Introduction

The measurement techniques of physical quantities are highly dependent on their magnitude. Interestingly gamma spectrometry is reasonably sensitive over a wide range of magnitude, i.e., from fraction of Bq to tens of MBq, just by adjusting the source to detector distance and time of counting. In the measurement of naturally occurring radioactive materials (NORM), the count rate may come down to ~1–2 Bq in a single experimental sample. Measurement of this minuscule amount of radioactivity thus demands careful consideration of the minute details of the counting protocols as small error in the measurement would be multiplied manifolds while expressing the end result in normalized form like Bq kg−1. The above statement is further supported in a paper by Durec et al. [1]. An inter comparison exercise on quantitative assessment of various natural radionuclides in river sediments were carried out by 25 laboratories of Europe. It was found that although gamma spectrometry was the most implemented technique rather than beta counting or alpha spectrometry, but all the results from gamma spectrometry were not acceptable. The activity of 226Ra reported by different laboratories on same sample varied between 21 and 106 Bq kg−1. This paper clearly demonstrates that for better understanding of low-level NORM measurement various parameters like proper photopeaks selection, counting time, sample geometry, detector efficiency and resolution, appropriate use of standards, etc., need to be rationalized. In a recent publication we have worked extensively on one of these parameters, selection of proper photopeaks, for measurement of 238U and 232Th. We could show that among more than 200 photopeaks in 238U, 232Th series, only few photopeaks from the daughter products of the corresponding series are reliable to get a good estimate of the level of uranium and thorium in NORM [2]. However, to the best of our knowledge till date no such attempt has been made to investigate minimum acceptable counting time for measurement of ultra-low level activity in natural samples based on the standard deviations between the activity values under different photopeaks.

In this connection, it is noteworthy to mention recent debates on the issue of selection of appropriate counting time required for measurement of NORM (238U, 232Th and 40K). A research group assessed the activity of 238U, 232Th and 40K radionuclides for 36000 s in sediment, soil and water samples collected from the lower basin of river Pra in Ghana [3]. In a letter to the editor a researcher objected that 36000 s was too small duration and the data should have been taken for at least 24 h [4]. The original authors in their response admitted that 24 h counting time would be better but 10 h was also not bad [5]. In all these three references, arguments on the choice of counting time were based on their experience and personal belief. Therefore it is clear that though counting time is one of the important parameters which influences the end result, never a thorough critical assessment was done on the minimum counting time required for NORM measurement.

The counting error decreases with increasing counting time. However, too long counting time also restricts the number of samples that can be analyzed by a detector. Therefore depending upon the practicality, different researchers have selected different counting time for low-level NORM measurement. A glance at literature shows that there is lot of arbitrariness in the selection of counting time irrespective of detector efficiency and type of detector [3, 6–60]. Table 1 lists the counting time taken by various researchers in different experimental set-up.

Table 1 Counting time taken by different research groups for NORM measurement (HPGe detectors have been used in all the following cases)
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In NORM measurement it is better to take average of the activity values under few photopeaks of daughter products belonging to decay series, 238U and 232Th rather than to conclude the amount of U/Th/Ra in a given sample from the activity value under a single photopeak. In principle, the activity obtained under different photopeaks for same or different isotopes of a particular series should be equal (after attaining secular equilibrium). However in practice this is never observed because of several factors like intrinsic property of the detector, noise due to electronics, background, Compton edge of the other photopeaks, etc.

It is customary to express the activity in the form of x ± Δx, where x represents activity of the sample and Δx represents the counting error (CE), which is equal to √x according to Poisson distribution. If the average activity obtained from different photopeaks is taken as x, then Δx is represented by \(\sqrt {\frac{{\Delta x_{1}^{2} + \Delta x_{2}^{2} + \ldots + \Delta x_{N}^{2} }}{N - 1}}\), where Δx 1, …Δx N denotes counting error for x 1 …x N. It has been shown by Naskar et al. [2] that apart from the counting error, standard deviation (SD) between the activity values obtained under different photopeaks is also a measure of error and sometime more significant than the counting error. Therefore merely representing the average value and its error in terms of counting error is not proper. Estimation of both CE and SD should be done, and the higher one should be reported as error.

