dashed line is the accumulation mode (subscript i = 2); and the dashed-dotted line is the coarse mode (subscript i = 3).
2. Aerosol model and the calculation method Liquid aerosol particles are almost always spherical. While the solid ones may have more complex shapes, these are usually approximated as spherical in the literature on aerosol research (Reist, 1993). Atmospheric aerosols with solid particles may combine with droplets of water vapor. Spherical and ellipsoidal shapes of the resulting aerosol particles, consisting either of pure silicon dioxide (SiO2 ) or its mixture with vapor (SiO2 + H2 O), are analyzed in this paper. They are illustrated in Fig. 2 together with the atoms of radon progeny that are either attached to the surface or distributed within the volume of the aerosol particle, representing the surface or volume contamination, respectively. DiKerent outcomes are possible as the 214 Pb attached to the aerosol particle undergoes recoil. Five of them are shown in Fig. 3 where the thick and thin arrows represent velocity directions for the recoil nucleus and alpha particle, respectively. The 218 Po is assumed to be on the surface of the aerosol particle in the cases labeled 1, 2 and 3. In case 1, an alpha particle is emitted toward the aerosol particle and the nucleus therefore recoils out of it and detaches. The opposite is assumed in case 2 where the nucleus does not detach as it does not have suHcient energy to pass across much of the aerosol particle that stops it. The detachment occurs in case 3 where the 214 Pb nucleus cannot be stopped. For surface contamination, the recoil factor is not less than 0.5. It is assumed in cases 4 and 5 in Fig. 3 that the 218 Po decays inside the volume of the aerosol. This can happen if the aerosol growth occurs subsequent to the attachment of the 218 Po on it. The recoil factor can be less than 0.5 if the diameter of the aerosol particle is large enough. Depending on the recoil direction and range, the 214 Pb nucleus may detach from (case 5), or get trapped within (case 4), the aerosol particle. The ranges of the recoil nuclei 214 Pb in aerosol material (water and SiO2 ) were calculated by SRIM2003 code (Zeigler, 2003). SRIM program calculates the stopping power as a function of
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Fig. 2. Aerosol models considered in this work: spherical SiO2 (A); spherical SiO2 kernel surrounded by H2 O (B); ellipsoidal SiO2 (C); ellipsoidal SiO2 kernel surrounded by H2 O (D). The points on the surface or within the volume represent the surface/volume type of contamination by 214 Pb. (R stands for radius).
Fig. 3. Cases of 214 Pb recoil analyzed in this work. Thin arrows represent alpha particles’ velocity directions and thick ones those of recoil nuclei.
energy in the form of a table for the particle energy larger than 10 keV. Ranges of recoil nuclei 214 Pb are found as Rq = 0:048 m in SiO2 , and Rw = 0:093 m in water. The directions of recoil nuclei were generated by the Monte Carlo method. While the geometric details of the simulation are omitted, the issues of physics pertaining to the energy and range calculation are summarized. The energy of recoil nucleus after traveling a distance x in some material was calculated from the following expression for the stopping power of the recoil nucleus: dE (2) f(E) = − ; dx where the function f(E) was determined by curve 2tting the SRIM2003 data and extrapolating it linearly below 10 keV. Then, the distance x is obtained by integration: x E0 Ex dE dE x= = ; (3) dx = − f(E) f(E) 0 E0 Ex where E0 is initial energy of the nucleus, and Ex is energy after the distance x. The known quantities in Eq. (3) are the distance x traveled and initial energy E0 . The function f(E) is known from the
N. Stevanovi-c et al. / Aerosol Science 35 (2004) 1041 – 1050
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SRIM program. The unknown quantity is energy Ex of the nuclei after traveling the distance x. It is the lower limit of integration in Eq. (3). The solution for Ex is obtained by an iterative numerical integration procedure. The range R of the 214 Pb in the aerosol material was calculated from: E0 dE R= : (4) f(E) 0 During the Monte Carlo simulation, the directions of recoil nuclei are sampled randomly up to N = 1 × 105 times and the number of detached nuclei, Ndtch was scored. In all cases the statistical calculation error is below 1%. The probability for detachment p(di ) = Ndtch =N is then determined for a given diameter p(di ) calculated for aerosols with the particle diameter from 0.001 up to 5 m. The data obtained are given in the Results section. The average detachment probability pQ (i.e. the recoil factor) was calculated from p(di ) f(di ) pQ = i ; (5) i f(di ) where, p(di ) is the detachment probability at the diameter di and f(di ) is the total size distribution. 