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Development of an absolute method for efficiency calibration of a coaxial HPGe detector for large volume sources
Nuclear Instruments and Methods in Physics Research A ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Development of an absolute method for efficiency calibration of a coaxial HPGe detector for large volume sources P.C. Ortiz Departamento de Física, Facultad de Ciencias, Universidad de Chile, Chile
art ic l e i nf o
a b s t r a c t
Article history: Received 7 December 2014 Received in revised form 11 March 2015 Accepted 18 March 2015
In this work an absolute methodology for the determination of the full energy peak efficiency of a gamma spectroscopy system for voluminous sources is presented. The method was tested for a highresolution coaxial HPGe detector and cylindrical homogeneous volume source. The volume source is represented by a set of point sources filling its volume. We found that the absolute efficiency of a volume source can be determined as the average over its volume of the absolute efficiency of each point source. Experimentally, we measure the intrinsic efficiency as a function upon source-detector position. Then, considering the solid angle and the attenuations of the gamma rays emitted to the detector by each point source, considered as embedded in the source matrix, the absolute efficiency for each point source inside of the volume was determined. The factor associate with the solid angle and the selfattenuation of photons in the sample was deduced from first principles without any mathematical approximation. The method was tested by determining the specific activity of 137Cs in cylindrical homogeneous sources, using IAEA reference materials with specific activities between 14.2 Bq/kg and 9640 Bq/kg at the moment of the experimentation. The results obtained shown a good agreement with the expected values. The relative difference was less than 7% in most of the cases. The main advantage of this method is that it does not require of the use of expensive and hard to produce standard materials. In addition it does not require of matrix effect corrections, which are the main cause of error in this type of measurements, and it is easy to implement in any nuclear physics laboratory. & 2015 Elsevier B.V. All rights reserved.
Keywords: Efficiency calibration Coaxial HPGe detector Volume source Specific activity
1. Introduction The gamma spectroscopy is one of the techniques used mostly in the determination and quantification of radionuclides, naturals and artificials, in different activities, for instance, in environmental samples analysis [1–3] or in activation experiment [4,5]. In the case of analyzing environmental samples or thick activated samples, the quantification of the radiation emitted is a difficult task unaccompanied some time by large uncertainty, due to the matrix effects (self-attenuation) present in source of the voluminous type. Such effect is the main source of uncertainty and its reduction is still an open subject. The standard methodology for radioisotope concentration quantification is a relative one based on reference material whose matrix (chemical composition, density, shape and size) must be identical to the matrix of the material to be analyzed [1,2]. These can be bought or prepared in the laboratory, in this case well
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trained technicians are employed [6]. A large number of methods, experimental and theoretical, have been developed to correct the differences between the matrix of the reference material and the sample [3,7–10]. Monte Carlo method and mathematical approaches have been utilized for both intrinsic and absolute efficiency determination [3,11–15]. The main advantage of absolute method is that for spectrometer calibration a standard material is not required being these most of time expensive and hard to obtain. Additionally, these methods avoid the accumulation of unnecessary radioactive materials in the laboratory. The aim of this work is to develop an absolute experimental method for coaxial HPGe detector calibration efficiency when large volume sample has to be measured. The characterization of the detector is carried out by means of the measurement of the intrinsic full energy peak efficiency as a function upon sourcedetector position and no calculation is needed about the detector. Then, this method can be considered as experimental. The only calculations are directed to the determination of the geometric efficiency.
http://dx.doi.org/10.1016/j.nima.2015.03.053 0168-9002/& 2015 Elsevier B.V. All rights reserved.
Please cite this article as: P.C. Ortiz, Nuclear Instruments & Methods in Physics Research A (2015), http://dx.doi.org/10.1016/j. nima.2015.03.053i
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2. Theory 2.1. Detection of full energy peak (FEP) efficiency In general there are two types of efficiency parameters in gamma spectrometry [16]: the absolute FEP efficiency defined as
ϵabs ¼
fNo of photons recorded in the FEPg ; fNo of photons emitted by sourceg
ð1Þ
which depends upon the detector characteristics and the material between source to detector, and the intrinsic FEP efficiency, given by
ϵint ¼
fNo of photons recorded in the FEPg ; fNo of photons incident on the detectorg
ð2Þ
which depends only upon detector physical and chemical characteristics. Considering the source emission probability per gamma decay. Both efficiencies are related by
ϵabs ¼ ϵint f ;
ð3Þ
where f is called geometric efficiency and it represents the fraction of photons of the emitted by the source that reach the detector. f depends upon the source-detector position and the attenuation of photons. 2.2. Spatial FEP efficiency distribution Let us consider a point source in the vicinity of a radiation detector, as is shown in Fig. 1. It has been shown that the FEP efficiency depends upon the relative source-detector position [17–19] so call it spatial FEP efficiency distribution. On the other hand, the number of detected photons Ndet and the geometric efficiency f also are sourcedetector position dependent. With those considerations, the relation (2) expressed in terms of the position of a point source is given by N ðr 0 ; z0 Þ ϵint ðr 0 ; z0 Þ ¼ det 0 0 ; ntot f ðr ; z Þ
ð4Þ
where ntot is the total number of photons emitted by the point source during the data acquisition time. The azimuthal angle dependence has been omitted due to the axial symmetry of the system. If the visible area of the detector from the source is A, then the geometric efficiency is given by Z cos ðαÞ f¼ ∏e μi datt;i dA; ð5Þ 2 A 4π d i
where Ω is the solid angle subtended by the source on the detector sensitive zone, furthermore f Ω is the geometric efficiency when only the solid angle is considered. The intrinsic efficiency determined under this consideration was defined by Moens as virtual intrinsic efficiency [11] therefore the attenuations due to the materials that cover the detector sensitive zone can be omitted because they are always present. To determine the solid angle, we must consider two cases: (i) the solid angle subtended by the front face of the detector sensitive zone and (ii) the solid angle subtended by the mantle of the detector sensitive zone. For the front face of the detector sensitive zone we can see, from Fig. 1, that cos ðαÞ ¼ z0 =d and that d corresponds to d ¼ r 2d 2r d r 0 cos ðθÞ þ r 02 þz02 ; 2
P z’
DETECTOR END−CUP
α
d y
r d1
r’ θ
x
0
dA
ð7Þ
where r d represents the radial position of the detector element area dA. Replacing Eq. (7) in Eq. (6), we obtain
ΩFace ðr 0 ; z0 Þ ¼
Z
Rd
Z 2π
0
0
z0 r d dθ dr d ðr 2d 2r d r 0
cos ðθÞ þr 02 þ z02 Þ3=2
:
ð8Þ
Whereas that for the mantle of the detector sensitive zone, we will ! use the vectorial notation and the scalar product. If r 1 is a perpendicular vector to the area element dA, for instance, ! r 1 ¼ ðRd cos ðθÞ; Rd sin ðθÞ; 0Þ; where Rd represent the detector radius, θ the angular position of ! the area element dA and r 2 a vector that goes from dA to P, i.e., ! r 2 ¼ ðr 0 Rd cos ðθÞ; Rd sin ðθÞ; z0 zdÞ; where zd is the axial position of the area element dA, the solid angle subtended by the mantle of the detector sensitive zone is
ΩMantle ðr0 ; z0 Þ ¼
z
A
where d is the distance from source to an element area on the detector active zone, μi is the attenuation coefficient of each material between the source and the detector sensitive zone, and finally the datt;i is the distance traveled by the photons in those materials. α is the angle between a perpendicular vector to the area element dA and a vector that go from dA to P. When the medium is air, the transmission of photons is considered to be 1 for close position to detector and also assuming that there are not attenuations between detector end-cup to detector sensitive zone, we can write the geometric efficiency for the point source in air as Z cos ðαÞ Ω fΩ ¼ dA ¼ ; ð6Þ 2 4π A 4π d
Z
0
Z θðr0 Þ
hd
θ
ðr0 Þ
! ! r1 r2 ! ! 3 Rd dθ dzd ; ‖r 1 ‖‖r 2 ‖
ð9Þ
where hd is the height of the detector and θðr 0 Þ is the maximum angular position of the subtended angle on the mantle of the detector sensitive zone for a point source positioned in r 0 as shown in Fig. 2 and corresponds to R θðr 0 Þ ¼ arccos d0 : r Finally, considering the relations (8) and (9), we have that the geometric efficiency when air is considered between the detector and the source, is given by
r’cos( θ) DETECTOR SENSITIVE ZONE
Fig. 1. Radioactive point source, P, positioned front to a detector in an arbitrary position. It is shown a possible path for photons emitted by the source striking the detector in a differential element of area, dA.
