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Optimization of the marinelli beaker dimensions using genetic algorithm
Journal of Environmental Radioactivity 172 (2017) 81e88
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Journal of Environmental Radioactivity journal homepage: www.elsevier.com/locate/jenvrad
Optimization of the marinelli beaker dimensions using genetic algorithm Seyed Mehrdad Zamzamian*, Seyed Abolfazl Hosseini, Mohammad Samadfam Department of Energy Engineering, Sharif University of Technology, Tehran Zip code: 8639-11365, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 August 2016 Received in revised form 1 March 2017 Accepted 10 March 2017
A computational code, based on the genetic algorithm and MCNPX version 2.6 code was developed and used to investigate the effects of some important parameters of HPGe detector (such as Al cap thickness, dead-layer thickness and Ge hole size) on optimum dimensions of marinelli beaker. In addition, the effects of detector material on optimal beaker dimensions were also investigated. Finally, the optimized beaker dimensions at various beaker volumes (300, 500, 700, 1000 and 1500 cm3) were determined for some conventional Ge detectors with different crystal sizes (16 sizes). These sets of data then were used to drive mathematical formulas (obtained by best fitting to data sets). The results showed that, there is no meaningful correlation between the optimum dimensions of the beaker and each of the dead-layer thickness, Al cap thickness and the Ge-crystal hole size. On the other hand, the optimum beaker radius increases with decreasing the density of the detector material while the beaker height decreases. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Genetic algorithm Marinelli beaker HPGe detector MCNPX2.6
1. Introduction Gamma-ray spectrometry is a common technique widely used for identification and quantification of radionuclides in samples of different matrices and geometries. The use of High Purity Germanium (HPGe) detectors with high energy resolution allows identification and activity measurement of natural and artificial radionuclides in biological, geological and environmental samples, with the highest possible sensitivity (Azbouche et al., 2015). To analysis the environmental samples with low radioactive concentrations, a sample with high mass is required to obtain accurate and precise results. In order to increase the net counts due to the samples, in addition to a good shielding method, the various measurement parameters should be optimally balanced. The use of marinelli beakers in environmental samples permits one to obtain greater net count by positioning the sample volume as close as possible to active volume of the detector. In a constant sample volume, the detector efficiency varies at different geometrical dimensions of the beaker due to self-attenuation of photon in sample and out-scattering from the detection region. Therefore, dimensions of the marinelli beaker should be optimum to increase
* Corresponding author. E-mail address: [email protected] (S.M. Zamzamian). http://dx.doi.org/10.1016/j.jenvrad.2017.03.020 0265-931X/© 2017 Elsevier Ltd. All rights reserved.
the efficiency of the detector (Ahmed et al., 2009; Shweikani et al., 2014). Several researchers have tried to develop convenient methods for determining the optimum dimensions of the marinelli beaker. Chung et al. (1991) have measured the optimum geometry of marinelli-like samples (0.1e4 L) using a 20% relative efficiency HPGe detector. They reported that the optimum geometry of larger samples (5e60 L) held in marinelli beaker can be extrapolated from their work. Sima (1990) used a Monte Carlo algorithm and auto absorption (self-absorption) factors to determine the semiconductor detector efficiency for marinelli geometry. Ahmed et al. (2009) have studied optimization of marinelli beaker for increasing efficiency of an Ortec GMX S Gamma-X HPGe detector using MCNP code. They recommended that the optimum dimensions for 200e500 mL sample volumes should be used for better measurement. Melquiades and Appoloni (2001) have measured self-absorption factors vs. density for five marinelli beakers containing powdered milk samples with 40K and 208Tl. They have verified the relation y ¼ 1.96608x-74813 for 40K line of 1460.8 keV and y ¼ 1.4484x-0.83258 for 208Tl line of 2614.47 keV, where x and y were density (g/cm3) and self-absorption factor, respectively. Debertin and Jianping (1989) calculated selfabsorption in marinelli beakers mathematically. Azbouche et al. (2015) developed a computational procedure for gamma ray spectroscopy of large volume sample based on achieving
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Abbreviations DL GA HPGe r h1 h2 rmax hmax
Dead-Layer Genetic Algorithm High Purity Germanium Beaker Radius Cylinder Height (height of the cylinder part of the beaker) Ring Height (height of the ring part of the beaker surrounding the detector) Maximum Limit of r Maximum Limit of h2
agreement between 152E source packed in marinelli beaker and MCNP5 simulation. They concluded that their method can be used in determination of the specific activities of radionuclides in soil. In the framework of environmental measurements, an application based on efficiency trfpansfer (ET) method have been described by Vidmar et al. (2010) as a means of calculating the full-energy-peak efficiencies (FEPE) and the ET method have been validated for marinelli geometries by Ferreux et al. (2013). After developing genetic algorithm (GA) by john Holland and his colleagues, the algorithm was considered as one of the best optimization tools in various scientific applications (Do and Nguyen, 2007; Goldberg, 1989; Haupt and Haupt, 2004; Pham and Karaboga, 2000). A combination of GA and MCNP simulation were used to obtain the maximum detector efficiency by Huy et al. (2012). They reported the optimum dimensions of a 450 cm3 marinelli beaker for the GC1518 HPGe detector. They suggested that the effects of gamma energy, chemical composition of the sample and the sample density on optimum dimensions are negligible. In this study, we also used a combination of GA and MCNP simulation to investigate the effects of some important detector parameters, such as thickness of Al cap, dead-layer (DL) and crystal size, on optimum dimensions of marinelli beaker.
Fig. 1. The HPGe detector and marinelli beaker.
Table 1 Specifications of the Canberra GC1518 HPGe detector (after Huy et al., 2007). Parameter
Dimension (cm)
Aluminum Cap thickness Aluminum Cap diameter Germanium dead-layer thickness End cap to crystal distance Germanium crystal height Germanium crystal diameter Core hole height Core hole diameter
0.15 7.62 0.116 0.5 3.2 5.4 1.7 0.7
2. MCNPX simulation of the detector and marinelli beaker
at a distance of 15 cm from detector surface. The peak absolute efficiencies of the detector with specifications given in Table 1, was extracted by running MCNPX for each energy, ranging from 60 to 1332 keV. Only the bare detector as illustrated in Fig. 1 without any shielding was simulated. Since the normalized count under full energy peak area for mono-energetic sources was considered as output of the MCNPX, definition of the surrounding materials (shield) were not included in the simulation results. The simulation results in the present study together with the results reported by
The MCNPX version 2.6 (Pelowitz, 2008) was used for the simulation of the HPGe detector and marinelli beaker. This computational code is routinely used by many researchers to obtain the efficiency of the HPGe detector (Azli, 2015; Chham et al., 2015; Elanique et al., 2012; Salgado et al., 2006; Saraiva et al., 2016). The HPGe detector model was based on Canberra GC1518 HPGe detector, as schematically shown in Fig. 1. The materials and dimensions of the detector (Table 1) were described by Huy et al. (2007). It was assumed that the dimensions of the sample are equivalent to those of the marinelli beaker. The marinelli beaker consist of the three dimensional parameters: radius, cylinder height and ring height (r, h1 and h2, respectively, in Fig. 1). The optimization of these dimensional parameters will be resulted in maximum efficiency of the detector (Ahmed et al., 2009). The pulse height tally (F8) was used to obtain the absolute efficiency. The output was binned by E card in order to give the peak absolute efficiency (Knoll, 2010). For validation of the simulation results, the energy-dependent detector efficiency curve (obtained from the MCNP simulation) was compared with the simulation results reported by Huy et al. (2012) for the same model of the HPGe detector. The energydependent detector efficiency curve was obtained by defining the mono-energetic point sources located at the central axis of detector
Fig. 2. Energy-dependent detector efficiencies; comparison between the simulated results obtained in this study with those reported by Huy et al. (2012).