3. Results 3.1. Surface contaminated aerosol 3.1.1. Spherical SiO2 aerosol The detachment probability was calculated for aerosol radii from 0.001 to 2:5 m with the step size of 0:001 m. The results are given in Fig. 4 (upper curve). One can observe that the detachment probability is close to 1 for small diameters and it decreases rapidly with the increasing diameter. For aerosols with the particle diameter of over 0:5 m, the detachment probability is about 0.55. By averaging according to Eqs. (1) and (5) for the parameters listed in Fig. 1, we obtained pQ = 0:697. This value is signi2cantly lower than 0.83 given by Mercer (1976). We have repeated Mercer’s analytical approach (with our data for the range and stopping power) and results are given as the scatter diagram in Fig. 4. There is a virtually complete match between the data for the detachment probability by Monte Carlo Method used here and the analytical calculations. This is an important test of our Monte Carlo program that is further used for more complex geometries. The diKerence between our and Mercer’s result (1976) is entirely due to the diKerent range for 214 Pb in SiO2 used: 0:048 m and 0.057, respectively. 3.1.2. Spherical SiO2 + H2 O aerosol Aerosols of our interest consist of a solid SiO2 kernel surrounded by water vapor. Both are assumed to be spherical (in this section of the paper). From the geometric point of view, the two concentric spheres represent the aerosol. The starting point for the recoil nuclei has been sampled on the surface of the outer sphere. The radius of SiO2 kernel was 14 of the aerosol particle radius. The detachment probability, as a function of the equivalent diameter, is given in Fig. 5 (the upper curve).
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Fig. 4. Detachment probability vs. diameter for the surface and volume contamination of the spherical SiO2 aerosol. The scatter data have been obtained by the Mercer’s analytical approach using the updated range and stopping power data reported since his work.
Fig. 5. Detachment probability vs. diameter for the surface and volume contamination of the spherical SiO2 + H2 O aerosol.
The average value of the recoil factor for spherical SiO2 + H2 O combination was found to be pQ = 0:728. This is slightly larger than for the pure SiO2 spherical aerosol, because the water density is lower than that for SiO2 ; the range of the recoil nucleus is almost twice as large in water. 3.1.3. Ellipsoidal SiO2 aerosols The detachment probability of 214 Po as a function of the equivalent diameter of the ellipsoidal aerosol particles is given in Fig. 6 (upper curve). The shape of the ellipsoid was varied by adjusting one of its major axis, A, in the range A = (0:005–2:5) m and by coupling the other two, B and C,
N. Stevanovi-c et al. / Aerosol Science 35 (2004) 1041 – 1050
1047
Fig. 6. Detachment probability vs. diameter for the surface and volume contamination of the ellipsoidal SiO2 aerosol.
by following expressions: B = (0:5A-A) m and C = (0:5B-B). The equivalent diameter de is then calculated as de = 2(ABC)1=3 :
(6)
This equivalent diameter varied in the interval de = (0:004–2:63) m. Because diKerent combinations of the ellipsoid’s axes (A; B and C) can produce the same equivalent diameter in Eq. (6), some scattering of data can be observed in Fig. 6 (upper curve). Hence, the average value of the detachment probability was calculated for the ellipsoidal aerosol: pQ = 0:748. This value is larger than that for a sphere, indicating that the shape of the aerosol particle inRuences the recoil factor. Nevertheless, it is still below 0.83 given by Mercer (1976). 3.1.4. Ellipsoidal aerosol combined of SiO2 + H2 O The results for the combined SiO2 + H2 O are given in Fig. 7 (upper curve). The starting points of the recoil nuclei were on the outer part of the aerosol particle. The water layer thickness was 0:05 m and this parameter has not been changed during the calculation. The internal SiO2 ellipsoid was varied systematically like in Section 3.1.3. The equivalent diameter was calculated from the outer ellipsoid’s parameters. When the equivalent diameter was larger than 0:5 m, the recoil factor was about 0.5. With the decreasing equivalent diameter, the recoil factor increases up to 0.8. The average value is pQ = 0:630. 3.2. Volumetric contamination of aerosols 3.2.1. Spherical SiO2 aerosol The emitting nucleus is assumed to be within the volume of the aerosol particle that grew after the attachment process. This growth may have occurred by coagulation, or simply by the joining of additional water vapor. The results are given in Fig. 4 (lower curve).
