Applied Radiation and Isotopes 200 (2023) 110946
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Applied Radiation and Isotopes
journal homepage: www.elsevier.com/locate/apradiso
A semi–empirical method for efficiency calibration of an HPGe detector
against different sample densities
Islam M. Nabil a,b , K.M. El-Kourghly c ,∗, A.F. El Sayed d
a
Radiation Measurements Department, Main Chemical Laboratories, Cairo, Egypt
Physics Department, Faculty of Science, Fayoum University, Fayoum, Egypt
Nuclear Safeguards and Physical Protection Department, Egyptian Atomic Energy Authority (EAEA), Cairo, Egypt
d
Physics Department, Faculty of Science, Cairo University, Cairo, Egypt
b
c
ARTICLE
INFO
Keywords:
Gamma spectrometry
ANGLE-3
Semi-empirical method
Peak efficiency equation
ABSTRACT
In this work, a semi-empirical equation in terms of 𝛾-energy, and sample density is derived, proposed,
benchmarked, and applied for the peak efficiency calibration of an HPGe detector with respect to an axial
source-to-detector configuration. The samples are in the form of cone-shaped Marinelli beakers of different
densities in the range 0.7–1.6 g∕cm3 . The method employs the experimental measurements with the ANGLE-3
software calculations using the efficiency transfer method. The peak efficiency curve of an HPGe detector is
calculated using the experimental measurements of point-like sources (133 Ba, 137 Cs, and 60 Co). The ANGLE-3
software is then used to calculate the peak efficiency curves for samples with different densities in the 𝛾-energy
range 81–1332 keV. The peak efficiency curves are then fitted to get the energy coefficient; in addition, a
linear relationship is then constructed between the energy coefficients and sample densities to get the density
coefficients, and the derived equation as well.
The derived equations are benchmarked using the peak efficiency curves by ANGLE-3 software in
comparison with that the equation results. The results are found to be in agreement with an average relative
error of about 1.5%. In addition, the derived equations are applied to estimate the activity concentration of
radionuclides present in 5 cone-shaped samples with different densities using experimental measurements. The
activity results are found to be in agreement with the certified values with an average relative error of about
2%. The limitation of the proposed equations is also discussed with respect to different material densities and
different chemical compositions and correction factors for material composition self-attenuation for various
materials are also presented.
1. Introduction
In the field of nuclear safeguards, nuclear security, and radiation
safety, gamma spectroscopy is used for quantitative and qualitative
analysis. It could be used for characterization and in-field verification
of nuclear materials and radioactive sources that could give rise to
radiation risk as well. Furthermore, it could be used in the field of
nuclear forensics, to find out the material’s origin, date, and place
of production, the age of the material, i.e. the date when it was last
chemically processed or purified, and if possible, the intended use of
the material.
The HPGe detector is preferred for such analysis due to its superior
energy resolution compared to other detector types (e.g., NaI, CZTe,
. . . etc.) (ALMisned et al., 2022; Hossain et al., 2012; Zakaly et al.,
2019). Quantitative analysis to estimate the isotopic activity/mass
content requires peak efficiency calibration curves. Such calibration
could be obtained either using relative or absolute methods.
The absolute methods (e.g., General Monte Carlo (MC) code) account for varieties of geometries and matrices, especially in the nuclear
fuel cycle material could exist in many forms ranging from powder
barrels in the mining and milling processes up to heterogeneous materials in the form of fresh and spent fuel assemblies. Accurate results
with good statistics obtained using the MC code require much time
and advanced computer systems which is not in reach to everyone.
The hybrid-analytical method proposed by El-Gammal et al. (2020) and
its application (El-Kourghly et al., 2021) employs the MC calculation
with a derived analytical formula to calculate the peak efficiency of
different source shapes and positions with respect to the detector.
The later one effectively reduces the run-time with good statistics as
mentioned by the authors. Unfortunately, the calibration accuracy in
∗ Corresponding author.
E-mail addresses: eaea_nsncrc@yahoo.com, kamel.elkourghly@icloud.com (K.M. El-Kourghly).
https://doi.org/10.1016/j.apradiso.2023.110946
Received 22 March 2023; Received in revised form 6 July 2023; Accepted 13 July 2023
Available online 23 July 2023
0969-8043/© 2023 Elsevier Ltd. All rights reserved.
