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Calculation of the solid angle subtended by a cylinder at a point
Applied Radiation and Isotopes 70 (2012) 2466–2470
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Calculation of the solid angle subtended by a cylinder at a point S. Tryka n University of Life Sciences in Lublin, Department of Agricultural Sciences, Laboratory of Physics, Szczebrzeska 102, PL-22-400 Zamosc, Poland
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 March 2010 Accepted 2 July 2012 Available online 9 July 2012
An analytical formula for calculating the solid angle subtended by a cylinder at a point has been derived from the general solid angle equation. The formula is expressed by double line integrals and by single integrals of simple elementary functions. These functions were then integrated and the formula was represented by products of some elementary functions and the incomplete Lagrange–Jacobi elliptic integrals of the third kind. The final formula was used to calculate representative values of solid angles and compare them with literature data. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Solid angle Isotropic point source Cylindrical detector Source–detector system Elliptic integrals
1. Introduction Solid angle formulas are necessary in a wide class of problems in nuclear physics (Kovarik and Adams, 1932; Berne, 1951; Jaffey, 1954; Masket et al., 1957a,b; Masket 1959; Rowlnds, 1961; Gardner and Verghese, 1971; Bland, 1984; Vega-Carrillo, 1996; Selim and Abbas, 1996; Prata, 2003, 2004; Conway, 2006; Galiano and Rodrigues, 2006; Whitcher, 2006; Cipolla, 2007), optical radiometry (Olivier and Gagnon, 1993; Olivier et al., 1993; Tryka, 1997, 1999, 2000, molecular chemistry (Guzei and Wendt, 2006; Taverner, 1996; David et al., 1995), and other fields of science and technology (Maxwell, 1873; Webster, 1927; Zaluzec, 2009). For instance, when the power of isotropic radiation is measured, it is directly proportional to the solid angle defined by the source–detector system. Similarly, the calculation of particle beams passing through a given aperture, or incident on a given object, requires an expression of a solid angle. In addition, several important electromagnetic problems must be solved using solid angle equations (Rowlnds, 1961; Maxwell, 1873; Webster, 1927). It is well known that the solid angle depends on the geometric configuration of the specific point relative to a given body and on the shape of the body. Therefore, there is a need to calculate solid angles for bodies of different shapes and at any position with respect to the given point. One of many cases, mainly related to the needs of nuclear physics and lighting technology, is the solid angle at a point subtended by a cylinder. Several research papers have been devoted to this problem and various more or less complicated numerical and analytical methods have been presented (Masket et al., 1957a,b; Selim and Abbas, 1996; Prata, 2003,
n
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2004; Whitcher, 2006). Undoubtedly some of these methods are inaccurate and give only approximate results. Therefore, it is necessary to present an analytical solution to the general solid angle equation which can be used for accurate calculation of the solid angle values. The aim of this paper is to provide a simple analytical formula for calculating the solid angle at a point subtended by a cylinder. The formula was obtained by solving the solid angle equation in the Cartesian coordinate system. The solution obtained was first expressed by double definite line integrals and then by single definite integrals of simple elementary functions. This was then integrated into the form of the several Lagrange–Jacobi incomplete elliptic integrals of the third kind. Then some data were computed for selected geometrical configurations of a point with respect to a cylinder. These computed data were compared to the data presented in the literature, illustrated graphically and tabulated.
2. Solid angle equation If the Cartesian coordinate system 0xyz is erected at a point, P, the solid angle, OPS , at point P, subtended by a planar surface, S, which is perpendicular to the z-axis at a distance z from the point P, is defined as follows: ZZ z dx dy OPS ¼ : ð1Þ 3=2 S ðx2 þ y2 þz2 Þ Sometimes, however, the surface S has its symmetry axis parallel to the axis z of the main coordinate system 0xyz. In this case we can introduce an additional Cartesian coordinate system 01x1y1z1 parallel to the main coordinate system 0xyz, and erected at the point 0c, being the center of symmetry of the surface S.
S. Tryka / Applied Radiation and Isotopes 70 (2012) 2466–2470
In these coordinates the solid angle equation becomes ZZ ðzc þ z1 Þdx1 dy1 OPS ðxc ,yc ,zc Þ ¼ , 2 2 2 3=2 ½ðx þx Þ þðy S c 1 c þ y1 Þ þ ðzc þ z1 Þ
ð2Þ
where xc, yc, and zc are the coordinates of the point 0c.
3. General solution to the solid angle equation Applying Eq. (2) to the system shown in Fig. 1, the solid angle,
OP-C (xc, yc, zc, R, h), at the point P subtended by a cylinder with a radius R40 and height hZ0 can be expressed as
OPC ðxc ,yc ,zc ,R,hÞ 8 > > OPS1 ðxc ,yc ,zc ,R,hÞ þ OPS2 ðxc ,yc ,zc ,RÞ > > > > < þ OPS3 ðxc ,yc ,zc ,R,hÞ, ¼ p, > > > > > 2p, > :O Pdisk ðxc ,yc ,zc ,RÞ,
0 o R o d,
arbitrary zc ,
0 o R ¼ d, 0 r d o R,
zc ¼ 0, zc ¼ 0,
0 r d o R,
zc a 0,
It is worth noting that Eq. (3) is also suitable for calculating the solid angle OPC(xc, yc, zc, R, h) in the case when the bottom base of the cylinder, or both the bottom and upper bases of a cylinder, lie under the plane 0xy in Fig. 1. According to Eq. (2), the solid angle OPC(xc, yc, zc, R, h) defined by Eq. (3) can be calculated by introducing an additional Cartesian coordinate system 01x1y1z1 at a distance d from the main coordinate system 0xyz (Fig. 1). In these coordinates the solid angles OPS1 ðxc ,yc ,zc ,R,hÞ, OPS2 ðxc ,yc ,zc ,R,hÞ, OPS3 ðxc ,yc ,zc ,R,hÞ, and OPdisk ðxc ,yc ,zc ,RÞ in Eq. (3) are expressed by ZZ ðhþ zc Þdx1 dy1 OPS1 ðxc ,yc ,zc ,R,hÞ ¼ , ð5aÞ 2 2 3=2 2 S1 ½ðd þ x1 Þ þ y1 þ ðh þ zc Þ
OPS2 ðxc ,yc ,zc ,RÞ ¼
ZZ
ð3Þ
The solid angle OPdisk ðxc ,yc ,zc ,RÞ in Eq. (3) is subtended at the point P by the complete bottom base of a cylinder and must be calculated when R4d. This calculation can be performed in the same way as for the solid angle at a point subtended by a circular disk and many analytical or numerical formulas for calculation of OPdisk ðxc ,yc ,zc ,RÞ can be adopted from the reports presented elsewhere (Jaffey, 1954; Masket, 1959; Gardner and Verghese, 1971; Bland, 1984; Olivier and Gagnon, 1993; Tryka, 1997, 1999).
