Mineral grains within sediment or rock absorb a radiation
dose from the decay of radionuclides in the host matrix. For the beta dose
component, the estimated dose rate must be adjusted for the attenuation of
beta particles within the mineral grains. Standard calculations, originally
designed for thermoluminescence dating of pottery, assume that the grain is
embedded in a homogenous medium. However, most current applications of
trapped-charge dating concern sand- or silt-sized dosimeters embedded in
granular sediment. In such cases, the radionuclide sources are not
homogeneous but are localised in discrete grains or held on grain surfaces. We show here that the mean dose rate to dosimeter grains in a granular matrix is dependent on the grain-size distributions of the source grains and of the bulk sediment, in addition to the grain size of the dosimeters. We further argue that U and Th sources are likely to be held primarily on grain surfaces, which causes the dose rate to dosimeter grains to be significantly higher than for sources distributed uniformly throughout grains. For a typical well-sorted medium sand, the beta dose rates derived from surface U and Th sources are higher by ∼ 20 % and ∼ 30 %, respectively, compared to a homogenous distribution of sources. We account for these effects using an expanded model of beta attenuation – including the effect of moisture – and validate the model against Monte Carlo radiation transport simulations within a geometry of packed spheres.
Introduction
Trapped-charge dating methods require an estimate of the radiation dose rate
to a natural dosimeter, usually embedded in sediment or rock. The dose rate
is provided largely through alpha, beta, and gamma radiation arising from the
decay of naturally occurring radionuclides (U and Th series and 40K).
The decay rates of the radionuclides and the amounts of energy released are
relatively well known. By using nuclear data summaries, the measured
activity concentrations of a sample can be converted to an infinite matrix
(IM) dose rate, i.e. the average dose rate to the bulk sample assuming that
the amount of energy absorbed per unit mass equals the amount released. The
different components of the dose rate are attenuated by sediment moisture,
and require a correction based on the presumed average moisture content
during burial. The quantity of interest, however, is the dose rate received
by the natural dosimeters in the matrix, which are usually grains of quartz
or feldspar. For the beta component, there is an additional grain-size-dependent attenuation factor because the range of beta particles in
sediment is comparable to the size of the grains. The beta dose rate to the
dosimeter grains, D˙β (Gy ka-1), can then be defined by
D˙β=D˙β,IM⋅catten⋅cwater+D˙β,internal,
where D˙β,IM is the dry IM beta dose rate (Gy ka-1) and catten and cwater are correction factors for attenuation by grain size and water, respectively. D˙β,internal (Gy ka-1) is
an additional component of the beta dose derived from radionuclides within
the dosimeter grain. This formulation implicitly assumes that the
contribution to the IM dose rate from the internal activity of any one
dosimeter grain is negligible, which is usually true.
Calculation of catten derives from the self-dose values of the
dosimeter grains (Bell, 1979; Mejdahl, 1979; Aitken, 1985, Guérin et
al., 2012). For an active grain within a homogenous inactive matrix, the
self-dose fraction, φ, is the proportion of the energy emitted by
the grain that is self-absorbed or, equivalently, the beta dose rate to the
grain as a proportion of the dose rate to an infinitely large grain. The
value of φ is dependent on the grain size and elemental
composition, and on the beta energy spectrum (i.e. the radionuclide source).
By symmetry, its complement gives the relative attenuation of dose for a
non-active grain in a homogenous, active matrix, hence catten=1-φ (see Aitken, 1985, Appendix C therein).
The key assumption in the use of 1-φ is that the matrix surrounding
the grain is homogenous at the range of beta radiation. This assumption is
likely to be valid in some circumstances, such as for quartz grains imbedded
in fine-grain pottery – and this was indeed the dominant application of
trapped-charge dating in the 1960s and 1970s when the original formulation
was developed. However, since the development of Optically Stimulated
Luminescence (OSL) dating (Huntley et al., 1985) and especially since the
development of single-aliquot OSL protocols (Duller, 1994; Murray
and Wintle, 2000), the vast majority of dating applications concern
sediment with sand-sized or silt-sized grains. In such cases, the size of
the grains is comparable with the range of beta particles. Beta sources in
such sediments are localised – either uniformly distributed throughout the
volume of some individual mineral grains, or in secondary mineral coatings
formed on grain surfaces – and so their distribution in the matrix is
heterogeneous. In such cases it is not clear that the assumption of a
homogenous matrix is reasonable (Guérin et al., 2012).
Our aim here is to re-assess the beta dose rate calculation for dosimeters
in granular sediment, and to propose a model of beta attenuation that is
sufficiently simple for routine application. The model (Sect. 2) seeks to
achieve this by modifying the 1-φ model that is currently in use,
taking account of the variable grain-size distributions of sources,
dosimeters, and bulk sediment, and the possibility of sources being held
on grain surfaces. As input to the model, Sect. 3 provides revised and
extended values of φ for each category of source (K, U, Th) for
both whole-grain and surface sources. These φ values take into
account a recent revision to the 40K beta spectrum and the effect of
etching on the self-dose values of dosimeter grains. In Sect. 4, the
simple model of beta attenuation is tested against detailed
Monte Carlo-based radiation transport simulations with a geometry of closely
packed spheres. Section 5 seeks to incorporate the effect of moisture into
the revised attenuation model. Section 6 shows empirical evidence of
surface-held radionuclides in typical samples.
Balanced Energy Model
Consider first a single dosimeter grain in a homogenous matrix (Fig. 1a). It
is assumed that the grain is too small to have any effect on the matrix
activity concentration. An infinitely small grain will receive the IM dose
rate, which in this case equals the dose rate in the homogenous matrix. A
dosimeter with real mass will receive less than the IM dose rate due to
attenuation (i.e. self-shielding). The attenuation is dependent on grain
size and is the complement of the self-dose, giving the following equation:
catten=1-φdosimeter.
This definition of attenuation has been in common use since Mejdahl (1979).