Applied Radiation and Isotopes 200 (2023) 110946
I.M. Nabil et al.
Table 1
𝛾-standard point like sources specification.
the aforementioned methods mainly depends on the input parameters
provided by the manufacturer or the detector operational conditions
which is difficult to be obtained in most cases. So different methods
are used to account for these parameters including, X-ray computed
tomography (Khedr et al., 2019; Lee et al., 2023; Kaya et al., 2022;
Zhang et al., 2022), and trial and error method to optimize some
detector parameters (Abdelati et al., 2018). Optimizing the detector
parameters is a tedious process and may take a long time due to the
availability of the used techniques.
Furthermore, ISOCS software is used for efficiency calibration purposes to estimate the isotopic mass content (Abdelati et al.), and the
U-235 enrichment (Tekin et al., 2022). But due to the long operation conditions some parameters of the detector could be changed
(e.g., dead layer), unfortunately, the ISOCS software could not account
for it which results in discrepancies between the assayed sample and
the estimated data.
The relative calibration technique overcomes some of the demerits
of absolute calibration methods, it provides accurate results as well.
However, the lack of standard sources with the same characteristics as
the assayed sample could lead to a significant inaccuracy in the efficiency calibration curve. Consequently, the Efficiency Transfer Method
(ETM) is introduced by Moens et al. (1981) and applied by many
authors (Moens and Hoste, 1983; Mihaljević et al., 1993; Wang et al.,
1995, 1997; Jiang et al., 1998) to calculate the peak efficiency curves of
cylindrical detectors with respect to different source shapes including
point, disk, cylindrical and Marinelli-beaker. The ETM combines the derived analytical formula of the effective solid angle with the measured
peak efficiencies for point-like sources.
ANGLE-3 software (Jovanovic et al., 2010) is introduced as an
application of Meon’s method (Moens et al., 1981) for peak efficiency
calibration of the detector with respect to different source geometries
and different matrices to overcome the lack of standard calibration
sources.
Although the ANGLE-3 software’s interactive window is simple to
use, providing direct derived equations that could be easily used for
peak efficiency calibration could save a lot of time and, in some cases,
replace the presence of the software itself.
In the present work, the ANGLE-3 software calculations are used
in collaboration with experimental measurements to derive the peak
efficiency equations for an HPGe with respect to Marinelli beaker
samples. The proposed equation will account for the matrix density and
chemical composition of the sample with different densities in the range
0.7–1.6 g/cm3 . The derived equation is benchmarked with the aid of
the ANGLE-3 software, it is also validated using experimental measurements. The limitations of the derived equations are also discussed in
the present paper.
Nuclide
Half-Life
(Year)
133
Ba
10.5
137
Cs
30.17
60
Co
5.27
𝛾-Energy
(keV)
𝐼𝛾 (%)
81.0
302.9
356
661.7
1173.2
1332.5
34.06
18.30
62.05
85.10
99.97
99.99
Activity
(kBq)
Production
date
106.5
111
22 May 2012
98.6
The standard point-like sources with about 3 mm active diameter
of radionuclides 133 𝐵𝑎, 137 𝐶𝑠, and 60 𝐶𝑜 produced by ritverc-online
(2015) and Abdelati et al. (2018) and encapsulated in Al of (25 mm
Diameter × 3 mm thickness) embedded into polyamides foiled with
a total thickness of 100 ± 10 𝑚𝑢m (Nakazawa et al., 2010). Table 1
presents the specification of the source including, their half-life, significant energies with their relative abundance, activities, and production
date. These sources are used to energy calibrate an HPGe detector, as
well as the reference peak efficiency calculations 𝛺𝑟𝑒𝑓 to be utilized
for efficiency transfer purposes. The sources are measured using an
axial S–D configuration at 8 cm from the detector front facet of the
detector. The measurement lifetime was about 3600s. Furthermore,
for the purpose of validating the derived peak efficiency equations a
total of five-volume sources in the form of a Marinelli beaker of 1 mm
polyethylene thickness containing multi 𝛾-emitters (137 𝐶𝑠, and 60 𝐶𝑜)
impeded in an epoxy matrix. The beakers are in the form of coneshaped cups with a 7.6 cm height, 7.9 cm top diameter, and 3.6 cm
bottom diameter.