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Huy et al. (2012) are shown in Fig. 2. The differences between the two simulated results are less than 8%. 3. Genetic algorithm As shown in Fig. 1, the calculations should be performed to optimize three variables of beaker radius r, cylinder height h1 and ring height h2. Since the sample volume (¼ beaker volume) is given by:
V ¼ p r 2 r02 h2 þ pr 2 h1
(1)
Therefore, at a constant volume, if two variables are known, the third one can be obtained from Eq. (1). So, only two independent variables should be considered for optimization. In this work, the beaker radius r and the ring height h2 were chosen. A genetic algorithm was developed using MATLAB software (The MathWorks, 2015) for optimization of the beaker dimensions at constant sample volumes. It was assumed that the inner radius of the beaker (r0) is constant and equal to 4.06 cm. Obviously, the beaker radius cannot take values smaller than r0. A two-objective binary genetic algorithm was used to solve the problem. A 20 variables chromosome string (Nvar ¼ 20) was defined as the input variable:
where P1, P2, …P20 are the random binaries. The variable P1, P2, … P10 belong to r (gene 1) and P11, P12, …P20 to h2 (gene 2). The detector full peak absolute efficiency (ε) was considered as the Cost function defined by the following equation:
Cost ¼ εðr; h1 ; h2 Þ
where rmax and h2max are the maximum limits for beaker radius and ring height, respectively. The values of rmax and h2max should be properly (as small as possible) determined in order to speed up the GA convergence. The optimum value of r when h2 ¼ 0 and the optimum value of h2 when h1 ¼ 0 were chosen as the maximum limits for beaker radius r and ring height h2, respectively (Fig. 3). Needless to say that these optimum values of r and h2 are obtained at special condition where h1 or h2 is assumed to be zero. It should be noted that the ultimate optimum values of r and h2 (when the h2 or h1 has non-zero values) will be definitely smaller than the optimum values determined from Fig. 3. One can simply imagine when h1 or h2 just starts to take a non-zero value, some of the sample at far most position (with regard to detector's active volume) will be relocated to a much closer positions, rendering a higher detector efficiency. The values of rmax and h2max, determined from Fig. 3 as ~6.5 cm and ~7.5 cm, respectively. An initial population of chromosome should be generated and decoded by the following formulas (Haupt and Haupt, 2004):
rquant ¼
XNgene1
(2)
with
r0 ð ¼ 4:06Þ r rmax
(3)
r0 ð ¼ 4:06Þ r rmax
(4)
m¼1
gene1ðmÞ2m þ 2ðMþ1Þ
r ¼ rquant ðrmax rmin Þ þ rmax h2quant ¼
chromosome ¼ ½P1 ; P2 ; …; P20
83
XNvar m¼Ngene1
gene2ðmÞ2m þ 2ðMþ1Þ
r ¼ h2quant ðhmax hmin Þ þ hmax
(5) (6) (7) (8)
where m is length of the string for each gene. Gene1(m) and Gene2(m) are binary versions of the chromosomes. The MCNPX computational code for each decoded string was run and the outputs was considered as cost values. A simple natural selection was used to sort cost values in a decreasing order and generate mating pool. Two of the most valuable costs were considered as elite. An arbitrary number of top costs survived and others was discarded. From top to bottom of the pool mating (with the exception of two elite), two successive mate was selected to create offsprings using a randomly crossover point and then a mutation probability of 0.01 (¼m) was used. Finally, a new
Fig. 3. (a): The detector efficiency versus the ring height h2 when h1 ¼ 0. (b): The detector efficiency versus the beaker radius r when h2 ¼ 0.
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generation, consisting of non-mutated parents (elite), parents, mutated parents, offsprings and mutated offsprings were produced. The standard deviation of r, h1 and h2 were considered as convergence limitation and successive generation continued while all of the deviations were more than 0.1. A path through the components of the mentioned procedures is shown as a flowchart in Fig. 4.
4. Results and discussion The optimum dimensions of a 450 cm3 marinelli beaker, with detector specifications given in Table 1, were obtained as: r ¼ 5.37 cm, h1 ¼ 2.26 cm and h2 ¼ 6.29 cm. These optimum values are in good agreement with those reported by Huy et al. (2012) as r ¼ 5.38 cm, h1 ¼ 2.18 cm and h2 ¼ 6.44 cm. The convergence of the searching process was achieved after ~200 generations for all the 3 variables of r, h1 and h2. The r and h2 had the highest and lowest convergence speed, respectively, as illustrated in Fig. 5. In order to assess the effect of detector DL on optimum dimensions of beaker, the optimum dimensions were calculated for 7 different DL thicknesses. The results are given in Table 2 together with their standard deviations. These results shows that there is almost no correlation between the optimum dimensions and the DL thickness of the detector. The arithmetic means of the optimum r, h1 and h2 (also given in Table 2) have standard deviations of 0.0335, 0.0490 and 0.1895, respectively. These deviations are almost in the same order of the standard deviations for the individual optimized dimensions at different DL thicknesses, implying that the effect of DL change (in the range between 0.035 cm and 0.146 cm) on optimum dimensions is within the inherent errors of the GA searching process. Contour plots of the detector efficiency are given in Fig. 6. The cost function is more sensitive to r than h1 and h2 (Fig. 6-a and Fig. 6-b) and more sensitive to h1 than h2 (Fig. 6-c). This implies that a small deviation from optimum value for r can cause a considerable decrease in detector efficiency while the similar deviation from optimum value for h2 may have a negligible effect on detector efficiency. The optimized beaker dimensions at various Al cap thickness (0.03e0.15 cm) and various sizes of the Ge hole are given in Table 3 and Table 4, respectively. These results shows that, similar to the DL effect (Table 2), the optimum dimensions of the beaker did not show any meaningful correlation neither with the Al cap thickness nor with the size of the Ge-crystal hole. The standard deviations of the arithmetic mean of the optimum values are in the same order of the standard deviations for the individual optimized values. These results together with the results given in Table 2 suggest that an optimally designed marinelli beaker for a specific detector can be equally used for detectors with different DL thickness, different Al cap thickness and different Ge-crystal hole size without any significant loss in efficiency.
Fig. 4. Flowchart of the binary GA.
Fig. 5. Convergence of 3 parameters of the beaker as fitness function versus chromosome generation in searching progresses.
Table 2 The variation of optimum dimensions of the marinelli beaker in various DL thickness. Other detector properties are same as Table 1. The sample emits a gamma line at 364.3 keV. Dead-Layer thickness (cm)
r (cm)
0.035 0.05 0.07 0.09 0.116 0.13 0.146 mean
5.4021 5.3770 5.4395 5.3991 5.3719 5.3402 5.3376 5.3811
h1 (cm) ± ± ± ± ± ± ± ±
0.0247 0.0077 0.0337 0.0174 0.0137 0.0139 0.0150 0.0335
2.2753 2.2868 2.1718 2.1825 2.2655 2.3123 2.2416 2.2480
± ± ± ± ± ± ± ±
h2 (cm) 0.0429 0.0365 0.0879 0.0622 0.0680 0.0402 0.0629 0.0490
6.0508 6.2051 6.0297 6.2865 6.2925 6.4245 6.6118 6.2716
± ± ± ± ± ± ± ±
Efficiency 0.0860 0.0319 0.0472 0.0939 0.0722 0.0628 0.0844 0.1895
0.0481 0.0469 0.0453 0.0438 0.0419 0.0410 0.0399
± ± ± ± ± ± ±
6.6911e-06 1.9457e-05 1.9100e-05 4.6163e-06 1.5463e-05 1.7737e-05 3.8957e-06
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The effect of detector material on optimum beaker dimensions were also investigated. The results for some common semiconductor materials are listed in Table 5. It was found that there is a meaningful correlation between the detector material density and optimum beaker dimensions. The beaker radius r increases with decreasing the density of the detector material while both the h1 and h2 decrease. It should be noted that the dependency of the optimum dimensions to the detector material density is not very considerable (although it is not negligible). A decrease in detector material density from 6.36 g/cm3 to 2.33 g/cm3 results only in an increase of r from 5.2621 to 5.6561, and decrease of h1 and h2 from 2.4959 to 6.6162 to 1.9038 and 5.3090, respectively. This implies that an optimally designed marinelli beaker for a specific detector can also be used for other detectors (with different materials) without significant difference in detector efficiency. For example, if a 450 cm3 beaker with dimensions optimized for a Ge detector (the 3rd row in Table 5) is used with other detectors with different materials, the difference between the obtained non-optimal efficiency and the corresponding optimum efficiency will be less than 4% (Table 6). The optimized values of r, h1 and h2 for different beaker volumes are listed in Table 7. Obviously, the optimum beaker dimensions increases and the detector efficiency decreases by increasing the sample volume. Similar calculations were performed for some conventional Ge detectors with different crystals sizes (16 sizes) at various sample volumes (300, 500, 700, 1000 and 1500 cm3). The results are summarized in Fig. 7 (r) and Fig. 8 (h1). It should be noted that all other detector properties including Al cap thickness, DL thickness and Ge hole dimensions are the same as those mentioned in Table 1. We suggested a best-fitted formula to each set of results in Figs. 7 and 8. Eqs. (9) and (10), both linear functions of sample volume, stand for beaker radius r and cylinder height h1, respectively. No formula proposed for the ring height h2 because it can be calculated by using Eq. (1). These equations were obtained after trying more than a dozen of different types of mathematical functions. In Eqs. (9) and (10), D (cm), H (cm) and V (cm3) are the Al cap diameter, crystal height and beaker volume, respectively.