zc dx1 dy1 h i3=2 , ðd þ x1 Þ2 þ y21 þ z2c
S2
OPS,disk ðxc ,yc ,zc ,RÞ ¼
where OPS1 ðxc ,yc ,zc ,R,hÞ, OPS2 ðxc ,yc ,zc ,R,hÞ, and OPS3 ðxc ,yc ,zc ,R,hÞ are the solid angles at the point P subtended by the surfaces S1, S2 and S3, respectively, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ d ¼ x2c þy2c :
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ZZ
zc dx1 dy1 S2 ,disk
S3
h
,
ð5cÞ
i3=2 ,
ð5dÞ
½ðd þ y1 Þ2 þy21 þ z2c 3=2
ZZ
OPS3 ðxc ,yc ,zc ,R,hÞ ¼
ð5bÞ
ðdbÞdy1 dz1 2
ðdbÞ þ y21 þðzc þz1 Þ2
where, from the geometrical relationships shown in Fig. 1, one can see that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð6Þ b ¼ R2 =d ¼ R2 = x2c þ y2c :
3.1. Double integral solid angle representation Transforming the solid-angle surface integrals in Eqs. (5a)– (5d) to line integrals and introducing the boundary conditions for the external contours of S1, S2, S3, and for the disk, being the complete bottom base of the cylinder in Fig. 1, we obtain pffiffiffiffiffiffiffiffiffiffi Z Z b R2 x21 ðhþ zc Þdy1 OPS1 ðxc ,yc ,zc ,R,hÞ ¼ dx1 pffiffiffiffiffiffiffiffiffiffi , 2 2 3=2 2 R2 x21 ½ðd þ x1 Þ þy1 þ ðhþ zc Þ R ð7aÞ
OPS2 ðxc ,yc ,zc ,RÞ ¼
Z
R
dx1
Z
OPdisk ðxc ,yc ,zc ,RÞ ¼
R
dx1
Z
h
dz1 0
pffiffiffiffiffiffiffiffiffiffi 2 R x21
zc dy1 ½ðd þx1 Þ2 þ y21 þz2c 3=2
pffiffiffiffiffiffiffiffiffiffi 2
Z
R x21
R
OPS3 ðxc ,yc ,zc ,R,hÞ ¼
R x21
b
Z
pffiffiffiffiffiffiffiffiffiffi 2
Z
pffiffiffiffiffiffiffiffiffiffi 2 R x21
c c
,
zc dy1 ½ðd þx1 Þ2 þ y21 þz2c 3=2 ðdbÞdy1
½ðdbÞ2 þ y21 þ ðzc þ z1 Þ2 3=2
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R x2c þy2c R2 R d R2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c¼ ¼ d x2c þ y2c
ð7bÞ
,
ð7cÞ
,
ð7dÞ
ð8Þ
is calculated from the geometrical relationships shown by the perpendicular projection of the cylinder on the 0xy plane in Fig. 1. 3.2. Single integral solid angle representation Evaluating the inner integrals in Eqs. (7a)–(7d) one can obtain
OPS1 ðxc ,yc ,zc ,R,hÞ Fig. 1. Perspective view of a cylinder at a distance d from a point P in the case when d 4R and the perpendicular projection of the cylinder on the 0xy plane, illustrating the geometrical dependencies among b, c, d and R.
Z
b
¼ R
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðh þzc Þ R2 x21 dx1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½ðd þx1 Þ2 þ ðh þ zc Þ2 d þ R2 þ2dx1 þðh þ zc Þ2
ð9aÞ
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S. Tryka / Applied Radiation and Isotopes 70 (2012) 2466–2470
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 y21 dx1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi OPS2 ðxc ,yc ,zc ,RÞ ¼ 2 b ½ðd þ x1 Þ2 þz2c d þ R2 þ 2dx1 þ z2c Z
R
2zc
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 y21 dx1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi OPdisk ðxc ,yc ,zc ,RÞ ¼ 2 R ½ðd þx1 Þ2 þ z2c d þR2 þ2dx1 þz2c Z
ð9bÞ
Table 1 Comparison of the solid angles OP C(xc, yc, zc, R, h) obtained by Masket et al. (1957b) and Whitcher (2006) at R ¼1 and h ¼2 with the results computed from Eqs. (3) and (10a)–(10d). The solid angle values were computed here for R, d, and h expressed in relative units. d/R
Masket et al. (1957b)
Whitcher (2006)
Eqs. (3) and (10a)–(10d)
1.2 1.4 1.6 18 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
1.95124 1.54292 1.27070 1.06851 0.91060 0.78374 0.68003 0.59420 0.52251 0.46217 0.41103 0.36742
1.95126 1.54287 1.27052 1.06847 0.91058 0.78371 0.68003 0.59430 0.52246 0.46223 0.41096 0.36741
1.9512354491095338 1.5429096802080182 1.2706886437362856 1.0685028185569470 0.9105899731346520 0.7837372513015295 0.6800269235407871 0.5941989458550273 0.5225080119295502 0.4621632897824739 0.4110233660987604 0.3674135828159619
OPS3 ðxc ,yc ,zc ,R,hÞ ¼
2zc
R
Z
h 0
ð9cÞ
2cðdbÞdz1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðdbÞ þ ðz1 þ zc Þ2 c2 þðdbÞ2 þ ðz1 þ zc Þ2 2
ð9dÞ Eqs. (9a)–(9d) represent general single integral solution to Eqs. (5a)–(5d) for the case when both axes z and z1 of the main and additional coordinate systems 0xyz and 01x1y1z1 are parallel to each other. This solution is very simple and is directly applicable for numerical evaluations of the solid angle values. Thus, the solid angle OP C(xc, yc, zc, R, h) can be calculated by substituting Eqs. (9a)– (9d) into (3) and using one of many algorithms for single definite integrals (Davis and Polonsky, 1972). 3.3. Solid angle representation by elliptic integrals Although Eqs. (9a)–(9d) look very simple, only Eq. (9d) can be integrated into elementary functions whereas Eqs. (9a)–(9d) must
Fig. 2. Solid angle OP C as a function of d and h at R¼ 1 cm and zc ¼ 0 (a), as a function of d and zc at R¼1 cm and h¼1 cm (b), as a function of h and zc at R¼ 1 cm and d¼2 cm (c), as a function of h and R at d¼ 1 cm and zc ¼2 cm (d), as a function of d and R at h¼ 1 cm and zc ¼ 1 cm (e), and as a function of R and zc at d¼ 1 cm and h¼ 1 cm (f). Note that these same results will be obtained when the variables R, d, H, and h are expressed in meters or other dimensional units applied to all variables simultaneously.