The source matrix is epoxy. The specifications of the source are
presented in Table 2 which includes their half-life, energies with their
relative abundance, and initial activities. The samples are measured
using the configuration described in Fig. 1 for 86,400 s.
The experimental setup configuration as well as the accusation time
for all the aforementioned measurements are optimized in such a way
that the dead time is about 2%, while the net peak count rate statistics
do not exceed 1.5%.
The Gamma-Vision software is then used to analyze the familiar gamma-ray spectra for point sources for efficiency calibration,
taking into account the corrected activity. The efficiency curve for
the point sources is then employed as a reference to estimate the
efficiency curve for other sample geometries using the ANGLE-3 software (Nakazawa et al., 2010). Furthermore, the obtained 𝛾-spectra
for volume sources were then analyzed to get the net count rates of
the 𝛾-energy lines 661.7,1173.2, and 1332.5 keV corresponding to the
radionuclides (137 𝐶𝑠 and 60 𝐶𝑜) are employed to estimate the activities
based on ANGLE-3 and the derived equations. The activities of the
radionuclides are corrected in consideration of the production date and
initial activities according to the following equation (Grabska et al.,
2022; Knoll, 2010),
2. Experimental measurements
The experimental measurements are used for the derivation and
validation of the peak efficiency equations using the net count rates
of 𝛾-standard sources (point, and cone-shaped) located axially and ⟂ to
the symmetry axis of an HPGe detector.
An ORTEC P-type HPGe 100% of cylindrical coaxial crystal configuration is used for the current experimental measurements. The detector
model is GEM100-PA95 with a crystal dimension of 79.4 mm diameter
93.4 mm height, 100% relative efficiency, and 1.32 keV FWHM with respect to the 122 keV 𝛾-energy line. It operates with +4600 bias voltage
in the 𝛾-energy range 15–3000 keV with a multi-channel analyzer unit
(DSPEC-jr.2) (Elsayed et al., 2021a,b; Nabil et al., 2022). It is connected
to the electric cooling system and shielded with a low background
shield. The spectrometer is controlled using the acquisition and data
analysis software Gamma Vision 6.09 (Ortec-online, 2012). The HPGe
detector is used for measuring the 𝛾-spectrum of different radionuclides
present in point and volume standard sources.
𝐴 = 𝐴0 × 𝑒−𝜆𝑡
(1)
where,
A is the corrected activity for time t, 𝐴0 is the production activity,
and 𝜆 is the decay constant for the particular nuclide.
3. ANGLE-3 calculations
The ANGLE-3 software is used for the full energy peak count calculation based on the ETM. The calculations are performed for an HPGe
with respect to a cone-shaped Marinelli beaker axially located and ⟂ to
the symmetry axis of the detector (Ho et al., 2022).
The experimental peak efficiency 𝜀𝑝,𝑟 curve as a function of 𝛾-energy
is estimated using point-like sources (133 𝐵𝑎, 137 𝐶𝑠, and 60 𝐶𝑜) positioned
2
Applied Radiation and Isotopes 200 (2023) 110946
I.M. Nabil et al.
Fig. 1. Experimental setup arrangement for an HPGe with respect to an axial V-s and P-s.
Table 2
𝛾-standard Cone-shaped Marinelli beakers specification.
Sample
ID
𝑆1
𝑆2
𝑆3
𝑆4
𝑆5
Material
Epoxy
Density
(g/cm3 )
Nuclide
Half-life
(Year)
𝛾-Energy
(keV)
𝐼𝛾
(%)
Activity
(Bq)±U (%)
0.7
1
1.17
1.25
1.6
137
30.17
661.7
85.10
4403±3.2%
1173.2
99.86
5076.4±3.1%
1332.5
99.98
5076.4±3.1%
Cs
60 Co
5.272
Fig. 3. Peak efficiency curves Vs. energy as calculated by the ANGLE-3 for different
sample densities.
Fig. 2. Peak efficiency curve vs. energy as calculated experimentally for point-like
sources.
4. Method
at 8 cm away from the front facet of the detector, and the peak
efficiency based on experimental measurements presented in Fig. 2 is
used as an input parameter in the ANGLE software.
