The Application of Full Spectrum Analysis to NaI(Tl) Gamma Spectrometry for the Determination of Burial Dose Rates

The configuration of the experimental setup used in this study is essentially the same as the one reported in Bu et al. 2018. Briefly, a 3″×3″ NaI(Tl) scintillation crystal (Harshaw) and photomultiplier tube (PMT) assembly is connected to a digital tube base (TB-5, Amptek); this contains all necessary electronics for the acquisition of gamma spectra. The entire detector (NaI(Tl) crystal, PMT and digital tube base) is contained within a 10 cm thick lead shield. DppMCA software from Amptek is used to acquire the gamma spectrum. In order to characterise the NaI(Tl) detector, and to evaluate its performance and the accuracy of the algorithm we employ for spectrum analysis (see below), samples were also measured using our standard high resolution HPGe gamma facility (Murray et al., 2018). Because the NaI(Tl) detector and the HPGe detector we used in this study have different dimensions, samples (finely ground sediment mixed with wax and cast in a cup-shape geometry) of different dimensions were prepared for counting on the two different detectors; in each case, these two cups were prepared simultaneously from the same homogenised sediment sample mixture. During data collection, temperatures at the top surface of the NaI(Tl) crystal and in the air in the lead shield were continuously measured by K-type thermocouples and monitored by a thermocouple data logger (USB TC-08, Pico Technology, UK).

Because of atmospheric ²²²Rn, there can be a correlation between background count rate and atmospheric pressure, season and room ventilation (Mentes and Eper-Pápai, 2015; Schubert et al., 2018). To minimise this effect, we have filled the cavity around the detector with polystyrene foam (Bu et al., 2018); this minimises the volume available to ²²²Rn contamination. Using this configuration, we have been unable to detect a significant change in background count rate in 6 measurements over a period of 33 weeks.

All natural samples were ignited at 450°C for 24 hours (to determine organic content) before grinding to <200 μm, and then mixing with high viscosity wax (Bottle wax, blend 1944, British Wax Refining Company) on a hotplate at ∼80°C. A typical wax:sample mass ratio of ∼1:2 gives a typical sample mass of 250–300 g in the final counting geometry. Sample and calibration standards are cast in the same cup-shaped geometry, with a wall thickness of 10 mm, an internal diameter of 80 mm and an internal height of 60 mm. Samples are stored for >20 days before counting, to allow ²²²Rn to reach secular equilibrium with its parent ²²⁶Ra. Inverted, the sample cups are placed over the top of the NaI detector for counting. Murray et al. (2018) report that the dependence of self-attenuation on sample mass for all likely mixtures in the sample cup is <2% over the energy range of interest, i.e., from ∼160 keV to ∼2.87 MeV.

In order to prepare uranium and thorium calibration standards, the appropriate IAEA certified reference material BL-5 (7.09 ± 0.03% U, NRCAN-1) or OKA-2 (2.893 ± 0.058% Th, NRCAN-2) was diluted in low activity quartz-rich sand (M32, Sibelco Belgium; ²²⁶Ra 1.39 ± 0.08; ²³²Th 0.85 ± 0.06, ⁴⁰K 6.8 ± 0.7 Bq·kg⁻¹, measured by an HPGe gamma spectrometer), ground to <200 μm. The diluted reference materials were then mixed with wax to give individual parent activities of ∼800 Bq per cup. For potassium calibration standards, analytical grade K₂SO₄ (14.20 Bq·g⁻¹ assuming stoichiometry, purity given by the manufacturer as 100.4%) was mixed directly with wax to give ∼2700 Bq per cup. From the calculation, the expected uncertainties arising from the weighing process are <<1%. Three standards were prepared from separate dilutions for each radionuclide (named as K1-3, U1-3 and Th1-3, respectively). A background sample cup was prepared by casting pure wax. In order to correct for the gain drift caused by temperature changes during counting the algorithm described in Bu et al. (2018) employed a common reference spectrum containing all characteristic peaks from ⁴⁰K, ²³⁸U and ²³²Th. This spectrum was derived from a mixed sample of ⁴⁰K, ²³⁸U and ²³²Th (named KUTh reference), prepared by diluting BL-5 and OKA-2 in K₂SO₄ to give a sample cup containing ∼2700 Bq of ⁴⁰K and ∼70 Bq of ²³⁸U and ²³²Th, respectively. The activity ratios of the three nuclides in this KUTh reference sample are chosen to be similar to the average ratio of ⁴⁰K, ²³⁸U and ²³²Th activities in natural geological samples (e.g., Ankjærgaard and Murray, 2007 and references therein).