S. Tryka / Applied Radiation and Isotopes 70 (2012) 2466–2470
2469
be represented by products of elementary and special functions. Therefore, after integration of Eqs. (9a)–(9d) we obtain
and the parameters n11, n12, n21, and n22 given by
4R OPS1 ðxc ,yc ,zc ,R,hÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pðn11 , f1 9m1 ÞPðn12 , f1 9m1 Þ , 2 2 ðdRÞ þ ðh þ zc Þ
m1 ¼
ð10aÞ 4R
OPS2 ðxc ,yc ,zc ,RÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pðn21 , f2 9m2 ÞPðn22 , f2 9m2 Þ , ðd þ RÞ2 þ z2c
m3 ¼
ðd þ RÞ2 þðh þzc Þ2 2
ðdRÞ þ ðh þ zc Þ
ðdRÞ2 þ z2c þ4dR
4R OPdisk ðxc ,yc ,zc ,RÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pðn21 , f2 9m2 ÞPðn12 , f2 9m2 Þ ðd þ RÞ2 þ z2c 4R 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pðn1 21 , f1 9m3 ÞPðn22 , f1 9m3 Þ , 2 2 ðdRÞ þ zc
ð10cÞ cðhþ zc Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðdbÞ c þ ðdbÞ2 þ ðh þ zc Þ2 czc qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 arctan ðdbÞ c2 þðdbÞ2 þ z2c
ðdRÞ2 þz2c
pffiffiffiffiffiffiffiffiffiffi ! Rd p ffiffiffiffiffiffiffiffiffiffiffi , f1 ¼ arcsin R þd
ð10bÞ
2
n11 ¼
ðh þzc Þ þiðd þ RÞ , ðhþ zc Þ þ iðdRÞ
n21 ¼
ðdRÞizc , ðd þ RÞizc
,
m2 ¼
ðd þ RÞ2 þz2c 4dR ðd þRÞ2 þ z2c
,
, pffiffiffiffiffiffiffiffiffiffiffi! Rþd f2 ¼ arcsin pffiffiffiffiffiffiffiffiffiffi , Rd n12 ¼
n22 ¼
ðd þ RÞ þiðh þzc Þ , ðdRÞ þ iðh þ zc Þ
ðdRÞ þ izc : ðd þ RÞ þizc
Eqs. (10a)–(10d) can be substituted into Eq. (3) to compute the solid angle OP-C (xc, yc, zc, R, h) by using high-language programs such as Mathematica (Wolfram, 1993) with built-in algorithms for elliptic integrals.
OPS3 ðxc ,yc ,zc ,R,hÞ ¼ 2 arctan
4. Examples of the computed results ð10dÞ
where P(n11,f19m1), P(n12,f19m1), P(n21,f29m2), P(n22,f29m2), P(n21,f29m2), P(n12,f29m2), P(n11,f19m3), and P(n11,f19m3) are the incomplete Lagrange–Jacobi elliptic integrals of the third kind expressed by the moduli m1, m2, and m3, amplitudes f1 and f2,
The results presented in this section were obtained from Eqs. (3) and (10a)–(10d) using a simple computer program written in Mathematica (Wolfram, 1993). The computer program consisted of only a small number of commands because Mathematica has built in algorithms for elliptic integrals. The solid angle values were calculated from Eqs. (3) and (10a)–(10d) with an accuracy to 16
Table 2 The values of the solid angle OP-C(xc, yc, zc, R, h) calculated from Eqs. (3) and (10a)–(10d). Note that the values of OP-C(xc, yc, zc, R, h) were obtained for the geometrical variables R, d, zc, and h expressed in cm but the same results are obtained when the variables are expressed in meters or in other dimensional units applied to all variables simultaneously. R (cm)
zc (cm)
h (cm)
d (cm)
0.5
0.0
0.0
0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0
0.5
1.0
2.0
1.0
0.0
0.5
1.0
2.0
2.0
0.0
0.5
1.0
2.0
O P-C (sr)
R (cm)
zc (cm)
p
1.0
0.0
0 0
h (cm)
0.0
p 0.6949701106890006 0.1522403756710433
0.5
p
1.0
0.9105899731346517 0.2697690643604855
p 1.0070196461588579 0.3959078586566225 0.5195350992976788 0.2827082992746628 0.0729295036827604 0.5195350992976788 0.3510759061625970 0.1504679991696342 0.5195350992976788 0.3791379722988689 0.1990682979788974 0.5195350992976788 0.4008328287315142 0.2498058639518810 0.1727677067450237 0.1378472883942611 0.0698040996965741 0.1727677067450237 0.1517355675640833 0.1005174284312810 0.1727677067450237 0.1595421448269064 0.1205416656695577 0.1727677067450237 0.1674980329646975 0.1435674439439300
2.0
1.0 0.0
0.5
1.0
2.0 2.0 0.0
0.5
1.0
2.0
d (cm)
O P-C (sr)
0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0
2p
p 0 2p
p 0.4267018374220097 2p
p 0.6949701106890006 2p
p 0.9105899731346517 1.6371035493454218 1.1226868336113744 0.3258002484534522 1.6371035493454218 1.1226868336113744 0.4665989650184542 1.6371035493454218 1.1226868336113744 0.5414201108991034 1.6371035493454218 1.1226868336113744 0.6097877177870376 0.6228560679715039 0.5195350992976788 0.2827082992746628 0.6228560679715038 0.5195350992976788 0.3252075274876703 0.6228560679715038 0.5195350992976788 0.3510759061625970 0.6228560679715038 0.5195350992976788 0.3791379722988689
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S. Tryka / Applied Radiation and Isotopes 70 (2012) 2466–2470
decimal places within computation time always less than 0.01 s when an IBM PC with 3 GHz CPU was used. However, when another computer program, with no built in algorithms for elliptic integrals, is used, the values of solid angle may be calculated from Eqs. (3) and (9a)–(9d) by applying one of many simple procedures for single definite integrals (Davis and Polonsky, 1972). To demonstrate the applicability of Eqs. (3) and (10a)–(10d), the results computed by Mathematica were compared first in Table 1 with the solid angles reported by Masket et al (1957b) and Whitcher (2006). It is seen that the results are approximately identical to an accuracy of five decimal places. Fig. 2(a)–(f) shows some computer simulations illustrated by three-dimensional surface plots. Generally we observe that the solid angle OP-C (xc, yc, zc, R, h) increases when the radius R increases and the distances d decreases. In addition, the solid angle OP-C (xc, yc, zc, R, h) clearly increases and then remains constant when the height h of the cylinder increases. On the other hand, from Fig. 2(b), (c), and (f), it can be seen that the solid angle OP-C(xc, yc, zc, R, h) increases with decreasing distance zc until zc 4h. Table 2 presents the results computed from Eqs. (3) and (10a)–(10d) to illustrate the dependence of the solid angle OP-C(xc, yc, zc, R, h) on the geometrical variables d, h, R, and zc. These results may serve as reference data for researchers applying existing formulae and tables or checking new formulae and programs for calculation of the solid angle OP-C(xc, yc, zc, R, h). 