Applied Radiation and Isotopes 70 (2012) 2695–2702

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Applied Radiation and Isotopes

0969-80

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A study for improving detection efficiency of an HPGe detector based gammaspectrometer using Monte Carlo simulation and genetic algorithms

Ngo Quang Huy a,n, Do Quang Binh b, Vo Xuan An a

a Faculty of Fundamental Sciences, Ho Chi Minh City University of Industry, 12 Nguyen Van Bao Street, Ward 4, Go Vap District, Ho Chi Minh City, Vietnamb Faculty of Foundation Sciences, University of Technical Education Ho Chi Minh City, 1 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Vietnam

H I G H L I G H T S

c Improvement of the detection efficiency of an HPGe detector based gamma spectrometer.c Application of Monte Carlo simulation and genetic algorithms.c Optimization of geometrical dimensions of a Marinelli beaker.c Effects of gamma energy, sample chemical composition and sample density are taken into account.

a r t i c l e i n f o

Article history:

Received 6 January 2010

Received in revised form

6 August 2012

Accepted 4 September 2012Available online 13 September 2012

Keywords:

Marinelli beaker

Optimal sizes

HPGe detector

Genetic algorithm

MCNP5 code

43/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.apradiso.2012.09.003

esponding author. Mobilephone: þ84 908 39

ail addresses: hlchau@hcm.vnn.vn, huyhanq@

a b s t r a c t

This work aims at improving the detection efficiency of an HPGe detector based gamma spectrometer

for measurements of environmental radioactivity sample. Application of a simple genetic algorithm and

the Monte Carlo simulation computer code MCNP5 allows to search for optimal dimensions of the

Marinelli beaker typed source geometry that maximizes the detector efficiency for a fixed configuration

of the gamma spectrometer. The interested gamma energies are in the range of 255–1926 keV.

Optimization calculation was repeated several times to deduce average dimensions of an optimal

Marinelli beaker typed sample with a volume of 450 cm3. Effects of gamma energy, sample chemical

composition and sample density on the optimal dimensions were also investigated. Calculated results

showed that the effects were negligible. A validated experiment with arrangements using an optimal

beaker and three other ones was carried out to verify calculated results. It is shown that experimental

and calculated results of the detector efficiency are in a good agreement.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays radioactivity measurements have been frequentlycarried out in order to evaluate and control environmentalradioactive background at many locations in the world. One ofthe most powerful tools for this kind of work is an HPGe detectorbased gamma spectrometry due to its high sensitivity and energyresolution, especially for low-level activity material. However, ascompared to different detectors like NaI(Tl) ones, HPGe detectorsoften have a lower full energy peak efficiency, therefore researchworks for improving the detection efficiency of HPGe baseddetector gamma spectrometers are of much interest at manylaboratories. In many cases for environmental measurements, alarge amount of samples is often prepared in a uniform materialcontained in Marinelli beakers (Hill et al., 1950), where the

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sample is distributed around the surfaces of the detector head.Although there was an American National Standard (IEEE, 1978)for this beaker for use with germanium detectors, which results ina uniform thickness of sample above and around the sides of thedetector, ones would have desired better beakers that improvesmore the counting efficiency of the detector. Bonfanti and DellaDora (1981) performed an experimental study to determineoptimal counting geometries of large gamma sources for true-coaxial and closed-end germanium detectors. They reported thattheir optimal sample geometries for 500 cm3 and 1000 cm3

allowed for large efficiency gains of 12–41% compared to theconventional geometries with the same volume. However, a studyof Hemingway (1986) reached a contrast conclusion that the largeincreases in efficiency when using the optimal Marinelli beakersdeduced from the report of Bonfanti and Della Dora (1981)compared to the conventional beakers were unlikely to beattainable. Besides, he suggested a spherical container of theMarinelli type, which enhanced the counting efficiency comparedto the conventional Marinelli beakers. However, the modified

N.Q. Huy et al. / Applied Radiation and Isotopes 70 (2012) 2695–27022696

Marinelli beaker has not been used in practice and the traditionalMarinelli beaker remains also preferable nowadays. Thoughresearchers have been interested in the problem, it is reckonedthat the difficulty in deducing an analytical function describingthe dependence of the detector efficiency upon dimensions ofmeasuring geometry prevents a development in research into thisoptimization problem.

Fortunately, recent advances in Monte Carlo simulation ofradiation transport process and genetic algorithm method openeda prospect of solving this problem. Validated Monte Carlo simulationcomputer codes such as MCNP (Hemingway, 1986; Briesmeister,2000; Forster et al., 2004) allow precisely simulating the responsefunction of germanium detectors (Sood et al., 2004; Ewa et al.,2001; Laborie et al., 2000; Vargas et al., 2002; Hurtado et al.,2004; Hardy et al., 2002; Rodenas et al., 2003; Helmer et al., 2004;Huy et al., 2007; Kamboj and Kahn, 2003; Salgado et al., 2006;Maleka and Maucec, 2005; Boson et al., 2008; Fabien Courtineet al., 2008; Budjas et al., 2009). Using these codes one cancalculate the detector efficiency as a function of geometricalparameters of components surrounding the detector, say dimen-sions of a Marinelli beaker used with the gamma spectrometer.Ahmed et al. (2009) used the MCNP5 code to simulate the peakefficiency of an ORTEC GMX S Gamma –X HPGe detector,investigating the dependence of detector efficiency on geometricdimensions of Marinelli beaker source sample to find out theoptimum values of sample height on detector cap and well radiussurrounding detector of Marinelli beaker. They recommended theoptimum dimensions of the Marinelli beakers with volumes of500–2000 ml, for gamma energy within a range of 0.1–2.0 MeVwhile keeping the beaker well height unchanged. The develop-ment of genetic algorithm (Goldberg, 1989; Yang et al., 2005; Doand Nguyen, 2007; Park, 1990) provides a mathematical appara-tus for effectively solving optimization problem. So the detectionefficiency optimization problem can be solved by combination ofthese two methods.

