International Journal of Instrumentation Science 2013, 2(2): 13-24 DOI: 10.5923/j.instrument.20130202.01
Calculation of Full Energy Peak Efficiency of NaI (Tl) Detectors by New Analytical Approach for Cylindrical Sources Ahmed M. El-Khatib, Mohamed S. Badawi, Mona M. Gouda* , Sherif S. Nafee, Ekram A. El-Mallah Physics Department, Faculty of Science, Alexandria University, Alexandria, 21511, Egypt
Abstract A new analytical approach for calcu lation of the full-energy peak efficiency of NaI (Tl) is deduced. In addition, self attenuation of the source matrix, the attenuation by the source container and the detector housing materials are considered in the mathematical treat ment. Results are co mpared with those measured by two cylindrical NaI (Tl) detectors with Resolution (FWHM ) at 662 keV equal to 7.5% and 8.5%. 152 Eu aqueous radioactive sources covering the energy range fro m 121 keV to 1408 keV were used. By comparison, the calculated and the measured full-energy peak efficiency values were in a good agreement.
Keywords Nai (Tl) Scintillation Detectors, Cy lindrical Sources, Fu ll-Energy Peak Efficiency, Self-Attenuation
1. Introduction NaI (Tl) detectors are co mmon ly used to ident ify and measure activ it ies of lo w-level radioact ive sources. They h ave h igh det ect ion efficien cy an d o p erat e at roo m temperature[1]. The determination of the activity for each radionuclide requ ires p rior kno wledge o f the fu ll-energy peak efficiency at each photon energy for a given measuring g eo met ry , wh ich mus t b e o bt ain ed by an efficien cy calibrat ion using known standard sources of exact ly the same geometrical dimensions, density, and chemical co mposition of the sample under study. Standard radioactive samples, if availab le, are cost ly and wou ld n eed to b e renewed , especially when the radionuclides have short half-lives[2]. In addition, the influence of the source matrix on the counting efficiency can be exp ressed by the self-attenuation factor, describing the fraction of gamma-rays not registered in the full-energy peak due to scattering or absorption within the sample[3]. An effect ive tool to overco me these problems cou ld b e th e use o f co mp ut at ion al t ech n iqu es[4-9] to comp lete calib rat ion of the gamma spectro met ry system. Also, Selim and Abbas [10-14] so lved th is prob lem by derived direct analytical integrals of the detector efficiencies (t ot al and fu ll-energy peak) fo r any so u rce-det ecto r configuration and imp lemented these analytical expressions into a numerical integration computer program. Moreover, * Corresponding author:
[email protected] (Mona M. Gouda) Published online at http://journal.sapub.org/instrument Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved
they introduced a new theoretical approach[15-17] based on that Direct Statistical method to determine the detector efficiency for an isotropic radiating point source at any arbitrary position fro m a cylindrical detector, as well as the extension of this approach to evaluate the volumetric sources. In the present work, authors will mod ify this simp lified approach to determine the full-energy peak efficiency of the co-axial detector with respect to point and volumetric sources, taking into account the attenuation by the dead layer (reflector) and the end-cap material o f the detector, the self attenuation by the source matrix and finally, the attenuation by the source container material.
2. Mathematical Viewpoint 2.1. The Case of a Non-axi al Point Source Consider a right circular cylindrical (2R × L), detector and an arbitrarily positioned isotropic radiating point source located at a distance h fro m the detector top surface, and at a lateral d istance ρ fro m its axis. The efficiency of the detector with respect to point source is given as follows[16]:
ε po int = f att ε g ε i
(1)
Where εi and εg are the intrinsic and the geometrical efficiencies wh ich are derived by Abbas et al., [16]. fatt is the attenuation factor of the detector dead layer and end cap material. In section 2.1.2, this factor will be recalculated by a new method which is dependent on calculating the average path length within these materials.
14
Ahmed M . El-Khatib et al.: Calculation of Full Energy Peak Efficiency of NaI (Tl) Detectors by New Analytical Approach for Cylindrical Sources
2.1.1. The Intrinsic ( εi ) and the Geo metrical ( εg ) Efficiencies The intrinsic and geometrical efficiencies are represented by Eqs. (2) and (3) respectively.