5. Conclusion An analytical solution to the solid angle equation in the Cartesian coordinate system is presented for the case of the solid angle at a point subtended by a cylinder. This solution was expressed by double line integrals, by single definite integrals, and by some incomplete Lagrange–Jacobi elliptic integrals of the third kind. The third solution was then used for calculation of representative solid angle values for comparison with literature data and graphical illustration. Selected results are listed in Table 2 as highly accurate reference data for potential users. Based on the comparison of the results presented, it is shown that the formulae obtained give very accurate results and are suitable for practical applications. References Berne, E., 1951. The calculation of the geometrical efficiency of end-window ¨ Geiger–Muller tubes. Rev. Sci. Instrum. 22, 509–512. Bland, C.J., 1984. Tables of the geometrical factor for various source detector configurations. Nucl. Instrum. Methods Phys. Res. 223, 602–606. Cipolla, S.J., 2007. Calculating the solid angle for different source shapes and orientations as viewed by a detector with a cylindrical collimator. Nucl. Instrum. Methods Phys. Res. A 579, 268–271. Conway, J.T., 2006. Generalizations of Ruby’s formula for the geometric efficiency of a parallel-disk source and detector system. Nucl. Instrum. Methods Phys. Res. A 562, 146–153.
David, P., White, D.P., Leach, P.G.L., Coville, N.J., 1995. Solid angles 5. Chiral molecules and the case of face selectivity. J. Math. Chem. 18, 99–114. Davis, P.J., Polonsky, I., 1972. Numerical interpolation, differentiation, and integration. 10th ed. In: Abramowicz, M., Stegum, J.A. (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series, vol. 55. National Bureau of Standards, Washington, DC, pp. 875–899. Galiano, E., Rodrigues, M., 2006. A comparison of different analytical methods of determining the solid angle of a circular coaxial source–detector system. Appl. Radiat. Isot. 64, 497–501. Gardner, R.P., Verghese, K., 1971. On the solid angle subtended by a circular disc. Nucl. Instrum. Methods 93, 163–167. Guzei, I., Wendt, M., 2006. An improved method for the computation of ligand steric effects based on solid angles. Dalton Trans. 33, 3991–3999. Jaffey, A.H., 1954. Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables. Rev. Sci. Instrum. 25 (4), 349–354. Kovarik, A.F., Adams Jr., N.I., 1932. A new determination of the disintegration constant of uranium by the method of counting a-particles. Phys. Rev. 40, 718–726. Masket, A.V., 1959. Solid angle subtended by a circular disk. Rev. Sci. Instrum. 30 (10), 950. Masket, A.V.H., Macklin, R.L., Schmitt, H.W., 1957a. Cylinder radiation and activation. Rev. Sci. Instrum. 28 (3), 189–190. Masket, A.V., Macklin, R.L., Schmitt, H.W., 1957b. Tables of solid angles values and activations. ORNL-2170 (USAEC) Oak Ridge National Laboratory, Oak Ridge, Tennessee. Maxwell J.C., 1873. A Treatise on Electricity and Magnetism, Third Edition., vol. 2. Geoffrey Cumberlege, Oxford University Press, London, pp. 33–46, 331–346. Olivier, P., Gagnon, D., 1993. Mathematical modeling of the solid angle function, Part I: approximation in homogeneous medium. Opt. Eng. 32 (9), 2261–2265. Olivier, P., Rioux, S., Gagnon, D., 1993. Mathematical modeling of the solid angle function, Part II: transmission through refractive media. Opt. Eng. 32 (9), 2266–2270. Prata, M.J., 2003. Analytical calculation of the solid angle defined by a cylindrical detector and a point cosine source with orthogonal axes. Radiat. Phys. Chem. 66, 387–395. Prata, M.J., 2004. Analytical calculation of the solid angle defined by a cylindrical detector and a point cosine source with parallel axes. Radiat. Phys. Chem. 69, 273–279. Rowlnds, G., 1961. Solid angle calculations. Int. J. Appl. Radiat. Isot. 10, 86–93. Selim, Y.S., Abbas, M.I., 1996. Direct calculation of the total efficiency of cylindrical scintillation detectors for extended circular sources. Radiat. Phys. Chem. 48 (1), 23–27. Taverner, B.C., 1996. Improved algorithm for accurate computation of molecular solid angles. J. Comput. Chem. 17 (14), 1612–1623. Tryka, S., 1997. Angular distribution of the solid angle at a point subtended by a circular disk. Optics Commun. 137, 317–333. Tryka, S., 1999. A method for calculating the average solid angle subtended by a circular disk from uniformly distributed points within a coaxial circular plane. Rev. Sci. Instrum. 70, 3915–3920. Tryka, S., 2000. Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on planar circular surface coaxial to the disc. J. Mod. Opt. 47, 769–791. Vega-Carrillo, H.R., 1996. Geometrical efficiency for a parallel disk source and detector. Nucl. Instrum. Methods Phys. Res. A 371, 535–537. Webster, A.G., 1927. Partial Differential Equations of Mathematical Physics. G.E. Stechert &CO, NY, pp. 367–368.. Whitcher, R., 2006. A Monte Carlo method to calculate the average solid angle subtended by a right cylinder to a source that is circular or rectangular, plane or thick, at any position. Radiat. Prot. Dosi. 118 (4), 459–474. Wolfram, S., 1993. Mathematica—a System for Doing Mathematics by Computer. Addison-Wesley, Reading, pp. 44–186, 683–707. Zaluzec, N.J., 2009. Detector solid angle formulas for use in X-ray energy dispersive spectrometry. Microsc. Microanal. 15, 93–98.