In this work, the problem was formulated in the form of anoptimization problem, which improves the detection efficiency ofthe gamma spectrometer as high as possible depending ondimensions of the Marinelli beaker geometry. The MCNP5 com-puter code was used to simulate gamma spectra and geneticalgorithm optimization was performed to search for optimalsolutions. In the following, Section 2 presents how to set up theproblem. Section 3 describes how to precisely model the detectorin MCNP simulation. Section 4 describes the application ofMCNP5 code and genetic algorithm for solving the optimizationproblem. Section 5 is a validation of calculated results by experi-ment and conclusion is presented in Section 6.

Fig. 1. Vertical section view of the lead shield with dimension

2. Setting up the problem

The gamma spectrometer to be studied in this work is installedat the Center for Nuclear Techniques Ho Chi Minh City, Vietnam.It is a typical kind of gamma spectrometer for environmentalradioactive measurements. The detector nominal characteristicsare as follows: 15% relative efficiency; 1.8 keV energy resolutionand 45:1 peak/Compton ratio at 1332 keV of gamma ray of 60Coisotope. The GC1518 detector, produced by Canberra IndustriesInc., is placed inside a cylindrical lead shield as seen in Fig. 1. Thedimensions and material compositions of detector componentsand lead shield were described in our previous work (Huy et al.,2007). According to Huy et al. (2007) it is emphasized that thethickness of dead layer of the detector is taken as 1.16 mminstead of 0.35 mm as provided by the manufacturer in 1996. Inalmost cases of usage, all the components of the spectrometer arefixed except the sample geometry. This is because the configura-tion of the spectrometer should be fixed so that its characteristicsare unchanged in measurements. Hence, changing dimensions ofthe beaker to find out the most relevant one is the mostconvenient way in order to improve the detection efficiency ofthe spectrometer. The problem then aims at finding optimaldimensions of Marinelli beakers to maximize the detector effi-ciency while the other components of the spectrometer are fixed.

In addition, each measurement requires a specific volume ofsample that should be in conformity with laboratory conditionsand measurement purposes. So the sample volume will be fixed ata certain value. Fig. 2 presents the arrangement of a Marinellibeaker in the spectrometer. As seen in Fig. 2 the Marinelli beakercan be divided into two parts with different volumes V1 and V2,where V1¼pr2h1 is the volume of the upper cylinder with radius r

and height h1 and V2¼p(r2�d2)h2 is the volume of the ring part

surrounding the detector side with an outer radius r, an innerradius d and a height h2. The inner radius d of the beaker isdetermined by the detector radius, therefore it is fixed. Thevolume of the beaker V¼V1þV2 is a function of three independentvariables r, h1 and h2. In case the volume of measuring sample is apredetermined value, say V0, then

pr2h1þpðr2�d2Þh2 ¼ V0 ð1Þ

Therefore, the Marinelli beaker is defined by only two independentvariables out of the three above mentioned variables. If the outerradius r of the beaker and the height h2 of the part surroundingthe detector side are chosen as independent variables, the thirdparameter h1 of the beaker can be calculated by using Eq. (1).

s in cm (a) and the detector with dimensions in mm (b).

Fig. 2. Schematic layout of Marinelli beaker.

N.Q. Huy et al. / Applied Radiation and Isotopes 70 (2012) 2695–2702 2697

From the above standpoint, the improvement of the detectionefficiency of the gamma spectrometer becomes a problem ofoptimization of Marinelli beaker dimensions containing a con-stant volume of sample. The optimization problem can beformulated as follows.

Decision variables are the outer radius r and the height h2 ofthe part surrounding the detector side.

The objective function is the detector efficiency of the gammaspectrometer:

e¼ f ðr,h2Þ-max ð2Þ

subject to the following constraints:

drrrR, ð3Þ

0rh2rH, ð4Þ

V2 ¼ pðr22d2Þh2rV0 ð5Þ

where R is the outer radius of the sample volume, H is thedistance from the upper detector surface to the bottom of thesample room, V0 is a predetermined volume value of sample.

The first and second constraints on the variables r and h2 aredue to the limit of the sample room space inside the lead shield,and the third constraint is the requirement on the fixed volume ofmeasured sample. It is recognized that this is a nonlinearoptimization problem and it is impossible to establish an explicitanalytical formula describing the dependence of the detectorefficiency on the decision variables. It means that the problemmight not be solved by analytical calculation. However, if thedetector efficiency can be evaluated as a function of the decisionvariables by a certain means, we can search for optimal solutionsby several methods.

