Annals of Nuclear Energy 38 (2011) 112–117

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Annals of Nuclear Energy

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GEANT4 simulation of photo-peak efficiency of small high puritygermanium detectors for nuclear power plant applications

Shakeel Ur Rehman, Sikander M. Mirza, Nasir M. Mirza ⇑, Muhammad Tariq SiddiqueDepartment of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad 45650, Pakistan

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 February 2010Received in revised form 11 August 2010Accepted 18 August 2010Available online 15 September 2010

Keywords:GEANT4High purity germaniumPhoto-peak efficiencyGamma spectroscopy

0306-4549/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.anucene.2010.08.010

⇑ Corresponding author. Tel.: +92 51 2207380 3; faE-mail addresses: nmm@pieas.edu.pk, nasirmm@y

GEANT4 – based Monte Carlo simulations have been carried out for the determination of photo-peak effi-ciency of heavily shielded small high purity germanium detector (HPGe) used for monitoring radiationlevels in nuclear power plants. The GEANT4 simulated values of HPGe detector efficiency for point as wellas for disk sources, for two different values of collimator diameter, have been found in good agreementwith the corresponding published results obtained by using the MCNP code. The work has been extendedto study the effect of radial displacement of a source relative to a detector on photo-peak efficiency forboth point and disk source, and at various values of c-ray energies. Also the effect of disk source radiuson photo-peak efficiency has been studied. Besides the results of different available physics models inGEANT4 have also been compared. The computed values of efficiency for point as well as for disk sourcesusing the Penelope and Livermore physics models have been found correspondingly consistent for vari-ous values of c-ray energies while some differences (e.g., Penelope model yields 6.3% higher values ofphoto-peak efficiency for Ec = 1.332 MeV, 10 mm collimator diameter) have been observed in the corre-sponding valued obtained by using the Standard physics model.

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1. Introduction

Measurement of radioactive contamination, caused dominantlyby corrosion and activation products, is essential for estimation ofaverage dose received by workers carrying out routine repair andmaintenance tasks in nuclear power plants. This is accomplishedby direct on-site measurements using radiation detectors insteadof taking physical samples for laboratory analysis. The equipmentgenerally consists of a heavily shielded high purity (HP) Germa-nium semiconductor detector along with compatible fast electron-ics suitable for high counting rates. The analysis using such directmeasurements poses problem for the detector calibration, becausethe required measurements have to be performed in difficult-to-access plant locations having high dose rates. The correspondingexperimental mockup of the source is quite complicated andexpensive. Therefore, various computer simulation techniquesincluding the Monte Carlo method have been widely used for thedetector efficiency calibration (Rodenas et al., 2000; Rehmanet al., 2009). The knowledge of efficiency of the detector is alsoessential for the absolute measurements of the strength of radioac-tive materials (Abbas, 2006).

In past, a number of research efforts have been devoted towardsthe calibration of the detector efficiency (Ashrafi et al., 1999;

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x: +92 51 2208070.ahoo.com (N.M. Mirza).

Anagnostakis and Simopoulos, 1996; Wang et al., 1997; Sudarshanand Singh, 1991; Noguchi et al., 1981). The precise determinationof the detector efficiency still remains an important issue (Helmeret al., 2004; Hurtado et al., 2004a; Saegusa et al., 2000; Hardy et al.,2002; Karamanis et al., 2002; Abbas et al., 2002; Presler et al.,2002; Ludington and Helmer, 2000; Bruggeman et al., 2000; Hay-ashi et al., 2000; Korun and Vidmar, 2000; García-Talavera et al.,2000). The direct experimental measurement of the detector effi-ciency requires the use of a large number of standard sources con-taining single gamma ray emitters with energies covering therange of interest (El-Gharbawy et al., 2005).