8 ΩFace ðr 0 ; z0 Þ > > < 4π f Ω ðr 0 ; z0 Þ ¼ Ω ðr 0 ; z0 Þ þ ΩMantle ðr 0 ; z0 Þ > Face > : 4π
r≦Rd r 4 Rd
:
ð10Þ
Please cite this article as: P.C. Ortiz, Nuclear Instruments & Methods in Physics Research A (2015), http://dx.doi.org/10.1016/j. nima.2015.03.053i
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Replacing Eq. (10) in Eq. (4), the intrinsic spatial efficiency distribution is determined by
ϵint ðr 0 ; z0 Þ ¼
Ndet ðr 0 ; z0 Þ : ntot f Ω ðr 0 ; z0 Þ
ð11Þ
If we ignore the geometric efficiency, the absolute spatial efficiency distribution, from Eq. (3), is given by
ϵabs ðr0 ; z0 Þ ¼
Ndet ðr 0 ; z0 Þ : ntot
ð12Þ
2.3. Absolute FEP efficiency for a volume source using the intrinsic spatial FEP efficiency distribution (Intrinsic spatial efficiency method) In general, for a point source embedded in a volume material of linear attenuation coefficient μ, as is shown in Fig. 3, the number of detected photons and collected by the spectroscopy system in
3
the FEP, from Eq. (4), is given by Ndet ðr 0 ; z0 Þ ¼ ntot ϵint ðr 0 ; z0 Þf Ω;μ ðr 0 ; z0 Þ;
ð13Þ
where f Ω;μ corresponds to the geometric efficiency that considers the attenuations in the volume of the photons directed to the detector. Now, if we want to determine the total number of γ photons emitted by a volume source with volume V', N TOT , we must add all contributions from each point source part of the volume source. NTOT is associated with the emission of a point source part of ! ! the volume source at the position r , ntot ð r Þ, by the next expression, ! ! N TOT ntot ð r Þ ¼ gð r Þ 0 ; V ! where gð r Þ is a distribution function of the radioactive material in the volume source. With those considerations, the number of γ photons detected and located under FEP, N, is given by Z N TOT !0 0 gð r Þϵint ðr 0 ; z0 Þf Ω;μ ðr 0 ; z0 Þ dV : ð14Þ N¼ 0 V V0 Then, from Eq. (14) and considering definition (1), the absolute efficiency for a volume source, with volume V 0 and attenuation coefficient μ, can be written as R !0 0 0 0 0 0 0 gð r Þϵint ðr ; z Þf Ω;μ ðr ; z Þ dV ϵμ;V 0 ¼ V : ð15Þ 0 V
r’ θ’
This corresponds to the average over the volume of the source of the absolute spatial efficiency distribution for a point source embedded in a volume material of attenuation coefficient μ and ! volume V 0 (see relation (3)) weighted for gð r Þ distribution. Fig. 2. Subtended angle by a point source on the mantle of a cylinder for radial positions greater than the radius of the cylinder. We can see that while we move away radially, the angle θ0 increases.
z
Now, we consider the case of a cylindrical volume source, as is shown in Fig. 4. In this case the geometric efficiency, from Eq. (5), corresponds to Z Ωμ cos ðαÞ μdatt f Ω;μ ¼ e dA ¼ ; ð16Þ 2 4π A 4π d
POINT SOURCE
VOLUME SOURCE P z’
2.4. Absolute FEP efficiency for a cylindrical volume source radioactively homogeneous
(r’,z’)
where we have defined Ωμ as the solid angle weighted for the probability of absorption in the matrix of the source, we will call effective solid angle. In Fig. 4 we can see that the distance datt is
Ω
z
DETECTOR END−CUP
Rm
z2
P r’
r
DETECTOR END−CUP
z’
d att
β
d3
DETECTOR SENSITIVE ZONE
d4
z1
r
θ
0 d1 DETECTOR SENSITIVE ZONE
Fig. 3. Radioactive point source, P, embedded in a volume material of attenuation coefficient μ, subtending a solid angle Ω on the detector sensitive zone of a detector of radiation.
x
dA Rd
Fig. 4. Radioactive point source, P, embedded in a cylindrical volume material positioned front to a detector in an arbitrary position. It is shown a possible path for photons emitted by the source striking the detector in a differential element of area, dA, and the distance traveled under the attenuation effect, datt .
Please cite this article as: P.C. Ortiz, Nuclear Instruments & Methods in Physics Research A (2015), http://dx.doi.org/10.1016/j. nima.2015.03.053i
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given by datt ¼ ðz z1Þ= cos ðβÞ; 0
ð17Þ
where β, for a cylindrical source coaxial to the detector, corresponds to the angle between a perpendicular vector to the frontal face of the detector with a vector that go from dA to P. To determine the contribution of the face of the detector sensitive zone to the effective solid angle, we note that cos ðβ Þ ¼ cos ðαÞ ¼ z0 =d, therefore datt corresponds to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð18Þ datt ¼ ðz0 z1Þ 1 þ 02 ðr 2d 2r d r 0 cos ðθÞ þ r 02 Þ: z Then, using (18) and (8) in (16), we have that the effective solid angle for the frontal face of the sensitive detector zone is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Z Rd Z 2π z0 e μðz0 z1 Þ 1 þ ðr 2 2r d r cos ðθÞ þ r 02 Þr d dθ dr d d 02 z Ωμ;Face ¼ : 2 2r r 0 cos ðθ Þ þ r 02 þ z02 Þ3=2 ðr 0 0 d d
Both, the absolute efficiency, ϵ, and the specific activity, As , are related by the following equation: As ¼
N
ϵΔtmIγ
f 1f 2f 3…
ð23Þ
where N represents the Full Energy Peak (FEP) area, Δt is the counting time, m is the mass of the sample and I γ is the γ emission probability. The terms f i are correction factors due to electronic (true coincidence, pile-up and dead time), dry mass, waiting or cooling time, variations of the source activity during the data acquisition, self-attenuation, shape and size. The difference between the traditional method and the method presented in this work, denominated intrinsic spatial efficiency methodology, is that corrections due to self-attenuation, shape and size of the volume source (matrix effects correction) are not necessary.