The difficulty arises when we consider a sediment containing more than one
grain. Figure 1b and c illustrate a matrix containing many grains within
a homogenous source region. The mass of the grains is 50 % of the total
mass in both cases, but the size of the grains is different. Let us assume
that these grains are inert: they have no radioactive sources. In these
scenarios, the IM dose rate equals half the dose rate in the active matrix, and when using the standard approach the dose rate to a dosimeter grain
is the same in each case. In reality, however, the presence of other grains
in the sediment has an effect on the dose rate to the dosimeter grain.
Because the grains are inert, the lowest dose rates in the sediment are
found in the middle of those other grains, where the dose is attenuated due
to shielding. The dosimeter grain is excluded from these low-dose regions,
and so the average dose rate to the dosimeter must be higher than the IM
dose rate. The larger the size of those other grains, the stronger the
effect.
Schematic illustration of some idealised sediments discussed in
Sect. 2. Red indicates a source region or grain, blue indicates a dosimeter
grain, and white indicates a grain or region of no radioactivity.
Accounting for this effect requires the inclusion of a new parameter in the
attenuation calculation. This parameter, θ, must describe the
relative efficiency of dose absorption for each grain or object in the
sediment. It must be dependent on the shape and composition of the grains
but be independent of mass; it must also have a value of 1 when averaged
over all parts of the sediment so that conservation of energy is
maintained. In fact, we already have a parameter that nearly fits this
description: 1-φ. Although 1-φ has been defined above as
the complement of the self-dose, it can also be thought of as the efficiency
of dose absorption in an object relative to a perfect absorber. For
example, if a grain has a 1-φ of 0.9, it receives 90 % of the
dose to a hypothetical perfect absorber. Of interest here, however, is the
efficiency of dose absorption relative to all objects in the matrix. To accomplish this, the 1-φ value
appropriate to the dosimeter grain should be normalised to the mass-weighted
average 1-φ for the matrix:
θ=1-φdosimeter1-φmatrix.
Note that the “matrix” contains everything in the sample, including the
dosimeter of interest. For the idealised scenarios of Fig. 1b and c, the
attenuation to a dosimeter placed within the matrix would be
catten=θ (in calculating 1-φmatrix, the homogenous
source region is a perfect absorber for which 1-φ=1 ). However, in
a realistic granular matrix it is not just the dosimeters that exist as
discrete grains but also the sources (Fig. 1d). Source grains have their own
self-dose, which is dependent on their size and is described by φsource. Only the portion of energy that leaves the source grains
(=1-φsource) is available to dosimeters, and all of that energy must be absorbed by the matrix. We can now describe the attenuation
coefficient for the portion of the dose derived from whole-grain sources as
cw=1-φsourcew1-φdosimeterw1-φmatrixw,
where the superscript w indicates that we are considering the φ of a
whole-grain source (as opposed to a surface source, discussed below). The
relevant φw values are the mass-weighted averages for the
distribution in question (source, dosimeter, or (entire) matrix). For a
grain-size distribution of n bins, the weighted average is
φw=∑i=1nφiwmi∑i=1nmi,
where m is the total mass of the grains in each bin. In practice then, the grain-size distribution of dosimeter grains is usually restricted to a narrow
range through grain-size separation in the laboratory. In this case, a
weighted average is not necessary for φdosimeterw, and
instead its value can be drawn directly from Table 1. If the dosimeter
grains have been etched, then this should be reflected in φdosimeterw by consulting one of the “etched” columns in Table 1.
Quartz self-dose values for whole-grain sources, φw.
With θ written out in full in Eq. (4), it is apparent that if φsourcew=φmatrixw then the equation reduces to cw=1-φdosimeterw, i.e. the standard definition for a
grain in a homogenous matrix. This reduction is valid for the idealised
scenario of Fig. 1a; it might also be relevant for some natural sediments if
sources are held in whole grains (not on surfaces) and the grain-size
distribution of source grains is the same as the bulk sediment, e.g.
quartz or feldspar grains in a well-sorted sand with no secondary
mineralisation. Note that we only consider average dose rates here, not
variations in dose rate from grain to grain; this might be considerable in
this example (e.g. Mayya et al., 2006; Cunningham et al., 2012; Guérin
et al., 2015).
With slight modification, the φ system can be extended to account
for sources that are held on grain surfaces. In this case the self-dose to
source grains is substantially reduced (Table 2), meaning that the
proportion of energy available to external grains is greater than for
whole-grain sources. The variables accounting for attenuation in the
dosimeter and the matrix remain unchanged, so we can define cs as the attenuation coefficient for the portion of the dose rate derived from
surface sources:
cs=1-φsource_effs1-φdosimeterw1-φmatrixw.
In this case, however, the activity concentration is not uniform across the
source grains. If we assume that activity per surface area is constant, it
follows that smaller grains have a greater activity concentration due to
their greater surface-to-volume ratio. Effectively, the mean grain size of
the sources is reduced, which can be accounted for by re-weighting φsource according to the surface-to-volume ratio of the grains:
φsource_effs=∑i=1nφismisvr‾i∑i=1nmisvr‾i,
where svr‾ is the surface-to-volume ratio. If we continue to assume grains are spherical, then
svr‾=4πr243πr3.
Finally, the beta attenuation coefficient for a radionuclide is the weighted
average of the two components (whole-grain sources and surface sources):
catten=pcw+1-pcs,
where p is the proportion of activity held in whole-grain sources.
The preceding discussion of surface sources has concerned those external to
the dosimeter grain. If activity is held on all surfaces, there is an
additional component of the dose rate that derives from the surface of the
dosimeter grain itself. Once again, the activity of the dosimeter grain is
influenced by the surface-to-volume ratio of the grain relative to other
grains in the sediment:
D˙β,internals=βIM1-psvr‾dosimetersvr‾sedimentφdosimetersurf,
where D˙β,internals has units of dose rate (e.g. Gy ka-1) and βIM is the infinite matrix dose rate of the
radionuclide in the bulk sediment. In practice, only the nuclides in the U
and Th chains are likely to be important as surface sources (see Sects. 6
and 7). K may be held on surfaces, e.g. in adhering clay grains, but the
self-dose from K surface sources is much smaller than for U and Th (see
Table 2).