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Fig. 7. Detachment probability vs. diameter for the surface and volume contamination of the ellipsoidal SiO2 + H2 O aerosol.
The detachment probability is much lower in this case than for aerosols with only surface contamination. This is particularly important for aerosol particles with diameters larger than 0:5 m, where p(di ) is much less than 0.5. By averaging the detachment probability, the value of the recoil factor of pQ = 0:482 is obtained. 3.2.2. Spherical SiO2 + H2 O aerosols The starting points were sampled on the surface of the internal sphere. The radius of this SiO2 sphere is 14 of the outer radius. The results are given in Fig. 5 (lower curve). For aerosols with the diameter in excess of 1 m, the detachment probability was as low as 0.25. It increased as the diameter decreased. The average value was pQ = 0:398. 3.2.3. Ellipsoidal SiO2 aerosol The results are given in Fig. 6 (lower curve). As the diameter increases, the detachment probability decreases from 1 down to 0.1. The average value is pQ = 0:58. 3.2.4. Ellipsoidal SiO2 + H2 O aerosol With all other parameters remaining the same as in the previous case, the thickness of the water layer was kept constant. The results are given in Fig. 7 (lower curve). It can be observed that the detachment probability drops rapidly to 0.01 for the equivalent diameter of about 0:5 m. The average value is pQ = 0:116. The average values of the detachment probability (i.e. the recoil factor) calculated for diKerent aerosol models are summarized in Table 1.
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Table 1 The average value of the recoil factor for diKerent shape, material and contamination type of the aerosol particle Aerosol material
SiO2 SiO2 + H2 O
Spherical aerosol particle shape
Ellipsoidal aerosol particle shape
Surface contamination
Volume contamination
Surface contamination
Volume contamination
0.697 0.728
0.482 0.398
0.748 0.630
0.580 0.116
4. Conclusion Radon gas can accumulate within the interior of buildings. As its short-lived 2rst progeny 218 Po decays, the descendant nuclei of the 214 Pb undergo recoil. This may cause them to detach from the indoor aerosol that had carried the 218 Po. The average fraction of the nuclei detached in this manner is called the “recoil factor”. Traditionally presumed to be a constant between 0.8 and 0.83, this factor is an important parameter in the calculation of the indoor radon-progeny concentration. Based on more recent advancements reported in the literature, we recalculated it in this paper by Monte Carlo simulation for eight combinations of three aerosol features: with spherical or ellipsoidal particle shape, consisting of either pure SiO2 or mixed with water vapor, and including a surface or volume contamination by the radon progeny. We found that the recoil factor depends strongly on all three of these features in addition to the size of the aerosol particles. The values calculated can be much smaller than those published by Mercer (1976). It is emphasized that Mercer used a diKerent expression for the stopping power and range, and considered only the spherical SiO2 aerosols. Under those speci2c conditions, our calculations con2rm his 2ndings (the value of 0.83). However, since we showed a strong dependence of the recoil factor on the shape, material, and contamination type of the aerosol particles, we cannot recommend a single value for it. Its range is between approximately 0.1 and 0.8. It must therefore be taken as a variable in the Jacobi’s predictive model for the radon progeny concentration and the related lung-dose calculations. The range of values that the recoil factor can take broadens with the hygroscopic growth of the aerosol particles. Acknowledgements The present research was supported by the Serbian Ministry of Science Technology and Development through the project No. 1425. References Fleischer, R. L. (1997). Radon: overview of properties, origin and transport. In S. A. Durrani, & R. Ili"c (Eds.), Radon measurements by etched track detectors—Application in radiation protection earth sciences and the environment (pp. 3–20). Singapore: World Scienti2c.
