On the stochastic dependence between photomultipliers in the TDCR method

Applied Radiation and Isotopes 70 (2012) 770–780

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On the stochastic dependence between photomultipliers in the TDCR method C. Bobin n, C. Thiam, B. Chauvenet, J. Bouchard CEA, LIST, Laboratoire National Henri Becquerel (LNE-LNHB), F-91191 Gif-sur-Yvette Cedex, France

a r t i c l e i n f o

a b s t r a c t

Article history: Received 28 July 2011 Received in revised form 2 December 2011 Accepted 15 December 2011 Available online 31 December 2011

The TDCR method (Triple to Double Coincidence Ratio) is widely implemented in National Metrology Institutes for activity primary measurements based on liquid scintillation counting. The detection efficiency and thereby the activity are determined using a statistical and physical model. In this article, we propose to revisit the application of the classical TDCR model and its validity by introducing a prerequisite of stochastic independence between photomultiplier counting. In order to support the need for this condition, the demonstration is carried out by considering the simple case of a monoenergetic deposition in the scintillation cocktail. Simulations of triple and double coincidence counting are presented in order to point out the existence of stochastic dependence between photomultipliers that can be significant in the case of low-energy deposition in the scintillator. It is demonstrated that a problem of time dependence arises when the coincidence resolving time is shorter than the time distribution of scintillation photons; in addition, it is shown that this effect is at the origin of a bias in the detection efficiency calculation encountered for the standardization of 3H. This investigation is extended to the study of geometric dependence between photomultipliers related to the position of light emission inside the scintillation vial (the volume of the vial is not considered in the classical TDCR model). In that case, triple and double coincidences are calculated using a stochastic TDCR model based on the Monte-Carlo simulation code Geant4. This stochastic approach is also applied to the standardization of 51Cr by liquid scintillation; the difference observed in detection efficiencies calculated using the standard and stochastic models can be explained by such an effect of geometric dependence between photomultiplier channels. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Radioactivity metrology TDCR method Liquid scintillation Geant4 code Stochastic independence

1. Introduction Developed for radionuclide standardization using liquid scintillation, the TDCR (Triple to Double Coincidence Ratio) method is applied using coincidence counting obtained with a specific system composed of three photomultiplier tubes (PMT) (Broda et al., 2007). At LNE-LNHB, double and triple coincidences between PMTs have been processed so far by the MAC3 module (Bouchard and Cassette, 2000). In order to avoid the excess counting due to after-pulses that characterize liquid scintillation, this home-made device is based on the live-time technique using extendable dead-times. Applying the same process, an FPGA-based digital system has been recently developed at LNE-LNHB (Bobin et al., 2010a); an adjustable resolving time has also been implemented in the digital device. In 2009, the digital device was used for the standardization of 3 H in the framework of an international comparison. The influence of the resolving time on coincidence counting was also investigated. A significant variation of coincidence counting was observed when the resolving time was increased beyond the

n

Corresponding author. Tel.: þ33 1 69 08 29 64; fax: þ 33 1 69 08 26 19. E-mail address: [email protected] (C. Bobin).

0969-8043/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2011.12.035

value set to 40 ns in the MAC3 module. In addition, it appears that the activity calculation obtained with the classical TDCR model does not completely compensate for these counting differences. From measurements carried out according to various resolving times, a residual difference of about 0.5% was obtained between the activities calculated at 40 ns and 250 ns (Bobin et al., 2010a). The explanation proposed here for this effect is that when the number of scintillation photons generated per disintegration is low (as for low-energy emitters like 3H), coincidence counting becomes sensitive to the time distribution of photons between PMTs. Additional measurements carried out with various radionuclides are presented in this article to strengthen the experimental evidence of this assumption. In order to investigate the influence of the resolving time on the classical TDCR model, probabilistic relations are developed in the simple case of a monoenergetic deposition in the scintillation cocktail. The assumptions underlying the classical TDCR model are revisited by introducing a condition of stochastic independence between PMT channels. From this probabilistic approach, it is assumed that a problem of time dependence between PMTs is at the origin of the variation of the 3H activity measurement with the resolving-time duration. To confirm this assertion, a simulation of the time distribution of photoelectrons detected in PMTs has been

C. Bobin et al. / Applied Radiation and Isotopes 70 (2012) 770–780

implemented to explore the influence of the resolving time on the variation of coincidence counting and the stochastic dependence between the channels of the TDCR setup. In the classical TDCR model, scintillation photons are distributed between the three PMTs without considering the position of the light emission inside the vial. The possibility of stochastic dependence between PMTs due to geometry is considered for low-energy depositions in the scintillation cocktail. To sustain this assumption, triple and double coincidences are calculated in the case of monoenergetic depositions by simulating the propagation of scintillation photons inside the optical chamber of the TDCR counter. For that purpose, the stochastic modeling based on the Monte-Carlo code Geant4 (Agostinelli et al., 2003; Allison et al., 2006), which was previously implemented for Cherenkov-TDCR measurements, has been extended to liquid scintillation (Thiam et al., 2010a; Bobin et al., 2010b). Simulated coincidences are compared with those given by the monoenergetic classical TDCR model in order to study the impact of geometrical effects inside the vial on the determination of the detection efficiency. The consequence of this phenomenon on the standardization of 51Cr is investigated as an explanation of the difference observed on detection efficiencies calculated using the standard and stochastic models. The TDCR results are also compared with the activity given by 4p(LS)b  g coincidence counting based on liquid scintillation (LS).

2. The TDCR model

771

counting in the different channels: PE ðx1 \ x2 Þ ¼ P E ðx1 ÞP E ðx2 9x1 Þ,

ð1Þ

with PE(x29x1), the conditional probability of a detection in channel 2, given a detection in channel 1 from the same disintegration PE ðx1 \ x2 \ x3 Þ ¼ PE ðx1 ÞPE ðx2 9x1 ÞPE ðx3 9x1 \ x2 Þ,

ð2Þ

with PE(x39x1\x2), the conditional probability of a detection in channel 3, given a double coincidence in channels 1 and 2 from the same disintegration. The general probabilistic relations for a TDCR system are given by expressions (3) and (4) where RT(E) denotes the detection efficiency to triple coincidences and RD(E) the detection efficiency to double coincidences RT ðEÞ ¼ P E ðx1 ÞP E ðx2 9x1 ÞP E ðx3 9x1 \ x2 Þ ¼ P E ðx1 \ x2 \ x3 Þ,

ð3Þ

RD ðEÞ ¼ P E ðx1 \ x2 Þ þP E ðx1 \ x3 Þ þ PE ðx2 \ x3 Þ2P E ðx1 \ x2 \ x3 Þ:

ð4Þ

2.3. Classical relations and their prerequisites The application of the classical TDCR model requires the condition of stochastic independence of the detection of a disintegration in the three counting channels. If the above condition applies, we have the following equalities: PE ðx2 9x1 Þ ¼ P E ðx2 Þ,PE ðx3 9x1 Þ ¼ PE ðx3 Þ and PE ðx2 9x3 Þ ¼ P E ðx2 Þ,

ð5Þ

and 2.1. General background

PE ðx3 9x1 \ x2 Þ ¼ PE ðx3 Þ:

The TDCR method is carried out using a specific detection system based on a symmetrical apparatus with three PMTs surrounding an optical chamber. Triple and double coincidences between PMTs are counted to measure the TDCR value as an experimental indicator of the detection efficiency. For the activity determination, a free-parameter physical and statistical model is constructed to establish a mathematical relation between the experimental TDCR value and the detection efficiency to double coincidences. The usual hypotheses underlying the implementation of the classical TDCR are largely described in the literature (Broda et al., 2007). Knowing the decay scheme of the radionuclide, the TDCR model calculates the number of photoelectrons detected in the PMTs using a Poisson distribution. In addition, Birks’ formula describes the nonlinear light emission due to ionization quenching of the energy transferred in the scintillation cocktail.

