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Theoretical estimates of the solid angle subtended by a dual diaphragm–detector assembly for alpha sources
ARTICLE IN PRESS
Applied Radiation and Isotopes 61 (2004) 1349–1351
Theoretical estimates of the solid angle subtended by a dual diaphragm–detector assembly for alpha sources Julio C. Aguiara, Eduardo Galianob,* b
a Departamento de Postgrado, Universidad Tecnologica Nacional, Buenos Aires, Argentina Department of Physics, Laurentian University, Ramsey Lake Rd. Sudbury, Ont., Canada P3E 2C6
Received 17 December 2003; received in revised form 17 December 2003; accepted 15 March 2004
Abstract Knowledge of the solid angle subtended by a detector is essential for the determination of the activity of a radioactive source. This work investigates the influence of diaphragm and source diameter, and source to diaphragm distance, on the solid angle subtended by a dual opposed a diaphragm–detector assembly. Expressions published by Segre, Ruby, and others, and the Monte Carlo method are used. The analytical expressions consistently produced higher estimates than the Monte Carlo method for the solid angle. r 2004 Elsevier Ltd. All rights reserved. Keywords: Radiation detector; Solid angle; Geometric efficiency; Monte Carlo method
where by definition
1. Introduction Knowledge of the solid angle subtended by a diaphragm–detector assembly is essential for the determination of the absolute activity of a radioactive source (Profio, 1976). This is especially true when working with planar circular sources and ion-implanted semiconductor detectors for which the detection efficiency is unity. In order to accurately determine the subtended solid angle, certain geometrical parameters such as the diaphragm diameter r, the source to diaphragm distance d, and the source diameter x, must all be known with an accuracy at least equal to that of the desired accuracy for the solid angle. A simplifying limiting case is that of a point source, i.e. the case where x=0. In this case the geometric efficiency eG is simply defined as the fractional solid angle subtended by the detector (Knoll, 2000) eG ¼ Oi =4p;
ð1Þ
*Corresponding author. Tel.: +1-705-675-1151; fax: +1705-675-4868. E-mail address: [email protected] (E. Galiano).
dOi ¼ dS=d 2 :
ð2Þ
dS is the differential area element subtended by the detector’s diaphragm at the detector surface. Integrating (2) we get Z 4p Z y 1 d 2 sin y dydf 2 d 0 0 0 0 n h r io ¼ 4p 1 cos arctan : 2d
O1 ¼
1 d2
Z
4p
Z
y
dS ¼
ð3Þ
This geometry is represented in Fig. 1. In practice however, all radioactive sources have finite dimensions. In this work, the influence of the geometric parameters r; d; and x; on the subtended solid angle are investigated. An early historical reference on the subject was published in 1959 by Segre, in which the following expression is derived for the subtended solid angle (Segre, 1959): (
"
O2 ¼ 4p 1 cos y0 1 þ
0969-8043/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2004.03.061
#) 3 x2 2y0 2 sin ; 4 2 d2
ð4Þ
ARTICLE IN PRESS J.C. Aguiar, E. Galiano / Applied Radiation and Isotopes 61 (2004) 1349–1351
1350
2. Materials and methods
Fig. 1. Standard geometry for a single diaphragm–detector assembly with a point source.