ε i = 1 − e− µ d εg =
(2)
Ω 4π
(3)
Where d is the average path length travelled by a photon through the detector, Ω is the solid angle subtended by the source-detector and they are represented by Eqs. (4) and (5) respectively. μ is the attenuation coefficient of the detector material. ∫ d (θ ,ϕ ) d Ω ∫ ∫ d (θ ,ϕ )sin θ dθ dϕ
d =
ϕθ
Ω =
∫
ϕθ
There are two main cases to be considered for calculat ing the intrinsic and geometrical efficiencies of the detector with respect to point source, viz., (i) the lateral d isplacement of the source is smaller than or equa0l the detector circular face’s radius (ρ≤R) and (ii) the lateral distance of the source is greater than the detector circular face’s radius (ρ>R). The two cases have been treated by Abbas et al.,[16]. The values of the polar and the azimuthal angles based on the source to detector configuration are shown in Table 1.
(4)
Ω
dΩ
Where θ and φ are the polar and the azimuthal angles respectively. d (θ,φ) is the possible path length travelled by the photon within the detector active volume. Ω =∫ ∫ sin θ dθ dϕ (5)
Ω
Table 1. The Values of the Polar and the Azimuthal Angles Based on the Source to Detector Configuration The polar angles
The azimuthal angles
R−ρ h+ L R−ρ θ 2 = tan −1 h θ3 = tan −1 R + ρ h+L θ 4 = tan −1 R + ρ h
θ1 = tan −1
θc = tan −1
2 2 2 2 ϕmax = cos−1 ρ − R + h tan θ 2ρ h tan θ 2 ρ − R2 + (h + L)2 tan 2 θ ϕmax ′ = cos−1 2ρ (h + L) tan θ
h + L 2 2 ρ − R h
ρ 2 − R2
θc′ = tan −1
ϕc = sin −1
R
ρ
2 2 −1 ρ − R = θT tan = (ϕmax ϕmax ′ ) h (h + L)
2.1.2. The Attenuation Factor ( fatt ) The attenuation factor fatt is expressed as:
f att = f lay f cap
(6)
where f lay and f cap are the attenuation factors of the detector dead layer and end cap material respectively and they are given by:
where While
µ
−µ
lay
and
µ
δ
lay lay f lay e= , f cap e = cap
δ lay and δ cap
− µcap δ cap
(7)
are the attenuation coefficients of the detector dead layer and the end cap material, respectively.
are the average path length travelled by a photon through the detector dead layer and end cap
material, respectively. They are represented as follow:
International Journal of Instrumentation Science 2013, 2(2): 13-24
= δ lay
15
∫ ∫ t′(θ ,ϕ )sin θ dθ dϕ ϕ∫ θ∫ t′(θ ,ϕ )sin θ dθ dϕ
ϕθ
= ∫ ∫ sin θ dθ dϕ
Ω
ϕθ
δ cap =
(8)
∫ ∫ t′′(θ ,ϕ )sin θ dθ dϕ ϕ∫ θ∫ t′′(θ ,ϕ )sin θ dθ dϕ ϕθ = ∫ ∫ sin θ dθ dϕ
Ω
ϕθ
Table 2. The Possible Path Lengths and the Average Path Length Traveled by the Photon Within the Dead Layer for Cases ρ≤R and ρ>R ρ≤R
ρ>R
t1′ =
t t1′ = DF cos θ
≅
δ lay =
Z1 I2
ρ cos ϕ +
t2′ = −
Z1 = +
∫0 ∫0 t1′ sin θ dθ dϕ
(R +t ) DS
2
− ρ 2 sin 2 ϕ
sin θ
ρ cos ϕ +
( R)
2
− ρ 2 sin 2 ϕ
sin θ
ρ 2 sin 2 ϕ 1+ 2R2
tD S
sin θ
δ lay =
= Z3 π θ2
t DF cos θ
+
ϕmax θ4
Z3 I4
θ2 ϕmax ′
∫ ∫0
θ1
t2′ sin θ dϕ dθ +
∫ ∫0
θ2
θc ϕc
θ4 ϕmax
θc′
θc
∫ ∫0 t1′ sin θ dϕ dθ + ∫ ∫0 θc′ ϕmax ′
θ dθ dϕ ∫0 ∫ t1′ sin = Z3 ∫ ∫0 θ2 θ
t2′ sin θ dϕ dθ +
1
+
θc′ ϕmax ′
θ2 ϕc
∫ ∫0 t2′ sin θ dϕ dθ
θc′
θ4 ϕmax
θ2
θc
∫ ∫0 t1′ sin θ dϕ dθ + ∫ ∫0
thickness tD S , (see Figure 1). The possible path lengths and the average path length travelled by the photon within the dead layer for cases (ρ≤R) and (ρ >R) are shown in Table 2, where t1′ and t2′ represents the photon path length through the upper and the side surface of the dead layer respectively. Consider the thickness of upper and side surface of the detector end cap material is ta and tw respectively, as shown in Figure 1. The possible path lengths and the average path
( θ2 ≥ θ′ c )
t1′ sin θ dϕ dθ
θc ϕc
Where t ′(θ , ϕ ) and t ′′(θ , ϕ ) are the possible path lengths travelled by the photon within the detector dead layer and end cap material, respectively. Consider the detector has a dead layer by covering its upper surface with thickness t DF and its side surface with
t1′ sin θ dϕ dθ
( θ2 < θ′ c)
t1′ sin θ dϕ dθ
length travelled by the photon within the detector end cap material for cases (ρ≤R) and (ρ>R) are shown in Table 3, and t2′′ represents the photon path length where t′′ 1 through the upper and the side surface of the detector end cap material, respectively. Fro m Table 4 we observe that, the case in which (ρ>R) has two sub cases which are (R < ρ ≤ Ra) and (ρ > Ra), where Ra is the inner radius of the detector end cap. There is a very important polar angle (θcap ) which must be considered when we study the case in wh ich (ρ > Ra) which is θcap and this is given by: ρ − Ra h−k
θcap = tan −1
(9)
Where k is the d istance between the detector end cap and the detector upper surface.