3. Precise modeling of the detector in MCNP simulation

In this study, the MCNP5 simulation computer code is used tosimulate the gamma spectra of radioactive sample contained inMarinelli beakers so that the detector efficiency of the HPGedetector based gamma spectrometer can be evaluated for mea-suring geometries; therefore precise modeling of the detector andmeasuring geometry in Monte Carlo simulation is very important.The MCNP computer code has been widely applied to modelingand simulation of HPGe detector gamma spectrometry (Kambojand Kahn, 2003; Salgado et al., 2006; Maleka and Maucec, 2005);however, recent research works (Hardy et al., 2002; Rodenaset al., 2003; Huy et al., 2007; Maleka and Maucec, 2005; Bosonet al., 2008) showed that calculated results were not completelyin agreement with experimental data. One of the reasons is the

difference between actual parameters of the detector and datasupplied by manufacturers, especially the dead layer thickness. Inour past work (Huy et al., 2007), we could precisely model theconsidered gamma spectrometer and reached a good agreementbetween MCNP simulation and experiment. In that work, wereckoned that an increase in the detector dead layer resulted in adecrease in the detector efficiency. By making a comparisonbetween the simulated detector efficiencies and experimentaldata of a point like radioactive source with the energy range from511 keV to 1332 keV, we also determined that the actual thick-ness of the detector dead layer is 1.16 mm. Based on the aboveresult, we determine the geometric parameters of the detectorand assembled components of the considered spectrometer asdescribed below.

The gamma spectrometer consists of the Canberra HPGeGC1518 detector, a pre-amplifier, a spectroscopy amplifier, anMCA and a high-voltage supply. The detector is placed in acylindrical aluminum endcap of 3.81 cm radius and 8.40 cmheight inside a lead shield. The lead shield is a hollow cylinderof 10 cm thickness, 25 cm external radius and 50 cm height. Thecylindrical hollow has a radius of 15 cm and a height of 30 cm.A 0.3 cm tin and 0.1 cm copper layers are lined at the lowersurface of the lead cover; a 0.8 cm copper layer at the uppersurface of the lead bottom; a 0.8 cm tin, 6.25 cm paraffin in thelower half and 4.75 cm in the upper half; and 0.6 cm copperlayers at the inner vertical surfaces of the lead shield. Thestructure of the coaxial detector and the geometric parametersand materials of the detector is shown in Fig. 1. The maincomponent of the detector is a germanium crystal with an innerhole. The cylindrical crystal has an outer diameter of 54 mm and aheight of 32 mm; and the inner cylindrical hole has a diameter of7 mm and a height of 17 mm. The nþ contact at the outer surfaceof the crystal (the outer electrode), which is considered to becompletely inactive, consists of a layer of diffused lithium atoms.The thickness of this layer is 1.16 mm of germanium equivalent.The boron ion implanted pþ contact of the inner hole surface (theinner electrode) has the thickness of 3 mm of germanium equiva-lent. The front window is an outer electrode covered by an 8.5 mmmetalized Mylar and a 0.1 mm kapton layer. The germaniumcrystal is placed in an aluminum holder of 0.76 mm thickness,which is covered by the aluminum endcap of 6.2 mm diameterand 1.5 mm thickness. The endcap-to-crystal distance is 5 mm.

With this model of the detector and the lead shield, wesimulated gamma spectra of a point like radioactive source tocalculate the energy dependent efficiency curve of the detector.The calculated results were verified by experimental data. Inexperiment, the source emitting gamma rays of 22Na (511 and1274 keV), 137Cs (662 keV), 54Mn (834 keV) and 60Co (1173 and1332 keV) was placed at a distance of 15 cm above the detectorwindow on its center axis. Relative uncertainties of measuredefficiencies were about 3%, which were mainly determined by theuncertainties of reference source activities, while statistical errorswere lower than 1%. In MCNP simulation, the F8 tally was used toextract the pulse height distribution of gamma spectra, the fullenergy peak efficiency was then calculated corresponding todifferent energy peaks. The experimental and simulated gammaspectra for 137Cs and 60Co nuclides are shown in Fig. 3. The energydependent efficiency curve of the detector is presented in Fig. 4.The agreement between the simulated curves and experimentdata in Figs. 3 and 4 is an indication that the model of the detectorand surrounding geometry was adequate. To save computingtime, mono-energetic source simulations were used. Hence, aseparate case run of MCNP was for each energy peak in thesource. This also neglects the influence of higher energy nuclideson the lower number of histories. The total number of histories ineach run must be large enough to obtain tally results with the

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Fig. 3. Experimental and simulated gamma spectra for 137Cs (3a, 3b) and 60Co (3c, 3d). The curves are normalized to have the equal areas under the 661.6 keV (3a) and

1332.5 keV (3c) full energy peaks.

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Fig. 4. Energy dependent curve of the detector efficiency.

N.Q. Huy et al. / Applied Radiation and Isotopes 70 (2012) 2695–27022698

minimum uncertainty related to the calculated efficiency. Simulationrelative error was kept under 0.003 if the number of source particleswas set as 3,000,000.

4. Genetic algorithm optimization

4.1. Genetic algorithm procedure

In this work a simple genetic algorithm was applied to solvethe problem. The decision variables r and h2 are encoded into astring of bits 1 or 0. A full string of length l consists of twosequential substrings of length l1 and l2, l¼ l1þ l2, of which thefirst substring represents the variable r and the second representsthe variable h2. The length of each substring is determined by theprecision and limited range of variable values. The precisions mr

and mh2 of r and h2 are evaluated by the following formula:

mr ¼rmax�rmin

2l1�1, mh2

¼h2max�h2min

2l2�1ð6Þ

where rmin and rmax are the limits of r and h2min and h2max are thelimits of h2.