Contents lists available at SciVerse ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Calculation of the solid angle subtended by a cylinder at a point S. Tryka n University of Life Sciences in Lublin, Department of Agricultural Sciences, Laboratory of Physics, Szczebrzeska 102, PL-22-400 Zamosc, Poland
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 March 2010 Accepted 2 July 2012 Available online 9 July 2012
An analytical formula for calculating the solid angle subtended by a cylinder at a point has been derived from the general solid angle equation. The formula is expressed by double line integrals and by single integrals of simple elementary functions. These functions were then integrated and the formula was represented by products of some elementary functions and the incomplete Lagrange–Jacobi elliptic integrals of the third kind. The final formula was used to calculate representative values of solid angles and compare them with literature data. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Solid angle Isotropic point source Cylindrical detector Source–detector system Elliptic integrals
1. Introduction Solid angle formulas are necessary in a wide class of problems in nuclear physics (Kovarik and Adams, 1932; Berne, 1951; Jaffey, 1954; Masket et al., 1957a,b; Masket 1959; Rowlnds, 1961; Gardner and Verghese, 1971; Bland, 1984; Vega-Carrillo, 1996; Selim and Abbas, 1996; Prata, 2003, 2004; Conway, 2006; Galiano and Rodrigues, 2006; Whitcher, 2006; Cipolla, 2007), optical radiometry (Olivier and Gagnon, 1993; Olivier et al., 1993; Tryka, 1997, 1999, 2000, molecular chemistry (Guzei and Wendt, 2006; Taverner, 1996; David et al., 1995), and other fields of science and technology (Maxwell, 1873; Webster, 1927; Zaluzec, 2009). For instance, when the power of isotropic radiation is measured, it is directly proportional to the solid angle defined by the source–detector system. Similarly, the calculation of particle beams passing through a given aperture, or incident on a given object, requires an expression of a solid angle. In addition, several important electromagnetic problems must be solved using solid angle equations (Rowlnds, 1961; Maxwell, 1873; Webster, 1927). It is well known that the solid angle depends on the geometric configuration of the specific point relative to a given body and on the shape of the body. Therefore, there is a need to calculate solid angles for bodies of different shapes and at any position with respect to the given point. One of many cases, mainly related to the needs of nuclear physics and lighting technology, is the solid angle at a point subtended by a cylinder. Several research papers have been devoted to this problem and various more or less complicated numerical and analytical methods have been presented (Masket et al., 1957a,b; Selim and Abbas, 1996; Prata, 2003,
n
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2004; Whitcher, 2006). Undoubtedly some of these methods are inaccurate and give only approximate results. Therefore, it is necessary to present an analytical solution to the general solid angle equation which can be used for accurate calculation of the solid angle values. The aim of this paper is to provide a simple analytical formula for calculating the solid angle at a point subtended by a cylinder. The formula was obtained by solving the solid angle equation in the Cartesian coordinate system. The solution obtained was first expressed by double definite line integrals and then by single definite integrals of simple elementary functions. This was then integrated into the form of the several Lagrange–Jacobi incomplete elliptic integrals of the third kind. Then some data were computed for selected geometrical configurations of a point with respect to a cylinder. These computed data were compared to the data presented in the literature, illustrated graphically and tabulated.
2. Solid angle equation If the Cartesian coordinate system 0xyz is erected at a point, P, the solid angle, OPS , at point P, subtended by a planar surface, S, which is perpendicular to the z-axis at a distance z from the point P, is defined as follows: ZZ z dx dy OPS ¼ : ð1Þ 3=2 S ðx2 þ y2 þz2 Þ Sometimes, however, the surface S has its symmetry axis parallel to the axis z of the main coordinate system 0xyz. In this case we can introduce an additional Cartesian coordinate system 01x1y1z1 parallel to the main coordinate system 0xyz, and erected at the point 0c, being the center of symmetry of the surface S.