In the case of K-feldspar dosimeters, there is a significant internal dose
rate derived from the 40K source, which in our notation is
D˙β,internalw=Dβ,IMgrainφdosimeterw,
where D˙β,internalw also has units of dose rate (e.g. Gy ka-1). Dβ,IMgrain is the infinite matrix dose rate of the
feldspar grain, and φdosimeterw should be selected from Table 3 (feldspars) using an “etch” column if appropriate.
Potassium feldspar self-dose values for whole-grain sources,
φw, for 40K sources only.
In Sect. 4 this model is tested against simulations using the Monte Carlo
radiation transport code, but first we update and extend tabulated values of
φ.
Self-dose tables
Revised estimates of φ for whole-grain sources (φw) are given in Table 1, calculated using Monte Carlo radiation transport
software MCNP6.2 (Goorley et al., 2012). The model geometry is a spherical
quartz grain (density 2.65 g cm-3) embedded in a larger, low-density
quartz sphere that represents a matrix. Beta electrons are generated
uniformly within the quartz grain; energy deposition in the grain is
recorded and given here as a proportion of the total energy of the starting
particles. The self-dose is defined separately for sources of 40K, the
238U (+235U) series, and the 232Th series. The spectra for
the U and Th series are taken from Guérin et al. (2012) and include
internal conversion and auger emissions. These spectra were used by
Guérin et al. (2012) for equivalent calculations for unetched grains,
and the φw values presented here for unetched U and Th are indistinguishable from theirs. The 40K spectrum used here is the Leutz et al. (1965) curve provided by Cresswell et al. (2018). The Leutz et al. (1965) experimental spectrum has a mean energy of 584 keV, roughly 4 % greater than the spectrum used by Guérin et al. (2012). Cresswell et al. (2018) argue that the higher energy spectrum is a better approximation of the decay transition in 40K, and the result is a modest reduction in
self-dose fraction for a K source compared to Guérin et al. (2012) (but
a significant increase in the IM dose rate).
Table 1 also shows the self-dose fractions for etched quartz, modelled using
concentric spheres of radius 10 and 20 µm less than the
grain radius. These estimates use a slightly different definition of
φ: they are calculated as the dose (energy per mass) received by the “etched” spheres as a proportion of the IM dose for the grain (as opposed to
the usual definition of energy absorbed to energy emitted; the two
definitions are equivalent for an unetched grain). The consequence of
etching is to increase the self-dose to the grain, because the outer,
lower-dose regions have been removed. The relationship is inverted when
considering an external dose: for a dosimeter grain in a homogenous matrix
source, the effect of etching is to reduce the dose to the grain. However,
the effect is relatively small, amounting to a reduction in external dose
rate of between ∼ 0.6 % for K and ∼ 2.8 %
for Th for a 200 µm grain relative to the IM dose. There is little
dependence on grain size, and little difference between a 10 µm etch and 20 µm etch. These results are broadly consistent with the
isolated estimates given by Bell (1979) and Brennan (2003).
The self-dose values are also shown for the case of a surface source,
φs (Table 2). Here, the simulations have been repeated but with the source particles generated within 1 µm of the grain boundary. The
self-dose to the grains is much lower than for whole-grain sources. The
effect of etching is to reduce the self-dose, because the etched regions are
located closer to the source. Table 3 shows the φw values for K-feldspar (KAlSi3O8) with density of 2.60 g cm-3, required for calculation of the internal dose rate to feldspar grains.
Monte Carlo simulations
The Balanced Energy Model (BEM) described in Sect. 2 is validated here
using Monte Carlo simulations of dose deposition in a simulated matrix
consisting of closely packed spherical grains. Three packing configurations
have been prepared for discrete grain-size distributions using the PackLSD
software (Donev et al., 2005), with each using a total of 5000 spheres with
a volumetric packing density of 60 %. The first configuration, “Geometry A” (Fig. 2), uses three grain sizes: 100, 140, and 400 µm diameter, with 50 % of the volume accounted for by the 400 µm grains. This
would be an unusual grain-size distribution to observe in nature, and it is
used here to test the application of beta attenuation models in extreme
cases. Geometries B and C (Fig. 3) are more realistic distributions,
corresponding broadly to a loessic silt and a well-sorted medium sand,
respectively.
Testing beta attenuation models against Monte Carlo radiation
transport simulations using MCNP6 with PackLSD geometry. Geometry
configuration A is the geometrical model for two tests, A1 and A2, with an
atypical grain-size distribution (colours used in the packing illustration
are to help visualise grain sizes). In test A1, the source grains are all
400 µm in diameter, with particles initiated homogenously throughout the grain (“whole grain”), calculated separately for 40K, U-series, and Th-series sources. The mean attenuation factor is shown for 100 and 140 µm dosimeters and compared against (a)1-φdosimeter and the (b) Balanced Energy Model. Test A2 shows equivalent results in (c) and (d) when sources
are all 100 µm grains.
Testing beta attenuation models for realistic grain-size
distributions approximating loess (mean grain size = 47 µm) and
well-sorted sand (mean grain size = 250 µm), using Monte Carlo
simulations with MCNP6 in a close-packed geometry. The grain-size
distribution of sources is the same as the matrix. Colours used in the
packing illustration are to help visualise grain sizes. Panels (a) and (c) show the performance of the 1-φdosimeter model; panels (b) and (d) show the performance of the Balanced Energy Model. Results are shown for both whole-grain and surface sources, with data normalised to the IM dose rate.
