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The dependence of the counting efficiency of Marinelli beakers for environmental samples on the density of the samples
ARTICLE IN PRESS
Applied Radiation and Isotopes 63 (2005) 87–92 www.elsevier.com/locate/apradiso
The dependence of the counting efficiency of Marinelli beakers for environmental samples on the density of the samples Z.B. Alfassia,, N. Lavib a Ben Gurion University, Nucl. Eng. Department, 84105 Beer Sheva, Israel Environment Services Company Ltd., Environment Radiation Laboratory, 67212 Tel Aviv, Israel
b
Abstract The effect of the density of the radioactive material packed in a Marinelli beaker on the counting efficiency was studied. It was found that for all densities (0.4–1.7 g/cm3) studied the counting efficiency () fits the linear log–log dependence on the photon energy (E) above 200 keV, i.e. obeying the equation ¼ aE b (a, b—parameters). It was found that for each photon energy the counting efficiency is linearly dependent on the density (r) of the matrix. ¼ a br (a, b—parameters). The parameters of the linear dependence are energy dependent (linear log–log dependence), leading to a final equation for the counting efficiency of Marinelli beaker involving both density of the matrix and the photon energy: ¼ a1 E b1 a2 E b2 r: r 2005 Elsevier Ltd. All rights reserved.
1. Introduction Environmental samples are usually of relatively low activity. In order to increase the number of counts measured by g ray spectrometry and hence the statistical accuracy, large samples should be counted. Larger samples mean that the additional amount of sample is located farther from the detector and hence their contribution is smaller. To increase the number of counts due to the additional material, it is usually suggested to use special beakers which cover the detector from all directions (called Marinelli beakers or also inverted-well beakers (IEBB, 1997)), rather than using a conventional cylindrical beaker, in which the sample is only over the detector cap. To obtain quantitative, accurate results, the detector must be calibrated with a radioactive standard source homogenously distributed in a beaker of the same Corresponding author.
E-mail address: [email protected] (Z.B. Alfassi).
dimensions and preferably containing the same material, or material with a similar composition or at least with the same density, as the environmental sample. Densities of environmental samples vary appreciably, typically from 0.5 g/cm3 (dry vegetation) to 1.7 g/cm3 (concrete). A radioactive standard material in a Marinelli beaker is available commercially, e.g. from Du Pont (mixture of several radionuclides with g-ray energies from 81 to 1836 keV) or a Marinelli standard containing 226Ra (with g-ray energies from 186 to 2400 keV) supplied by Isotrak of AEA Technology plc (previously marketed by Amersham International plc). However, these commercial reference materials are quite expensive and available only with one value of density. In the case of a synthetic mixture of radionuclides, most of them are short-lived and after about 2 yr only the energies of 60, 662, 1173 and 1332 keV are still detectable. Debertin and Jianping (1989) show how from a calibration of a detector with a Marinelli beaker filled with one material the calibration for another density can be calculated. However, it is best to calibrate
0969-8043/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2005.01.007
ARTICLE IN PRESS Z.B. Alfassi, N. Lavi / Applied Radiation and Isotopes 63 (2005) 87–92
88 Table 1 Main gamma-ray lines of daughters
232
Th at secular equilibrium with its
Energy (keV)
Radionuclide
Intensity (%)
209.4 270.3 328 338.4 463.0 911.1 968.9 1459.2 1587.9 583.1 860.5 2614.5 727.17 238.6
228
4.55 3.77 3.36 12.01 4.64 29.0 17.46 0.92 3.71 30.9 4.31 35.8 11.8 43.1
Ac Ac 228 Ac 228 Ac 228 Ac 228 Ac 228 Ac 228 Ac 228 Ac 208 Tl 228 Ac 228 Ac 212 Bi 212 Pb 228
experimentally the beaker for the required density. This is the reason why it is so important to be able to prepare a calibrated Marinelli secondary standard in each laboratory. In previous papers (Lavi and Alfassi 2004, 2005), we described a procedure for preparation of a homogenous mixture of natural thorium oxide (ThO2) (232Th at secular equilibrium with its daughters) with either CaCO3 or dried milk. The advantage of using natural thorium at secular equilibrium with its daughters is the multitude of g rays from 75 to 2614 keV (Reus and Westmeier, 1983). However, we limit ourselves to g lines above 238 keV (shown in Table 1). In this paper, we describe the preparation of similar radioactive sources with various densities in order to study the dependence of the efficiencies of g counting of a Ge detector on the density of the matrix for 450 and 1000 ml Marinelli beakers.
2. Experimental studies 2.1. Preparation of environmental standards containing ThO2 A homogeneous calibrated mixture of ThO2 powder (at secular equilibrium with its daughters due to the long shelf life, more than 40 yr, of this bottle) with CaCO3 was prepared as described previously (Lavi and Alfassi, 2005). The activity concentration of this mixture for 232 Th is 1.7570.07 Bq/cm3. The equilibrium was ascertained by the measurement with a calibrated HPGe detector of a 5 g sample.