A string is decoded into values of r and h2 following theformula:

h2 ¼ h2minþDecðm1,m2,:::,ml1 Þh2max�h2min

2l2�1ð7Þ

r¼ rminþDecðml1þ1,ml1þ2,:::,mlÞrmax�rmin

2l1�1, ð8Þ

N.Q. Huy et al. / Applied Radiation and Isotopes 70 (2012) 2695–2702 2699

where Decðm1,m2, :::,ml1 Þ is the decimal number of the first sub-string and Decðml1þ1,ml1þ2,:::,mlÞ is the decimal number of thesecond substring.

Because genetic algorithms use the information of a fitnessfunction when selecting the fittest individuals of the currentgeneration for the next generation and the fitness function isrequired to be nonnegative, it is necessary to map the objectivefunction to a fitness function form (Goldberg, 1989). According tothe penalty method (Do and Nguyen, 2007), to process constraint(5) the fitness function is established as follows:

f itðr,h2Þ ¼ e�Max 1�V0

V2

� �e,0

� �ð9Þ

In this work, selection is performed by the roulette wheel spinmethod that chooses the fittest individuals in the current gen-eration to create a breeding pool for the next generation. Inaddition, an elitism strategy is incorporated into the geneticsearch by transferring the best solutions in terms of the fitnessfound during the search process to the next generation (Yanget al., 2005; Do and Nguyen, 2007). Crossover is performed by theone-point method that mixes parts of the two parent solutions inthe breeding pool to create two offsprings. And mutation isperformed to maintain the diversity of the population by ran-domly altering the value of a bit between 1 and 0 with a smallprobability. The genetic algorithm search process is carried out bythe following steps:

Step 1: Create the initial population at random.Step 2: Each string in the current generation is decoded into areal number in the intervals [rmin, rmax] for r and [h2min, h2max]for h2.Step 3: Assign the real number of r and h2 to the MCNP5 input.Step 4: Run the MCNP5 code to simulate the gamma spectrumfor the measuring geometry corresponding the values of r andh2. Based on the obtained spectrum the detector efficiency iscalculated.Step 5: Evaluate the fitness corresponding to that stringaccording to Eq. (9).Step 6: Repeat from step 2 to step 5 until all the individuals inthe current generation are evaluated.

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Step 7: Check the stopping condition. The search process stopswhen the population average fitness value in several sequen-tial generations does not improve.Step 8: Perform the selection to update the breeding pool fornew generation.Step 9: Perform the crossover for individuals in the breedingpool.Step 10: Perform the mutation for every bit of each string inthe breeding pool.Step 11: Update the new population and repeat the searchprocess from step 2.

4.2. Optimal solutions of genetic algorithm

The problem was first solved by finding optimal solutions for thecase of the sample volume V0¼450 cm3 and the gamma energy of364 keV. In simulation and optimization calculation, Marinellibeaker has 1 mm thick vessels made of polyvinyl chloride, and itsinside volume is the volume of sample containing in the beaker sothe dimensions of sample is just the internal ones of the Marinellibeaker. The decision variables r and h2 vary in the range determinedby dimensions of the sample room inside the lead shield so that themargin values of r are rmin¼4.06 cm and rmax¼7.2 cm, and themargin values of h2 are h2min¼0 and h2max¼12 cm. Genetic algo-rithm parameters are set up as follows: the crossover probability of0.6, the mutation probability of 0.01, these parameters were provedworking well in a wide range of optimization problems (Park, 1990),and the population size of 50.

Calculated results with a search process running over 100generations in four separated times are presented in Fig. 5. Popula-tion average fitness curves in four different runs all increase withgeneration, strongly vary at the beginning stage and reach anequilibrium value of about 0.04 after about 60 generations. Thisproves that the genetic algorithm improves the detector efficiencywith generation and the search process makes a good progress. Thedetector efficiency in each run reaches its maximum value of0.04134 corresponding to decision variable values of r¼5.41 cm,h2¼6.23 cm, 0.04132 corresponding to r¼5.40 cm, h2¼6.46 cm,0.04131 corresponding to r¼5.40 cm, h2¼6.49 cm, and 0.04133

50 60 70 80 90 100eration

ing over 100 generations in four separated runs.

N.Q. Huy et al. / Applied Radiation and Isotopes 70 (2012) 2695–27022700

corresponding to r¼5.38 cm, h2¼6.44 cm. It can be recognized thatthe search process approaches the optimal solutions with r in theinterval [5.38, 5.41] cm and h2 in the interval [6.23, 6.49] cm. It isclear that the search process reaches near the global optimumsolution.

Analyzing the solutions with fitness values bigger than 0.04stored in the search process indicates that the search process canfind out an optimal solution in each run. This optimal solutionmight not be exactly the global optimum but in the ‘‘optimalregion’’ consisting of the global optimum. From this point of view,we can conclude that for the sample volume of V0¼450 cm3 andthe gamma energy of 364 keV, the optimal solution found by thegenetic algorithm for 4 runs is in a region defined by r¼(5.38–5.41) cm and h2¼(6.23–6.49) cm, where 5.40 cm and 6.40 cm areaverage values of r and h2, respectively.