Because our NaI(Tl) scintillation detector is insensitive to low energy photons, photons emitted by ²²²Rn short-lived daughters dominate the U-series spectrum; given secular equilibrium between ²²²Rn and ²²⁶Ra (ensured by casting in wax and storing), our U analyses are actually ²²⁶Ra analyses.

The full spectrum analysis algorithm used in this study requires considerably more stringent energy stability across the entire spectrum than does the three-window algorithm. After the measurement, spectra from unknown samples, and background and ⁴⁰K, ²³⁸U and ²³²Th calibration standards were all energy corrected to the common reference spectrum from the KUTh reference sample before spectrum analysis. The linear correction algorithm (with an intercept) corrects two specific peaks in each spectrum to the two corresponding peaks in the reference spectrum. The correction algorithm is described in Bu et al. (2018), but we have now improved the selection of the correction reference peaks, especially for correction of ⁴⁰K calibration standard spectra (in which other peaks are only derived from the background, and so are poorly defined).

The gamma spectra from the background, ⁴⁰K, ²³⁸U and ²³²Th calibration standards and unknown natural samples all have characteristic peaks of very different absolute and relative intensities; as a result, the most useful reference peaks are dependent on the spectrum to be drift corrected. For example, natural samples usually contain significant activity concentrations of ⁴⁰K, ²³⁸U and ²³²Th, and so drift correction employs the ubiquitous ∼85 keV peak (X-ray) from U and Th, and the 1.46 MeV gamma-ray peak from ⁴⁰K. The same is true of background spectra although all the intensities are, of course, much weaker. However, this choice is impractical when we select reference peaks for the gamma spectra from calibration standards, e.g., in the ⁴⁰K standard spectrum, the X-ray peak is not detectable, and so the 1.46 MeV from ⁴⁰K emission and its Compton backscatter peak (measured at 226 keV) are used. This is in contrast to the more conventional choice of Bu et al. (2018), where the 1.46 MeV peak (⁴⁰K) and the background 2.61 MeV peak from the ²³²Th series were employed. We now prefer the backscattered peak because the background 2.61 MeV peak is too weak to allow accurate determination of the peak centroid. For the ²³⁸U and ²³²Th calibration standards, we selected the same peaks as those described in Bu et al. (2018), i.e., for the ²³⁸U calibration standard with negligible ⁴⁰K, we employ the X-ray peak and the ²¹⁴Bi peak (1.76 MeV, U series); for ²³²Th calibration standard, we use the X-ray peak and the ²⁰⁸Tl peak (2.61 MeV).

Fig. 1 illustrates an example of all corrected spectra from one set of ⁴⁰K, ²³⁸U and ²³²Th calibration standards, a corrected background spectrum, and the KUTh mixed reference spectrum used for drift correction. The shaded region in Fig. 1 shows the region of interest (ROI) for the full spectrum analysis; this is discussed in more detail in Section 2 – Full spectrum analysis and calculation of activity.

Fig. 1

Cups from all three sets of ⁴⁰K, ²³⁸U and ²³²Th calibration standards (9 in total) were each counted for 20 h, then each spectrum was drift corrected to the reference spectrum, and normalised to counting time to give the count rate in each channel (counts·ks⁻¹). The background was counted three times (each for 20 h); after correction and normalisation, these three spectra were averaged. This averaged background spectrum was then subtracted from each ⁴⁰K, ²³⁸U and ²³²Th normalised spectrum. The detection efficiency (counts·ks⁻¹·Bq⁻¹) of each nuclide was derived by dividing the count rate in each channel with the known activity (Bq). Finally, the detection efficiencies resulting from three standard spectra were averaged to reduce the contribution to scatter resulting from manufacturing (mainly dilution and casting) of the calibration standards.

In FSA, the shape of a large part of the sample spectrum is analysed, and decomposed to determine the contributions from the three individual ⁴⁰K, ²³⁸U and ²³²Th standard spectra and the background spectrum. The counts in the first prominent peak (as shown in Fig. 1) are mostly X-ray emissions (from U and Th), while the counts above the 2.61 MeV peak are mostly from cosmic rays; these two energy peaks (∼85 keV and 2.61 MeV) provide limits to the range of energies employed in the FSA. Here, we select a region of interest (ROI) from channel 70 to 960 (in total M = 891 channels), corresponding to an energy region from ∼160 keV to ∼2.87 MeV.