ð19Þ To determine the effective solid angle for the mantle of the ! detector sensitive zone, we can define r 3 as a perpendicular vector to the face front of the detector, for instance, ! r 3 ¼ ð0; 0; z0 Þ: Then, the effective solid angle subtended for the mantle of the detector sensitive zone is ! !! Z 0 Z θðrÞ ! ! μ ðz0 z1 Þ‖! r 3 ‖ ‖ r 2 ‖= r 3 r 2 r1 r2 e Ωμ;Mantle ðr0 ; z0 Þ ¼ Rd dθ dzd : ! ! hd θ ðrÞ ‖r 1 ‖ ‖r 2 ‖3 ð20Þ Therefore, using Eqs. (19) and (20) in Eq. (16), we have that the geometric efficiency for the source point embedded in a cylindrical volume material of attenuation coefficient μ is 8 Ωμ;Face ðr0 ; z0 Þ > > ; r≦Rd < 4π ð21Þ f Ω;μ ðr 0 ; z0 Þ ¼ 0 0 0 0 Ω ðr ; z Þ þ Ωμ;Mantle ðr ; z Þ > > : μ;Face ; r 4 Rd 4π On the other hand, if the distribution of radioactive material in ! the source is homogeneous, this implies that gð r Þ ¼ 1, then, the absolute FEP efficiency for a cylindrical volume source radioactively homogeneous, is given by R 0 0 0 0 0 0 ϵint ðr ; z Þf Ω;μ ðr ; z Þ dV ϵμ;V ¼ V ; ð22Þ 0 V
3. Material and methods 3.1. Spectroscopy system In this work we have used a spectroscopy system, which consist of a coaxial HPGe detector (Ortec GEM-10195), shielded with a cylindrical Lead wall, 10 cm thickness, and two foils of Cadmium and Copper of 1 mm thickness each one. The structure and dimensions of the detector, given on the manufacturer technical note, are shown in Fig. 5. The data acquisition system is composed of an amplifier Ortec 572 and a MCA Nucleus controlled by the PCA-II software.
3.2.
137
Cs radioactive source
A 137Cs planar source with an activity at the moment of the experimentation of 31,857 Bq distributed in a disc of 3.5 mm diameter was used.
SYMMETRY AXIS 35,0 mm 23,7 mm
where f Ω;μ corresponds to Eq. (21).
4,97 mm
2.5. Determination of the specific activity of an environmental cylindrical volume source: standard methodology vs intrinsic spatial efficiency methodology Let us consider the case of a radioactive volume source on a radiation detector, such as is shown in Fig. 4. Among all photons emitted by the source only a fraction will be detected due to the geometry (solid angle), the matrix effects (selfattenuation) and intrinsic characteristics of the detector (intrinsic efficiency). Due to the complications to determine these factors, independently, is convenient to use the absolute efficiency, because this considers all effects simultaneously. In this case, the method to determine experimentally the specific activity corresponds to the relative method. Once the absolute efficiency of the detector is known, the specific activity of the sample material can be determined.
38,4 mm DETECTOR SENSITIVE ZONE
Fig. 5. Scheme of the detector HPGe GEM-10195. The dimensions of the detector sensitive zone and end-cup are shown.
Please cite this article as: P.C. Ortiz, Nuclear Instruments & Methods in Physics Research A (2015), http://dx.doi.org/10.1016/j. nima.2015.03.053i
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5
z
3.3. Reference materials In order to test the proposed method in this work, four matrix reference materials supplied by IAEA were used and their specifications are reported in Table 1. 3.4. Volume source
SOURCE
Cylindrical containers were made with PVC pipes of 2 mm thickness and height according to the amount available of the reference material. These were filled with the reference material and sealed in both faces with Mylar foil of 6.4 μm. The geometric characteristics, radius and height, and the mass of the cylindrical volume sources are informed in Table 2.
z’
END−CUP
3.5. Determination of the spatial FEP efficiency distribution The determination of the spatial efficiency distribution was performed by mapping the FEP area on the r z plane (Fig. 6) using the planar source described in Section 3.2. Due to the rather small radius of the planar source, in first approximation it has been considered as a point source. Angular variations were not necessary due to the axial symmetry of the system. The response of the spectroscopy system to different source positions was measured. Each spectrum was acquired during 180 s and repeated five times. To avoid dead time corrections the time of measurement was adjusted to the live time of system. In mapping, four radial positions, between 0 and 39 mm, and 14 axial positions, between 8.47 and 63.47 mm, were considered. The source position was taken at the planar source center. The FEP area in each position is presented in Fig. 7. We can see that the FEP area decreases as we move away in both radial and axial directions. Also, for each radial position, the variation of the FEP area is large for close position to the detector becoming small as we move away along the z-axis. Due to the source activity pile up correction was applied. The continuous function, N det ðr 0 ; z0 Þ, was obtained by the interpolation of the data. The total number of gamma photons emitted Table 1 Matrix and specific activity of the reference materials, given by the factory. Material matrix
Material title
Specific activity
Reference materials Liquid Soil Moss–soil Grass
Sample 3 Sample 1 IAEA-447 IAEA-372
16.7 7 0.1 52.6 71.1 425 7 10 11,320 7360
[20] [20] [21] [22]
137
Cs Bq/kg
r’
r
DETECTOR SENSITIVE ZONE
Fig. 6. Experimental setting for the determination of the spatial efficiency distribution. The lattice in the grid on the r z plane represents the different source positions to carry out the mapping of the FEP area.
Fig. 7. Graphic of the FEP area along the z-axis for each radial position.
by the source during the acquisition was determined considering the activity of the source informed in Section 3.2 while the solid angle subtended by the source on the detector f Ω ðr 0 ; z0 Þ was computed using Eq. (6). Then using Eqs. (11) and (12) the intrinsic and absolute spatial FEP efficiency distribution were obtained. 3.6. Determination of the attenuation coefficient
Table 2 Geometric characteristics and mass of the cylindrical sources prepared for this study. References material
Number source
Radius (mm)
Height (mm)
Mass (g)
The determination of the attenuation coefficient was carried out by the known transmission method [23] using a NaI(Tl) detector. The set-up is shown in Fig. 8. 3.7. Determination of the specific activity
Cylindrical volume source Sample 3 1 2
29.5 70.1 29.5 70.1
18.7 7 0.1 25.9 7 0.1
48.53 7 0.01 69.63 70.01
Sample 1
1 2
29.5 70.1 29.5 70.1
17.5 7 0.1 18.7 7 0.1
38.94 70.01 43.87 70.01
IAEA-447
1 2
22.6 70.1 29.4 70.1
25.17 0.1 15.0 70.1
29.447 0.01 34.80 70.01
IAEA-372
1 2 3 4
29.4 70.1 29.4 70.1 29.5 70.1 29.5 70.1
24.370.1 24.370.1 20.4 7 0.1 15.17 0.1
33.637 0.01 27.79 7 0.01 24.507 0.01 18.97 70.01
The cylindrical volume sources were positioned coaxially and closest to the detector over a Marinelli beaker used as a sampler holder, as is shown in Fig. 9. In Table 3, the time length of the measurements and the FEP area obtained are shown. Due to the low efficiency of the detector used in these measurements, around 10% relative to a 3 in: 3 in: NaI(Tl), we can see that for the source with a specific activity of order of 17 Bq/ kg, Sample 3, the obtained uncertainties of the FEP area are greater than 12%, even after of two days of measurement. For the sources
Please cite this article as: P.C. Ortiz, Nuclear Instruments & Methods in Physics Research A (2015), http://dx.doi.org/10.1016/j. nima.2015.03.053i
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with specific activities greater than 52 Bq/kg the FEP area is more precise. With relative uncertainties less than 8%, 2% and 1% for the materials sample 1, IAEA-447 and IAEA-372, respectively, were obtained in one day (86,400 s) of measurement. Taking into account the dimensions of the cylindrical source, the results obtained for the intrinsic spatial efficiency and the linear attenuation coefficient, the absolute efficiency of the spectroscopy system for 661.65 keV was determined by means of Eq. (22). Then, using Eq. (23), the specific activity of the analyzed material was determined. The uncertainties of the absolute efficiency and the specific activity were determined by the error propagation equation for uncorrelated variables [24].