4.3. Influence of gamma energy, sample chemical composition and

sample density on the optimal sizes of Marinelli beaker

The optimization calculation was carried out for a large rangeof gamma ray energies and for various chemical compositions anddensities of sample material in order to evaluate the influence ofthese parameters on optimal sizes of Marinelli beaker.

4.3.1. Influence of gamma energy

In order to investigate the influence of gamma energy onoptimal dimensions of Marinelli beaker, the optimization calcula-tion was carried for 10 energy values from 255 keV to 1926 keV.Table 1 presents the calculated optimal dimensions for the case ofalluvial soil with the density of 1 g/cm3. It is seen from Table 1that the optimal dimensions of Marinelli beaker are not sensitiveto the gamma energies. This is expressed by the fact that therelative standard deviations are 0.41% for r, 2.60% for h1, 1.89% forh2 and 1.64% for t.

4.3.2. Influence of chemical composition of sample material

A set of 9 samples of various chemical compositions includingwater, hay powder, rye flower powder, milk powder, cabbagepowder, animal blood, alluvial soil, red soil and limestonepowder, as presented in Table 2, was used for analysis of theeffect of chemical composition on the optimal dimensions ofMarinelli beaker. Optimization calculations were carried out forthe above samples with the same 1 g/cm3 density and for 10energies from 255 keV to 1926 keV. Table 3 presents calculatedresults for optimal values of r, h1, h2 and t parameters, averagedover 10 gamma energy values (as in Table 1). The average values,standard deviations and relative standard deviations for the9 samples are shown in the last three rows. It follows from

Table 1Calculated optimal dimensions r, h1, h2 and t for 10 energy values in the case of

alluvial soil with the density of 1 g/cm3.

Gamma energy (keV) r (cm) h1 (cm) h2 (cm) t (cm)

255 5.32 2.20 6.85 1.26

364 5.42 2.14 6.25 1.36

392 5.40 2.14 6.40 1.34

662 5.38 2.23 6.33 1.32

834 5.44 2.13 6.09 1.38

1173 5.43 2.10 6.26 1.37

1274 5.43 2.15 6.16 1.37

1332 5.43 2.13 6.16 1.37

1661 5.39 2.27 6.12 1.33

1926 5.42 2.21 6.04 1.36

Average value 5.42 2.17 6.16 1.36

Standard deviation 0.02 0.06 0.12 0.02

Relative standard deviation (%) 0.41 2.60 1.89 1.64

Table 3 that the optimal values of r, h1, h2 and t fluctuate with therelative standard deviations of 0.30%, 1.12%, 1.46% and 1.23%,respectively. These relative standard deviations are equivalent tothe ones in Table 1, so the influence of chemical compositions onoptimal dimensions of Marinelli beaker is insignificant.

4.3.3. Influence of sample material density

The alluvial soil with 5 density values from 0.8 to 1.6 g/cm3

was used for analysis of the effect of material density on theoptimal dimensions of Marinelli beaker. Table 4 presents theoptimal values of r, h1, h2 and t for 10 energies. The average sizesof r, h1, h2, t, their standard deviations and relative standarddeviations are presented in the last three rows of the table. Asseen in Table 4, the relative standard deviations are also as smallas the ones in the above cases. It means that there is no influenceof density on the optimal sizes of Marinelli beaker.

From the above results, it is obvious that there is no discern-able trend with gamma energy, sample chemical composition andsample density for the optimal sizes of Marinelli beaker. There-fore setting the averages of all the r, h1, h2 and t parameters inTables 1, 3, and 4 as the optimal dimensions of Marinelli beakerfor a wide range of sample with different energies, chemicalcompositions and material densities for using with the HPGedetector based gamma spectrometer is acceptably an adequatemanner. In conclusion, for a Marinelli beaker with a volume of450 cm3 and an 8.12 cm hollow diameter the optimal dimensionsare as follows r¼5.40 cm, h1¼2.18 cm, h2¼6.27 cm andt¼1.34 cm. Furthermore, an investigation of the values of r, h1,h2 and t in the above tables shows that the optimal Marinellibeaker geometries have the depth of the reentrant cavity and thetop radius approximately twice the length and the radius of thegermanium crystal of the detector respectively, and the ratio ofthe top layer to the side layer approximately equal to 1.5. It isnoted that on the detector top surface is a thin capton window,which weakly absorbs gamma rays, but the detector side is madeof aluminum, which stronger absorbs gamma rays emitted fromthe source; hence the fact that the thickness of the top layer ofsample is bigger than the thickness of the side layer of sample isreasonable.

5. Experimental validation of calculated results

An experiment was carried out to verify the calculated results.The internal dimensions of experimental beakers are presented inTable 5 together with the full energy peak efficiencies. The sampleused for measurements was an aqueous 131I solution, which hasthree main gamma lines of 284.4 keV, 364.3 keV and 636.5 keV.The four various beakers include an optimal beaker (called beaker 1),dimensions of which were calculated for 364.3 keV in Section 4.2,a conventional beaker (beaker 2) (IEEE, 1978) and two non-optimal beakers (beakers 3, 4). The last column in Table 5presents the ratio of the detector efficiency corresponding cases,in which beakers 2–4 were used, to that for beaker 1.