where
y0 ¼ arctan
A vertically oriented, dual opposed diaphragm– detector assembly has been built for possible applications in the calibration of a emitting radioactive sources. Both circular detectors are of the passivated implanted planar silicon (PIPS) type and the diaphragm for each is located at a distance d from the planar circular alpha source of diameter x. Mechanical provisions have been made which allow the independent variation of the source to diaphragm distance for each detector, but in this investigation those distances were kept equal at all times. Diaphragms of 2.0, 4.0 and 6.0 cm diameters have been built with both Al and PVC. The diaphragm– detector system is enclosed in a cylindrical vacuum chamber with a height of 30 cm and a diameter of 15 cm, in order to reduce interactions between the a particles and air molecules. The background radiation field has been measured at 7 cts/day. A Soloist Ortec module was used consisting of an EG&G Ortec model 919 Ethernim and a multichannel alpha spectrometer (EG&G Ortec, Oak Ridge, TN). The advantage of such a geometry with respect to a conventional single detector is that for a given source to diaphragm distance, the geometric efficiency is automatically doubled with an improvement in statistics. The system layout is illustrated in Fig. 2. It is important when designing a detector for absolute activity measurements to know with a high degree of
r : 2d
Previous to that, Burtt had proposed a different analytical expression for solid angle based on absolute activity measurements utilizing a diaphragm equipped, window-type Geiger–Mueller counter. According to Poksheva, this work appears to have received little attention (Burtt, 1949; Poksheva, 1965). In 1995, Ruby proposed the following expression for solid angle, which incorporates some concepts developed earlier by Burtt and Tsoulfanidis (Ruby, 1995; Tsoulfanidis, 1983)
r2 3 x 2 r 2 5 x 4 r 4 1 þ þ þ 8d 2 4 d 2d 8 d 2d x2 x 2 35x6 r 6 þ3 þ d 2d 64 d 2d x2 r 2 x2 r 2 þ : ð5Þ þ6 d 2d d 2d
O3 ¼ 4p
Based on Burtt’s and Ruby’s work, Vega investigated the applications of Monte Carlo methods for determining solid angles. In particular, he used a FORTRAN 77 code with 500,000 histories (Vega, 1996).
Fig. 2. Geometry of the dual diaphragm–detector assembly with a planar disk source enclosed in its vacuum chamber.
ARTICLE IN PRESS J.C. Aguiar, E. Galiano / Applied Radiation and Isotopes 61 (2004) 1349–1351
accuracy the detector’s subtended solid angle. The purpose of this work is the theoretical computation of the subtended solid angle by this dual diaphragm– detector system using the three different expressions given above and the Monte Carlo method. A VISUAL FORTRAN 6.0 Monte Carlo code based on earlier work by Wielopolski was written to compute the subtended solid angle of the system (Wielopolski, 1977). For each calculation 500,000 individual histories were used. Preliminary and limited measurements with NISTtraceable planar a sources were conducted in order to gain a better understanding of the operational parameters of the diaphragm–detector system. These sources were mounted on a thin film of organic material in the form of a copolymer of polyvinyl chloride and polyvinyl acetate called VYNS, following the method outlined by Lally (1984). These measurements indicate that VYNS mounting does not significantly alter the quality or intensity of a emissions. When completed, the results of these measurements will be presented elsewhere.
1351
Table 1 The solid angles subtended by the dual diaphragm–detector assembly for different geometric configurations as estimated by the different methods r (cm)
x (cm)
d (cm)
O1 (sr)
O2 (sr)
O3 (sr)
O4 (sr)
4 4 4 4 4 2 2 2 2 2 6 6 6 6 6
1.5 1.5 1.5 1.5 1.5 0.7 0.7 0.7 0.7 0.7 0.4 0.4 0.4 0.4 0.4
3.31 8.2 12.1 15.0 20.0 3.31 8.2 12.1 15.0 20.0 3.31 8.2 12.1 15.0 20.0
3.6218 0.7158 0.3364 0.2204 0.1248 1.074 0.1848 0.0854 0.0556 0.0314 6.5106 1.5300 0.7386 0.4880 0.2780
4.1672 0.7198 0.3368 0.2206 0.1248 1.0858 0.1848 0.0854 0.0556 0.0314 6.6210 1.5312 0.7388 0.4882 0.2780
3.4298 0.6990 0.3326 0.2188 0.1246 1.0416 0.1838 0.0852 0.0556 0.0314 7.7650 1.5270 0.7378 0.4878 0.2780
3.6008 0.7054 0.3314 0.2178 0.1238 1.0572 0.1814 0.0838 0.0550 0.0310 6.4992 1.5200 0.7328 0.4850 0.2778
