Research paper2D modelling: A Monte Carlo approach for assessing heterogeneous beta dose rate in luminescence and ESR dating: Paper I, theory and verification
Introduction
In ESR and luminescence dating, it is essential to determine the dose rate for age calculation. Even if samples are often heterogeneous at different scales, in respect of mineralogical structure and/or distribution of radioactive elements in the sample and its surroundings (Nathan et al., 2003; Guérin et al., 2012a; Urbanova et al., 2015), it is usual to determine the U, Th, K contents (by beta counting, gamma spectrometry or ICPMS measurements), apply the infinite matrix assumption and consider attenuation factors based on the individual grain size and the bulk moisture content (Fleming, 1973; Zimmerman, 1971; Mejdahl, 1979) as well as, eventually, disequilibrium in the natural radioactive decay series (Krbetschek et al., 1994). However, the beta dose rate received by the mineral phase containing the dated grains (e.g., quartz or feldspar phases in rocks, clay in sediment or pottery, and etc.) can be significantly different from that of the bulk beta dose rate, depending on the sizes and proportions of the different mineral phases, and the contrasts in the radioactive element contents (Nathan et al., 2003). Determining the specific beta dose rate in the phase used for the equivalent dose determination is then of paramount importance in heterogeneous samples, but remains delicate due to the complexity and variety of the structures observable in these samples.
One option to tackle this problem is directly map the dose distribution (Rufer and Preusser, 2009; Guérin et al., 2012b) but since beta particles have a range of a few millimeters, this approach is rather difficult. An alternative is to develop numerical simulations (Nathan et al., 2003; Mayya et al., 2006; Guérin, 2011; Cunningham et al., 2012; Martin, 2015; Martin et al., 2015a) and this approach has already shown promising results for investigating beta microdosimetry. Some of them show that, in heterogeneous samples, in particular when the presence of high radioactivity contrasts between different minerals and/or coarse materials (millimeter or centimeter size) is attested, the standard method may lead to biased estimations that can sometimes be more than twice the simulated beta dose rate (Nathan et al., 2003). However, it has to be noted that only few of these studies have received an experimental validation that would be necessary for confirming the order of magnitude of the observed differences. In addition, a numerical approach usually requires the creation of 3D models, what remains a complex task, as well as the necessary characterisation of the structure and composition of the sample, used for the creation of the numerical model itself.
The recent development of the DosiVox software, which includes the Geant4 database for particles-to-matter interactions and allows users to compute dose rate simulations without programming skills, makes this approach more accessible. Although this tool has simplified the modelling process, the major issues are still remaining on acquiring data of sample's structure, mineral's chemical composition and its radioactivity. Even when it is possible to extrapolate the three dimensional structure from a slice analysis (for instance obtained using LA-ICP-MS or X-ray fluorescence mapping) (Martin et al., 2015b), a 3D approach is still time consuming as it requires several days or weeks of calculation on standard computers (without considering the time for the analysis and processing). Another possibility is to model a sediment or a rock sample by packing randomly spherical grains distributed in a compact structure (Nathan et al., 2003; Guérin et al., 2012a). This approach is limited by the efficiency of the algorithm (Donev et al., 2005) and the representativeness of the geometric construction. The shape of the grain may also have an impact on the calculated beta dose rate, depending on their size compared to beta particle range since a significant effect can be observed for grains close or larger than the beta average range in sediment (i.e. few millimeters) (Fain et al., 1999; Nathan, 2010). If dating are usually performed on smaller grains – few hundred of micrometers for the quartz inclusion technique (Fleming, 1973) or less than 20 μm for the fine grain method (Zimmerman, 1971) – the shape of larger fractions that can be present in the sample (for instance, calcite or granite grains or small stones) is likely to have a significant effect on the beta dose rate, especially if the contrast between the coarse material and the mineral phases containing the dated grains is strong (Nathan et al., 2003). In addition, random grain distributions cannot represent agglomerated grains or layered samples with different granulometric distributions.
X-ray computed tomography (CT) may be an alternative since it allows the visualization and quantification of the internal structure of objects (Aso et al., 2007; Mena et al., 2015). However, CT equipments are not easy accessible and sample scanning is expensive. More importantly, it cannot be used for samples rich in quartz and plagioclases since these two minerals are almost impossible to be distinguished in CT images (Boone et al., 2011).
Plachy and Sutton (1982) used a dose point kernel method to integrate the beta dose rate contributions to quartz from the main mineral phases present in a granite. They used cathodoluminescence photography to create 2D maps of the different phases while abrasing the sample slice after slice. They showed in particular that one can obtain the equivalent beta dose rate in applying their calculation method in two dimensions (to the individual images) or three dimensions (to the 3D images reconstituted by assembling the 2D images). This observation can probably be related to the isotropy of the mineral distribution in the sample, as stereology can predict the conservation of size ratios (volume fractions, surface fractions and intercepted length fractions of the different phases) and the mean intercept length between two and three dimensions for an isotropic texture (Underwood, 1970; Degallaix and Ilschner, 2007). As beta dose rate distribution is related to the size ratio and distance (it is in fact the basic of the dose point kernel method) between different mineral phases, the conservation of the beta dose rate between two and three dimension can be seen as a consequence of the isotropy of the sample, according to stereology.