16
Ahmed M . El-Khatib et al.: Calculation of Full Energy Peak Efficiency of NaI (Tl) Detectors by New Analytical Approach for Cylindrical Sources
Where I1 and I2 are the numerator and the denominator of
d equation obtained by Abbas et al.(2006) for non axial point source (ρ≤R). α is the angle between the lateral distance ρ and the detector’s major axis. The geometrical efficiency εg is given by: H + ho 2π S
∫ ∫ ∫I
εg =
ho
2
( ρ ≤ R) ρ d ρ dα dh
0 0
(12)
2π
The new forms of the average path length travelled by the photon through the detector dead layer and the detector end cap material are given by Eqs. (13) and (14) respectively. H + ho 2π S
∫ ∫ ∫ Z1 ρ d ρ dα dh
δ lay =
ho 0 0 H + ho 2π S
∫ ∫ ∫ I 2 ( ρ ≤ R) ρ d ρ dα dh
Figure 1. A diagram of a cylindrical –type detector with a non-axial point source (ρ>R)
2.2. The Case of a Co-axial Cylindrical Source The efficiency of a cylindrical detector with rad ius R and height L using a cylindrical source with radius S and height H is given by:
ε cyl =
S self S sc f att ε i ε g
(10)
ho
2.2.1. Cy lindrical Source with Radius Smaller than or Equal to the Detector’s Radius (S≤R) The intrinsic and geometrical efficiencies are as identified before in Eqs. (2) and (3) respectively, but the average path length d traveled by the photon through the detector active volume and the solid angle will have new forms due to the geometry of the volu metric source, as shown in Figure 2. The average path length is expressed as: H + ho 2π S
∫ ∫ ∫ I ( ρ ≤ R) ρ d ρ dα dh h
Z1
is as identified befo re in Table 2.
where
Z1′
is as identified befo re in Table 3.
In the case of a co-axial cy lindrical source with rad ius S smaller than the detector radius R, there are t wo photon possible path lengths to leave the source as follow: ⅰ. To exit fro m the base
t1 =
o
0
0
H + ho 2π S
∫ ∫
ho
0
∫ I 2 ( ρ ≤ R) ρ d ρ dα dh 0
h − ho cos θ
(15)
ⅱ. To exit fro m the side t2 =
ρ cos ϕ + S 2 − ρ 2 sin 2 ϕ sin θ
(16)
Where h o is the distance between the source active volume and the detector upper surface. The polar and the azimuthal angles will take the values: S−ρ h − ho S+ρ θ3′ = tan −1 h − ho ρ 2 − S 2 + (h − ho )2 tan 2 θ ϕs = cos−1 2(h − ho ) ρ tan θ
θ1′ = tan −1
(R2 − S 2 ) (R2 − ρ 2 ) − θT = tan −1 ho (h − ho ) h(h − ho )
(17)
(ϕS == ϕmax at θ θT )
Where φs is the maximu m azimuthal angle for the photon to emerge fro m the bottom of the source. H + ho 2π S
∫ ∫ ∫ Z1′ ρ d ρ dα dh
1
d =
0 0
where
V