5.1. Beakers

The four experimental beakers all have a cylindrical hollowwith the diameter of 8.12 cm. It is suitable for the HPGe GC1518detector having a 7.62 cm aluminum end cap diameter (Huy et al.,2007). All the beakers are made of the same material; the outer(indicated by number 1 in Fig. 6) and inner (number 2) vessels aremade from 0.4 mm thick polyvinyl chloride sheets, whilst hor-izontal layers (numbers 3–5) are 2 mm thick polyacrylic plates.The materials were cut by a laser-cutting machine then joinedby cyanoacrylate glue with the total tolerance of about 0.5 mm.

Table 2Chemical compositions of 9 samples used for analysis of the effect of chemical composition on the optimal dimensions of Marinelli beaker.

Sample Chemical composition

Water H2O (100%)

Hay powder (mg/kg) (IAEA, 2002) Ba (6), Br (9), Ca (21,000), Cd (0.03), Co (0.13), Cr (6.5), Cs (0.017), Cu (9.4), Eu (0.0024), Fe (186), Hg (0.013), K (21,000), La (0.07),

Mg (1360), Mn (47), Mo (0.09), Na (500), Ni (4.2), P (2300), Pb (1.6), Rb (7.6), Sb (0.0019), Sc (0.0014), Se (0.022), Sr (40), Zn (24).

Rye flour powder (mg/kg)

(IAEA, 2002)

As (0.1), Ba (11), Br (0.38), Ca (149), Cd (0.017), Cl (570), Co (0.0023), Cs (0.002), Cu (0.95), Fe (4.1), K (1925), Mg (121), Mn (2.06),

Mo (0.12), Na (2.6), P (592), Rb (0.48), Sb (0.0026), Zn (2.53).

Milk powder (mg/kg) (IAEA, 2002) Br (12.32), Ca (12,870), Cu (0.57), Fe (2.53), K (17,620), Mg (1060), Mn (0.19), Mo (0.31), Na (4180), P (10,100), Rb (14.03),

Sr (4.09), Zn (39.56).

Cabbage powder (mg/kg)

(IAEA, 2002)

As (0.01), Ba (11), Ca (18,500), Cd (0.12), Cr (1.3), Cu (5.67), Fe (148), Hg (0.013), K (32,500), Mg (2160), Mn (31.9), Na (580),

Ni (1.05), Se (0.12), Sr (49.2), Zn (38.6).

Animal blood (mg/kg) (IAEA, 2002) Br (22), Ca (286), Cu (4.3), Fe (2400), K (2500), Mg (99), Na (12,600), Ni (1), P (940), Pb (0.18), Rb (2.3), S (6500), Se (0.24), Zn (13).

Alluvial soil (%) (Huy and Luyen,

2004)

SiO2 (89.32), Al2O3 (4.42), Fe2O3 (0.50), FeO (0.97), TiO2 (0.50), MnO (0.05), MgO (0.04), CaO (0.74), Na2O (0.12), K2O (0.21), P2O5

(0.04), other (1.58)

Red soil (%) (Huy and Luyen, 2004) SiO2 (33.08), Al2O3 (25.46), Fe2O3 (18.06), FeO (2.32), TiO2 (4.52), MnO (0.22), MgO (0.49), Na2O (0.11), K2O (0.14), P2O5 (0.08),

other (14.56)

Limestone powder (%) (Huy and

Luyen, 2004)

CaCO3 (71.33), MgCO3 (24.84), other (3.83)

Table 3The optimal values of r, h1, h2 and t parameters for 9 samples of various chemical

compositions, averaged over 10 energies.

Sample r (cm) h1 (cm) h2 (cm) t (cm)

Water 5.39 2.18 6.40 1.33

Hay powder 5.39 2.22 6.26 1.33

Rye flower powder 5.40 2.17 6.32 1.34

Milk powder 5.37 2.23 6.38 1.31

Cabbage powder 5.37 2.25 6.34 1.31

Animal blood 5.38 2.21 6.38 1.32

Alluvial soil 5.39 2.20 6.33 1.33

Red soil 5.43 2.18 6.11 1.37

Limestone powder 5.40 2.22 6.23 1.34

Average value 5.39 2.21 6.31 1.33

Standard deviation 0.02 0.02 0.09 0.02

Relative standard deviation (%) 0.30 1.12 1.46 1.23

Table 4Five sets of the optimal values of r, h1, h2 and t of the Marinelli beaker for 5 various

densities of alluvial soil averaged for 10 gamma energies from 255 keV to

1926 keV.

Density (g/cm3) r (cm) h1 (cm) h2 (cm) t (cm)

0.8 5.38 2.24 6.30 1.32

1.0 5.40 2.18 6.27 1.34

1.2 5.41 2.18 6.20 1.35

1.4 5.39 2.13 6.46 1.33

1.6 5.41 2.09 6.42 1.35

Average value 5.40 2.16 6.33 1.34

Standard deviation 0.01 0.06 0.11 0.01

Relative standard deviation (%) 0.24 2.59 1.72 0.96

Table 5Dimensions of 4 measured Marinelli beakers and calculated values of the full

energy peak efficiency at the gamma energy of 364.3 keV.

