Introduction

Building materials are derived from Earth’s resources, such as soil or rocks, and thus they contain naturally occurring radionuclides from the 238U and 232Th decay series, as well as 40K and, as a result, their use carries a risk of radiation exposure. This exposure can be external from direct gamma radiation exposure or internal from inhalation of radioactive radon/thoron [1]. The latter can be especially harmful in an indoor environment and can contribute to development of lung cancer, if there is no appropriate ventilation [2]. For this reason a number of regulations have been introduced providing so-called hazard indices. For example, to ensure safety regarding the dose acquired exclusively from building materials, activity concentration index value of 1 can be used as a conservative screening tool [3]. Similar mathematical formulation can be seen in other hazard indices, namely: radium equivalent activity, representative level index, absorbed and annual gamma dose rates, respectively, gamma effective indices, as well as external and internal radiation hazard indices.

The issue of building materials radioactivity is well documented in various works, as listed in Table 1. However, because the μDOSE system is a relatively new setup for estimating 40K as well as 238U, 235U and 232Th decay chain members activities [4], no research on hazard indices has been done so far. A 2022 study [5] on the accuracy of the μDOSE system shows good agreement with well-established methods of dosimetry, such as HRGS or thick source alpha counting. The study also mentions that the correlation of results provided by the μDOSE system improves precision of dose rate estimation; however, this correlation has not been investigated on its own and nor has its significance in comparison to the HRGS method.

Table 1 Radiation hazard indices; names, parameters and descriptions of 7 hazard indices, along with references to other works where the indices were studied
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The aim of this work is to test how uncertainties of hazard indices are influenced, if radionuclide estimates are correlated. This is tested with two independent setups of HRGS and the μDOSE system that provide uncorrelated and correlated estimates respectively. In addition, advantages of reducing hazard indices uncertainties are investigated by using correlated radionuclide estimates and drawbacks as errors are induced by disequilibrium in radioactive decay chains.

Experimental

Materials

All 7 samples that were investigated in this work are frequently used building materials of known origin: clay bricks (SiO2, feldspars), beach rock (sand and/or gravel, CaCO3) and sand (SiO2). Samples were dried in a drying chamber at an elevated temperature of 80 °C for several days. During drying sample masses were monitored to ensure water was removed. Every sample material was divided into two subsamples for μDOSE and HRGS measurement. Detailed descriptions of sample preparation for these systems is provided below.

Subsamples and reference materials for μDOSE system measurements were ground in a Fritsch Pulverisette 6 planetary mill for 45 min at 200 rpm each to a fine powder, then 3.00 g of prepared material was placed onto 70 mm diameter sample discs and measured using the μDOSE system.

Subsamples and reference materials for HRGS measurements were put and sealed in γBEAKERS [21, 22] (ca. 100 g samples) and then stored for a period of about 30 days to obtain secular equilibrium to avoid bias that arises from 222Rn emanation [23,24,25,26,27]. Subsamples were measured for 24–48 h, times varying depending on the activities of each individual sample for obtaining optimal count rate statistics.

μDOSE system—α/β and delayed coincidence counting

For α/β particle measurements μDOSE system was used, described in detail in [4, 5]. It is designed for detecting α-α (220Rn/216Po, 219Rn/215Po) and β-α (212Bi/212Po, 214Bi/214Po) decay pairs and identifying the isotope pair the particles come from, based on the characteristic time intervals between the subsequent particle emissions. These pairs can later be used to determine 238U, 235U, 232Th and 40K content. This is done with assumption of secular equilibrium, where the activities of the decay pairs are equal to the activities of corresponding decay chains parent radioisotopes: 238U, 235U or 232Th. The remainder of the emitted β particles are assigned to 40K. The system was calibrated using IAEA-RGU-1, IAEA-RGTh-1 and IAEA-RGK-1 [28] reference materials obtained from the International Atomic Energy Agency. In this work a system that was calibrated for 3.00 g was used.

Background activity was obtained for a 3.00 g background plastic disc placed accordingly in the sample holder. The background measurement lasted ca. 48 h and was performed in the same laboratory conditions as were the samples.

High resolution gamma spectrometry

Gamma spectrometry was performed with a Canberra HPGe detector with FWHM of 1.8 keV and relative efficiency 40% at 1332 keV. The HRGS system was calibrated with IAEA-RGU-1, IAEA-RGTh-1 and IAEA-RGK-1 [28] reference materials. Activities were calculated for selected energy lines: 295.2 keV, 351.9 keV (214Pb) and 609.3 keV (214Bi) for uranium series; 338.3 keV, 911.2 keV (228Ac) and 583.2 keV (208Tl) for thorium series; and 1460.8 keV (40K) (data obtained from NuDat 3.0).