where V is the volu me of the cylindrical source (V = πS2 H), S self is the self attenuation factor of the source matrix and S sc is the attenuation factor of the source container material. If the source is over the surface of the detector, one has to consider two main cases. The first one is the case where the radius of the source is smaller than or equal to the detector circular face’s radius (S≤R) and the second one is the case where the radius of the source is greater than to the detector circular face’s rad ius (S>R), so that, one will d iscuss Eq. (10) for each case separately as follo w:
(13)
(11)
δ cap =
0 0 ho H + ho 2π S
∫ ∫0 ∫0 I 2 ( ρ ≤ R) ρ d ρ dα dh
ho
(14)
International Journal of Instrumentation Science 2013, 2(2): 13-24
1
Table 3. The Possible Path Lengths and the Average Path Length Traveled by the Photon Within the Detector End Cap Material for Cases ρ≤R and ρ>R ρ≤R
ρ>R
R < ρ ≤ Ra
t1′′ = ta cos θ
ta t1′′ = cos θ
δ cap =
Z1′ I2
δ cap =
Z 3′ I4
π θ2
Z1′ = ∫ 0
+
∫0 t1′′sinθ dθ dϕ
ϕmax θ4
∫0 θ∫ t1′′sinθ dθ dϕ 2
′ θ c′ ϕ max
∫ ∫
θ1
+
∫ ∫ t1′′sin θ dϕ dθ
θ c′ 0
+
θ 4 ϕ max
∫ ∫ θ
∫ ∫
θc
0
0
θ 4 ϕmax
∫ ∫ θ c
= Z 3′ t1′′sin θ dϕ dθ
2
2
ρ2
2 Ra 2 sin θ
≅
sin 2 ϕ
θ c ϕc
t1′′sin θ dϕ dθ + ∫ ∫ t1′′sin θ dϕ dθ
θ1 ≥ θcap
θ c′ 0
t1′′sin θ dϕ dθ
0
θcap ϕmax ′
∫ ∫ θ 1
tw 1 +
− ρ sin ϕ 2
Z 3′ I4
′ θ c′ ϕmax
+
θ c ϕc
− ρ 2 sin 2 ϕ
sin θ
1
0
( Ra )
ρ cos ϕ +
δ cap =
t1′′sin θ dϕ dθ
2
sin θ
= Z 3′ Z 3′ =
( Ra + tw )
ρ cos ϕ +
t2′′ =
−
ta cos θ
t2′′ sin θ dϕ dθ +
0
′ θc′ ϕmax
∫ ∫ θ cap
θ c ϕc
θ 4 ϕmax
θ c′ 0
θc
+ ∫ ∫ t1′′sin θ dϕ dθ + ∫
∫
International Journal of Instrumentation Science 2013, 2(2): 13-24
t1′′ =
ρ > Ra
t1′′sin θ dϕ dθ
0
θ
′ c > θcap
t1′′sin θ dϕ dθ
0
17
Ahmed M . El-Khatib et al.: Calculation of Full Energy Peak Efficiency of NaI (Tl) Detectors by New Analytical Approach for Cylindrical Sources
′ θ c′ ϕmax
θ1
+
θ c ϕc
θ 4 ϕmax
1
0
∫ ∫
∫ ∫
∫ ∫ t ′′ sin θ dϕ dθ 2
θ 4 ϕmax
∫ ∫
t2′′ sin θ dϕ dθ +
∫ ∫ θ 0
t2′′ sin θ dϕ dθ
θ4> θcap ≥ θc
t1′′sin θ dϕ dθ
0
θ c ϕc
∫θ ∫ t ′′ sin θ dϕ dθ 2
c′
0
θ 4 ϕmax
c
θ c ϕc
θ cap
′ θ c′ ϕmax
1
+
t2′′ sin θ dϕ dθ +
0
∫ ∫ θ
t1′′sin θ dϕ dθ
θ c′ 0
0
θ cap ϕmax θc
t2′′ sin θ dϕ dθ +
θcap ≥ θ′c
0
c
′ θ c′ ϕmax
= Z 3′
2
0
t ′′sin θ dϕ dθ + ∫ ∫ ∫ ∫ θ θ
θ1
+
∫ ∫ t ′′ sin θ dϕ dθ
θ c′
0
cap
= Z 3′
t2′′ sin θ dϕ dθ +
θ cap ϕc
0
θcap ≥ θ4
Ahmed M. El-Khatib et al.: Calculation of Full Energy Peak Efficiency of NaI (Tl) Detectors by New Analytical Approach for Cylindrical Sources
∫ ∫
= Z 3′
18
2
International Journal of Instrumentation Science 2013, 2(2): 13-24
19
The self attenuation factor of the source matrix is given by: −µ t s (18) S =e self Where μ s is the attenuation coefficient of the source matrix and t is the average path length travelled by a photon inside the source and is given by: H + ho π S
∫ ∫0 ∫0 g1 ρ d ρ dα dh
t = H +h ho oπ S
∫ ∫0 ∫0 I 2 ( ρ ≤ R) ρ d ρ dα dh
(19)
ho
with
g1 = ∫ ∫ t (θ ,ϕ ) sin θ dθ dϕ ϕθ
(20)
Figure 2. The possible cases of the photon path lengths through source – detector system (S≤R)
