A direct mathematical method to calculate the efficiencies
of 4πNaI (Tl) scintillation detector
Salam Noureddeena, Mahmoud I. Abbas1, a, b, Mahmoud Koreka
a) Physics Department, Faculty of Science, Beirut Arab University, Beirut, Lebanon
b) Physics Department, Faculty of Science, Alexandria University, 21511 Alexandria, Egypt
Abstract
A direct mathematical method is applied to calculate the total and geometrical
efficiencies of a 4π NaI(Tl) scintillation detector for an arbitrarily positioned radiating
point source. The 4π NaI(Tl) scintillation detector is made up of 8 square large NaI(Tl)
detectors[1]. The central square is left void, see figure (1), so the radiating source can be
moved easily. The results are compared with previous experimental treatment.
Keywords: 4π NaI(Tl) gamma detectors; Geometrical efficiency; Total efficiency and
Extended sources.
1. Introduction
Scintillation detectors are widely applied for gamma-ray detection. Gamma rays could
be emitted from a point, line, disc or cylindrical sources. The detection of these radiations
is done by different detectors [1-12]. These detectors were designed with different
efficiencies, depending on the shape of the detector and on the material of its active
medium. The geometrical and total efficiencies for the source – detector system were
determined experimentally and theoretically [1]. In the present work we will take into
consideration the photon path length through the detector active medium in order to
determine the efficiency of the 4πNaI (Tl) scintillation detector [11]. The work described
below involves the use of straightforward analytical formulae for the computation of the
4πNaI(Tl) detector array geometrical and total efficiencies. Section 2 presents direct
mathematical formulae for the geometrical and total efficiencies in the case of isotropic
radiating axial point, non-axial point, plane and volumetric sources. Section 3 contains
comparisons between the calculated efficiency using the formulae derived in this work
1
Corresponding author, email: mabbas@physicist.net
with the published experimental and simulated values illustrating the validity of the
present mathematical formulae. The conclusions are presented in section 4.
2. Mathematical viewpoint
The position of the isotropic point source is defined by (ρ,h), where,  is the lateral
displacement between the source and the detector axis, whereas h is the height of the
source from its center. The polar (θ) and azimuthal (φ) angles represent the direction of
the photon when it enters the detector active volume. The effective rays which enter the
detector active volume traverse a distance d until it emerges from detector [10].
2.1. Axial point source placed inside the detector void part
The incident photon may enter the detector’s inner side and emerges from see figure (2)
i)
lower base 1(LB1):
d =
1
ii)
(1)
detector side (1):
d =
2
iii)
a - a1
2 sin(θ ) cos(φ)
(2)
lower base (2) (LB2):
d =
3
iv)
a
c
2 sin(θ ) cos(φ) cos(θ )
b
c
2 sin(θ ) cos(φ) cos(θ )
(3)
detector side (2):
d =
4
b - b1
2 sin(θ ) cos(φ)
(4)
The polar angle θ takes the steps
a
  tan - 1 1
1
(5)
a
(6)
2c
θ 2 = tan - 1
θ3 = tan
2c
- 1 b1
(7)
2c
b
θ 4 = tan - 1 1
2c
(8)
The azimuthal angle φ takes the steps
2
Φ
1
= tan - 1
b
a
(9)
Φ
π
-1 a
=
+
tan
2
2
b
Φ
= π + tan - 1
3
a
Φ4
(10)
b
=
(11)
a
3π
+ tan - 1
2
b
(12)
The total efficiency can be calculated by
1 

 
1 2

 
  /2
1 / 2
2 4
2
∫ ∫f1dd  ∫ ∫f 2dd  ∫ ∫f3dd  ∫ ∫f 4dd
 3
 4
0
0 
1
1
1
1



 
3  / 2
3  /2
4 4
4 4
f dd  ∫ ∫f dd  ∫ ∫f dd  ∫ ∫f dd
∫ ∫
4
1
3
3
 1
 2
 3
 3
2
1
3
3
2  2
(13)
2  / 2
 ∫∫f1dd  ∫ ∫f 2dd
 