on these results, the calculated geometric efficiencies ranged from 0.0024 to 0.618. 3. Results and discussion The results of calculations for distinct configurations by the different methods are presented in Table 1. Note that O1 is insensitive to the source diameter since x does not appear in Eq. (3). Eqs. (3)–(5), produce differing results, these differences becoming more important when r/d>1. Based on preliminary and limited measurements with calibrated sources, it appears that in this regime the Monte Carlo method produces the most accurate results—and is therefore to be preferred over the other methods. However we find that for the condition r/do1, the most reasonable results are obtained by taking an average of Eqs. (3)–(5). For given values of r and x, the agreement between the methods—including the Monte Carlo method— increases with increasing d. The reduced spread in the results of O1 ; O2 ; and O3 ; with increasing d may be partially explained by the diminishing value of the term r/2d which appears in all three expressions. It is not clear why the Monte Carlo method would also be in better agreement with the other three methods with increasing d: However, with respect to the Monte Carlo method, the other three methods generally produced higher estimates for the subtended solid angle. Settling whether the Monte Carlo method underestimates the solid angle—or the other methods overestimate it—will have to await the results of a more complete set of measurements. Finally—as expected from simple geometric considerations—the largest solid angles are obtained using the smallest values of d and the largest values of r. Conversely, the smallest solid angles are obtained using the largest values of d and the smallest values of r. Based
Acknowledgements The authors wish to acknowledge the assistance of the Laboratorio de Metrologia-Centro Atomico Ezeiza, and of Mr. Pablo Arenillas. References Burtt, B.J., 1949. Absolute beta counting. Nucleonics 5, 28. Knoll, G.F., 2000. Radiation Detection and Measurement, 3rd Edition. Wiley, New York, pp. 116–118. Lally, A.E., Glover, K.M., 1984. Source preparation in alpha spectrometry. Nucl. Instrum. Methods Phys. Res. 223, 259– 265. Poksheva, J.G., 1965. Lawrence Radiation Laboratory, Report UCRL-16061, pp. 100–137. Profio, A.E., 1976. Experimental Reactor Physics. Wiley, New York, pp. 137–138. Ruby, L., 1995. Further comments on the geometrical efficiency of a parallel disk-source and detector system. Nucl. Instrum. Methods Phys. Res. A 337, 531–533. Segre, E., 1959. Experimental Nuclear Physics, Vol III. John Wiley & Sons, New York, pp. 435–438. Tsoulfanidis, N., 1983. The defined solid angle method. In: Measurement and Detection of Radiation. Hemisphere Publishers Inc., Washington, DC. Vega, H., 1996. Geometrical efficiency for a parallel disk source and detector. Nucl. Instrum. Methods Phys. Res. A 371, 535–537. Wielopolski, L., 1977. The Monte Carlo calculation of the average solid angle subtended by a right circular cylinder by distributed sources. Nucl. Instrum. Methods Phys. Res. 143, 577–581.
Applied Radiation and Isotopes 61 (2004) 1349–1351
Theoretical estimates of the solid angle subtended by a dual diaphragm–detector assembly for alpha sources Julio C. Aguiara, Eduardo Galianob,* b
a Departamento de Postgrado, Universidad Tecnologica Nacional, Buenos Aires, Argentina Department of Physics, Laurentian University, Ramsey Lake Rd. Sudbury, Ont., Canada P3E 2C6
Received 17 December 2003; received in revised form 17 December 2003; accepted 15 March 2004
Abstract Knowledge of the solid angle subtended by a detector is essential for the determination of the activity of a radioactive source. This work investigates the influence of diaphragm and source diameter, and source to diaphragm distance, on the solid angle subtended by a dual opposed a diaphragm–detector assembly. Expressions published by Segre, Ruby, and others, and the Monte Carlo method are used. The analytical expressions consistently produced higher estimates than the Monte Carlo method for the solid angle. r 2004 Elsevier Ltd. All rights reserved. Keywords: Radiation detector; Solid angle; Geometric efficiency; Monte Carlo method
where by definition
1. Introduction Knowledge of the solid angle subtended by a diaphragm–detector assembly is essential for the determination of the absolute activity of a radioactive source (Profio, 1976). This is especially true when working with planar circular sources and ion-implanted semiconductor detectors for which the detection efficiency is unity. In order to accurately determine the subtended solid angle, certain geometrical parameters such as the diaphragm diameter r, the source to diaphragm distance d, and the source diameter x, must all be known with an accuracy at least equal to that of the desired accuracy for the solid angle. A simplifying limiting case is that of a point source, i.e. the case where x=0. In this case the geometric efficiency eG is simply defined as the fractional solid angle subtended by the detector (Knoll, 2000) eG ¼ Oi =4p;
ð1Þ
*Corresponding author. Tel.: +1-705-675-1151; fax: +1705-675-4868. E-mail address: [email protected] (E. Galiano).