The present study makes an earnest attempt to investigate the nature of SD and CE with respect to different counting time to get more meaningful result from low-level radioactivity measurements.

Experimental

High-purity Germanium (HPGe) detector with 50% relative efficiency and 3.1 keV resolution at 1332 keV was used to measure 238U and 232Th activities in four soil samples collected from West Bengal and Punjab state of India. The detector was well shielded with 10 cm thick lead shield. The entire detector and shielding arrangement has been described elsewhere [6]. Samples used for measurement were air-dried, grinded, weighed (50 g each) and sealed in airtight petri-plates for more than one month to maintain secular equilibrium. Energy calibration was done using point sources of 152Eu, 137Cs, 133Ba and 60Co. In addition to unknown soil samples, two 238U samples of known strength, 2 Bq and 5 Bq were prepared from weighed amount of IAEA uranium ore (Pitchblende) standard (0.14 and 0.35 g correspond to 2 and 5 Bq respectively). Also two 232Th samples, 2 and 5 Bq were prepared using weighed amount of thorium acetate, [Th(CH3COO)4] (0.995 and 2.49 mg correspond to 2 and 5 Bq respectively). For validation of result, in both cases of U and Th measurement, 2 Bq samples were used respectively as U and Th standards, whereas the samples having strength 5 Bq were used as samples of known activity. The samples were counted for 5000, 10000, 20000, 30000, 50000, 75000, 100000, 125000 and 150000 s. The background activity was also measured at all the above-mentioned counting times. The respective background activity was stripped from corresponding spectrum prior to analysis. The activities of 238U and 232Th in different samples have been taken as the average of activities obtained under different photopeaks mentioned in Table 2 as it has been observed by our group that the said peaks provide the most reliable data [2]. It is noteworthy to mention here that these photopeaks essentially represent 226Ra in 238U series as the equilibrium is established between 226Ra and its daughter products after the samples are hermetically sealed for 30 days or more. It is further assumed that the equilibrium between 238U and 226Ra also exist in natural samples. However, in some cases due to geochemical cycles, this equilibrium may not exist. In the experimental samples we have observed that 238U and 226Ra are in equilibrium. This has been confirmed by measuring activities from 63.29 keV (4.8%) and 92.38 (2.81%), 92.80 keV (2.77%) photopeaks of 234Th, which were found similar to the activities calculated from the photopeaks shown in Table 2.

Table 2 Sets of photopeaks considered for the present study [2]
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Results and discussion

Tables 3, 4, 5 and 6 represents observed activity of U and Th for samples SB1, SB2, PU1, PU2 at different counting times along with their relative counting error \(\left( {{\text{RCE}} = \frac{Counting \,error}{mean\, value} \times 100} \right)\) and Relative Standard Deviation \({\text{RSD}} = \frac{Standard\, deviation}{mean\, value} \times 100\). Table 7 shows the observed activity of sample of known strength of U and Th (5 Bq) along with their RCE and RSD. The two errors, namely, RCE and RSD related to measured activities of 238U and 232Th of the four samples have been plotted in Fig. 1a–d. Similarly the errors related to U and Th activities of the samples with known strengths have been plotted in Fig. 1e.

Table 3 Observed activities and errors of sample SB1 at different counting times
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Table 4 Observed activities and errors of sample SB2 at different counting times
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Table 5 Observed activities and errors of sample PU1 at different counting times
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Table 6 Observed activities and errors of sample PU2 at different counting times
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Table 7 Observed activity of 5 Bq known activity of 238U and 5 Bq known activity of 232Th along with their respective errors
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Fig. 1
figure 1

a Variation of RCE and RSD with counting time for sample SB1. b Variation of RCE and RSD with counting time for sample SB2. c Variation of RCE and RSD with counting time for sample PU1. d Variation of RCE and RSD with counting time for sample PU2. e Variation of RCE and RSD with counting time for standards of 5 Bq known activity