r ¼ gðD; HÞ þ dðD; HÞV
(9)
gðD; HÞ ¼ 0:69444 þ 0:44279D þ 0:14149H dðD; HÞ ¼ ð413 28:4731D 7:34345HÞ 105 h1 ¼ aðD; HÞ þ bðD; HÞV
(10)
aðD; HÞ ¼ 2:83312 0:09193D 0:17008H bðD; HÞ ¼ ð21:3 þ 1:05081D 3:54647HÞ 104 The relative differences between the optimized dimensions of r and h1 obtained by GA (data points in Figs. 7 and 8) with those predicted by the proposed equations of 9 and 10 are shown in Fig. 9 and Fig. 10 for r and h1, respectively. The relative difference for beaker radius r rarely exceeds 5% and those of cylinder height are always less than 25%.
Fig. 6. Contour plot of the detector efficiency (ε) as a function of a) r and h1, b) r and h2 and c) h1 and h2. The white region shows restricted area due to beaker volume limitation.
5. Conclusion A genetic algorithm (GA), developed in MATLAB and coupled with MCNPX computational code, was used as a searching tool to determine the optimized dimensions of the marinelli beaker. The optimum dimensions of a 450 cm3 marinelli beaker for a Canberra
GC1518 HPGe detector, were obtained as: r ¼ 5.37 cm, h1 ¼ 2.26 cm and h2 ¼ 6.29 cm, in good agreement with those reported by Huy et al. (2012) as r ¼ 5.38 cm, h1 ¼ 2.18 cm and h2 ¼ 6.44 cm. It was concluded that the sensitivity of the detector efficiency to
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Table 3 The variation of optimal dimensions of the marinelli beaker in various Aluminum cap thickness. Other detector properties are same as Table 1. The sample emits a gamma line at 364.3 keV. Al cap thickness (cm)
r (cm)
0.03 0.05 0.07 0.09 0.11 0.13 0.15 mean
5.3507 5.3170 5.3216 5.3052 5.3208 5.3446 5.3719 5.3331
h1 (cm) ± ± ± ± ± ± ± ±
0.0080 0.0033 0.0167 0.0106 0.0112 0.0122 0.0137 0.0233
2.4378 2.4999 2.4310 2.3782 2.3564 2.3624 2.2655 2.3902
h2 (cm)
± ± ± ± ± ± ± ±
0.0176 0.0448 0.0191 0.0244 0.0190 0.0133 0.0680 0.0748
6.0470 6.1562 6.2865 6.5439 6.4706 6.2713 6.2925 6.2954
Efficiency
± ± ± ± ± ± ± ±
0.0758 0.0815 0.0822 0.0967 0.0522 0.0645 0.0722 0.1704
0.0448 0.0443 0.0438 0.0433 0.0429 0.0424 0.0419
± ± ± ± ± ± ±
3.4343e-06 1.4824e-05 1.2935e-06 9.7256e-06 1.2952e-05 1.6499e-06 1.5463e-05
Table 4 The variation of optimal dimensions of the marinelli beaker in various Ge hole height. Other detector properties are same as Table 1. The sample emits a gamma line at 364.3 keV. Ge hole dimensions (cm)
r (cm)
h1 (cm)
h2 (cm)
Efficiency
r r r r r
¼ ¼ ¼ ¼ ¼
0.35, 0.35, 0.35, 0.35, 0.35,
h h h h h
¼ ¼ ¼ ¼ ¼
0.4 0.7 1.0 1.3 1.7
5.3665 5.3561 5.3336 5.2645 5.3719
± ± ± ± ±
0.0157 1.8279e-15 0.0155 0.0260 0.0137
2.3796 2.3754 2.3818 2.4014 2.2655
± ± ± ± ±
0.0464 0.0138 0.0213 0.0514 0.0680
6.0666 6.1529 6.3104 6.8307 6.2925
± ± ± ± ±
0.0456 0.0325 0.0730 0.0932 0.0722
0.0420 0.0421 0.0420 0.0419 0.0419
± ± ± ± ±
1.5378e-05 5.5684e-06 1.1448e-06 1.5692e-05 1.5463e-05
r r r r r
¼ ¼ ¼ ¼ ¼
0.15, 0.20, 0.25, 0.30, 0.35,
h h h h h
¼ ¼ ¼ ¼ ¼
1.7 1.7 1.7 1.7 1.7
5.3208 5.3558 5.3253 5.3041 5.3719
± ± ± ± ±
0.0112 0.0009 0.0155 0.0009 0.0137
2.3564 2.3949 2.4446 2.4334 2.2655
± ± ± ± ±
0.0190 0.0030 0.0209 0.0240 0.0680
6.4706 6.1095 6.2252 6.4190 6.2925
± ± ± ± ±
0.0522 0.0001 0.0715 0.0600 0.0722
0.0421 0.0421 0.0421 0.0420 0.0419
± ± ± ± ±
1.0856e-05 1.3905e-05 1.2376e-05 5.0514e-06 1.5463e-05
r r r r r
¼ ¼ ¼ ¼ ¼
0, h ¼ 0 (no hole) 0.15, h ¼ 0.3 0.20, h ¼ 0.7 0.25, h ¼ 1.0 0.30, h ¼ 1.3
5.3531 5.3022 5.4069 5.3298 5.3767
± ± ± ± ±
0.0099 0.0123 0.0303 0.0104 0.0291
2.2250 2.4150 2.1465 2.2478 2.3481
± ± ± ± ±
0.0452 0.0171 0.0601 0.0444 0.0863
6.5298 6.4799 6.3153 6.6585 6.0666
± ± ± ± ±
0.0302 0.0895 0.0905 0.0238 0.0456
0.0421 0.0421 0.0420 0.0420 0.0420
± ± ± ± ±
9.5828e-07 9.2179e-06 1.6867e-05 1.3168e-05 1.6949e-05
mean
5.3426 ± 0.0361
2.3387 ± 0.0875
6.3480 ± 0.2199
Table 5 The effect of the detector material on the detector efficiency and optimum dimension of the beaker (detector dimensions are based on those ones in Table 1 and the sample emits a gamma line at 364.3 keV). Detector material
r (cm)
HgI2 (r ¼ 6.36 g.cm-3) CdTe (r ¼ 5.85 g.cm-3) Ge (r ¼ 5.323 g.cm-3) GaAs (r ¼ 5.317 g.cm-3) Si (r ¼ 2.33 g.cm-3)
5.2621 5.3491 5.3719 5.4154 5.6561
h1 (cm) ± ± ± ± ±
0.0201 0.0216 0.0137 0.0224 0.0075
2.4959 2.2814 2.2655 2.1968 1.9038
± ± ± ± ±
h2 (cm) 0.0629 0.0392 0.0680 0.0566 0.0289
6.6162 6.4293 6.2925 6.1383 5.3090
± ± ± ± ±
efficiency 0.0488 0.0798 0.0722 0.0346 0.0253
0.0828 0.0710 0.0419 0.0417 0.0026
± ± ± ± ±
2.7251e-05 8.8016e-06 1.5463e-05 2.2377e-05 1.6529e-06
Table 6 Comparison between the optimal and non-optimal efficiencies. Non-optimal efficiencies are obtained by a beaker that its dimensions were optimized for a Ge detector (hence, its dimensions are not optimized for any of the 4 detectors in the table). Detector material
Non-optimal efficiency
Optimum efficiency
%difference
HgI2 (r ¼ 6.36 g cm3) CdTe (r ¼ 5.85 g cm3) GaAs (r ¼ 5.317 g cm3) Si (r ¼ 2.33 g cm3)
0.0827 0.0709 0.0417 0.0025
0.0828 0.0710 0.0417 0.0026
0.12 0.14 0 3.85
beaker dimensions of r, h1 and h2 follows the order r > h1 > h2. No meaningful correlation was observed between the optimum dimensions of the beaker and any of the dead-layer thickness, Al cap thickness and the Ge-crystal hole size. On the other hand, a weak correlation, does exist between the detector material density and optimum beaker dimensions. The beaker radius increases with decreasing the density of the detector material while the beaker height decreases.