Background activity was obtained through a 167 h measurement of an empty γBEAKER in an identical laboratory setting in which samples were measured.

Theoretical

Radiation hazard indices

To determine whether building materials meet set standards, are within established norms and are safe to use, a number of hazard indices were devised. In many cases hazard indices are estimated from 226Ra, 232Th and 40K concentrations and provide a simplified information for a given risk factor. Frequently used indices are calculated as a linear combination of 226Ra, 232Th and 40K radionuclides contents in the following way:

$$HI={S}_{\mathrm{Ra}}{A}_{\mathrm{Ra}}+{S}_{\mathrm{Th}}{A}_{\mathrm{Th}}+{S}_{\mathrm{K}}{A}_{\mathrm{K}}$$
(1)

where HI is the hazard index, ARa, ATh and AK are 226Ra, 232Th and 40K concentrations expressed in Bq·kg−1 respectively, SRa, STh and SK are parameters for a given hazard index. List of hazard indices, SRa, STh, SK parameters and a brief summary is provided in Table 1.

Uncertainty propagation for uncorrelated and correlated inputs

For uncorrelated inputs given hazard index uncertainty (σTh) can be is calculated as:

$${\sigma }_{HI}^{2}={({S}_{\mathrm{Ra}}{\sigma }_{\mathrm{Ra}})}^{2}+{({S}_{\mathrm{Th}}{\sigma }_{\mathrm{Th}})}^{2}+{({S}_{\mathrm{K}}{\sigma }_{\mathrm{K}})}^{2}$$
(2)

where σRa, σTh and σK are uncertainties of 226Ra, 232Th and 40K activities. Usually HRGS measurements are considered to be uncorrelated. However for correlated inputs hazard index uncertainty should be calculated using the formula that takes into account correlations:

$$\sigma_{HI}^{2} = \left[ {\begin{array}{*{20}c} {S_{{{\text{Ra}}}} } & {S_{{{\text{Th}}}} } & {S_{{\text{K}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {\sigma_{{{\text{Ra}}}}^{2} } \hfill & {\sigma_{{{\text{Ra}} - {\text{Th}}}} } \hfill & {\sigma_{{{\text{Ra}} - {\text{K}}}} } \hfill \\ {\sigma_{{{\text{Th}} - {\text{Ra}}}} } \hfill & {\sigma_{{{\text{Th}}}}^{2} } \hfill & {\sigma_{{{\text{Th}} - {\text{K}}}} } \hfill \\ {\sigma_{{{\text{K}} - {\text{Ra}}}} } \hfill & {\sigma_{{{\text{K}} - {\text{Th}}}} } \hfill & {\sigma_{{\text{K}}}^{2} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {S_{{{\text{Ra}}}} } \\ {S_{{{\text{Th}}}} } \\ {S_{{\text{K}}} } \\ \end{array} } \right]$$
(3)

where non-diagonal σ elements are covariances of elements given in subscripts. Depending on sign and value of non-diagonal elements the final HI uncertainty will give different values. For uncorrelated inputs non diagonal σ are zeros and Eqs. 2 and 3 provide the same values.

Results and discussion

Correlated 226Ra, 232Th and 40K estimates

The μDOSE system provides correlated 226Ra, 232Th and 40K estimates. This arises because the system detects α particles that can be emitted from decay chains while β particles are emitted from decay chains and 40K. Four decay pairs 220Rn/216Po, 219Rn/215Po, 212Bi/212Po, 214Bi/214Po detected by the μDOSE system, provide information on relative 226Ra, 232Th and 40K content. Detailed calculation procedure is provided in [29]. Correlations of 226Ra, 232Th and 40K contents are visualised in Fig. 1 where 100 k points were drawn with respect to correlations determined by α, β and four decay pairs counting statistics.

Fig. 1
figure 1

Two 226Ra, 232Th and 40K compositions. Isolines are showing projected 2-D PDF contour-plots drawn from a multivariate normal distribution for the μDOSE system for 1 h (a) and 10 h (b) measurements

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Hazard indices and their uncertainties

Hazard indices and their uncertainties were estimated for the μDOSE system and HRGS using radionuclide content. Numerical values are provided in Table 2. In the case of the μDOSE system uncertainties are calculated using two approaches: first neglects correlations (Eq. 2) and the second approach takes correlations into account (Eq. 3). This is illustrated on Figs. 2, 3, 4, 5, 6, 7 and 8 which provide comparison of hazard indices values and their uncertainties.

