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Calibration of the 4π γ-ray spectrometer using a new numerical simulation approach
ARTICLE IN PRESS Applied Radiation and Isotopes 68 (2010) 1746–1753
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Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Review
Calibration of the 4p g-ray spectrometer using a new numerical simulation approach Sherif S. Nafee a,b,n,1, Mohamed S. Badawi a, Ali M. Abdel-Moneim a, Seham A. Mahmoud a a b
Physics Department, Faculty of Science, Alexandria University, 21121 Alexandria, Egypt Physics Department, Faculty of Science, King Abdullaziz University, 21589 Jeddah, Saudi Arabia
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 July 2009 Received in revised form 20 February 2010 Accepted 20 February 2010
The 4p g-counting system is well suited for analysis of small environmental samples of low activity because it combines advantages of the low background and the high detection efficiency due to the 4p solid angle. A new numerical simulation approach is proposed for the HPGe well-type detector geometry to calculate the full-energy peak and the total efficiencies, as well as to correct for the coincidence summing effect. This method depends on a calculation of the solid angle subtended by the source to the detector at the point of entrance, (Abbas, 2006a). The calculations are carried out for nonaxial point and cylindrical sources inside the detector cavity. Attenuation of photons within the source itself (self-attenuation), the source container, the detector’s end-cap and the detector’s dead layer materials is also taken into account. In the Belgium Nuclear Research Center, low-activity aqueous solutions of 60Co and 88Y in small vials are routinely used to calibrate a g-ray p-type well HPGe detector in the 60–1836 keV energy range. Efficiency values measured under such conditions are in good agreement with those obtained by the numerical simulation. & 2010 Elsevier Ltd. All rights reserved.
Keywords: HPGe well-type detector Cylindrical sources Self-attenuation Coincidence summing effect
Contents 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . Numerical simulation model. . . . . . . . The coincidence summing corrections Experimental setup . . . . . . . . . . . . . . . Results and discussion. . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Methods to correct for the loss of counts caused by the coincidence summing effects are very important in g-spectrometry, especially when g-ray detectors are used for very low-level radioactive samples (Abbas, 2001a). In such measurements, a well-type detector is frequently used to analyze low-activity small-volume samples. Advantages of such measurements are very high efficiency, reduced
n Corresponding author at: Physics Department, Faculty of Science, Alexandria University, 21121 Alexandria, Egypt. Tel.: +20123528946. E-mail addresses: [email protected], [email protected], [email protected] (S.S. Nafee). 1 Current address at Faculty of Science, Physics Department, King Abdullaziz University, Bld. 115, Rm 261, 21589 Jeddah, Saudi Arabia. Tel.: + 966565813180.
0969-8043/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2010.02.013
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dependence of the efficiency on the measurements geometry, and small self-attenuation corrections (Sima, 2000). Angular correlations can be neglected because of the large solid angle of the detection (Laorie et al., 2002). It is supposed that the low-energy part of the spectrum below 60 keV is not important and that charge collection problems, if present, do not affect the count rate in the measurements, as reported by Sima (2000). The coincidence correction factors, which are relevant to radionuclides emitting multiple cascade radiations, have been ~ addressed by several authors (Debertin and Schotzig, 1979; McCallum and Coote, 1975; Morel et al., 1983; Blaauw, 1993, 1998; Sima and Arnold, 1996; Wang et al., 1999). When two or more g-rays are emitted in coincidence in a decay of the same nucleus and are recorded as one pulse within the resolution time of the detecting system, they are detected also coincidently
ARTICLE IN PRESS S.S. Nafee et al. / Applied Radiation and Isotopes 68 (2010) 1746–1753
~ (Debertin and Schotzig, 1979). Accordingly, the apparent fullenergy peak efficiency for a given peak (uncorrected for coincidence effects) must be multiplied by the coincidence correction factor to result in the true full-energy peak efficiency (Abbas, 2001a). The problem is difficult for extended sources, especially for small-volume sources inside 4p g-counting systems, because one needs to know the spatial dependence of the detector efficiencies within the detector volume (Sima, 2000). The coincidence effects are significant for geometries with a source ~ and a detector in close proximity (Debertin and Schotzig, 1979), although they can be neglected when distances between them are longer than 15 cm (Xilei and Heydorn, 1991). Ignoring these effects inside a well-type detector typically results in errors of a factor of 2 in measured activities of 60Co and 88Y (Laorie et al., 2002). In this work, a p-type well-type HPGe detector was calibrated in the Laboratory of g-spectrometry of the Belgium Nuclear Research Center (SCK.CEN, Mol, Belgium). Cylindrical 1.0 and 1.5- mL containers filled with radioactive aqueous solutions to 70% and 80% of their capacities, respectively, were used. The total, the full-energy peak efficiency values and the coincidence correction factors were calculated for the source–detector geometries by numerical simulations. The simulations were based on the direct mathematical method reported by Selim and Abbas (1994, 1995), which has been successfully applied to various source–detector systems, such as disk source (Selim and Abbas, 1996, 2000; Selim et al., 1998), cylindrical source (Abbas, 2001a, 2001b, 2001c), Marinelli beaker source (Abbas, 2001a; Nafee et al., 2009), parallelepiped detector (Abbas, 2001b, 2006b), NIST gas sphere source (Pibida et al., 2007), NIST bar source (Nafee and Abbas, 2008) and well-type detector (Abbas, 2001c, 2006a; Abbas and Selim, 2002).
2. Numerical simulation model The detection efficiency epoint (Well) of a well-type detector (Fig. 1) of outer radius R, inner radius R1, base height (L+ L2), and depth (L1 + L2) can be calculated by dividing the detector into two parts: the upper and the lower ones. So, there will be two cases to consider for a photon emitted from an isotropically radiating non-axial point source P arbitrarily positioned inside the well cavity. Accordingly, the efficiency will be presented as a sum of
1747
two terms
epointðWellÞ ¼ epointðupper partÞ þ epointðlower partÞ ,
ð1Þ
were epoint is the efficiency of the detector in the case of a point source. It was reported (Selim and Abbas, 1994, 1995) to be Z Z 1 epoint ¼ fatt ð1emdi Þsin y dj dy, ð2Þ 4p y j where y and j are the polar and azimuthal angles, respectively, m is the attenuation coefficient of the detector active medium for gray photons with energy Eg, and di is the possible photon path length within the detector active volume, which will be discussed in detail below. The factor fatt, which accounts for the photon attenuation by the source container, holder, dead layer, and the detector end-cap materials, can be expressed (Hamzawy, 2004) as P ð3Þ fatt ¼ e i mi di , where, mi, is the attenuation coefficient for the ith absorber for photons with energy Eg and di is the photon path length through the ith absorber. The value [1 exp( mdi)] is the probability of interaction when the photon traverses a thickness equal to di inside the detector active medium (consequently, the intrinsic efficiency). The total efficiency eT can be calculated by replacing m in Eq. (2) with the total attenuation coefficient mT for the detector material at the g-ray energy Eg, (the coherent scattering part excluded), as reported by Hubbell and Seltzer (1995). On the other hand, in the computation of the full-energy peak efficiency eP, m in the same equation is replaced with the full-energy peak attenuation coefficient mP for the detector material, which represents only the part contributing to the full-energy peak (photoelectric coefficient plus the fractions of the incoherent and pair production coefficients leading to the full-energy peak) (Selim et al., 2007). The factor fatt is applicable to the full-energy peak efficiency (because any early interaction would remove the count from the full-energy peak), but not to the total efficiency (where Compton-scattered photons still contribute) (Abbas, 2001c). The quantities r and L1 specify the location of an arbitrarily positioned point source P, while the polar (y) and the azimuthal (j) angles define the direction of a g-ray photon. The effective rays traverse distance d in the detector’s active volume before they leave the crystal.
Fig. 1. A diagram of a well-type detector with a non-axial point source.
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There are five possibilities for photons to enter and leave the 4p g-counting detector in the upper and lower parts with the corresponding different traverse distances d1, d2, d3, d4 and d5. 1. The photon may enter via Base 1 and exit via Base 2 of the detector, propagating a distance L L1 d1 ¼ : ð4Þ cos y cos y 2. It may enter via Side 1 and exit via Base 2 of the detector, propagating a distance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos j þ D R1 2 r2 sin2 j L : ð5Þ d2 ¼ sin y cos y 3. It may enter via Base 1 and exit via Side 2 of the detector, propagating a distance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos j þ D R2 r2 sin2 j L1 : ð6Þ d3 ¼ cos y sin y 5. It may enter via Side 1 and exit via Side 2 of the detector, propagating a distance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos j þ D R2 r2 sin2 j r cos j þ D R1 2 r2 sin2 j : d4 ¼ sin y sin y ð7Þ 7. It may enter via Side 1 and exit via the top of the detector, propagating a distance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos j þ D R1 2 r2 sin2 j L2 d5 ¼ : ð8Þ sin y 9cos y9
In these five equations, D depends on the relation between the transfer angle yT and the incident angle y ( 1 y o yT ð9Þ D¼ 1 y Z yT The extreme values of the polar angle based on the source-todetector configuration are given by the equations Rr R r , , y1 ¼ tan1 y01 ¼ tan1 1 L L1 Rþr R þr , , y2 ¼ tan1 y02 ¼ tan1 1 L L1 ð10Þ Rþr R1 þ r 0 , y3 ¼ ptan1 , y3 ¼ ptan1 L2 L2 0 1 Rr 1 R1 r , y4 ¼ ptan , y4 ¼ ptan L2 L2 The maximal azimuthal angles j for the photon to exit via Bases 1 and 2 for the lower part and from the top of the upper part, respectively, are given by
jLO ¼ cos1
j0LO ¼ cos1
r2 R2 þL2 tan2 y , 2rLtan y ! r2 R1 2 þL1 2 tan2 y , 2rL1 tan y
ð10Þ
jUP ¼ cos
j0UP ¼ cos1
2
!