Two mathematical formulas (obtained by best fitting to values determined by GA) was suggested to predict the optimized beaker dimensions at various beaker sizes (300, 500, 700, 1000 and 1500 cm3) for some conventional Ge detectors of different sizes. The relative differences between the optimized dimensions obtained by GA with those predicted by the proposed mathematical formulas rarely exceeds 5% for beaker radius r and are always less than 25% for cylinder height h1.
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Table 7 The detector efficiency and the optimal dimensions of the beaker for various beaker volumes. The detector properties are reported in Table 1. The sample emits gamma line at 364.3 keV. Marinelli volume (L)
r (cm)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.3 1.5
4.9720 5.2731 5.4154 5.7104 5.9569 5.9769 6.1297 6.1391 6.5280 6.7230 7.0749
h1 (cm) ± ± ± ± ± ± ± ± ± ± ±
0.0218 0.0123 0.0202 0.0110 0.0246 0.0083 0.0044 0.0114 0.0197 0.0101 0.0203
1.9036 2.0716 2.4512 2.6054 2.3713 3.1221 3.4601 3.7538 3.7419 3.7313 3.6339
Fig. 7. Variation of the optimum beaker radius versus the volume for 16 conventional Ge detectors with different crystals sizes. D: diameter of the aluminum cap and H: height of the Ge crystal (other properties like Al cap thickness, DL thickness, Ge hole size and the gap between the Al cap and Ge crystal are the same as Table 1).
Fig. 8. Variation of the Cylinder Height h1 versus the volume for 16 conventional Ge detectors with different crystals sizes. D: diameter of the aluminum cap and H: height of the Ge crystal (other properties like Al cap thickness, DL thickness, Ge hole size and the gap between the Al cap and Ge crystal are the same as Table 1).
± ± ± ± ± ± ± ± ± ± ±
h2 (cm) 0.0841 0.0401 0.0624 0.0437 0.0498 0.0469 0.0156 0.0145 0.0618 0.0435 0.0715
5.8809 6.1584 6.7965 6.5754 7.2993 7.4385 7.4194 8.3395 7.2976 8.5373 8.8047
± ± ± ± ± ± ± ± ± ± ±
efficiency 0.0712 0.0034 0.0195 0.0302 0.0695 0.0734 0 0.0525 0.0464 0.0110 0.0203
0.0483 0.0438 0.0404 0.0373 0.0349 0.0328 0.0311 0.0295 0.0282 0.0260 0.0242
± ± ± ± ± ± ± ± ± ± ±
1.1848e-05 1.6803e-05 2.1750e-05 2.9856e-05 2.4166e-05 4.0584e-006 2.8748e-006 1.6735e-05 2.1394e-05 2.6078e-06 1.1098e-05
Fig. 9. Relative difference of the radius. Numbers are for D ¼ 7.6 H ¼ 3.6 cm, D ¼ 8.3 H ¼ 3.6 cm, D ¼ 8.9 H ¼ 3.6 cm, D ¼ 9.5 H ¼ 3.6 cm, D ¼ 7.6 H ¼ 3.9 cm, D ¼ 8.3 H ¼ 3.9 cm, D ¼ 8.9 H ¼ 3.9 cm, D ¼ 9.5 H ¼ 3.9 cm, D ¼ 7.6 H ¼ 4.5 cm, D ¼ 8.3 H ¼ 4.5 cm, D ¼ 8.9 H ¼ 4.5 cm, D ¼ 9.5 H ¼ 4.5 cm, D ¼ 7.6 H ¼ 5.2 cm, D ¼ 8.3 H ¼ 5.2 cm, D ¼ 8.9 H ¼ 5.2 cm, D ¼ 9.5 H ¼ 5.2 cm and V ¼ 300, 500, 700, 1000, 1500 respectively for each dimension.
Fig. 10. Relative difference of the h1. Sequence of the numbers is as Fig. 9.
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References Ahmed, A.S., Capello, K., Chiang, A., Cardenas-Mendez, E., Kramer, G.H., 2009. Optimization of geometric parameters for Marinelli beaker to maximize the detection efficiency of an HPGe detector. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrom. Detect. Assoc. Equip. 610, 718e723. Azbouche, A., Belgaid, M., Mazrou, H., 2015. Monte Carlo calculations of the HPGe detector efficiency for radioactivity measurement of large volume environmental samples. J. Environ. Radioact. 146, 119e124. Azli, T., 2015. Performance revaluation of a N-type coaxial HPGe detector with front edges crystal using MCNPX. Appl. Radiat. Isot. 97, 106e112. Chham, E., García, F.P., El Bardouni, T., Ferro-García, M.A., Azahra, M., Benaalilou, K., Krikiz, M., Elyaakoubi, H., El Bakkali, J., Kaddour, M., 2015. Monte Carlo analysis of the influence of germanium dead layer thickness on the HPGe gamma detector experimental efficiency measured by use of extended sources. Appl. Radiat. Isot. 95, 30e35. Chung, C., Lee, C.-J., Shiee, S.-F., 1991. Optimum geometry of radioactive aqueous sample container for measurement with a germanium detector. Int. J. Radiat. Appl. Instrum. Part A. Appl. Radiat. Isot. 42, 541e545. Debertin, K., Jianping, R., 1989. Measurement of the activity of radioactive samples in Marinelli beakers. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrom. Detect. Assoc. Equip. 278, 541e549. Do, B.Q., Nguyen, L.P., 2007. Application of a genetic algorithm to the fuel reload optimization for a research reactor. Appl. Math. Comput. 187, 977e988. Elanique, A., Marzocchi, O., Leone, D., Hegenbart, L., Breustedt, B., Oufni, L., 2012. Dead layer thickness characterization of an HPGe detector by measurements and Monte Carlo simulations. Appl. Radiat. Isot. 70, 538e542. py, M.-C., 2013. Validation of efficiency transfer Ferreux, L., Pierre, S., Thanh, T.T., Le for Marinelli geometries. Appl. Radiat. Isot. 81, 67e70. Goldberg, D.E., 1989. Genetic Algorithms in Search. Optimization and Machine Learning ADDISON-WESLEY.