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Hinds, W. C. (1998). Aerosol technology, properties, behavior and measurements of airborne particles. New York: Wiley. D Ili"c, R., & Sutej, T. (1997). Radon monitoring devices based on etched track detectors. In S. A. Durrani, & R. Ili"c (Eds.), Radon measurements by etched track detectors —Application in radiation protection earth sciences and the environment (pp. 103–128). Singapore: World Scienti2c. Jacobi, W. (1972). Activity and potential energy of 222 Rn and 220 Rn daughters in diKerent air atmosphere. Health Physics, 22, 441–450. Marsh, J. W., & Birchall, A. (2000). Sensitivity analysis of the weighted equivalent lung dose per unit exposure from radon progeny. Radiation Protection Dosimetry, 87, 167–178. Marsh, W., Birchall, A., Butterweck, G., Dorrian, M.-D., Huet, C., Ortega, X., Reineking, A., Tymen, G., Schuler, Ch., Vargas, A., Vessu, G., & Wendt, J. (2002). Uncertainty analysis of the weighted equivalent lung dose per unit exposure to radon progeny in the home. Radiation Protection Dosimetry, 102, 229–248. Mercer, T. T. (1976). The eKect of particle size on the escape of recoiling RaB atoms from particulate surfaces. Health Physics, 31, 173–174. PorstendEorfer, J. (1984). Behaviour of radon daughter products in indoor air. Radiation Protection Dosimetry, 7, 107–113. PorstendEorfer, J., & Reineking, A. (2000). Radon characteristics related to dose for di9erent living places of the human. Proceedings of the IRPA10, T-9-1, Hiroshima, Japan. Reineking, A., Knutson, E. A., George, A. C., Solomon, S. B., Kesten, J., Butterweck, G., & PorstendEorfer, J. (1994). Size distribution of unattached and aerosol-attached short-lived radon decay products: Some results of inter-comparison measurements. Radiation Protection Dosimetry, 56(1–4), 113–118. Reist, P. C. (1993). Aerosol Science and Technology (2nd ed.). New York: McGraw-Hill. Zeigler, J. F. (2003). The stopping and range of ions in matter. Available on www.srim.org.
The recoil factor of
214
Pb
N. Stevanovi"ca , D. Nikezi"ca;∗ , A. Djordjevichb a
Physics Faculty of Science, University of Kragujevac, R. Domanovic 12, 34000 Kragujevac Serbia and Montenegro b City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong Received 19 November 2003; received in revised form 16 February 2004; accepted 17 February 2004
Abstract The majority of the 218 Po atoms (the 2rst progeny of radon) are attached to natural aerosols in the air. After their alpha decay, the newly formed nuclei of 214 Pb may detach from the carrier aerosol due to recoil. The average fraction of the nuclei detached in this manner is called the “recoil factor”. It is an important parameter in the calculation of the radon-progeny concentration and the related radiation dose in lungs. We determine this parameter for eight common aerosol types and show that it cannot be treated as a 2xed constant since it varies strongly with the aerosol particle size, shape, material, and surface/volume type of contamination. The range of values that the recoil factor can take broadens with the hygroscopic growth of the aerosol particles, to the maximum range between approximately 0.1 and 0.8. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Aerosol; Radon progeny;
214
Pb; Recoil
1. Introduction Radium (226 Ra) is present in the materials used for the construction of most buildings and the terrain underneath. A progeny of its radioactive decay is radon (222 Rn). This chemically inert gas may migrate gradually to the buildings’ interior. Its accumulation there represents a signi2cant radiation hazard. A short-lived progeny of its decay is polonium (218 Po). Some of the 218 Po may remain as a free or unattached fraction that is in the atomic form or bound to small species of molecular dimensions. However, most of it is likely to attach to natural aerosols normally present in the air, giving rise to the “attached fraction” of radon progeny. The rather complex behavior of both fractions contributes to the uncertainty of measurement of the total indoor radon-progeny concentration ∗
Corresponding author. Tel.: +381-34-336223; fax: +381-34-335040. E-mail address: [email protected] (D. Nikezi"c).