In turn, we obtain the following simplified relations for the coincidence detection efficiencies: RT ðEÞ ¼ P E ðx1 ÞP E ðx2 ÞP E ðx3 Þ,

ð6Þ

ð7Þ

RD ðEÞ ¼ P E ðx1 ÞP E ðx2 Þ þ P E ðx2 ÞP E ðx3 Þ þ PE ðx1 ÞP E ðx3 Þ2P E ðx1 ÞP E ðx2 ÞP E ðx3 Þ:

ð8Þ The problem of asymmetry due to non-identical PMTs is treated in the literature (Broda et al., 2007). Since the question of stochastic independence studied below is of another nature, the assumption of the symmetry between the three counting channels is made for the present study and leads to the following equalities: PE ðx1 Þ ¼ P E ðx2 Þ ¼ PE ðx3 Þ ¼ P E ðxÞ:

ð9Þ

As a result, RT(E) and RD(E) can be expressed as a function of the probability PE(x): RT ðEÞ ¼ P E ðxÞ3 ,

ð10Þ

RD ðEÞ ¼ 3P E ðxÞ2 2P E ðxÞ3 :

ð11Þ

2.2. General probabilistic relations of the TDCR model As a first step of this study, it is of interest to establish the general probabilistic relations underlying TDCR measurements before any assumptions that can lead to the classical model. For that purpose, a detection system with three PMTs is considered without any hypothesis concerning its symmetry. Considering the disintegrations of a radionuclide that result in an energy deposition E in the scintillation cocktail, the event ‘‘detection in counting channel i of a disintegration’’ is denoted xi, and PE(xi) is the corresponding probability of occurrence. The joint probability of events x1 and x2 is denoted as PE(x1\x2); this expression corresponds to the detection in the two channels 1 and 2 for the same disintegration in the scintillator, i.e. a double coincidence. Similarly, the joint probability of events x1, x2 and x3 corresponding to the detection of a triple coincidence is denoted as PE(x1\x2\x3). Using the formalism of conditional probabilities, the following classical relations are established when no particular hypothesis is made regarding the stochastic independence between the

For an energy deposited E in the scintillation cocktail, the number of emitted photons is distributed according to a Poisson distribution with a mathematical expectation equal to m(E). The probability PE(x) to count at least one photoelectron is given by expression (12), which is a function of the Poisson null-probability depending on the photocathode quantum efficiency n (in fact, this factor incorporates other features such as the spectral sensitivity) PE ðxÞ ¼ 1expðmðEÞn=3Þ

ð12Þ

The relations obtained so far apply to monoenergetic depositions per disintegration in the scintillation cocktail. When the energy depositions are distributed according to a probability density function S(E), the former relations have to be integrated over the total energy to obtain the final expressions of RT, RD and TDCR as follows: Z Emax SðEÞP E ðxÞ3 dE ð13Þ RT ¼ 0

RD ¼

C. Bobin et al. / Applied Radiation and Isotopes 70 (2012) 770–780

Z

Emax

SðEÞð3PE ðxÞ2 2P E ðxÞ3 ÞdE

ð14Þ

0

TDCR ¼

Z

Emax

.Z SðEÞPE ðxÞ3 dE

0

Emax

SðEÞð3P E ðxÞ2 2PE ðxÞ3 ÞdE

ð15Þ

0

2.4. Specific relations obtained for a monoenergetic TDCR model In the present study on stochastic dependence between PMTs, a simplified classical TDCR model is proposed for the ideal case of monoenergetic depositions in the scintillation cocktail. In that case, the detection efficiencies RT and RD are equal to RT(E) and RD(E), respectively, and P(x) is equal to PE(x). From previous relations (10) and (11), the following expression is obtained: RT =RD ¼ TDCR ¼ PðxÞ=ð322PðxÞÞ,

ð16Þ

leading to a simple analytic relation between RD and TDCR RD ¼ 27TDCR2 =ð1 þ2TDCRÞ3 :

ð17Þ

2.5. Comment on the condition of stochastic independence between photomultipliers It is worth pointing out that the condition of stochastic independence between PMTs has never been explicitly mentioned among the usual assumptions specified for the application of the TDCR model. However, it is equivalent in the 4pb  g coincidence method to the assumption of independence of the beta and gamma counting channels that makes it possible to express the coincidence counting rate as the activity multiplied by the product of the detection efficiencies eb and eg of the beta and gamma channels (Campion, 1959; Baerg, 1966). If this condition is not fulfilled, stochastic dependence between PMTs arises as a consequence of the nonequality between expressions (3) and (7) for triple coincidences and expressions (4) and (8) for double coincidences that entails the nonvalidity of the application of the classical TDCR model. Possible effects that can lead to the loss of independence between counting channels are studied in the following sections. In particular, this problem could be at the origin of the influence of the resolving time in the activity measurement of 3H. To confirm this assertion, counting simulations have been carried out to show that the stochastic dependence between PMTs is sensitive to the variation of the resolving-time duration in the case of low-energy depositions (Section 3). Besides, the influence on the detection efficiency calculation is also investigated using the monoenergetic TDCR model. Based on the simulation code Geant4, a similar methodology has been applied to the problem of geometric dependence between PMTs due to the position of light emission inside the scintillation vial, which is not considered in the standard TDCR model (Section 4).

3. Influence of the resolving time on the stochastic independence between photomultipliers 3.1. Experimental background For a reliable determination of the detection efficiency, the application of the TDCR method requires the equivalence between the settings of the detection system and the classical model used. For instance, the photomultiplier threshold has to be adjusted before the single photoelectron pulse to express the probability P(x) as a simple function of the Poisson null-probability. Depending on the radionuclide to be measured, it has been shown that a systematic rejection of a fraction of single

photoelectron pulses can lead to significant discrepancies in the activity determination (Mo et al., 2006). The duration of the resolving time can also have an influence on TDCR coincidence counting (Steele et al., 2009). In the case of the standardization of 3H, complementary studies revealed that the classical TDCR model does not completely compensate for the observed variation of coincidence counting with the resolvingtime duration (Bobin et al., 2010a; Mo et al., 2010). The proposed explanation of this effect is that when the number of scintillation photons is low due to a low-energy transfer to the liquid scintillator, triple and double coincidence counting becomes sensitive to the time distribution of photons between photomultipliers that, in turn, introduces stochastic dependence between the three TDCR channels. To support this assertion, complementary measurements have been carried out for various radionuclides having different energy emissions, using a digital platform dedicated to TDCR measurements (Bobin et al., 2010a). The results presented in Fig. 1 are obtained for 18F that emits high-energy positrons (Eb þ max.  633 keV) measured in the commercial scintillation cocktail UltimaGold. Unlike the results obtained with 3H in the same scintillator (Bobin et al., 2010a), no particular variation of the double coincidence rate with the resolving time is observed; indeed, a resolving time equal to about 16 ns is sufficient to avoid coincidence losses. An interesting result is also obtained in the case of 99mTc in UltimaGold. The plots in Fig. 2 represent the variation of the coincidence counting rates as a function of the resolving time, taking the result obtained for 40 ns as a reference. The rather odd fact that triple coincidences are less sensitive to the resolvingtime variation is a consequence of the decay scheme of 99mTc. Indeed, this radionuclide decays mainly through a cascade of two gamma transitions (a highly-converted one of about 2 keV followed by a second one of about 140 keV). Double coincidences are mainly due to 2-keV electrons that produce a low number of scintillation photons; therefore, double coincidences are sensitive to the time distribution of photons between PMTs. On the contrary, triple coincidences are predominantly caused by 140-keV photons that produce a large amount of scintillation photons. Based on measurements of 3H (Eb  max.  18.59 keV) in UltimaGold, the plots displayed in Fig. 3 compare the evolution of double coincidence counting rates according to TDCR experimental values obtained on one hand by photomultiplier defocusing and, on the other hand, by decreasing the resolving time. The difference obtained between both procedures clearly shows the influence of 1020 1010 1000 Counts.s-1

772

990 980 970 960 950 940 0

20

40

60

80

100

120

140

160

180

Resolving time / ns Fig. 1. Evolution of experimental double-coincidence counting rates according to increasing resolving times in the case of high-energy depositions given by 18F positron emission in UltimaGold.