There are various general purpose Monte Carlo codes availablefor radiation transport simulation namely MCNP, PENELOPE,GEANT4, etc. In the past Rodenas et al. (2000) have carried out acomparative study for HP Germanium detector using MCNP simu-lations as well as experimental measurements. Their reported re-sults show increasing trend of discrepancy in calculated-to-experimental (C/E) ratios with radial displacement of source andit reached near 10% for 12 cm radial displacement. In the case ofabsolute efficiency, the discrepancy as high as 39% in the C/E valueshave been found for Cs-137 source. They also observe that discrep-ancies are higher for smaller values of collimator diameter. Simi-larly, Karamanis et al. (2002) have carried out experimental aswell as Monte Carlo simulation study of efficiency of 40% HPGedetectors for point sources, employing both MCNP4B and GEANT3.21 codes. Their reported results show generally good agreement

S.U. Rehman et al. / Annals of Nuclear Energy 38 (2011) 112–117 113

between simulation and experimental data. However, in the lowenergy range large discrepancies are reported with experimentaldata to simulation ratio varying in 0.85–1.8 range below about200 keV which need systematic investigation. More recently, Rod-enas and Gallardo (2007) have carried out MCNP simulations forthe estimation of detection efficiency of HP Germanium detectorin lead shield with collimator at 30� viewing angle from a contam-inated pipe. The reported results show some overlap in the effi-ciency curves for 60 cm and 70 cm source-to-detector distances.They recommend that validation of their simulated results shouldbe done with some experimental measurements.

In the present work, we report HPGe detector efficiency deter-mined using the GEANT4.9.2 code, referred to as GEANT4 in thisarticle. This includes energy as well as the source-to-detector dis-tance dependence of efficiency for small sized, lead shielded HPGedetectors generally used as radiation monitors in nuclear powerplants. The results for both point and disk sources are included.We present a comparison of the GEANT4 simulations with dataavailable in literature. These simulations have been carried outusing the Standard physics model and two low energy physicsmodels PENELOPE and LIVERMORE available in GEANT4 and thecorresponding comparisons are included.

Fig. 1. A three dimensional view of the high purity germanium detector and leadshield with collimator as modeled in GEANT4.

1.1. GEANT4 software

The GEANT4 program is widely used toolkit for carrying outMonte Carlo based particle transport calculations. In order to builda simulation package the user implements several classes to de-scribe the detector geometry which includes the materials used,detector sensitive components, shielding, etc., the primary particlegenerator which includes particle type, energy, position and direc-tion distributions etc., along with the relevant particles and physicsprocesses. The GEANT4 code is based on object-oriented program-ming and allows user to derive classes and to overload standardGEANT4 functionality (Hurtado et al., 2004b). It uses Monte Carlomethods and a number of theoretical models to fully simulatethe passage of particles through matter. In its low energy exten-sions, the physics processes cover the energy range from 250 eVup to 100 GeV. Energy spectrum for photons detected by the detec-tor is calculated by simulating all relevant physical processes andinteractions (Isakar et al., 2007). Three physics models are avail-able in the GEANT4 viz. Standard Physics and low energy physicsLivermore and Penelope. Since no approximations are made, theuncertainties in GEANT4 Monte Carlo simulations as low as lessthan 1% are attainable (Agostinelli et al., 2003).

2. Monte Carlo model and simulations

A typical small sized HPGe detector, used as radiation monitorsin nuclear power plants has been modeled in this work (Rodenaset al., 2000, 2005; Rodenas and Gallardo, 2007). The detector,essentially composed of 10 mm thick high purity Germanium crys-tal, is placed in the heavy lead shield assembly. There is a collima-tor provided in the shield which allows a narrow beam of gammarays to reach the detector. Fig. 1 shows detector-shield geometryalong with physical location of radiation source. Two types of gam-ma ray sources have been used in these simulations: a point isotro-pic source and a planar disk source.