S. Tryka / Applied Radiation and Isotopes 70 (2012) 2466–2470
In these coordinates the solid angle equation becomes ZZ ðzc þ z1 Þdx1 dy1 OPS ðxc ,yc ,zc Þ ¼ , 2 2 2 3=2 ½ðx þx Þ þðy S c 1 c þ y1 Þ þ ðzc þ z1 Þ
ð2Þ
where xc, yc, and zc are the coordinates of the point 0c.
3. General solution to the solid angle equation Applying Eq. (2) to the system shown in Fig. 1, the solid angle,
OP-C (xc, yc, zc, R, h), at the point P subtended by a cylinder with a radius R40 and height hZ0 can be expressed as
OPC ðxc ,yc ,zc ,R,hÞ 8 > > OPS1 ðxc ,yc ,zc ,R,hÞ þ OPS2 ðxc ,yc ,zc ,RÞ > > > > < þ OPS3 ðxc ,yc ,zc ,R,hÞ, ¼ p, > > > > > 2p, > :O Pdisk ðxc ,yc ,zc ,RÞ,
0 o R o d,
arbitrary zc ,
0 o R ¼ d, 0 r d o R,
zc ¼ 0, zc ¼ 0,
0 r d o R,
zc a 0,
It is worth noting that Eq. (3) is also suitable for calculating the solid angle OPC(xc, yc, zc, R, h) in the case when the bottom base of the cylinder, or both the bottom and upper bases of a cylinder, lie under the plane 0xy in Fig. 1. According to Eq. (2), the solid angle OPC(xc, yc, zc, R, h) defined by Eq. (3) can be calculated by introducing an additional Cartesian coordinate system 01x1y1z1 at a distance d from the main coordinate system 0xyz (Fig. 1). In these coordinates the solid angles OPS1 ðxc ,yc ,zc ,R,hÞ, OPS2 ðxc ,yc ,zc ,R,hÞ, OPS3 ðxc ,yc ,zc ,R,hÞ, and OPdisk ðxc ,yc ,zc ,RÞ in Eq. (3) are expressed by ZZ ðhþ zc Þdx1 dy1 OPS1 ðxc ,yc ,zc ,R,hÞ ¼ , ð5aÞ 2 2 3=2 2 S1 ½ðd þ x1 Þ þ y1 þ ðh þ zc Þ
OPS2 ðxc ,yc ,zc ,RÞ ¼
ZZ
ð3Þ
The solid angle OPdisk ðxc ,yc ,zc ,RÞ in Eq. (3) is subtended at the point P by the complete bottom base of a cylinder and must be calculated when R4d. This calculation can be performed in the same way as for the solid angle at a point subtended by a circular disk and many analytical or numerical formulas for calculation of OPdisk ðxc ,yc ,zc ,RÞ can be adopted from the reports presented elsewhere (Jaffey, 1954; Masket, 1959; Gardner and Verghese, 1971; Bland, 1984; Olivier and Gagnon, 1993; Tryka, 1997, 1999).
zc dx1 dy1 h i3=2 , ðd þ x1 Þ2 þ y21 þ z2c
S2
OPS,disk ðxc ,yc ,zc ,RÞ ¼
where OPS1 ðxc ,yc ,zc ,R,hÞ, OPS2 ðxc ,yc ,zc ,R,hÞ, and OPS3 ðxc ,yc ,zc ,R,hÞ are the solid angles at the point P subtended by the surfaces S1, S2 and S3, respectively, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ d ¼ x2c þy2c :
2467
ZZ
zc dx1 dy1 S2 ,disk
S3
h
,
ð5cÞ
i3=2 ,
ð5dÞ
½ðd þ y1 Þ2 þy21 þ z2c 3=2
ZZ
OPS3 ðxc ,yc ,zc ,R,hÞ ¼
ð5bÞ
ðdbÞdy1 dz1 2
ðdbÞ þ y21 þðzc þz1 Þ2
where, from the geometrical relationships shown in Fig. 1, one can see that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð6Þ b ¼ R2 =d ¼ R2 = x2c þ y2c :
3.1. Double integral solid angle representation Transforming the solid-angle surface integrals in Eqs. (5a)– (5d) to line integrals and introducing the boundary conditions for the external contours of S1, S2, S3, and for the disk, being the complete bottom base of the cylinder in Fig. 1, we obtain pffiffiffiffiffiffiffiffiffiffi Z Z b R2 x21 ðhþ zc Þdy1 OPS1 ðxc ,yc ,zc ,R,hÞ ¼ dx1 pffiffiffiffiffiffiffiffiffiffi , 2 2 3=2 2 R2 x21 ½ðd þ x1 Þ þy1 þ ðhþ zc Þ R ð7aÞ
OPS2 ðxc ,yc ,zc ,RÞ ¼
Z
R
dx1
Z
OPdisk ðxc ,yc ,zc ,RÞ ¼
R
dx1
Z
h
dz1 0
pffiffiffiffiffiffiffiffiffiffi 2 R x21
zc dy1 ½ðd þx1 Þ2 þ y21 þz2c 3=2
pffiffiffiffiffiffiffiffiffiffi 2
Z
R x21
R
OPS3 ðxc ,yc ,zc ,R,hÞ ¼
R x21
b
Z
pffiffiffiffiffiffiffiffiffiffi 2
Z
pffiffiffiffiffiffiffiffiffiffi 2 R x21
c c
,
zc dy1 ½ðd þx1 Þ2 þ y21 þz2c 3=2 ðdbÞdy1
½ðdbÞ2 þ y21 þ ðzc þ z1 Þ2 3=2
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R x2c þy2c R2 R d R2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c¼ ¼ d x2c þ y2c
ð7bÞ
,
ð7cÞ
,
ð7dÞ
ð8Þ
is calculated from the geometrical relationships shown by the perpendicular projection of the cylinder on the 0xy plane in Fig. 1. 3.2. Single integral solid angle representation Evaluating the inner integrals in Eqs. (7a)–(7d) one can obtain
OPS1 ðxc ,yc ,zc ,R,hÞ Fig. 1. Perspective view of a cylinder at a distance d from a point P in the case when d 4R and the perpendicular projection of the cylinder on the 0xy plane, illustrating the geometrical dependencies among b, c, d and R.
