The dose rate of the 90Sr / 90Y beta source used in most luminescence readers is a laboratory key parameter. If the source's calibration is incorrect, results for equivalent dose and age are also incorrect. The significance of beta-source calibration is therefore well known and has been subject to interlaboratory comparison studies (e.g. Pernicka and Wagner, 1979; Göksu et al., 1995).

Past studies have established that charge build-up, attenuation and backscatter constitute the physical mechanisms controlling the dose absorbed in the sample's mineral grain. The interplay of these mechanisms depends on mineral type (Aitken, 1985), on grain transparency (Bell and Mejdahl, 1981), beta-source-to-grain distance (Wintle and Aitken, 1977), grain size (Goedicke, 2007; Armitage and Bailey, 2005; Mauz and Lang, 2004) and the sample carrier's substrate (Greilich et al., 2008; Armitage and Bailey, 2005; Mauz and Lang, 2004; Wintle and Aitken, 1977). In addition, accuracy and/or precision of the calibration value depend on the measurement protocol (Guérin and Valladas, 2014; Kadereit and Kreutzer, 2013), the atomic numbers (Z) of mineral and sample carrier (Hansen et al., 2018) and the accuracy of the gamma dose to mineral calculation (Burbidge et al., 2016; Tribolo et al., 2019). Despite this well-established body of knowledge, Hansen et al. (2015) note an unexplained 3 % dispersion of their calibration data, subsequently investigated by Autzen et al. (2017). They show that overdispersed calibration data result from attenuation and backscattering, which change in response to changing grain shape and changing sample-carrier material (Autzen et al., 2017). As a consequence, the beta-dose rate should decrease for grain sizes > 100 µm (Wintle and Aitken, 1977) because with increasing grain size the contribution of low-energy backscatter decreases and the primary energy spectrum is more attenuated (Hansen et al., 2018; Greilich et al., 2008). While this has improved our understanding of calibration data significantly, some details are still not fully explained. Here, we test the hypothesis that, in addition to grain size and disc substrate, aliquot size and beta-source shape influence the dose rate. We carried out experiments using three quartz calibration samples characterised by three different grain-size fractions arranged in aliquots of different sizes and compared the experimental data with simulated data obtained from GEANT4 (Agostinelli et al., 2003) and MCNP6 (Werner, 2017; Werner et al., 2018). The results from experiments should allow identifying the impact of grain size, aliquot size and beta-source shape on the dose rate. The simulations should provide a more complete picture of the impact of individual parameters that is hard to achieve with experimental data due to experimental uncertainties being typically above 5 %. We show here that grain size and aliquot size impact on the absorbed dose in response to the irradiation geometry and that this interplay should be reflected in the design of calibration measurements.

The simulation of the irradiation in the lexsyg SMART was performed using the GEANT4 and MCNP6.2 toolboxes. The irradiation geometry simulated (Fig. 2) was adopted from the technical description of the manufacturer and from the sample-carrier description (Fig. 1) with the sample placed in the centre of the cup. Source and housing including the fixing screws were represented as one stainless-steel cylinder surrounded by a stainless-steel shield. The quartz grains were not considered individually but represented as a cylinder, the size of which was modified according to the grain size (height) and aliquot diameter to be simulated. For simulating the dose rate as a function of depth in a given aliquot, the “sample cylinder” was subdivided into 5 or 10 µm thick layers depending on the grain size to be modelled. The material was SiO2 with a density of 1.8 g cm-3 which represents the packing of sand- and silt-sized spherical grains mounted as aliquots. A 5 µm layer of silicon oil was added between the sample and the sample carrier for the simulation of coarse-grain aliquots (grain sizes from 25 to 250 µm). The spectra of the 90Sr / 90Y beta source were simulated using the GEANT4 radioactive decay function (Hauf et al., 2013). Then 108 disintegrations of 90Sr were simulated in each run, and three runs were carried out for each aliquot configuration. The Penetration and Energy Loss of Positrons and Electrons (PENELOPE) code for low-energy particle physics (Baró et al., 1995; Ivanchenko et al., 2011) was employed to calculate path and interaction of the beta particle with the structures presented in the model. The dose deposited in the SiO2 target was recorded in the whole sample cylinder, and a dose-rate profile was constructed as a function of depth in the sample. For simulating small aliquots, the MCNP6 code was used: the target was split into seven spherical cells (Fig. 2b) and the F6 tally was used to simulate the energy deposition averaged over the target cell for electrons and photons separately (see the Supplement for details). The output files produced by the MCNP6 code were used to quantify photon and electron production originating from the interaction mechanisms between beta particle and matter (for details, see the Supplement). The precision of the GEANT4-derived result was calculated for each aliquot configuration at the 95 % confidence level (0.95 CL), based on the standard deviation between the results of the three runs per simulation. The uncertainty of the MCNP6-derived result was obtained from the fractional standard deviation calculated by the Monte Carlo routine.