The effective solid angle is then calculated for the reference experimental setup 𝛺𝑟 and also for a given S–D distance 𝛺𝑥 in this way, the
peak efficiency curve as a function of 𝛾-energy could be constructed
for the considered configuration assuming a virtual peak-to-total ratio
using the following equation,
𝜀𝑝,𝑥
𝛺
= 𝜀𝑝,𝑟 𝑥
𝛺𝑟𝑒𝑓
The proposed methodology is based on the semi-empirical calibration techniques using the ETM in which, the efficiency curve for
point-like sources is constructed and then transferred to a volume
source using the ANGLE-3 software. The efficiency curves are calculated considering a total of 5 cone-shaped Marinelli beakers of different
densities located symmetrically and ⟂ the symmetry axis of an HPGe.
(2)
The obtained efficiency curves are then fitted using an exponential
second-order degree polynomial. Two equations are generated for each
The peak efficiency curves are calculated for samples with different
densities in the range 0.7–1.6 g/cm3 to be used for derivation, and
benchmarking the proposed equations.
sample density considering two energy regions above and below the
knee point (Liye et al., 2006). The peak efficiency 𝜖 at a given energy
3
Applied Radiation and Isotopes 200 (2023) 110946
I.M. Nabil et al.
Fig. 4. An example that demonstrates the linear fitting polynomial function for the sample density vs. the energy coefficients.
Table 3
The equation coefficient (𝜒𝑖 ) of samples efficiency created by ANGLE-3.
line (E in keV) could be expressed as follows;
𝐿𝑛(𝜖) =
𝑛
∑
𝜒𝑖 (𝜌)(𝑙𝑛(𝐸))𝑖
Density (g/cm3 )
(3)
𝜒𝑜
𝑖=0
0.70
1.00
1.17
1.25
1.60
where, i = 0, 1, . . . n is the Polynomial order, and 𝜒𝑖 (𝜌) is the polynomial
coefficients that differ with samples density, this coefficient could be
obtained for each density by fitting Eq. (3) to the mathematically
calculated efficiency values for that particular density. the coefficient
𝜒𝑖 (𝜌) could be expressed in a polynomial form as a function of the
sample density as below,
𝜒𝑖 (𝜌) =
𝑙
∑
𝜆𝑖𝑘 𝜌𝑘
(4)
k is the polynomial, (g/cm3 )
𝑛 ∑
𝑙
∑
𝜆𝑖𝑘 𝜌𝑘 (𝑙𝑛(𝐸))𝑖
𝜒2
Below
Above
Below
Above
Below
−3.347
−3.648
−3.799
−3.866
−4.129
−41.426
−41.766
−41.926
−41.990
−42.247
0.490
0.540
0.563
0.573
0.610
15.180
15.230
15.249
15.257
15.275
−0.083
−0.085
−0.086
−0.086
−0.087
−1.490
−1.490
−1.490
−1.490
−1.487
Table 4
The values of the density coefficient (𝜆𝑖𝑘 ) obtained using a linear polynomial fitting.
HH k
0
1
0
1
i
Above
Below
H
H
0
-0.868
-2.765
0.910
-40.830
1
0.132
0.404
0.106
15.117
2
-0.004
-0.081
0.003
-1.493
𝑘=0
𝐿𝑛(𝜖) =
𝜒1
Above
(5)
𝑖=0 𝑘=0
energy range with a single polynomial equation. The polynomial coefficients are determined for each energy region separately using the
following equation:
The polynomial order i and k could be obtained from the polynomial
fit of the calculated data. Thus knowing the constants 𝜆𝑖𝑘 the peak
efficiency could be obtained for a wide range of 𝛾-energies for different matrix densities using the general efficiency Eq. (5). The derived
equation is then benchmarked using the ANGLE-3. Furthermore, the
derived equation is validated by estimating the activity of the sample
based on the proposed equation as well as experimental measurements
and comparing the results to that of the certified values.
𝜖 = exp [
2
∑
𝜒𝑖 (𝜌)(𝑙𝑛(𝐸))𝑖 ]
(6)
𝑖=0
and the energy coefficient values 𝜒𝑖 are presented in Table 3.