r2 R1 þ L2 tan2 y : 2rL2 9tan y9
epointðupperpartÞ ¼
j¼4 1 X Y, 4p j ¼ 1 j
where Y2 ¼ Ve,
ð14Þ
R y4 R 2p R y04 R j0UP R y 0 R 2p f5 dj dy, Y3 ¼ y4 4 0 f5 dj dy Y1 ¼ p=2 0 f4 dj dy, Y2 ¼ y03 0 R y R j Rj Y4 ¼ y34 0 UP f5 dj dy 0 UP f4 dj dy : ð15Þ In these equations, fi ¼ fatt ð1emdi Þsin y,
i ¼ 4 and 5:
ð16Þ
There are two cases to consider for the lower part of the detector according to the relation between the extreme values of the photon angles yi. In the first case, y0 2 r y2 o p/2, there are two subcases depending on the relation between y0 1, y1 and y0 2: either y0 1 r y1 o y0 2 or y0 1 r y0 2 o y1. In this case, the efficiency of the lower part is given by
epointðlowerpartÞ ¼
j¼5 1 X Y, 4p j ¼ 1 j
ð17Þ
where R y0 R 2p R y R 2p R p=2 R 2p Y1 ¼ 0 1 0 f1 dj dy, Y2 ¼ y011 0 f2 dj dy, Y3 ¼ y1 0 f4 dj dy h i R y0 R y0 R j0 Y4 ¼ y012 0 LO f1 dj dy 0 LO f2 dj dy R y R j Rj Y5 ¼ y12 0 LO f2 dj dy 0 LO f4 dj dy , ð18Þ and fi ¼ fatt ð1emdi Þsin y,
i ¼ 1,2 and 4:
There are also two subcases in the second case, y2 r y0 2 o p/2, depending on the relation between y0 1, y1 and the transfer angle yT: y1 r y0 1 and y0 1 r y1 o yT. Accordingly, the efficiency of the lower part of the detector in the two subcases can be expressed by Eqs. (20)–(22) and Eqs. (23)–(25), respectively. The former case
epointðlower partÞ ¼
j¼5 1 X Y, 4p j ¼ 1 j
ð20Þ
where R y R 2p R y0 R 2p R p=2 R 2p Y1 ¼ 0 1 0 f1 dj dy, Y2 ¼ y1 1 0 f3 dj dy, Y3 ¼ y01 0 f4 dj dy R y0 R j0 R j0 Y4 ¼ y012 0 LO f3 dj dy 0 LO f4 dj dy R y R j Rj Y5 ¼ y12 0 LO f1 dj dy 0 LO f3 dj dy :
In these equations, fi ¼ fatt ð1em:di Þsin y,
r2 R2 þL2 2 tan2 y , 2rL2 9tan y9 2
The efficiency of the upper part of the detector can be expressed as
ð21Þ
!
1
There is a special case for the polar angle y ¼ yT, which is called the transfer angle 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 2 1 @ R L1 þ r ðLL1 ÞR1 LA ðwhen jLO ¼ jLO Þ: yT ¼ tan ð12Þ LL1 ðLL1 Þ
i ¼ 1,3 and 4:
ð22Þ
In the latter case, ð11Þ
epointðlowerpartÞ ¼
j¼6 1 X Y, 4p j ¼ 1 j
ð23Þ
ARTICLE IN PRESS S.S. Nafee et al. / Applied Radiation and Isotopes 68 (2010) 1746–1753
where R y0 R 2p R p=2 R 2p Y1 ¼ 0 1 0 f1 dj dy, Y2 ¼ y02 0 f4 dj dy i R y hR j0 R 2p Y3 ¼ 01 0 LO f1 dj dy þ j0LO f2 dj dy y1 i R y0 hR j0 R 2p Y4 ¼ y2 2 0 LO f3 dj dy þ j0LO f4 dj dy i R y hR j0 Rj R 2p Y5 ¼ y1T 0 LO f1 dj dy þ j0LO f2 dj dy þ j f4 dj dy LO LO i R y hR j R j0 R 2p Y6 ¼ yT2 0 LO f1 dj dy þ j LO f3 dj dy þ j0LO f4 dj dy :
where ms is the source medium attenuation coefficient and ds is the length of the path of the photon in the source material. The latter depends on the polar and azimuthal angles (y, j) inside the source itself and can be expressed as ð24Þ
LO
Here, fi ¼ fatt ð1emdi Þsin y,
i ¼ 1,2,3 and 4:
ð25Þ
Setting r ¼0 in the case of a non-axial point source in the welltype detector will lead to the case of an arbitrarily positioned isotropically radiating axial point source. The double integrals can be evaluated numerically using the trapezoidal rule. A computer program has been written to evaluate the efficiency of a well-type detector with respect to such source at any source–detector separation. Although the accuracy of the integration improves with increase in interval number n, the convergence is already very good at n ¼20. A cylindrical source can be considered consisting of a group of uniformly distributed point sources P (Fig. 2). The efficiency of the detector with respect to each of them is epoint (Well), and the efficiency of the well-type detector with respect to an extended source is given (Pibida et al., 2007) as R f Se dV ð26Þ eV ¼ V att f pointðWellÞ , V where V is the volume of the source. Any element dV of the volume can be expressed in cylindrical coordinates (Pibida et al., 2007) as dV ¼ r dr da dh,
ds ¼
8 > > > ds1 ¼ > > > > > > > > ds1 ¼ > > > > > > > > > > > ds2 ¼ > > > > > > > > > > < ds2 ¼
L1 h0 cos y L1 h0 cos y
In the case of an isotropically radiating extended source, not all the photons emitted from its radioactive nuclei leave the source volume with the same energy, and, sometimes, some of them get totally absorbed in the source itself (Pibida et al., 2007). This effect is taken into account with the so-called self-absorption factor, Sf, (Abbas et al., 2001)
for y r y5 for y5 o y r y6 and j r jslo
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos j þ S2 r2 sin2 j
for y5 o y r y6 and j 4 jslo
y qsin ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r cos j þ S2 r2 sin2 j
for y 4 y6
sin y
L3 ðL1 h0 Þ > ds1 ¼ > > > 9cos y9 > > > > > L ðL 1 h0 Þ >d ¼ 3 > > s1 > > y9 9cos > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > r cos j þ S2 r2 sin2 j > > > > ds2 ¼ sin y > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > r cos j þ S2 r2 sin2 j > :d ¼ s2
:
for y r y8 for y8 o y r y7 and j r jsup for y8 o y r y7 and j 4 jsup for y 4 y8
sin y
ð30Þ In these equations Sr , y5 ¼ tan1 L1 h0 Sþr , y7 ¼ ptan1 L3 ðL1 h0 Þ
y6 ¼ tan1
Sþr L1 h0
y8 ¼ ptan1
Sr , L3 ðL1 h0 Þ
ð31Þ
!
jslo ¼ cos
r2 S2 þ ðL1 ho Þ2 tan2 y 2rðL1 h0 Þtan y
1
ð27Þ
where r is the lateral distance from the detector axis and a is the angle between the photon track and the detector major axis. Eq. (26) can now be rewritten as R R R fatt Sf epointðWellÞ r dr da dh : ð28Þ evolume ¼ h a r V
Sf ¼ ems ds ,
1749
ð32Þ !
jsup ¼ cos1
r2 S2 þ ðL3 ðL1 h0 ÞÞ2 tan2 y , 2rðL3 ðL1 h0 ÞÞ9tan y9
where y5 through y8 are the extreme polar angles of the source, jsup and jslo are the maximal azimuthal angles for photons exiting the source in its upper and lower parts, respectively, and h0 is the source–detector separation. Thus, the full-energy peak efficiency eCyl of a well-type detector with respect to a cylindrical source of radius S and height L3 can be expressed as R h0 þ L3 R 2p R S h0
eCyl ¼
0
e
0 fatt Sf pointðupper partÞ
r dr da þ pS2 L3
R 2p R S 0
e
0 fatt Sf pointðlower partÞ
ð29Þ
r dr da dh :
ð33Þ Table 1 lists the three different photon path lengths through the cylindrical container corresponding to five main cases of the photon travel.
Table 1 The lengths of photon paths, dti, in the source–detector system. The values d1, d3 are the possible lengths of the photon path inside the detector active medium when the photon enters the detector via Base 1 and leaves it via Base 2 and Side 2, respectively. The values d2, d4, d5 are the possible lengths of the photon path inside the detector active medium when the photon enters the detector via Side 1 and leaves it via Base 2, Side 2 and the top surface of the detector, respectively. The values t1, t2 and t3 are the thicknesses of the top surface, window, and the side wall of the detector, respectively (Fig. 2).
Fig. 2. Geometrical parameters of a well-type detector system with a cylindrical source.
d
Dt1
dt2
dt3
d1, d3 d2, d4, d5
t1/cos y t1/sin y
t2/cos y t2/sin y
t3/cos y t3/sin y
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g10, g21 and g20, respectively, can be given by
3. The coincidence summing corrections For an illustration of the idea of the correction for the coincidence effects, consider a simple case of a radionuclide point source that emits two g-photons with energies g10 and g21 produced by transitions 1-0 and 2-1 between the energy states shown in Fig. 3. The rate of the pulses (events) resulted from the full-energy absorption of photons with energies g10 and g21 in the absence of the summing effect is given by the following equations (Knoll, 2000): N10 ¼ AP10 e10P ,
ð34Þ
N21 ¼ AP21 e21P ,
ð35Þ
C10 ¼ ð1P21 e21T Þ1 C21 ¼ ð1P10 e10T Þ1 P10 P21 e10P e21P 1 C20 ¼ 1þ : P20 e20P
ð42Þ
When an extended source (currently a cylinder) is used to calibrate a cylindrical detector, eijT and eijP are the total and the full-energy peak efficiencies given by Eq. (33) above.