0021-8502/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jaerosci.2004.02.006
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D (Ili"c & Sutej, 1997). This concentration depends on several factors including the radon source strength, the ventilation rate in the environment, the aerosol concentration and its size distribution, the progeny’s rate of attachment to the aerosol particles, and these particles’ deposition rates (PorstendEorfer, 1984; Fleischer, 1997; Jacobi, 1972). A model that accounts for these factors and calculates the concentration of the indoor radon progeny has been reported (Jacobi, 1972). The 2ne-tuning of a parameter in this model known as the “recoil factor” is the objective of this paper. Attached to an aerosol particle, the 218 Po radon progeny decays by the emission of an alpha particle. The remaining nucleus of lead (214 Pb) undergoes recoil with energy of 117 keV. This energy is suHcient to detach the 214 Po nucleus from the aerosol particle on which 218 Po had been attached. The average fraction of detached nuclei is called the recoil factor, denoted by p. Mercer (1976) has estimated it at p ≈ 0:8. This value, assumed to be a constant between 0.8 and 0.83, has since been used in virtually all reported calculations. Research reported subsequent to Mercer’s work has provided new information about the stopping power of nuclei in diKerent media (Zeigler, 2003) and their size distribution (Reineking et al., 1994). Based on such additional insight, we re-evaluate the recoil factor in this paper. Our motivation to do that stems from this factor’s signi2cance in the calculation of the indoor radon-progeny concentration (Jacobi, 1972). The behavior and characteristics of aerosols depend strongly on their particle size. The diameter d is used as a measure of that size when a spherical shape may be assumed. An equivalent diameter de = (6V=)1=3 is evaluated for irregularly shaped aerosol particles with a volume of V (Hinds, 1998). Typically, the particle size varies from 0.001 to 100 m according to the log-normal distribution. It was thought previously that the aerosol distribution had only one maximum. Such single-mode distribution was assumed by Mercer (1976) when calculating the recoil factor. However, a second maximum has been detected in the meantime (PorstendEorfer & Reineking, 2000). The corresponding distribution would therefore have to be more than just a single-mode distribution. Detailed measurements have actually identi2ed three distribution modes. They were referred to as the nucleation, accumulation and course modes (Reineking et al., 1994). Each had its own diameter AMTDi (median of aerosol thermodynamic diameter) and geometric standard deviation ( gi ) (i = 1; 2; 3). The notion of there being three modal distributions of attached radon progeny has been accepted widely. For example, the dose conversion factor has been calculated on that basis (Marsh & Birchall, 2000; Marsh et al., 2002). We too adopt the three-mode distribution concept in our calculations in this work. The three-mode distribution of the attached progeny is described by the following equation (PorstendEorfer & Reineking, 2000): f(d) =
3 i=1
2
e−(ln d−ln AMTDi ) =2 ln √ fi 2 ln( gi ) d
2
( gi )
;
(1)
where fi is a fraction of the mode i; i = 1 for the nucleation mode, i = 2 for the accumulation mode and i = 3 for the coarse mode. Numerical values in Eq. (1) recommended by Marsh and Birchall (2000) in their “sensitivity analysis” are: AMTD1 = 0:05 m, g1 = 2:0 f1 = 0:28; AMTD2 = 0:25 m g2 = 2:0 f2 = 0:70; AMTD3 = 1:50 m, g3 = 1:5 f3 = 0:02. The corresponding three modes and the cumulative distribution curve are shown in Fig. 1.
N. Stevanovi-c et al. / Aerosol Science 35 (2004) 1041 – 1050
1043
10 AMTD 1 =0.05 σg1 =2.0 f1 =0.28 AMTD 2 =0.25 σg2 =2.0 f2 =0.70 AMTD 3 =1.50 σg3 =1.5 f3 =0.02
DISTRIBUTIONS
1
0.1
0.01
0.001 0.001
0.01
0.1
1
10
Equivalent diameter (µm)
Fig. 1. Radioactive aerosol distribution: solid line is the total distribution; dotted line is the nucleation mode (subscript i = 1); dashed line is the accumulation mode (subscript i = 2); and the dashed-dotted line is the coarse mode (subscript i = 3).
2. Aerosol model and the calculation method Liquid aerosol particles are almost always spherical. While the solid ones may have more complex shapes, these are usually approximated as spherical in the literature on aerosol research (Reist, 1993). Atmospheric aerosols with solid particles may combine with droplets of water vapor. Spherical and ellipsoidal shapes of the resulting aerosol particles, consisting either of pure silicon dioxide (SiO2 ) or its mixture with vapor (SiO2 + H2 O), are analyzed in this paper. They are illustrated in Fig. 2 together with the atoms of radon progeny that are either attached to the surface or distributed within the volume of the aerosol particle, representing the surface or volume contamination, respectively. DiKerent outcomes are possible as the 214 Pb attached to the aerosol particle undergoes recoil. Five of them are shown in Fig. 3 where the thick and thin arrows represent velocity directions for the recoil nucleus and alpha particle, respectively. The 218 Po is assumed to be on the surface of the aerosol particle in the cases labeled 1, 2 and 3. In case 1, an alpha particle is emitted toward the aerosol particle and the nucleus therefore recoils out of it and detaches. The opposite is assumed in case 2 where the nucleus does not detach as it does not have suHcient energy to pass across much of the aerosol particle that stops it. The detachment occurs in case 3 where the 214 Pb nucleus cannot be stopped. For surface contamination, the recoil factor is not less than 0.5. It is assumed in cases 4 and 5 in Fig. 3 that the 218 Po decays inside the volume of the aerosol. This can happen if the aerosol growth occurs subsequent to the attachment of the 218 Po on it. The recoil factor can be less than 0.5 if the diameter of the aerosol particle is large enough. Depending on the recoil direction and range, the 214 Pb nucleus may detach from (case 5), or get trapped within (case 4), the aerosol particle. The ranges of the recoil nuclei 214 Pb in aerosol material (water and SiO2 ) were calculated by SRIM2003 code (Zeigler, 2003). SRIM program calculates the stopping power as a function of
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Fig. 2. Aerosol models considered in this work: spherical SiO2 (A); spherical SiO2 kernel surrounded by H2 O (B); ellipsoidal SiO2 (C); ellipsoidal SiO2 kernel surrounded by H2 O (D). The points on the surface or within the volume represent the surface/volume type of contamination by 214 Pb. (R stands for radius).