The density factor is introduced into Eq. (6), using a linear polynomial fitting function between the energy coefficients and the sample
densities as shown in Fig. 4.
5. Results and discussion:
𝜒𝑖 (𝜌) =
In this work, the peak efficiency equation is derived based on experimental measurements as well as mathematical calculations. The peak
efficiency is calculated for 5 cone-shaped Marinelli beakers of different
densities in the 𝛾-energy range 0–1332 keV. ANGLE-3 software is used
to calculate the peak efficiency for the S–D configuration described in
Section 3. Fig. 3 shows the peak efficiency calibration curves in the
𝛾-energy range 0–1332 keV at different sample densities in the range
0.7–1.6 g∕cm3 .
The curves are then fitted using two quadratic polynomial equations
considering two energy regions; below (𝐸 ≤ 120 keV) and above (𝐸 >
120 keV) the knee point due to the difficulty of covering the selected
1
∑
𝜆𝑖𝑘 𝜌𝑘
(7)
𝑘=0
𝜆𝑖𝑘 indicates the density coefficients and their values are presented in
Table 4.
Thus, using Eq. (7) into Eq. (6) the general equation for peak
efficiency calculation can be written as follows,
𝜖 = exp [
2 ∑
1
∑
𝜆𝑖𝑘 𝜌𝑘 (𝑙𝑛(𝐸))𝑖 ]
(8)
𝑖=0 𝑘=0
Eq. (8) could be used for the peak efficiency calculation of an HPGe
detector with respect to a cone-shaped Marinelli beaker in terms of
4
Applied Radiation and Isotopes 200 (2023) 110946
I.M. Nabil et al.
Fig. 5. Peak efficiency vs. Energy comparison between the ANGLE-3 software and derived equations.
Table 5
The estimated activity’s relative difference based on ANGLE-3 and the derived equation
in comparison with the certified values.
Sample ID
𝑆1
𝑆2
𝑆3
𝑆4
𝑆5
Relative difference, (%)(
𝐶𝑎 −𝐶𝑒
𝐶𝑒
×100)
661.7 keV
1173.2 keV
1332.5 keV
ANGLE
Derived eqn.
ANGLE
Derived eqn.
ANGLE
Derived eqn.
3.98
0.56
−1.93
−2.66
−0.15
2.24
−1.74
−4.39
−5.19
−2.92
2.86
−3.42
−3.01
−2.84
−2.62
2.36
−4.33
−4.12
−4.01
−4.01
2.35
−2.58
−2.91
−3.41
−3.64
3.40
−2.02
−2.53
−3.11
−3.56
lines corresponding to the 137 Cs, and 60 Co isotopes are acquainted and
the activity (𝐴𝑐 ) is then estimated based on the following equation:
𝐴𝑐 =
𝐶𝑅
𝐼𝛾 𝜖𝑎𝑏𝑠
(9)
𝐶𝑅 indicates the net count rate obtained using experimental measurements, 𝐼𝛾 is the 𝛾-energy line branching ratio, and 𝜖𝑎𝑏𝑠 is the absolute
Peak efficiency at a certain energy based on Eq. (8).
The estimated activities based on Eq. (8) are then compared to the
ANGLE-based, and certified values as shown in Fig. 6. The comparison
results show an agreement between the estimated activity values based
on the derived equation and that based on ANGLE-3, and certified
values. The average relative error is about 2.3, 2.06, and 1.54% for the
𝛾-energy lines 661.7, 1173.2, and 1332.5 keV respectively with overall
an average relative error of about 1.9% as presented in Table 5, while
the uncertainties of certified activity, efficiency curves, and peak count
rates are about 3.2, 1.7, and 0.0014%. The obtained difference could
be the results from the detector input parameters (e.g., dead layer)
introduced to the ANGLE-3 software as well as the ANGLE-3 software
limitation itself.
Fig. 6. The estimated activities based on ANGLE-3 and the derived equation in
comparison with the certified values.
𝛾-energy, and sample density using the coefficients (𝜆𝑖𝑘 ) presented in
Table 4.
This equation could be applied in the 𝛾-energy range 0–1332 keV,
using a two-region concept (below and above the knee).