Double Coincidence Rate / kCounts.s-1

C. Bobin et al. / Applied Radiation and Isotopes 70 (2012) 770–780

5 0

%

-5 -10 -15

Double coincidences Triple coincidences

-20 -25 0

100

200 300 Resolving time / ns

400

500

Double Coincidence Rate / kCounts.s-1

Fig. 2. Evolution of experimental triple- and double-coincidence counting rates according to increasing resolving times for energy depositions given by 99mTc radiations in UltimaGold. Percentages are calculated using the respective coincidence rate obtained for a resolving time equal to 40 ns as a reference. Double coincidences are mainly produced by 2-keV electrons in the scintillation cocktail.

1850

1800

Photomultiplier defocusing (240 ns) Resolving time variation / focus A Resolving time variation / focus B 240 ns 120 ns

1750

40 ns

16 ns 24 ns 240 ns

1700

160 ns 80 ns

1650

40 ns

16 ns

32 ns 24 ns

1600 0.44

0.46

0.48 TDCR

773

1.16e+4 1.14e+4

Photomultiplier defocusing (240 ns) Resolving time variation

1.12e+4

56 ns

1.10e+4

40 ns 32 ns

1.08e+4

128 ns 24 ns

1.06e+4

112 ns 80 ns

1.04e+4 1.02e+4

16 ns

1.00e+4 9.80e+3 9.60e+3 9.40e+3 0.25

0.26

0.27

0.28 TDCR

0.29

0.30

0.31

Fig. 4. 51Cr measurements in UltimaGold (the associated uncertainties correspond to counting statistics at one standard deviation). Comparison between two techniques to decrease the detection efficiency to double coincidences: on one hand by photomultiplier defocusing (with a resolving time equal to 240 ns) and, on the other hand, by decreasing the resolving time for a fixed focusing voltage.

Furthermore, contrarily to 3H measurements displayed in Fig. 3, the decrease of the resolving-time duration leads to higher detection efficiencies to double coincidences for a same experimental TDCR value. It is worth pointing that the residual difference of 0.5% on the 3 H activity calculation first interpreted by a non-compensation of the TDCR model (Bobin et al., 2010a) is directly observed to have a TDCR value at 40 ns between the counting rates to double coincidences obtained by PMT defocusing and by resolving-time variation. From the different plots displayed in Figs. 3 and 4, it can be assumed that the decrease of the resolving-time duration acts differently on the variation of triple- and double-coincidence rates depending on the energy distribution of the radiation emitted by the radionuclide. This phenomenon could be at the origin of the difference observed between the two procedures applied to reduce the detection efficiency.

0.50

Fig. 3. 3H measurements in UltimaGold (the associated uncertainties correspond to counting statistics at one standard deviation). Comparison between two experimental techniques to alter the detection efficiency to double coincidences: on one hand by photomultiplier defocusing (with a resolving time equal to 240 ns) and, on the other hand, by decreasing the resolving time (according to two different focus voltages denoted A and B).

the resolving time on coincidence counting: on one side, the plots corresponding to photomultiplier defocusing have been obtained using a resolving time equal to 240 ns and on the other side, measurements have been obtained by decreasing the resolving time (down to 16 ns) and two different focusing voltage settings corresponding to, respectively, high (focus A) and intermediate (focus B) detection efficiencies. It can be noted that for the two focus settings, the coincidence measurements obtained with a resolving time equal to 16 ns fit those obtained by PMT defocusing with a resolving time equal to 240 ns. This behavior cannot be generalized because, as depicted in Fig. 4, the evolution of double coincidence counting rates according to TDCR experimental values is different in the case of 51Cr measurements when the resolving time is decreased (this radionuclide is a discrete-energy emitter with maximum energies of x-ray photons and Auger electrons comprised between 4 keV and 6 keV). Indeed, the results obtained by applying both procedures to decrease detection efficiencies show significant differences only for resolving times lower than 32 ns.

3.2. Coincidence calculations based on the time-distribution simulation of photoelectrons in photomultipliers The objective of simulating the time distribution of photoelectrons is to investigate the impact of the resolving-time duration on the stochastic dependence between PMTs in the case of low-energy depositions per disintegration and its consequence on the detectionefficiency calculation. Using MATLABs software, the time-sequence simulation of photoelectrons in PMTs and coincidence calculations are implemented on the base of the monoenergetic TDCR model described in Section 2.4. In order to take into account the variability of the number of emitted photons, the simulation executes successive monoenergetic depositions in the scintillation cocktail from which time sequences of detected photoelectrons are generated for each PMT. The timestamp of the first photoelectron detected in each PMT represents the quantity of interest drawn out from the simulated time sequences. This result is used afterwards to calculate double and triple coincidences between PMTs according to different resolving-time durations. Considered as after-pulses, subsequent photoelectrons detected after the first one created in the same PMT are not taken into account for coincidence calculations. It can be reminded that the problem of after-pulses is properly accounted for by the use of extendable dead times of suitable duration.

774

C. Bobin et al. / Applied Radiation and Isotopes 70 (2012) 770–780

Table 1 Evolution of the stochastic dependence between simulated PMT countings according to increasing resolving times. Resolving time (ns)

PE(x1\x2)

PE(x1)PE(x2)

1  [PE(x1\x2)/PE(x1)PE(x2)] (%)

PE(x1\x2\x3)

PE(x1)PE(x2)PE(x3)

1  [PE(x1\x2\x3)/PE(x1) PE(x2)PE(x3)] (%)

8 16 24 32 40 80 160 240 320

0.2627(2) 0.3367(2) 0.3602(2) 0.3691(3) 0.3735(2) 0.3827(1) 0.3920(1) 0.3961(1) 0.3980(2)

0.2884(2) 0.3472(2) 0.3663(2) 0.3737(3) 0.3774(2) 0.3852(1) 0.3933(1) 0.3966(1) 0.3981(2)

8.9 3.0 1.7 1.2 1.0 0.6 0.3 0.1 o0.1

0.1269(1) 0.1923(1) 0.2151(2) 0.2235(2) 0.2277(2) 0.2364(1) 0.2452(1) 0.2494(1) 0.2510(1)

0.1551(1) 0.2045(1) 0.2217(2) 0.2284(2) 0.2317(2) 0.2390(1) 0.2465(1) 0.2499(1) 0.2511(1)

18.2 6.0 3.0 2.1 1.7 1.1 0.5 0.2 o 0.1

To obtain time sequences of photoelectrons in each PMT per disintegration, the simulation is programmed according to the following description:

 The number of emitted photons and their associated time-

 

stamps are randomly generated according to two possible de-excitation modes corresponding to fast and delayed lifetimes (as usually described for the scintillation process). For each component, the number of photons is defined according to a Poisson distribution; the timestamp associated with each photon is subsequently given by a second trial based on an exponential decay law; Following the classical TDCR model, scintillation photons are distributed between the three PMTs using a multinomial generator with a probability equal to 1/3; From the sequence of photons that enter each PMT, detected photoelectrons are determined using binomial trials with a quantum efficiency n equal to 0.25.