The homogeneity of this standard was checked by splitting 1000 g of this mixture to ten portions of 100 g each. Each of these portions was counted in a conventional beaker and it was found that all ten portions give the same count rate within the counting uncertainty. As the environmental materials themselves have some natural radioactivity, we measured each material separately in the same Marinelli beaker. The activity of these environmental materials is shown in Table 2. Five Marinelli beaker standards were prepared by mixing 100 cm3 of the Th/CaCO3 calibrated mixture with 900 cm3 of environmental materials, grass, milk powder, calcium carbonate, soil and concrete. The mixing was done in steps, each time adding 100 cm3 of the environmental material and mixing it thoroughly. The densities of the final standards vary from 0.4 to 1.7 g/cm3 according to the environmental material. The homogeneity of each final mixture was checked by splitting the entire mixture into several portions of similar weights and counting separately each of these portions with a HPGe detector (EG&G Ortec, now a division of URS) with a 42% relative efficiency (to NaI (Tl) of 300 300 at 1332 keV). The specific count rates of the various portions (counts per second per gram) were found to be within the statistical counting uncertainty, leading to the conclusion that all these standards are homogenous. The problem of summing up of peaks due to the small distance between the Marinelli beaker and the detector was checked by looking for the sum-up peaks. It was found that at most the sum peaks are about 0.3% of the parent peak. The losses from the main peaks due to sum of two photons in the detector are higher than what is found in the sum peaks as it is the total detection efficiency over the whole spectrum (Compton+photopeak) that causes the loss (Debertin and Helmer, 1988; Debertin and Jianping, 1989). Counts in the photopeak from one transition may be detected in coincidence with a photon in the same cascade, detected in either the photopeak or the Compton continuum. The sum peaks detected in the spectrum are only from the detection of both transitions in the photo peaks. Thus, the total correction could be estimated to be about 3–5% (Debertin and Helmer, 1988; Debertin and Jianping, 1989). If the activities of the calibrating mixture and the unknown sample are very different, then the loss by the summing of two photons should be calculated via the value of the sum peaks. In any case, the loss probably does not vary significantly with the sample density, so there is a little effect on the thesis of the paper. Another check of homogeneity and the lack of need for correction due to summing peaks is the determination of known reference materials using our mixture for calibration. Table 3 shows the results obtained for 134Cs,
ARTICLE IN PRESS Z.B. Alfassi, N. Lavi / Applied Radiation and Isotopes 63 (2005) 87–92
89
Table 2 Levels of radioactivity measured in some environmental materials in Israel (Bq/kg) (dry weight) Material
232
226
40
Density [g cm3]
Elite milk powder Grass Soil Concrete
1.1–1.4 3.4–5.5 3.8–4.6 2.5–3.8
— 1.2–2.2 3.5–4.5 26–33
252 490 125–140 25–33
0.67 0.47 1.40 1.70
Table 3 Determination of
Th
134
Cs,
137
Cs and
Ra
K
40
K in reference standard materials and comparison with literature values
Reference material
134
137
40
IAEA-373 grassa This workb IAEA-152, Milk powderc This work
1167 1136751.3 764758 749731
12350 121187478 2129 2120783
432 428716 539 528723
Cs (Bq kg1)
Cs (Bq kg1)
K (Bq kg1)
a For determination of radioactivity, 88.77 g of IAEA-373-Grass was mixed and homogenized with 371.23 g grass and transferred into a Marinelli beaker of 1000 ml. b The given uncertainty is two standard deviations. c For determination of radioactivity, 128.18 g of IAEA-152-Milk powder was mixed and homogenized with 550 g of milk powder (product of Israel) and transferred into a Marinelli beaker of 1000 ml.
137
3. Results and discussion Figs. 1 and 2 (Fig. 1 for Marinelli beakers of 1000 ml and Fig. 2 for Marinelli beakers of 450 ml) show the efficiency vs. photon energy curves in a log–log form for
-1.2
log efficiency
Cs and 40K in grass and milk powder IAEA reference materials calculated from our Marinelli standards and comparison with literature values. The agreement of our results with the literature data validates our Marinelli standards and is another indication of the homogeneity of our calibrated mixtures. After the Marinelli beaker was filled with our calibrated mixture to the required volume (either 450 or 1000 ml), it was covered with melted bees-wax (thickness of 2–3 mm) in order to maintain the same geometrical shape. Finally, the Marinelli beakers were hermetically sealed using silicon glue. The Marinelli beaker filled with a calibrated mixture was counted on the above-mentioned HPGe detector and the counting efficiency was calculated from the ratio of the number of counts at the full energy peak (photopeak) to the number of disintegrations and the intensity of the gamma line (Table 1). Besides the gamma lines given in Table 1, the Marinelli beaker showed also a peak of 511 keV. However, the counts in this energy can be also due to pair production in the sample and hence this line was not taken into consideration in our calibration calculations.
-1.7 Marinelli beaker 0.45 L Vegetation Soil Concrete -2.2
2.3
2.7
3.1
3.5
log energy (keV)
Fig. 1. Log efficiency () vs. log photon energy (E) for Marinelli beaker of 450 ml containing matrices of various densities (vegetation, soil or concrete).
three Marinelli beakers standards (grass, soil and concrete) having different densities (from 0.4 to 1.7 g/ cm3) with a constant amount of 232Th (total activity of 17577 Bq). It can be seen that the regular linear correlation of log (efficiency) vs. log (photon energy) known for point sources (Knoll, 1989) exists also for Marinelli beakers of various densities, as was found previously for densities of 1.0 and 0.7 g/cm3 in 1000 ml Marinelli beakers (Lavi and Alfassi, 2004, 2005). Table 4 shows the linear regression formula for the linear log–log correlation together with the correlation coefficients. All correlation coefficients are larger than 0.997 (450 ml) and 0.999 (1000 ml), indicating a very good correlation.
