ARTICLE IN PRESS Applied Radiation and Isotopes 64 (2006) 1048–1056 www.elsevier.com/locate/apradiso Analytical calculations of the solid angles subtended by a well-type detector at point and extended circular sources Mahmoud I. Abbas Physics Department, Faculty of Science, Alexandria University, 21121 Alexandria, Egypt Received 8 March 2006; received in revised form 15 April 2006; accepted 28 April 2006 Abstract Knowledge of the solid angle (and consequently, the geometrical efficiency) is essential in all absolute measurements of the strengths of radioactive materials and to calibrate detectors. The method of high-efficiency g counting by means of well-type HPGe and NaI (Tl) detectors is widely used and has proved a powerful tool, particularly when low-activity, small-volume environmental samples are to be analyzed by g-ray spectrometry. In the present work, we introduce a direct analytical method for calculating the solid angle subtended by a well-type detector at axial point, non-axial point, extended circular disk and cylindrical sources. The validity of the derived analytical expressions was successfully confirmed by the comparisons with some published data (experimental and Monte Carlo). r 2006 Elsevier Ltd. All rights reserved. Keywords: Solid angle; Geometrical efficiency; Point; Circular disk and cylindrical sources; Well-type detector 1. Introduction The solid angle is widely used during absolute methods to calibrate detectors or to determine (using absolute methods) the activity of a radioactive source. Several efforts have been reported previously to deal with treatments of the efficiencies (total eT, photopeak ep and geometrical eg ¼ O/4p; O is the solid angle subtended by the detector at the source point) of right circular cylindrical detectors for point, circular disk and volumetric sources (Grosswendt and Waibel, 1976; Wielopolski, 1977; Noguchi et al., 1981; Tsoulfanidis, 1983; Wicham, 1991; Ruby, 1995; Vega, 1996; Jiang et al., 1998; Selim and Abbas, 1994, 1995, 2000; Selim et al., 1998; Abbas, 1995, 2001a; Aguiar and Galiano, 2004; Pommé, 2004. Also, Burtt (1949) has presented an expression to calculate the geometrical efficiency; this expression is valid for a wide selection of the parameters except for large sources close to the detector. In addition, Prata (2003, 2004a,b) derived analytical expressions for the solid angle subtended by a cylindrical detector at a point source, a circular disk Tel.: +20 3 5853282; fax: +20 3 3911794. E-mail address: mabbas@physicist.net. 0969-8043/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2006.04.010 detector at a point cosine source and a cylindrical detector at a point cosine source with parallel axes, respectively. Recently, Galiano and Rodrigues (2006) presented seven different analytical expressions for the solid angle subtended by a circular detector for a co-axial circular source. These expressions have been published by different investigators over the course of the last half century. Furthermore, the geometrical efficiency of a parallelepiped detector for an arbitrarily positioned point source has been calculated by using an efficient Monte Carlo algorithm, (Wielopolski, 1984) and a direct analytical expression (Abbas, 1995, 2001b). Finally, the method of highefficiency g counting by means of well-type HPGe and NaI (Tl) detectors is widely used and has proved a powerful tool, particularly when low-activity, smallvolume environmental samples are to be analyzed by g-ray spectrometry. Treatment of the detection efficiencies (total and full-energy peak efficiencies) of well-type detectors has been given in previous works (Abbas, 2001c; Abbas and Selim, 2002). In this paper, we present a direct mathematical method to calculate the solid angle subtended by a well-type detector at point and extended circular sources. The arrangement of this paper is as follows. Section 2 presents direct mathematical formulae ARTICLE IN PRESS 1049 M.I. Abbas / Applied Radiation and Isotopes 64 (2006) 1048–1056 for the solid angle, and consequently the geometrical efficiency, in four different cases (axial point, non-axial point, extended circular disk and cylindrical sources). Section 3 contains comparisons between the calculated solid angle using the formulae derived in this work with the published values illustrating the validity of the present mathematical formulae. Conclusions are presented in Section 4. Ro Ri ( − π ) 2 P(0, h)