The parameters related to the random distributions used to calculate the number of emitted photons and the timestamps have been defined to fit approximately the experimental behavior of the double coincidences according to increasing resolving time as displayed in Fig. 2. The mean number of emitted photons is set to 12 photons corresponding to a mean value of photoelectrons per PMT equal to 1. Considering the simulation based on Geant4 described in the following section, this number corresponds approximately to an energy deposition of 5 keV. A correct fit is obtained using the following values: in the case of the fast component, a mean number of emitted photons Nfast ¼11 photons with a lifetime equal to 7 ns and for the delayed component, a mean number of emitted photons Ndelayed ¼1 photon with a lifetime equal to 100 ns (the majority of photons are emitted according to the fast component). As shown in Table 1, the existence of a stochastic dependence between channels is clearly revealed by the difference between PE(x1\x2) and PE(x1)PE(x2) for double coincidences and by the difference between PE(x1\x2\x3) and PE(x1)PE(x2)PE(x3) for triple coincidences. These differences diminish and vanish when the resolving time is increased. Taken from the same simulation, Table 2 displays detection efficiencies given by the simulated double coincidences; these values are compared with the calculation results obtained with the analytical expression (17) established for the monoenergetic model using the simulated TDCR value. As for detection probabilities, these results clearly show that the difference between calculated and simulated values vanishes when extending the resolving time. From Tables 1 and 2, a link is distinctly established between the existence of stochastic dependence between PMTs, the discrepancy between calculated and simulated double-coincidence detection efficiencies and the duration of the resolving time. Furthermore, this tendency depends on the mean number of simulated photoelectrons effectively detected per PMT during the resolving time. As expected in the

Table 2 Comparison between simulated detection efficiencies to double coincidences and calculated ones using the monoenergetic TDCR model according to increasing resolving-time durations. Resolving time (ns)

Simulated TDCR

Difference between Calculated Simulated calc. and sim. double double double coincidences coincidences coincidences (%)

8 16 24 32 40 80 160 240 320

0.2374(1) 0.3076(2) 0.3306(3) 0.3386(2) 0.3425(2) 0.3501(1) 0.3576(1) 0.3614(2) 0.3630(1)

0.5346(2) 0.6250(2) 0.6505(2) 0.6601(2) 0.6648(1) 0.6753(1) 0.6856(1) 0.6901(1) 0.6916(2)

0.4744(2) 0.6063(3) 0.6437(5) 0.6561(4) 0.6620(4) 0.6734(2) 0.6842(2) 0.6897(1) 0.6919(2)

11.2 3.0 1.0 0.6 0.4 0.3 0.2 o 0.1 o 0.1

present simulation, this number converges to 1 when extending the resolving-time duration; for instance, it is equal to 0.947 at 40 ns (with a variance of about 0.94) and is equal to 0.736 at 8 ns (with a variance of about 0.68). 3.3. Interpretation As observed in Tables 1 and 2, the existence of time dependence between PMTs cannot be disregarded in the case of low-energy deposition. When the condition of stochastic independence is not fulfilled, this effect can have an impact on the activity determination. As described in Section 2.2, this problem is properly taken into account in the TDCR model when conditional probabilities between PMT channels are used. For instance, as illustrated in Fig. 5, PE(x29x1) is no longer equal to PE(x2) when the resolving time is shorter than the time distribution of photoelectrons. The stochastic dependence is the result of systematic losses of photoelectrons detected in a PMT after the end of the resolving time triggered by another PMT from the same energy deposition. As described in the appendix, the probability of counting in one channel has to be expressed as a function of the resolving-time duration. Compared with the expected behavior given by the classical TDCR model, this phenomenon can affect differently the detection efficiencies to triple and double coincidences.

4. Influence of the geometric dependence between photomultipliers in the TDCR model 4.1. Extension to liquid scintillation of the TDCR-Cherenkov model based on the Geant4 code In the classical TDCR model, the possibility to be affected by geometrical effects on coincidences between PMTs due to the

C. Bobin et al. / Applied Radiation and Isotopes 70 (2012) 770–780

775

1

Time-distribution of photoelectrons detected in channels 1 and 2 2

Short resolving time: Double coincidence lost

First photoelectron detected in channel 2 not counted: P(x2|x1) not equal to P(x2)

Long resolving time: Double coincidence counted

First photoelectron detected in channel 2 counted: P(x2|x1) equal to P(x2) Triggering of the extendable dead-time

Fig. 5. Influence of the resolving time on double-coincidence counting. For a resolving time shorter than the time distribution of photoelectrons in each PMT, a double coincidence can be lost when the photoelectrons generated in channel 2 are not counted. This systematic effect leads to a problem of stochastic dependence between photomultipliers expressed by the inequality between PE(x1\x2) and PE(x1)PE(x2).

position of light emission inside the vial is not taken into account. This possibility has been investigated using an extension to liquid scintillation of a Geant4 stochastic model initially developed for activity measurements by means of the TDCR-Cherenkov method. This stochastic TDCR model is based on a complete modeling of the optical-chamber geometry (including the standard borosilicate 20 mL vial) implemented to simulate the propagation of Cherenkov photons from their creation to the production of photoelectrons in photomultipliers. The detector components and their optical properties have already been described in previous studies that include the standardization of 90Y using the TDCR-Cherenkov technique (Bobin et al., 2010b). It is worth noting that the meniscus existing at the interface constituted by the liquid surface and the inner vial wall is also considered in the modeling (Thiam et al., 2010b). In view of extending this stochastic model to liquid scintillation, the borosilicate vial is filled with 10 mL of UltimaGold using the atomic composition available in the literature (Broda et al., 2007). The energy released subsequently by a disintegration is simulated in the whole volume corresponding to the scintillation cocktail. A refractive index equal to 1.5 is applied for the scintillator and a mean wavelength value equal to 430 nm is set for the spectral response of scintillation photons. The refractive index of the bialkali photocathode (XP2020Q PMT) is taken from Harmer et al. (2006). As for the classical TDCR model (Broda et al., 2007), the nonlinearity of light emission due to ionization quenching is simulated using Birks’ formula (with a kB factor equal to 0.01 cm/MeV). The Geant4 implementation of the lowenergy electromagnetic physics models developed for the PENELOPE code was chosen for the transport of electrons, x-ray and g-photons (Baro´ et al., 1995; Apostolakis et al., 1999). Unlike the simulation of Cherenkov photons, the scintillation yield expressed in terms of photons per keV transferred to the scintillation cocktail, cannot be directly calculated. In our case, the scintillation yield in UltimaGold has been experimentally deduced from the detection efficiencies to double coincidences given by the 4pb  g coincidence method applied to 54Mn and 60Co (Campion, 1959). The 4p(LS)b  g detection system used combines the TDCR setup in the b-channel with a HPGe semiconductor detector (Canberra GX1518) in the g-channel; coincidence counting is performed using a livetimed anticoincidence system based on extendable dead-times (Bobin et al., 2007). For both radionuclides 54Mn and 60Co, the detection efficiencies to double coincidences are estimated from the ratio Nc/Ng between the coincidence (Nc) and g-count (Ng) rates using a g-window centered on photopeaks ( 835 keV for 54Mn, 1173 keV and  1332 keV for 60Co).