dOi ¼ dS=d 2 :
ð2Þ
dS is the differential area element subtended by the detector’s diaphragm at the detector surface. Integrating (2) we get Z 4p Z y 1 d 2 sin y dydf 2 d 0 0 0 0 n h r io ¼ 4p 1 cos arctan : 2d
O1 ¼
1 d2
Z
4p
Z
y
dS ¼
ð3Þ
This geometry is represented in Fig. 1. In practice however, all radioactive sources have finite dimensions. In this work, the influence of the geometric parameters r; d; and x; on the subtended solid angle are investigated. An early historical reference on the subject was published in 1959 by Segre, in which the following expression is derived for the subtended solid angle (Segre, 1959): (
"
O2 ¼ 4p 1 cos y0 1 þ
0969-8043/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2004.03.061
#) 3 x2 2y0 2 sin ; 4 2 d2
ð4Þ
ARTICLE IN PRESS J.C. Aguiar, E. Galiano / Applied Radiation and Isotopes 61 (2004) 1349–1351
1350
2. Materials and methods
Fig. 1. Standard geometry for a single diaphragm–detector assembly with a point source.
where
y0 ¼ arctan
A vertically oriented, dual opposed diaphragm– detector assembly has been built for possible applications in the calibration of a emitting radioactive sources. Both circular detectors are of the passivated implanted planar silicon (PIPS) type and the diaphragm for each is located at a distance d from the planar circular alpha source of diameter x. Mechanical provisions have been made which allow the independent variation of the source to diaphragm distance for each detector, but in this investigation those distances were kept equal at all times. Diaphragms of 2.0, 4.0 and 6.0 cm diameters have been built with both Al and PVC. The diaphragm– detector system is enclosed in a cylindrical vacuum chamber with a height of 30 cm and a diameter of 15 cm, in order to reduce interactions between the a particles and air molecules. The background radiation field has been measured at 7 cts/day. A Soloist Ortec module was used consisting of an EG&G Ortec model 919 Ethernim and a multichannel alpha spectrometer (EG&G Ortec, Oak Ridge, TN). The advantage of such a geometry with respect to a conventional single detector is that for a given source to diaphragm distance, the geometric efficiency is automatically doubled with an improvement in statistics. The system layout is illustrated in Fig. 2. It is important when designing a detector for absolute activity measurements to know with a high degree of
r : 2d
Previous to that, Burtt had proposed a different analytical expression for solid angle based on absolute activity measurements utilizing a diaphragm equipped, window-type Geiger–Mueller counter. According to Poksheva, this work appears to have received little attention (Burtt, 1949; Poksheva, 1965). In 1995, Ruby proposed the following expression for solid angle, which incorporates some concepts developed earlier by Burtt and Tsoulfanidis (Ruby, 1995; Tsoulfanidis, 1983)
r2 3 x 2 r 2 5 x 4 r 4 1 þ þ þ 8d 2 4 d 2d 8 d 2d x2 x 2 35x6 r 6 þ3 þ d 2d 64 d 2d x2 r 2 x2 r 2 þ : ð5Þ þ6 d 2d d 2d
O3 ¼ 4p
Based on Burtt’s and Ruby’s work, Vega investigated the applications of Monte Carlo methods for determining solid angles. In particular, he used a FORTRAN 77 code with 500,000 histories (Vega, 1996).
Fig. 2. Geometry of the dual diaphragm–detector assembly with a planar disk source enclosed in its vacuum chamber.
ARTICLE IN PRESS J.C. Aguiar, E. Galiano / Applied Radiation and Isotopes 61 (2004) 1349–1351
accuracy the detector’s subtended solid angle. The purpose of this work is the theoretical computation of the subtended solid angle by this dual diaphragm– detector system using the three different expressions given above and the Monte Carlo method. A VISUAL FORTRAN 6.0 Monte Carlo code based on earlier work by Wielopolski was written to compute the subtended solid angle of the system (Wielopolski, 1977). For each calculation 500,000 individual histories were used. Preliminary and limited measurements with NISTtraceable planar a sources were conducted in order to gain a better understanding of the operational parameters of the diaphragm–detector system. These sources were mounted on a thin film of organic material in the form of a copolymer of polyvinyl chloride and polyvinyl acetate called VYNS, following the method outlined by Lally (1984). These measurements indicate that VYNS mounting does not significantly alter the quality or intensity of a emissions. When completed, the results of these measurements will be presented elsewhere.