Grains are randomly assigned to be sources, dosimeters, or neither,
according to a chosen probability distribution, and simulations were
conducted using MCNP6 (Goorley et al., 2012). All grains were given a
composition SiO2 and density 2.65 g cm-3 (note that there is little
difference in electron stopping power between the main silicate minerals;
e.g. compare Tables 1 and 3 for whole-grain K sources); the remaining space
is defined as air with density 1.205 × 10-3 g cm-3. Of the six outer
boundaries, four are periodic and two are reflective; hence, all energy is
conserved within the box geometry. Simulations were run separately for each
source (K, U, Th) using the same energy spectra described in Sect. 2. The
beta dose (MeV g-1) was recorded in up to 999 dosimeter cells and is
expressed here as a proportion of the IM dose rate. A total of four tests
are described below using three geometries.
Test A1 uses Geometry A, with sources restricted to 400 µm grains.
The total source mass is 50 %, meaning that every 400 µm grain is a source. Simulations are shown separately for K, U, and Th sources, with beta
particles initiated homogenously throughout the source grains (“whole grain”
sources). The performance of the beta attenuation models is shown in Fig. 2a
and b, which plot the estimated beta dose against that observed in the MCNP
simulations. The standard model of beta attenuation assumes that the
dosimeter grain is embedded in a homogenous matrix and is defined here as
1-φdosimeter. In this test, the standard model overestimates the dose to dosimeter grains by 6 %–11 %. Source grains are large and have a
large self-dose, hence there is less energy available to be deposited in
dosimeter grains. The Balanced Energy Model takes this into account and
gives accurate estimates of attenuation for each source (Fig. 2b).
Test A2 uses Geometry A, with whole-grain sources restricted to 100 µm grains. The total mass of sources is 6.5 %, corresponding to 918 out of the 3837 100 µm grains. In this case, the self-dose to the source grains is small, yet a large proportion of the mass is comprised of 400 µm grains. The dosimeter grains (100 and 140 µm), which are
smaller than the sediment average, receive a larger-than-average dose. Note
that when the sources are 40K the attenuation is greater than 1,
meaning that the dose received by the dosimeters is greater than the IM beta
dose for the bulk sediment (this is balanced by the lower-than-average dose
received by the 400 µm grains). The 1-φdosimeter model underestimates the attenuation by 6 %–10 % depending on the source (Fig. 2c), while the BEM is consistent with the Monte Carlo simulations (Fig. 2d).
Test B1 uses Geometry B, which has a grain-size distribution broadly
corresponding to a coarse silt. Source grains comprise 19 % of the total
and have the same size distribution as the matrix. Separate simulations have
been run for whole-grain and surface sources. Dosimeter grains are 20 and 40 µm in diameter. In the case of whole-grain sources, the attenuation
estimated by 1-φdosimeter and the BEM are identical and
correspond closely to the Monte Carlo simulation (Fig. 3a and b). The two
models are not equivalent in the case of surface sources for three reasons.
Firstly, the self-dose of the sources is reduced because the sources are
not generated in the centre of the grain. Secondly, the effective mean grain
size of the sources is also reduced because of the greater
surface-to-volume ratio of smaller grains. Thirdly, the dosimeter grains
receive a self-dose from sources held on their surface. For surface sources
of U and Th, the 1-φdosimeter model of attenuation
underestimates the simulated beta dose by ∼ 7 %. In
contrast, the BEM remains accurate.
Test C1 uses Geometry C, corresponding to a well-sorted sand with mean grain
size of 250 µm. Again, source grains (19 %) have the same grain-size
distribution as the matrix. The simulated attenuation is shown for 200 µm dosimeters (Fig. 3c and d), for whole-grain and surface sources. For grain sizes of 250 µm there is a significant self-dose to source
grains (see Table 1), which is much reduced if sources are located on grain
surfaces (see Table 2). As such, the energy available to dosimeters is
greater when sources are located on grain surfaces. In addition, there is a
significant self-dose to dosimeter grains from surface-held sources. These
effects are not accounted for by the 1-φdosimeter model of
attenuation, leading to an underestimate of simulated beta dose by 9 %,
17 %, and 23 %, for K, U, and Th sources, respectively. The effect is
accounted for in the BEM, which gives accurate estimates of attenuation.
Moisture
Moisture in sediment absorbs radiation. Energy thus absorbed is unavailable
to the sediment, reducing the dose rate received by dosimeter grains. The
strength of the required correction is largely defined by the average mass
of water during the burial period. For beta radiation, the coefficient
of attenuation by water, cwater, is derived from Zimmerman (1971):
cwater=11+xW,
where W is the water-content fraction, expressed as [mass of water / mass of
dry sediment]. The dimensionless parameter x describes the degree to which
water is more effective at absorbing beta radiation than the bulk sediment.
The value of x has been estimated as 1.25 (Zimmerman, 1971) or 1.19 (Aitken
and Xie, 1990) using the ratios of the stopping powers of Al (Zimmerman) or SiO2 (Aitken and Xie) compared to water over the relevant range of beta
energy. Values close to 1.20 have been confirmed using Monte Carlo radiation
software (Nathan and Mauz, 2008; Guérin et al., 2012). However, these
estimates effectively assume that the sediment is a homogenous mixture of
elements. Guérin et al. (2012) have questioned the accuracy of x for
granular sediment, noting that variations in pore size and moisture content
may lead to a different value for x.
Simulated beta dose to 200 µm grains in a sandy matrix (mean grain size 250 µm) plotted against the water content. Simulations are shown for K, U, and Th sources, either whole-grain or surface-held sources. The external beta dose is expressed as a fraction of the IM beta dose rate. Also shown are predicted dose rates using three different derivations for x.