Calculating or modelling the beta dose rate in 2D, as Plachy and Sutton (1982) did, has a strong potential for dating heterogeneous samples, as it is easier to obtain data about composition, radioactivity and spatial distribution of the different mineral phases from two dimensional analysis than three dimensional analysis, or mineral separation. In order to take advantage of this method and provide a user-friendly tool for the dosimetric dating community, the DosiVox-2D software has been developed (Martin, 2015, chapter 11). Contrary to DosiVox, DosiVox-2D allows 2D-modelling but is only applicable to heterogeneous sediments and rock samples whose minerals are randomly distributed in space (isotropic), i.e. the probability of encountering a particular mineral phase is equal in all the three directions. However, this property has only to apply at the beta particle range (around 2 mm) and at the sampling size required for the dating (usually from a millimeter to a centimeter). Sample slices are used to obtain mineral compositions, distributions and radioactive element concentrations for subsequent simulations.
In this paper, we first explain the concept of 2D modelling, introduce the DosiVox-2D software, and then examine this new modelling approach with virtual and natural examples. The results are compared to those obtained with voxelised 3D models run with DosiVox (Martin et al., 2015a), grains packing models (Nathan et al., 2003; Guérin et al., 2012a), and the standard approach based on the infinite matrix dose rate concept (Fleming, 1973; Zimmerman, 1971; Aitken, 1985). The shape of the beta dose rate distributions obtained with the 3D and 2D modellings are also compared. The application of tabulated attenuation factors for calculating the beta dose rate to the modelled mineral phases is also discussed. At this stage of demonstration of the concept and its potential, we made the approximation that all samples considered here are isotropic. The effect of anisotropy on the modelling results will only be slightly discussed in this paper, and more developed in Fang et al. (2018).
Section snippets
Principles and theory of 2D modelling
Like previously explain, the principle of 2D-modelling relies on the conservation of several properties between 2D and 3D in isotropic samples. In particular, the surface fractions of the different phases on a representative 2D image are equal to the volume fractions (in the 3D), which ensure that the mineral proportions are respected. The mean intercept length, which can be defined as the average chord length of a particular phase measured in random directions, is also conserved in 2D if the
Software description
DosiVox-2D is a Geant4 based software for creating fast dosimetric models. It simulates particles-to-matter interactions by a Monte-Carlo approach (Agostinelli et al., 2003; Allison et al., 2006, 2016), calculating the paths and interactions of each primary particle and secondary particles, step by step, according to interaction probabilities in the material. This process displays the particle's behavior in the matter, and has already shown its usefulness for dose rate calculation in dosimetric
2D-modelling in virtual sample
To investigate the applicability of DosiVox-2D, four mineralogical environments (Table 1 and Fig. 2) were generated in DosiSed (Martin, 2015, chapter 13), a software that generates distributions of randomly packed spherical grains of different minerals and radioactivities, and simulates their dose rates (Nathan et al., 2003; Guérin et al., 2012a). In this test, the results of 2D and 3D model are compared with that of standard approach (infinite matrix dose and attenuation factors).
For each
Comparison of the beta dose rate distribution shape between 3D and 2D models
The use of the single grain dating technique (Duller et al., 1999) often leads to the observation of large equivalent doses overdispersions in sediment samples. Various factors may be involved like incomplete bleaching or sensibility changes and beta dose rate variations at the grain scale (Nathan et al., 2003; Guérin et al., 2012a). These variations may be visualized and investigated for different grain sizes and mineral phases of a sample with DosiVox and DosiVox-2D (Martin, 2015, chapter 11,
Beta dose rate in non-apparent grains
Luminescence and ESR measurements are usually not performed on bulk material or on the millimeter or larger fraction, but rather on grain size fractions ranging from few micrometers (fine grains) to few hundreds of micrometers (coarse grains). These grains may not explicitly appear on the mapping analysis for various reasons: they can be smaller than the resolution of the analysis, or impossible to deconvolve because of the probing range, or indistinguishable from the surrounding material (like
Comparison between 3D and 2D modelling in a granite sample
The following simulations are used to compare the results and accuracy of 3D and 2D modelling of a natural granite sample (migmatite type) from Kullu valley, Northwestern Indian Himalaya. However, we assumed for this demonstration that the effect of this slight anisotropy does not affect significantly the beta dose rate. The sample is mainly composed of quartz (30% in mass), plagioclase (39% in mass), K-feldspar (18% in mass) and mica (13% in mass). It has been determined by ICPMS and ICPOES
Conclusions
We verified that 2D-modelling provides similar results as 3D modelling (and also as spheres packing modelling for the first cases) for both the virtual sediment cases and the granite sample. By integrating many more parameters about the distribution of minerals and radioactivity, this method allows calculating beta dose rates in heterogeneous samples for which the standard calculation method and attenuation factors are not fully reliable, or for which the uncertainty relative to their
Acknowledgments
This work was funded by the research programme LaScArBx (Labex Sciences Archéologiques de Bordeaux) supported by the ANR, and the Nouvelle-Aquitaine region. The authors are grateful to Alan Guitard for the adaptation of DosiVox interface and the development of DosiVox-2D interface. We are thankful to Prof. Gordon Lister and Dr. Marnie Forster for providing the rock sample. We also thank Prof. Tim Senden and Dr. Michael Turner for CT scanning, Mr. Holger Averdunk and Mrs. Jill Middleton for
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