ARTICLE IN PRESS Z.B. Alfassi, N. Lavi / Applied Radiation and Isotopes 63 (2005) 87–92
90
In Fig. 3, it can be seen that for the 1000 ml Marinelli beaker there exists a linear dependence of the counting efficiency on the density of the calibrated mixture for each photon energy, leading to an equation of the form ¼ a br,
(1)
where stands for the counting efficiency, r denotes the density of the material in the Marinelli beaker, and a and b are constants (the regression parameters). Eq. (1) can be approximated theoretically by assuming a very small-radius radioactive cylinder and a very small-radius detector. For a small-radius cylinder and detector, we can slice the cylinder to thin slices and all photons coming from the same slice are passing through the same thickness of the sample and hence have the same self-absorption. Denoting the rate of emission of photons from the whole cylinder of height H in the direction of the detector by I 0 ; the rate of emission from a slice of thickness dh without considering self-
absorption is dI ¼
Considering the self-absorption, assuming that the slice is at a distance h from the detector (i.e. the slice distance from the detector is between h and h þ dh), dI ¼
dh I 0 emh . H
Here m is the linear absorption coefficient. For the whole column we should integrate dI from h ¼ 0 to H; leading to the self-absorption factor f (Cushall et al., 1983; McMahon et al., 2004): f ¼
I 1 emH . ¼ I0 mH
Developing the numerator to Taylor series of the second power yields the following for the self-absorption factor: f ¼
-1.20
dh I 0. H0
I mH 0:5m2 H 2 ¼ 1 0:5Hm. ¼ I0 mH
Marienelli beaker 1L
Vegetation Soil Concrete
Counting efficiency
log efficiency
0.060
-1.60
-2.00
-2.40 2.3
2.6
2.9
3.2
3.5
0.030
0.000 0.00
238 keV 300 keV 338 keV 463 keV 860 keV 911 keV 2614 keV
0.50
1.00
1.50
2.00
3
Density (g/cm )
log energy (keV)
Fig. 2. Log efficiency () vs. log photon energy (E) for Marinelli beaker of 1000 ml containing matrices of various densities (vegetation, soil or concrete).
Fig. 3. Counting efficiency () vs. the density (r) of the matrix for 1000 ml Marinelli beaker for photon energies of 238, 300,338, 463, 860, 911 and 2614 keV.
Table 4 Linear regression equations and correlation coefficients for log efficiency () vs. log photon energy (E) for Marinelli beaker of 450 and 1000 ml, containing matrices of various densities (the regression was done for the range 238–2614 keV) Standard material
Density (g/cm3)
Linear log–log regression equation
Correlation coefficient (R2 )
450 ml Vegetation Milk powder Calcium carbonate Soil Concrete
0.47 0.70 1.05 1.40 1.70
Log(e) ¼ 0.820342 log(E)+0.725149 Log(e) ¼ 0.811292 log(E)+0.689298 Log(e) ¼ 0.796028 log(E)+0.632371 Log(e) ¼ 0.782634 log(E)+0.579723 Log(e) ¼ 0.771199 log(E)+0.535264
0.9984 0.9980 0.9977 0.9974 0.9971
1000 ml Vegetation Milk powder Calcium carbonate Soil Concrete
0.47 0.70 1.05 1.40 1.70
Log(e) ¼ 0.776331 log(E)+0.454474 Log(e) ¼ 0.762528 log(E)+0.398993 Log(e) ¼ 0.738395 log(E)+0.308923 Log(e) ¼ 0.718119 log(E)+0.228404 Log(e) ¼ 0.676519 log(E)+0.095340
0.9997 0.9995 0.9996 0.9994 0.9993
ARTICLE IN PRESS Z.B. Alfassi, N. Lavi / Applied Radiation and Isotopes 63 (2005) 87–92
Of course this is only an approximation of the real case, since in a real situation we do not have simple slices for which all the photons transverse the same distance in the matrix (Miller, 1987). It should be emphasized that Compton scattering is not necessarily the main process of gamma absorption for all energies and for all absorbing materials. For low-energy photons or large atomic number (Z) absorbing materials, the main process will be the photoelectric effect, which will yield a dependence on the density of the third up to the fourth power of the density. However, for environmental samples, which consist mainly of low-Z elements (up to Z ¼ 14 of Si), and for photon energies of the main contaminants that are above 200 keV, Compton scattering is the main process. The values of the regression parameters a and b together with the correlation coefficients for the various densities are summarized in Table 5 for the 450 ml and the 1000 ml Marinelli beakers. It can be seen that for the 450 ml beakers the fit is always better than R2 ¼ 0:99; while for the 1000 ml beakers the lower energy fit is also very good. However, for the higher energies, the correlation coefficient is slightly less than 0.99, although Fig. 3 shows still a quite good linearity. The main reason for the slightly worse fit is that the change of f with r is
Table 5 Linear regression equations and correlation coefficients of counting efficiency (e) vs. density (r) of the matrix for 450 and 1000 ml Marinelli beaker for various photon energies Energy (keV)
Linear regression equation
Correlation coefficient (R2 )
0.45 L
log slope x 10000
f ¼ 1 br.