Using the framework of the BEM, we can attempt a fuller explanation of the
role of water in beta attenuation. Noting again that x is a ratio, we need to define the efficiency of dose absorption in pore water and in the dry
matrix. In fact, the latter has already been defined in Sect. 2 as
1-φmatrix. This value is calculated as the weighted average of the 1-φw values for all grains in the matrix. For fine-grained matrices, the self-dose of the grains is very small, and thus 1-φmatrix∼ 1. As the grain size increases, the value of 1-φmatrix becomes significantly less than 1. To calculate x, we need to include a similar parameter for the efficiency of dose absorption by pore water so that
x=1-φwatereff1-φmatrix,
where 1-φwatereff is the effective 1-φ of the pore water. The evaluation of 1-φwatereff is more difficult, but there are some constraints on its value. First, we can recognise that pore water attenuates beta particles, and so it is a non-perfect absorber of radiation definable by 1-φwater,
i.e. dependent on the shape and density of the pore water. We must also
account for the difference in elemental stopping power between the dry
matrix and water, previously assessed as ∼ 1.20. We might
expect that 1-φwatereff∼1.21-φwater. In the limiting case of a homogenous matrix, where both 1-φmatrix and 1-φwater equal 1, then x=1.20. As the grain size (and pore size) increases, 1-φmatrix decreases faster than 1-φwater (because the density
of water is always much less than mineral grains). In consequence, the ratio
x increases with grain size. In contrast, when the water content increases
then so does its self-dose, thus 1-φwater decreases and x decreases. However, for sandy and silty sediments, it is likely that the
self-dose of the water is very close to zero even for saturated sediment.
In practice then, a reasonable approximation for x is
x∼1.201-φmatrix.
The application of this model can be seen in Fig. 4, in which Monte Carlo
simulations have been repeated using geometry C1 (well-sorted sand, mean = 250 µm) with pore space occupied by water at densities corresponding to 1 %–30 % W. The external beta dose to 200 µm grains is shown to decrease as W increases, at a faster rate than predicted using x= 1.20. When x is instead estimated from Eq. (14), the predicted dose is much closer to
that observed from the simulation results. The use of Eq. (14) introduces no
error for K sources and only small errors in the case of U and Th. Firstly,
for U and Th sources the pore water in this simulation has a non-negligible
self-dose, and thus Eq. (14) leads to a slight overestimation of x (Fig. 4b–c). In contrast, Eq. (14) slightly underestimates x for U and Th sources when they are held on grain surfaces (Fig. 4e–f). This is more difficult to explain but
might be caused by differences in the beta energy spectra between
whole-grain and surface sources (significant proportions U and Th series
beta energy is carried by very low-energy electrons, which would never leave
a whole-grain source).
Estimates of the water-content parameter x for four well-sorted
grain-size distributions. Values depend on the radionuclide source, its
location (whole-grain or surface), and on the W (mass of
water over dry matrix). Estimated using Monte Carlo simulations with a geometry of packed grains.
An alternative to Eq. (14) is to estimate x using a Monte Carlo simulation with a representative grain-size distribution.
Table 4 provides estimates of x for four grain geometries, each built from a 1000-grain packed cube, with 60 % volumetric packing density. The
grain-size distributions represent well-sorted silt or sand, with mean grain
size ranging from 50 to 300 µm, including the “silt”
and “sand” distributions used in Sect. 2. Beta particles are simulated
within the grains or on grain surfaces, with the grains once again specified
to be SiO2 with density of 2.65 g cm-3. Pore water is homogenous,
with a density set to reflect the chosen water content from 1 %–30 %. The value of x is
defined as the dose to water divided by the (average) dose of the grains.
For this calculation, only the external beta dose is considered
(i.e. 1-φsource) because the internal dose rates (in both
sources and dosimeters) are not affected by water. The x values listed in
Table 4 have several significant features. First, it is clear that x is
always greater than 1.20 for granular matrices. For silt-sized matrices, x
values are only slightly greater than 1.20 and show little dependence on
the water content or radionuclide source. For sand-sized matrices, the
differences in x are much starker, particularly for surface-held sources of U and Th. For the coarsest simulation (mean grain size of 305 µm), x ranges from 1.34 (K sources with 30 % W) to 2.37 (surface-held Th and 1 % W). However, the most extreme values of x occur when the water content is
very low (e.g. 1 %), in which case the attenuation calculation is not
very sensitive to x. In practice, selecting x from a range of simulated values (Table 4) offers little improvement when compared with the simplicity of Eq. (14), even when (as here) the grain-size distribution is known exactly (see Fig. 4).
The water-content correction can be achieved using Eq. (12), after either
selecting an appropriate value of x for each radionuclide from Table 4 or
using the simple approximation in Eq. (14). However, there is an alternative
formulation to obtain the same result, which we note here to show the
integration of the water-content correction within the BEM. Via Eqs. (3)–(5), we explained that beta attenuation to dosimeter grains depends on how the efficiency of dose absorption in the dosimeters (1-φdosimeter)
compares to the efficiency of all absorbers in the (dry) matrix (1-φmatrix). In the wet matrix, pore water functions as an additional
absorber and could be included in the denominator of Eq. (4). For example,
when considering the whole-grain sources of a radionuclide, the attenuation
coefficient to dosimeters in the wet matrix, cw,wet would be
cw,wet=1-φsourcew1-φdosimeterw1-φmatrix,wetw11+W,
which is similar to Eq. (4) but with a mass correction for the water
(1/1+W), and the 1-φ of the wet matrix in the denominator.
The latter is defined by including 1-φwatereff in the weighted average of 1-φ values of all objects in the matrix (Eq. 5). This can be written explicitly as
1-φmatrix,wetw=1-φmatrix(dry)w+W1-φwatereff1+W,
noting again that (to a first approximation) 1-φwatereff∼1.20 for silt-sized and sand-sized matrices. This
formulation is exactly equivalent to Eqs. (12) and (13).