4. Experimental setup where A is the activity of the source, P10 and P21 are the emission probabilities corresponding to the g-lines g10 and g21, e10P and e21P are the full-energy peak efficiencies corresponding to the two lines. If the summing effect occurs, new pulses DN20 will be recorded under the peak g20 with the energy equal to the sum of the two energies (g10 and g21) (Knoll, 2000). This count can be obtained using the equation
DN20 ¼ AP10 P21 e10P e21P :
ð36Þ
The summing effect can be of two types: summing-in, which adds events under the individual g-ray full-energy peaks g10 and g21, and summing-out, which subtracts events there. In the latter case, the number of counts will be decreased by DN20: 0 ¼ N10 DN20 ¼ N10 ð1P21 e21T Þ, N10
ð37Þ
0 ¼ N21 DN20 ¼ N21 ð1P10 e10T Þ, N21
ð38Þ
where e21T, and e10T are the total efficiencies. That reduces the counts under the peaks g10 and g20. For the peak g20, the number of counts N20 will be increased by DN20 P10 P21 e10P e21P 0 , ð39Þ N20 ¼ N20 þ DN20 ¼ N20 1 þ P20 e20P
The absolute full-energy peak efficiencies were calculated for p-types Canberra HPGe well-type detector (Model GCW6023; active volume 300 cm3, relative efficiency at 1.33 MeV was 68.2%). Fig. 4 shows its dimensions, and Table 2 lists its detailed specifications. The sources were standard high-density polyethylene plastic vials of D (1 mL) and R (1.5 mL) types supplied by the Department of Finemechanies of the Biological Laboratory of the Vrije Universiteit (Amsterdam). They contained 0.7 and 1.2 mL of aqueous solutions of radioisotopes, respectively. Several radionuclides (241Am, 109Cd, 57Co, 123mTe, 113Sn, 85Sr, 137Cs, 51Cr, 88Y and 60Co) from Eckert & Ziegler Isotope Products Laboratories (IPL, USA Source #1160-56, reference date 2006-05-01 21H00) were used. The diameters, heights, volumes, side and bottom thicknesses of the sources are listed in Table 3. Table 4 provides the activities of the sources along with their uncertainties. All sources were positioned directly on the entrance window inside the welltype detector so that the source–detector separations would be
where the counts N20 can be defined as N20 ¼ AP20 e20P :
ð40Þ
In Eq. (40), P20, and e20P are the emission probability and the fullenergy peak efficiency for the g20 line, respectively. According to Piton et al. (2000), the correction factor Cij for each peak is Cij ¼
Count without summing N ¼ 0, Count with summing N
ð41Þ
where i and j are the upper and the lower levels between which the transition occurs. Therefore, the correction factors for g-lines
Fig. 3. A simple decay scheme illustrating the concept of coincidence summing correction factors. See discussion in Section 3.
Fig. 4. Technical drawing of the well-type detector of Model GCW6023 provided by the manufacturer.
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Table 2 Characteristics of the used well-type HPGe detector. Manufacturer
Canberra Industries
Serial number Detector model Geometry Mounting Relative efficiency P/C ratio Active volume Resolution at 133.5 keV Voltage bias Amplifier ADC MCA and range Shaping time Shaping mode Detector type Correction for dead-time
b 06013 GCW6023 Co-axial open-end Vertical 68.2% 70.2 300 cm3 1.98 keV ( +) 4500 V ICB 9615 ICB 9633 AIM 556 (8192) 4 ms Gaussian HPGe (P-type) LFC- ND 599
area calculation functions. Changes in the peak fits were also made using the interactive peak fit interface when that was necessary for reducing residuals and errors in the peak areas. The peak areas, the live time, the run time and the start time for each spectrum were entered into spreadsheets that were subsequently used to perform calculations necessary for generating efficiency curves. Numerical simulation was used to calculate the coincidence summing factors for correcting the measured full-energy peak efficiencies in order to obtain the true efficiencies of measurements as discussed above.
5. Results and discussion The full-energy peak efficiency values for the p-type HPGe well-type detector were measured as a function of the photon energy using the following equation
eðEÞ ¼ Table 3 Radioactive sources used in the measurements. Composition
Volume
Geometry
(HDPE) (HDPE)
70% 80%
1 ml 1.5 ml
D-type R-type
Geometry
Diameter
Height
Thickness
D-type R-type
0.93 1.13
1.37 1.62
Bottom
Side
0.06 0.08
0.07 0.08
Table 4 Activities of the used radioactive sources. Nuclide
41
Am Cd 57 Co 123m Te 51 Cr 113 Sn 85 Sr 137 Cs 60 Co 88 Y 109
Activity (Bq) 1 mL source
1.5 mL source
Uncertainty (k¼ 3r)
239 2119 80 105 2755 429 518 362 433 815
615 5457 207 269 7093 1104 1333 933 1114 2099
3.0 3.1 3.1 3.1 3.0 3.0 3.0 3.0 3.0 3.0
very small and the angular correlation effects could be neglected ~ (Debertin and Schotzig, 1979). The spectra were acquired with multichannel analyzers and processed with ISO 9001 Genie 2000 data acquisition and analysis software (Canberra Industries). The acquisition time was long enough to get at least 20,000 counts and, thus, keep the uncertainties below 0.5% (Knoll, 2000). The peaks were fitted using Gaussians with low-energy tails appropriate for germanium detectors. In order to avoid dead-time and pile-up effects, the activity of each radionuclide in the sources did not exceed a few Bq ~ (Debertin and Schotzig, 1979). That kept short-distance count rates within acceptable range regardless of the length of the necessary counting times at long distances. Acquired spectra were analyzed with GammaVision software (ORTEC, Oak Ridge, TN) using its automatic peak search and peak
1751
NðEÞ Y Ci , TAS PðEÞ
ð43Þ
where N(E) is the number of counts in the full-energy peak, which could be obtained using the Genie 2000 software, T is the measuring time (in s), P(E) is the photon emission probability at energy E, AS is the radionuclide activity, and Ci are correction factors for the dead-time effect, coincidence summing effect and radionuclide decay. The calibration source decay correction Cd can be obtained from the decay constant l and the interval DT between the reference time and the run time (Pibida et al., 2007): Cd ¼ elDT :
ð44Þ
Count rates were kept below 800 counts/s to avoid pulse pile-up effects. The uncertainty of the full-energy peak efficiency was obtained by uncertainty propagation with an assumption that all measured quantities were independent. Assuming that the only correction made was for the source decay, the uncertainty of the full-energy peak efficiency se is given by equation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 @e @e @e @e @e se ¼ s2N þ s2T þ s2A þ s2P þ s2l , @N @T @A @P @l ð45Þ where sN, sT, sA, sP, and sl, are the uncertainties of the quantities N(E), T, A, P(E), and l, respectively (Pibida et al., 2007). As the measurements were repetitive, the uncertainty in the full-energy peak efficiency se was combined with the statistical component to produce the combined standard uncertainty as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ðe eÞ2 , ð46Þ ue ¼ s2e þ i i nðn1Þ where the second term under the radical is the standard deviation of the mean value of the measured full-energy peak efficiency, n is the number of the measurements, e is the efficiency mean, and se is given by Eq. (45) (Pibida et al., 2007; Perez-Andujar and Pibida, 2004). The uncertainties in the measured efficiencies were close to 1.00% for 60Co gamma lines (1173 and 1332 KeV) and to 1.05% for 88Y gamma lines (898 and 1836 KeV); however, they reached 2.3% for 51Cr and 113Sn gamma lines. Coincidence summing effects in the reference measurement geometries were not negligible. The true full-energy peak efficiencies have been estimated by applying the coincidence correction factors given by Eq. (42). Table 5 lists calculated coincidence correction factors for 60Co and 88Y lines from 1 and 1.5 mL extended sources. In Figs. 5–8, the full-energy peak efficiencies calculated by our method (solid lines) are compared with the measured values (symbols). The calculated efficiencies
ARTICLE IN PRESS S.S. Nafee et al. / Applied Radiation and Isotopes 68 (2010) 1746–1753
Table 5 Calculated coincidence correction factors for sources. 60
Geometry
Co and
88
Y in two extended
88
Co
D-type, 1 mL R-type, 1.5 mL
60
Y
1173.3 keV
1332.65 keV
898.04 keV
1836.42 keV
1.001621 1.001289
1.001883 1.001302
1.001254 1.001267
1.00896 1.007656
Full- Energy peak Efficiency %
1.0
Full- Energy peak Efficiency %
1752
0.8 0.7 0.6 0.5 0.4 0.3 0.2 Measured efficiency values for 1.5mL True calculated efficiency
0.1 0.0
0.8
10
100 Photon Energy (KeV)
1000
0.6 Fig. 7. True full-energy peak efficiencies calculated with our simualtion method (solid line) and the measured efficiencies (dots and intervals) as functions of the photon energy for the R-type source (1.5 mL).
0.4 0.2
4
Measured efficiency values for 1mL True calculated efficiency values 100 Photon Energy (KeV)
1000
Fig. 5. True full-energy peak efficiencies calculated with our simualtion method (solid line) and measured efficiencies (dots and intervals) as functions of the photon energy for the D-type source (1 mL).
Discrepancy %
10
D-Type
3
0.0
2 1 0 -1 -2
4
Discrepancy %
-3
D-Type
3
-4 0
2 1
200 400 600 800 1000 1200 1400 1600 1800 2000 Photon Energy (KeV)
Fig. 8. Percent difference (D%) between the measured and the calculated true full-energy peak efficiencies as a function of the photon energy for the R-type geometry.