Formalising the above, in each channel, the measured unknown sample spectrum S is made up of the sum of the ⁴⁰K, ²³⁸U and ²³²Th detection efficiencies DE_k (k=1, 2, 3), each multiplied by the activity A_k contained in the unknown sample, plus an averaged background spectrum (Bg), i.e.: (2.1) S=∑k=13DEkAk+Bg S=\sum\nolimits_{k=1}^{3}{D{{E}_{k}}{{A}_{k}}+Bg}

As we know the total number of channels in the energy range of interest is M, in Eq. 2.1, there are then M linear equations in total with three unknown variables A_k (k=1, 2, 3). We use least-square linear regression to estimate the ⁴⁰K, ²³⁸U and ²³²Th activities by minimising the chi-squared: (2.2) χ2=∑i=1M[S(i)−∑k=13DEk(i)Ak−Bg(i)]2 {{\chi }^{2}}=\sum\nolimits_{i=1}^{M}{{{\left[ S\left( i \right)-\sum\nolimits_{k=1}^{3}{D{{E}_{k}}\left( i \right){{A}_{k}}-Bg\left( i \right)} \right]}^{2}}}

After obtaining the first estimate of the activities A_k,0, the estimation accuracy is improved by a second least-squares linear regression following the inclusion of a weight factor W(i) into Eq. 2.2: (2.3) χ2=∑i=1M[S(i)−∑k=13DEk(i)Ak−Bg(i)]2/W(i) {{\chi }^{2}}=\sum\nolimits_{i=1}^{M}{{{\left[ S\left( i \right)-\sum\nolimits_{k=1}^{3}{D{{E}_{k}}\left( i \right){{A}_{k}}-Bg\left( i \right)} \right]}^{2}}/W\left( i \right)}

In Eq. 2.3, instead of employing the weight factor W(i) = S(i) as described in Hendriks et al. (2001), or as a more complicated form involving estimated activities and uncertainties on all relevant spectra (Jeong et al., 2014), we simply use the absolute value of the fitting error in each channel generated in the previous step as the weight factor W(i).

In this way, those channels with a higher fitting error have a higher weight factor, giving a faster convergence. We carried out this process iteratively by continuously using a newly obtained W(i) in the current step (defined as step n) as the weight factor for the next step (step n+1) of chi-squared minimisation, until the changes in all of the estimated activities reach the condition: (2.5) |Ak,n+1−Ak,n|≤1e−5Bq \left| {{A}_{k,n+1}}-{{A}_{k,n}} \right|\le 1{{e}^{-5}}Bq

This process delivers a fitting error contribution of ∼0.001 Bq·kg⁻¹ for a typical >100 g sample.

Fig. 2 shows an example using the natural sample (LD791); the original sample spectrum and the fitted spectrum overlap can be seen to agree very well over the entire energy range; the weak structure visible in the residuals presumably arises from errors in the drift corrections.

Fig. 2

The uncertainty analysis on the burial dose rate of natural samples is not a trivial task; there are many sources of uncertainty, both correlated and uncorrelated, contributing to the final propagated uncertainty on the burial dose rate. In this study, as usual, we categorise them into systematic and random uncertainties. The systematic uncertainties depend mainly on the standards we use to calibrate the ⁴⁰K, ²³⁸U and ²³²Th detection efficiencies on our NaI gamma detector. The random uncertainty includes the uncertainties contributed by the finite counts in each channel (counting statistics) and the fitting uncertainty arising from the fitting algorithm used to determine the activity concentrations. The details of the derivation of systematic and random uncertainties are summarised in the Supplementary Material.

Once the activity concentrations of ⁴⁰K, ²³⁸U and ²³²Th in the unknown sample are determined, we derive beta and gamma dose rates based on the dose rate conversion factors published by Guérin et al. (2011) and Liritzis et al. (2013). In addition, we updated various parameters including the atomic weights, abundances and decay constants (and their uncertainties) of ⁴⁰K, and all U and Th decay series from the IAEA (IAEA Live Chart of Nuclides, March 2018). The revised activity concentrations per ppm are listed in Table 1. In order to determine the uncertainties in beta and gamma dose rates, updated uncertainties of all relevant quantities, e.g., half-lives, energies, branching ratios, atomic weights, atomic abundances etc., were taken into account and combined during the error propagation process.