Radioactivity vs. grain size
The Balanced Energy Model seeks to account for the size distribution of
sources, dosimeters, and the matrix, as well as for the location of radionuclides
in the source grain. It was observed, however, that the model simplifies to
1-φdosimeter when two conditions are met: (1) the source
grain-size distribution is the same as the matrix, and (2) radionuclides are
distributed homogenously within source grains. These conditions can be
tested by measuring the radionuclide concentrations for different grain-size
fractions of a sample. If sources are held on grain surfaces, the
radionuclide concentrations would be proportional to surface-to-volume ratio
(e.g. Olley, 1994). Similarly, any difference in the grain-size distributions
of source and sediment would show up as a grain-size dependence in
radionuclide concentrations. Therefore, if conditions (1) and (2) are met,
the radionuclide concentrations would be independent of grain size and the
Balanced Energy Model could be simplified to 1-φdosimeter.
The dependence of radionuclide concentration on grain size is tested here
for selected samples from sand-sized and silt-sized sediments:
Sample 178108, a coarse sand from Pleistocene fluvial terrace of the Tejo River, Portugal;
Sample 178110, a fine sand from coastal aeolianite (rich in quartz and containing some carbonate
bioclasts and pedogenic nodules), Oitavos region,
Portugal;
Sample 178113, a late Pleistocene loess from Dunaszekcső, Hungary
(Újvári et al., 2018);
Sample 191597, a late Holocene loess from Adventdalen, Svalbard (Gilbert et al., 2018).
Samples were separated into grain-size fractions by wet sieving and Stokes
settling and then ashed, ground, and embedded in wax casts in preparation for
gamma spectrometry. Measurements were performed on a number of HpGe gamma
spectrometers at the DTU laboratory, following the procedures described in
Murray et al. (1987, 2018): briefly, the 40K gamma
emission is measured directly, 226Ra is defined by the gamma emissions
of its progeny 214Pb and 214Bi and forms the most precise
definition of U series activity (we make no judgement here on the state of
disequilibrium in nature), and 232Th is defined by the gamma emissions of its
progeny 228Ac and 212Bi, assuming secular equilibrium.
Activity concentrations for 40K (left column), 226Ra
(middle), and 232Th (right) for various grain-size fractions of four
sedimentary samples measured using HpGe gamma spectrometry and plotted
against the surface-to-volume ratio of the grains. The error bars in the y
direction indicate the 1σ random uncertainty in measurement. The
horizontal width in the error-bar head and tail indicate the range of the
grain-size fraction. Grain-size fractions are given on the secondary x axis. (a–c) Sample 178108, aeolian sand; (d–f) Sample 178110, aeolian sand; (g–i) Sample 178113, loess; and (j–l) Sample 191597, loess. Note that the lower two
samples are plotted on a log scale. Proportionality is illustrated with
curved lines.
Activity concentrations are plotted in Fig. 5 as a function of the
surface-to-volume ratio of the grains in each fraction. The grain-size
fractions are indicated on the secondary x axis. For sand-sized samples (Fig. 5a–f) there is a clear difference between the trends for 40K and for 226Ra and 232Th. For 226Ra and 232Th, the concentrations
are proportional to the surface-to-volume ratio when grain sizes are less
than 250 µm. This proportionality is lost for the larger grain-size
fractions. The trend is a strong indication that U and Th sources are held
on grain surfaces. The non-proportionality in larger fractions could
reflect the carbonate component: fragments of shell or pedogenic carbonate
nodules are generally larger than the silicate grains and may have U and
Th sources that are independent of grain size. In the case of 40K,
there is no evidence of proportionality with surface-to-volume ratio, and
for sample 178108 there is little dependence on grain size. There is a peak in the
40K content at the most dominant grain-size fraction, which is more
pronounced for sample 178110. The association of peak location with dominant
grain size suggests that the 40K source lies in the silicate fraction
– presumably K-feldspar – with the concentrations diluted by non-silicate
minerals (carbonates) at the extremes of the grain-size distribution. The
trends are broadly similar for the loess samples (Fig. 5g–l). The dominant
grain-size fraction of these samples is in the 20–90 µm range. In
this range, the 226Ra and 232Th activity concentrations are
proportional to surface-to-volume ratio; the relationship is largely absent for
40K. For the loess sample 191597 (Fig. 5j–l), there is also an increase
in activity towards larger grain-size fractions; these fractions are very
minor by quantity and presumably represent a locally sourced non-loessic
component of the soil.
Discussion
Gamma spectrometry measurements on grain-size fractions (Sect. 6) show
that, for samples considered here, activity concentrations of 226Ra and
232Th are largely proportional to surface-to-volume ratio of the
grains. These observations are in line with previous findings (e.g. Megumi
et al., 1982; Olley, 1994) and readily explained by the patterns of
mobility exhibited by the different elements. The igneous sources of U and
Th lie in granites and pegmatites. U and Th can form the major or minor
component of accessory minerals (zircon, monazite) but are also found in
significant concentrations adsorbed to crystal or grain boundaries and as
trace elements in the main silicate minerals (Gascoyne, 1992). Under
oxidising conditions at the Earth's surface, the weathering of igneous rock
leads to the mobilisation of U and Th. Uranium is oxidised into its stable
6+ state, forming the soluble uranyl ion, and also readily adheres to
particulates. Thorium is relatively insoluble but is significantly
mobilised as colloids or adsorbed on grain surfaces. In soils and sediments,
U, Th, and Ra are likely to be bound to iron and manganese oxide crusts,
deposited through co-precipitation or ion exchange. Mobilisation processes
are influenced by properties of the mineral host (e.g. surface area),
the solution (e.g. pH, salinity), and the chemical properties of the
species (Chabaux et al., 2003; Carvalho et al., 2014). Note also that the
chemical differences between U, Th, and Ra make it unlikely that the
surface-bound radionuclides are in secular equilibrium (for example, the
excess of 226Ra observed by Olley, 1994).