0 -1 -2 -3 -4 0
200 400 600 800 1000 1200 1400 1600 1800 2000 Photon Energy (KeV)
Fig. 6. Percent difference (D%) between the measured and the calculated true full-energy peak efficiencies values as a function of the photon energy for the Dtype geometry.
were obtained using Eq. (33), in which the self-absorption factor Sf, the photon attenuation fatt, and the volumes of the sources were used to get the final corrected values. The discrepancies between the calculated and the measured true full-energy peak efficiencies were calculated as follows:
D% ¼
ecalculated emeasured 100: ecalculated
ð47Þ
The actual measured photopeak efficiencies for 60Co and 88Y gamma lines, calculated as the number of counts under the fullenergy peak divided by the emission probability, are listed in Table 5 alongside the values calculated by us. The actual measured photopeak efficiencies for 1173 and 898 KeV gamma lines are greater than those for 1332 and 1836 KeV gamma lines,
respectively. This is in line with the higher number of counts under the photopeak. Table 5 shows discrepancies below 3% and below 2.5% for the R- and D-type sources, respectively. This means that the volume of the source has a noticeable effect on the actual measured photopeak efficiencies. The larger the volume of the source, the higher the discrepancies between the actual photopeak efficiencies and the calculated ones. This is consistent with the results of our mathematical modeling for complex and large extended sources reported in other publications (Pibida et al., 2007; Nafee and Abbas, 2008; Nafee et al., 2009). Tables 5 and 6 show that the coincidence corrections greatly affect the count rate under the photopeak and, consequently, the efficiency values. A correlation analysis has been performed using the GraphPad Prism program (Version 5.0) (Motulsky, 2007). We have found a negative correlation between the full-energy peak efficiencies and the coincidence correction factors. The correlation coefficients rxy calculated with equation n 1 X xi x yi y ð48Þ rxy ¼ n1 i ¼ 1 sx sy were approximately 0.8 and 0.75 for the D- and R-type sources, respectively. In other words, the lower the coincidence correction factor, the better the detection efficiency.
ARTICLE IN PRESS S.S. Nafee et al. / Applied Radiation and Isotopes 68 (2010) 1746–1753 Table 6 Actual photopeak efficiencies for 60
Co
88
Y
60
Co and
Gamma line (keV)
1173.3 1332.65 898.04 1836.42
1753
88
Y in two extended sources. [Counts/(emission probability)] measured
[Counts/(emission probability)] calculated
R-type, 1.5 mL
D-type, 1 mL
R-type, 1.5 mL
D-type, 1 mL
4.28E + 6 3.86E + 6 2.45E + 6 9.95E + 5
1.97E + 6 1.63E + 6 1.07E +6 4.75E + 5
4.38E + 6 3.77E + 5 2.38E + 6 1.02E +6
1.94E + 6 1.65E + 6 1.08E +6 4.65E + 5
The discrepancies between the true measured efficiency values and the calculated ones for 60Co gamma lines (1173 and 1332 KeV) were 1.79% and 1.74%, respectively, for the D-type geometry and 2.30% and 2.38, respectively, for the R-type geometry. This is because the two gamma lines have nearly the same emission probability and the same coincidence correction factors, as shown in Table 5. However, the discrepancies are much bigger for 88Y gamma lines (1.5% and 3.3% for the D-type while 2.84% and 1.91% for the R-type, respectively). This can be attributed to different emission probabilities for the two lines, which result in different coincidence correction factors listed in Table 5. As can be seen from Figs. 6 and 8, the discrepancies vary between 71.5% and 73.3%. This can be explained by the variations in the calculations of the full-energy peak attenuation coefficients.
6. Conclusions Numerical simulation has been successfully used to generate the full-energy peak efficiency curve for the co-axial p-type HPGe well-type detector calibrated with cylindrical sources inside the detector’s cavity. Discrepancies between the calculated and experimental results were below 3.5%. In addition, the coincidence correction and the source self-absorption factors for the efficiency values have been calculated. This technique is good for calibrations of 4p counting systems with low-activity radioactive sources.
Acknowledgement The authors would like to thank the Authorities of the Belgium Nuclear Research Center (SCK.CEN), Laboratory for Gamma-ray Spectrometry, Boeretang 200, B-2400 Mol, Belgium, for allowing M.S. Badawi to carry out the experimental measurements. References Abbas, M.I., 2001a. HPGe detector photopeak efficiency calculation including selfabsorption and coincidence corrections for Marinelli beaker sources using compact analytical expressions. Appl. Radiat. Isot. 54, 761. Abbas, M.I., 2001b. A direct mathematical method to calculate the efficiencies of a parallelepiped detector for an arbitrarily positioned point source. Radiat. Phys. Chem. 60, 3. Abbas, M.I., 2001c. Analytical formulae for well-type NaI(Tl) and HPGe detectors efficiency computation. Appl. Radiat. Isot. 55, 245. Abbas, M.I., Selim, Y.S., Bassiouni, M., 2001. HPGe detector photopeak efficiency calculation including self-absorption and coincidence corrections for cylindrical sources using compact analytical expressions. Radiat. Phys. Chem. 61, 429. Abbas, M.I., Selim, Y.S., 2002. Calculation of relative full-energy peak efficiencies of well-type detectors. Nucl. Instrum. Method A 480, 651. Abbas, M.I., 2006a. Analytical calculations of the solid angles subtended by a welltype detector at point and extended circular sources. Appl. Radiat. Isot. 64, 1048. Abbas, M.I., 2006b. Validation of analytical formulae for the efficiency calibration of gamma detectors used in laboratory and in-situ measurements. Appl. Radiat. Isot. 64, 1661.
Blaauw, M., 1993. The use of sources emitting coincident g-rays for determination of absolute efficiency curves of highly efficient Ge detectors. Nucl. Instrum. Method A 332, 493. Blaauw, M., 1998. Calibration of the well-type germanium gamma-ray detector employing two gamma-ray spectra. Nucl. Inst. Method A 419, 146. Canberra Industries web page, /http://www.canberra.com/products/839.aspS. ~ Debertin, K., Schotzig, U., 1979. Coincidence summing corrections in Ge(Li)spectrometry at low source-to-detector distances. Nucl. Instrum. Method 158, 471. Hamzawy, A., 2004. Accurate direct mathematical determination of the efficiencies of gamma detectors arising from radioactive sources of different shapes. Ph.D. Dissertation. Alexandria University, Egypt. Hubbell, J.H., Seltzer, S.M., 1995. Tables of X-ray mass attenuation coefficients a nd mass energy absorption coefficients 1 KeV to 20 MeV for elements Z¼1 to 92 and 48 additional substances of dosimetric interest. NISTIR 5632, USA. Knoll, F.G., 2000. Radiation Detection and Measurement 3rd ed. Wiley, New York 802pp. Laorie, J.M., Le Petit, G., Abt, D., Girard, M., 2002. Monte Carlo calculation of the efficiency response of low-background well-type HPGe detector. Nucl. Instrum. Method A 479, 618. McCallum, G.J., Coote, G.E., 1975. Influence of source-detector distance on relative intensity and angular correlation measurements with Ge(Li) spectrometers. Nucl. Instrum. Method 130, 189. Morel, J., Chauvenet, B., Kadachi, A., 1983. Coincidence summing corrections in gamma-ray spectrometry for normalized geometries. Int. J. Appl. Radiat. Isot. 34, 1115. Motulsky, H.J., 2007. GraphPad Prism 0.5 Software, GraphPad Sotware, Inc. Nafee, S.S., Abbas, M.I., 2008. Calibration of closed-end HPGe detectors using bar (parallelepiped) sources. Nucl. Instrum. Method A 592, 80. Nafee, S.S., Badawi, M.S., Abdel-Moneim, A.M., Mahmoud, S.A., Abbas, M.I., 2009. An analytical approach for the calibration of HPGe cylindrical detector using an inverted well-beaker. ANS Trans. 101, 645. Perez-Andujar, A., Pibida, L., 2004. Performance of CdTe, HPGe and NaI(Tl) detectors for radioactivity measurements. Appl. Radiat. Isot. 60, 41. Pibida, L., Nafee, S.S., Unterweger, M., Hammond, M.M., Karam, L., Abbas, M.I., 2007. Calibration of HPGe gamma-ray detectors for measurement of radioactive noble gas sources. Appl. Radiat. Isot. 65, 225. Piton, F., Lepy, M.C., Be, M.M., Plagnard, J., 2000. Efficiency transfer and coincidence summing corrections for gamma-ray spectrometry. Appl. Radiat. Isot. 52, 791. Selim, Y.S., Abbas, M.I., 1994. Source-detector geometrical efficiency. Radiat. Phys. Chem. 44, 1. Selim, Y.S., Abbas, M.I., 1995. Direct calculation of the total efficiency of cylindrical scintillation detectors for non-axial point sources. Egypt. J. Phys. 26, 79. Selim, Y.S., Abbas, M.I., 1996. Direct calculation of the total efficiency cylindrical scintillation detectors for extended circular sources. Radiat. Phys. Chem. 48, 23. Selim, Y.S., Abbas, M.I., Fawzy, M.A., 1998. Analytical calculation of the efficiencies of gamma scintillators. Part I: Total efficiency of coaxial disk sources. Radiat. Phys. Chem. 53, 589. Selim, Y.S., Abbas, M.I., 2000. Analytical calculations of gamma scintillators efficiencies. Part II: Total efficiency for wide coaxial disk sources. Radiat. Phys. Chem. 58, 15. Selim, Y.S., Elzaher, M.A., El-Katib, A.M., Bassiouny, M.M., 2007. Contribution of pair production process in full-energy peak efficiency calculation of semiconductor detectors for axial point sources with high-energy gamma-rays. Radiat. Phys. Chem. 76, 1. Sima, O., Arnold, D., 1996. Self-attenuation and coincidence summing corrections calculated by Monte Carlo simulations for gamma-spectrometric measurements with well-type germanium detectors. Appl. Radiat. Isot. 47, 889. Sima, O., 2000. Accurate calculation of the total efficiency of Ge well-type detectors suitable for efficiency calibration using common standard sources. Nucl. Instrum. Method A 450, 98. Wang, T.K., Hou, I.M., Tseng, C.L., 1999. Well-type HPGe detector absolute peakefficiency calibration and true-coincidence correction. Nucl. Instrum. Method A 425, 504. Xilei, L., Heydorn, K., 1991. Correction for X- and g-Ray Coincidence Effect in the Efficiency Calibration of an n-Type Germanium Detector. Risø-R-593 National Laboratory, Roskilde, Denmark.