Radioactive Source
Collimator
Table 3 Table of data of the FEP area obtained in the measurement of each cylindrical volume source. The FEP area corresponds to the emissions of 137Cs and a energy of 661.65 keV. The net and rate FEP area are shown. u represents the relative uncertainty given by the data acquisition software. Reference material
Number source
FEP area
Acquisition time (s)
Net counts
Rate counts/s
u (%)
Sample 3
1 2
304 836
0.00422 0.00581
29 12
72,000 144,000
Sample 1
1 2
2275 1036
0.01053 0.01199
7 8
216,000 86,400
IAEA-447
1 2
13,131 6446
0.060791 0.08139
1 2
216,000 79,200
IAEA-372
1 2 3 4
68,891 113,215 64,664 74,579
1.4547 1.3104 1.2802 1.0405
0.4 0.3 0.4 0.4
43,200 86,400 50,400 71,674
4. Result and discussion 4.1. Spatial efficiency distribution of the HPGe detector h
Volume Sample
Collimator
Detector
Fig. 8. Experimental set-up to determine the attenuation coefficient by transmission method.
AXIAL SYMMETRY AXIS
Rm
Following the procedure described in Section 3.5, the intrinsic and absolute spatial efficiency distribution of a HPGe detector was determined. Figs. 10 and 11 show the intrinsic and absolute spatial efficiency distribution, respectively. We can see that the intrinsic efficiency increases while we move away axially from the detector as well as when the radial position increases. The first case is due to the fact that while the source moves away from the detector, the photons impact more perpendicular on the face of the detector, then, the path length average increases and therefore the probability of interaction. For the second case, the same argument should apply but it is not clear. On the other hand the fast radial growth around 23 mm, can be associated to the radius of the detector, 23.7 mm, an intrinsic characteristic of the detector. The absolute spatial efficiency distribution, Fig. 11, gives us information on the optimum dimensions, in statistical terms, of voluminous sources to be used in this spectroscopy system. For instance, we can see that the major contribution to the FEP area is obtained for axial positions less than 10 mm and radial position less than 20 mm. For positions greater than 23 and 35 mm, axial and radial respectively, the contribution is reduced to one half. 4.2. Specific activity of the sources
SAMPLE HOLDER (MARINELLI)
hm
6,97 mm
DETECTOR END−CUP
DETECTOR SENSITIVE ZONE
Fig. 9. Experimental setting for the determination of the specific activity of a cylindrical volume source of height hm y radius Rm .
By means of the procedure described in Section 3.6 the linear attenuation coefficient was determined for all reference materials. Comparing the result obtained for the water matrix the relative difference was less than 1% compared with the value given by XCOM [25]. Then, by means of the procedure described in Section 3.7 the absolute efficiency for the voluminous sources and their specific activity was determined. The results of the specific activity are presented in Table 4. In general these results are satisfactory, specially for the materials with specific activities greater than 52 Bq/kg. The best results, with relative differences less than 3%, were obtained with the reference material Sample 1, which has an activity of 52 Bq/kg. While that the worst, with relative differences between 2% and 12%, were for the sources with the reference material IAEA Sample 3, which has a nominal activity of 16.7 Bq/kg but only 14.2 Bq/kg at moment of the measurements. However, the uncertainties of the measurements give account of the expectation value. Respect to
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7
Fig. 10. Graphics of the intrinsic spatial efficiency distribution of the HPGe detector, GEM-10195, for 661.7 keV.
Fig. 11. Graphics of the absolute spatial efficiency distribution of the HPGe detector, GEM-10195, for 661.7 keV.
Table 4 Results of the absolute efficiency and the specific activity obtained for the different volume sources used in this work. In the first column, the number in parenthesis corresponds to the certified activity given by the supplier, in Bq/kg. Reference material
Number source
Absolute efficiency
Specific activity obtained (Bq/kg)
Relative difference (%)
Table comparative of the results Sample 3 1 7037 8 (16.727 0.08) 2 6147 6
17.17 5.0 18.8 7 2.2
2.1 12
Sample 1 (52.6 7 1.1)
1 2
7367 9 7187 8
51.9 73.6 53.8 74.1
1.4 2.2
IAEA-447 (425 7 10)
1 2
741 7 10 7607 10
399 7 7 403 7 9
6.2 5.3
6717 7 6787 7 7187 8 785 7 9
10,7247 118 10,5577 113 10,868 7126 10,593 7 142
5.2 6.7 3.9 6.4
(10 5 )
IAEA-372 1 (11,320 7 360) 2 3 4
the other references materials analyzed, all show relative differences less than 7%. In the present work, to illustrate how the method works, we have only used a single energy to determine the efficiency, however, for a complete efficiency calibration either a set of single gamma source or a multi-gamma source can be used. In the former case, our method can be applied directly for each source separately, while for a multi-gamma source the true coincidence correction should be considered.
5. Conclusions We have presented a method to determine the efficiency of a coaxial HPGe detector for volume sources and we have successfully
applied it in cylindrical volume sources radioactively homogeneous, Eq. (22), by means of the determination of the specific activity of 137Cs. The range of the specific activity measured was between 14.2 Bq/kg and 9640 Bq/kg at the moment of the experimentation. For this reason, the method developed in this work can be applied adequately to environmental radioactivity analysis, due to that these types of sources are considered as radioactively homogeneous. To emphasize this fact the reference materials used in this work have been soil, water and grass. Finally, a general expression to determine the absolute efficiency for a volume source radioactively non-homogeneous was obtained, Eq. (15), however experiments with this type of sources were not conducted in the present work. In the future, more studies should be dedicated to this type of source.
Acknowledgments The author would like to acknowledge the assistance given by Alejandro J. Martinez in the development of the numerical tools needed for the application of the method and to Dr. José Roberto Morales for the assistance and advices for writing this paper. Experimental support from Center For Experimental Physics (CEFEX) of the University of Chile is recognized. Also, I would like to thank the two anonymous referees for providing constructive and helpful comments to improve the quality of the manuscript. References [1] IAEA, Measurement of radionuclides in food and the environment: a guidebook, Technical Reports No. 295, IAEA, Vienna, 1989. [2] Michael F. L'Annunziata, Handbook of Radioactivity Analysis, Third Edition, Elsevier, Amsterdam, 2012. [3] Gordon L. Gilmore, Practical and Gamma Ray Spectrometry, Second Edition, John Wiley & Sons, Chichester, 2008.