The GEANT4 program based Monte Carlo simulations initiatewith a cycle that starts with the generation of a c-ray photon fromthe source and then these are followed by the tracking of photonsin various regions of modeled geometry. The tracking of a singlephoton is stopped when it leaves the volume of interest or whenthe energy of photon becomes lower than a specified thresholdvalue called cut-off energy. This cycle, also called sampling, is re-

peated for large enough number of times in order to reduce the sta-tistical fluctuations in the parameters of interest below thanprescribed limits. In these simulations, sampling of over 106 pho-tons has been carried out for each data point for good statistics.To improve the efficiency of simulation, source biasing techniquehas been used.

In the first case, a point isotropic c-ray source has been consid-ered. The source-to-detector distance was taken as 1 m and thecollimator diameter was 10 mm. Values of photo-peak efficiencyof the detector for different gamma ray energies ranging from662 keV to 1332 keV are calculated. Next a 5 cm diameter planardisk source was considered. The source-to-detector distance waskept 1 m and diameter of the collimator was 10 mm. The valuesof photo-peak efficiency for various c-ray energies are calculated.The full simulations of point and planar disk source were repeatedwith collimator diameter set to 20 mm.

After the validation of our GEANT4 detector model, the simula-tions of point and planar disk sources for various gamma energieshave been repeated with low energy physics models available inGEANT4 program namely PENELOPE and LIVERMORE and the re-sults are compared with GEANT4 standard physics results.

To see the effect of radial position of the gamma ray source rel-ative to the detector, the radial distance from the central detectoraxis of both point and disk source was varied from 0 to 10 cm. Sim-ilarly the effect of diameter of the planar disk source was also stud-ied by varying the diameter of the disk source from 0.0002 cm to20 cm.

3. Results

The first round of simulations was performed for point source.The diameter of the collimator was set to 10 mm and standardphysics model was used. The gamma rays of various energies rang-ing from 662 keV to 1.332 MeV were simulated. Photo-peak effi-ciency was calculated and the values were compared with thealready published results of MCNP Monte Carlo code (Rodenaset al., 2000). The results are in good agreement with MCNP results(Table 1).

Table 1Values of photo-peak efficiencies of HPGe detector for a point isotropic c-ray sourcewith 10 and 20 mm collimator diameter.

c-ray energy(MeV)

GEANT4 (this work) Rodenas et al. (2000)

Value Relativeerror

MCNP Experiment

Collimator diameter: 10 mm0.662 2.68 � 10�7 1.21 � 10�8 2.76 � 10�7 1.99 � 10�7

0.834 1.93 � 10�7 3.83 � 10�9 1.98 � 10�7 1.53 � 10�7

1.173 1.17 � 10�7 5.23 � 10�9 1.30 � 10�7 1.23 � 10�7

1.332 9.50 � 10�8 5.58 � 10�9 1.11 � 10�7 1.06 � 10�7

Collimator diameter: 20 mm0.662 1.04 � 10�6 1.43 � 10�8 9.70 � 10�7 8.53 � 10�7

0.834 7.14 � 10�7 2.24 � 10�9 7.27 � 10�7 6.13 � 10�7

1.173 4.33 � 10�7 5.25 � 10�9 4.58 � 10�7 4.17 � 10�7

1.332 3.69 � 10�7 3.57 � 10�9 4.01 � 10�7 3.57 � 10�7

Fig. 2. Variation of photo-peak efficiency of HPGe detector with radial displace-ment of point isotropic c-ray source.

114 S.U. Rehman et al. / Annals of Nuclear Energy 38 (2011) 112–117

In the second round of simulations the collimator diameter wasset to 20 mm and the rest of the parameters were kept the same.Photo-peak efficiency values were calculated and compared withthe published values of MCNP code for the same geometry. Againthe agreement between the values was very good (Table 1).