There are three cases for the values of g 1 according to the values of the polar angle θi as follo w: ⅰ. The case in which (θ4 > θ3 ′ and θ2 > θ1 ′) θ1′
θ3′
θ2 π
0
θ1′
θ1′ 0
g1 =π ∫ t1 sinθ dθ + ∫ ϕs t1 sinθ dθ + −
θ3′ ϕs
θ4 ϕmax
∫ ∫ t2 sinθ dϕ dθ + ∫ ∫
θ1′ 0
θ2
ⅱ. The case in which (θ3 ′ ≥ θ4 and θ1 ′ ≥ θ2 )
= g1 π
θ2
∫ ∫ t2 sinθ dϕ dθ (21)
t2 sinθ dϕ dθ
0
θ4
∫0 t1 sin θ dθ + θ∫ ϕmax t1 sin θ dθ
(22)
2
ⅲ. The case in which (θ 3 ′ ≥ θ 4 and θ 1 ′ < θ 2 ) θ1′ θT
g1 = π ∫ t1 sinθ dθ + 0
+
θ2 π
θ4
∫ ϕs t1 sinθ dθ + θ∫ ϕmax t1 sinθ dθ
θ1′
T
θT ϕs
θT ϕmax
∫ ∫ t2 sinθ dϕ dθ − θ∫ ∫0 t2 sinθ dϕ dθ + θ∫ ∫0
θ1′ 0
1′
(23)
t2 sinθ dϕ dθ
2
If t B is the source container bottom thickness and t K is the source container side thickness, so, there are two photon possible path lengths to exit fro m the source container as follo w: ⅰ. Ⅰ. To exit fro m the base
t1′′′= ⅱ. Ⅱ. To exit fro m the side
tB cosθ
(24)
20
Ahmed M . El-Khatib et al.: Calculation of Full Energy Peak Efficiency of NaI (Tl) Detectors by New Analytical Approach for Cylindrical Sources
( S + tK )
ρ cos ϕ +
= t2′′′
2
− ρ 2 sin 2 ϕ
−
sin θ
≅
tK 1 +
(S )
ρ cos ϕ +
2
− ρ 2 sin 2 ϕ
sin θ
ρ sin 2 ϕ 2
(25)
2
2S sin θ
The polar and the azimuthal angles can take the values:
− ρ h − h0
S +t
θ1′′= tan −1
K
S +t + ρ K θ3′′ = tan −1 h − h0 ρ 2 − (S + t )2 + (h − h )2 tan 2 θ o K ϕs′ = cos−1 2(h − ho ) ρ tanθ ( R 2 − ( S + t K )2 ) ( R 2 − ρ 2 ) tan −1 θT′ = − ho (h − ho ) h(h − ho )
(26)
(ϕS′ = ϕmax at θ = θT′ )
Where φs ' is the maximu m azimuthal angle for the photon to emerge fro m the bottom of the source container. The attenuation factor of the container material is given by:
Ssc = e
− µc tc
(27)
Where μ c is the attenuation coefficient of the source container material and
tc is the average path length travelled by a
photon inside the source container and is expressed as:
H + ho π S
tc =
∫ ∫ ∫ g1c ρ d ρ dα dh
ho 0 0 H + ho π S
(28)
∫ ∫0 ∫0 I 2 ( ρ ≤ R) ρ d ρ dα dh
ho
with
g1c = ∫ ∫ t ′′′(θ ,ϕ )sin θ dθ dϕ
(29)
ϕθ
There are three cases for the values of g 1c according to the values of the polar angles θi as follo w: ⅰ. The case in which (θ4 > θ3 ″ and θ2 > θ1 ″) θ3′′ θ1′′ θ2 π
g1c =π ∫ t1′′′sinθ dθ + ∫ ϕs′ t1′′′sinθ dθ + θ1′′
0
θ3′′ϕs′
−∫
θ4 ϕmax
∫ t2′′′ sinθ dϕ dθ + ∫ ∫
θ1′′ 0
θ2
0
ⅱ. The case in which (θ3 ″ ≥ θ4 and θ1 ″ ≥ θ2 )
= g1c π
θ4
0
θ2
ⅲ. The case in which (θ3 ″ ≥ θ4 and θ1 ″ < θ2 )
+
∫ ϕmax t1′′′sin θ dθ
θ1′′
θT′
θ4
0
θ2 π
θ1′′ θT′ ϕs′
θT′
θ1′′ 0
1′′
g1c =π ∫ t1′′′sinθ dθ +
(30)
t2′′′ sinθ dϕ dθ
θ2
∫ t1′′′sin θ dθ +
∫ ∫ t2′′′ sinθ dϕ dθ
θ1′′ 0
(31)
∫ ϕs′ t1′′′sinθ dθ + ∫ ϕmax t1′′′sinθ dθ θT′ ϕmax
∫ ∫ t2′′′ sinθ dϕ dθ − θ∫ ∫0 t2′′′ sinθ dϕ dθ + θ∫ ∫0 2
(32)
t2′′′ sinθ dϕ dθ
International Journal of Instrumentation Science 2013, 2(2): 13-24
Z1 and Z 3 are as identified before in Table 2.