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[16] G.F. Knoll, Radiation Detection and Measurement, Second Edition, John Wiley & Sons, New York, 1988. [17] P.M. Grant, Nuclear Instruments and Methods in Physics Research 127 (1971) 371. [18] R.G. Helmer, International Journal of Applied Radiation and Isotopes 34 (8) (1983) 1105. [19] M. Korun, A. Likar, T. Vidmar, Nuclear Instruments and Methods in Physics Research Section A 390 (1997) 203. [20] A. Shakhahiro, U. Sansone, Radionuclide activity measurements in environmental samples of water, soil and grass: CCRI(II)-S4 comparison report, Technical Supplement, vol. 46, 2009, p. 06004 CRI(II)-S4. 〈http://www.iop. org/EJ/abstract/0026-1394/46/1A/06004/〉. [21] A. Shakhahiro, S. Tarjan, A. Cecattelli, G. Kis-Benedek, M. Betti, Applied Radiation and Isotopes 70 (2012) 1632. [22] A. Shakhahiro, U. Sansone, D. Arnold, P. Dirak, Jerome J. La Rosa, S.M. Jerome, M. Makarewics, J. Mentcheva, K. Sato, S. Tarjan, Applied Radiation and Isotopes 66 (2008) 1718. [23] D. Demir, A. Un, M. Ozgul, Y. Sahin, Applied Radiation and Isotopes 66 (2008) 1834. [24] R. Philip, Bevington, Data Reduction and Error Analysis for the Physical Sciences, Third Edition, McGraw-Hill, New York, 2003. [25] M.J. Berger, J.H. Hubble, X-COM: Photon Cross Section on a Personal Computers, 1987.
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Development of an absolute method for efficiency calibration of a coaxial HPGe detector for large volume sources P.C. Ortiz Departamento de Física, Facultad de Ciencias, Universidad de Chile, Chile
art ic l e i nf o
a b s t r a c t
Article history: Received 7 December 2014 Received in revised form 11 March 2015 Accepted 18 March 2015
In this work an absolute methodology for the determination of the full energy peak efficiency of a gamma spectroscopy system for voluminous sources is presented. The method was tested for a highresolution coaxial HPGe detector and cylindrical homogeneous volume source. The volume source is represented by a set of point sources filling its volume. We found that the absolute efficiency of a volume source can be determined as the average over its volume of the absolute efficiency of each point source. Experimentally, we measure the intrinsic efficiency as a function upon source-detector position. Then, considering the solid angle and the attenuations of the gamma rays emitted to the detector by each point source, considered as embedded in the source matrix, the absolute efficiency for each point source inside of the volume was determined. The factor associate with the solid angle and the selfattenuation of photons in the sample was deduced from first principles without any mathematical approximation. The method was tested by determining the specific activity of 137Cs in cylindrical homogeneous sources, using IAEA reference materials with specific activities between 14.2 Bq/kg and 9640 Bq/kg at the moment of the experimentation. The results obtained shown a good agreement with the expected values. The relative difference was less than 7% in most of the cases. The main advantage of this method is that it does not require of the use of expensive and hard to produce standard materials. In addition it does not require of matrix effect corrections, which are the main cause of error in this type of measurements, and it is easy to implement in any nuclear physics laboratory. & 2015 Elsevier B.V. All rights reserved.
Keywords: Efficiency calibration Coaxial HPGe detector Volume source Specific activity
1. Introduction The gamma spectroscopy is one of the techniques used mostly in the determination and quantification of radionuclides, naturals and artificials, in different activities, for instance, in environmental samples analysis [1–3] or in activation experiment [4,5]. In the case of analyzing environmental samples or thick activated samples, the quantification of the radiation emitted is a difficult task unaccompanied some time by large uncertainty, due to the matrix effects (self-attenuation) present in source of the voluminous type. Such effect is the main source of uncertainty and its reduction is still an open subject. The standard methodology for radioisotope concentration quantification is a relative one based on reference material whose matrix (chemical composition, density, shape and size) must be identical to the matrix of the material to be analyzed [1,2]. These can be bought or prepared in the laboratory, in this case well
E-mail address: [email protected]
trained technicians are employed [6]. A large number of methods, experimental and theoretical, have been developed to correct the differences between the matrix of the reference material and the sample [3,7–10]. Monte Carlo method and mathematical approaches have been utilized for both intrinsic and absolute efficiency determination [3,11–15]. The main advantage of absolute method is that for spectrometer calibration a standard material is not required being these most of time expensive and hard to obtain. Additionally, these methods avoid the accumulation of unnecessary radioactive materials in the laboratory. The aim of this work is to develop an absolute experimental method for coaxial HPGe detector calibration efficiency when large volume sample has to be measured. The characterization of the detector is carried out by means of the measurement of the intrinsic full energy peak efficiency as a function upon sourcedetector position and no calculation is needed about the detector. Then, this method can be considered as experimental. The only calculations are directed to the determination of the geometric efficiency.
http://dx.doi.org/10.1016/j.nima.2015.03.053 0168-9002/& 2015 Elsevier B.V. All rights reserved.
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2. Theory 2.1. Detection of full energy peak (FEP) efficiency In general there are two types of efficiency parameters in gamma spectrometry [16]: the absolute FEP efficiency defined as
ϵabs ¼
fNo of photons recorded in the FEPg ; fNo of photons emitted by sourceg
ð1Þ
which depends upon the detector characteristics and the material between source to detector, and the intrinsic FEP efficiency, given by
ϵint ¼
fNo of photons recorded in the FEPg ; fNo of photons incident on the detectorg
ð2Þ
which depends only upon detector physical and chemical characteristics. Considering the source emission probability per gamma decay. Both efficiencies are related by
ϵabs ¼ ϵint f ;
ð3Þ
where f is called geometric efficiency and it represents the fraction of photons of the emitted by the source that reach the detector. f depends upon the source-detector position and the attenuation of photons. 2.2. Spatial FEP efficiency distribution Let us consider a point source in the vicinity of a radiation detector, as is shown in Fig. 1. It has been shown that the FEP efficiency depends upon the relative source-detector position [17–19] so call it spatial FEP efficiency distribution. On the other hand, the number of detected photons Ndet and the geometric efficiency f also are sourcedetector position dependent. With those considerations, the relation (2) expressed in terms of the position of a point source is given by N ðr 0 ; z0 Þ ϵint ðr 0 ; z0 Þ ¼ det 0 0 ; ntot f ðr ; z Þ
ð4Þ
where ntot is the total number of photons emitted by the point source during the data acquisition time. The azimuthal angle dependence has been omitted due to the axial symmetry of the system. If the visible area of the detector from the source is A, then the geometric efficiency is given by Z cos ðαÞ f¼ ∏e μi datt;i dA; ð5Þ 2 A 4π d i
where Ω is the solid angle subtended by the source on the detector sensitive zone, furthermore f Ω is the geometric efficiency when only the solid angle is considered. The intrinsic efficiency determined under this consideration was defined by Moens as virtual intrinsic efficiency [11] therefore the attenuations due to the materials that cover the detector sensitive zone can be omitted because they are always present. To determine the solid angle, we must consider two cases: (i) the solid angle subtended by the front face of the detector sensitive zone and (ii) the solid angle subtended by the mantle of the detector sensitive zone. For the front face of the detector sensitive zone we can see, from Fig. 1, that cos ðαÞ ¼ z0 =d and that d corresponds to d ¼ r 2d 2r d r 0 cos ðθÞ þ r 02 þz02 ; 2
P z’
DETECTOR END−CUP
α
d y
r d1
r’ θ
x
0
dA
ð7Þ
where r d represents the radial position of the detector element area dA. Replacing Eq. (7) in Eq. (6), we obtain
ΩFace ðr 0 ; z0 Þ ¼
Z
Rd
Z 2π
0
0
z0 r d dθ dr d ðr 2d 2r d r 0
cos ðθÞ þr 02 þ z02 Þ3=2
:
ð8Þ
Whereas that for the mantle of the detector sensitive zone, we will ! use the vectorial notation and the scalar product. If r 1 is a perpendicular vector to the area element dA, for instance, ! r 1 ¼ ðRd cos ðθÞ; Rd sin ðθÞ; 0Þ; where Rd represent the detector radius, θ the angular position of ! the area element dA and r 2 a vector that goes from dA to P, i.e., ! r 2 ¼ ðr 0 Rd cos ðθÞ; Rd sin ðθÞ; z0 zdÞ; where zd is the axial position of the area element dA, the solid angle subtended by the mantle of the detector sensitive zone is
ΩMantle ðr0 ; z0 Þ ¼
z
A
where d is the distance from source to an element area on the detector active zone, μi is the attenuation coefficient of each material between the source and the detector sensitive zone, and finally the datt;i is the distance traveled by the photons in those materials. α is the angle between a perpendicular vector to the area element dA and a vector that go from dA to P. When the medium is air, the transmission of photons is considered to be 1 for close position to detector and also assuming that there are not attenuations between detector end-cup to detector sensitive zone, we can write the geometric efficiency for the point source in air as Z cos ðαÞ Ω fΩ ¼ dA ¼ ; ð6Þ 2 4π A 4π d
Z
0
Z θðr0 Þ
hd
θ
ðr0 Þ
! ! r1 r2 ! ! 3 Rd dθ dzd ; ‖r 1 ‖‖r 2 ‖
ð9Þ
where hd is the height of the detector and θðr 0 Þ is the maximum angular position of the subtended angle on the mantle of the detector sensitive zone for a point source positioned in r 0 as shown in Fig. 2 and corresponds to R θðr 0 Þ ¼ arccos d0 : r Finally, considering the relations (8) and (9), we have that the geometric efficiency when air is considered between the detector and the source, is given by
r’cos( θ) DETECTOR SENSITIVE ZONE
Fig. 1. Radioactive point source, P, positioned front to a detector in an arbitrary position. It is shown a possible path for photons emitted by the source striking the detector in a differential element of area, dA.