During the derivation of field beta and gamma dose rates, radon loss in the field is assumed to be 20% ± 10%; this is the default assumption used in our laboratory. In addition, we assume ²²⁶Ra to be in equilibrium with its series parent ²³⁸U. Both these assumptions can be varied on a site-specific basis.

The total dry burial dose rate was then calculated by summing the beta and gamma dose rates. Since the activity concentrations of the ²³⁸U and ²³²Th daughter nuclides are all dependent on activity concentrations of ²³⁸U and ²³²Th, respectively, the contributions from their individual uncertainties to the total beta or gamma uncertainties are correlated. In addition, beta and gamma dose rates are also correlated because they are all derived from the same ⁴⁰K, ²³⁸U and ²³²Th activity concentrations. During the summing of beta and gamma dose rates, propagation of these correlated uncertainties are handled by the uncertainty analysis tool Metas.UncLib (Zeier et al., 2012), in the same manner as the uncertainties for the activities (see Supplementary Material).

MDLs for ⁴⁰K, ²³⁸U and ²³²Th activities (or activity concentrations for a sample with a certain weight) are one of the most important specifications of a gamma spectrometer intended for the measurement of dose rates. In this study, we used the same procedures as described in Bu et al. (2018) to measure these MDLs experimentally. In short, we measured the same KUTh mixed sample cup for ten various lengths of time t: 30, 60, 120, 300, 600, 1200, 2400, 4500, 9000 and 72000 s. The background cup was counted for 72000, 71970, 71940, 71880, 71700, 71400, 70800, 69600, 67500, 63000 s. After drift correction, we combine a sample spectrum (e.g. that counted for 30 s) with the corresponding background spectrum (in this case, that counted for 71970 s) to give a spectrum corresponding to a total effective counting time of 72000 s (20 hours). This gives us in total ten synthesised spectra equivalent to those that would have been obtained from ten samples of well-known relative activity concentrations; all counted for 20 h. The expected activity concentration (A_EXP) corresponding to the counting time t of each synthesised sample was calculated from the activity concentration measured for 20 h (A_{MEAS_20h}): (3.1) AEXP=AMEAS_20h×t/72000 {{A}_{EXP}}={{A}_{MEAS\_20h}}\times t/72000

These ten synthesised sample spectra were analysed by FSA and the apparent ⁴⁰K, ²³⁸U and ²³²Th activities (A_MEAS) derived. Here we define the MDL as the value of A_EXP expected to have a corresponding random uncertainty of 30% on A_MEAS. Fig. 3a–3c summarises the FSA analysis results, and gives MDLs for ⁴⁰K, ²³⁸U and ²³²Th activities of 1.1, 0.09 and 0.07 Bq, respectively. By dividing by the maximum practical sample mass in our cup geometry (0.23 kg), the corresponding MDLs on activity concentrations for ⁴⁰K, ²³⁸U and ²³²Th are 4.8, 0.4 and 0.3 Bq·kg⁻¹, respectively. Table 2 compares these FSA MDLs with those from the three-window approach (Bu et al., 2018) from a NaI(Tl) spectrometer, and MDLs from a high resolution HPGe gamma spectrometer (Murray et al., 2018). Table 2 shows FSA MDLs are one order of magnitude lower than those achieved by three-window spectrum analysis, and are one order of magnitude higher than those from an HPGe detector. This indicates the FSA approach is a significant improvement on the results obtainable from a low resolution NaI(Tl) gamma spectrometer, presumably because of much greater statistical precision. The total number of counts recorded in the energy region of interest in FSA is orders of magnitude greater than that in the windows-of-interest in the three-window approach; the FSA approach makes use of almost all statistical information contained in the sample spectrum.

Fig. 3

The accuracy of the measurement and analyses can be examined in Fig. 3d–3f, in which the shaded regions in all three subpanels represent the activity ranges above MDLs, for ⁴⁰K, ²³⁸U and ²³²Th, respectively. The deviations A_MEAS – A_EXP between the measured and expected activities for ⁴⁰K, ²³⁸U and ²³²Th, are normalised to the expected activity A_EXP, as shown in Fig. 3d–3f, as a function of A_EXP. The error bars shown in Fig. 3d–3f are derived from the random uncertainties of the expected and measured activities. Within these shaded regions, ⁴⁰K, ²³⁸U, and ²³²Th analyses are all consistent with the expected results, without any obvious systematic error, even when the expected activities approach the MDLs.