where
2.2.2. Cy lindrical Source with Radius Greater than the Detector’s Radius (S>R)
21
H + ho 2π R
δ cap = H +h
∫
S
∫ ∫0 ( ∫0 Z1′ ρ d ρ + R∫ Z3′ ρ d ρ ) dα dh h o
o 2π
R
S
0
0
R
ho
(36)
∫ ( ∫ I2 (ρ ≤ R)ρ d ρ + ∫ I2 (ρ > R)ρ d ρ ) dα dh
Where Z1′ and Z 3′ are as identified befo re in Table 3. In the case of a co-axial cylindrical source with radius greater than the radius of the detector, there are t wo probabilit ies to be considered; the first probability that the lateral d istance of the source is smaller than the detector circular face rad ius, i.e. ρ ≤ R and the second probability that the lateral distance of the source is greater than the detector circular face rad ius, i.e. ρ > R and in the two cases, there is only one path to the photon for the way out fro m the source which is exit fro m the base and is given by:
t1 =
h − ho cosθ
(37)
Where h o is the distance between the source active volume and the detector upper surface. The self attenuation factor of the source matrix S self is as identified before in Eq. (18) and the average path length source is given by:
Figure 3. The possible cases of the photon path lengths through source – detector system (S>R)
ho + H π R
ho + H π S
ho 0 0
ho 0 R ho + H π S
∫
=t t (= t1)
t travelled by a photon inside the
ho + H π R
∫ ∫ g1 ρ d ρ dα dh +
∫ ∫ I2 (ρ ≤ R)ρ d ρ dα dh +
∫
ho 0 0
The average path length d travelled by the photon where through the detector active volume and the solid angle will take new forms due to the geomet ry of the volu metric= source, g1 as shown in Figure 3. The average path length is expressed as: H + ho 2π R
S
∫ ∫0 ( ∫0 I1 (ρ ≤ R)ρ d ρ + R∫ I1 (ρ > R)ρ d ρ ) dα dh
d = H +h ho
o 2π R
S
0
R
∫ ∫0
ho
g2= (33)
( ∫ I 2 ( ρ ≤ R) ρ d ρ + ∫ I 2 ( ρ > R) ρ d ρ ) dα dh
=
Where I1 and I2 are the numerator and the denominator of d equation obtained by Abbas et al.(2006) for non axial point source. α is the angle between the lateral distance ρ and the detector’s majo r axis. The geo metrical efficiency εg is given by: H + ho 2π R
εg =
S
∫ ∫0 ( ∫0 I2 (ρ ≤ R)ρ d ρ + R∫ I2 (ρ > R)ρ d ρ ) dα dh (34)
ho
2π
The new forms of the average path length travelled by the photon through the detector dead layer and the detector end cap material are given by Eqs. (35) and (36) respectively.
