calc_AliquotSize(grain.size, sample.diameter, packing.density = 0.65,
MC = TRUE, grains.counted, plot = TRUE, ...)numeric (required): mean grain size (microns) or a
range of grain sizes from which the mean grain size is computed
(e.g. c(100,200)).numeric (required): diameter (mm) of the targeted
area on the sample carrier.numeric (with default) empirical value for mean packing
density.
If packing.density = "inf" a hexagonal structure on an
infinite plane with a packing density of $0.906\ldots$ is assumed.logical (optional): if TRUE the function performs
a monte carlo simulation for estimating the amount of grains on the
sample carrier and assumes random errors in grain size distribution
and numeric (optional) grains counted on a sample carrier. If
a non-zero positive integer is provided this function will calculate the
packing density of the aliquot.
If more than one value is provided the logical (with default): plot output
(TRUE/FALSE)main, xlab, MC.iter).RLum.Results get_RLum.Resultsn:
$$n = (\pi*x^2)/(\pi*y^2)*d$$
where x is the radius of the aliquot size (microns), y
is the mean radius of the mineral grains (mm) and d is the packing
density (value between 0 and 1).
Packing density
The default value for packing.density is 0.65, which is the mean of
empirical values determined by Heer et al. (2012) and unpublished data from
the Cologne luminescence laboratory. If packing.density = "inf"
a maximum density of $\pi/\sqrt12 = 0.9068\ldots$ is used. However,
note that this value is not appropriate as the standard preparation
procedure of aliquots resembles a PECC ("Packing Equal Circles in a
Circle") problem where the maximum packing density is asymptotic to about
0.87.
Monte Carlo simulation
The number of grains on an aliquot can be estimated by Monte Carlo
simulation when setting MC = TRUE.
Each of the parameters necessary to calculate n (x, y, d)
are assumed to be normally distributed with means $\mu_x, \mu_y, \mu_d$
and standard deviations $\sigma_x, \sigma_y, \sigma_d$.
For the mean grain size random samples are taken first from $N(\mu_y, \sigma_y)$,
where $\mu_y = mean.grain.size$ and
$\sigma_y = (max.grain.size-min.grain.size)/4$ so that 95% of
all grains are within the provided the grain size range. This effectively takes into
account that after sieving the sample there is still a small chance of having grains
smaller or larger than the used mesh sizes. For each random sample the mean grain size
is calculated, from which random
subsamples are drawn for the Monte Carlo simulation.
The packing density is assumed to be normally distributed with an empirically
determined $\mu = 0.65$ (or provided value) and $\sigma = 0.18$. The
normal distribution is truncated at d = 0.87 as this is approximately the
maximum packing density that can be achieved in PECC problem.
The sample diameter has $\mu = sample.diameter$ and $\sigma = 0.2$ to
take into account variations in sample disc preparation (i.e. applying silicon spray
to the disc). A lower truncation point at x = 0.5 is used, which assumes
that aliqouts with smaller sample diameters of 0.5 mm are discarded. Likewise,
the normal distribution is truncated at 9.8 mm, which is the diameter of the
sample disc.
For each random sample drawn from the normal distributions the amount of
grains on the aliquot is calculated. By default, 10^5 iterations are used, but
can be reduced/increased with MC.iter (see ...). The results
are visualised in a bar- and boxplot together with a statistical summary.## Estimate the amount of grains on a small aliquot
calc_AliquotSize(grain.size = c(100,150), sample.diameter = 1)
## Calculate the mean packing density of large aliquots
calc_AliquotSize(grain.size = c(100,200), sample.diameter = 8,
grains.counted = c(2525,2312,2880))Run the code above in your browser using DataLab