Next planar disk source of 5 cm diameter was considered. Thesource was placed on the central detector axis. The source wasplaced 1 m away from the detector face. The diameter of the collima-tor was set to 10 mm. The simulations were carried out and photo-peak efficiencies were calculated for various gamma ray energies

Table 2Values of photo-peak efficiencies of HPGe detector for a 5 cm diameter planar disk c-ray source with 10 and 20 mm collimator diameter.

c-ray energy(MeV)

GEANT4 (this work) Rodenas et al. (2000)

Value Relative error MCNP Experiment

Collimator diameter: 10 mm0.662 2.28 � 10�7 2.30686 � 10�8 2.35 � 10�7 1.99 � 10�7

0.834 1.90 � 10�7 1.84794 � 10�8 1.63 � 10�7 1.53 � 10�7

1.173 1.10 � 10�7 2.14417 � 10�8 1.11 � 10�7 1.23 � 10�7

1.332 8.18 � 10�8 1.90265 � 10�9 8.67 � 10�8 1.06 � 10�7

Collimator diameter: 20 mm0.662 9.63 � 10�7 1.87 � 10�8 9.92 � 10�7 8.53 � 10�7

0.834 6.94 � 10�7 2.99 � 10�8 6.83 � 10�7 6.13 � 10�7

1.173 4.37 � 10�7 2.23 � 10�8 4.41 � 10�7 4.17 � 10�7

1.332 3.48 � 10�7 2.03 � 10�8 3.86 � 10�7 3.56 � 10�7

Table 3Values of photo-peak efficiencies of various indicated physics models used in GEANT4 for point isotropic c-rays source.

c-ray energy (MeV) 10 mm collimator diameter 20 mm collimator diameter

Standard Penelope Livermore Standard Penelope Livermore

0.662 2.68 � 10�7 2.67 � 10�7 2.68 � 10�7 1.04 � 10�6 1.02 � 10�6 1.02 � 10�6

0.834 1.93 � 10�7 1.93 � 10�7 1.93 � 10�7 7.14 � 10�7 7.04 � 10�7 7.04 � 10�7

1.173 1.17 � 10�7 1.15 � 10�7 1.17 � 10�7 4.33 � 10�7 4.38 � 10�7 4.38 � 10�7

1.332 9.50 � 10�8 1.01 � 10�7 9.50 � 10�8 3.69 � 10�7 3.65 � 10�7 3.65 � 10�7

Table 4Photo-peak efficiencies of various indicated physics models available in GEANT4 for a 5 cm diameter planar disk source.

c-ray energy (MeV) 10 mm collimator diameter 20 mm collimator diameter

Standard Penelope Livermore Standard Penelope Livermore

0.662 2.28 � 10�7 2.28 � 10�7 2.28 � 10�7 9.63 � 10�7 9.63 � 10�7 9.63 � 10�7

0.834 1.90 � 10�7 1.90 � 10�7 1.90 � 10�7 6.94 � 10�7 6.96 � 10�7 6.96 � 10�7

1.173 1.10 � 10�7 1.10 � 10�7 1.10 � 10�7 4.37 � 10�7 4.29 � 10�7 4.29 � 10�7

1.332 8.18 � 10�8 8.18 � 10�8 8.18 � 10�8 3.48 � 10�7 3.42 � 10�7 3.42 � 10�7

Fig. 3. Variation of photo-peak efficiency of HPGe detector with radial displace-ment of planar disk c-ray source of 5 cm diameter.

Fig. 4. Variation of photo-peak efficiency of HPGe detector with radius of a planerdisk c-ray source placed concentric with and perpendicular to the detector axis.

S.U. Rehman et al. / Annals of Nuclear Energy 38 (2011) 112–117 115

(Table 2). The GEANT4 simulation based values were found to be ingood agreement with the published values of MCNP for same

Fig. 5. Variation of effective solid angle subtended by detector on

experimental setup (Rodenas et al., 2005). The simulations for disksource were repeated with collimator diameter set to 20 mm andthe values of photo-peak efficiencies were calculated. Again goodagreement was found between the values of photo-peak efficiencycomputed by using GEANT4 program and MCNP code (Table 2).