1351
Table 1 The solid angles subtended by the dual diaphragm–detector assembly for different geometric configurations as estimated by the different methods r (cm)
x (cm)
d (cm)
O1 (sr)
O2 (sr)
O3 (sr)
O4 (sr)
4 4 4 4 4 2 2 2 2 2 6 6 6 6 6
1.5 1.5 1.5 1.5 1.5 0.7 0.7 0.7 0.7 0.7 0.4 0.4 0.4 0.4 0.4
3.31 8.2 12.1 15.0 20.0 3.31 8.2 12.1 15.0 20.0 3.31 8.2 12.1 15.0 20.0
3.6218 0.7158 0.3364 0.2204 0.1248 1.074 0.1848 0.0854 0.0556 0.0314 6.5106 1.5300 0.7386 0.4880 0.2780
4.1672 0.7198 0.3368 0.2206 0.1248 1.0858 0.1848 0.0854 0.0556 0.0314 6.6210 1.5312 0.7388 0.4882 0.2780
3.4298 0.6990 0.3326 0.2188 0.1246 1.0416 0.1838 0.0852 0.0556 0.0314 7.7650 1.5270 0.7378 0.4878 0.2780
3.6008 0.7054 0.3314 0.2178 0.1238 1.0572 0.1814 0.0838 0.0550 0.0310 6.4992 1.5200 0.7328 0.4850 0.2778
on these results, the calculated geometric efficiencies ranged from 0.0024 to 0.618. 3. Results and discussion The results of calculations for distinct configurations by the different methods are presented in Table 1. Note that O1 is insensitive to the source diameter since x does not appear in Eq. (3). Eqs. (3)–(5), produce differing results, these differences becoming more important when r/d>1. Based on preliminary and limited measurements with calibrated sources, it appears that in this regime the Monte Carlo method produces the most accurate results—and is therefore to be preferred over the other methods. However we find that for the condition r/do1, the most reasonable results are obtained by taking an average of Eqs. (3)–(5). For given values of r and x, the agreement between the methods—including the Monte Carlo method— increases with increasing d. The reduced spread in the results of O1 ; O2 ; and O3 ; with increasing d may be partially explained by the diminishing value of the term r/2d which appears in all three expressions. It is not clear why the Monte Carlo method would also be in better agreement with the other three methods with increasing d: However, with respect to the Monte Carlo method, the other three methods generally produced higher estimates for the subtended solid angle. Settling whether the Monte Carlo method underestimates the solid angle—or the other methods overestimate it—will have to await the results of a more complete set of measurements. Finally—as expected from simple geometric considerations—the largest solid angles are obtained using the smallest values of d and the largest values of r. Conversely, the smallest solid angles are obtained using the largest values of d and the smallest values of r. Based
Acknowledgements The authors wish to acknowledge the assistance of the Laboratorio de Metrologia-Centro Atomico Ezeiza, and of Mr. Pablo Arenillas. References Burtt, B.J., 1949. Absolute beta counting. Nucleonics 5, 28. Knoll, G.F., 2000. Radiation Detection and Measurement, 3rd Edition. Wiley, New York, pp. 116–118. Lally, A.E., Glover, K.M., 1984. Source preparation in alpha spectrometry. Nucl. Instrum. Methods Phys. Res. 223, 259– 265. Poksheva, J.G., 1965. Lawrence Radiation Laboratory, Report UCRL-16061, pp. 100–137. Profio, A.E., 1976. Experimental Reactor Physics. Wiley, New York, pp. 137–138. Ruby, L., 1995. Further comments on the geometrical efficiency of a parallel disk-source and detector system. Nucl. Instrum. Methods Phys. Res. A 337, 531–533. Segre, E., 1959. Experimental Nuclear Physics, Vol III. John Wiley & Sons, New York, pp. 435–438. Tsoulfanidis, N., 1983. The defined solid angle method. In: Measurement and Detection of Radiation. Hemisphere Publishers Inc., Washington, DC. Vega, H., 1996. Geometrical efficiency for a parallel disk source and detector. Nucl. Instrum. Methods Phys. Res. A 371, 535–537. Wielopolski, L., 1977. The Monte Carlo calculation of the average solid angle subtended by a right circular cylinder by distributed sources. Nucl. Instrum. Methods Phys. Res. 143, 577–581.