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Review
Calibration of the 4p g-ray spectrometer using a new numerical simulation approach Sherif S. Nafee a,b,n,1, Mohamed S. Badawi a, Ali M. Abdel-Moneim a, Seham A. Mahmoud a a b
Physics Department, Faculty of Science, Alexandria University, 21121 Alexandria, Egypt Physics Department, Faculty of Science, King Abdullaziz University, 21589 Jeddah, Saudi Arabia
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 July 2009 Received in revised form 20 February 2010 Accepted 20 February 2010
The 4p g-counting system is well suited for analysis of small environmental samples of low activity because it combines advantages of the low background and the high detection efficiency due to the 4p solid angle. A new numerical simulation approach is proposed for the HPGe well-type detector geometry to calculate the full-energy peak and the total efficiencies, as well as to correct for the coincidence summing effect. This method depends on a calculation of the solid angle subtended by the source to the detector at the point of entrance, (Abbas, 2006a). The calculations are carried out for nonaxial point and cylindrical sources inside the detector cavity. Attenuation of photons within the source itself (self-attenuation), the source container, the detector’s end-cap and the detector’s dead layer materials is also taken into account. In the Belgium Nuclear Research Center, low-activity aqueous solutions of 60Co and 88Y in small vials are routinely used to calibrate a g-ray p-type well HPGe detector in the 60–1836 keV energy range. Efficiency values measured under such conditions are in good agreement with those obtained by the numerical simulation. & 2010 Elsevier Ltd. All rights reserved.
Keywords: HPGe well-type detector Cylindrical sources Self-attenuation Coincidence summing effect
Contents 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . Numerical simulation model. . . . . . . . The coincidence summing corrections Experimental setup . . . . . . . . . . . . . . . Results and discussion. . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Methods to correct for the loss of counts caused by the coincidence summing effects are very important in g-spectrometry, especially when g-ray detectors are used for very low-level radioactive samples (Abbas, 2001a). In such measurements, a well-type detector is frequently used to analyze low-activity small-volume samples. Advantages of such measurements are very high efficiency, reduced
n Corresponding author at: Physics Department, Faculty of Science, Alexandria University, 21121 Alexandria, Egypt. Tel.: +20123528946. E-mail addresses: [email protected], [email protected], [email protected] (S.S. Nafee). 1 Current address at Faculty of Science, Physics Department, King Abdullaziz University, Bld. 115, Rm 261, 21589 Jeddah, Saudi Arabia. Tel.: + 966565813180.
0969-8043/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2010.02.013
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1746 1747 1750 1750 1751 1753 1753 1753
dependence of the efficiency on the measurements geometry, and small self-attenuation corrections (Sima, 2000). Angular correlations can be neglected because of the large solid angle of the detection (Laorie et al., 2002). It is supposed that the low-energy part of the spectrum below 60 keV is not important and that charge collection problems, if present, do not affect the count rate in the measurements, as reported by Sima (2000). The coincidence correction factors, which are relevant to radionuclides emitting multiple cascade radiations, have been ~ addressed by several authors (Debertin and Schotzig, 1979; McCallum and Coote, 1975; Morel et al., 1983; Blaauw, 1993, 1998; Sima and Arnold, 1996; Wang et al., 1999). When two or more g-rays are emitted in coincidence in a decay of the same nucleus and are recorded as one pulse within the resolution time of the detecting system, they are detected also coincidently
ARTICLE IN PRESS S.S. Nafee et al. / Applied Radiation and Isotopes 68 (2010) 1746–1753
~ (Debertin and Schotzig, 1979). Accordingly, the apparent fullenergy peak efficiency for a given peak (uncorrected for coincidence effects) must be multiplied by the coincidence correction factor to result in the true full-energy peak efficiency (Abbas, 2001a). The problem is difficult for extended sources, especially for small-volume sources inside 4p g-counting systems, because one needs to know the spatial dependence of the detector efficiencies within the detector volume (Sima, 2000). The coincidence effects are significant for geometries with a source ~ and a detector in close proximity (Debertin and Schotzig, 1979), although they can be neglected when distances between them are longer than 15 cm (Xilei and Heydorn, 1991). Ignoring these effects inside a well-type detector typically results in errors of a factor of 2 in measured activities of 60Co and 88Y (Laorie et al., 2002). In this work, a p-type well-type HPGe detector was calibrated in the Laboratory of g-spectrometry of the Belgium Nuclear Research Center (SCK.CEN, Mol, Belgium). Cylindrical 1.0 and 1.5- mL containers filled with radioactive aqueous solutions to 70% and 80% of their capacities, respectively, were used. The total, the full-energy peak efficiency values and the coincidence correction factors were calculated for the source–detector geometries by numerical simulations. The simulations were based on the direct mathematical method reported by Selim and Abbas (1994, 1995), which has been successfully applied to various source–detector systems, such as disk source (Selim and Abbas, 1996, 2000; Selim et al., 1998), cylindrical source (Abbas, 2001a, 2001b, 2001c), Marinelli beaker source (Abbas, 2001a; Nafee et al., 2009), parallelepiped detector (Abbas, 2001b, 2006b), NIST gas sphere source (Pibida et al., 2007), NIST bar source (Nafee and Abbas, 2008) and well-type detector (Abbas, 2001c, 2006a; Abbas and Selim, 2002).
2. Numerical simulation model The detection efficiency epoint (Well) of a well-type detector (Fig. 1) of outer radius R, inner radius R1, base height (L+ L2), and depth (L1 + L2) can be calculated by dividing the detector into two parts: the upper and the lower ones. So, there will be two cases to consider for a photon emitted from an isotropically radiating non-axial point source P arbitrarily positioned inside the well cavity. Accordingly, the efficiency will be presented as a sum of
1747
two terms
epointðWellÞ ¼ epointðupper partÞ þ epointðlower partÞ ,
ð1Þ
were epoint is the efficiency of the detector in the case of a point source. It was reported (Selim and Abbas, 1994, 1995) to be Z Z 1 epoint ¼ fatt ð1emdi Þsin y dj dy, ð2Þ 4p y j where y and j are the polar and azimuthal angles, respectively, m is the attenuation coefficient of the detector active medium for gray photons with energy Eg, and di is the possible photon path length within the detector active volume, which will be discussed in detail below. The factor fatt, which accounts for the photon attenuation by the source container, holder, dead layer, and the detector end-cap materials, can be expressed (Hamzawy, 2004) as P ð3Þ fatt ¼ e i mi di , where, mi, is the attenuation coefficient for the ith absorber for photons with energy Eg and di is the photon path length through the ith absorber. The value [1 exp( mdi)] is the probability of interaction when the photon traverses a thickness equal to di inside the detector active medium (consequently, the intrinsic efficiency). The total efficiency eT can be calculated by replacing m in Eq. (2) with the total attenuation coefficient mT for the detector material at the g-ray energy Eg, (the coherent scattering part excluded), as reported by Hubbell and Seltzer (1995). On the other hand, in the computation of the full-energy peak efficiency eP, m in the same equation is replaced with the full-energy peak attenuation coefficient mP for the detector material, which represents only the part contributing to the full-energy peak (photoelectric coefficient plus the fractions of the incoherent and pair production coefficients leading to the full-energy peak) (Selim et al., 2007). The factor fatt is applicable to the full-energy peak efficiency (because any early interaction would remove the count from the full-energy peak), but not to the total efficiency (where Compton-scattered photons still contribute) (Abbas, 2001c). The quantities r and L1 specify the location of an arbitrarily positioned point source P, while the polar (y) and the azimuthal (j) angles define the direction of a g-ray photon. The effective rays traverse distance d in the detector’s active volume before they leave the crystal.
Fig. 1. A diagram of a well-type detector with a non-axial point source.
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There are five possibilities for photons to enter and leave the 4p g-counting detector in the upper and lower parts with the corresponding different traverse distances d1, d2, d3, d4 and d5. 1. The photon may enter via Base 1 and exit via Base 2 of the detector, propagating a distance L L1 d1 ¼ : ð4Þ cos y cos y 2. It may enter via Side 1 and exit via Base 2 of the detector, propagating a distance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos j þ D R1 2 r2 sin2 j L : ð5Þ d2 ¼ sin y cos y 3. It may enter via Base 1 and exit via Side 2 of the detector, propagating a distance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos j þ D R2 r2 sin2 j L1 : ð6Þ d3 ¼ cos y sin y 5. It may enter via Side 1 and exit via Side 2 of the detector, propagating a distance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos j þ D R2 r2 sin2 j r cos j þ D R1 2 r2 sin2 j : d4 ¼ sin y sin y ð7Þ 7. It may enter via Side 1 and exit via the top of the detector, propagating a distance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos j þ D R1 2 r2 sin2 j L2 d5 ¼ : ð8Þ sin y 9cos y9
In these five equations, D depends on the relation between the transfer angle yT and the incident angle y ( 1 y o yT ð9Þ D¼ 1 y Z yT The extreme values of the polar angle based on the source-todetector configuration are given by the equations Rr R r , , y1 ¼ tan1 y01 ¼ tan1 1 L L1 Rþr R þr , , y2 ¼ tan1 y02 ¼ tan1 1 L L1 ð10Þ Rþr R1 þ r 0 , y3 ¼ ptan1 , y3 ¼ ptan1 L2 L2 0 1 Rr 1 R1 r , y4 ¼ ptan , y4 ¼ ptan L2 L2 The maximal azimuthal angles j for the photon to exit via Bases 1 and 2 for the lower part and from the top of the upper part, respectively, are given by
jLO ¼ cos1
j0LO ¼ cos1
r2 R2 þL2 tan2 y , 2rLtan y ! r2 R1 2 þL1 2 tan2 y , 2rL1 tan y
ð10Þ
jUP ¼ cos
j0UP ¼ cos1
2
!