4
1.0 L 3.5
3
2.5 2.2
2.7
e ¼ 0.007998r+0.060987 e ¼ 0.006268r+0.051407 e ¼ 0.005517r+0.046971 e ¼ 0.004002r+0.036945 e ¼ 0.001684r+0.021889 e ¼ 0.001646r+0.020954 e ¼ 0.000417r+0.008159
0.9964 0.9953 0.9955 0.9946 0.9939 0.9965 0.9963
1000 ml 238 300 338 463 860 911 2614
e ¼ 0.007781r+0.043550 e ¼ 0.006169r+0.036602 e ¼ 0.005402r+0.033340 e ¼ 0.003982r+0.026309 e ¼ 0.001631r+0.015679 e ¼ 0.001591r+0.015081 e ¼ 0.000492r+0.007107
0.9916 0.9916 0.9917 0.9899 0.9885 0.9880 0.9827
3.7
Fig. 4. Linear log–log dependence of the slope of the density curve on the energy of the photons.
5
4.5
0.45 L 1L 4
3.5 2.2
2.7
3.2
3.7
log energy (keV)
Fig. 5. Linear log–log dependence of the intercept of the density curve on the energy of the photons.
so small that it is almost within the experimental uncertainty of determination of the efficiency. It should also be recalled that neglecting higher terms of the Taylor series is less justified for the larger r. Figs. 4 and 5 show that a and b for both 450 and 1000 ml beakers fulfill a linear log–log correlation with the photon energy. Thus, it is possible to write a ¼ a1 E b1 ; b ¼ a2 E b2 .
450 ml 238 300 338 463 860 911 2614
3.2 log energy (keV)
log intercept x 10000
As the linear absorption coefficient m for Compton scattering of light elements is proportional to the density of the matrix, r; we find that the self-absorption factor f can be given by
91
(2)
The value of a is for the case where the density r ¼ 0, which means no self-absorption. In this case, we expect to have the same dependence as for a point source in which there is no self-absorption. However, it should be reminded that the value of a is for an extended source with no self-absorption, while the usual equation is only for a point source. Substituting Eq. (2) in Eq. (1) yields a general equation which gives the counting efficiency as a function of both energy and density: ¼ a1 E b1 a2 E b2 r.
(3)
The values of the regression parameters a and b are given in Table 6. The four parameters of Eq. (3) are characteristic for each detector–Marinelli beaker setup. For photon
ARTICLE IN PRESS Z.B. Alfassi, N. Lavi / Applied Radiation and Isotopes 63 (2005) 87–92
92
Table 6 Value of parameters for Eq. (3) Volume (ml)
a1
b1
a2
b2
450 1000
6.226 2.863
–0.8394 –0.7658
7.661 5.119
–1.2440 1.1800
energies above 200 keV, the effect of self-absorption is usually smaller than the effect of distance and hence the parameters can be quite different for various setups. As can be seen from Table 5 and from Figs. 4 and 5, the slope of the dependence on the density is quite similar for the 450 and 1000 ml Marinelli beakers, and the main difference is due to the intercept.
References Cushall, N.H., Larsen, I.L., Olsen, C.R., 1983. Direct analysis of 210Pb in sediment samples: self-absorption corrections. Nuclear Instrum. Methods B 206, 309–312. Debertin, K., Helmer, R.G., 1988. g- and X-ray Spectrometry with Semiconductor Detectors. Elsevier, New York.
Debertin, K., Jianping, R., 1989. Measurement of the activity of radioactive samples in Marinelli beakers. Nuclear Instrum. Methods A 278, 541–549. IEBB, 1997. IEEE standard test procedures for Germanium Gamma Ray Detector. The Institute of Electrical and Electronics Engineers Inc. Standard 325, 1996, p. 46. Knoll, G.F., 1989. Radiation Detection and Measurement, second ed. Wiley, New York, p. 436. Lavi, N., Alfassi, Z.B., 2004. Development and application of Marinelli beaker standards for monitoring radioactivity in dairy-products by gamma-ray spectrometry. Appl. Radiat. Isot. 61, 1437–1441. Lavi, N., Alfassi, Z.B., 2005. Development of Marinelli beaker standards containing thorium oxide and application for measurements of radioactive environmental samples. Radiat. Meas., in press. McMahon, C.A., Fegan, M.F., Wong, J., Long, S.C., Ryan, T.P., Colgan, P.A., 2004. Determination of self-absorption corrections for gamma analysis of environmental samples: comparing gamma-absorption curves and spiked matrixmatched samples. Appl. Radiat. Isot. 60, 571–577. Miller, K.M., 1987. Self-absorption corrections for gamma ray spectral measurements of 210Pb in environmental samples. Nuclear Instrum. Methods A 258, 281–285. Reus, U., Westmeier, W., 1983. Catalog of gamma rays from radioactive decay. Atom. Data Nucl. Data Tables 29, 193–406.