Z
b
¼ R
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðh þzc Þ R2 x21 dx1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½ðd þx1 Þ2 þ ðh þ zc Þ2 d þ R2 þ2dx1 þðh þ zc Þ2
ð9aÞ
2468
S. Tryka / Applied Radiation and Isotopes 70 (2012) 2466–2470
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 y21 dx1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi OPS2 ðxc ,yc ,zc ,RÞ ¼ 2 b ½ðd þ x1 Þ2 þz2c d þ R2 þ 2dx1 þ z2c Z
R
2zc
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 y21 dx1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi OPdisk ðxc ,yc ,zc ,RÞ ¼ 2 R ½ðd þx1 Þ2 þ z2c d þR2 þ2dx1 þz2c Z
ð9bÞ
Table 1 Comparison of the solid angles OP C(xc, yc, zc, R, h) obtained by Masket et al. (1957b) and Whitcher (2006) at R ¼1 and h ¼2 with the results computed from Eqs. (3) and (10a)–(10d). The solid angle values were computed here for R, d, and h expressed in relative units. d/R
Masket et al. (1957b)
Whitcher (2006)
Eqs. (3) and (10a)–(10d)
1.2 1.4 1.6 18 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
1.95124 1.54292 1.27070 1.06851 0.91060 0.78374 0.68003 0.59420 0.52251 0.46217 0.41103 0.36742
1.95126 1.54287 1.27052 1.06847 0.91058 0.78371 0.68003 0.59430 0.52246 0.46223 0.41096 0.36741
1.9512354491095338 1.5429096802080182 1.2706886437362856 1.0685028185569470 0.9105899731346520 0.7837372513015295 0.6800269235407871 0.5941989458550273 0.5225080119295502 0.4621632897824739 0.4110233660987604 0.3674135828159619
OPS3 ðxc ,yc ,zc ,R,hÞ ¼
2zc
R
Z
h 0
ð9cÞ
2cðdbÞdz1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðdbÞ þ ðz1 þ zc Þ2 c2 þðdbÞ2 þ ðz1 þ zc Þ2 2
ð9dÞ Eqs. (9a)–(9d) represent general single integral solution to Eqs. (5a)–(5d) for the case when both axes z and z1 of the main and additional coordinate systems 0xyz and 01x1y1z1 are parallel to each other. This solution is very simple and is directly applicable for numerical evaluations of the solid angle values. Thus, the solid angle OP C(xc, yc, zc, R, h) can be calculated by substituting Eqs. (9a)– (9d) into (3) and using one of many algorithms for single definite integrals (Davis and Polonsky, 1972). 3.3. Solid angle representation by elliptic integrals Although Eqs. (9a)–(9d) look very simple, only Eq. (9d) can be integrated into elementary functions whereas Eqs. (9a)–(9d) must
Fig. 2. Solid angle OP C as a function of d and h at R¼ 1 cm and zc ¼ 0 (a), as a function of d and zc at R¼1 cm and h¼1 cm (b), as a function of h and zc at R¼ 1 cm and d¼2 cm (c), as a function of h and R at d¼ 1 cm and zc ¼2 cm (d), as a function of d and R at h¼ 1 cm and zc ¼ 1 cm (e), and as a function of R and zc at d¼ 1 cm and h¼ 1 cm (f). Note that these same results will be obtained when the variables R, d, H, and h are expressed in meters or other dimensional units applied to all variables simultaneously.
S. Tryka / Applied Radiation and Isotopes 70 (2012) 2466–2470
2469
be represented by products of elementary and special functions. Therefore, after integration of Eqs. (9a)–(9d) we obtain
and the parameters n11, n12, n21, and n22 given by
4R OPS1 ðxc ,yc ,zc ,R,hÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pðn11 , f1 9m1 ÞPðn12 , f1 9m1 Þ , 2 2 ðdRÞ þ ðh þ zc Þ
m1 ¼
ð10aÞ 4R
OPS2 ðxc ,yc ,zc ,RÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pðn21 , f2 9m2 ÞPðn22 , f2 9m2 Þ , ðd þ RÞ2 þ z2c
m3 ¼
ðd þ RÞ2 þðh þzc Þ2 2
ðdRÞ þ ðh þ zc Þ
ðdRÞ2 þ z2c þ4dR
4R OPdisk ðxc ,yc ,zc ,RÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pðn21 , f2 9m2 ÞPðn12 , f2 9m2 Þ ðd þ RÞ2 þ z2c 4R 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pðn1 21 , f1 9m3 ÞPðn22 , f1 9m3 Þ , 2 2 ðdRÞ þ zc
ð10cÞ cðhþ zc Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðdbÞ c þ ðdbÞ2 þ ðh þ zc Þ2 czc qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 arctan ðdbÞ c2 þðdbÞ2 þ z2c
ðdRÞ2 þz2c
pffiffiffiffiffiffiffiffiffiffi ! Rd p ffiffiffiffiffiffiffiffiffiffiffi , f1 ¼ arcsin R þd
ð10bÞ
2
n11 ¼
ðh þzc Þ þiðd þ RÞ , ðhþ zc Þ þ iðdRÞ
n21 ¼
ðdRÞizc , ðd þ RÞizc
,
m2 ¼
ðd þ RÞ2 þz2c 4dR ðd þRÞ2 þ z2c
,
, pffiffiffiffiffiffiffiffiffiffiffi! Rþd f2 ¼ arcsin pffiffiffiffiffiffiffiffiffiffi , Rd n12 ¼
n22 ¼
ðd þ RÞ þiðh þzc Þ , ðdRÞ þ iðh þ zc Þ
ðdRÞ þ izc : ðd þ RÞ þizc
Eqs. (10a)–(10d) can be substituted into Eq. (3) to compute the solid angle OP-C (xc, yc, zc, R, h) by using high-language programs such as Mathematica (Wolfram, 1993) with built-in algorithms for elliptic integrals.