h K 2. Mathematical viewpoint The solid angle subtended by a surface detector at an isotropic radiating point source can be defined by Z Z O¼ sin y df d y; (1) y L f where (y) and (f) are the polar and the azimuthal angles, respectively. The work described below involves the use of straightforward analytical formulae for the computation of the solid angle subtended by a well-type detector at axial point, non-axial point, extended circular disk and cylindrical sources. For each source we have two cases for the calculation of the solid angle to be considered. The first one to calculate the solid angle when the isotropic radiating source lies inside the detector well and the second case to calculate the solid angle when the isotropic radiating source lies outside the detector well. P (0, h)  h (i) The case of an isotropic radiating axial point source P(0, h) and a well-type detector (Fig. 1). The polar angle (y) takes the values b ¼ tan
1 Ri h
K and
ðK
hÞ cos b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðK
hÞ2 þ R2i K (2) L g ¼ tan
1 Ro h
K and ðh
KÞ cos g ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ðh
KÞ2 þ R2o (3) The geometrical notations of K, h, Ri and Ro are as shown in Fig. 1. The azimuthal angle (f) takes always the value 2p for all the values of the polar angle (y). Taking these situations into consideration, the final expression of the solid angle subtended by a well-type detector at an axial point source is given by (a) the axial point source lies inside the detector well: Z bZ p Oaxial sin y df dy ¼ 2pð1
cos bÞ. (4) ¼ 2 in 0 0 (b) the axial point source lies outside the detector well: Z gZ p Oaxial sin y df dy ¼ 2pð1
cos gÞ. (5) ¼ 2 out 0 0 0 Fig. 1. Axial point source–detector configuration. (ii) The case of an arbitrarily positioned isotropic radiating point source P(r,h) and a well-type detector. The quantities (r,h) specify the location of the non-axial point source (Fig. 2). The polar angle (y) takes the steps a ¼ tan
1 Ri  r h h cos a ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , h2 þ ðRi  rÞ2 (6a) ARTICLE IN PRESS 1050 M.I. Abbas / Applied Radiation and Isotopes 64 (2006) 1048–1056 + (β − π ðbÞ Ro Ri P (, h) (-− ) 2 f ¼ p for aþ pypbþ ðcÞ f ¼ 2fmax ðh; Ri Þ for a
pypaþ , π ) 2 ðdÞ + K r2
y2 þ x2 tan2 y . 2xr tany ¼ 2p Z a
sin y dy 0 aþ Z þ2 - þp a
Z bþ fmax ðh; Ri Þ sin y dy sin y dy þ p aþ K (8) Considering the previous cases we ultimately get the final expression of the solid angle subtended by a welltype detector at a non-axial point source as (a) the non-axial point source lies inside the detector well: non
axial Oin P (, h) (7d) where f(x,y) is the maximum azimuthal angle and is defined as fmax ðx; yÞ ¼ cos
1 L (7b) (7c) f ¼ 2fmax ððh
KÞ; R0 Þ for bþ pypgþ , - + a
pypb
, and Z b
sin y dy; ð9aÞ a
 1 non
axial Oin ¼ 2p 1 þ cos aþ
cos a
2  
cos bþ
cos b
Z aþ þ2 fmax ðh; Ri Þ sin y dy. ð9bÞ a
L (b) the non-axial point source lies outside the detector well: Z non
axial Oout ¼ Oinnon
axial þ 2 0
Fig. 2. Non-axial point source–detector configuration. Ri  r b ¼ tan
1 h
K
ðK
hÞ cos b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðK
hÞ2 þ ðRi  rÞ2 (6b) Ro  r . (6c) h
K For a certain value of the polar angle (y), the azimuthal angle (f) takes the values g ¼ tan
1 ðaÞ f ¼ 2p for 0pypa
, (7a) Z b
b
fmax ððh
KÞ; Ro Þ sin y dy ! bþ fmax ððh
KÞ; Ro Þ sin y dy , Onon
axial ¼ Onon
axial þ2 out in Onon
axial in gþ Z ð10aÞ gþ bþ fmax ððh
KÞ; Ro Þ sin y dy, (10b) where is the maximum azimuthal angle and is defined asis as identified before in Eq. (9b). The integral in Eqs. (9b) and (10b) are elliptic integrals and does not have a closed solution. Then a numerical solution is obtained using the Simpson’s rule. Although the accuracy of the integration increases with increasing the number of intervals n, the integration converges well at n ¼ 10. The numerical ARTICLE IN PRESS 1051 M.I. Abbas / Applied Radiation and Isotopes 64 (2006) 1048–1056 values of the elliptic integral in Eq. (10b) are always less than zero (i.e. –ve). (iii) The case of an isotropic radiating co-axial circular disk source and a well-type detector (Fig. 3), the solid angle is given by Ro (a) the disk source lies inside the detector well: Z 2 S non
axial Odisk ¼ Oin r dr, in S2 0 is as identified before in Eq. (9b) where Onon