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In all the Fig. 1a–e, RCE decreases with increasing counting time, which is as expected. However, this is not the case for RSD, which varies rather randomly in case of lower counting time i.e. sometime RSD is very high and sometime it is too low in magnitude. The RSD is found to follow a systematic trend only beyond certain counting time. In lower counting time, relatively low RSD is explained by the fact that by chance the activity values under different peaks come closer although they may be far from the true value. This argument is further strengthened and empirically demonstrated from Fig. 2 wherein activity of 238U and 232Th under individual photopeaks has been plotted for known sample strength of 5 Bq. For example, in 5000 s counting time, RSD for 5 Bq 238U standard is as high as 20.8%, and the individual activity values under the photopeaks of 295.2, 351.9 and 609.3 keV are 6.24, 6.53, 9 Bq respectively. The RSD decreased to only 4.2% at 30000 s with the individual activity values under the above peaks 6.58, 6.97, 6.42 Bq respectively. In both cases the average values are far from true value (5 Bq). A similar trend can be seen in the case of 232Th series. RSD value for 5000 s was as high as 53.6%, which is accounted for large difference in the activity values of different daughter products of 232Th.

Fig. 2
figure 2

Variation of activity under different photopeaks with true value for standards of 5 Bq known activity

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It can be safely stated that as the counting time is increased, the data becomes more reliable. For example, in 150000 s both RCE and RSD values are smaller. However, one cannot go for very high counting time due to several restrictions. It would be a good idea for experimenters involved in low-level radioactivity measurements to empirically select a counting time for at least 1–2 samples which satisfies the conditions that RSD values come closer to RCE values and also RSD values do not fluctuate a lot during different counting time intervals, i.e. a regular trend in it is observed. In the present case, 75,000–100,000 s could be a reasonable minimum counting time, which have been indicated in the Fig. 1a–e with an arrow. However, to be in safe side one can choose 100,000 s.

All the above discussions have been made by taking average of three photopeaks from 238U and 232Th series as depicted in Table 2. Further we have checked the nature of RSD values with variation of counting time and by introducing more number of photopeaks. For 238U we have consecutively added 1764.49 keV (15.4%) and 2204.21 keV (5.08%) photopeaks, both from 214Bi. Similarly for 232Th, 727.33 keV (6.58%) photopeak from 212Bi and 860.56 (12.42%) from 208Tl were consecutively added. It has been observed that addition of photopeaks deteriorates the situation in terms of both randomness and magnitude of RSD. To illustrate with an example, RSD values of sample SB1 for 3,4 and 5 photopeak combinations at different counting times have been tabulated in Tables 8 and 9 for 238U and 232Th respectively. The addition of photopeaks increased RSD values as high as 91% and 118% for 238U and 232Th. The five photopeaks combinations for both 238U and 232Th show that even 100,000 s counting time may not be sufficient to stabilize the RSD values.

Table 8 RSD (%) for different photopeaks combination of 238U for sample SB1 at different counting times
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Table 9 RSD (%) for different photopeaks combination of 232Th for sample SB1 at different counting times
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Further, we have normalized the RSD values with respect to number of photopeaks, i.e., RSD values of three photopeaks combinations have been divided by 3, four photopeaks combinations have been divided by 4 and so on. The normalized values, i.e., RSD/photopeak have been plotted in Fig. 3. The figure strengthens our early data and shows that with increasing number of photopeaks RSD/photopeak also increases. Figure 3 also strongly supports the choice of three most reliable photopeaks each from 238U and 232Th series as described in Table 2 and our earlier paper [2].

Fig. 3
figure 3

Variation of RSD/photopeak with counting time and different photopeak combinations

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Apart from proper selection of gamma lines, parameters like type and efficiency of detector, calibration standard and geometry, etc., may also influence the minimum counting time required for NORM measurement. Therefore it is recommended that the experimenter should consider RCE and RSD values to select minimum reasonable counting time in their own system to get more reliable results.

Conclusion

In the study of NORM employing gamma ray spectrometric setup it is important to have prior idea of both standard deviation of activities under different photopeaks selected for NORM measurement and the counting error to determine minimum counting time required for set of similar samples to obtain meaningful result. It is generally observed that relative standard deviations (RSD) will be small in magnitude and will merge with relative counting errors (RCE) as counting time becomes longer. The recommendation of empirically determining a reasonable minimum counting time based of RSD and RCE values will also minimize the ambiguity and arbitrariness in selection of counting time in getting more consistent output by different researchers.