δlay =
H + ho 2π R
S
0 0
R S
∫
ho H + ho 2π R
∫ (∫ Z1 ρ d ρ + ∫ Z3 ρ d ρ ) dα dh
∫ ∫0 (∫0 I2 (ρ ≤ R) ρ d ρ + R∫ I2 (ρ > R) ρ d ρ ) dα dh
ho
π
θc′ ϕmax ′
∫ ∫
θ1 0
θ2
θ4
0
θ2
∫ t1 sin θ dθ +
∫ ∫ ∫ g2 ρ d ρ dα dh ∫
(38)
∫ ∫ I2 (ρ > R)ρ d ρ dα dh
ho 0 R
∫ ϕmax t1 sin θ dθ (39)
θc ϕc
θ4 ϕmax
θc′ 0
θc o
t1 sinθ dϕ dθ + ∫ ∫ t1 sinθ dϕ dθ + ∫
θc′
θc
θ4
θ1
c′
c
∫
t1 sinθ dϕ dθ (40)
′ t1 sinθ dθ + ∫ ϕc t1 sinθ dθ + ∫ ϕmaxt1 sinθ dθ ∫ ϕmax θ θ
If t B is the source container bottom thickness, so, there is only one path of the photon for the way out fro m the source container which is the exit fro m the base and is given by:
t1′′′=
tB cosθ
(41)
The attenuation factor of the container material S sc is as identified before in Eq. (27) and the average path length tc travelled by a photon inside the source container is exp ressed as: tc = t (t1′′′) (42)
3. Experimental Setup 35)
The full-energy peak efficiency values are carried out for two NaI (Tl) detectors with resolutions 8.5% and 7.5% at the 662 keV peaks of 137 Cs labeled as Det.1 and Det.2
22
Ahmed M . El-Khatib et al.: Calculation of Full Energy Peak Efficiency of NaI (Tl) Detectors by New Analytical Approach for Cylindrical Sources
respectively. The manufacturer parameters and the setup values are shown in Table 4. The sources are polypropylene (PP) plastic v ials of volumes 25 mL and 400 mL filled with an aqueous solution containing 152 Eu radionuclide which emits γ-ray in the energy range fro m 121 keV to 1408 keV, Table 5 shows sources dimensions. The efficiency measurements are carried out by positioning the sources over the end cap of the detector. In order to minimize the dead time, the activity of the sources is prepared to be (5048 ± 49.98 Bq). The measurements are carried out to obtain statistically significant main peaks in the spectra that are recorded and processed by winTMCA 32 software made by ICx Technologies. Measured spectrum which saved as spectrum ORTEC files can be opened by ISO 9001 Genie 2000 data acquisition and analysis software made by Canberra. The acquisition time is high enough to get at least the number of counts 20,000, wh ich make the statistical uncertainties less than 0.5%. The spectra are analy zed with the program using its automatic peak search and peak area calculat ions, along with changes in the peak fit using the interactive peak fit interface when necessary to reduce the residuals and error in the peak area values. The peak areas, the live time, the run time and the start time for each spectrum are entered in the spreadsheets that are used to perform the calcu lations necessary to generate the efficiency curves. Table 4. The Manufacturer Parameters and the Setup Values Items Manufacturer Serial Number Detector Model Type Mounting Resolution (FWHM) at 662 keV Cathode to Anode voltage Dynode to Dynode Cathode to Dynode T ube Base Shaping Mode Detector Type Weight (kg) Crystal Volume(cm3) Crystal Diameter (mm) Crystal Length (mm) Top cover Thickness(mm) Side cover Thickness(mm) Reflector – Oxide (mm) Outer Diameter(mm) Outer Length(mm)
De t.1 Canberra 09L 654 802 Cylindrical Vertical 8.5% +1100 V dc +80 V dc +150 V dc Model 2007 Gaussian NaI (T l) 0.77 103 50.8 50.8 Al (0.5) Al (0.5) 2.5 57.2 53.9
De t.2 Canberra 09L 652 802 Cylindrical Vertical 7.5% +1100 V dc +80 V dc +150 V dc Model 2007 Gaussian NaI (T l) 1.8 347.64 76.2 76.2 Al (0.5) Al (0.5) 2.5 80.9 79.4
Table 5. Parameters of the Sources Items Outer diameter (mm) Height (mm) Wall thickness (mm) Activity (Bq)
Source Volume (mL) 25 400 32.1 113.89 36.21 42.25 1.2 2.03 5048 ± 49.98 5048 ± 49.98
4. Results and Discussion The full-energy peak efficiency values for all NaI (Tl) detectors are measured as a function of the photon energy using the following equation ε (E) =
N(E) ∏Ci T A S P(E)
(43)
Where N(E) is the number of counts under the full-energy peak that is determined using Genie 2000 software, T is the measuring time (in second), P(E) is the photon emission probability at energy E, A S is the radionuclide activity and Ci are the correction factors due to dead time and radionuclide decay. In these measurements of low activ ity sources, the dead time always less than 3%. So the corresponding factor is obtained simp ly using ADC live time. The statistical uncertainties of the net peak areas are s maller than 0.5% since the acquisition time is long enough to get the number counts at least 20,000 counts. The background subtraction is done. The decay correction Cd fo r the calibration source fro m the reference time to the run time is given by:
Cd = eλ ΔT
(44)