8 ΩFace ðr 0 ; z0 Þ > > < 4π f Ω ðr 0 ; z0 Þ ¼ Ω ðr 0 ; z0 Þ þ ΩMantle ðr 0 ; z0 Þ > Face > : 4π
r≦Rd r 4 Rd
:
ð10Þ
Please cite this article as: P.C. Ortiz, Nuclear Instruments & Methods in Physics Research A (2015), http://dx.doi.org/10.1016/j. nima.2015.03.053i
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Replacing Eq. (10) in Eq. (4), the intrinsic spatial efficiency distribution is determined by
ϵint ðr 0 ; z0 Þ ¼
Ndet ðr 0 ; z0 Þ : ntot f Ω ðr 0 ; z0 Þ
ð11Þ
If we ignore the geometric efficiency, the absolute spatial efficiency distribution, from Eq. (3), is given by
ϵabs ðr0 ; z0 Þ ¼
Ndet ðr 0 ; z0 Þ : ntot
ð12Þ
2.3. Absolute FEP efficiency for a volume source using the intrinsic spatial FEP efficiency distribution (Intrinsic spatial efficiency method) In general, for a point source embedded in a volume material of linear attenuation coefficient μ, as is shown in Fig. 3, the number of detected photons and collected by the spectroscopy system in
3
the FEP, from Eq. (4), is given by Ndet ðr 0 ; z0 Þ ¼ ntot ϵint ðr 0 ; z0 Þf Ω;μ ðr 0 ; z0 Þ;
ð13Þ
where f Ω;μ corresponds to the geometric efficiency that considers the attenuations in the volume of the photons directed to the detector. Now, if we want to determine the total number of γ photons emitted by a volume source with volume V', N TOT , we must add all contributions from each point source part of the volume source. NTOT is associated with the emission of a point source part of ! ! the volume source at the position r , ntot ð r Þ, by the next expression, ! ! N TOT ntot ð r Þ ¼ gð r Þ 0 ; V ! where gð r Þ is a distribution function of the radioactive material in the volume source. With those considerations, the number of γ photons detected and located under FEP, N, is given by Z N TOT !0 0 gð r Þϵint ðr 0 ; z0 Þf Ω;μ ðr 0 ; z0 Þ dV : ð14Þ N¼ 0 V V0 Then, from Eq. (14) and considering definition (1), the absolute efficiency for a volume source, with volume V 0 and attenuation coefficient μ, can be written as R !0 0 0 0 0 0 0 gð r Þϵint ðr ; z Þf Ω;μ ðr ; z Þ dV ϵμ;V 0 ¼ V : ð15Þ 0 V
r’ θ’
This corresponds to the average over the volume of the source of the absolute spatial efficiency distribution for a point source embedded in a volume material of attenuation coefficient μ and ! volume V 0 (see relation (3)) weighted for gð r Þ distribution. Fig. 2. Subtended angle by a point source on the mantle of a cylinder for radial positions greater than the radius of the cylinder. We can see that while we move away radially, the angle θ0 increases.
z
Now, we consider the case of a cylindrical volume source, as is shown in Fig. 4. In this case the geometric efficiency, from Eq. (5), corresponds to Z Ωμ cos ðαÞ μdatt f Ω;μ ¼ e dA ¼ ; ð16Þ 2 4π A 4π d
POINT SOURCE
VOLUME SOURCE P z’
2.4. Absolute FEP efficiency for a cylindrical volume source radioactively homogeneous
(r’,z’)
where we have defined Ωμ as the solid angle weighted for the probability of absorption in the matrix of the source, we will call effective solid angle. In Fig. 4 we can see that the distance datt is
Ω
z
DETECTOR END−CUP
Rm
z2
P r’
r
DETECTOR END−CUP
z’
d att
β
d3
DETECTOR SENSITIVE ZONE
d4
z1
r
θ
0 d1 DETECTOR SENSITIVE ZONE
Fig. 3. Radioactive point source, P, embedded in a volume material of attenuation coefficient μ, subtending a solid angle Ω on the detector sensitive zone of a detector of radiation.
x
dA Rd
Fig. 4. Radioactive point source, P, embedded in a cylindrical volume material positioned front to a detector in an arbitrary position. It is shown a possible path for photons emitted by the source striking the detector in a differential element of area, dA, and the distance traveled under the attenuation effect, datt .
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given by datt ¼ ðz z1Þ= cos ðβÞ; 0
ð17Þ
where β, for a cylindrical source coaxial to the detector, corresponds to the angle between a perpendicular vector to the frontal face of the detector with a vector that go from dA to P. To determine the contribution of the face of the detector sensitive zone to the effective solid angle, we note that cos ðβ Þ ¼ cos ðαÞ ¼ z0 =d, therefore datt corresponds to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð18Þ datt ¼ ðz0 z1Þ 1 þ 02 ðr 2d 2r d r 0 cos ðθÞ þ r 02 Þ: z Then, using (18) and (8) in (16), we have that the effective solid angle for the frontal face of the sensitive detector zone is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Z Rd Z 2π z0 e μðz0 z1 Þ 1 þ ðr 2 2r d r cos ðθÞ þ r 02 Þr d dθ dr d d 02 z Ωμ;Face ¼ : 2 2r r 0 cos ðθ Þ þ r 02 þ z02 Þ3=2 ðr 0 0 d d
Both, the absolute efficiency, ϵ, and the specific activity, As , are related by the following equation: As ¼
N
ϵΔtmIγ
f 1f 2f 3…
ð23Þ
where N represents the Full Energy Peak (FEP) area, Δt is the counting time, m is the mass of the sample and I γ is the γ emission probability. The terms f i are correction factors due to electronic (true coincidence, pile-up and dead time), dry mass, waiting or cooling time, variations of the source activity during the data acquisition, self-attenuation, shape and size. The difference between the traditional method and the method presented in this work, denominated intrinsic spatial efficiency methodology, is that corrections due to self-attenuation, shape and size of the volume source (matrix effects correction) are not necessary.