Further a comparison of the various physics models available inGEANT4 program was carried out. For this purpose, simulationswere carried out for point source as well as for disk source using col-limator diameter of 10 mm and 20 mm respectively, using Standard,Penelope and Livermore physics models. Photo-peak efficiencies forvarious gamma ray energies were calculated and compared. Tables 3and 4 contain all the calculated values. It is clear from the tables thatcalculated efficiencies for the three models are very close to eachother and the differences between them are negligible. Also, in somecases the values based on Penelope and on Livermore are exactly thesame. This is because of the reason that the low energy physics mod-els in GEANT4 code are primarily for simulation of X- andc-rays withenergies less than 100 keV. In higher energy ranges these modelshave little or no effects on simulations results in comparison withstandard model results.

After validation of the GEANT4 model and comparison of differ-ent physics models of GEANT4, the effect of gamma ray sourceposition relative to detector on photo-peak efficiency was explored.For this purpose two simulation runs were carried out. In both of the

a point source with radial position of the gamma ray source.

116 S.U. Rehman et al. / Annals of Nuclear Energy 38 (2011) 112–117

runs the axial distance between the source and detector face waskept 1 m. In the first run point gamma ray source was consideredand its radial position relative to detector was varied from 0 to10 cm. The values of photo-peak efficiencies for various gammaray energies were calculated. In the second run 5 cm diameter planardisk source was considered and its radial position was varied fromcentral axis to 10 cm. The values of photo-peak efficiencies were cal-culated. The results are shown as Figs. 2 and 3 respectively.

It is clear from Fig. 2 that the decrease in photo-peak efficiencywith radial displacement is slow up to 10 mm radial displacementand then the decrease is steep. The trend is same for various gammaray energies but is more pronounced for lower gamma ray energiesthan higher energies. This behavior can be explained by consideringFig. 5, where three positions A–C of a point isotropic gamma sourceare shown in simplified collimator–detector geometry. It can be eas-ily seen that as the source is moved outwards in radial direction, thearea of the detector directly visible to source, may also be calledeffective solid angle, reduces meaning thereby that the number ofun-scattered gamma rays reaching the detector reduces which is re-flected in the reduction of photo-peak efficiency values, as only un-scattered gamma rays with the exception of gamma rays scatteredthrough Rayleigh scattering, contributes to photo-peak efficiencyof the detector. From Fig. 5 it can be visualized that if the source isdisplaced from point A within the dimensions of collimator (dottedline), then the reduction in the number of un-scattered gamma raysreaching the detector is small. This is reflected in the values of photo-peak efficiency for radial displacement of up to 10 mm (Fig. 2) wherethe decrease in the values with displacement is small.

Fig. 3 shows the plot of photo-peak efficiency values for planardisk source of 5 cm diameter with radial displacement. Similarbehavior as in Fig. 2 can be readily seen with the difference thatthe photo-peak efficiency values remains almost the same up to20 mm radial displacement and the reduces sharply with radialdistance. This can be explained if the planar disk source is imag-ined to be a collection of point isotropic source placed on the sur-face of the disk. In Fig. 5, it can be imagined that the disk sourceextends on both sides of point A up to point B. Now as discussedabove the region of source nearer point A will contribute the mostin photo-peak efficiency values and outer regions will contributeless. As the source is moved outwards in radial direction, the regionaround point A will shift upward and the area of the source aroundpoint B will shift towards point C, so now the central region will becontributing less, the upper region near point C will contribute theleast and lower region will contribute the most. Therefore the neteffect of moving the source upwards will be smaller comparedwith point source. As the lower region, that was contributing lessbefore displacement will be contributing more after displacement.For the small values of the radial displacement of source, thechange in source-to-detector solid angle remains small, as the re-sult, the corresponding change in the photo-peak efficiency is alsosmall. However, the converse is true for large values of radial dis-placement of source and in this case, large decrease in photo-peakefficiency is found as shown in Fig. 3.

The Fig. 4 represents the effect of increasing the planar disksource radius on the photo-peak efficiency values. In this case also,a decreasing trend in the photo-peak efficiency values is observedwith increasing radius of the planar disk source. Again if the disksource is considered to be collection of points isotropic sourcesthen as the radius of the disk source is increased from 0, the num-ber of gamma rays emitted by the source is also increased. But asdiscussed above with reference to Fig. 5, the fraction of un-scat-tered gamma rays reaching detector is greatest for central regionand as we move outwards the fraction reduces and is negligiblysmall for the regions lying around point C (Fig. 5). Therefore asthe radius of the disk source is increased the photo-peak efficiencyvalues decrease.