The procedure used to establish the relation between the scintillation yield and the experimental detection efficiency to double coincidences is implemented differently from the usual TDCR technique. In the case of TDCR-Cherenkov measurements described in a previous work (Bobin et al., 2010b), the calculation of triple and double coincidences is performed by applying a binomial trial to the simulated photoelectrons produced at each PMT photocathode; the binomial generator is based on a focusing parameter representing the probability for a photoelectron to reach the first dynode of the PMT. At least one success is needed to obtain a count in a PMT. To calculate the detection efficiency to double coincidences, the optimization procedure is performed by adjusting the focusing parameter in order to fit the experimental TDCR value. The TDCRCherenkov measurements have shown that the maximum detection efficiency is obtained for a focusing parameter comprised between 0.95 and 1. For the estimation of the scintillation yield in UltimaGold, the optimization procedure has been performed by adjusting this physical parameter in order to obtain the detection efficiency to double coincidences given by 4p(LS)b  g coincidence measurements using a focusing parameter equal to 0.97. 60 Co disintegrates by b  emissions to excited levels of 60Ni; the de-excitation leads mainly to the emission of two g-ray photons in cascade ( 1173 keV and 1332 keV). From 4p(LS)b  g coincidence measurements, the maximum detection efficiency to double coincidences is close to 0.97. With regard to the simulation, this high-detection efficiency is considered as mainly due to the b-decay with electron energies distributed according to a b-spectrum (Eb  max.  317 keV) calculated from the data available in the Table of radionuclides (Be´ et al., 2006). A scintillation yield comprised between 7 and 8 photons/keV is deduced from the optimization procedure using the experimental detection efficiency to double coincidences. 54 Mn disintegrates through electron capture to the  835 keV excited level of 54Cr; the de-excitation leads to the emission of a single g-ray photon. From 4p(LS)b  g coincidence measurements, the maximum detection efficiency given by the Nc/Ng ratio is equal to about 0.54. For the detection-efficiency calculation according to the scintillation yield with the Geant4 model, the energy released in the scintillator (x-ray photon and Augerelectron maximum energies comprised between 4 keV and 6 keV) is calculated using the KLM rearrangement model described in the literature (Kossert and Grau Carles, 2006; Broda et al., 2007). The data relative to the 54Mn decay scheme are taken from the Table of radionuclides (Be´ et al., 2006). Because the detection efficiency is measured by coincidence counting

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using a g-window centered on a photopeak, the energy transferred by the 835 keV gamma transition in the scintillation cocktail is not simulated. From the Geant4 calculation, a scintillation yield comprised between 8 and 8.5 photons/keV is obtained. Due to the nonlinearity of the light emission for low-energy deposition, 54Mn calculations are more sensitive to the scintillation yield estimation; consequently, a scintillation yield equal to 8.2 (3) photons/keV is adopted for the calculations presented in the next sections. It is worth noting that this estimation is in agreement with the experimental data available in the literature (Fuchs and Laustriat, 1970). Furthermore, the scintillation yield has been tested by comparing the TDCR and 4p(LS)b  g coincidence activity measurement results of 60Co. For that purpose, the simulation of g-photon interactions in the detector (scintillation cocktail, glass vial, photomultiplier windows) generated by the cascade of the two gamma transitions has been added to the b-emission in the Geant4 model. From the experimental TDCR value equal to 0.971, the detection efficiency to double coincidences is calculated using the same procedure as described for the standardization of 90Y (Bobin et al., 2010b). The activity equal to 3343 (4) Bq given by the Geant4 TDCR model is in good agreement with the result equal to 3346 (3) Bq obtained with the 4p(LS)b  g coincidence method. 4.2. Existence of geometric dependence between photomultipliers for monoenergetic depositions From the Geant4 simulation applied to different monoenergetic depositions (5 keV, 8 keV, 12 keV and 20 keV), the probabilities PE(x1\x2), PE(x1)PE(x2), PE(x1\x2\x3) and PE(x1)PE(x2)PE(x3) are calculated to investigate the influence of geometry on the stochastic dependence between PMT channels. As for the study of the influence of the resolving time, the TDCR and the detection

efficiency to double coincidences calculations are also performed for comparison with the result given by the analytical monoenergetic TDCR model. The different calculations are conducted with a scintillation yield equal to 8.2 photons/keV; a focusing parameter equal to 0.97 is used to determine the number of photoelectrons that reach the first dynode of each PMT. As a first step, monoenergetic electrons are randomly generated in the whole volume corresponding to the scintillation cocktail; in the second step, the emission of monoenergetic electrons is limited to the vial height axis at the center of the scintillator volume. The calculations obtained for electrons deposited in the whole volume are displayed in Table 3. The differences between PE(x1\x2) and PE(x1)PE(x2) for double coincidences and between PE(x1\x2\x3) and PE(x1)PE(x2)PE(x3) for triple coincidences clearly reveal a problem of stochastic dependence. These differences diminish for increasing energies and vanish for E¼ 20 keV. Linked to this effect, the calculated and simulated detection efficiencies to double coincidences show an increasing discrepancy with decreasing energy. It has to be noted that contrarily to the effect of time dependence, calculated detection efficiencies with the monoenergetic TDCR expression (17) are higher than those simulated. Moreover, despite the stochastic independence between PMTs obtained for an energy deposit of 20 keV, the significant difference between the mean number of simulated photoelectrons per PMT and its variance does not comply with the hypothesis of a distribution described by a Poisson distribution. However, a good agreement between the simulated and calculated detection efficiencies to double coincidences is obtained due to the large number of photoelectrons created in each PMT. In Table 4, the same calculations are displayed when the emission of monoenergetic electrons is limited to the height axis at the center of the scintillator volume. Most of the differences

Table 3 Application of the Geant4 TDCR model for energy depositions due to monoenergetic electrons in the whole scintillator. Energy deposit in the LS volume

E ¼5 keV

E ¼8 keV

E ¼12 keV

E ¼20 keV

Simulated TDCR value Sim. double coinc. detection efficiency Calc. double coinc. detection efficiency Relative deviation (%) PE(x1\x2) PE(x1)PE(x2) 1 [PE(x1\x2)/PE(x1)PE(x2)] (%) PE(x1\x2\x3) PE(x1)PE (x2)PE(x3) 1 [PE(x1\x2\x3)/PE(x1)PE(x2)PE(x3)] (%) Mean number of photoelectrons in PMTs Variance Relative deviation (%)

0.3273 (4) 0.6094 (6) 0.6386 (8) 4.8 0.3364 (4) 0.3315 (4) 1.5 0.1995 (3) 0.1908 (3) 4.6  0.884  0.994  12

0.5587 (5) 0.8682 (7) 0.8878 (7) 2.3 0.6130 (6) 0.6086 (6) 0.73 0.485 (4) 0.475 (4) 2.1  1.6  1.8  12

0.7765 0.9719 0.9783 0.66 0.8271 0.8252 0.23 0.7547 0.7496 0.68  2.65  3.1  17

0.9491 0.9985 0.9990 o 0.1 0.9644 0.9642 o 0.1 0.9476 0.9468 o 0.1  4.8  6.4  33

(6) (7) (8) (7) (7) (6) (6)

(7) (7) (7) (7) (7) (7) (7)

Table 4 Application of the Geant4 TDCR model for the emission of monoenergetic electrons limited to the height axis at the center of the scintillator volume. Energy deposit along the central axis

E ¼ 5 keV

E ¼8 keV

E ¼12 keV

E ¼20 keV

Simulated TDCR Sim. double coinc. detection efficiency Calc. double coinc. detection efficiency Relative deviation PE(x1\x2) PE(x1)PE(x2) 1  [PE(x1\x2)/PE(x1)PE(x2)] PE(x1\x2\x3) PE(x1)PE(x2)PE(x3) 1  [PE(x1\x2\x3)/PE(x1)PE(x2)PE(x3)] Mean number of photoelectrons in PMTs Variance Relative deviation