5.1. Benchmarking and application of the derived calibration equations
5.1.1. Benchmarking
The derived peak efficiency equations are benchmarked by calculating the detector peak efficiency with respect to samples of different
densities at 0.4–1.8 g/cm3 . The calculated peak efficiencies are then
compared to that obtained based on the ANGLE-3 calculations as shown
in Fig. 5. The calculated peak efficiencies agreed with those estimated
based on the ANGLE-3 software With a relative difference of about
1.5%.
5.2. Limitation of the proposed method and future work
In the field of radiation measurements, different factors could affect
activity calculations. These factors include the solid angle (S–D configuration, distance, source, and detector geometry), and the attenuation
factors due to the matrix material to be assayed, and the container
itself. The proposed equation accounts for the solid angle parameters
since the calibration would be carried out using a fixed S–D distance
for an axial S–D configuration and a fixed source geometry. It also
accounts for the attenuation factors due to the container wall, and
material density.
For accurate measurements, the attenuation factor due to the material composition has to be considered. For that, the effect of attenuation
5.1.2. Application
The derived equation is applied for the activity estimation of samples with different densities ranging from 0.7–1.6 g∕cm3 . Experimental
measurements are carried out as described in Section 2 for a total of
5 cone-shaped Marinelli beakers. The net count rates of the 𝛾-energy
5
Applied Radiation and Isotopes 200 (2023) 110946
I.M. Nabil et al.
Fig. 7. Peak efficiency curves of different chemical compositions at 𝜌 = 0.7 g/cm3 using ANGLE software.
6. Conclusion
due to the material composition is studied using the ANGLE-3 software.
The peak efficiency curves are generated assuming different material
compositions (e.g., epoxy, Sand, Aluminium, Fish-bone, and Phosphate)
and compared to the epoxy to estimate the deviation and introduce a
correction factor if needed. 𝑍𝑒𝑓 𝑓 of the selected material is in the range
of 11.6 to 78.6.
In this work, the peak efficiency equations were derived based on
the efficiency transfer method using the ANGLE software in the 𝛾energy range 81–1332 keV. The peak efficiency curves for cone-shaped
Marinelli beakers of different densities are calculated and then fitted using the quadratic polynomial in order to obtain the energy coefficients.
In addition, a linear polynomial equation was used to fit the relation
between the sample densities and the energy coefficients. The final
peak efficiency equations account for the sample densities in the range
0.7–1.6 g/cm3 at a given 𝛾-energy line for a specific S–D configuration,
and distance. The derived equations are then benchmarked for peak
efficiency calculations and applied for activity estimation of samples
with different densities. Results show a good agreement between the
derived equations and the ANGLE software, even with the activitycertified values. Furthermore, the self-attenuation correction factor
was estimated for different material compositions using the ANGLE
software with respect to the Epoxy material. The proposed method
is very simple, time-saving, and could be used for peak efficiency
calibration of HPGe detectors without the need for simulation codes.
In the future, this work would extend to introduce the S–D distance
parameter to the derived equations that would be useful for optimization of the experimental setup parameters e.g., dead time. In addition,
the self-attenuation correction factor due to a wide range of chemical
compositions would also be discussed later on.
Fig. 7 shows the calculated peak efficiency curves of the aforementioned compositions, and Figs. 8 shows the relative difference between
the different compositions with respect to the epoxy. It is firmly from
the figure and the table listed in Appendix A that the attenuation factor
due to material composition is remarkable in low 𝛾-energy region and
could reach an error of about 43.3%, while a moderate deviation in the
range of 3.68% is noticed in the 𝛾-energy range 200–1400 keV. For that,
the self-attenuation correction factor (𝜇𝑠 ) due to material composition
has to be considered.
The self-attenuation correction factor (𝜇𝑠 ) due to material composition is calculated for different material compositions with respect to the
Epoxy using the ANGLE software by applying the following equation,
𝜇𝑠 =
𝜀𝑐
𝜀𝑟
(10)
𝜀𝑐 , 𝜀𝑟 are the sample efficiency with a given chemical composition to
the reference efficiency of the Epoxy material.
The values of 𝜇𝑠 for different chemical compositions in the energy
range 0–1332 keV are presented in Appendix B.
