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A series expansion of Conway's generalised solid-angle formulas
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 579 (2007) 272–274 www.elsevier.com/locate/nima
A series expansion of Conway’s generalised solid-angle formulas Stefaan Pomme´, Jan Paepen European Commission, Joint Research Centre, Institute for Reference Materials and Measurements, Retieseweg 111, B-2440 Geel, Belgium Available online 6 April 2007
Abstract A series expansion is presented of Conway’s solid-angle formula for two parallel, non-coaxial discs. Similar equations for non-coaxial, parallel ring and point sources are verified, as well as a quickly converging integral representation for the solid angle of two coaxial discs. Their performance is tested by comparison with results from a numerical integration method. r 2007 Elsevier B.V. All rights reserved. PACS: 02.70.c; 06.20.f; 02.30.Lt; 02.30.Mv Keywords: Solid angle; Efficiency; Absolute counting
1. Introduction
2. Conway’s equations
Solid-angle calculations have applications in numerous fields, radioactivity measurements being one of them. They are often required for the absolute calibration of a radioactivity measurement system or, alternatively, one can apply measurements at a ‘defined solid angle’ to obtain the activity of a source. Source and detector are often modelled as coaxial discs. For this basic type of configuration, Ruby and co-worker [1,2] presented an elegant formula for the geometry factor, involving an integral of the product of two Bessel functions. Recently, Conway [3] presented a generalisation of this formula for two parallel, non-coaxial discs. The equation involves the product of three Bessel functions. Additionally, he presented a useful integral representation of the coaxial case, which converges very fast to high accuracy. In this work, a series expansion of the generalised formula of Conway is presented and the performance of the mentioned formulas is tested by comparison with results from a numerical-integration algorithm [4].
The source is represented by a flat disc with a radius RS, characterised by a homogeneous distribution of the active material and an isotropic emission of the particles. The detector window is assumed to be perfectly circular, with radius RD, positioned parallel with the source, having a distance h between both planes and a distance a between the symmetry axes of the discs. A schematic representation of such configuration is shown in Fig. 1. Conway [3] has shown that the corresponding geometry factor, G ¼ O=4p, can be calculated from the following equation: Z RD 1 esh G¼ ds. (1) J 0 ðsaÞJ 1 ðsRD ÞJ 1 ðsRS Þ RS 0 s
Corresponding author. Tel.: +32 14 571 289.
E-mail address: [email protected] (S. Pomme´). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.04.054
He also produced a solid-angle formula for a non-coaxial (infinitely thin) ring-shaped source with radius RS, Z RD 1 G¼ J 0 ðsaÞJ 1 ðsRD ÞJ 0 ðsRS Þ esh ds. (2) 2 0 Taking into consideration that J0(0) ¼ 1 and J1(0) ¼ 0, one can easily derive solid-angle formulas for coaxial discs (a ¼ 0), for coaxial or non-coaxial point sources (RS ¼ 0) and for coaxial ring sources (a ¼ 0). The first case derives exactly to Ruby’s solid-angle equation [1,2]. The equations for a coaxial ring source and a non-coaxial point source are
ARTICLE IN PRESS S. Pomme´, J. Paepen / Nuclear Instruments and Methods in Physics Research A 579 (2007) 272–274
RD
h
RS
a
Fig. 1. Schematic representation of a source with radius RS parallel to a detector with radius RD at a distance h. The distance between the symmetry axes of the discs is a.