0.4076 (5) 0.7492 (6) 0.7450 (9)  0.1% 0.4530 (5) 0.4530 (5) Negligible 0.3054 (5) 0.3050 (5)  0.1%  1.19  1.2  1%

0.6874 (6) 0.9520 (7) 0.9523 (7) Negligible 0.7537 (6) 0.7535 (6) Negligible 0.6544 (6) 0.6540 (6) Negligible  2.03  2.03 Negligible

0.9000 (7) 0.9962 (7) 0.9963 (7) Negligible 0.9303 (7) 0.9303 (7) Negligible 0.8974 (7) 0.8975 (7) Negligible  3.34  3.35 Negligible

0.9932 (7) 1 1 No 0.9954 (7) 0.9954 (7) Negligible 0.9931 (7) 0.9931 (7) Negligible  6.09  6.11 Negligible

C. Bobin et al. / Applied Radiation and Isotopes 70 (2012) 770–780

5. Standardization of

51

Cr using liquid scintillation

5.1. 4p(LS)b  g coincidence measurements 51 Cr disintegrates though electron capture either to the ground state of 51V (90.11 (5)%) or to its 320-keV excited level (9.89 (5)%) from which the subsequent de-excitation produces a single g-ray photon. Liquid scintillation sources were prepared in standard 20 mL glass vials containing the radioactive solution aliquot mixed with 10 mL of UltimaGold. The 4p(LS)b  g detection setup (Bobin et al., 2007) used for the standardization of 51Cr has already been described in Section 4.1. The detection efficiencies have been estimated from the Nc/Ng ratio calculated using a g-window centered on the photopeak at 320 keV. The extrapolation technique has been performed by defocusing the PMTs (Baerg, 1967); the detection efficiencies to double coincidences corresponding to the different settings of the focus voltages ranged from 0.34 to 0.49. Using the function Nb ¼F(1 Nc/Ng), a linear fit based on least squares has been applied to the results of seventeen measurements corresponding to different b-efficiencies (R2  0.999). From the measurements of six radioactive sources, the activity concentration of 51Cr has been estimated to 22.96 (11) MBq/g at the reference date. The associated uncertainty budget is listed in Table 5. It has to be reminded that regarding the 4pb  g coincidence method, 51Cr is considered as a triangular decay-scheme radionuclide due to the decay branch, which goes directly to the ground state of 51V. Consequently, the extrapolated activity measurement is corrected

Table 5 Uncertainty budget for the oincidence counting.

51

Cr standardization based on 4p(LS)b  g antic-

Uncertainty components Statistics Weighing Background Extrapolation Live-time technique Decay correction Accidental coincidences between PMTs Decay-scheme correction

Activity concentration / kBq.g-1

mentioned above vanish or residuals are not significant. These results indicate that the condition of stochastic independence is fulfilled when the light emission is emitted from the center of the optical chamber. Furthermore, the differences between the mean number of photoelectrons detected in PMTs and its variance are strongly reduced; this result complies with the usual hypothesis in the classical TDCR model that the number of detected photoelectrons follows a Poisson distribution. As for the time dependence described in Section 3.3, the comparison between the results displayed in Tables 3 and 4 confirms the influence of geometrical effects on the distribution of photons between PMTs due to the position of the light emission inside the scintillator volume. Indeed, the consequence on stochastic dependence between PMTs and the related effect on the calculation of the detection efficiency to double coincidences cannot be disregarded for energies lower than 20 keV. This phenomenon can be interpreted as a consequence of systematic effects that depend on the light emission position combined with reflection and refraction processes of scintillation photons at the different interfaces constituting the optical chamber (vial/air, air/PMT-window). The complexity of these processes can be properly accounted for using a stochastic simulation of photon propagation; as a result, the approximation of a light emission from the center of the optical chamber is no longer valid for energy depositions lower than 20 keV. Besides, significant differences (greater than 10%) are obtained between the mean number of photoelectrons per PMT and its variance. This problem of a distribution of the number of photoelectrons that does not comply with a Poisson distribution has been investigated in previous works (Broda et al., 2007). However, when the number of simulated photoelectrons per photomultiplier is high (as for E¼20 keV), this effect does not have a significant impact on the activity determination. Finally, from the above considerations, it comes out that the stochastic independence between PMTs (as required by the classical TDCR model) is achieved with an isotropic light emission from the center of the optical chamber.

777

% Uniform distribution applied on 6 sources 0.2 Gravimetric measurements using the 0.1 pycnometer method 0.2 Linear fit applied on 17 coincidence 0.35 measurements Frequency clock used for the live-time 0.01 technique 0.1 T1/2(51Cr): 27.703 (3) d Conservative estimation 0.1 PK correction on the extrapolation 0.05 technique Relative combined standard uncertainty 0.48

23100 23050 23000 22950 22900 22850 0.26

0.27

0.28

0.29

0.30

0.31

TDCR Fig. 6. 51Cr activity concentrations calculated with the Geant4 TDCR model according to different TDCR experimental values (the associated uncertainties correspond to counting statistics at one standard deviation).

for by a factor equal to 0.999 due to the difference between the K-shell electron-capture probabilities PK of each decay branch (Funck and Nylandstedt Larsen, 1983). 5.2. TDCR activity measurements using the Geant4 model The same sources were measured using the TDCR method. Based on the experience resulting from 3H activity measurements (Bobin et al., 2010a), the resolving time is set to 240 ns. Displayed in Fig. 4, the behavior of double coincidence counting rates according to the experimental TDCR value is discussed in Section 3.1 (the experimental TDCR values are comprised between 0.26 and 0.31). The counting measurements are obtained by PMT defocusing applied to three different radioactive sources. As described in Section 4.1, the Geant4 model is based on an extension to liquid scintillation of a Geant4 model previously developed for activity measurements by means of Cherenkov counting (Bobin et al., 2010b). It is reminded that the link between the experimental TDCR value and the simulated detection efficiency to double coincidences is obtained using binomial trials based on a focusing parameter applied to simulated photoelectrons in PMTs. The simulation of the energy transfer in the liquid scintillator is based on the KLM rearrangement model (Broda et al., 2007). The corresponding parameters are calculated from the 51Cr decay data available in the Table of radionuclides (Be´ et al., 2006). The activity concentrations displayed in Fig. 6 are calculated with the Geant4 model from the double coincidence counting

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Table 6 Uncertainty budget for the 51Cr standardization based on TDCR measurements using the Geant4 model for the determination of the activity. Uncertainty components Statistics Weighing Background Geant4 model Live-time technique Decay correction PMT asymmetry kB factor Decay-scheme parameters

%

Uniform distribution applied on 3 sources Gravimetric measurements using the pycnometer method Variation of the parameters influencing the photoelectrons production Frequency clock used for the live-time technique

0.2 0.1

This underestimation of the activity measurements can be related to the observations reported by Simpson et al. (2010) for the standardization of 55Fe (which is also a discrete low-energy emitter). Using the classical TDCR model, a higher activity concentration was obtained with polyethylene vials compared with glass vials. This result can be ascribed to the diffusing property of polyethylene vials that modifies reflection and refraction processes occurring at the wall/air interface of glass vials.

