calc_AliquotSize: Estimate the amount of grains on an aliquot

In R-Lum/Luminescence: Comprehensive Luminescence Dating Data Analysis

Estimate the amount of grains on an aliquot

Description

This function can be used to either estimate the number of grains on an aliquot or to compute the packing density, depending on the arguments provided.

The following formula is used to estimate the number of grains n:

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 (a 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 (2012) and unpublished data from the Cologne luminescence laboratory.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 / \sqrt{12} \approx 0.9069 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. All 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 a 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 aliquots with sample diameter smaller than 0.5 mm are discarded. Likewise, the normal distribution is truncated at the diameter of the sample carrier (9.8 mm by default, but controllable via the sample_carrier.diameter argument).

For each random sample drawn from the normal distribution, the amount of grains on the aliquot is calculated. By default, 10^4 iterations are used, but the can be controlled with option MC.iter (see ...). Results are visualised in a bar- and boxplot together with a statistical summary.

Usage

calc_AliquotSize(
  grain.size,
  sample.diameter,
  packing.density = 0.65,
  MC = TRUE,
  grains.counted,
  sample_carrier.diameter = 9.8,
  plot = TRUE,
  ...
)

Arguments

grain.size	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)).
sample.diameter	numeric (required): diameter (mm) of the targeted area on the sample carrier.
packing.density	numeric (with default): empirical value for the mean packing density. If packing.density = Inf, a hexagonal structure on an infinite plane with a packing density of \pi / \sqrt{12} \approx 0.9069 is assumed.
MC	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 packing density. Requires grain.size to be specified as a 2-element vector with the range (min and max) of grain sizes. For more information see details.
grains.counted	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 mean packing density and its standard deviation are calculated. Note that this overrides packing.density.
sample_carrier.diameter	numeric (with default): diameter (mm) of the sample carrier.
plot	logical (with default): enable/disable the plot output.
...	further arguments and graphical parameters to be passed. In particular, MC.iter to control the number of Monte Carlo iterations, main, xlab, col, line_col, line_lwd, cex, rug (TRUE/FALSE), boxplot (TRUE/FALSE), summary (TRUE/FALSE), legend (TRUE/FALSE).

Value

Returns a terminal output. In addition an RLum.Results object is returned containing the following element:

.$summary	data.frame summary of all relevant calculation results.
.$args	list used arguments
.$call	call the function call
.$MC	list results of the Monte Carlo simulation

The output should be accessed using the function get_RLum.

Function version

0.33

Author(s)

Christoph Burow, University of Cologne (Germany)
Marco Colombo, Institute of Geography, Heidelberg University (Germany)

References

Duller, G.A.T., 2008. Single-grain optical dating of Quaternary sediments: why aliquot size matters in luminescence dating. Boreas 37, 589-612.

Heer, A.J., Adamiec, G., Moska, P., 2012. How many grains are there on a single aliquot?. Ancient TL 30, 9-16.

Further reading

Chang, H.-C., Wang, L.-C., 2010. A simple proof of Thue's Theorem on Circle Packing. https://arxiv.org/pdf/1009.4322v1, 2013-09-13.

Graham, R.L., Lubachevsky, B.D., Nurmela, K.J., Oestergard, P.R.J., 1998. Dense packings of congruent circles in a circle. Discrete Mathematics 181, 139-154.

Huang, W., Ye, T., 2011. Global optimization method for finding dense packings of equal circles in a circle. European Journal of Operational Research 210, 474-481.

Examples

## Estimate the amount of grains on a small aliquot
calc_AliquotSize(grain.size = c(100,150), sample.diameter = 1, MC.iter = 100)

## Calculate the mean packing density of large aliquots
calc_AliquotSize(grain.size = c(100,200), sample.diameter = 8,
                 grains.counted = c(2525,2312,2880), MC.iter = 100)

R-Lum/Luminescence documentation built on June 9, 2025, 2:40 p.m.