equivalent, as could be expected for reasons of axial symmetry Z RD 1 G¼ J 1 ðsRD ÞJ 0 ðsRS Þ esh ds, (3) 2 0 in which the source radius RS can be replaced by the displacement a of the point source from the axis of symmetry. Conway [3] also presented a quickly converging integral representation of the solid angle for coaxial discs. It can be transformed into the following form: Z RD 1 p sin2 F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dF, (4) G¼ RS 2p 0 x cos Fð y þ x cos FÞ
273
and the source and detector radii, i.e., h4a+RS+RD. For smaller source–detector distances, the series involves the summation of both small and large terms with opposite sign and does not easily converge to realistic values. Also Conway’s solid-angle formula for a ring-shaped source (Eq. (2)) can be expanded in a similar series as Eq. (7), differing only in the factor m!(m+1)!, which is to be replaced by m!2. The resulting series can also be applied for the cases dealt with by Eq. (3), by introducing, e.g., a ¼ 0 or RS ¼ 0. Alternatively, the following explicit series expansion of Eq. (3) can be used " 2 X q X q RD 1 lþm RD 2l RS 2m G¼ 4 2h l¼0 m¼0 h h ½2ðl þ mÞ þ 1! for q ! 1. ð8Þ l!ðl þ 1Þ!m!2 The integral representation (Eq. (4)) has been implemented in software as a summation G
n RD 1 X sin2 j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , RS 2n i¼1 x cos jð y þ x cos jÞ
(9)
in which j ¼ (i0.5)p/n and n is an integer value of choice (e.g., n ¼ 50). 4. Benchmark
in which we define x¼
R2S þ R2D þ h2 2RS RD
and y ¼
The validity of the presented equations has been checked by comparison with the results of a numerical integration method [4]. The solid angle subtended by a circular detector for a point source is calculated as the sum of two terms
h2 . 2RS RD
3. Series expansion A series expansion of Eqs. (1)–(3) can be derived in a similar way as has been done for Ruby’s equations [5]. Considering that the Bessel function Jn(x) can be expressed as a power series, 1 X ð1Þk xnþ2k (5) J n ðxÞ ¼ k!ðk þ nÞ! 2 k¼0 and that Z 1 ey yn dy ¼ n!,
(6)
0
one readily finds a complete expansion of Eq. (1), which can easily be implemented in a software routine: " 2 X q X q X q RD 1 kþlþm a2k RD 2l G¼ 4 h 2h k¼0 l¼0 m¼0 h # 2m RS ½2ðk þ l þ mÞ þ 1! for q ! 1. ð7Þ h k!2 l!ðl þ 1Þ!m!ðm þ 1Þ! In practice, this serial expansion is an elegant and precise algorithm for the solid-angle calculation of non-coaxial disc source and detector configurations for cases in which the distance h is higher than the sum of the lateral displacement a
OP ðrÞ ¼ O1 þ O2 , ( 2pð1 cos y1 Þ for aoRD ; O1 ¼ 0 for aXRD ; Z
(10) (11)
y2
2ðp jðyÞÞ sin y dy.
O2 ¼
(12)
y1
The angles are defined as [4] R D a , y1 ¼ arctan h RD þ a y2 ¼ arctan , h jðyÞ ¼ arccos
2 RD fa2 þ h2 tan2 yg . 2ah tan y
(13)
(14)
(15)
The solid angle for a voluminous source can be calculated from an average of the solid angle in all points of the source. Conway’s equations (Eqs. (1)–(3)) were evaluated using the software packages Mathcads 8 Professional and Mathematicas 5.2. In Mathcad it was required to set the built-in tolerance constant (TOL) as low as possible, in order to increase the precision to which integrals are
ARTICLE IN PRESS 274
S. Pomme´, J. Paepen / Nuclear Instruments and Methods in Physics Research A 579 (2007) 272–274
Table 1 Geometry factor G for different source and detector configurations, and the equations that can be used to obtain the factor a/h
Geometry Point source
Coaxial Non-coaxial
0 0 0.5 1 10 10
Disk source
0 0 0 0 0 0
RD/h 0.5 1 0.5 1 1 10
Coaxial
0
0.5
0.5
Non-coaxial
0 0.1
1 0.5
1 0.5
1 10 10 Ring source
RS/h
Coaxial
Non-coaxial
1 1 1
1 1 10
0
0.5
0.5
0 0.1 1 10 10
1 0.5 1 1 1
1 0.5 1 1 10
G ¼ O/4p
Equations
0.052786405 0.14644661 0.041343290 0.04137 0.089340580 0.00024902173 0.21514574
(1–3), (7) for qX15, (8) for qX15, (10–15) (1–3), (10–15) (1–3), (10–15) (7) for q ¼ 42 (the maximum in Mathcad) (1–3), (10–15) (1–3), (10–15) (1–3), (10–15)
0.046753040 0.0467536 0.11610539 0.046368218 0.04636816 0.079731212 0.00025183135 0.21420895
(1), (4), (9) for nX6, (10–15) (7) for q ¼ 42 (the maximum in Mathcad) (1), (4), (9) for nX9, (10–15) (1), (10–15) (7) for q ¼ 28 (the maximum in Mathcad) (1), (10–15) (1), (10–15) (1), (10–15)
0.041343290 0.04137 0.0413437 0.089340580 0.041083322 0.070727685 0.00025467016 0.21332800
(2–3), (10–15) (8) for q ¼ 42 (the maximum in Mathcad) (8) for q ¼ 700 (Mathematica) (2–3), (10–15) (2) (2) (2) (2)
The accuracy of the results from the series expansions (Eqs. (7) and (8)) when a+RS+RD approaches h, is limited and the number of correct digits depends on the capability of handling large numbers by the mathematical software packages.