The presence of radionuclides on grain surfaces has an influence on the
average beta dose rate to dosimeter grains. Source grains have a self-dose, which is dependent on the size of the grain, and that portion of their total
emitted energy is therefore unavailable to dosimeter grains. When
radionuclides are held on the surface of source grains, the self-dose is
reduced compared to a whole-grain source, leading to an increase in the dose
available to dosimeters. In Sect. 3, the calculation of beta dose rates
has been revised by expanding the φ system, with whole-grain and
surface sources accounted for separately. The model requires an estimate,
for each radionuclide category, of the proportion of activity that is held
in whole-grain sources (p in Eq. 9). In the case of 40K, this should
normally equal 1 because it is likely that for most sediments the majority
of the K is held in the primary mineral form (K-feldspar). The choice is
more difficult in the case of U and Th, because both whole-grain and surface
sources are possible. We suspect that surface sources predominate for most
dating applications (so p= 0), as is the case for the samples tested in Sect. 6. However, for some sediments the U and Th sources are located in
resistant minerals – principally zircon and monazite. These minerals can
become concentrated as placer deposits in beach sediment, most often in
tropical regions where rates of erosion are high (Gascoyne, 1992). In such
cases, concentrations of U and Th would correlate with heavy minerals and
associated elements (e.g. Murray and Mohanti, 2006).
The increase in dose rate due to surface-held sources is most significant
for sand-sized sediments. In model C1, for example, the U and Th beta dose
rates are increased by ∼ 20 % and ∼ 30 %,
respectively, compared to whole-grain sources of the same size (see Sect. 4). For silt-sized sediments, the increase is roughly 7 % for U and Th,
although likely to be sensitive to dosimeter grain size. At present,
trapped-charge dating methods are most commonly applied to sandy or silty
sediments, hence the BEM has immediate relevance to dating applications.
Samples with high (U + Th) : K ratios will be most affected: the U and Th radionuclides are most likely to be held on surfaces, and the consequences
of this are also more severe than for 40K. However, the estimated mean
energy of 40K has recently been revised upwards (Cresswell et al.,
2018). Taken together, the BEM and the Cresswell et al. (2018) conversion
factors will tend to increase the estimated dry beta dose rates by roughly
10 %–20 % for sandy sediments, and ∼ 6 % for silty
sediments, compared to current practice. However, current practice also
underestimates the attenuation of the beta dose rate by water, particularly
if sources are held on grain surfaces. The parameter x (describing the degree to which water is a more effective absorber of dose than the dry matrix) is not a constant but is dependent on the grain-size distribution and somewhat dependent on the water content. The dependence of x on grain size, and the increased dose rate from surface-held sources, have both been observed in a previous simulation study (Guerin et al., 2012). The BEM is able to go further: it provides an explanatory model for the effects by using
1-φ as a parameter to describe the efficiency of dose absorption, and
it provides a means of accounting for the effects that is simple enough for
routine analysis (see Supplement).
We have focused discussion on the way surface-held sources alter the
attenuation calculation, but the BEM also departs from 1-φdosimeter when the grain-size distribution of sources differs from that of the bulk matrix. This might occur in sediment with a complex
provenance and/or a broad grain-size distribution: for example, a sand with
a loessic component, or a coastal sediment containing large shell fragments.
The BEM provides a framework for the beta dose rate calculation in such
cases; all that is needed is the grain-size distributions of source and
sediment, and the average φ value for each distribution. The
φ value for a shell, for example, can be obtained relatively simply
with radiation transport codes – far simpler than building a full geometry
for a mixed sediment (cf. Cunningham et al., 2011; Cunningham, 2016).
Finally, we should note that the average beta dose rate is unaffected by the
number of source grains (although the dispersion certainly is).
Conclusion
For a single grain in a homogenous matrix, grain-size attenuation of the
beta dose rate is accurately defined by 1-φdosimeter. In a
granular matrix, 1-φdosimeter is only accurate on average if the size distribution of source grains is the same as the bulk matrix. This assumption fails, for example, if radionuclides are held on grain surfaces, which is likely in the case of the U and Th series. The Balanced Energy Model of beta attenuation, described here, modifies and extends the φ system so that it is applicable to granular sediment. The BEM has been successfully tested against radiation transport modelling and is able to
account for surface-held sources, for differences in the grain-size
distributions of source and sediment, and for the grain-size dependence of
the water-content correction. Its use is likely to improve the accuracy of
beta dose rate estimates for dating applications.
Data availability
A spreadsheet implementation of the Balanced Energy Model is included in the Supplement. All data in this paper is given in the figures, tables, or Supplement.
The supplement related to this article is available online at: https://doi.org/10.5194/gchron-4-517-2022-supplement.
Author contributions
JPB and ASM initiated and guided the project. ACC designed the model, conducted simulations and measurements, and prepared the manuscript.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
Our thanks to Pedro Cunha, Gábor Újvári, and Christian
Rasmussen for providing samples. We are very grateful to Guillaume
Guérin and Svenja Riedesel for their thoughtful reviews of the
manuscript and to Barbara Mauz for commenting on the preprint.
Financial support
This research has been supported by the European Research Council, H2020 European Research Council (RELOS, grant no. 639904).
Review statement
This paper was edited by Julie Durcan and reviewed by Guillaume Guérin and Svenja Riedesel.
References
Aitken, M. J.: Thermoluminescence dating, Academic Press,
ISBN-13: 978-0120463817, 1985.
Aitken, M. J. and Xie, J.: Moisture correction for annual gamma dose,
Ancient-TL, 8, 6–9, 1990.
Bell, W. T.: Attenuation Factors for the Absorbed Radiation Dose in Quartz
Inclusions for Thermoluminescence Dating, Ancient-TL, 2–13, 1979.Brennan, B. J.: Beta doses to spherical grains, Radiat. Meas., 37, 299–303,
10.1016/S1350-4487(03)00011-8, 2003.
Carvalho, F., Chambers, D., Fernandes, S., Fesenko, S., Goulet, R., Howard,
B., Kim, C. K., Martin, P., Moore, W. S., and Phaneuf, M.: The environmental
behaviour of radium: revised edition, International Atomic Energy Agency,ISBN-13: 978-9201433107, 2014.