Applied Radiation and Isotopes 63 (2005) 87–92 www.elsevier.com/locate/apradiso
The dependence of the counting efficiency of Marinelli beakers for environmental samples on the density of the samples Z.B. Alfassia,, N. Lavib a Ben Gurion University, Nucl. Eng. Department, 84105 Beer Sheva, Israel Environment Services Company Ltd., Environment Radiation Laboratory, 67212 Tel Aviv, Israel
b
Abstract The effect of the density of the radioactive material packed in a Marinelli beaker on the counting efficiency was studied. It was found that for all densities (0.4–1.7 g/cm3) studied the counting efficiency () fits the linear log–log dependence on the photon energy (E) above 200 keV, i.e. obeying the equation ¼ aE b (a, b—parameters). It was found that for each photon energy the counting efficiency is linearly dependent on the density (r) of the matrix. ¼ a br (a, b—parameters). The parameters of the linear dependence are energy dependent (linear log–log dependence), leading to a final equation for the counting efficiency of Marinelli beaker involving both density of the matrix and the photon energy: ¼ a1 E b1 a2 E b2 r: r 2005 Elsevier Ltd. All rights reserved.
1. Introduction Environmental samples are usually of relatively low activity. In order to increase the number of counts measured by g ray spectrometry and hence the statistical accuracy, large samples should be counted. Larger samples mean that the additional amount of sample is located farther from the detector and hence their contribution is smaller. To increase the number of counts due to the additional material, it is usually suggested to use special beakers which cover the detector from all directions (called Marinelli beakers or also inverted-well beakers (IEBB, 1997)), rather than using a conventional cylindrical beaker, in which the sample is only over the detector cap. To obtain quantitative, accurate results, the detector must be calibrated with a radioactive standard source homogenously distributed in a beaker of the same Corresponding author.
E-mail address: [email protected] (Z.B. Alfassi).
dimensions and preferably containing the same material, or material with a similar composition or at least with the same density, as the environmental sample. Densities of environmental samples vary appreciably, typically from 0.5 g/cm3 (dry vegetation) to 1.7 g/cm3 (concrete). A radioactive standard material in a Marinelli beaker is available commercially, e.g. from Du Pont (mixture of several radionuclides with g-ray energies from 81 to 1836 keV) or a Marinelli standard containing 226Ra (with g-ray energies from 186 to 2400 keV) supplied by Isotrak of AEA Technology plc (previously marketed by Amersham International plc). However, these commercial reference materials are quite expensive and available only with one value of density. In the case of a synthetic mixture of radionuclides, most of them are short-lived and after about 2 yr only the energies of 60, 662, 1173 and 1332 keV are still detectable. Debertin and Jianping (1989) show how from a calibration of a detector with a Marinelli beaker filled with one material the calibration for another density can be calculated. However, it is best to calibrate
0969-8043/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2005.01.007
ARTICLE IN PRESS Z.B. Alfassi, N. Lavi / Applied Radiation and Isotopes 63 (2005) 87–92
88 Table 1 Main gamma-ray lines of daughters
232
Th at secular equilibrium with its
Energy (keV)
Radionuclide
Intensity (%)
209.4 270.3 328 338.4 463.0 911.1 968.9 1459.2 1587.9 583.1 860.5 2614.5 727.17 238.6
228
4.55 3.77 3.36 12.01 4.64 29.0 17.46 0.92 3.71 30.9 4.31 35.8 11.8 43.1
Ac Ac 228 Ac 228 Ac 228 Ac 228 Ac 228 Ac 228 Ac 228 Ac 208 Tl 228 Ac 228 Ac 212 Bi 212 Pb 228
experimentally the beaker for the required density. This is the reason why it is so important to be able to prepare a calibrated Marinelli secondary standard in each laboratory. In previous papers (Lavi and Alfassi 2004, 2005), we described a procedure for preparation of a homogenous mixture of natural thorium oxide (ThO2) (232Th at secular equilibrium with its daughters) with either CaCO3 or dried milk. The advantage of using natural thorium at secular equilibrium with its daughters is the multitude of g rays from 75 to 2614 keV (Reus and Westmeier, 1983). However, we limit ourselves to g lines above 238 keV (shown in Table 1). In this paper, we describe the preparation of similar radioactive sources with various densities in order to study the dependence of the efficiencies of g counting of a Ge detector on the density of the matrix for 450 and 1000 ml Marinelli beakers.
2. Experimental studies 2.1. Preparation of environmental standards containing ThO2 A homogeneous calibrated mixture of ThO2 powder (at secular equilibrium with its daughters due to the long shelf life, more than 40 yr, of this bottle) with CaCO3 was prepared as described previously (Lavi and Alfassi, 2005). The activity concentration of this mixture for 232 Th is 1.7570.07 Bq/cm3. The equilibrium was ascertained by the measurement with a calibrated HPGe detector of a 5 g sample.