evaluated. The function NIntegrate was used in Mathematica with the options PrecisionGoal-30, WorkingPrecision-60 and MaxRecursion-15. For any combination of the parameters RS, RD, a and h, both packages gave the same result, up to at least nine digits. Eqs. (7)–(9) were evaluated by Visual Basics routines in Excels as well as directly in Mathematica and Mathcad. Some numerical examples are shown in Table 1. All distance parameters are expressed as multiples of the distance between source and detector plane. 5. Conclusions Conway’s equations (Eqs. (1)–(3)) yield the same results as the numerical integration method (Eqs. (10)–(15)) [4].
The series expansions (Eqs. (7) and (8)) give correct and precise results if the distance between source and detector plane is sufficiently large (rule of thumb: h4a+RS+RD). Eqs. (4) or (9) is indeed a quickly converging solution that reaches high accuracy already for modest values of n.
References [1] [2] [3] [4]
L. Ruby, J.B. Rechen, Nucl. Instr. and Meth. 58 (1968) 345. L. Ruby, Nucl. Instr. and Meth. A 337 (1994) 531. J.T. Conway, Nucl. Instr. and Meth. A 562 (2006) 146. S. Pomme´, L. Johansson, G. Sibbens, B. Denecke, Nucl. Instr. and Meth. A 505 (2003) 286. [5] S. Pomme´, Nucl. Instr. and Meth. A 531 (2004) 616.
Nuclear Instruments and Methods in Physics Research A 579 (2007) 272–274 www.elsevier.com/locate/nima
A series expansion of Conway’s generalised solid-angle formulas Stefaan Pomme´, Jan Paepen European Commission, Joint Research Centre, Institute for Reference Materials and Measurements, Retieseweg 111, B-2440 Geel, Belgium Available online 6 April 2007
Abstract A series expansion is presented of Conway’s solid-angle formula for two parallel, non-coaxial discs. Similar equations for non-coaxial, parallel ring and point sources are verified, as well as a quickly converging integral representation for the solid angle of two coaxial discs. Their performance is tested by comparison with results from a numerical integration method. r 2007 Elsevier B.V. All rights reserved. PACS: 02.70.c; 06.20.f; 02.30.Lt; 02.30.Mv Keywords: Solid angle; Efficiency; Absolute counting
1. Introduction
2. Conway’s equations
Solid-angle calculations have applications in numerous fields, radioactivity measurements being one of them. They are often required for the absolute calibration of a radioactivity measurement system or, alternatively, one can apply measurements at a ‘defined solid angle’ to obtain the activity of a source. Source and detector are often modelled as coaxial discs. For this basic type of configuration, Ruby and co-worker [1,2] presented an elegant formula for the geometry factor, involving an integral of the product of two Bessel functions. Recently, Conway [3] presented a generalisation of this formula for two parallel, non-coaxial discs. The equation involves the product of three Bessel functions. Additionally, he presented a useful integral representation of the coaxial case, which converges very fast to high accuracy. In this work, a series expansion of the generalised formula of Conway is presented and the performance of the mentioned formulas is tested by comparison with results from a numerical-integration algorithm [4].
The source is represented by a flat disc with a radius RS, characterised by a homogeneous distribution of the active material and an isotropic emission of the particles. The detector window is assumed to be perfectly circular, with radius RD, positioned parallel with the source, having a distance h between both planes and a distance a between the symmetry axes of the discs. A schematic representation of such configuration is shown in Fig. 1. Conway [3] has shown that the corresponding geometry factor, G ¼ O=4p, can be calculated from the following equation: Z RD 1 esh G¼ ds. (1) J 0 ðsaÞJ 1 ðsRD ÞJ 1 ðsRS Þ RS 0 s
Corresponding author. Tel.: +32 14 571 289.