axial in and S is the radius of the circular disk source. From Eq. (9b), the above expression can be rewritten as Ri Odisk in ¼ S 2 S2 S Z 2p þ pðcos aþ
cos a

cos bþ
cos b
Þ 0 þ2 ρ K ¼ 2p þ þ L ! aþ Z fmax ðh; Ri Þ sin y dy r dr a
h (11a)  2p  þ f ðhÞ
f
ðhÞ
f þ ðh
KÞ
f
ðh
KÞ S2 4 S2 Z 0 S Z aþ a
fmax ðh; Ri Þ sin y dy r dr, ð11bÞ where ða ; cos a Þ, ðb ; cos b Þ and fmax ðh; Ri Þ are as identified before in Eqs. (6a), (6b) and (8), respectively. Whereas f  ðhÞ is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f  ðhÞ ¼ h h2 þ ðRi  SÞ2
h h2 þ R2i  Ri h 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 h þ Ri  Ri B C ð12Þ  ln@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A. 2 2 h þ ðRi  SÞ þ ðS  Ri Þ (b) the disk source lies outside the detector well: Z 2 S non
axial Odisk ¼ Oout r dr, (13a) out S2 0 S non
axial where Oout is as identified before in Eq. (10b) and S is the radius of the circular disk source. From Eq. (10b), the above expression can be rewritten as Z Z þ 4 S g disk fmax ððh
KÞ; Ro Þ sin y dy r dr, Odisk out ¼ Oin þ 2 S 0 bþ (13b) ρ h K L Fig. 3. Co-axial disk source–detector configuration. where b+, g+, fmax((h
K),R0) and Odisk are as in identified before in Eqs. (6b), (6c), (8), and (11b), respectively. The integral in Eqs. (11b) and (13b) are elliptic integrals and does not have a closed solution. Then a numerical solution is obtained using the Simpson’s rule. Although the accuracy of the integration increases with increasing the number of intervals n, the integration converges well at n ¼ 15. The numerical values of the elliptic integral in Eq. (13b) are always less than zero (i.e.
ve). (iii) The case of an isotropic radiating co-axial cylindrical source and a well-type detector (Fig. 4), the solid angle is given by (a) the cylindrical source lies inside the detector well: Z 1 Hþho disk cyl Oin ¼ Oin dh, (14a) H ho ARTICLE IN PRESS 1052 M.I. Abbas / Applied Radiation and Isotopes 64 (2006) 1048–1056 Eq. (11b), ho is the distance between the cylindrical source circular base and the bottom of the crystal well, and H is the height of the cylindrical source. From Eq. (11b), the above expression can be rewritten as Ro Ri S Z Hþho  þ 2p f ðhÞ
f
ðhÞ 2 S :H ho 
f þ ðh
KÞ
f
ðh
KÞ dh Z Hþho Z S Z aþ 4 þ 2 fmax ðh; Ri Þ S H ho 0 a
 sin y dy r dr dh, Ocyl in ¼ 2p þ H ρ K h ho ð14bÞ where a7, fmax (h,Ri) and f7(h) are as identified before in Eqs. (6a), (8) and (12), respectively. (b) the cylindrical source lies outside the detector well: L Ocyl out ¼ S 1 H Z Hþho ho Odisk out dh; (15a) where Odisk out , the solid angle subtended by a well-type detector at an isotropic radiating co-axial circular disk source lies outside the detector well, is as identified before in Eq. (13b), ho is the distance between the cylindrical source lower face and the bottom of the crystal well, and H is the height of the cylindrical source. From Eq. (13b), the H ρ Table 1 Parameters of the well-type NaI (Tl) detector used by Snyder (1965) Crystal radius (Ro) Crystal length (K+L) Radius of the crystal well (Ri) Length of the crystal well (K) h 2.2225 cm 5.08 cm 0.9575 cm 3.81 cm ho K 1 L Fig. 4. Co-axial cylindrical source–detector configuration. Geometrical efficiency Present Work Snyder (1965) 0.9 0.8 0.7 0.6 1 1.5 2 2.5 3 3.5 4 h cm where Odisk in , the solid angle subtended by a well-type detector at an isotropic radiating co-axial circular disk source lies inside the detector well, is as identified before in Fig. 5. Variation of the geometrical efficiency of a NaI (Tl) well-type detector for an axial point source as a function of the distance between the source and the bottom of the crystal well h, solid line is the present work, symbols represent the Monte Carlo values of Snyder (1965). ARTICLE IN PRESS 1053 M.I. Abbas / Applied Radiation and Isotopes 64 (2006) 1048–1056 above expression can be rewritten as 4 cyl Ocyl out ¼ Oin þ 2 S H Z Hþho Z S Z 0 ho gþ bþ fmax ððh
KÞ; Ro Þ sin y dy r dr dh, ð15bÞ where b+, g+, fmax((h
K),R0) and Ocyl in are as identified before in Eqs. (6b), (6c), (8), and (14b), respectively. The numerical evaluation of the multiple integrals in Eqs. (14b) and (15b) is performed using the Simpson’s rule. Although the accuracy of the integration increases with increasing the number of intervals n, the integration converges well at n ¼ 20. The numerical values of the elliptic integral in Eq. (15b) are always less than zero (i.e.