Fig. 3. Cases of 214 Pb recoil analyzed in this work. Thin arrows represent alpha particles’ velocity directions and thick ones those of recoil nuclei.
energy in the form of a table for the particle energy larger than 10 keV. Ranges of recoil nuclei 214 Pb are found as Rq = 0:048 m in SiO2 , and Rw = 0:093 m in water. The directions of recoil nuclei were generated by the Monte Carlo method. While the geometric details of the simulation are omitted, the issues of physics pertaining to the energy and range calculation are summarized. The energy of recoil nucleus after traveling a distance x in some material was calculated from the following expression for the stopping power of the recoil nucleus: dE (2) f(E) = − ; dx where the function f(E) was determined by curve 2tting the SRIM2003 data and extrapolating it linearly below 10 keV. Then, the distance x is obtained by integration: x E0 Ex dE dE x= = ; (3) dx = − f(E) f(E) 0 E0 Ex where E0 is initial energy of the nucleus, and Ex is energy after the distance x. The known quantities in Eq. (3) are the distance x traveled and initial energy E0 . The function f(E) is known from the
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SRIM program. The unknown quantity is energy Ex of the nuclei after traveling the distance x. It is the lower limit of integration in Eq. (3). The solution for Ex is obtained by an iterative numerical integration procedure. The range R of the 214 Pb in the aerosol material was calculated from: E0 dE R= : (4) f(E) 0 During the Monte Carlo simulation, the directions of recoil nuclei are sampled randomly up to N = 1 × 105 times and the number of detached nuclei, Ndtch was scored. In all cases the statistical calculation error is below 1%. The probability for detachment p(di ) = Ndtch =N is then determined for a given diameter p(di ) calculated for aerosols with the particle diameter from 0.001 up to 5 m. The data obtained are given in the Results section. The average detachment probability pQ (i.e. the recoil factor) was calculated from p(di ) f(di ) pQ = i ; (5) i f(di ) where, p(di ) is the detachment probability at the diameter di and f(di ) is the total size distribution. 3. Results 3.1. Surface contaminated aerosol 3.1.1. Spherical SiO2 aerosol The detachment probability was calculated for aerosol radii from 0.001 to 2:5 m with the step size of 0:001 m. The results are given in Fig. 4 (upper curve). One can observe that the detachment probability is close to 1 for small diameters and it decreases rapidly with the increasing diameter. For aerosols with the particle diameter of over 0:5 m, the detachment probability is about 0.55. By averaging according to Eqs. (1) and (5) for the parameters listed in Fig. 1, we obtained pQ = 0:697. This value is signi2cantly lower than 0.83 given by Mercer (1976). We have repeated Mercer’s analytical approach (with our data for the range and stopping power) and results are given as the scatter diagram in Fig. 4. There is a virtually complete match between the data for the detachment probability by Monte Carlo Method used here and the analytical calculations. This is an important test of our Monte Carlo program that is further used for more complex geometries. The diKerence between our and Mercer’s result (1976) is entirely due to the diKerent range for 214 Pb in SiO2 used: 0:048 m and 0.057, respectively. 3.1.2. Spherical SiO2 + H2 O aerosol Aerosols of our interest consist of a solid SiO2 kernel surrounded by water vapor. Both are assumed to be spherical (in this section of the paper). From the geometric point of view, the two concentric spheres represent the aerosol. The starting point for the recoil nuclei has been sampled on the surface of the outer sphere. The radius of SiO2 kernel was 14 of the aerosol particle radius. The detachment probability, as a function of the equivalent diameter, is given in Fig. 5 (the upper curve).