4. Conclusions

Following conclusions may be drawn from the GEANT4 basedMonte Carlo evaluation of photo-peak efficiency of the small HPGedetectors commonly used as radiation level monitors in nuclearpower plants:

� For various values of c-ray energies in 0.662–1.332 MeV range,the GEANT4 computed values of photo-peak efficiency for bothpoint as well as disk sources are in good agreement with thecorresponding results obtained from MCNP simulations as wellas experimental data.� Both Penelope and Livermore physics models in GEANT4 pro-

gram have been found to yield consistent values of photo-peakefficiency while differences up to 2% have been found in thecase of the standard physics model particularly towards thehigh energy values and for larger diameter collimator.� For both point as well as disk sources, the GEANT4 simulated

values of photo-peak efficiency show decreasing trend withincrease in radial distance of the source, consistent with theexpected behavior.� The GEANT4 simulated values of efficiency have a decreasing

behavior with increase in the radius of disk source which isagain consistent with the expected behavior.

Acknowledgment

Shakeel-ur-Rehman gratefully acknowledges the financialsupport from the Higher Education Commission (HEC), Pakistanfor the Ph.D. (Indigenous) fellowship under scholarship No. 17-5-I(P-160)/HEC/Sch/2004.

References

Abbas, M.I., 2006. Analytical calculation of the solid angles subtended by a well-type detector at point and extended circular sources. Appl. Radiat. Isot. 64 (9),1048–1056.

Abbas, K., Simonelli, F., D’Alberti, F., Forte, M., Stroosnijder, M.F., 2002. Reliability oftwo calculation codes for efficiency calibrations of HPGe detectors. Appl. Radiat.Isot. 56, 703–709.

Agostinelli, S., Allison, J., Amako, K., Apostolakis, J., Araujo, H., Arce, P., Asai, M., Axen,D., Banerjee, S., Barrand, G., Behner, F., Bellagamba, L., Boudreau, J., Broglia, L.,Brunengo, A., Burkhardt, H., Chauvie, S., Chuma, J., Chytracek, R., Cooperman, G.,et al., 2003. GEANT4—a simulation toolkit. Nucl. Instrum. Methods A 506, 250–303.

Anagnostakis, M.J., Simopoulos, S.E., 1996. An experimental/numerical method forthe efficiency calibration of low-energy germanium detectors. Environ. Int. 22,93–99.

Ashrafi, S., Likar, A., Vidmar, T., 1999. Precise Modeling of HPGe detectors. Nucl.Instrum. Methods A 438, 421–428.

Bruggeman, M., Solidang, Carchon R., 2000. A computer code for the computation ofthe effective solid angle and correction factors for gamma spectroscopy-basedwaste assay. Appl. Radiat. Isot. 52, 771–776.

El-Gharbawy, H.A., Metwally, S.M., Sharshar, T., Elnimr, T., Badran, H.M., 2005.Establishment of HPGe detector efficiency for point source including truecoincidence correction. Nucl. Instrum. Methods A 550, 201–211.

García-Talavera, M., Nader, H., Daza, M.J., Quintana, B., 2000. Towards a propermodeling of detector and source characteristics in Monte Carlo simulations.Appl. Radiat. Isot. 52, 777–783.

Hardy, J.C., Iacob, V.E., Sanchez-Vega, M., Effinger, R.T., Lipnik, P., Mayes, V.E., Willis,D.K., Helmer, R.G., 2002. Precise efficiency calibration of an HPGe detector:source measurements and Monte Carlo calculations with sub-percent precision.Appl. Radiat. Isot. 56, 65–69.