The homogeneity of this standard was checked by splitting 1000 g of this mixture to ten portions of 100 g each. Each of these portions was counted in a conventional beaker and it was found that all ten portions give the same count rate within the counting uncertainty. As the environmental materials themselves have some natural radioactivity, we measured each material separately in the same Marinelli beaker. The activity of these environmental materials is shown in Table 2. Five Marinelli beaker standards were prepared by mixing 100 cm3 of the Th/CaCO3 calibrated mixture with 900 cm3 of environmental materials, grass, milk powder, calcium carbonate, soil and concrete. The mixing was done in steps, each time adding 100 cm3 of the environmental material and mixing it thoroughly. The densities of the final standards vary from 0.4 to 1.7 g/cm3 according to the environmental material. The homogeneity of each final mixture was checked by splitting the entire mixture into several portions of similar weights and counting separately each of these portions with a HPGe detector (EG&G Ortec, now a division of URS) with a 42% relative efficiency (to NaI (Tl) of 300 300 at 1332 keV). The specific count rates of the various portions (counts per second per gram) were found to be within the statistical counting uncertainty, leading to the conclusion that all these standards are homogenous. The problem of summing up of peaks due to the small distance between the Marinelli beaker and the detector was checked by looking for the sum-up peaks. It was found that at most the sum peaks are about 0.3% of the parent peak. The losses from the main peaks due to sum of two photons in the detector are higher than what is found in the sum peaks as it is the total detection efficiency over the whole spectrum (Compton+photopeak) that causes the loss (Debertin and Helmer, 1988; Debertin and Jianping, 1989). Counts in the photopeak from one transition may be detected in coincidence with a photon in the same cascade, detected in either the photopeak or the Compton continuum. The sum peaks detected in the spectrum are only from the detection of both transitions in the photo peaks. Thus, the total correction could be estimated to be about 3–5% (Debertin and Helmer, 1988; Debertin and Jianping, 1989). If the activities of the calibrating mixture and the unknown sample are very different, then the loss by the summing of two photons should be calculated via the value of the sum peaks. In any case, the loss probably does not vary significantly with the sample density, so there is a little effect on the thesis of the paper. Another check of homogeneity and the lack of need for correction due to summing peaks is the determination of known reference materials using our mixture for calibration. Table 3 shows the results obtained for 134Cs,
ARTICLE IN PRESS Z.B. Alfassi, N. Lavi / Applied Radiation and Isotopes 63 (2005) 87–92
89
Table 2 Levels of radioactivity measured in some environmental materials in Israel (Bq/kg) (dry weight) Material
232
226
40
Density [g cm3]
Elite milk powder Grass Soil Concrete
1.1–1.4 3.4–5.5 3.8–4.6 2.5–3.8
— 1.2–2.2 3.5–4.5 26–33
252 490 125–140 25–33
0.67 0.47 1.40 1.70
Table 3 Determination of
Th
134
Cs,
137
Cs and
Ra
K
40
K in reference standard materials and comparison with literature values
Reference material
134
137
40
IAEA-373 grassa This workb IAEA-152, Milk powderc This work
1167 1136751.3 764758 749731
12350 121187478 2129 2120783
432 428716 539 528723
Cs (Bq kg1)
Cs (Bq kg1)
K (Bq kg1)
a For determination of radioactivity, 88.77 g of IAEA-373-Grass was mixed and homogenized with 371.23 g grass and transferred into a Marinelli beaker of 1000 ml. b The given uncertainty is two standard deviations. c For determination of radioactivity, 128.18 g of IAEA-152-Milk powder was mixed and homogenized with 550 g of milk powder (product of Israel) and transferred into a Marinelli beaker of 1000 ml.
137
3. Results and discussion Figs. 1 and 2 (Fig. 1 for Marinelli beakers of 1000 ml and Fig. 2 for Marinelli beakers of 450 ml) show the efficiency vs. photon energy curves in a log–log form for
-1.2
log efficiency
Cs and 40K in grass and milk powder IAEA reference materials calculated from our Marinelli standards and comparison with literature values. The agreement of our results with the literature data validates our Marinelli standards and is another indication of the homogeneity of our calibrated mixtures. After the Marinelli beaker was filled with our calibrated mixture to the required volume (either 450 or 1000 ml), it was covered with melted bees-wax (thickness of 2–3 mm) in order to maintain the same geometrical shape. Finally, the Marinelli beakers were hermetically sealed using silicon glue. The Marinelli beaker filled with a calibrated mixture was counted on the above-mentioned HPGe detector and the counting efficiency was calculated from the ratio of the number of counts at the full energy peak (photopeak) to the number of disintegrations and the intensity of the gamma line (Table 1). Besides the gamma lines given in Table 1, the Marinelli beaker showed also a peak of 511 keV. However, the counts in this energy can be also due to pair production in the sample and hence this line was not taken into consideration in our calibration calculations.