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Fig. 4. Detachment probability vs. diameter for the surface and volume contamination of the spherical SiO2 aerosol. The scatter data have been obtained by the Mercer’s analytical approach using the updated range and stopping power data reported since his work.
Fig. 5. Detachment probability vs. diameter for the surface and volume contamination of the spherical SiO2 + H2 O aerosol.
The average value of the recoil factor for spherical SiO2 + H2 O combination was found to be pQ = 0:728. This is slightly larger than for the pure SiO2 spherical aerosol, because the water density is lower than that for SiO2 ; the range of the recoil nucleus is almost twice as large in water. 3.1.3. Ellipsoidal SiO2 aerosols The detachment probability of 214 Po as a function of the equivalent diameter of the ellipsoidal aerosol particles is given in Fig. 6 (upper curve). The shape of the ellipsoid was varied by adjusting one of its major axis, A, in the range A = (0:005–2:5) m and by coupling the other two, B and C,
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Fig. 6. Detachment probability vs. diameter for the surface and volume contamination of the ellipsoidal SiO2 aerosol.
by following expressions: B = (0:5A-A) m and C = (0:5B-B). The equivalent diameter de is then calculated as de = 2(ABC)1=3 :
(6)
This equivalent diameter varied in the interval de = (0:004–2:63) m. Because diKerent combinations of the ellipsoid’s axes (A; B and C) can produce the same equivalent diameter in Eq. (6), some scattering of data can be observed in Fig. 6 (upper curve). Hence, the average value of the detachment probability was calculated for the ellipsoidal aerosol: pQ = 0:748. This value is larger than that for a sphere, indicating that the shape of the aerosol particle inRuences the recoil factor. Nevertheless, it is still below 0.83 given by Mercer (1976). 3.1.4. Ellipsoidal aerosol combined of SiO2 + H2 O The results for the combined SiO2 + H2 O are given in Fig. 7 (upper curve). The starting points of the recoil nuclei were on the outer part of the aerosol particle. The water layer thickness was 0:05 m and this parameter has not been changed during the calculation. The internal SiO2 ellipsoid was varied systematically like in Section 3.1.3. The equivalent diameter was calculated from the outer ellipsoid’s parameters. When the equivalent diameter was larger than 0:5 m, the recoil factor was about 0.5. With the decreasing equivalent diameter, the recoil factor increases up to 0.8. The average value is pQ = 0:630. 3.2. Volumetric contamination of aerosols 3.2.1. Spherical SiO2 aerosol The emitting nucleus is assumed to be within the volume of the aerosol particle that grew after the attachment process. This growth may have occurred by coagulation, or simply by the joining of additional water vapor. The results are given in Fig. 4 (lower curve).
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Fig. 7. Detachment probability vs. diameter for the surface and volume contamination of the ellipsoidal SiO2 + H2 O aerosol.
The detachment probability is much lower in this case than for aerosols with only surface contamination. This is particularly important for aerosol particles with diameters larger than 0:5 m, where p(di ) is much less than 0.5. By averaging the detachment probability, the value of the recoil factor of pQ = 0:482 is obtained. 3.2.2. Spherical SiO2 + H2 O aerosols The starting points were sampled on the surface of the internal sphere. The radius of this SiO2 sphere is 14 of the outer radius. The results are given in Fig. 5 (lower curve). For aerosols with the diameter in excess of 1 m, the detachment probability was as low as 0.25. It increased as the diameter decreased. The average value was pQ = 0:398. 3.2.3. Ellipsoidal SiO2 aerosol The results are given in Fig. 6 (lower curve). As the diameter increases, the detachment probability decreases from 1 down to 0.1. The average value is pQ = 0:58. 3.2.4. Ellipsoidal SiO2 + H2 O aerosol With all other parameters remaining the same as in the previous case, the thickness of the water layer was kept constant. The results are given in Fig. 7 (lower curve). It can be observed that the detachment probability drops rapidly to 0.01 for the equivalent diameter of about 0:5 m. The average value is pQ = 0:116. The average values of the detachment probability (i.e. the recoil factor) calculated for diKerent aerosol models are summarized in Table 1.