r2 R1 þ L2 tan2 y : 2rL2 9tan y9
epointðupperpartÞ ¼
j¼4 1 X Y, 4p j ¼ 1 j
where Y2 ¼ Ve,
ð14Þ
R y4 R 2p R y04 R j0UP R y 0 R 2p f5 dj dy, Y3 ¼ y4 4 0 f5 dj dy Y1 ¼ p=2 0 f4 dj dy, Y2 ¼ y03 0 R y R j Rj Y4 ¼ y34 0 UP f5 dj dy 0 UP f4 dj dy : ð15Þ In these equations, fi ¼ fatt ð1emdi Þsin y,
i ¼ 4 and 5:
ð16Þ
There are two cases to consider for the lower part of the detector according to the relation between the extreme values of the photon angles yi. In the first case, y0 2 r y2 o p/2, there are two subcases depending on the relation between y0 1, y1 and y0 2: either y0 1 r y1 o y0 2 or y0 1 r y0 2 o y1. In this case, the efficiency of the lower part is given by
epointðlowerpartÞ ¼
j¼5 1 X Y, 4p j ¼ 1 j
ð17Þ
where R y0 R 2p R y R 2p R p=2 R 2p Y1 ¼ 0 1 0 f1 dj dy, Y2 ¼ y011 0 f2 dj dy, Y3 ¼ y1 0 f4 dj dy h i R y0 R y0 R j0 Y4 ¼ y012 0 LO f1 dj dy 0 LO f2 dj dy R y R j Rj Y5 ¼ y12 0 LO f2 dj dy 0 LO f4 dj dy , ð18Þ and fi ¼ fatt ð1emdi Þsin y,
i ¼ 1,2 and 4:
There are also two subcases in the second case, y2 r y0 2 o p/2, depending on the relation between y0 1, y1 and the transfer angle yT: y1 r y0 1 and y0 1 r y1 o yT. Accordingly, the efficiency of the lower part of the detector in the two subcases can be expressed by Eqs. (20)–(22) and Eqs. (23)–(25), respectively. The former case
epointðlower partÞ ¼
j¼5 1 X Y, 4p j ¼ 1 j
ð20Þ
where R y R 2p R y0 R 2p R p=2 R 2p Y1 ¼ 0 1 0 f1 dj dy, Y2 ¼ y1 1 0 f3 dj dy, Y3 ¼ y01 0 f4 dj dy R y0 R j0 R j0 Y4 ¼ y012 0 LO f3 dj dy 0 LO f4 dj dy R y R j Rj Y5 ¼ y12 0 LO f1 dj dy 0 LO f3 dj dy :
In these equations, fi ¼ fatt ð1em:di Þsin y,
r2 R2 þL2 2 tan2 y , 2rL2 9tan y9 2
The efficiency of the upper part of the detector can be expressed as
ð21Þ
!
1
There is a special case for the polar angle y ¼ yT, which is called the transfer angle 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 2 1 @ R L1 þ r ðLL1 ÞR1 LA ðwhen jLO ¼ jLO Þ: yT ¼ tan ð12Þ LL1 ðLL1 Þ
i ¼ 1,3 and 4:
ð22Þ
In the latter case, ð11Þ
epointðlowerpartÞ ¼
j¼6 1 X Y, 4p j ¼ 1 j
ð23Þ
ARTICLE IN PRESS S.S. Nafee et al. / Applied Radiation and Isotopes 68 (2010) 1746–1753
where R y0 R 2p R p=2 R 2p Y1 ¼ 0 1 0 f1 dj dy, Y2 ¼ y02 0 f4 dj dy i R y hR j0 R 2p Y3 ¼ 01 0 LO f1 dj dy þ j0LO f2 dj dy y1 i R y0 hR j0 R 2p Y4 ¼ y2 2 0 LO f3 dj dy þ j0LO f4 dj dy i R y hR j0 Rj R 2p Y5 ¼ y1T 0 LO f1 dj dy þ j0LO f2 dj dy þ j f4 dj dy LO LO i R y hR j R j0 R 2p Y6 ¼ yT2 0 LO f1 dj dy þ j LO f3 dj dy þ j0LO f4 dj dy :
where ms is the source medium attenuation coefficient and ds is the length of the path of the photon in the source material. The latter depends on the polar and azimuthal angles (y, j) inside the source itself and can be expressed as ð24Þ
LO
Here, fi ¼ fatt ð1emdi Þsin y,
i ¼ 1,2,3 and 4:
ð25Þ
Setting r ¼0 in the case of a non-axial point source in the welltype detector will lead to the case of an arbitrarily positioned isotropically radiating axial point source. The double integrals can be evaluated numerically using the trapezoidal rule. A computer program has been written to evaluate the efficiency of a well-type detector with respect to such source at any source–detector separation. Although the accuracy of the integration improves with increase in interval number n, the convergence is already very good at n ¼20. A cylindrical source can be considered consisting of a group of uniformly distributed point sources P (Fig. 2). The efficiency of the detector with respect to each of them is epoint (Well), and the efficiency of the well-type detector with respect to an extended source is given (Pibida et al., 2007) as R f Se dV ð26Þ eV ¼ V att f pointðWellÞ , V where V is the volume of the source. Any element dV of the volume can be expressed in cylindrical coordinates (Pibida et al., 2007) as dV ¼ r dr da dh,
ds ¼
8 > > > ds1 ¼ > > > > > > > > ds1 ¼ > > > > > > > > > > > ds2 ¼ > > > > > > > > > > < ds2 ¼
L1 h0 cos y L1 h0 cos y
In the case of an isotropically radiating extended source, not all the photons emitted from its radioactive nuclei leave the source volume with the same energy, and, sometimes, some of them get totally absorbed in the source itself (Pibida et al., 2007). This effect is taken into account with the so-called self-absorption factor, Sf, (Abbas et al., 2001)
for y r y5 for y5 o y r y6 and j r jslo
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r cos j þ S2 r2 sin2 j
for y5 o y r y6 and j 4 jslo
y qsin ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r cos j þ S2 r2 sin2 j
for y 4 y6
sin y
L3 ðL1 h0 Þ > ds1 ¼ > > > 9cos y9 > > > > > L ðL 1 h0 Þ >d ¼ 3 > > s1 > > y9 9cos > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > r cos j þ S2 r2 sin2 j > > > > ds2 ¼ sin y > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > r cos j þ S2 r2 sin2 j > :d ¼ s2
:
for y r y8 for y8 o y r y7 and j r jsup for y8 o y r y7 and j 4 jsup for y 4 y8
sin y
ð30Þ In these equations Sr , y5 ¼ tan1 L1 h0 Sþr , y7 ¼ ptan1 L3 ðL1 h0 Þ
y6 ¼ tan1
Sþr L1 h0
y8 ¼ ptan1
Sr , L3 ðL1 h0 Þ
ð31Þ
!
jslo ¼ cos
r2 S2 þ ðL1 ho Þ2 tan2 y 2rðL1 h0 Þtan y
1
ð27Þ
where r is the lateral distance from the detector axis and a is the angle between the photon track and the detector major axis. Eq. (26) can now be rewritten as R R R fatt Sf epointðWellÞ r dr da dh : ð28Þ evolume ¼ h a r V
Sf ¼ ems ds ,
1749
ð32Þ !
jsup ¼ cos1
r2 S2 þ ðL3 ðL1 h0 ÞÞ2 tan2 y , 2rðL3 ðL1 h0 ÞÞ9tan y9
where y5 through y8 are the extreme polar angles of the source, jsup and jslo are the maximal azimuthal angles for photons exiting the source in its upper and lower parts, respectively, and h0 is the source–detector separation. Thus, the full-energy peak efficiency eCyl of a well-type detector with respect to a cylindrical source of radius S and height L3 can be expressed as R h0 þ L3 R 2p R S h0
eCyl ¼
0
e
0 fatt Sf pointðupper partÞ
r dr da þ pS2 L3
R 2p R S 0
e
0 fatt Sf pointðlower partÞ
ð29Þ
r dr da dh :
ð33Þ Table 1 lists the three different photon path lengths through the cylindrical container corresponding to five main cases of the photon travel.
Table 1 The lengths of photon paths, dti, in the source–detector system. The values d1, d3 are the possible lengths of the photon path inside the detector active medium when the photon enters the detector via Base 1 and leaves it via Base 2 and Side 2, respectively. The values d2, d4, d5 are the possible lengths of the photon path inside the detector active medium when the photon enters the detector via Side 1 and leaves it via Base 2, Side 2 and the top surface of the detector, respectively. The values t1, t2 and t3 are the thicknesses of the top surface, window, and the side wall of the detector, respectively (Fig. 2).
Fig. 2. Geometrical parameters of a well-type detector system with a cylindrical source.
d
Dt1
dt2
dt3
d1, d3 d2, d4, d5
t1/cos y t1/sin y
t2/cos y t2/sin y
t3/cos y t3/sin y
ARTICLE IN PRESS 1750
S.S. Nafee et al. / Applied Radiation and Isotopes 68 (2010) 1746–1753
g10, g21 and g20, respectively, can be given by
3. The coincidence summing corrections For an illustration of the idea of the correction for the coincidence effects, consider a simple case of a radionuclide point source that emits two g-photons with energies g10 and g21 produced by transitions 1-0 and 2-1 between the energy states shown in Fig. 3. The rate of the pulses (events) resulted from the full-energy absorption of photons with energies g10 and g21 in the absence of the summing effect is given by the following equations (Knoll, 2000): N10 ¼ AP10 e10P ,
ð34Þ
N21 ¼ AP21 e21P ,
ð35Þ
C10 ¼ ð1P21 e21T Þ1 C21 ¼ ð1P10 e10T Þ1 P10 P21 e10P e21P 1 C20 ¼ 1þ : P20 e20P
ð42Þ
When an extended source (currently a cylinder) is used to calibrate a cylindrical detector, eijT and eijP are the total and the full-energy peak efficiencies given by Eq. (33) above.