Fig. 9 shows that there are transition ranges with a quick change
in 𝑍𝑒𝑓 𝑓 in between these locations. The photoelectric effect is the
primary photon interaction process in the low-energy range, 81–88
keV. This energy contains the compound’s highest value l of 𝑍𝑒𝑓 𝑓
which is given a heavy weight by the photoelectric absorption cross
section’s 𝑍𝑒𝑓 𝑓 dependence and so the correction factor decrease. On
the contrary, 𝑍𝑒𝑓 𝑓 falls to a lower value characteristic for Compton
scattering which is the primary interaction process at intermediate
energies in the transition area at 1173.2–1332.5 keV at which 𝑍𝑒𝑓 𝑓
is once more almost constant Thus, the correction factor is almost
constant.
CRediT authorship contribution statement
Islam M. Nabil: Writing – original draft, Validation, Methodology,
Formal analysis, Conceptualization. K.M. El-Kourghly: Writing – review & editing, Validation, Supervision, Methodology, Investigation,
Conceptualization. A.F. El Sayed: Supervision, Methodology, Investigation, Conceptualization.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
In future work, we aspire to include the S–D distance parameter
in the derived equations to be more efficient. In addition, other parameters affecting the peak efficiency calibration including the detector
dead layer, and the self-attenuation correction due to a wide range of
chemical compositions for accurate measurements to be discussed later.
Data availability
No data was used for the research described in the article.
6
Applied Radiation and Isotopes 200 (2023) 110946
I.M. Nabil et al.
Fig. 8. Peak efficiency deviation percentages created between ANGLE and NE for different material compositions with different densities in the range 0.7–1.6 g/cm3 .
Appendix A. Effect of sample matrix(composition, and density) on
the calculated peak efficiency as a function of 𝜸-energy
Appendix B. Self-absorption correction factor
See Table A.6.
See Table B.7.
7
Applied Radiation and Isotopes 200 (2023) 110946
I.M. Nabil et al.
Fig. 9. The effect of the self attenuation correction factor of phosphate with respect to epoxy at 𝜌 = 1.17 g/cm3 .
Table A.6
Peak efficiency relative difference due to sample matrix as a function of 𝛾-energy with respect to epoxy.
Energy (keV)
Relative difference (%) (
𝜀𝑀𝑎𝑡𝑒𝑟𝑖𝑎𝑙 −𝜀𝐸𝑝𝑜𝑥𝑦
𝜀𝐸𝑝𝑜𝑥𝑦
× 100) at different density (g/cm3 )
0.7
81
88
122.1
279.2
319.7
450
550
661.7
800
898
1000
1173.2
1332.5
Energy (keV)
81
88
122.1
279.2
319.7
450
550
661.7
800
898
1000
1173.2
1332.5
1
Sand
Aluminium
Fish bone
Phosphate
Sand
Aluminium
Fish bone
Phosphate
−0.70
−0.22
0.85
1.18
1.14
1.02
0.95
0.88
0.81
0.77
0.73
0.68
0.64
−1.33
−0.52
1.27
1.87
1.81
1.62
1.50
1.39
1.29
1.22
1.16
1.08
1.01
−19.14
−15.30
−5.52
0.84
0.99
1.04
1.00
0.95
0.89
0.85
0.81
0.75
0.69
−29.03
−29.83
−20.76
−1.20
−0.35
0.60
0.81
0.88
0.90
0.89
0.88
0.83
0.79
−0.92
−0.29
1.13
1.60
1.55
1.40
1.30
1.21
1.12
1.06
1.02
0.94
0.88
−1.74
−0.69
1.70
2.54
2.47
2.22
2.06
1.92
1.78