Marinelli beaker r (cm) h1 (cm) h2 (cm) t (cm) Efficiency Ratio

Beaker 1 (Optimal) 5.38 2.18 6.44 1.32 0.04133 1.0000

Beaker 2 (Conventional) 5.57 1.50 6.66 1.50 0.03994 0.9664

Beaker 3 (Non-optimal) 5.65 0.56 8.12 1.59 0.03646 0.8822

Beaker 4 (Non-optimal) 5.69 0.24 8.54 1.63 0.03423 0.8282

h1

2 mm

h2

r

0.4 mmd = 7.72 cm

12

3

4

5

Fig. 6. Dimensions of experimental Marinelli beakers.

N.Q. Huy et al. / Applied Radiation and Isotopes 70 (2012) 2695–2702 2701

The vertical length of the aqueous 131I solution layer h1 above thedetector was measured with an error of about 0.5 mm. The totalheight of the beakers is larger than the sum of h1þh2, thereforethe volume of beakers is larger than the sample volume.

The internal dimensions of the beakers are just the ones ofsample as presented in Table 5, except for h1.

5.2. Sample and counting efficiency

An aqueous 131I solution was prepared with a specific activityso that the relative uncertainty of counts reached several partsper thousand and the dead time of the counting system was lessthan 5%. The beakers were all filled with a 450 cm3 solution andmeasured by the gamma spectrometer. Photopeak area at284.4 keV, 364.3 keV and 636.5 keV gamma lines was obtainedand the full energy peak efficiency at those energies was thenevaluated. Ratios of the detector efficiencies in arrangementsusing the other beakers to the ones using the optimal beakerare presented in Table 6. It means that the values correspondingto the optimal beaker geometry are unity. It follows from Table 6that calculated results are in a good agreement with measureddata.

It is also noticed that the efficiencies corresponding to theconventional beaker geometry approach to 97–98% of the onescorresponding to the optimal beaker geometry. It means that theconventional beaker is also a near optimal case for the HPGeGC1518 detector.

6. Conclusion

In this work, the MCNP computer code was used to model andsimulate precisely the response function of the HPGe detectorof a gamma spectrometer used for environmental radioactivitymeasurements. Based on simulated gamma spectra, the detector

Table 6Calculated and measured values of the detector efficiency for arrangements with three Marinelli beakers relative to the optimal beaker geometry.

Beaker 284.2 keV 364.3 keV 636.5 keV

Calculation Experiment Calculation Experiment Calculation Experiment

1 (Optimal) 1.000 1.000 1.000 1.000 1.000 1.000

2 (Convent.) 0.97570.005 0.98670.005 0.96670.005 0.98070.005 0.97870.005 0.97070.005

3 (Non-opt.) 0.89270.005 0.90370.010 0.88270.005 0.90470.010 0.88870.005 0.88970.010

4 (Non-opt.) 0.83570.005 0.83770.010 0.82870.005 0.82270.010 0.83370.005 0.81770.010

N.Q. Huy et al. / Applied Radiation and Isotopes 70 (2012) 2695–27022702

efficiency was evaluated as a function of dimensions of Marinellibeaker typed source geometries and the problem of measuringsample dimension optimization for a constant volume wascompletely formulated. A simple genetic algorithm in combina-tion with MCNP5 code was used to search for optimal Marinellibeakers with a relevant computing time. While other componentsof the spectrometer are fixed, the optimal Marinelli beakergeometry has the dimensions in the ranges of (5.32–5.44) cmfor outer radius r, (2.09–2.27) cm for the height of the samplelayer above the detector h1, (6.04–6.85) cm and (1.26–1.38) cmfor the height and the thickness of the sample part surroundingthe detector side h2 and t, respectively. The experimental andcalculated results show that the optimal geometry of r¼5.38 cm,h1¼2.18 cm, h2¼6.44 cm and t¼1.32 cm for the sample volumeof 450 cm3 makes the detector efficiency of the spectrometer tobe increased 2–3% compared to using a conventional Marinellibeaker (IEEE, 1978) with the same source volume. Calculatedresults also proved that there had no significant influence ofgamma spectrum energy in the range 255–1926 keV, samplechemical composition and material density on the optimaldimensions of the Marinelli beaker geometry. In practice, aMarinelli beaker with dimensions taken from the optimal regionis a good choice for measurements of low-level radioactiveenvironmental sample using HPGe detector based gammaspectrometers.

Acknowledgments

The authors express their sincere thanks to Dr. Tran VanLuyen, M.Sc. Thai My Phe and Mr. Nguyen Van Mai for their helpin use of the HPGe GC1518 detector based gamma spectrometerand valuable discussions.

This work was completed with financial support from Vietnam’sNational Foundation for Science and Technology Development(NAFOSTED), code 103.04.01.09.

References

Ahmed, A.S., Capello, K., Chiang, A., Mendez, E.C., Kramer, G.H., 2009. Optimizationof geometric parameters for Marinelli beaker to maximize the detectionefficiency of an HPGe detector. Nucl. Instrum. Methods A 610, 718–723.

Bonfanti, G., Della Dora, G., 1981. Optimum counting geometries of uniform andlarge g-sourcses for Ge(Li) detectors. An experimental study. Radiochem.Radioanal. Lett. 49, 215–230.