4. Experimental setup where A is the activity of the source, P10 and P21 are the emission probabilities corresponding to the g-lines g10 and g21, e10P and e21P are the full-energy peak efficiencies corresponding to the two lines. If the summing effect occurs, new pulses DN20 will be recorded under the peak g20 with the energy equal to the sum of the two energies (g10 and g21) (Knoll, 2000). This count can be obtained using the equation
DN20 ¼ AP10 P21 e10P e21P :
ð36Þ
The summing effect can be of two types: summing-in, which adds events under the individual g-ray full-energy peaks g10 and g21, and summing-out, which subtracts events there. In the latter case, the number of counts will be decreased by DN20: 0 ¼ N10 DN20 ¼ N10 ð1P21 e21T Þ, N10
ð37Þ
0 ¼ N21 DN20 ¼ N21 ð1P10 e10T Þ, N21
ð38Þ
where e21T, and e10T are the total efficiencies. That reduces the counts under the peaks g10 and g20. For the peak g20, the number of counts N20 will be increased by DN20 P10 P21 e10P e21P 0 , ð39Þ N20 ¼ N20 þ DN20 ¼ N20 1 þ P20 e20P
The absolute full-energy peak efficiencies were calculated for p-types Canberra HPGe well-type detector (Model GCW6023; active volume 300 cm3, relative efficiency at 1.33 MeV was 68.2%). Fig. 4 shows its dimensions, and Table 2 lists its detailed specifications. The sources were standard high-density polyethylene plastic vials of D (1 mL) and R (1.5 mL) types supplied by the Department of Finemechanies of the Biological Laboratory of the Vrije Universiteit (Amsterdam). They contained 0.7 and 1.2 mL of aqueous solutions of radioisotopes, respectively. Several radionuclides (241Am, 109Cd, 57Co, 123mTe, 113Sn, 85Sr, 137Cs, 51Cr, 88Y and 60Co) from Eckert & Ziegler Isotope Products Laboratories (IPL, USA Source #1160-56, reference date 2006-05-01 21H00) were used. The diameters, heights, volumes, side and bottom thicknesses of the sources are listed in Table 3. Table 4 provides the activities of the sources along with their uncertainties. All sources were positioned directly on the entrance window inside the welltype detector so that the source–detector separations would be
where the counts N20 can be defined as N20 ¼ AP20 e20P :
ð40Þ
In Eq. (40), P20, and e20P are the emission probability and the fullenergy peak efficiency for the g20 line, respectively. According to Piton et al. (2000), the correction factor Cij for each peak is Cij ¼
Count without summing N ¼ 0, Count with summing N
ð41Þ
where i and j are the upper and the lower levels between which the transition occurs. Therefore, the correction factors for g-lines
Fig. 3. A simple decay scheme illustrating the concept of coincidence summing correction factors. See discussion in Section 3.
Fig. 4. Technical drawing of the well-type detector of Model GCW6023 provided by the manufacturer.
ARTICLE IN PRESS S.S. Nafee et al. / Applied Radiation and Isotopes 68 (2010) 1746–1753
Table 2 Characteristics of the used well-type HPGe detector. Manufacturer
Canberra Industries
Serial number Detector model Geometry Mounting Relative efficiency P/C ratio Active volume Resolution at 133.5 keV Voltage bias Amplifier ADC MCA and range Shaping time Shaping mode Detector type Correction for dead-time
b 06013 GCW6023 Co-axial open-end Vertical 68.2% 70.2 300 cm3 1.98 keV ( +) 4500 V ICB 9615 ICB 9633 AIM 556 (8192) 4 ms Gaussian HPGe (P-type) LFC- ND 599
area calculation functions. Changes in the peak fits were also made using the interactive peak fit interface when that was necessary for reducing residuals and errors in the peak areas. The peak areas, the live time, the run time and the start time for each spectrum were entered into spreadsheets that were subsequently used to perform calculations necessary for generating efficiency curves. Numerical simulation was used to calculate the coincidence summing factors for correcting the measured full-energy peak efficiencies in order to obtain the true efficiencies of measurements as discussed above.
5. Results and discussion The full-energy peak efficiency values for the p-type HPGe well-type detector were measured as a function of the photon energy using the following equation
eðEÞ ¼ Table 3 Radioactive sources used in the measurements. Composition
Volume
Geometry
(HDPE) (HDPE)
70% 80%
1 ml 1.5 ml
D-type R-type
Geometry
Diameter
Height
Thickness
D-type R-type
0.93 1.13
1.37 1.62
Bottom
Side
0.06 0.08
0.07 0.08
Table 4 Activities of the used radioactive sources. Nuclide
41
Am Cd 57 Co 123m Te 51 Cr 113 Sn 85 Sr 137 Cs 60 Co 88 Y 109
Activity (Bq) 1 mL source
1.5 mL source
Uncertainty (k¼ 3r)
239 2119 80 105 2755 429 518 362 433 815
615 5457 207 269 7093 1104 1333 933 1114 2099
3.0 3.1 3.1 3.1 3.0 3.0 3.0 3.0 3.0 3.0
very small and the angular correlation effects could be neglected ~ (Debertin and Schotzig, 1979). The spectra were acquired with multichannel analyzers and processed with ISO 9001 Genie 2000 data acquisition and analysis software (Canberra Industries). The acquisition time was long enough to get at least 20,000 counts and, thus, keep the uncertainties below 0.5% (Knoll, 2000). The peaks were fitted using Gaussians with low-energy tails appropriate for germanium detectors. In order to avoid dead-time and pile-up effects, the activity of each radionuclide in the sources did not exceed a few Bq ~ (Debertin and Schotzig, 1979). That kept short-distance count rates within acceptable range regardless of the length of the necessary counting times at long distances. Acquired spectra were analyzed with GammaVision software (ORTEC, Oak Ridge, TN) using its automatic peak search and peak
1751
NðEÞ Y Ci , TAS PðEÞ
ð43Þ
where N(E) is the number of counts in the full-energy peak, which could be obtained using the Genie 2000 software, T is the measuring time (in s), P(E) is the photon emission probability at energy E, AS is the radionuclide activity, and Ci are correction factors for the dead-time effect, coincidence summing effect and radionuclide decay. The calibration source decay correction Cd can be obtained from the decay constant l and the interval DT between the reference time and the run time (Pibida et al., 2007): Cd ¼ elDT :
ð44Þ
Count rates were kept below 800 counts/s to avoid pulse pile-up effects. The uncertainty of the full-energy peak efficiency was obtained by uncertainty propagation with an assumption that all measured quantities were independent. Assuming that the only correction made was for the source decay, the uncertainty of the full-energy peak efficiency se is given by equation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 @e @e @e @e @e se ¼ s2N þ s2T þ s2A þ s2P þ s2l , @N @T @A @P @l ð45Þ where sN, sT, sA, sP, and sl, are the uncertainties of the quantities N(E), T, A, P(E), and l, respectively (Pibida et al., 2007). As the measurements were repetitive, the uncertainty in the full-energy peak efficiency se was combined with the statistical component to produce the combined standard uncertainty as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ðe eÞ2 , ð46Þ ue ¼ s2e þ i i nðn1Þ where the second term under the radical is the standard deviation of the mean value of the measured full-energy peak efficiency, n is the number of the measurements, e is the efficiency mean, and se is given by Eq. (45) (Pibida et al., 2007; Perez-Andujar and Pibida, 2004). The uncertainties in the measured efficiencies were close to 1.00% for 60Co gamma lines (1173 and 1332 KeV) and to 1.05% for 88Y gamma lines (898 and 1836 KeV); however, they reached 2.3% for 51Cr and 113Sn gamma lines. Coincidence summing effects in the reference measurement geometries were not negligible. The true full-energy peak efficiencies have been estimated by applying the coincidence correction factors given by Eq. (42). Table 5 lists calculated coincidence correction factors for 60Co and 88Y lines from 1 and 1.5 mL extended sources. In Figs. 5–8, the full-energy peak efficiencies calculated by our method (solid lines) are compared with the measured values (symbols). The calculated efficiencies
ARTICLE IN PRESS S.S. Nafee et al. / Applied Radiation and Isotopes 68 (2010) 1746–1753
Table 5 Calculated coincidence correction factors for sources. 60
Geometry
Co and
88
Y in two extended
88
Co
D-type, 1 mL R-type, 1.5 mL
60
Y
1173.3 keV
1332.65 keV
898.04 keV
1836.42 keV
1.001621 1.001289
1.001883 1.001302
1.001254 1.001267
1.00896 1.007656
Full- Energy peak Efficiency %
1.0
Full- Energy peak Efficiency %
1752
0.8 0.7 0.6 0.5 0.4 0.3 0.2 Measured efficiency values for 1.5mL True calculated efficiency
0.1 0.0
0.8
10
100 Photon Energy (KeV)
1000
0.6 Fig. 7. True full-energy peak efficiencies calculated with our simualtion method (solid line) and the measured efficiencies (dots and intervals) as functions of the photon energy for the R-type source (1.5 mL).
0.4 0.2
4
Measured efficiency values for 1mL True calculated efficiency values 100 Photon Energy (KeV)
1000
Fig. 5. True full-energy peak efficiencies calculated with our simualtion method (solid line) and measured efficiencies (dots and intervals) as functions of the photon energy for the D-type source (1 mL).
Discrepancy %
10
D-Type
3
0.0
2 1 0 -1 -2
4
Discrepancy %
-3
D-Type
3
-4 0
2 1
200 400 600 800 1000 1200 1400 1600 1800 2000 Photon Energy (KeV)
Fig. 8. Percent difference (D%) between the measured and the calculated true full-energy peak efficiencies as a function of the photon energy for the R-type geometry.