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Table 1 The average value of the recoil factor for diKerent shape, material and contamination type of the aerosol particle Aerosol material
SiO2 SiO2 + H2 O
Spherical aerosol particle shape
Ellipsoidal aerosol particle shape
Surface contamination
Volume contamination
Surface contamination
Volume contamination
0.697 0.728
0.482 0.398
0.748 0.630
0.580 0.116
4. Conclusion Radon gas can accumulate within the interior of buildings. As its short-lived 2rst progeny 218 Po decays, the descendant nuclei of the 214 Pb undergo recoil. This may cause them to detach from the indoor aerosol that had carried the 218 Po. The average fraction of the nuclei detached in this manner is called the “recoil factor”. Traditionally presumed to be a constant between 0.8 and 0.83, this factor is an important parameter in the calculation of the indoor radon-progeny concentration. Based on more recent advancements reported in the literature, we recalculated it in this paper by Monte Carlo simulation for eight combinations of three aerosol features: with spherical or ellipsoidal particle shape, consisting of either pure SiO2 or mixed with water vapor, and including a surface or volume contamination by the radon progeny. We found that the recoil factor depends strongly on all three of these features in addition to the size of the aerosol particles. The values calculated can be much smaller than those published by Mercer (1976). It is emphasized that Mercer used a diKerent expression for the stopping power and range, and considered only the spherical SiO2 aerosols. Under those speci2c conditions, our calculations con2rm his 2ndings (the value of 0.83). However, since we showed a strong dependence of the recoil factor on the shape, material, and contamination type of the aerosol particles, we cannot recommend a single value for it. Its range is between approximately 0.1 and 0.8. It must therefore be taken as a variable in the Jacobi’s predictive model for the radon progeny concentration and the related lung-dose calculations. The range of values that the recoil factor can take broadens with the hygroscopic growth of the aerosol particles. Acknowledgements The present research was supported by the Serbian Ministry of Science Technology and Development through the project No. 1425. References Fleischer, R. L. (1997). Radon: overview of properties, origin and transport. In S. A. Durrani, & R. Ili"c (Eds.), Radon measurements by etched track detectors—Application in radiation protection earth sciences and the environment (pp. 3–20). Singapore: World Scienti2c.
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Hinds, W. C. (1998). Aerosol technology, properties, behavior and measurements of airborne particles. New York: Wiley. D Ili"c, R., & Sutej, T. (1997). Radon monitoring devices based on etched track detectors. In S. A. Durrani, & R. Ili"c (Eds.), Radon measurements by etched track detectors —Application in radiation protection earth sciences and the environment (pp. 103–128). Singapore: World Scienti2c. Jacobi, W. (1972). Activity and potential energy of 222 Rn and 220 Rn daughters in diKerent air atmosphere. Health Physics, 22, 441–450. Marsh, J. W., & Birchall, A. (2000). Sensitivity analysis of the weighted equivalent lung dose per unit exposure from radon progeny. Radiation Protection Dosimetry, 87, 167–178. Marsh, W., Birchall, A., Butterweck, G., Dorrian, M.-D., Huet, C., Ortega, X., Reineking, A., Tymen, G., Schuler, Ch., Vargas, A., Vessu, G., & Wendt, J. (2002). Uncertainty analysis of the weighted equivalent lung dose per unit exposure to radon progeny in the home. Radiation Protection Dosimetry, 102, 229–248. Mercer, T. T. (1976). The eKect of particle size on the escape of recoiling RaB atoms from particulate surfaces. Health Physics, 31, 173–174. PorstendEorfer, J. (1984). Behaviour of radon daughter products in indoor air. Radiation Protection Dosimetry, 7, 107–113. PorstendEorfer, J., & Reineking, A. (2000). Radon characteristics related to dose for di9erent living places of the human. Proceedings of the IRPA10, T-9-1, Hiroshima, Japan. Reineking, A., Knutson, E. A., George, A. C., Solomon, S. B., Kesten, J., Butterweck, G., & PorstendEorfer, J. (1994). Size distribution of unattached and aerosol-attached short-lived radon decay products: Some results of inter-comparison measurements. Radiation Protection Dosimetry, 56(1–4), 113–118. Reist, P. C. (1993). Aerosol Science and Technology (2nd ed.). New York: McGraw-Hill. Zeigler, J. F. (2003). The stopping and range of ions in matter. Available on www.srim.org.