E-mail address: [email protected] (S. Pomme´). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.04.054
He also produced a solid-angle formula for a non-coaxial (infinitely thin) ring-shaped source with radius RS, Z RD 1 G¼ J 0 ðsaÞJ 1 ðsRD ÞJ 0 ðsRS Þ esh ds. (2) 2 0 Taking into consideration that J0(0) ¼ 1 and J1(0) ¼ 0, one can easily derive solid-angle formulas for coaxial discs (a ¼ 0), for coaxial or non-coaxial point sources (RS ¼ 0) and for coaxial ring sources (a ¼ 0). The first case derives exactly to Ruby’s solid-angle equation [1,2]. The equations for a coaxial ring source and a non-coaxial point source are
ARTICLE IN PRESS S. Pomme´, J. Paepen / Nuclear Instruments and Methods in Physics Research A 579 (2007) 272–274
RD
h
RS
a
Fig. 1. Schematic representation of a source with radius RS parallel to a detector with radius RD at a distance h. The distance between the symmetry axes of the discs is a.
equivalent, as could be expected for reasons of axial symmetry Z RD 1 G¼ J 1 ðsRD ÞJ 0 ðsRS Þ esh ds, (3) 2 0 in which the source radius RS can be replaced by the displacement a of the point source from the axis of symmetry. Conway [3] also presented a quickly converging integral representation of the solid angle for coaxial discs. It can be transformed into the following form: Z RD 1 p sin2 F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dF, (4) G¼ RS 2p 0 x cos Fð y þ x cos FÞ
273
and the source and detector radii, i.e., h4a+RS+RD. For smaller source–detector distances, the series involves the summation of both small and large terms with opposite sign and does not easily converge to realistic values. Also Conway’s solid-angle formula for a ring-shaped source (Eq. (2)) can be expanded in a similar series as Eq. (7), differing only in the factor m!(m+1)!, which is to be replaced by m!2. The resulting series can also be applied for the cases dealt with by Eq. (3), by introducing, e.g., a ¼ 0 or RS ¼ 0. Alternatively, the following explicit series expansion of Eq. (3) can be used " 2 X q X q RD 1 lþm RD 2l RS 2m G¼ 4 2h l¼0 m¼0 h h ½2ðl þ mÞ þ 1! for q ! 1. ð8Þ l!ðl þ 1Þ!m!2 The integral representation (Eq. (4)) has been implemented in software as a summation G
n RD 1 X sin2 j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , RS 2n i¼1 x cos jð y þ x cos jÞ
(9)
in which j ¼ (i0.5)p/n and n is an integer value of choice (e.g., n ¼ 50). 4. Benchmark
in which we define x¼
R2S þ R2D þ h2 2RS RD
and y ¼
The validity of the presented equations has been checked by comparison with the results of a numerical integration method [4]. The solid angle subtended by a circular detector for a point source is calculated as the sum of two terms
h2 . 2RS RD
3. Series expansion A series expansion of Eqs. (1)–(3) can be derived in a similar way as has been done for Ruby’s equations [5]. Considering that the Bessel function Jn(x) can be expressed as a power series, 1 X ð1Þk xnþ2k (5) J n ðxÞ ¼ k!ðk þ nÞ! 2 k¼0 and that Z 1 ey yn dy ¼ n!,
(6)
0
one readily finds a complete expansion of Eq. (1), which can easily be implemented in a software routine: " 2 X q X q X q RD 1 kþlþm a2k RD 2l G¼ 4 h 2h k¼0 l¼0 m¼0 h # 2m RS ½2ðk þ l þ mÞ þ 1! for q ! 1. ð7Þ h k!2 l!ðl þ 1Þ!m!ðm þ 1Þ! In practice, this serial expansion is an elegant and precise algorithm for the solid-angle calculation of non-coaxial disc source and detector configurations for cases in which the distance h is higher than the sum of the lateral displacement a
OP ðrÞ ¼ O1 þ O2 , ( 2pð1 cos y1 Þ for aoRD ; O1 ¼ 0 for aXRD ; Z
(10) (11)
y2
2ðp jðyÞÞ sin y dy.