ve). Table 2 Parameters of the well-type HPGe detector used by Wang et al. (1999) 3. Validation of the present method Crystal radius (Ro) Crystal length (K+L) Radius of the crystal well (Ri) Length of the crystal well (K) The geometrical efficiency calculated by the present model is tested against various data sets obtained by the experimental and Monte Carlo methods (for two lowenergy g lines 88 and 100 keV, respectively) as follows. We applied the present method to several distances of a point source from the detector well bottom and to a set of cylindrical sources with different volumes. The geometrical efficiency of a well-type NaI (Tl) detector, with parameters listed in Table 1, for an axial point source placed at different heights, h ¼ 1.5, 2.5 and 3.5 cm, above the bottom of the crystal well have been calculated and compared with the values obtained using the Monte Carlo method by Snyder (1965), as shown in Fig. 5. The percentage deviations between calculated efficiency values (using the present formulae and that published by Snyder, 1965) are less than (0.1%). The percentage deviation is given by    Presentwork 
Published g  g D¼  100%. (16) Presentwork g 2.65 cm 5.50 cm 1.05 cm 3.60 cm 10 Geometrical efficiency Present Work Wang et al. (1999) 1 0.1 0 0.2 0.4 0.6 0.8 1 H cm 1.2 1.4 1.6 1.8 Fig. 6. Variation of the geometrical efficiency of a HPGe well-type detector for a set of cylindrical sources with different volumes as a function of the cylindrical source height H, solid line is the present work, symbols represent the experimental values of Wang et al. (1999). The geometrical efficiency of a well-type HPGe detector, with parameters listed in Table 2, for a set of cylindrical sources with volumes 0.5, 1.0, 1.5 and 2.0 ml (the cylinders have equal radii, S ¼ 0.62 cm, and different heights, H ¼ 0.414, 0.822, 1.242 and 1.656 cm, respectively) have been calculated and compared with the experimental data published by Wang et al. (1999), as shown in Fig. 6. Geometrical Efficiency 1 Axial-point source 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 h cm Fig. 7. Variation of the geometrical efficiency of a well-type (Ri ¼ 1.05, Ro ¼ 2.65, K ¼ 3.6 and L ¼ 1.9 cm) detector for an axial point source as a function of the distance between the source and the bottom of the crystal well h. ARTICLE IN PRESS 1054 M.I. Abbas / Applied Radiation and Isotopes 64 (2006) 1048–1056 The percentage deviations between calculated and measured efficiency values for cylindrical sources are less than (4%). The results presented in Figs. 5 and 6 confirm the validity of the present mathematical formulae for the computation of the well-type detector geometrical efficiencies. In addition, systematic calculations of the geometrical efficiency of a bare well-type detector, with parameters listed in Table 1, for an axial point source as a function of the distance between the source and the bottom of the crystal well, h, were calculated and represented in Fig. 7. The geometrical efficiency of a bare well-type detector, with parameters listed in Table 1, for a non-axial point source placed at different heights, h ¼ 0, 0.5, 1, 5 and 10 cm, above the bottom of the crystal well as a function of the lateral distance r has been calculated and given in Fig. 8. Fig. 9 shows the calculated values of the geometrical efficiency of a bare well-type detector, with parameters listed in Table 1, for an isotropic radiating co-axial circular disk source placed at different heights, h ¼ 0, 0.5, 1, 5 and 10 cm, above the bottom of the crystal well as a function of the circular disk source radius S. Finally, the geometrical efficiency of a bare well-type detector, with parameters listed in Table 1, for a co-axial cylindrical source, with height H ¼ 1 cm, placed at different heights, ho ¼ 0, 0.5, 1, Non-axial point source h 0.8 Geometrical Efficiency Geometrical Efficiency 1 zero 0.5 cm 1.0 cm 5.0 cm 10.0 cm 0.6 1 0.98 0.96 0.94 0.92 0.9 0 0.2 0.4 0.6 0.8 1 Lateral Distance cm 0.4 0.2 0 0 0.5 1 1.5 Lateral Distance cm 2 3 2.5 Fig. 8. Variation of the geometrical efficiency of a well-type (Ri ¼ 1.05, Ro ¼ 2.65, K ¼ 3.6 and L ¼ 1.9 cm) detector for a non-axial