OPS3 ðxc ,yc ,zc ,R,hÞ ¼ 2 arctan
4. Examples of the computed results ð10dÞ
where P(n11,f19m1), P(n12,f19m1), P(n21,f29m2), P(n22,f29m2), P(n21,f29m2), P(n12,f29m2), P(n11,f19m3), and P(n11,f19m3) are the incomplete Lagrange–Jacobi elliptic integrals of the third kind expressed by the moduli m1, m2, and m3, amplitudes f1 and f2,
The results presented in this section were obtained from Eqs. (3) and (10a)–(10d) using a simple computer program written in Mathematica (Wolfram, 1993). The computer program consisted of only a small number of commands because Mathematica has built in algorithms for elliptic integrals. The solid angle values were calculated from Eqs. (3) and (10a)–(10d) with an accuracy to 16
Table 2 The values of the solid angle OP-C(xc, yc, zc, R, h) calculated from Eqs. (3) and (10a)–(10d). Note that the values of OP-C(xc, yc, zc, R, h) were obtained for the geometrical variables R, d, zc, and h expressed in cm but the same results are obtained when the variables are expressed in meters or in other dimensional units applied to all variables simultaneously. R (cm)
zc (cm)
h (cm)
d (cm)
0.5
0.0
0.0
0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0
0.5
1.0
2.0
1.0
0.0
0.5
1.0
2.0
2.0
0.0
0.5
1.0
2.0
O P-C (sr)
R (cm)
zc (cm)
p
1.0
0.0
0 0
h (cm)
0.0
p 0.6949701106890006 0.1522403756710433
0.5
p
1.0
0.9105899731346517 0.2697690643604855
p 1.0070196461588579 0.3959078586566225 0.5195350992976788 0.2827082992746628 0.0729295036827604 0.5195350992976788 0.3510759061625970 0.1504679991696342 0.5195350992976788 0.3791379722988689 0.1990682979788974 0.5195350992976788 0.4008328287315142 0.2498058639518810 0.1727677067450237 0.1378472883942611 0.0698040996965741 0.1727677067450237 0.1517355675640833 0.1005174284312810 0.1727677067450237 0.1595421448269064 0.1205416656695577 0.1727677067450237 0.1674980329646975 0.1435674439439300
2.0
1.0 0.0
0.5
1.0
2.0 2.0 0.0
0.5
1.0
2.0
d (cm)
O P-C (sr)
0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0 0.5 1.0 2.0
2p
p 0 2p
p 0.4267018374220097 2p
p 0.6949701106890006 2p
p 0.9105899731346517 1.6371035493454218 1.1226868336113744 0.3258002484534522 1.6371035493454218 1.1226868336113744 0.4665989650184542 1.6371035493454218 1.1226868336113744 0.5414201108991034 1.6371035493454218 1.1226868336113744 0.6097877177870376 0.6228560679715039 0.5195350992976788 0.2827082992746628 0.6228560679715038 0.5195350992976788 0.3252075274876703 0.6228560679715038 0.5195350992976788 0.3510759061625970 0.6228560679715038 0.5195350992976788 0.3791379722988689
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decimal places within computation time always less than 0.01 s when an IBM PC with 3 GHz CPU was used. However, when another computer program, with no built in algorithms for elliptic integrals, is used, the values of solid angle may be calculated from Eqs. (3) and (9a)–(9d) by applying one of many simple procedures for single definite integrals (Davis and Polonsky, 1972). To demonstrate the applicability of Eqs. (3) and (10a)–(10d), the results computed by Mathematica were compared first in Table 1 with the solid angles reported by Masket et al (1957b) and Whitcher (2006). It is seen that the results are approximately identical to an accuracy of five decimal places. Fig. 2(a)–(f) shows some computer simulations illustrated by three-dimensional surface plots. Generally we observe that the solid angle OP-C (xc, yc, zc, R, h) increases when the radius R increases and the distances d decreases. In addition, the solid angle OP-C (xc, yc, zc, R, h) clearly increases and then remains constant when the height h of the cylinder increases. On the other hand, from Fig. 2(b), (c), and (f), it can be seen that the solid angle OP-C(xc, yc, zc, R, h) increases with decreasing distance zc until zc 4h. Table 2 presents the results computed from Eqs. (3) and (10a)–(10d) to illustrate the dependence of the solid angle OP-C(xc, yc, zc, R, h) on the geometrical variables d, h, R, and zc. These results may serve as reference data for researchers applying existing formulae and tables or checking new formulae and programs for calculation of the solid angle OP-C(xc, yc, zc, R, h). 5. Conclusion An analytical solution to the solid angle equation in the Cartesian coordinate system is presented for the case of the solid angle at a point subtended by a cylinder. This solution was expressed by double line integrals, by single definite integrals, and by some incomplete Lagrange–Jacobi elliptic integrals of the third kind. The third solution was then used for calculation of representative solid angle values for comparison with literature data and graphical illustration. Selected results are listed in Table 2 as highly accurate reference data for potential users. Based on the comparison of the results presented, it is shown that the formulae obtained give very accurate results and are suitable for practical applications. References Berne, E., 1951. The calculation of the geometrical efficiency of end-window ¨ Geiger–Muller tubes. Rev. Sci. Instrum. 22, 509–512. Bland, C.J., 1984. Tables of the geometrical factor for various source detector configurations. Nucl. Instrum. Methods Phys. Res. 223, 602–606. Cipolla, S.J., 2007. Calculating the solid angle for different source shapes and orientations as viewed by a detector with a cylindrical collimator. Nucl. Instrum. Methods Phys. Res. A 579, 268–271. Conway, J.T., 2006. Generalizations of Ruby’s formula for the geometric efficiency of a parallel-disk source and detector system. Nucl. Instrum. Methods Phys. Res. A 562, 146–153.
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