Chabaux, F., Riotte, J., and Dequincey, O.: U-Th-Ra fractionation during
weathering and river transport, in: Uranium-series geochemistry, edited by: Bourdon, B., Henderson, G. M., Lundstom, C. C., and Turner, S. P., Mineralogical Society of America, Washington, DC, USA, TS14 ISBN-13: 978-0939950645, 2003.Cresswell, A. J., Carter, J., and Sanderson, D. C. W.: Dose rate conversion
parameters: Assessment of nuclear data, Radiat. Meas., 120, 195–201,
10.1016/j.radmeas.2018.02.007, 2018.
Cunningham, A. C.: External beta dose rates to mineral grains in
shell-rich sediment, Ancient TL, 34, 1–5, 2016.Cunningham, A. C., Bakker, M. A. J., van Heteren, S., van der Valk, B., van
der Spek, A. J. F., Schaart, D. R., and Wallinga, J.: Extracting storm-surge
data from coastal dunes for improved assessment of flood risk, Geology, 39,
1063–1066, 10.1130/G32244.1, 2011.Cunningham, A. C., DeVries, D. J., and Schaart, D. R.: Experimental and
computational simulation of beta-dose heterogeneity in sediment, Radiat.
Meas., 47, 1060–1067, 10.1016/j.radmeas.2012.08.009, 2012.
Donev, A., Torquato, S., and Stillinger, F. H.: Neighbor list
collision-driven molecular dynamics simulation for nonspherical hard
particles. I. Algorithmic details, J. Comput. Phys, 202, 737–764, 2005.Duller, G. A. T.: Luminescence dating of sediments using single aliquots:
New procedures, Quaternary Sci. Rev., 13, 149–156,
10.1016/0277-3791(94)90041-8, 1994.
Gascoyne, M.: Geochemistry of the actinides and their daughters, in:
Uranium-series disequilibrium: applications to earth, marine, and environmental sciences, edited by: Ivanovich, M. and Harmon, R. S., Oxford University Press, Oxford, ISBN: 978-0198542780, 1992.Gilbert, G. L., O'Neill, H. B., Nemec, W., Thiel, C., Christiansen, H. H.,
and Buylaert, J.-P.: Late Quaternary sedimentation and permafrost
development in a Svalbard fjord-valley, Norwegian high Arctic,
Sedimentology, 65, 2531–2558, 10.1111/sed.12476, 2018.
Goorley, T., James, M., Booth, T., Brown, F., Bull, J., Cox, L. J., Durkee,
J., Elson, J., Fensin, M., and Forster, R. A.: Initial MCNP6 release
overview, Nucl. Technol., 180, 298–315, 2012.
Guérin, G., Mercier, N., Nathan, R., Adamiec, G., and Lefrais, Y.: On
the use of the infinite matrix assumption and associated concepts: a
critical review, Radiat. Meas., 47, 778–785, 2012.Guérin, G., Jain, M., Thomsen, K. J., Murray, A. S., and Mercier, N.:
Modelling dose rate to single grains of quartz in well-sorted sand samples:
The dispersion arising from the presence of potassium feldspars and
implications for single grain OSL dating, Quat. Geochronol., 27,
52–65, 10.1016/j.quageo.2014.12.006, 2015.
Huntley, D. J., Godfrey-Smith, D. I., and Thewalt, M. L.: Optical dating of
sediments, Nature, 313, 105–107, 1985.Leutz, H., Schulz, G., and Wenninger, H.: The decay of potassium-40, Z.
Physik, 187, 151–164, 10.1007/BF01387190, 1965.Mayya, Y. S., Morthekai, P., Murari, M. K., and Singhvi, A. K.: Towards
quantifying beta microdosimetric effects in single-grain quartz dose
distribution, Radiat. Meas., 41, 1032–1039,
10.1016/j.radmeas.2006.08.004, 2006.
Megumi, K., Oka, T., Yaskawa, K., and Sakanoue, M.: Contents of natural
radioactive nuclides in soil in relation to their surface area, J. Gepphys.
Res.-Sol. Ea., 87, 10857–10860, 1982.
Mejdahl, V.: Thermoluminescence dating: beta-dose attenuation in quartz
grains, Archaeometry, 21, 61–72, 1979.Murray, A., Marten, R., Johnston, A., and Martin, P.: Analysis for naturally
occuring radionuclides at environmental concentrations by gamma
spectrometry, J. Radioanal. Nucl. Ch., 115, 263–288, 1987.
Murray, A. S. and Mohanti, M.: Luminescence dating of the barrier spit at
Chilika lake, Orissa, India, Radiat. Prot. Dosim., 119, 442–445,
10.1093/rpd/nci500, 2006.
Murray, A. S. and Wintle, A. G.: Luminescence dating of quartz using an
improved single-aliquot regenerative-dose protocol, Radiat. Meas., 32,
57–73, 2000.Murray, A. S., Helsted, L. M., Autzen, M., Jain, M., and Buylaert, J. P.:
Measurement of natural radioactivity: Calibration and performance of a
high-resolution gamma spectrometry facility, Radiat. Meas., 120, 215–220,
10.1016/j.radmeas.2018.04.006, 2018.Nathan, R. P. and Mauz, B.: On the dose-rate estimate of carbonate-rich
sediments for trapped charge dating, Radiat. Meas., 43, 14–25,
10.1016/j.radmeas.2007.12.012, 2008.Olley, J. M.: The use of 232U and 232Th decay series radionuclides in sediment tracing, Australian National University, PhD thesis, 1994.Újvári, G., Stevens, T., Molnár, M., Demény, A., Lambert,
F., Varga, G., Jull, A. J. T., Páll-Gergely, B., Buylaert, J.-P., and
Kovács, J.: Coupled European and Greenland last glacial dust activity
driven by North Atlantic climate, PNAS, 114, E10632–E10638,
10.1073/pnas.1712651114, 2017.
Zimmerman, D. W.: Thermoluminescent dating using fine grains from pottery,
Archaeometry, 13, 29–52, 1971.