Hayashi, N., Miyahara, H., Mori, C., 2000. Highly precise measurement of the relativegamma-ray detection efficiency curve. J. Nucl. Sci. Technol. 37, 139.

Helmer, R.G., Nica, N., Hardy, J.C., Iacob, V.E., 2004. Precise efficiency calibration ofan HPGe detector up to 3.5 MeV, with measurements and Monte Carlocalculations. Appl. Radiat. Isot. 60, 173–177.

Hurtado, S., García-León, M., García-Tenorio, R., 2004a. GEANT4 code for simulationof a germanium gamma-ray detector and its application to efficiencycalibration. Nucl. Instrum. Methods A 518, 764–774.

Hurtado, S., García-León, M., García-Tenorio, R., 2004b. Monte Carlo simulation ofthe response of a germanium detector for low-level spectrometrymeasurements using GEANT4. Appl. Radiat. Isot. 61, 139–143.

S.U. Rehman et al. / Annals of Nuclear Energy 38 (2011) 112–117 117

Isakar, K., Realo, K., Kiisk, M., Realo, E., 2007. Efficiency corrections in low-energygamma spectrometry. Nucl. Instrum. Methods A 580, 90–93.

Karamanis, D., Lacoste, V., Andriamonje, S., Barreau, G., Petit, M., 2002. Experimentaland simulated efficiency of a HPGe detector with point-like and extendedsources. Nucl. Instrum. Methods A 487, 477–487.

Korun, M., Vidmar, T., 2000. Monte Carlo calculations of the total-to-peak ratio ingamma-ray spectrometry. Appl. Radiat. Isot. 52, 785–789.

Ludington, M.A., Helmer, R.G., 2000. High accuracy measurements and Monte Carlocalculations of the relative efficiency curve of an HPGe detector from 433 to2754 keV. Nucl. Instrum. Methods A 446, 506–521.

Noguchi, M., Takeda, K., Higuchi, H., 1981. Semi-empirical c-ray peak efficiencydetermination including self-absorption correction based on numericalintegration. Appl. Radiat. Isot. 32, 17–22.

Presler, O., Peled, O., German, U., Leichter, Y., Alfassi, Z.B., 2002. Off-center efficiencyof HPGe detectors. Nucl. Instrum. Methods A 484, 444–450.

Rehman, S., Mirza, S. M., Mirza, N. M., 2009. A fast, primary-interaction Monte Carlomethodology for determination of total efficiency of cylindrical scintillationdetectors. Nucl. Technol. Radiat. Prot. 24(3), pp. 195–203. doi: 10.2298/NTRP0903195R.

Rodenas, J., Gallardo, S., 2007. Application of the MonteCarlo method to develop amodel to estimate the internal contamination of pipes in a nuclear reactor. Nucl.Instrum. Methods A 580, 134–136.

Rodenas, J., Martinavarro, A., Rius, V., 2000. Validation of the MCNP code for thesimulation of Ge-detector calibration. Nucl. Instrum. Methods A 450, 88–97.

Rodenas, J., Burgos, M.C., Zarza, I., Gallardo, S., 2005. Simulation of germaniumdetector calibration using the Monte Carlo method: comparison between pointand surface source models. Radiat. Prot. Dosim. 116, 55–58.

Saegusa, J., Oishi, T., Kawasaki, K., Yoshizawa, M., Yoshida, M., Sawahata, T., Honda,T., 2000. Determination of gamma-ray efficiency curves for volume samples bythe combination of Monte Carlo simulations and point source calibration. J.Nucl. Sci. Technol. 37, 1075.

Sudarshan, M., Singh, R., 1991. Applicability of the analytical functions and semi-empirical formulae to the full energy peak efficiency of large volume HPGegamma detectors. J. Phys. D: Appl. Phys. 24, 1693.

Wang, T., Mar, W., Ying, T., Tseng, C., Liao, C., Wang, M., 1997. HPGe detectorefficiency calibration for extended cylinder and Marinelli-beaker sources usingthe ESOLAN program. Appl. Radiat. Isot. 48, 83–95.