0.05 0.4 0.01

T1/2(51Cr): 27.703 (3) d Conservative estimation obtained from the Geant4 model Variation of the kB value between 0.01 cm/MeV and 0.008 cm/MeV Variation of the PK and oK values

0.1 0.2

Relative combined standard uncertainty

0.53

0.1 0.1

rates and their associated TDCR values presented in Fig. 4. The calculated detection efficiency to double coincidences corresponding to the maximum TDCR value (approximately equal to 0.305) is found to be equal to 0.496 (kB ¼0.01 cm/MeV). From the measurements of five sources, the activity concentration for the 51 Cr solution (22.98 (12) MBq/g at the reference date) is in good agreement with the result given by 4p(LS)b  g coincidence counting. The uncertainty budget is listed in Table 6. The component related to the Geant4 model is obtained by the variation of the distance of the photomultiplier windows to the center of the optical chamber (16–17 mm). A difference of about 0.1% is observed when varying the kB value to 0.008 cm/MeV. 5.3. Comparison with the classical TDCR model To be compared with the results presented above, a classical TDCR model based on a free parameter was constructed for the standardization of 51Cr. The main characteristics of the classical model are described in Section 2. The same decay parameters for the KLM rearrangement model as those implemented in the Geant4 model are used. The detection efficiencies to x-ray photons and the 320-keV g-spectrum in the scintillation cocktail are calculated using the Geant4 model of the optical chamber. Using the same measurements as those treated with the Geant4 model, the activity concentration of the 51Cr solution was found equal to 22.05 (12) MBq/g at the reference date with the classical TDCR model. In order to investigate the origin of the significant difference (equal to about 4%) with the results obtained by both the 4p(LS)b  g coincidence method and the Geant4 model, the same methodology described for the problem of geometric dependence between PMTs is applied. Most of the radiation emitted subsequently to 51Cr desintegrations is constituted of low-energy Auger electrons. The Geant4 model developed for the 51Cr standardization has been modified in order to limit the emission of those Auger-electrons along the height axis of the scintillator vial (for simulation convenience, the emission of photons remains implemented in the whole volume). The activity concentration given by the modified Geant4 simulation is equal to 22.46 (12) MBq/g. As described in Section 4.2, this result confirms the tendency of underestimating the activity for low-energy emitters when the approximation of an isotropic light emission from the center of the optical chamber is applied.

6. Conclusion The validity of the classical TDCR model for the standardization of low-energy emitters has already been studied (Broda and J˛eczmieniowski, 2004; Broda et al., 2007). Different probabilistic distributions describing the number of photoelectrons emitted in PMTs were investigated as an alternative to the Poisson distribution. For that purpose, binomial and Polya distributions were implemented in the TDCR model (with the drawback of introducing an additional free parameter). An alternative is proposed in this article. Supported by additional experimental evidences, the influence of the resolving time on double coincidence rates is clearly confirmed for low-energy emitters (3H, 99mTc and 51Cr). In addition, the consequence on the activity determination is shown through the differences observed between two techniques performed to decrease the detection efficiency: on one hand, by PMT defocusing and on the other hand, by reducing the resolving time. An interpretation of this problem is proposed by revisiting the classical TDCR model thanks to the introduction of stochastic independence between PMTs. This condition of stochastic independence between counting channels was never explicitly expressed for the valid application of the classical TDCR model. In the case of coincidence calculations, the influence of short resolving times on the stochastic dependence between channels has been clearly shown when a low number of photoelectrons per PMT is detected. In addition, the stochastic dependence between PMTs is linked to biases found for calculated detection efficiencies to double coincidences as given by the classical TDCR model. This phenomenon can explain the discrepancies encountered for the activity determination of 3H when decreasing the resolving time (Bobin et al., 2010a). Geometrical effects on the activity determination related to the position of light emission inside the scintillator volume are investigated. Geometric dependence between PMTs has been studied using a stochastic TDCR model based on the Geant4 code that allows the simulation of the photon propagation in the optical chamber of the TDCR detection setup (including the vial filled with the scintillation cocktail). As for time dependence, a link between geometric dependence between PMTs and discrepancies for calculated detection efficiencies to double coincidences with the monoenergetic TDCR model is observed. This problem of geometric dependence between PMTs is interpreted as a consequence of the position of light emission combined with reflection and refraction processes of scintillation photons at the different interfaces constituting the optical chamber. The same tendency of overestimating the detection efficiency to double coincidences is observed when standardizing 51Cr with the classical TDCR model whereas a good agreement has been achieved between both 4p(LS)b  g coincidence counting and the TDCR model based on Geant4. As a consequence of geometrical effects in the liquid scintillation vial, significant differences between the mean number of simulated photoelectrons per PMT and its variance are observed. This phenomenon does not comply with the Poisson distribution usually implemented in the classical TDCR model. As already

C. Bobin et al. / Applied Radiation and Isotopes 70 (2012) 770–780

mentioned, this problem has been studied as a consequence of a low mean number of photoelectrons per PMT (Broda and J˛eczmieniowski, 2004). On the contrary, Geant4 simulations show that the difference does not vanish when increasing the energy deposition in the scintillation cocktail. However, the impact on the stochastic dependence as well as on the calculation of the detection efficiencies to double coincidences is not significant due the increasing mean number of simulated photoelectrons in PMTs. Finally, it is shown that the condition of stochastic independence between PMTs has to be fulfilled in order to properly apply the classical TDCR model. Indeed, the present study points out that this condition is not always verified when standardizing low-energy emitters. Consequently, the experimental settings have to be adapted to the radionuclide to be standardized in order to match the classical TDCR model. For instance, in the case of 3H, the resolving-time duration has to be set to a value that depends on the fluorescence lifetimes corresponding to the scintillation cocktail used (about 200 ns in the case of UltimaGold). Regarding the geometric dependence between photomultipliers, it is shown that the stochastic independence is achieved when the light emission comes from the center of the optical chamber. Complementary studies with the Geant4 model are underway in order to find out the best conditions to keep valid this approximation.

Appendix A In order to illustrate in a simple way the problem of time dependence between photomultipliers in the classical TDCR method, the calculation of the detection probabilities is developed hereafter in the case of a symmetrical setup composed of 2 photomultiplier tubes (PMT). For the sake of simplicity, the time distribution of photon emissions following energy depositions in the scintillator is described by an exponential decreasing law of constant l. The same notations as in Section 2 are used. It is first assumed that the number of photons produced per energy deposit E follows a Poisson distribution of mathematical expectation m; in addition, these emitted photons are equiprobably distributed between the two PMTs according to a multinomial distribution of parameters (1/2, 1/2) and the subsequent creation of photoelectrons follows a binomial distribution of probability n. From these hypotheses, the probability of getting x photoelectrons in channel 1 and y photoelectrons in channel 2 is then equal to the following expression: ðmn=2Þx þ y mn e x!y!

ðA:1Þ

The first photoelectron detected in one of the 2 PMTs triggers the coincidence resolving time. Then the condition for detecting the energy deposit in the other channel is that at least one photoelectron has to be created in this channel before the end of the resolving time. Therefore, when x photoelectrons are created in a channel, the probability for the first one to occur within t and t þdt period after the energy deposit can be written as lt x

ðe

lxt

Þ lx dt ¼ e

lx dt:

ðA:2Þ

In turn the probability, when y photoelectrons are created in channel 2, for the first one to occur within the resolving time t triggered by channel 1 at time t after the energy deposit, is equal to Z tþt 0 ðelt Þy ly dt 0 ¼ elyt ð1elyt Þ: ðA:3Þ t

779

The probability of detecting a photoelectron in channel 1 includes two cases: (1) first photoelectron created in channel 1, (2) first photoelectron created in channel 2. In the first case, this probability is just equal to the probability for the first photoelectron to be in channel 1 and it is expressed by Z 1 Z 1 0 x : ðA:4Þ ðelt Þx lx ðelt Þy ly dt 0 dt ¼ xþy 0 t In the second case, the first photoelectron has to be created in channel 2, and in addition, at least one of the x photoelectrons in channel 1 has to be produced within the resolving time; the expression of this probability then becomes Z 1 Z tþt y 0 ð1elxt Þ: ðelt Þy ly ðelt Þx lx dt 0 dt ¼ ðA:5Þ x þy t 0 Finally, the probability of detecting a photoelectron in channel 1 is equal to 1

yelxt : x þy

ðA:6Þ

This expression clearly shows a dependence on the number of photoelectrons created in channel 2, which vanishes when the product xt is high enough to make the exponential term negligible. More generally, the detection probability P(x1) of the energy deposited E in channel 1 can be obtained by summing over the total distribution of photoelectrons x and y created in both channels:  1 X 1  X yelxt ðmn=2Þx þ y mn e Pðx1 Þ ¼ 1 xþy x!y! x¼1y¼0 ¼ 1emn=2 

1 X 1 X

y

x¼1y¼1

elxt ðmn=2Þx þ y mn e : xþy x!y!