The data presented here indicate that the dose rate of an individual beta source results from the interplay of a number of parameters. Most of these were identified by previous studies including grain-size-dependent build-up and attenuation of charge (e.g. Wintle and Aitken, 1977; Goedicke, 2007; Autzen et al., 2017). During SAR-based measurements using a 90Sr / 90Y beta source, incident beta particles penetrated the grain to a certain depth alongside backscattered electrons which had energies less than the initial source energy (Bell, 1980). Thus, the absorbed beta dose should decrease with increasing grain size (Wintle and Aitken, 1977; Goedicke, 2007; Hansen et al., 2018). That is why Hansen et al. (2018), building on findings of Greilich et al. (2008), attribute the undesirable overdispersion of their calibration value to variation of grain shape and volume because low-energy beta particles are increasingly attenuated in grains > 100 µm, as already described by Bell (1980). In our simulation, however, charge build-up overcompensates the effects of attenuation resulting in a sustained rise of absorbed beta dose in grains > 150 µm resting on material of relatively high Z (Fig. 8). As a consequence, the simulation shows a continued rise of dose for grains 10–300 µm (Fig. 7) with a flattening of the rise above ∼ 150 µm grain size. This is arguably different but not too dissimilar to datasets deduced from experiments: Geodicke (2007) show an initial rise of dose up to 25–50 µm grain size, followed by a dose plateau for grain sizes 40–130 µm and a decrease for grains >  200 µm and Armitage and Bailey (2005) show a rise to ∼ 40 µm followed by a “jump” to a dose plateau for 50–250 µm grains. Thus, the competing mechanisms of build-up and attenuation lead to divergent dose-rate results mainly for ∼ 50–200 µm grains, likely caused by the geometry of the irradiation field (Bell, 1980).

Large aliquots show the expected build-up of charge with increasing grain size towards secondary equilibrium and small aliquots show the expected larger absorbed dose (Figs. S3–S6) due to the radial increase of dose rate towards the centre of the sample carrier (e.g. Spooner and Allsop, 2000; Veronese et al., 2007). This aliquot-size effect was indeed already highlighted in earlier studies (e.g. Bailiff, 1980; Bell, 1980). Our study provides evidence for further differentiating the aliquot-size effect: the dose enhancement in small aliquots is not the same in the simulation and experiment and is not the same for all grain sizes. The differences are caused by different penetration depths in grains and by the changing effect of backscattered electrons. The interplay seems to have the most variable effect on 50–100 µm grains mounted as aliquots of < 8 mm size (Fig. S8). The dose enhancement is likely reversed when large grains (i.e. > 200 µm) are mounted as small aliquots because with this geometry the probability of backscattered beta particles hitting the edge grain is reduced. However, this edge effect remains to be investigated in greater detail because with changing sphericity of grains (e.g. Autzen et al., 2017) and with potentially changing density of grain packing when the ideal grain monolayer is not achieved, the probability of beta interaction changes as well.

We also show that the shape of the beta source controls the magnitude of the absorbed dose and hence the build-up of charge. The fact that the dose absorbed in small grains must be lower than the dose absorbed in large grains is masked by the ring sources for which fine and coarse grains may absorb the same dose depending on the size of the aliquot (Fig. 4a). The open-ring source shows differences that are statistically negligible for all geometries, suggesting that homogeneity of the source associated with special design reduces the effect of grain and aliquot size on the calibration value.

Autzen et al. (2017) recommend minimising shape and volume variation of sample grains used for calibration, but our data suggest using multiple grain-size fractions for calibration. We think that as long as the sample originates from a natural sedimentary deposit, either way it includes grains of various shape and form. We echo Goedicke (2007) in that the calibration procedure should employ small (4–20 µm), medium (20–80 µm), large (80–200 µm) and very large (200–300 µm) grain sizes. In addition, these grain-size fractions should be measured with small, medium and large aliquots. Calibrating all possible irradiation geometries of an individual beta source appears to be more important the more inhomogeneous a source is, and because source homogeneity is virtually unknown, the calibration procedure must take geometry into account. This will improve the accuracy of the calibration value with respect to the unknown natural sample.

Within the limits of the SAR protocol, the experimental uncertainty of the calibration value is usually reasonably low, thanks to the purpose-prepared sample material. However, with regard to beta-source calibration, a higher precision is desirable. Burbidge et al. (2016) show that parallel multiple-aliquot calibration transfer provides better accuracy and precision than single-aliquot measurements on single-dosed samples. Bos et al. (2006) show that the uncertainty can be reduced to 0.9 %. Their procedure envisages, first, a calculation of the administered gamma dose through Fricke solutions and, second, gamma irradiating several subsamples each with a different dose (e.g. 5, 10, ... 30 Gy). The determined beta De values (s) are then plotted versus the gamma doses (Gy) and the inverse slope of the fitted line gives the beta-dose rate (Bos et al., 2006). The total uncertainty is derived from the uncertainties of beta and gamma irradiation. We therefore say that the laboratory's key parameter can be improved in terms of accuracy and precision by including several grain sizes, several aliquot sizes and several gamma doses in the calibration experiments.

With the number of parameters in mind, it is clear that predicting the dose rate through a series of simulations is too laborious in comparison to a series of relatively simple SAR-based experiments. Here, indeed additional work is required to better estimate the impact of the edge effect on dose rate. If the experimental approach is the way forward, then effort should be made to improve accuracy and precision of the calibration value. Future work should therefore focus on gamma irradiating a calibration sample of several grain-size ranges with several gamma doses in order to determine the value from the regression line and not from a single data point.