0 -1 -2 -3 -4 0
200 400 600 800 1000 1200 1400 1600 1800 2000 Photon Energy (KeV)
Fig. 6. Percent difference (D%) between the measured and the calculated true full-energy peak efficiencies values as a function of the photon energy for the Dtype geometry.
were obtained using Eq. (33), in which the self-absorption factor Sf, the photon attenuation fatt, and the volumes of the sources were used to get the final corrected values. The discrepancies between the calculated and the measured true full-energy peak efficiencies were calculated as follows:
D% ¼
ecalculated emeasured 100: ecalculated
ð47Þ
The actual measured photopeak efficiencies for 60Co and 88Y gamma lines, calculated as the number of counts under the fullenergy peak divided by the emission probability, are listed in Table 5 alongside the values calculated by us. The actual measured photopeak efficiencies for 1173 and 898 KeV gamma lines are greater than those for 1332 and 1836 KeV gamma lines,
respectively. This is in line with the higher number of counts under the photopeak. Table 5 shows discrepancies below 3% and below 2.5% for the R- and D-type sources, respectively. This means that the volume of the source has a noticeable effect on the actual measured photopeak efficiencies. The larger the volume of the source, the higher the discrepancies between the actual photopeak efficiencies and the calculated ones. This is consistent with the results of our mathematical modeling for complex and large extended sources reported in other publications (Pibida et al., 2007; Nafee and Abbas, 2008; Nafee et al., 2009). Tables 5 and 6 show that the coincidence corrections greatly affect the count rate under the photopeak and, consequently, the efficiency values. A correlation analysis has been performed using the GraphPad Prism program (Version 5.0) (Motulsky, 2007). We have found a negative correlation between the full-energy peak efficiencies and the coincidence correction factors. The correlation coefficients rxy calculated with equation n 1 X xi x yi y ð48Þ rxy ¼ n1 i ¼ 1 sx sy were approximately 0.8 and 0.75 for the D- and R-type sources, respectively. In other words, the lower the coincidence correction factor, the better the detection efficiency.
ARTICLE IN PRESS S.S. Nafee et al. / Applied Radiation and Isotopes 68 (2010) 1746–1753 Table 6 Actual photopeak efficiencies for 60
Co
88
Y
60
Co and
Gamma line (keV)
1173.3 1332.65 898.04 1836.42
1753
88
Y in two extended sources. [Counts/(emission probability)] measured
[Counts/(emission probability)] calculated
R-type, 1.5 mL
D-type, 1 mL
R-type, 1.5 mL
D-type, 1 mL
4.28E + 6 3.86E + 6 2.45E + 6 9.95E + 5
1.97E + 6 1.63E + 6 1.07E +6 4.75E + 5
4.38E + 6 3.77E + 5 2.38E + 6 1.02E +6
1.94E + 6 1.65E + 6 1.08E +6 4.65E + 5
The discrepancies between the true measured efficiency values and the calculated ones for 60Co gamma lines (1173 and 1332 KeV) were 1.79% and 1.74%, respectively, for the D-type geometry and 2.30% and 2.38, respectively, for the R-type geometry. This is because the two gamma lines have nearly the same emission probability and the same coincidence correction factors, as shown in Table 5. However, the discrepancies are much bigger for 88Y gamma lines (1.5% and 3.3% for the D-type while 2.84% and 1.91% for the R-type, respectively). This can be attributed to different emission probabilities for the two lines, which result in different coincidence correction factors listed in Table 5. As can be seen from Figs. 6 and 8, the discrepancies vary between 71.5% and 73.3%. This can be explained by the variations in the calculations of the full-energy peak attenuation coefficients.
6. Conclusions Numerical simulation has been successfully used to generate the full-energy peak efficiency curve for the co-axial p-type HPGe well-type detector calibrated with cylindrical sources inside the detector’s cavity. Discrepancies between the calculated and experimental results were below 3.5%. In addition, the coincidence correction and the source self-absorption factors for the efficiency values have been calculated. This technique is good for calibrations of 4p counting systems with low-activity radioactive sources.
Acknowledgement The authors would like to thank the Authorities of the Belgium Nuclear Research Center (SCK.CEN), Laboratory for Gamma-ray Spectrometry, Boeretang 200, B-2400 Mol, Belgium, for allowing M.S. Badawi to carry out the experimental measurements. References Abbas, M.I., 2001a. HPGe detector photopeak efficiency calculation including selfabsorption and coincidence corrections for Marinelli beaker sources using compact analytical expressions. Appl. Radiat. Isot. 54, 761. Abbas, M.I., 2001b. A direct mathematical method to calculate the efficiencies of a parallelepiped detector for an arbitrarily positioned point source. Radiat. Phys. Chem. 60, 3. Abbas, M.I., 2001c. Analytical formulae for well-type NaI(Tl) and HPGe detectors efficiency computation. Appl. Radiat. Isot. 55, 245. Abbas, M.I., Selim, Y.S., Bassiouni, M., 2001. HPGe detector photopeak efficiency calculation including self-absorption and coincidence corrections for cylindrical sources using compact analytical expressions. Radiat. Phys. Chem. 61, 429. Abbas, M.I., Selim, Y.S., 2002. Calculation of relative full-energy peak efficiencies of well-type detectors. Nucl. Instrum. Method A 480, 651. Abbas, M.I., 2006a. Analytical calculations of the solid angles subtended by a welltype detector at point and extended circular sources. Appl. Radiat. Isot. 64, 1048. Abbas, M.I., 2006b. Validation of analytical formulae for the efficiency calibration of gamma detectors used in laboratory and in-situ measurements. Appl. Radiat. Isot. 64, 1661.
Blaauw, M., 1993. The use of sources emitting coincident g-rays for determination of absolute efficiency curves of highly efficient Ge detectors. Nucl. Instrum. Method A 332, 493. Blaauw, M., 1998. Calibration of the well-type germanium gamma-ray detector employing two gamma-ray spectra. Nucl. Inst. Method A 419, 146. Canberra Industries web page, /http://www.canberra.com/products/839.aspS. ~ Debertin, K., Schotzig, U., 1979. Coincidence summing corrections in Ge(Li)spectrometry at low source-to-detector distances. Nucl. Instrum. Method 158, 471. Hamzawy, A., 2004. Accurate direct mathematical determination of the efficiencies of gamma detectors arising from radioactive sources of different shapes. Ph.D. Dissertation. Alexandria University, Egypt. Hubbell, J.H., Seltzer, S.M., 1995. Tables of X-ray mass attenuation coefficients a nd mass energy absorption coefficients 1 KeV to 20 MeV for elements Z¼1 to 92 and 48 additional substances of dosimetric interest. NISTIR 5632, USA. Knoll, F.G., 2000. Radiation Detection and Measurement 3rd ed. Wiley, New York 802pp. Laorie, J.M., Le Petit, G., Abt, D., Girard, M., 2002. Monte Carlo calculation of the efficiency response of low-background well-type HPGe detector. Nucl. Instrum. Method A 479, 618. McCallum, G.J., Coote, G.E., 1975. Influence of source-detector distance on relative intensity and angular correlation measurements with Ge(Li) spectrometers. Nucl. Instrum. Method 130, 189. Morel, J., Chauvenet, B., Kadachi, A., 1983. Coincidence summing corrections in gamma-ray spectrometry for normalized geometries. Int. J. Appl. Radiat. Isot. 34, 1115. Motulsky, H.J., 2007. GraphPad Prism 0.5 Software, GraphPad Sotware, Inc. Nafee, S.S., Abbas, M.I., 2008. Calibration of closed-end HPGe detectors using bar (parallelepiped) sources. Nucl. Instrum. Method A 592, 80. Nafee, S.S., Badawi, M.S., Abdel-Moneim, A.M., Mahmoud, S.A., Abbas, M.I., 2009. An analytical approach for the calibration of HPGe cylindrical detector using an inverted well-beaker. ANS Trans. 101, 645. Perez-Andujar, A., Pibida, L., 2004. Performance of CdTe, HPGe and NaI(Tl) detectors for radioactivity measurements. Appl. Radiat. Isot. 60, 41. Pibida, L., Nafee, S.S., Unterweger, M., Hammond, M.M., Karam, L., Abbas, M.I., 2007. Calibration of HPGe gamma-ray detectors for measurement of radioactive noble gas sources. Appl. Radiat. Isot. 65, 225. Piton, F., Lepy, M.C., Be, M.M., Plagnard, J., 2000. Efficiency transfer and coincidence summing corrections for gamma-ray spectrometry. Appl. Radiat. Isot. 52, 791. Selim, Y.S., Abbas, M.I., 1994. Source-detector geometrical efficiency. Radiat. Phys. Chem. 44, 1. Selim, Y.S., Abbas, M.I., 1995. Direct calculation of the total efficiency of cylindrical scintillation detectors for non-axial point sources. Egypt. J. Phys. 26, 79. Selim, Y.S., Abbas, M.I., 1996. Direct calculation of the total efficiency cylindrical scintillation detectors for extended circular sources. Radiat. Phys. Chem. 48, 23. Selim, Y.S., Abbas, M.I., Fawzy, M.A., 1998. Analytical calculation of the efficiencies of gamma scintillators. Part I: Total efficiency of coaxial disk sources. Radiat. Phys. Chem. 53, 589. Selim, Y.S., Abbas, M.I., 2000. Analytical calculations of gamma scintillators efficiencies. Part II: Total efficiency for wide coaxial disk sources. Radiat. Phys. Chem. 58, 15. Selim, Y.S., Elzaher, M.A., El-Katib, A.M., Bassiouny, M.M., 2007. Contribution of pair production process in full-energy peak efficiency calculation of semiconductor detectors for axial point sources with high-energy gamma-rays. Radiat. Phys. Chem. 76, 1. Sima, O., Arnold, D., 1996. Self-attenuation and coincidence summing corrections calculated by Monte Carlo simulations for gamma-spectrometric measurements with well-type germanium detectors. Appl. Radiat. Isot. 47, 889. Sima, O., 2000. Accurate calculation of the total efficiency of Ge well-type detectors suitable for efficiency calibration using common standard sources. Nucl. Instrum. Method A 450, 98. Wang, T.K., Hou, I.M., Tseng, C.L., 1999. Well-type HPGe detector absolute peakefficiency calibration and true-coincidence correction. Nucl. Instrum. Method A 425, 504. Xilei, L., Heydorn, K., 1991. Correction for X- and g-Ray Coincidence Effect in the Efficiency Calibration of an n-Type Germanium Detector. Risø-R-593 National Laboratory, Roskilde, Denmark.