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
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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 tan1
Ri
hK
and
ðK hÞ
cos b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,
ðK hÞ2 þ R2i
K
(2)
L
g ¼ tan
1
Ro
hK
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 tan1
Ri r
h
h
cos a ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,
h2 þ ðRi rÞ2
(6a)
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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:
nonaxial
Oin
P (, h)
(7d)
where f(x,y) is the maximum azimuthal angle and is
defined as
fmax ðx; yÞ ¼ cos1
L
(7b)
(7c)
f ¼ 2fmax ððh KÞ; R0 Þ for bþ pypgþ ,
-
+
a pypb ,
and
Z
b
sin y dy;
ð9aÞ
a
1
nonaxial
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
nonaxial
Oout
¼ Oinnonaxial þ 2
0
Fig. 2. Non-axial point source–detector configuration.
Ri r
b ¼ tan1
hK
ðK hÞ
cos b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,
ðK hÞ2 þ ðRi rÞ2
(6b)
Ro r
.
(6c)
hK
For a certain value of the polar angle (y), the
azimuthal angle (f) takes the values
g ¼ tan1
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 ,
Ononaxial
¼ Ononaxial
þ2
out
in
Ononaxial
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
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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 nonaxial
Odisk
¼
Oin
r dr,
in
S2 0
is as identified before in Eq. (9b)
where Ononaxial
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 nonaxial
Odisk
¼
Oout
r dr,
(13a)
out
S2 0
S
nonaxial
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((hK),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
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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).
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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((hK),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.
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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.
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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.
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