0 
4 1
f i = (1 - e
- μdi
) sin( θ ) . e
-
2
μ1.t
sin( θ )
, i=1,2….10.
(14)
where, μ and μ1 are the total attenuation coefficient of the detector and its housing
[13](with thickness t cm) material for gamma-ray photon with energy Eγ, respectively.
2.2 Non-axial point source placed inside the detector void part
The incidence photon may enter the detector’s inner side and emerges from see
figure (3)
i) lower base 1 (LB1):
d1 =
a 2 ρ cos(Φ)
c
2 sin( θ ) cos(Φ) cos(θ )
(15)
ii) detector side (1):
a - a1
d2 =
2 sin( θ ) cos(Φ)
(16)
iii) lower base 2 (LB2):
d3 =
a + 2 ρ cos(Φ)
c
- 1
cos(θ ) 2 sin( θ ) cos(Φ)
(17)
3
iv) detector side 2
a - a1
2 sin( θ ) cos(Φ)
d4 =
(18)
v) detector side 3
b - b1
2 sin( θ ) cos(Φ)
d5 =
(19)
vi) lower base 3 (LB3)
d6 =
b1
c
ρ sin( α)
+
cos(θ ) 2 sin(θ ) sin( Φ) sin(θ )
(20)
vii) detector side 4
d7 =
b - b1
2 sin( θ ) cos(Φ)
viii)
(21)
lower base 4 (LB4)
b1
c
ρ sin( α)
cos(θ ) 2 sin( θ ) sin( Φ) sin( θ )
d8 =
(22)
The final expression of the total efficiency of a non-axial point source at different
positions is given by:
 
5 2
  /2
2 2
5
 2  ∫ ∫f dd  ∫ ∫
f 1 dd  ∫ ∫
f 2 dd
1
 
0 

0
8 1
1
 
  /2
2  / 2
7 8
7
 ∫ ∫f 2 dd  ∫ ∫f 3 dd  ∫ ∫f 4 dd
2

8


2
 
6 7
 
6 4
(23)

6 8
 
8 6

  /2
  /2
6
8
∫ ∫f 5 dd  ∫ ∫f 6 dd  ∫∫f 7 dd  ∫ ∫f 8 dd
 4
 3
 5
 
5
5
7
7 6
The polar angle θ takes the steps
a - 2 ρcos(α)
-1 1
θ1 = tan
θ 2 = tan - 1
θ 3 = tan - 1
(24)
2.c. cos(Φ)
a - 2 ρcos(α)
(25)
2.c. cos(Φ)
b1 - 2 ρsin( α)
(26)
2.c. sin( Φ)
4
θ 4 = tan - 1
θ5 = tan - 1
θ 6 = tan - 1
θ 7 = tan - 1
θ8 = tan - 1
b - 2 ρsin( α)
(27)
2.c. sin( Φ)
b1 + 2 ρsin( α)
(28)
2.c. sin( Φ)
b + 2 ρsin( α)
(29)
2.c. sin( Φ)
a1 + 2 ρ cos(α)
(30)
2.c. cos(Φ)
a + 2 ρ cos(α)
(31)
2.c. cos(Φ)
The azimuthal angle φ takes the steps
 a



 - sin( ) 
2
1


  tan
5
b

 - cos( ) 


2

(32)
b



   cos( ) 

2
1

   tan 
6 2
 b




sin(

)


 2

(33)



    tan - 1
7




(34)

a
  sin ( ) 
2


b
-  cos( ) 
2

 a



 -  cos( ) 
3
2
1


 
 tan
8 2
b

   sin ( ) 