ð19Þ To determine the effective solid angle for the mantle of the ! detector sensitive zone, we can define r 3 as a perpendicular vector to the face front of the detector, for instance, ! r 3 ¼ ð0; 0; z0 Þ: Then, the effective solid angle subtended for the mantle of the detector sensitive zone is ! !! Z 0 Z θðrÞ ! ! μ ðz0 z1 Þ‖! r 3 ‖ ‖ r 2 ‖= r 3 r 2 r1 r2 e Ωμ;Mantle ðr0 ; z0 Þ ¼ Rd dθ dzd : ! ! hd θ ðrÞ ‖r 1 ‖ ‖r 2 ‖3 ð20Þ Therefore, using Eqs. (19) and (20) in Eq. (16), we have that the geometric efficiency for the source point embedded in a cylindrical volume material of attenuation coefficient μ is 8 Ωμ;Face ðr0 ; z0 Þ > > ; r≦Rd < 4π ð21Þ f Ω;μ ðr 0 ; z0 Þ ¼ 0 0 0 0 Ω ðr ; z Þ þ Ωμ;Mantle ðr ; z Þ > > : μ;Face ; r 4 Rd 4π On the other hand, if the distribution of radioactive material in ! the source is homogeneous, this implies that gð r Þ ¼ 1, then, the absolute FEP efficiency for a cylindrical volume source radioactively homogeneous, is given by R 0 0 0 0 0 0 ϵint ðr ; z Þf Ω;μ ðr ; z Þ dV ϵμ;V ¼ V ; ð22Þ 0 V
3. Material and methods 3.1. Spectroscopy system In this work we have used a spectroscopy system, which consist of a coaxial HPGe detector (Ortec GEM-10195), shielded with a cylindrical Lead wall, 10 cm thickness, and two foils of Cadmium and Copper of 1 mm thickness each one. The structure and dimensions of the detector, given on the manufacturer technical note, are shown in Fig. 5. The data acquisition system is composed of an amplifier Ortec 572 and a MCA Nucleus controlled by the PCA-II software.
3.2.
137
Cs radioactive source
A 137Cs planar source with an activity at the moment of the experimentation of 31,857 Bq distributed in a disc of 3.5 mm diameter was used.
SYMMETRY AXIS 35,0 mm 23,7 mm
where f Ω;μ corresponds to Eq. (21).
4,97 mm
2.5. Determination of the specific activity of an environmental cylindrical volume source: standard methodology vs intrinsic spatial efficiency methodology Let us consider the case of a radioactive volume source on a radiation detector, such as is shown in Fig. 4. Among all photons emitted by the source only a fraction will be detected due to the geometry (solid angle), the matrix effects (selfattenuation) and intrinsic characteristics of the detector (intrinsic efficiency). Due to the complications to determine these factors, independently, is convenient to use the absolute efficiency, because this considers all effects simultaneously. In this case, the method to determine experimentally the specific activity corresponds to the relative method. Once the absolute efficiency of the detector is known, the specific activity of the sample material can be determined.
38,4 mm DETECTOR SENSITIVE ZONE
Fig. 5. Scheme of the detector HPGe GEM-10195. The dimensions of the detector sensitive zone and end-cup are shown.
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5
z
3.3. Reference materials In order to test the proposed method in this work, four matrix reference materials supplied by IAEA were used and their specifications are reported in Table 1. 3.4. Volume source
SOURCE
Cylindrical containers were made with PVC pipes of 2 mm thickness and height according to the amount available of the reference material. These were filled with the reference material and sealed in both faces with Mylar foil of 6.4 μm. The geometric characteristics, radius and height, and the mass of the cylindrical volume sources are informed in Table 2.
z’
END−CUP
3.5. Determination of the spatial FEP efficiency distribution The determination of the spatial efficiency distribution was performed by mapping the FEP area on the r z plane (Fig. 6) using the planar source described in Section 3.2. Due to the rather small radius of the planar source, in first approximation it has been considered as a point source. Angular variations were not necessary due to the axial symmetry of the system. The response of the spectroscopy system to different source positions was measured. Each spectrum was acquired during 180 s and repeated five times. To avoid dead time corrections the time of measurement was adjusted to the live time of system. In mapping, four radial positions, between 0 and 39 mm, and 14 axial positions, between 8.47 and 63.47 mm, were considered. The source position was taken at the planar source center. The FEP area in each position is presented in Fig. 7. We can see that the FEP area decreases as we move away in both radial and axial directions. Also, for each radial position, the variation of the FEP area is large for close position to the detector becoming small as we move away along the z-axis. Due to the source activity pile up correction was applied. The continuous function, N det ðr 0 ; z0 Þ, was obtained by the interpolation of the data. The total number of gamma photons emitted Table 1 Matrix and specific activity of the reference materials, given by the factory. Material matrix
Material title
Specific activity
Reference materials Liquid Soil Moss–soil Grass
Sample 3 Sample 1 IAEA-447 IAEA-372
16.7 7 0.1 52.6 71.1 425 7 10 11,320 7360
[20] [20] [21] [22]
137
Cs Bq/kg
r’
r
DETECTOR SENSITIVE ZONE
Fig. 6. Experimental setting for the determination of the spatial efficiency distribution. The lattice in the grid on the r z plane represents the different source positions to carry out the mapping of the FEP area.
Fig. 7. Graphic of the FEP area along the z-axis for each radial position.
by the source during the acquisition was determined considering the activity of the source informed in Section 3.2 while the solid angle subtended by the source on the detector f Ω ðr 0 ; z0 Þ was computed using Eq. (6). Then using Eqs. (11) and (12) the intrinsic and absolute spatial FEP efficiency distribution were obtained. 3.6. Determination of the attenuation coefficient
Table 2 Geometric characteristics and mass of the cylindrical sources prepared for this study. References material
Number source
Radius (mm)
Height (mm)
Mass (g)
The determination of the attenuation coefficient was carried out by the known transmission method [23] using a NaI(Tl) detector. The set-up is shown in Fig. 8. 3.7. Determination of the specific activity
Cylindrical volume source Sample 3 1 2
29.5 70.1 29.5 70.1
18.7 7 0.1 25.9 7 0.1
48.53 7 0.01 69.63 70.01
Sample 1
1 2
29.5 70.1 29.5 70.1
17.5 7 0.1 18.7 7 0.1
38.94 70.01 43.87 70.01
IAEA-447
1 2
22.6 70.1 29.4 70.1
25.17 0.1 15.0 70.1
29.447 0.01 34.80 70.01
IAEA-372
1 2 3 4
29.4 70.1 29.4 70.1 29.5 70.1 29.5 70.1
24.370.1 24.370.1 20.4 7 0.1 15.17 0.1
33.637 0.01 27.79 7 0.01 24.507 0.01 18.97 70.01
The cylindrical volume sources were positioned coaxially and closest to the detector over a Marinelli beaker used as a sampler holder, as is shown in Fig. 9. In Table 3, the time length of the measurements and the FEP area obtained are shown. Due to the low efficiency of the detector used in these measurements, around 10% relative to a 3 in: 3 in: NaI(Tl), we can see that for the source with a specific activity of order of 17 Bq/ kg, Sample 3, the obtained uncertainties of the FEP area are greater than 12%, even after of two days of measurement. For the sources
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with specific activities greater than 52 Bq/kg the FEP area is more precise. With relative uncertainties less than 8%, 2% and 1% for the materials sample 1, IAEA-447 and IAEA-372, respectively, were obtained in one day (86,400 s) of measurement. Taking into account the dimensions of the cylindrical source, the results obtained for the intrinsic spatial efficiency and the linear attenuation coefficient, the absolute efficiency of the spectroscopy system for 661.65 keV was determined by means of Eq. (22). Then, using Eq. (23), the specific activity of the analyzed material was determined. The uncertainties of the absolute efficiency and the specific activity were determined by the error propagation equation for uncorrelated variables [24].