ðA:7Þ

Similarly, we have in channel 2 Pðx2 Þ ¼ 1emn=2 

1 X 1 X

x

x¼1y¼1

elyt ðmn=2Þx þ y mn e : xþy x!y!

ðA:8Þ

As a result, the product of these two probabilities gives Pðx1 ÞPðx2 Þ ¼ ð1emn=2 Þ2 2ð1emn=2 Þ

1 X 1 X x¼1y¼1

þ

1 X 1 X x¼1y¼1

y

elxt ðmn=2Þx þ y mn e xþy x!y!

y

elxt ðmn=2Þx þ y mn e x þy x!y!

!2 :

ðA:9Þ

The detection of double coincidences occurs when, after the detection of a first photoelectron in one PMT, there is at least one photoelectron created in the other PMT within the resolving-time duration. Thereby the detection probability to double coincidences P(x1\x2) is expressed as following: Pðx1 \ x2 Þ ¼

1 X 1 X xð1elyt Þ þ yð1elxt Þ ðmn=2Þx þ y mn e xþy x!y! x¼1y¼1

¼ ð1emn=2 Þ2 2

1 X 1 X

y

x¼1y¼1

elxt ðmn=2Þx þ y mn e : x þy x!y!

ðA:10Þ

In turn, the difference between P(x1\x2) and P(x1)P(x2) can be calculated as Pðx1 \ x2 ÞPðx1 ÞPðx2 Þ ¼ 2



1 X 1 X

y ðmnelxt =2Þðmn=2Þy 3mn=2 e x þy x!y! x¼1y¼1 1 X 1 X x¼1y¼1

y

elxt ðmn=2Þx þ y mn e x þy x!y!

!2 : ðA:11Þ

This difference tends toward zero when the term exp( lxt) is close to zero. Therefore, the stochastic independence is obtained when the

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resolving time triggered by the first channel is long enough, or when the energy deposited (and thus the mean value of x, mn/2) is high enough, to render negligible the probability that all photoelectrons created in the second channel occur after the resolving time. References Agostinelli, S., et al., 2003. Geant4—a simulation toolkit. Nucl. Instrum. Methods A 506, 250–303. Allison, J., et al., 2006. Geant4 developments and applications. IEEE Trans. Nucl. Sci. 53 (1), 270–549. Apostolakis, J., Giani, S., Maire, M., Nieminen, P., Pia, M.G., Urban, L., 1999. Geant4 Low Energy Electromagnetic Models for Electrons and Photons, CERN-OPEN99-034. Geneva. Baerg, A.P., 1966. Measurement of radioactive disintegration rate by the coincidence method. Metrologia 2, 23–32. Baerg, A.P., 1967. Absolute measurement of radioactivity. Metrologia 3 (4), 105–108. Baro´, J., Sempau, J., Ferna´ndez-Varea, J.M., Salvat, F., 1995. PENELOPE: an algorithm for Monte Carlo simulation of the penetration and energy loss of electrons and positrons in matter. Nucl. Instrum. Methods B 100, 31–46. Be´, M.M., Chiste´, V., Dulieu, C., Browne, E., Baglin, C., Chechev, V., Kuzmenko, N., Helmer, R., Kondev, F., MacMahon, D., Lee, K.B., 2006. Table of radionuclides. Monographie, 3. Bobin, C., Bouchard, J., Hamon, C., Iroulart, M.G., Plagnard, J., 2007. Standardization of 67 Ga using a 4p(LS)b  g anti-coincidence system. Appl. Radiat. Isot. 65, 757–763. Bobin, C., Bouchard, J., Censier, B., 2010a. First results in the development of an online digital counting platform dedicated to primary measurements. Appl. Radiat. Isot. 68, 1519–1522. Bobin, C., Thiam, C., Bouchard, J., Jaubert, F., 2010b. Application of a stochastic TDCR model based on Geant4 for Cherenkov primary measurements. Appl. Radiat. Isot. 68, 2366–2371. Bouchard, J., Cassette, P., 2000. MAC3: an electronic module for the processing of pulses delivered by a three photomultiplier liquid scintillation counting system. Appl. Radiat. Isot. 52, 669–672.

Broda, R., Je˛ czmieniowski, A., 2004. Statistics of the LS-detector in the case of low counting efficiency. Appl. Radiat. Isot. 60, 453–458. Broda, R., Cassette, P., Kossert, K., 2007. Radionuclide metrology using liquid scintillation counting. Metrologia 44, S36–S52. Campion, P.J., 1959. The standardization of radioisotopes by the beta-gamma coincidence method using high efficiency detectors. Int. J. Appl. Radiat. Isot. 60, 232–248. Fuchs, C., Laustriat, G., 1970. Mesure du rendement absolu de radioluminescence des scintillateurs liquides a base de benze ne et de ses de´rive´s me´thyle´s. Rev. Phys. Appl. 5, 617–623. Harmer, S.W., Downey, R., Wang, Y., Towsend, P.D., 2006. Variation on optical constants between photocathodes. Nucl. Instrum. Methods A 564, 439–450. Kossert, K., Grau Carles, A., 2006. The LSC efficiency for Low-Z electron-capture nuclides. Appl. Radiat. Isot. 64, 1146–1453. Funck, E., Nylandstedt Larsen, A., 1983. The influence from low energy x-rays and Auger electrons on 4pb  g coincidence measurements of electron-capturedecaying nuclides. Appl. Radiat. Isot. 34, 565–569. Mo, L., Cassette, P., Baldock, C., 2006. The influence of rejection of a fraction of the single photoelectron peak in liquid scintillation counting. Nucl. Instrum. Methods A 558, 490–496. Mo, L., Bignell, L.J., Steele, T., Alexiev, D., 2010. Activity measurements of 3H using the TDCR method and observation of source stability. Appl. Radiat. Isot. 68, 1540–1542. Simpson, B.R.S., Van Wyngaardt, W.M., Lubbe, J., 2010. Fe-55 activity measurements at the NMISA revisited. Appl. Radiat. Isot. 68, 1529–1533. Steele, T., Mo, L., Bignell, L., Smith, M., Alexiev, D., 2009. FASEA: a FPGA acquisition system and software event analysis for liquid scintillation counting. Nucl. Instrum. Methods A 609, 217–220. Thiam, C., Bobin, C., Bouchard, J., 2010a. Simulation of Cherenkov photons emitted in photomultiplier windows induced by Compton diffusion using Monte Carlo code Geant4. Appl. Radiat. Isot. 68, 1515–1518. Thiam, C., Bobin, C., Bouchard, J., 2010b. Radiopharmaceutical 11C activity measurements by means of the TDCR-Cherenkov method based on a Geant4 stochastic modeling. In: Proceedings of the 2010 International Liquid Scintillation Conference (LSC2010), 6–10 September 2010, pp. 341–348.