point source placed at different heights, h ¼ 0, 0.5, 1, 5 and 10 cm, above the bottom of the crystal well as a function of the lateral distance r. Also shown in the inset an enlargement of the geometrical efficiency regions corresponding to the case of the non-axial point source lies inside the detector well. Circular disk source h 0.8 Geometrical Efficiency Geometrical Efficiency 1 zero 0.5 cm 1.0 cm 5.0 cm 10.0 cm 0.6 1 0.98 0.96 0.94 0.92 0.9 0.2 0 0.4 0.6 0.8 1 S cm 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 S cm Fig. 9. Variation of the geometrical efficiency of a well-type (Ri ¼ 1.05, Ro ¼ 2.65, K ¼ 3.6 and L ¼ 1.9 cm) detector for an isotropic radiating co-axial circular disk source placed at different heights, h ¼ 0, 0.5, 1, 5 and 10 cm, above the bottom of the crystal well as a function of the circular disk radius S. Also shown in the inset an enlargement of the geometrical efficiency regions corresponding to the case of the circular disk source lies inside the detector well. ARTICLE IN PRESS 1055 M.I. Abbas / Applied Radiation and Isotopes 64 (2006) 1048–1056 Cylindrical source ho 0.8 Geometrical Efficiency Geometrical Efficiency 1 zero 0.5 cm 1.0 cm 5.0 cm 10.0 cm 0.6 1 0.98 0.96 0.94 0.92 0.9 0.2 0 0.4 0.6 0.8 1 S cm 0.4 0.2 0 0 0.5 1 1.5 S cm 2 2.5 3 Fig. 10. Variation of the geometrical efficiency of a well-type (Ri ¼ 1.05, Ro ¼ 2.65, K ¼ 3.6 and L ¼ 1.9 cm) detector for a co-axial cylindrical source, with height H ¼ 1 cm, placed at different heights, ho ¼ 0, 0.5, 1, 5 and 10 cm, above the bottom of the crystal well as a function of the circular base radius S of the cylindrical source. Also shown in the inset an enlargement of the geometrical efficiency regions corresponding to the case of the cylindrical source lies inside the detector well. 5 and 10 cm, above the bottom of the crystal well as a function of the circular base radius S of the cylindrical source has been calculated and given in Fig. 10. 4. Conclusions Direct mathematical expressions to calculate geometrical efficiency of well-type NaI (Tl) and HPGe detectors have been derived in the case of axial point, non-axial point, extended circular disk and cylindrical sources. The agreement between the results calculated in this work and the published values is very good; the high discrepancies in the case of the point source were very small (0.1%). Meanwhile, in the case of the cylindrical sources the high discrepancies being less than 4.0%. This means that the present approach is efficient and sufficiently powerful to evaluate the geometrical efficiency of well-type detectors. References Abbas, M.I., 1995. Source–detector geometrical factor. Theoretical and experimental studies, Ph.D. Thesis. Abbas, M.I., 2001a. HPGe detector photopeak efficiency calculation including self-absorption and coincidence corrections for Marinilli beaker sources using compact analytical expressions. Appl. Radiat. Isoto. 54, 761. Abbas, M.I., 2001b. A direct mathematical method to calculate the efficiencies of a parallelepiped detector for an arbitrarily positioned point source. Radiat. Phys. Chem. 60, 3. Abbas, M.I., 2001c. 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A hybrid method for calculating absolute peak efficiency of germanium detectors. Nucl. Instrum. Methods A 413, 281–292. Noguchi, M., Takeda, K., Higuchi, H., 1981. Semi-empirical g-ray peak efficiency determination including self-absorption correction based on numerical integration. Int. J. Appl. Radiat. Isot 32, 17–22. Pommé, S., 2004. A complete series expansion of Ruby’s solid angle formula. Nucl. Instrum. Methods A 531, 616–620. Prata, M.J., 2004a. Analytical calculation of the solid angle subtended by a circular disc detector at a point cosine source. Nucl. Instrum. Methods A 521, 576–585. Prata, M.J., 2004b. Analytical calculation of the solid angle defined by a cylindrical detector at a point cosine source with parallel axes. Radiat. Phys. Chem. 69, 273–279. Prata, M.J., 2003. Solid angle subtended by a cylindrical detector at a point source in terms of elliptic integrals. Radiat. Phys. Chem. 67, 599–603. Ruby, L., 1995. 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