Radioactive Source
Collimator
Table 3 Table of data of the FEP area obtained in the measurement of each cylindrical volume source. The FEP area corresponds to the emissions of 137Cs and a energy of 661.65 keV. The net and rate FEP area are shown. u represents the relative uncertainty given by the data acquisition software. Reference material
Number source
FEP area
Acquisition time (s)
Net counts
Rate counts/s
u (%)
Sample 3
1 2
304 836
0.00422 0.00581
29 12
72,000 144,000
Sample 1
1 2
2275 1036
0.01053 0.01199
7 8
216,000 86,400
IAEA-447
1 2
13,131 6446
0.060791 0.08139
1 2
216,000 79,200
IAEA-372
1 2 3 4
68,891 113,215 64,664 74,579
1.4547 1.3104 1.2802 1.0405
0.4 0.3 0.4 0.4
43,200 86,400 50,400 71,674
4. Result and discussion 4.1. Spatial efficiency distribution of the HPGe detector h
Volume Sample
Collimator
Detector
Fig. 8. Experimental set-up to determine the attenuation coefficient by transmission method.
AXIAL SYMMETRY AXIS
Rm
Following the procedure described in Section 3.5, the intrinsic and absolute spatial efficiency distribution of a HPGe detector was determined. Figs. 10 and 11 show the intrinsic and absolute spatial efficiency distribution, respectively. We can see that the intrinsic efficiency increases while we move away axially from the detector as well as when the radial position increases. The first case is due to the fact that while the source moves away from the detector, the photons impact more perpendicular on the face of the detector, then, the path length average increases and therefore the probability of interaction. For the second case, the same argument should apply but it is not clear. On the other hand the fast radial growth around 23 mm, can be associated to the radius of the detector, 23.7 mm, an intrinsic characteristic of the detector. The absolute spatial efficiency distribution, Fig. 11, gives us information on the optimum dimensions, in statistical terms, of voluminous sources to be used in this spectroscopy system. For instance, we can see that the major contribution to the FEP area is obtained for axial positions less than 10 mm and radial position less than 20 mm. For positions greater than 23 and 35 mm, axial and radial respectively, the contribution is reduced to one half. 4.2. Specific activity of the sources
SAMPLE HOLDER (MARINELLI)
hm
6,97 mm
DETECTOR END−CUP
DETECTOR SENSITIVE ZONE
Fig. 9. Experimental setting for the determination of the specific activity of a cylindrical volume source of height hm y radius Rm .
By means of the procedure described in Section 3.6 the linear attenuation coefficient was determined for all reference materials. Comparing the result obtained for the water matrix the relative difference was less than 1% compared with the value given by XCOM [25]. Then, by means of the procedure described in Section 3.7 the absolute efficiency for the voluminous sources and their specific activity was determined. The results of the specific activity are presented in Table 4. In general these results are satisfactory, specially for the materials with specific activities greater than 52 Bq/kg. The best results, with relative differences less than 3%, were obtained with the reference material Sample 1, which has an activity of 52 Bq/kg. While that the worst, with relative differences between 2% and 12%, were for the sources with the reference material IAEA Sample 3, which has a nominal activity of 16.7 Bq/kg but only 14.2 Bq/kg at moment of the measurements. However, the uncertainties of the measurements give account of the expectation value. Respect to
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7
Fig. 10. Graphics of the intrinsic spatial efficiency distribution of the HPGe detector, GEM-10195, for 661.7 keV.
Fig. 11. Graphics of the absolute spatial efficiency distribution of the HPGe detector, GEM-10195, for 661.7 keV.
Table 4 Results of the absolute efficiency and the specific activity obtained for the different volume sources used in this work. In the first column, the number in parenthesis corresponds to the certified activity given by the supplier, in Bq/kg. Reference material
Number source
Absolute efficiency
Specific activity obtained (Bq/kg)
Relative difference (%)
Table comparative of the results Sample 3 1 7037 8 (16.727 0.08) 2 6147 6
17.17 5.0 18.8 7 2.2
2.1 12
Sample 1 (52.6 7 1.1)
1 2
7367 9 7187 8
51.9 73.6 53.8 74.1
1.4 2.2
IAEA-447 (425 7 10)
1 2
741 7 10 7607 10
399 7 7 403 7 9
6.2 5.3
6717 7 6787 7 7187 8 785 7 9
10,7247 118 10,5577 113 10,868 7126 10,593 7 142
5.2 6.7 3.9 6.4
(10 5 )
IAEA-372 1 (11,320 7 360) 2 3 4
the other references materials analyzed, all show relative differences less than 7%. In the present work, to illustrate how the method works, we have only used a single energy to determine the efficiency, however, for a complete efficiency calibration either a set of single gamma source or a multi-gamma source can be used. In the former case, our method can be applied directly for each source separately, while for a multi-gamma source the true coincidence correction should be considered.
5. Conclusions We have presented a method to determine the efficiency of a coaxial HPGe detector for volume sources and we have successfully
applied it in cylindrical volume sources radioactively homogeneous, Eq. (22), by means of the determination of the specific activity of 137Cs. The range of the specific activity measured was between 14.2 Bq/kg and 9640 Bq/kg at the moment of the experimentation. For this reason, the method developed in this work can be applied adequately to environmental radioactivity analysis, due to that these types of sources are considered as radioactively homogeneous. To emphasize this fact the reference materials used in this work have been soil, water and grass. Finally, a general expression to determine the absolute efficiency for a volume source radioactively non-homogeneous was obtained, Eq. (15), however experiments with this type of sources were not conducted in the present work. In the future, more studies should be dedicated to this type of source.
Acknowledgments The author would like to acknowledge the assistance given by Alejandro J. Martinez in the development of the numerical tools needed for the application of the method and to Dr. José Roberto Morales for the assistance and advices for writing this paper. Experimental support from Center For Experimental Physics (CEFEX) of the University of Chile is recognized. Also, I would like to thank the two anonymous referees for providing constructive and helpful comments to improve the quality of the manuscript. References [1] IAEA, Measurement of radionuclides in food and the environment: a guidebook, Technical Reports No. 295, IAEA, Vienna, 1989. [2] Michael F. L'Annunziata, Handbook of Radioactivity Analysis, Third Edition, Elsevier, Amsterdam, 2012. [3] Gordon L. Gilmore, Practical and Gamma Ray Spectrometry, Second Edition, John Wiley & Sons, Chichester, 2008.
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