Haupt, R.L., Haupt, S.E., 2004. Practical Genetic Algorithms. JOHN WILEY & SONS, New Jersey. Huy, N., Binh, D., An, V., 2007. Study on the increase of inactive germanium layer in a high-purity germanium detector after a long time operation applying MCNP code. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrom. Detect. Assoc. Equip. 573, 384e388. Huy, N.Q., Binh, D.Q., An, V.X., 2012. A study for improving detection efficiency of an HPGe detector based gamma spectrometer using Monte Carlo simulation and genetic algorithms. Appl. Radiat. Isot. 70, 2695e2702. Knoll, G.F., 2010. Radiation Detection and Measurement. John Wiley & Sons. Melquiades, F., Appoloni, C., 2001. Self-absorption correction for gamma spectrometry of powdered milk samples using Marinelli beaker. Appl. Radiat. Isot. 55, 697e700. Pelowitz, D.B., 2008. MCNPX USER’S MANUAL Version 2.6.0. Los Alamos National Laboratory Report. Pham, D.T., Karaboga, D., 2000. Intelligent Optimisation Techniques. Springer. Salgado, C.M., Conti, C.C., Becker, P.H., 2006. Determination of HPGe detector response using MCNP5 for 20e150keV X-rays. Appl. Radiat. Isot. 64, 700e705. Saraiva, A., Oliveira, C., Reis, M., Portugal, L., Paiva, I., Cruz, C., 2016. Study of the response of an ORTEC GMX45 HPGe detector with a multi-radionuclide volume source using Monte Carlo simulations. Appl. Radiat. Isot. 113, 47e52. Shweikani, R., Hasan, M., Tlas, M., Doubal, A., 2014. Determination of the optimal cylindrical geometry heights for gamma-ray spectrometric analysis. Radiat. Meas. 70, 34e38. Sima, O., 1990. Monte Carlo simulation versus semiempirical calculation of auto absorption factors for semiconductor detector calibration in complex geometries. Prog. Nucl. Energy 24, 327e336. The MathWorks, I., 2015. MATLAB and Statistics Toolbox Release R2015a (Massachusetts, United States). Vidmar, T., Vodenik, B., Ne cemer, M., 2010. Efficiency transfer between extended sources. Appl. Radiat. Isot. 68, 2352e2354.
Contents lists available at ScienceDirect
Journal of Environmental Radioactivity journal homepage: www.elsevier.com/locate/jenvrad
Optimization of the marinelli beaker dimensions using genetic algorithm Seyed Mehrdad Zamzamian*, Seyed Abolfazl Hosseini, Mohammad Samadfam Department of Energy Engineering, Sharif University of Technology, Tehran Zip code: 8639-11365, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 5 August 2016 Received in revised form 1 March 2017 Accepted 10 March 2017
A computational code, based on the genetic algorithm and MCNPX version 2.6 code was developed and used to investigate the effects of some important parameters of HPGe detector (such as Al cap thickness, dead-layer thickness and Ge hole size) on optimum dimensions of marinelli beaker. In addition, the effects of detector material on optimal beaker dimensions were also investigated. Finally, the optimized beaker dimensions at various beaker volumes (300, 500, 700, 1000 and 1500 cm3) were determined for some conventional Ge detectors with different crystal sizes (16 sizes). These sets of data then were used to drive mathematical formulas (obtained by best fitting to data sets). The results showed that, there is no meaningful correlation between the optimum dimensions of the beaker and each of the dead-layer thickness, Al cap thickness and the Ge-crystal hole size. On the other hand, the optimum beaker radius increases with decreasing the density of the detector material while the beaker height decreases. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Genetic algorithm Marinelli beaker HPGe detector MCNPX2.6
1. Introduction Gamma-ray spectrometry is a common technique widely used for identification and quantification of radionuclides in samples of different matrices and geometries. The use of High Purity Germanium (HPGe) detectors with high energy resolution allows identification and activity measurement of natural and artificial radionuclides in biological, geological and environmental samples, with the highest possible sensitivity (Azbouche et al., 2015). To analysis the environmental samples with low radioactive concentrations, a sample with high mass is required to obtain accurate and precise results. In order to increase the net counts due to the samples, in addition to a good shielding method, the various measurement parameters should be optimally balanced. The use of marinelli beakers in environmental samples permits one to obtain greater net count by positioning the sample volume as close as possible to active volume of the detector. In a constant sample volume, the detector efficiency varies at different geometrical dimensions of the beaker due to self-attenuation of photon in sample and out-scattering from the detection region. Therefore, dimensions of the marinelli beaker should be optimum to increase
* Corresponding author. E-mail address: [email protected] (S.M. Zamzamian). http://dx.doi.org/10.1016/j.jenvrad.2017.03.020 0265-931X/© 2017 Elsevier Ltd. All rights reserved.
the efficiency of the detector (Ahmed et al., 2009; Shweikani et al., 2014). Several researchers have tried to develop convenient methods for determining the optimum dimensions of the marinelli beaker. Chung et al. (1991) have measured the optimum geometry of marinelli-like samples (0.1e4 L) using a 20% relative efficiency HPGe detector. They reported that the optimum geometry of larger samples (5e60 L) held in marinelli beaker can be extrapolated from their work. Sima (1990) used a Monte Carlo algorithm and auto absorption (self-absorption) factors to determine the semiconductor detector efficiency for marinelli geometry. Ahmed et al. (2009) have studied optimization of marinelli beaker for increasing efficiency of an Ortec GMX S Gamma-X HPGe detector using MCNP code. They recommended that the optimum dimensions for 200e500 mL sample volumes should be used for better measurement. Melquiades and Appoloni (2001) have measured self-absorption factors vs. density for five marinelli beakers containing powdered milk samples with 40K and 208Tl. They have verified the relation y ¼ 1.96608x-74813 for 40K line of 1460.8 keV and y ¼ 1.4484x-0.83258 for 208Tl line of 2614.47 keV, where x and y were density (g/cm3) and self-absorption factor, respectively. Debertin and Jianping (1989) calculated selfabsorption in marinelli beakers mathematically. Azbouche et al. (2015) developed a computational procedure for gamma ray spectroscopy of large volume sample based on achieving
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Abbreviations DL GA HPGe r h1 h2 rmax hmax
Dead-Layer Genetic Algorithm High Purity Germanium Beaker Radius Cylinder Height (height of the cylinder part of the beaker) Ring Height (height of the ring part of the beaker surrounding the detector) Maximum Limit of r Maximum Limit of h2
agreement between 152E source packed in marinelli beaker and MCNP5 simulation. They concluded that their method can be used in determination of the specific activities of radionuclides in soil. In the framework of environmental measurements, an application based on efficiency trfpansfer (ET) method have been described by Vidmar et al. (2010) as a means of calculating the full-energy-peak efficiencies (FEPE) and the ET method have been validated for marinelli geometries by Ferreux et al. (2013). After developing genetic algorithm (GA) by john Holland and his colleagues, the algorithm was considered as one of the best optimization tools in various scientific applications (Do and Nguyen, 2007; Goldberg, 1989; Haupt and Haupt, 2004; Pham and Karaboga, 2000). A combination of GA and MCNP simulation were used to obtain the maximum detector efficiency by Huy et al. (2012). They reported the optimum dimensions of a 450 cm3 marinelli beaker for the GC1518 HPGe detector. They suggested that the effects of gamma energy, chemical composition of the sample and the sample density on optimum dimensions are negligible. In this study, we also used a combination of GA and MCNP simulation to investigate the effects of some important detector parameters, such as thickness of Al cap, dead-layer (DL) and crystal size, on optimum dimensions of marinelli beaker.
Fig. 1. The HPGe detector and marinelli beaker.