O2 ¼
(12)
y1
The angles are defined as [4] R D a , y1 ¼ arctan h RD þ a y2 ¼ arctan , h jðyÞ ¼ arccos
2 RD fa2 þ h2 tan2 yg . 2ah tan y
(13)
(14)
(15)
The solid angle for a voluminous source can be calculated from an average of the solid angle in all points of the source. Conway’s equations (Eqs. (1)–(3)) were evaluated using the software packages Mathcads 8 Professional and Mathematicas 5.2. In Mathcad it was required to set the built-in tolerance constant (TOL) as low as possible, in order to increase the precision to which integrals are
ARTICLE IN PRESS 274
S. Pomme´, J. Paepen / Nuclear Instruments and Methods in Physics Research A 579 (2007) 272–274
Table 1 Geometry factor G for different source and detector configurations, and the equations that can be used to obtain the factor a/h
Geometry Point source
Coaxial Non-coaxial
0 0 0.5 1 10 10
Disk source
0 0 0 0 0 0
RD/h 0.5 1 0.5 1 1 10
Coaxial
0
0.5
0.5
Non-coaxial
0 0.1
1 0.5
1 0.5
1 10 10 Ring source
RS/h
Coaxial
Non-coaxial
1 1 1
1 1 10
0
0.5
0.5
0 0.1 1 10 10
1 0.5 1 1 1
1 0.5 1 1 10
G ¼ O/4p
Equations
0.052786405 0.14644661 0.041343290 0.04137 0.089340580 0.00024902173 0.21514574
(1–3), (7) for qX15, (8) for qX15, (10–15) (1–3), (10–15) (1–3), (10–15) (7) for q ¼ 42 (the maximum in Mathcad) (1–3), (10–15) (1–3), (10–15) (1–3), (10–15)
0.046753040 0.0467536 0.11610539 0.046368218 0.04636816 0.079731212 0.00025183135 0.21420895
(1), (4), (9) for nX6, (10–15) (7) for q ¼ 42 (the maximum in Mathcad) (1), (4), (9) for nX9, (10–15) (1), (10–15) (7) for q ¼ 28 (the maximum in Mathcad) (1), (10–15) (1), (10–15) (1), (10–15)
0.041343290 0.04137 0.0413437 0.089340580 0.041083322 0.070727685 0.00025467016 0.21332800
(2–3), (10–15) (8) for q ¼ 42 (the maximum in Mathcad) (8) for q ¼ 700 (Mathematica) (2–3), (10–15) (2) (2) (2) (2)
The accuracy of the results from the series expansions (Eqs. (7) and (8)) when a+RS+RD approaches h, is limited and the number of correct digits depends on the capability of handling large numbers by the mathematical software packages.
evaluated. The function NIntegrate was used in Mathematica with the options PrecisionGoal-30, WorkingPrecision-60 and MaxRecursion-15. For any combination of the parameters RS, RD, a and h, both packages gave the same result, up to at least nine digits. Eqs. (7)–(9) were evaluated by Visual Basics routines in Excels as well as directly in Mathematica and Mathcad. Some numerical examples are shown in Table 1. All distance parameters are expressed as multiples of the distance between source and detector plane. 5. Conclusions Conway’s equations (Eqs. (1)–(3)) yield the same results as the numerical integration method (Eqs. (10)–(15)) [4].
The series expansions (Eqs. (7) and (8)) give correct and precise results if the distance between source and detector plane is sufficiently large (rule of thumb: h4a+RS+RD). Eqs. (4) or (9) is indeed a quickly converging solution that reaches high accuracy already for modest values of n.
References [1] [2] [3] [4]
L. Ruby, J.B. Rechen, Nucl. Instr. and Meth. 58 (1968) 345. L. Ruby, Nucl. Instr. and Meth. A 337 (1994) 531. J.T. Conway, Nucl. Instr. and Meth. A 562 (2006) 146. S. Pomme´, L. Johansson, G. Sibbens, B. Denecke, Nucl. Instr. and Meth. A 505 (2003) 286. [5] S. Pomme´, Nucl. Instr. and Meth. A 531 (2004) 616.