-1.7 Marinelli beaker 0.45 L Vegetation Soil Concrete -2.2
2.3
2.7
3.1
3.5
log energy (keV)
Fig. 1. Log efficiency () vs. log photon energy (E) for Marinelli beaker of 450 ml containing matrices of various densities (vegetation, soil or concrete).
three Marinelli beakers standards (grass, soil and concrete) having different densities (from 0.4 to 1.7 g/ cm3) with a constant amount of 232Th (total activity of 17577 Bq). It can be seen that the regular linear correlation of log (efficiency) vs. log (photon energy) known for point sources (Knoll, 1989) exists also for Marinelli beakers of various densities, as was found previously for densities of 1.0 and 0.7 g/cm3 in 1000 ml Marinelli beakers (Lavi and Alfassi, 2004, 2005). Table 4 shows the linear regression formula for the linear log–log correlation together with the correlation coefficients. All correlation coefficients are larger than 0.997 (450 ml) and 0.999 (1000 ml), indicating a very good correlation.
ARTICLE IN PRESS Z.B. Alfassi, N. Lavi / Applied Radiation and Isotopes 63 (2005) 87–92
90
In Fig. 3, it can be seen that for the 1000 ml Marinelli beaker there exists a linear dependence of the counting efficiency on the density of the calibrated mixture for each photon energy, leading to an equation of the form ¼ a br,
(1)
where stands for the counting efficiency, r denotes the density of the material in the Marinelli beaker, and a and b are constants (the regression parameters). Eq. (1) can be approximated theoretically by assuming a very small-radius radioactive cylinder and a very small-radius detector. For a small-radius cylinder and detector, we can slice the cylinder to thin slices and all photons coming from the same slice are passing through the same thickness of the sample and hence have the same self-absorption. Denoting the rate of emission of photons from the whole cylinder of height H in the direction of the detector by I 0 ; the rate of emission from a slice of thickness dh without considering self-
absorption is dI ¼
Considering the self-absorption, assuming that the slice is at a distance h from the detector (i.e. the slice distance from the detector is between h and h þ dh), dI ¼
dh I 0 emh . H
Here m is the linear absorption coefficient. For the whole column we should integrate dI from h ¼ 0 to H; leading to the self-absorption factor f (Cushall et al., 1983; McMahon et al., 2004): f ¼
I 1 emH . ¼ I0 mH
Developing the numerator to Taylor series of the second power yields the following for the self-absorption factor: f ¼
-1.20
dh I 0. H0
I mH 0:5m2 H 2 ¼ 1 0:5Hm. ¼ I0 mH
Marienelli beaker 1L
Vegetation Soil Concrete
Counting efficiency
log efficiency
0.060
-1.60
-2.00
-2.40 2.3
2.6
2.9
3.2
3.5
0.030
0.000 0.00
238 keV 300 keV 338 keV 463 keV 860 keV 911 keV 2614 keV
0.50
1.00
1.50
2.00
3
Density (g/cm )
log energy (keV)
Fig. 2. Log efficiency () vs. log photon energy (E) for Marinelli beaker of 1000 ml containing matrices of various densities (vegetation, soil or concrete).
Fig. 3. Counting efficiency () vs. the density (r) of the matrix for 1000 ml Marinelli beaker for photon energies of 238, 300,338, 463, 860, 911 and 2614 keV.
Table 4 Linear regression equations and correlation coefficients for log efficiency () vs. log photon energy (E) for Marinelli beaker of 450 and 1000 ml, containing matrices of various densities (the regression was done for the range 238–2614 keV) Standard material
Density (g/cm3)
Linear log–log regression equation
Correlation coefficient (R2 )
450 ml Vegetation Milk powder Calcium carbonate Soil Concrete
0.47 0.70 1.05 1.40 1.70
Log(e) ¼ 0.820342 log(E)+0.725149 Log(e) ¼ 0.811292 log(E)+0.689298 Log(e) ¼ 0.796028 log(E)+0.632371 Log(e) ¼ 0.782634 log(E)+0.579723 Log(e) ¼ 0.771199 log(E)+0.535264
0.9984 0.9980 0.9977 0.9974 0.9971
1000 ml Vegetation Milk powder Calcium carbonate Soil Concrete
0.47 0.70 1.05 1.40 1.70
Log(e) ¼ 0.776331 log(E)+0.454474 Log(e) ¼ 0.762528 log(E)+0.398993 Log(e) ¼ 0.738395 log(E)+0.308923 Log(e) ¼ 0.718119 log(E)+0.228404 Log(e) ¼ 0.676519 log(E)+0.095340
0.9997 0.9995 0.9996 0.9994 0.9993
ARTICLE IN PRESS Z.B. Alfassi, N. Lavi / Applied Radiation and Isotopes 63 (2005) 87–92
Of course this is only an approximation of the real case, since in a real situation we do not have simple slices for which all the photons transverse the same distance in the matrix (Miller, 1987). It should be emphasized that Compton scattering is not necessarily the main process of gamma absorption for all energies and for all absorbing materials. For low-energy photons or large atomic number (Z) absorbing materials, the main process will be the photoelectric effect, which will yield a dependence on the density of the third up to the fourth power of the density. However, for environmental samples, which consist mainly of low-Z elements (up to Z ¼ 14 of Si), and for photon energies of the main contaminants that are above 200 keV, Compton scattering is the main process. The values of the regression parameters a and b together with the correlation coefficients for the various densities are summarized in Table 5 for the 450 ml and the 1000 ml Marinelli beakers. It can be seen that for the 450 ml beakers the fit is always better than R2 ¼ 0:99; while for the 1000 ml beakers the lower energy fit is also very good. However, for the higher energies, the correlation coefficient is slightly less than 0.99, although Fig. 3 shows still a quite good linearity. The main reason for the slightly worse fit is that the change of f with r is
Table 5 Linear regression equations and correlation coefficients of counting efficiency (e) vs. density (r) of the matrix for 450 and 1000 ml Marinelli beaker for various photon energies Energy (keV)
Linear regression equation
Correlation coefficient (R2 )
0.45 L
log slope x 10000
f ¼ 1 br.