Table 1 Specifications of the Canberra GC1518 HPGe detector (after Huy et al., 2007). Parameter
Dimension (cm)
Aluminum Cap thickness Aluminum Cap diameter Germanium dead-layer thickness End cap to crystal distance Germanium crystal height Germanium crystal diameter Core hole height Core hole diameter
0.15 7.62 0.116 0.5 3.2 5.4 1.7 0.7
2. MCNPX simulation of the detector and marinelli beaker
at a distance of 15 cm from detector surface. The peak absolute efficiencies of the detector with specifications given in Table 1, was extracted by running MCNPX for each energy, ranging from 60 to 1332 keV. Only the bare detector as illustrated in Fig. 1 without any shielding was simulated. Since the normalized count under full energy peak area for mono-energetic sources was considered as output of the MCNPX, definition of the surrounding materials (shield) were not included in the simulation results. The simulation results in the present study together with the results reported by
The MCNPX version 2.6 (Pelowitz, 2008) was used for the simulation of the HPGe detector and marinelli beaker. This computational code is routinely used by many researchers to obtain the efficiency of the HPGe detector (Azli, 2015; Chham et al., 2015; Elanique et al., 2012; Salgado et al., 2006; Saraiva et al., 2016). The HPGe detector model was based on Canberra GC1518 HPGe detector, as schematically shown in Fig. 1. The materials and dimensions of the detector (Table 1) were described by Huy et al. (2007). It was assumed that the dimensions of the sample are equivalent to those of the marinelli beaker. The marinelli beaker consist of the three dimensional parameters: radius, cylinder height and ring height (r, h1 and h2, respectively, in Fig. 1). The optimization of these dimensional parameters will be resulted in maximum efficiency of the detector (Ahmed et al., 2009). The pulse height tally (F8) was used to obtain the absolute efficiency. The output was binned by E card in order to give the peak absolute efficiency (Knoll, 2010). For validation of the simulation results, the energy-dependent detector efficiency curve (obtained from the MCNP simulation) was compared with the simulation results reported by Huy et al. (2012) for the same model of the HPGe detector. The energydependent detector efficiency curve was obtained by defining the mono-energetic point sources located at the central axis of detector
Fig. 2. Energy-dependent detector efficiencies; comparison between the simulated results obtained in this study with those reported by Huy et al. (2012).
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Huy et al. (2012) are shown in Fig. 2. The differences between the two simulated results are less than 8%. 3. Genetic algorithm As shown in Fig. 1, the calculations should be performed to optimize three variables of beaker radius r, cylinder height h1 and ring height h2. Since the sample volume (¼ beaker volume) is given by:
V ¼ p r 2 r02 h2 þ pr 2 h1
(1)
Therefore, at a constant volume, if two variables are known, the third one can be obtained from Eq. (1). So, only two independent variables should be considered for optimization. In this work, the beaker radius r and the ring height h2 were chosen. A genetic algorithm was developed using MATLAB software (The MathWorks, 2015) for optimization of the beaker dimensions at constant sample volumes. It was assumed that the inner radius of the beaker (r0) is constant and equal to 4.06 cm. Obviously, the beaker radius cannot take values smaller than r0. A two-objective binary genetic algorithm was used to solve the problem. A 20 variables chromosome string (Nvar ¼ 20) was defined as the input variable:
where P1, P2, …P20 are the random binaries. The variable P1, P2, … P10 belong to r (gene 1) and P11, P12, …P20 to h2 (gene 2). The detector full peak absolute efficiency (ε) was considered as the Cost function defined by the following equation:
Cost ¼ εðr; h1 ; h2 Þ
where rmax and h2max are the maximum limits for beaker radius and ring height, respectively. The values of rmax and h2max should be properly (as small as possible) determined in order to speed up the GA convergence. The optimum value of r when h2 ¼ 0 and the optimum value of h2 when h1 ¼ 0 were chosen as the maximum limits for beaker radius r and ring height h2, respectively (Fig. 3). Needless to say that these optimum values of r and h2 are obtained at special condition where h1 or h2 is assumed to be zero. It should be noted that the ultimate optimum values of r and h2 (when the h2 or h1 has non-zero values) will be definitely smaller than the optimum values determined from Fig. 3. One can simply imagine when h1 or h2 just starts to take a non-zero value, some of the sample at far most position (with regard to detector's active volume) will be relocated to a much closer positions, rendering a higher detector efficiency. The values of rmax and h2max, determined from Fig. 3 as ~6.5 cm and ~7.5 cm, respectively. An initial population of chromosome should be generated and decoded by the following formulas (Haupt and Haupt, 2004):
rquant ¼
XNgene1
(2)
with
r0 ð ¼ 4:06Þ r rmax
(3)
r0 ð ¼ 4:06Þ r rmax
(4)
m¼1
gene1ðmÞ2m þ 2ðMþ1Þ
r ¼ rquant ðrmax rmin Þ þ rmax h2quant ¼
chromosome ¼ ½P1 ; P2 ; …; P20
83
XNvar m¼Ngene1
gene2ðmÞ2m þ 2ðMþ1Þ
r ¼ h2quant ðhmax hmin Þ þ hmax
(5) (6) (7) (8)
where m is length of the string for each gene. Gene1(m) and Gene2(m) are binary versions of the chromosomes. The MCNPX computational code for each decoded string was run and the outputs was considered as cost values. A simple natural selection was used to sort cost values in a decreasing order and generate mating pool. Two of the most valuable costs were considered as elite. An arbitrary number of top costs survived and others was discarded. From top to bottom of the pool mating (with the exception of two elite), two successive mate was selected to create offsprings using a randomly crossover point and then a mutation probability of 0.01 (¼m) was used. Finally, a new
Fig. 3. (a): The detector efficiency versus the ring height h2 when h1 ¼ 0. (b): The detector efficiency versus the beaker radius r when h2 ¼ 0.
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generation, consisting of non-mutated parents (elite), parents, mutated parents, offsprings and mutated offsprings were produced. The standard deviation of r, h1 and h2 were considered as convergence limitation and successive generation continued while all of the deviations were more than 0.1. A path through the components of the mentioned procedures is shown as a flowchart in Fig. 4.
4. Results and discussion The optimum dimensions of a 450 cm3 marinelli beaker, with detector specifications given in Table 1, were obtained as: r ¼ 5.37 cm, h1 ¼ 2.26 cm and h2 ¼ 6.29 cm. These optimum values are in good agreement with those reported by Huy et al. (2012) as r ¼ 5.38 cm, h1 ¼ 2.18 cm and h2 ¼ 6.44 cm. The convergence of the searching process was achieved after ~200 generations for all the 3 variables of r, h1 and h2. The r and h2 had the highest and lowest convergence speed, respectively, as illustrated in Fig. 5. In order to assess the effect of detector DL on optimum dimensions of beaker, the optimum dimensions were calculated for 7 different DL thicknesses. The results are given in Table 2 together with their standard deviations. These results shows that there is almost no correlation between the optimum dimensions and the DL thickness of the detector. The arithmetic means of the optimum r, h1 and h2 (also given in Table 2) have standard deviations of 0.0335, 0.0490 and 0.1895, respectively. These deviations are almost in the same order of the standard deviations for the individual optimized dimensions at different DL thicknesses, implying that the effect of DL change (in the range between 0.035 cm and 0.146 cm) on optimum dimensions is within the inherent errors of the GA searching process. Contour plots of the detector efficiency are given in Fig. 6. The cost function is more sensitive to r than h1 and h2 (Fig. 6-a and Fig. 6-b) and more sensitive to h1 than h2 (Fig. 6-c). This implies that a small deviation from optimum value for r can cause a considerable decrease in detector efficiency while the similar deviation from optimum value for h2 may have a negligible effect on detector efficiency. The optimized beaker dimensions at various Al cap thickness (0.03e0.15 cm) and various sizes of the Ge hole are given in Table 3 and Table 4, respectively. These results shows that, similar to the DL effect (Table 2), the optimum dimensions of the beaker did not show any meaningful correlation neither with the Al cap thickness nor with the size of the Ge-crystal hole. The standard deviations of the arithmetic mean of the optimum values are in the same order of the standard deviations for the individual optimized values. These results together with the results given in Table 2 suggest that an optimally designed marinelli beaker for a specific detector can be equally used for detectors with different DL thickness, different Al cap thickness and different Ge-crystal hole size without any significant loss in efficiency.