2

(35)
2.3 Line source placed inside the detector void part
The efficiency of a 4πNaI (Tl) scintillation detector arising from a line of length ℓ cm is
derived as [3]
h2
ξ1dh
∫
ξL =
h1
(36)
h2 - h1
where, ξ1 is the total efficiency of an axial point source as identified before in equation
13. The total efficiency of the line source is determined at different positions along the
axial axis of the detector.
5
2.4 Disc source placed inside the detector void part
The efficiency of 4πNaI (Tl) scintillation detector arising from a disc source is derived as
[3]
s
2 ∫
 2  d
D 
0
(37)
s2
Where ξ2 is the total efficiency of a non-axial point source as identified before in
equation 23. The total efficiency of the disc source is determined at different positions
along the axis of the detector.
2.5 Cylindrical source placed inside the detector void part
The efficiency of 4πNaI (Tl) scintillation detector arising from a cylindrical source is
derived as [3]
h2
ξ D dh
∫
ξC =
h1
(38)
h2 - h1
where ξD is the total efficiency of a disc source as identified before in equation 37.
3. Results
The direct mathematical method is applied for both geometrical and total efficiencies
for the 4πNaI(Tl) detector array using isotropic point source. The 4πNaI scintillation
detector is made up of 8 square large NaI(Tl) detectors as shown in Figure 1. The central
square is left void so the radiating source can be moved easily. The cross sectional size
and the length of a NaI(Tl) crystal are 10.2 cm×10.2 cm and 40.6 cm, respectively. The
housing of the NaI(Tl) crystal is 1 mm thick Stainless steel. The solid angle of the
detection geometry is high, so that more than 95% can reach the detector surface. The
total efficiency values of a point source placed at different positions along the axial axis
and at different energies have been calculated for 4πNaI (Tl) scintillator detector by the
direct mathematical method and compared with Byun et al. [1, 12] as shown in figures 4
and 5. The total efficiency is also determined at lateral distances ρ from the central axis
of the detector as shown in figure (6). The figure shows no variation of the total
6
efficiency as the lateral distance changes for the same height. Three different sources
non-axial point, line source of length 5 cm and disc source of radius 2 cm have been used
in the theoretical calculations for this detector. As shown in figures 7, 8 and 9 as the
height of the source from the center increases the total efficiency decreases for the
different sources which is consistent with the results obtained by Byun et al [1, 12]. In
our work the height is taken from zero which is the center to 20.3 cm which is at the
surface of the detector. The same calculations are repeated again on the lower side of the
detector and the graph is drawn between the efficiency and the position. It is clear also
that the total efficiency for disc source is greater than that of line and point sources. The
percentage difference between the calculated values and the measured ones is given by
Diff (%) 
 cal   meas  *100
(39)
 meas
4. Conclusions
Direct mathematical expressions to calculate the total and the full-energy peak
efficiencies of 4πNaI(Tl) scintillation detector have been derived in the case of axial
point, non-axial point and extended to line, disk and cylindrical sources. In addition, the
attenuation of photons by the source container and the detector housing materials is also
presented in simple straight forward mathematical expressions. The agreement between
the results calculated in this work and the published values is very good, the high
discrepancies being less than 4 % (point source).
7
Sample access
Fig (1)
TThreeThe 4π NaI(Tl) scintillation detector
is made up of 8 square large NaI(Tl) detectors
dimension
al view of
the
cylindrical
Phoswich
scintillatio
n detector
Dector
8
Fig (2) The position of point source for an axial point source
9
Fig (3) The position of point source for a non- axial point source.
10
Present work
Experimental work
1.0
Relative Total Efficiency
0.9
0.8
0.7
0.6
0.5
0.4
0.3
-20
-10
0
10
20
Source position in cm
Fig (4) The variation of the relative total efficiency with energy for a
point source moves around the center of the detector
.
Present work
Experimental
0.95
0.90
0.85
Total Efficiency
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Energy in Mev
Fig (5) The variation of the total efficiency with energy for a point source placed at the
center of the detector
11
0.5
Total effeciency
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
Lateral distance p (cm)
Fig (6) The variation of the total efficiency with lateral distance
Total efficiency
Relative total efficiency
1.00
0.95
Efficiency
0.90
0.85
0.80
0.75
0.70
0.65
0.60
-20
-10
0
10
20
Position in (cm)
Fig (7) The variation of the total and relative efficiency with the position of the
point source placed at the non-axial point
12
Total efficiency
Relative total efficiency
1.00
0.95
Efficiency
0.90
0.85
0.80
0.75
0.70
0.65
0.60
-20
-10
0
10
20
Position in (cm)
Fig (8) The variation of the total and relative efficiency with the position of the
line source placed at the axis of the detector
Total Effeciency
Relative Total Effeciency
1.0
Effeciency
0.9
0.8
0.7
0.6
0.5
-20
-10
0
10
20
Position in (cm)
Fig (9) The variation of the total and relative efficiency with the position of the disc source
placed at the axial point
13
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