4
1.0 L 3.5
3
2.5 2.2
2.7
e ¼ 0.007998r+0.060987 e ¼ 0.006268r+0.051407 e ¼ 0.005517r+0.046971 e ¼ 0.004002r+0.036945 e ¼ 0.001684r+0.021889 e ¼ 0.001646r+0.020954 e ¼ 0.000417r+0.008159
0.9964 0.9953 0.9955 0.9946 0.9939 0.9965 0.9963
1000 ml 238 300 338 463 860 911 2614
e ¼ 0.007781r+0.043550 e ¼ 0.006169r+0.036602 e ¼ 0.005402r+0.033340 e ¼ 0.003982r+0.026309 e ¼ 0.001631r+0.015679 e ¼ 0.001591r+0.015081 e ¼ 0.000492r+0.007107
0.9916 0.9916 0.9917 0.9899 0.9885 0.9880 0.9827
3.7
Fig. 4. Linear log–log dependence of the slope of the density curve on the energy of the photons.
5
4.5
0.45 L 1L 4
3.5 2.2
2.7
3.2
3.7
log energy (keV)
Fig. 5. Linear log–log dependence of the intercept of the density curve on the energy of the photons.
so small that it is almost within the experimental uncertainty of determination of the efficiency. It should also be recalled that neglecting higher terms of the Taylor series is less justified for the larger r. Figs. 4 and 5 show that a and b for both 450 and 1000 ml beakers fulfill a linear log–log correlation with the photon energy. Thus, it is possible to write a ¼ a1 E b1 ; b ¼ a2 E b2 .
450 ml 238 300 338 463 860 911 2614
3.2 log energy (keV)
log intercept x 10000
As the linear absorption coefficient m for Compton scattering of light elements is proportional to the density of the matrix, r; we find that the self-absorption factor f can be given by
91
(2)
The value of a is for the case where the density r ¼ 0, which means no self-absorption. In this case, we expect to have the same dependence as for a point source in which there is no self-absorption. However, it should be reminded that the value of a is for an extended source with no self-absorption, while the usual equation is only for a point source. Substituting Eq. (2) in Eq. (1) yields a general equation which gives the counting efficiency as a function of both energy and density: ¼ a1 E b1 a2 E b2 r.
(3)
The values of the regression parameters a and b are given in Table 6. The four parameters of Eq. (3) are characteristic for each detector–Marinelli beaker setup. For photon
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Table 6 Value of parameters for Eq. (3) Volume (ml)
a1
b1
a2
b2
450 1000
6.226 2.863
–0.8394 –0.7658
7.661 5.119
–1.2440 1.1800
energies above 200 keV, the effect of self-absorption is usually smaller than the effect of distance and hence the parameters can be quite different for various setups. As can be seen from Table 5 and from Figs. 4 and 5, the slope of the dependence on the density is quite similar for the 450 and 1000 ml Marinelli beakers, and the main difference is due to the intercept.
References Cushall, N.H., Larsen, I.L., Olsen, C.R., 1983. Direct analysis of 210Pb in sediment samples: self-absorption corrections. Nuclear Instrum. Methods B 206, 309–312. Debertin, K., Helmer, R.G., 1988. g- and X-ray Spectrometry with Semiconductor Detectors. Elsevier, New York.
Debertin, K., Jianping, R., 1989. Measurement of the activity of radioactive samples in Marinelli beakers. Nuclear Instrum. Methods A 278, 541–549. IEBB, 1997. IEEE standard test procedures for Germanium Gamma Ray Detector. The Institute of Electrical and Electronics Engineers Inc. Standard 325, 1996, p. 46. Knoll, G.F., 1989. Radiation Detection and Measurement, second ed. Wiley, New York, p. 436. Lavi, N., Alfassi, Z.B., 2004. Development and application of Marinelli beaker standards for monitoring radioactivity in dairy-products by gamma-ray spectrometry. Appl. Radiat. Isot. 61, 1437–1441. Lavi, N., Alfassi, Z.B., 2005. Development of Marinelli beaker standards containing thorium oxide and application for measurements of radioactive environmental samples. Radiat. Meas., in press. McMahon, C.A., Fegan, M.F., Wong, J., Long, S.C., Ryan, T.P., Colgan, P.A., 2004. Determination of self-absorption corrections for gamma analysis of environmental samples: comparing gamma-absorption curves and spiked matrixmatched samples. Appl. Radiat. Isot. 60, 571–577. Miller, K.M., 1987. Self-absorption corrections for gamma ray spectral measurements of 210Pb in environmental samples. Nuclear Instrum. Methods A 258, 281–285. Reus, U., Westmeier, W., 1983. Catalog of gamma rays from radioactive decay. Atom. Data Nucl. Data Tables 29, 193–406.