Fig. 4. Flowchart of the binary GA.
Fig. 5. Convergence of 3 parameters of the beaker as fitness function versus chromosome generation in searching progresses.
Table 2 The variation of optimum dimensions of the marinelli beaker in various DL thickness. Other detector properties are same as Table 1. The sample emits a gamma line at 364.3 keV. Dead-Layer thickness (cm)
r (cm)
0.035 0.05 0.07 0.09 0.116 0.13 0.146 mean
5.4021 5.3770 5.4395 5.3991 5.3719 5.3402 5.3376 5.3811
h1 (cm) ± ± ± ± ± ± ± ±
0.0247 0.0077 0.0337 0.0174 0.0137 0.0139 0.0150 0.0335
2.2753 2.2868 2.1718 2.1825 2.2655 2.3123 2.2416 2.2480
± ± ± ± ± ± ± ±
h2 (cm) 0.0429 0.0365 0.0879 0.0622 0.0680 0.0402 0.0629 0.0490
6.0508 6.2051 6.0297 6.2865 6.2925 6.4245 6.6118 6.2716
± ± ± ± ± ± ± ±
Efficiency 0.0860 0.0319 0.0472 0.0939 0.0722 0.0628 0.0844 0.1895
0.0481 0.0469 0.0453 0.0438 0.0419 0.0410 0.0399
± ± ± ± ± ± ±
6.6911e-06 1.9457e-05 1.9100e-05 4.6163e-06 1.5463e-05 1.7737e-05 3.8957e-06
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85
The effect of detector material on optimum beaker dimensions were also investigated. The results for some common semiconductor materials are listed in Table 5. It was found that there is a meaningful correlation between the detector material density and optimum beaker dimensions. The beaker radius r increases with decreasing the density of the detector material while both the h1 and h2 decrease. It should be noted that the dependency of the optimum dimensions to the detector material density is not very considerable (although it is not negligible). A decrease in detector material density from 6.36 g/cm3 to 2.33 g/cm3 results only in an increase of r from 5.2621 to 5.6561, and decrease of h1 and h2 from 2.4959 to 6.6162 to 1.9038 and 5.3090, respectively. This implies that an optimally designed marinelli beaker for a specific detector can also be used for other detectors (with different materials) without significant difference in detector efficiency. For example, if a 450 cm3 beaker with dimensions optimized for a Ge detector (the 3rd row in Table 5) is used with other detectors with different materials, the difference between the obtained non-optimal efficiency and the corresponding optimum efficiency will be less than 4% (Table 6). The optimized values of r, h1 and h2 for different beaker volumes are listed in Table 7. Obviously, the optimum beaker dimensions increases and the detector efficiency decreases by increasing the sample volume. Similar calculations were performed for some conventional Ge detectors with different crystals sizes (16 sizes) at various sample volumes (300, 500, 700, 1000 and 1500 cm3). The results are summarized in Fig. 7 (r) and Fig. 8 (h1). It should be noted that all other detector properties including Al cap thickness, DL thickness and Ge hole dimensions are the same as those mentioned in Table 1. We suggested a best-fitted formula to each set of results in Figs. 7 and 8. Eqs. (9) and (10), both linear functions of sample volume, stand for beaker radius r and cylinder height h1, respectively. No formula proposed for the ring height h2 because it can be calculated by using Eq. (1). These equations were obtained after trying more than a dozen of different types of mathematical functions. In Eqs. (9) and (10), D (cm), H (cm) and V (cm3) are the Al cap diameter, crystal height and beaker volume, respectively.
r ¼ gðD; HÞ þ dðD; HÞV
(9)
gðD; HÞ ¼ 0:69444 þ 0:44279D þ 0:14149H dðD; HÞ ¼ ð413 28:4731D 7:34345HÞ 105 h1 ¼ aðD; HÞ þ bðD; HÞV
(10)
aðD; HÞ ¼ 2:83312 0:09193D 0:17008H bðD; HÞ ¼ ð21:3 þ 1:05081D 3:54647HÞ 104 The relative differences between the optimized dimensions of r and h1 obtained by GA (data points in Figs. 7 and 8) with those predicted by the proposed equations of 9 and 10 are shown in Fig. 9 and Fig. 10 for r and h1, respectively. The relative difference for beaker radius r rarely exceeds 5% and those of cylinder height are always less than 25%.
Fig. 6. Contour plot of the detector efficiency (ε) as a function of a) r and h1, b) r and h2 and c) h1 and h2. The white region shows restricted area due to beaker volume limitation.
5. Conclusion A genetic algorithm (GA), developed in MATLAB and coupled with MCNPX computational code, was used as a searching tool to determine the optimized dimensions of the marinelli beaker. The optimum dimensions of a 450 cm3 marinelli beaker for a Canberra
GC1518 HPGe detector, were obtained as: r ¼ 5.37 cm, h1 ¼ 2.26 cm and h2 ¼ 6.29 cm, in good agreement with those reported by Huy et al. (2012) as r ¼ 5.38 cm, h1 ¼ 2.18 cm and h2 ¼ 6.44 cm. It was concluded that the sensitivity of the detector efficiency to
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Table 3 The variation of optimal dimensions of the marinelli beaker in various Aluminum cap thickness. Other detector properties are same as Table 1. The sample emits a gamma line at 364.3 keV. Al cap thickness (cm)
r (cm)
0.03 0.05 0.07 0.09 0.11 0.13 0.15 mean
5.3507 5.3170 5.3216 5.3052 5.3208 5.3446 5.3719 5.3331
h1 (cm) ± ± ± ± ± ± ± ±
0.0080 0.0033 0.0167 0.0106 0.0112 0.0122 0.0137 0.0233
2.4378 2.4999 2.4310 2.3782 2.3564 2.3624 2.2655 2.3902
h2 (cm)
± ± ± ± ± ± ± ±
0.0176 0.0448 0.0191 0.0244 0.0190 0.0133 0.0680 0.0748
6.0470 6.1562 6.2865 6.5439 6.4706 6.2713 6.2925 6.2954
Efficiency
± ± ± ± ± ± ± ±
0.0758 0.0815 0.0822 0.0967 0.0522 0.0645 0.0722 0.1704
0.0448 0.0443 0.0438 0.0433 0.0429 0.0424 0.0419
± ± ± ± ± ± ±
3.4343e-06 1.4824e-05 1.2935e-06 9.7256e-06 1.2952e-05 1.6499e-06 1.5463e-05
Table 4 The variation of optimal dimensions of the marinelli beaker in various Ge hole height. Other detector properties are same as Table 1. The sample emits a gamma line at 364.3 keV. Ge hole dimensions (cm)
r (cm)
h1 (cm)
h2 (cm)
Efficiency
r r r r r
¼ ¼ ¼ ¼ ¼