1.69
1.62
1.50
1.41
−23.82
−19.29
−7.22
1.14
1.34
1.43
1.38
1.30
1.22
1.17
1.12
1.04
0.97
−35.07
−36.01
−25.92
−1.62
−0.47
0.82
1.11
1.21
1.25
1.23
1.22
1.16
1.10
Sand
Aluminium
Fish bone
Phosphate
Sand
Aluminium
Fish bone
Phosphate
−1.02
−0.32
1.27
1.82
1.77
1.60
1.49
1.38
1.28
1.22
1.17
1.09
1.02
−1.94
−0.77
1.91
2.89
2.81
2.54
2.36
2.20
2.04
1.95
1.86
1.73
1.62
−25.89
−21.11
−8.04
1.30
1.53
1.63
1.58
1.49
1.41
1.34
1.29
1.20
1.11
−37.61
−38.61
−28.23
−1.84
−0.53
0.93
1.27
1.39
1.43
1.42
1.40
1.34
1.26
−1.07
−0.34
1.33
1.92
1.86
1.69
1.57
1.46
1.36
1.29
1.24
1.15
1.08
−2.03
−0.80
2.01
3.05
2.96
2.68
2.50
2.33
2.16
2.06
1.97
1.83
1.72
−26.75
−21.87
−8.40
1.37
1.61
1.72
1.67
1.58
1.49
1.42
1.37
1.27
1.18
−38.64
−39.66
−29.20
−1.93
−0.56
0.98
1.34
1.47
1.52
1.50
1.48
1.42
1.34
1.17
1.25
Energy (keV)
81
88
122.1
279.2
319.7
450
550
661.7
800
898
1000
1173.2
1332.5
1.6
Sand
Aluminium
Fish bone
Phosphate
−1.25
−0.39
1.58
2.31
2.25
2.05
1.92
1.80
1.67
1.60
1.53
1.43
1.44
−2.36
−0.94
2.38
3.68
3.59
3.27
3.06
2.86
2.67
2.55
2.44
2.27
2.23
−29.82
−24.63
−9.76
1.65
1.95
2.09
2.04
1.94
1.83
1.76
1.69
1.57
1.56
−42.20
−43.31
−32.67
−2.31
−0.67
1.20
1.64
1.80
1.87
1.85
1.83
1.76
1.76
8
Applied Radiation and Isotopes 200 (2023) 110946
I.M. Nabil et al.
Table B.7
Self-absorption correction factor (𝜇𝑠 ) as a function of 𝛾-energy, sample composition, and density with respect to epoxy.
Density
Material
(g/cm3 )
Self absorption correction factor (𝜇𝑠 ) at different energy (keV)
81
88
122.1
165.9
279.2
319.7
450
550
661.7
800
898
1000
1173.2
1332.5
0.7
Sand
Aluminium
Fish bone
Phosphate
0.99
0.99
0.81
0.71
1.00
0.99
0.85
0.70
1.01
1.01
0.94
0.79
1.01
1.02
0.99
0.91
1.01
1.02
1.01
0.99
1.01
1.02
1.01
1.00
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1
Sand
Aluminium
Fish bone
Phosphate
0.99
0.98
0.76
0.65
1.00
0.99
0.81
0.64
1.01
1.02
0.93
0.74
1.02
1.03
0.99
0.89
1.02
1.03
1.01
0.98
1.02
1.02
1.01
1.00
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.01
1.01
1.01
1.17
Sand
Aluminium
Fish bone
Phosphate
0.99
0.98
0.74
0.62
1.00
0.99
0.79
0.61
1.01
1.02
0.92
0.72
1.02
1.03
0.98
0.87
1.02
1.03
1.01
0.98
1.02
1.03
1.02
0.99
1.02
1.03
1.02
1.01
1.01
1.02
1.02
1.01
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.25
Sand
Aluminium
Fish bone
Phosphate
0.99
0.98
0.73
0.61
1.00
0.99
0.78
0.60
1.01
1.02
0.92
0.71
1.02
1.03
0.98
0.87
1.02
1.03
1.01
0.98
1.02
1.03
1.02
0.99
1.02
1.03
1.02
1.01
1.02
1.03
1.02
1.01
1.01
1.02
1.02
1.01
1.01
1.02
1.01
1.02
1.01
1.02
1.01
1.02
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.01
1.02
1.01
1.01
1.6
Sand
Aluminium
Fish bone
Phosphate
0.99
0.98
0.70
0.58
1.00
0.99
0.75
0.57
1.02
1.02
0.90
0.67
1.02
1.04
0.98
0.85
1.02
1.04
1.02
0.98
1.02
1.04
1.02
0.99
1.02
1.03
1.02
1.01
1.02
1.03
1.02
1.02
1.02
1.03
1.02
1.02
1.02
1.03
1.02
1.02
1.02
1.03
1.02
1.02
1.01
1.02
1.02
1.02
1.01
1.02
1.02
1.02
1.01
1.02
1.01
1.02
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