Boson, J., Agren, G., Johasson, L., 2008. A detailed investigation of HPGe detectorresponse for improved Monte Carlo efficiency calculations. Nucl. Instrum.Methods A 587, 304–314.

Briesmeister, J.F., 2000. MCNP-A General Monte-Carlo N-Particle Transport Code,Version 4C. Los Alamos, LA-13709-M.

Budjas, D., Heisel, M., Maneschg, W., Simgen, H., 2009. Optimization of theMC-model of a p-type Ge-spectrometer for the purpose of efficiency determi-nation. Appl. Radiat. Isot. 67, 706–710.

Courtine, F., Pilleyre, T., Sanzelle, S., Miallier, D., 2008. Ge well detector calibrationby means of a trial and error procedure using the dead layers as a unique

parameter in a Monte Carlo simulation. Nucl. Instrum. Methods A 596,229–234.

Do, Q.B., Nguyen, L.P., 2007. Application of a genetic algorithm to the fuel reloadoptimization for a research reactor. Appl. Math. Comput. 187, 977–988.

Ewa, I.O.B., Bodizs, D., Czifrus, Sz., Molnar, Zs., 2001. Monte Carlo determination offull energy peak efficiency for a HPGe detector. Appl. Radiat. Isot. 55, 103–108.

Forster, R.Arthur, et al., 2004. MCNPTM

Version 5. Nucl. Instrum. Methods B 213,82–86.

Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization, and MachineLearning. Addison-Wesley 1989.

Hardy, J.C., Jacob, 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 HPGedetector: source measurements and Monte Carlo calculations with sub-

percent precision. Appl. Radiat. Isot. 56, 65–69.Helmer, R.G., Nica, N., Hardy, J.C., Iacob, V.E., 2004. Precise efficiency calibration of

an HPGe detector up to 3.5 MeV with measurements and Monte-Carlocalculations. Appl. Radiat. Isot. 60, 173–177.

Hemingway, J.D., 1986. Investigation towards an improved Marinelli beaker for

gamma detectors. J. Radioanal. Nucl. Chem. 99, 299–306.Hill, R.F., Hine, G.J., Marinelli, L.D, 1950. The quantitative determination of gamma

radiation in biological research. Am. J. Roentgenol. Radium Ther. 63, 160–169.Hurtado, S., Garcia-Leon, M., Garcia-Tenorio, R., 2004. GEAN4 code for simulation

of a germanium gamma-ray detector and its application to efficiency calibra-tion. Nucl. Instrum. Methods A 518, 764–774.

Huy, N.Q., Binh, D.Q., An, V.X., 2007. Study on the increase of inactive germaniumlayer in a high-purity germanium detector after a long time operationapplying MCNP code. Nucl. Instrum. Methods A 573, 384–388.

Huy, N.Q., Luyen, T.V, 2004. A method to determine 238U activity in environmentalsoil samples by using 63.3-keV-photopeak-gamma HPGe spectrometer. Appl.

Radiat. Isot. 61, 1419–1424.IAEA, 2002. IAEA Analytical Quality Control Services Reference Materials Catalogue

2002–2003, A-1400 Vienna, Austria.IEEE, 1978. IEEE Standard Techniques for Determination of Germanium Semiconductor

Detector Gamma-Ray Efficiency using a Standard Marinelli (Re-entrant) BeakerGeometry, ANSI-IEEE Std 680-1978.

Kamboj, S., Kahn, B., 2003. Use of Monte Carlo simulation to examine gamma-rayinteractions in germanium detectors. Radiat. Meas. 37, 1–8.

Laborie, J.M., Le Petit, G., Abt, D., Girard, M., 2000. Monte Carlo calculation of theefficiency calibration curve and coincidence-summing corrections in low-levelgamma-ray spectrometry using well-type HPGe detectors. Appl. Radiat. Isot

53, 57–62.Maleka, P.P., Maucec, M., 2005. Monte Carlo uncertainty analysis of germanium

detector response to gamma rays with energies bellows 1 MeV 538,631–639Nucl. Instrum. Methods A 538, 631–639.

Park, G.T., 1990. Advances in optimization and their applicability to problems inthe field of nuclear science and engineering. Adv. Nucl. Sci. Technol. 21, 195.

Rodenas, J., Pascual, A., Zarza, I., Serradell, V., Ortiz, J., Ballesteros, L., 2003. Analysisof the influence of germanium dead layer on detector calibration simulationfor environmental radioactive samples using the Monte Carlo method. Nucl.

Instrum. Methods A 496, 390–399.Salgado, C.M., Conti, C.C., Becker, P.H.B., 2006. Determination of HPGe detector

response using MCNP5 for 20–150 keV X-rays. Appl. Radiat. Isot. 64, 639–664.Sood, Avneet, Forster, R.Arthur, Adams, Bryce J., White, Morgan C., 2004. Verifica-

tion of the pulse height tally in MCNP5. Nucl. Instrum. Methods B 213,167–171.

Vargas, M.Jurado, Fernandez Timon, A., Cornejo Diaz, N., Perez Sanchez, D., 2002.Monte Carlo simulation of the self-absorption corrections for natural samplesin gamma-ray spectrometry. Appl. Radiat. Isot. 57, 893–898.

Yang, S.M., Shao, D.G., Luo, Y.J., 2005. A novel evolution strategy for multiobjectiveoptimization problem. Appl. Math. Comput. 170, 850–873.