0.35, 0.35, 0.35, 0.35, 0.35,
h h h h h
¼ ¼ ¼ ¼ ¼
0.4 0.7 1.0 1.3 1.7
5.3665 5.3561 5.3336 5.2645 5.3719
± ± ± ± ±
0.0157 1.8279e-15 0.0155 0.0260 0.0137
2.3796 2.3754 2.3818 2.4014 2.2655
± ± ± ± ±
0.0464 0.0138 0.0213 0.0514 0.0680
6.0666 6.1529 6.3104 6.8307 6.2925
± ± ± ± ±
0.0456 0.0325 0.0730 0.0932 0.0722
0.0420 0.0421 0.0420 0.0419 0.0419
± ± ± ± ±
1.5378e-05 5.5684e-06 1.1448e-06 1.5692e-05 1.5463e-05
r r r r r
¼ ¼ ¼ ¼ ¼
0.15, 0.20, 0.25, 0.30, 0.35,
h h h h h
¼ ¼ ¼ ¼ ¼
1.7 1.7 1.7 1.7 1.7
5.3208 5.3558 5.3253 5.3041 5.3719
± ± ± ± ±
0.0112 0.0009 0.0155 0.0009 0.0137
2.3564 2.3949 2.4446 2.4334 2.2655
± ± ± ± ±
0.0190 0.0030 0.0209 0.0240 0.0680
6.4706 6.1095 6.2252 6.4190 6.2925
± ± ± ± ±
0.0522 0.0001 0.0715 0.0600 0.0722
0.0421 0.0421 0.0421 0.0420 0.0419
± ± ± ± ±
1.0856e-05 1.3905e-05 1.2376e-05 5.0514e-06 1.5463e-05
r r r r r
¼ ¼ ¼ ¼ ¼
0, h ¼ 0 (no hole) 0.15, h ¼ 0.3 0.20, h ¼ 0.7 0.25, h ¼ 1.0 0.30, h ¼ 1.3
5.3531 5.3022 5.4069 5.3298 5.3767
± ± ± ± ±
0.0099 0.0123 0.0303 0.0104 0.0291
2.2250 2.4150 2.1465 2.2478 2.3481
± ± ± ± ±
0.0452 0.0171 0.0601 0.0444 0.0863
6.5298 6.4799 6.3153 6.6585 6.0666
± ± ± ± ±
0.0302 0.0895 0.0905 0.0238 0.0456
0.0421 0.0421 0.0420 0.0420 0.0420
± ± ± ± ±
9.5828e-07 9.2179e-06 1.6867e-05 1.3168e-05 1.6949e-05
mean
5.3426 ± 0.0361
2.3387 ± 0.0875
6.3480 ± 0.2199
Table 5 The effect of the detector material on the detector efficiency and optimum dimension of the beaker (detector dimensions are based on those ones in Table 1 and the sample emits a gamma line at 364.3 keV). Detector material
r (cm)
HgI2 (r ¼ 6.36 g.cm-3) CdTe (r ¼ 5.85 g.cm-3) Ge (r ¼ 5.323 g.cm-3) GaAs (r ¼ 5.317 g.cm-3) Si (r ¼ 2.33 g.cm-3)
5.2621 5.3491 5.3719 5.4154 5.6561
h1 (cm) ± ± ± ± ±
0.0201 0.0216 0.0137 0.0224 0.0075
2.4959 2.2814 2.2655 2.1968 1.9038
± ± ± ± ±
h2 (cm) 0.0629 0.0392 0.0680 0.0566 0.0289
6.6162 6.4293 6.2925 6.1383 5.3090
± ± ± ± ±
efficiency 0.0488 0.0798 0.0722 0.0346 0.0253
0.0828 0.0710 0.0419 0.0417 0.0026
± ± ± ± ±
2.7251e-05 8.8016e-06 1.5463e-05 2.2377e-05 1.6529e-06
Table 6 Comparison between the optimal and non-optimal efficiencies. Non-optimal efficiencies are obtained by a beaker that its dimensions were optimized for a Ge detector (hence, its dimensions are not optimized for any of the 4 detectors in the table). Detector material
Non-optimal efficiency
Optimum efficiency
%difference
HgI2 (r ¼ 6.36 g cm3) CdTe (r ¼ 5.85 g cm3) GaAs (r ¼ 5.317 g cm3) Si (r ¼ 2.33 g cm3)
0.0827 0.0709 0.0417 0.0025
0.0828 0.0710 0.0417 0.0026
0.12 0.14 0 3.85
beaker dimensions of r, h1 and h2 follows the order r > h1 > h2. No meaningful correlation was observed between the optimum dimensions of the beaker and any of the dead-layer thickness, Al cap thickness and the Ge-crystal hole size. On the other hand, a weak correlation, does exist between the detector material density and optimum beaker dimensions. The beaker radius increases with decreasing the density of the detector material while the beaker height decreases.
Two mathematical formulas (obtained by best fitting to values determined by GA) was suggested to predict the optimized beaker dimensions at various beaker sizes (300, 500, 700, 1000 and 1500 cm3) for some conventional Ge detectors of different sizes. The relative differences between the optimized dimensions obtained by GA with those predicted by the proposed mathematical formulas rarely exceeds 5% for beaker radius r and are always less than 25% for cylinder height h1.
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Table 7 The detector efficiency and the optimal dimensions of the beaker for various beaker volumes. The detector properties are reported in Table 1. The sample emits gamma line at 364.3 keV. Marinelli volume (L)
r (cm)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.3 1.5
4.9720 5.2731 5.4154 5.7104 5.9569 5.9769 6.1297 6.1391 6.5280 6.7230 7.0749
h1 (cm) ± ± ± ± ± ± ± ± ± ± ±
0.0218 0.0123 0.0202 0.0110 0.0246 0.0083 0.0044 0.0114 0.0197 0.0101 0.0203
1.9036 2.0716 2.4512 2.6054 2.3713 3.1221 3.4601 3.7538 3.7419 3.7313 3.6339
Fig. 7. Variation of the optimum beaker radius versus the volume for 16 conventional Ge detectors with different crystals sizes. D: diameter of the aluminum cap and H: height of the Ge crystal (other properties like Al cap thickness, DL thickness, Ge hole size and the gap between the Al cap and Ge crystal are the same as Table 1).
Fig. 8. Variation of the Cylinder Height h1 versus the volume for 16 conventional Ge detectors with different crystals sizes. D: diameter of the aluminum cap and H: height of the Ge crystal (other properties like Al cap thickness, DL thickness, Ge hole size and the gap between the Al cap and Ge crystal are the same as Table 1).
± ± ± ± ± ± ± ± ± ± ±
h2 (cm) 0.0841 0.0401 0.0624 0.0437 0.0498 0.0469 0.0156 0.0145 0.0618 0.0435 0.0715
5.8809 6.1584 6.7965 6.5754 7.2993 7.4385 7.4194 8.3395 7.2976 8.5373 8.8047
± ± ± ± ± ± ± ± ± ± ±
efficiency 0.0712 0.0034 0.0195 0.0302 0.0695 0.0734 0 0.0525 0.0464 0.0110 0.0203
0.0483 0.0438 0.0404 0.0373 0.0349 0.0328 0.0311 0.0295 0.0282 0.0260 0.0242
± ± ± ± ± ± ± ± ± ± ±
1.1848e-05 1.6803e-05 2.1750e-05 2.9856e-05 2.4166e-05 4.0584e-006 2.8748e-006 1.6735e-05 2.1394e-05 2.6078e-06 1.1098e-05
Fig. 9. Relative difference of the radius. Numbers are for D ¼ 7.6 H ¼ 3.6 cm, D ¼ 8.3 H ¼ 3.6 cm, D ¼ 8.9 H ¼ 3.6 cm, D ¼ 9.5 H ¼ 3.6 cm, D ¼ 7.6 H ¼ 3.9 cm, D ¼ 8.3 H ¼ 3.9 cm, D ¼ 8.9 H ¼ 3.9 cm, D ¼ 9.5 H ¼ 3.9 cm, D ¼ 7.6 H ¼ 4.5 cm, D ¼ 8.3 H ¼ 4.5 cm, D ¼ 8.9 H ¼ 4.5 cm, D ¼ 9.5 H ¼ 4.5 cm, D ¼ 7.6 H ¼ 5.2 cm, D ¼ 8.3 H ¼ 5.2 cm, D ¼ 8.9 H ¼ 5.2 cm, D ¼ 9.5 H ¼ 5.2 cm and V ¼ 300, 500, 700, 1000, 1500 respectively for each dimension.
Fig. 10. Relative difference of the h1. Sequence of the numbers is as Fig. 9.
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