Table 2 Numerical data of 226Ra, 232Th and 40K radionuclides contents from μDOSE and HRGS
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Fig. 2
figure 2

Radium equivalent activity (Raeq) with uncertainties calculated for all samples

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Fig. 3
figure 3

Representative level index (RLI) with uncertainties calculated for all samples

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Fig. 4
figure 4

Absorbed gamma dose (DR) with uncertainties calculated for all samples

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Fig. 5
figure 5

Annual effective dose rate (HR) with uncertainties calculated for all samples

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Fig. 6
figure 6

Activity concentration index Iγ with uncertainties calculated for all samples

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Fig. 7
figure 7

External radiation hazard index (Hex) with uncertainties calculated for all samples

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Fig. 8
figure 8

Internal radiation hazard index (Hin) with uncertainties calculated for all samples

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Data presented in Figs. 2, 3, 4, 5, 6, 7 and 8 shows that the uncorrelated uncertainties for μDOSE measurements are greater than the correlated uncertainties. This shows that by including the correlation between the activities of 226Ra, 232Th and 40K in the estimation of hazard indices uncertainties, the provided results are more precise. This is due to the fact that several elements within the covariance matrices (Eq. 3) are negative which contributes to the lessening of the uncertainties. The correlated uncertainties from μDOSE measurements are also similar in value to the ones obtained from HRGS, thus proving that the measurement accuracy is maintained regardless of the chosen method.

Screening—measurement time, uncertainty and type II error

In some cases sample mass and sample throughput can be a limiting factor; therefore, an investigation on how uncertainty changes as a function of time was performed. The μDOSE system offers the possibility of measuring small samples and in this work the system was calibrated for 3.00 g samples, whereas HRGS was calibrated for 100 g. Figure 9 shows activity concentration index value and uncertainty as a function of measurement time for HRGS and μDOSE system. In both cases uncertainties encompass counting statistics, sample mass and reference materials uncertainties which are the main uncertainty contributors for prolonged measurements. Despite large mass differences, μDOSE uncertainties that take into account correlations are comparable with HRGS (Fig. 9). For the first few hours of the measurement activity concentration index uncertainty is relatively large for μDOSE. After 10 h (Fig. 9) uncertainties from the μDOSE and HRGS systems are at the same level, which further proves that the μDOSE system is a reliable screening tool and can be an alternative to conventional HRGS. Nevertheless, algorithms that are used for measuring net peak area do not operate well on poorly defined baseline and therefore first peak quantification, used for activity estimation, was available after 46 min of measurements and required manually adjusting regions for peak detection.

Fig. 9
figure 9

Activity concentration index value uncertainty as a function of measurement time for HRGS (100 g samples) and μDOSE system (3.00 g samples)

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In case of screening building materials for activity concentration index (Fig. 9) it can be observed that after ca. 1 h of measurement its value is well below recommended value 1 [3]. For shorter measurements HRGS counting statistics (discussed in the paragraph above) and μDOSE counting statistics do not provide definitive material classification. This finding has substantial implications for future studies, especially when sample mass or measurement time is a limiting factor. What is more, understanding the differences between correlated and uncorrelated uncertainties allows researchers to take advantage of correlated ones.

While screening HI there are two classification errors: type I error “false positive” and type II error “false negative”. If HI value is above threshold, additional and prolonged measurement should be made to resolve this. In case of type II error, this can be difficult to detect and can have more significant consequences. For example, given HI can falsely be assumed to be within limits. This can happen because there are several factors, not related to counting statistics, that contribute to this. For example 238U decay chain can be in disequilibrium, [25, 30, 31] or sample chemical composition can be unknown. This can cause issues in both HRGS as well as in the μDOSE system and unfortunately those factors are not controlled routinely due to cost of additional measurements. Nevertheless, HI screening limits can be set to reduce the risk of type II errors at the expense of type I error.

The proposed methodology and obtained findings allow to bridge a gap in the existing literature, especially in the context of improving the throughput with measurement systems like μDOSE. The precision of conducted measurements ensures that safety thresholds are reliably met, reducing risks associated with type I and II errors.

Conclusions

In this study, values of selected hazard indices of building materials were compared using the μDOSE system for 3.00 g samples against the HRGS calibrated for 100 g samples. The primary objective was to understand the significance of the correlation of activities of 226Ra, 232Th, and 40K provided uniquely by the μDOSE system. The findings indicated that the uncertainties in hazard indices from both the HRGS and μDOSE systems, when correlations are considered, are comparable in value. Furthermore, accounting for these correlations offers a substantial enhancement in the precision of results compared to when they are disregarded.