Where λ is the decay constant and ΔT is the time interval over which the source decays corresponding to the run time. The main source of uncertainty in the efficiency calcu lations is the uncertainties of the activities of the standard source solutions. Once the efficiencies have been fixed by applying the correction factors, the overall efficiency curve is obtained by fitting the experimental points to a polynomial logarith mic function of the fifth order using a non linear least square fit[18]. In th is way, the correlation between data points from the same calib rated source has been included to avoid the overestimation of the uncertainty in the measured efficiency. The uncertainty in the full-energy peak efficiency σε is given by: 2
2
2
∂ε ∂ε ∂ε 2 2 2 σ= ε σ A + σ P + σ N (45) ε A P N ∂ ∂ ∂
Where σA , σP and σN are the uncertainties associated with the quantities A S, P(E) and N(E) respectively. Figs. 4 and 5 show the full-energy peak efficiencies for both NaI (Tl) detectors (Det.1 and Det.2) wh ich include measured, calculated with S self and calculated without Sself for cylindrical sources (25 mL and 400 mL) p laced at the end cap of the detector as functions of the photon energy. Obviously, the non inclusion of the self attenuation factor in the calculations caused an increase in the fu ll energy peak efficiency values. So to get correct results; the self attenuation factor must be taken into consideration. The percentage deviations between the calculated (with and without S self ) and the measured full-energy peak efficiency values are calculated by:
Δ1 % =
εcal-with S -ε meas self ×100 εcal-with S self
(46)
International Journal of Instrumentation Science 2013, 2(2): 13-24
Δ2 % =
ε cal-without S -ε meas self ×100 ε cal-without S
23
(47)
self
where εcal-with Sself, εcal-without Sself and ε meas are the calculated with / without self attenuation factor and experimentally measured efficiencies, respectively. Table 6 shows the comparison between the percentage deviations Δ1 % and Δ2 % for different volu mes placed at the end cap of NaI (Tl) detectors.
Figure 5. The full-energy peak efficiencies of a NaI(Tl) detector (Det.2); measured, calculated with Sself and calculated without Sself for different cylindrical sources placed at the end cap of the detector as functions of the photon energy
5. Conclusions
Figure 4. The full-energy peak efficiencies of a NaI(Tl) detector (Det.1); measured, calculated with Sself and calculated without Sself for different cylindrical sources placed at the end cap of the detector as functions of the photon energy Table 6. The comparison between the percentage deviations Δ1% and Δ2% for different volumes placed at the end cap of different NaI (T l) detectors Source Volume (mL) De tector
Ene rgy (ke V)
De t.2
400
Δ1 %
Δ2 %
Δ1 %
Δ2 %
0.11
20.30
-0.20
30.45
244.69
0.25
16.91
0.71
26.10
344.28
-0.48
14.47
-1.31
21.91
443.97
1.33
14.80
0.10
21.19
778.90
-1.05
9.87
-0.54
16.43
121.78
De t.1
25
964.13
1.51
11.19
0.51
15.82
1112.11
0.44
9.60
-1.52
13.14
1408.01
0.34
8.50
0.48
13.32
121.78
0.03
20.31
-0.06
31.34
244.69
0.14
16.87
-0.43
25.94
344.28
0.01
14.93
0.02
23.57
443.97
-0.63
13.14
-0.55
21.27
778.90
0.10
10.93
-1.36
16.24
964.13
0.80
10.58
-0.07
15.78
1112.11
-0.39
8.88
-1.37
13.70
1408.01
0.40
8.58
0.45
13.67
A direct analytical approach for calcu lating the fu ll-energy peak efficiency has been derived. In addition, the self attenuation factor of source matrix, the attenuation factors of the source container and the detector housing materials have been calculated. The discrepancies between calculated with S self and measured full-energy peak efficiency values were found to be less than (1.5%) wh ile, between calculated without S self and measured full-energy peak efficiency values were found to be less than (32%). The examination of the present results as given in tables and figures reflects the importance of considering the self attenuation factor in studying the efficiency of any detector using volu metric sources.
ACKNOWLEDGEMENTS The authors would like to exp ress their sincere thanks to Prof. Dr. Mah moud. I. Abbas, Faculty of Science, Alexandria Un iversity, for the very valuable pro fessional guidance in the area of radiation physics and for his fruitful scientific co llaborations on this topic. Dr. Mohamed. S. Badawi would like to introduce a special thanks to The Physikalisch-Technische Bundesanstalt (PTB) in Braunschweig, Berlin, Germany for fru itful help in preparing the homemade volu metric sources.
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Ahmed M . El-Khatib et al.: Calculation of Full Energy Peak Efficiency of NaI (Tl) Detectors by New Analytical Approach for Cylindrical Sources
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