The precision and accuracy of the FSA algorithm were further evaluated by comparison of our analyses with those from a high resolution HPGe spectrometry facility (Murray et al., 2018; making the same assumptions concerning secular equilibrium), using 20 natural samples covering a range of activity concentrations. Fig. 4a–4c shows the deviation AC_NaI – AC_HPGe between activity concentrations obtained from the NaI(Tl) spectrometer and those from the HPGe spectrometer are normalised to AC_HPGe. The resulting dry dose rates for all 20 unknown samples are normalised in the same manner, and the normalised deviations are compared in Fig. 4d.

Fig. 4

In this study, all unknown samples, ⁴⁰K, ²³⁸U and ²³²Th calibration standard cups, mixed KUTh reference cup for drift correction and background cup were measured at a room temperature of ∼20–25°C for 20 hours. The maximum temperature drift during each measurement was ∼0.4°C.

As shown in Fig. 4a–4c, almost all normalised deviations (AC_NaI – AC_HPGe)/AC_HPGe of ⁴⁰K, ²³⁸U and ²³²Th for 20 samples lie within the shaded region of ±5%, with only one data point below −5% line in ²³⁸U panel and another two just above +5% line in ²³²Th panel. For the total beta and gamma dose rates (Fig. 4d), normalised deviations (DR_NaI – DR_HPGe)/DR_HPGe lie within the shaded region (±4%) for all 20 studied samples. The standard deviations (n=20) on these ratios are also listed in each subpanel in Fig. 4. All activity concentration ratios have a standard deviation of <3%, and the corresponding variability in dose rates is <2%.

In addition, Fig. 4a–4d show that there is no suggestion of a systematic deviation, either in the activity concentrations or in the total dry beta and gamma dose rates. This is also indicated in Table 3 by the average ratios of AC_NaI/AC_HPGe and DR_NaI/DR_HPGe, with low standard deviations (n = 20) of 0.4 to 0.6%.

Another important consideration when using a gamma spectrometer for measuring dose rates in OSL dating applications is the counting time. The required count time is dictated by the required dose rate uncertainty and the sample activity. As discussed above, the total uncertainty on dose rate is mainly derived from random and systematic uncertainty in ⁴⁰K, ²³⁸U and ²³²Th activity concentrations. While the systematic uncertainty is fixed (at ∼2%), the random uncertainty, no matter whether from the fitting error of FSA algorithm or from counting statistics, is mainly determined by the counts in each channel.

Quantitative investigation of the dependency on counting time was carried out by counting the same natural sample (LD962) for different lengths of time, i.e., 5, 10, 20, 30 min, and 1, 2, 5, 10, 20 hours. Fig. 5a–5c and (d) show the dependence of the measured activity concentrations of ⁴⁰K, ²³⁸U and ²³²Th, and total beta and gamma dose rates, respectively, on counting time. The results are compared with those measured on the HPGe spectrometer for 20 hours, i.e., 235, 8.6, and 8.3 Bq·kg⁻¹ for activity concentrations of ⁴⁰K, ²³⁸U and ²³²Th and 1.09 Gy·ka⁻¹ for total dose rate; these values are represented by the thick horizontal dashed lines in each sub-panel in Fig. 5. The shaded region in each subpanel links to the right vertical axis and shows the dependence of the corresponding normalised deviations between the results from the NaI(Tl) spectrometer and the HPGe spectrometer for activity concentrations (AC_NaI − AC_HPGe)/AC_HPGe (Fig. 5a–5c) and for corresponding derived dose rates (DR_NaI − DR_HPGe)/DR_HPGe (Fig. 5d). These data indicate the counting times required to achieve a specified level of precision in the activity concentrations of ⁴⁰K, ²³⁸U and ²³²Th and the derived dose rate, e.g. a 30 min count will lead to a ∼±10% uncertainty on dose rate.

Fig. 5

Fig. 6 shows the variation in the uncertainty on the total dose rate with the counting time, for a sample with 235, 8.6, and 8.3 Bq·kg⁻¹ of ⁴⁰K, ²³⁸U and ²³²Th, respectively and a dose rate of 1.09 Gy·ka⁻¹. These data allow the selection of the count time required to achieve a specified uncertainty. For example, even counting for only 1 to 2 hours gives a dose rate with a relative uncertainty ≤5%. A longer count of >5 hours brings down the uncertainty below ±3%. The sample used to illustrate these dependencies is amongst those with the lowest activity concentrations (see Fig. 4); one can expect to obtain even lower uncertainties with a shorter counting time when the samples with higher activity concentrations are measured.

Fig. 6