Luminescence (version 0.3.1)

calc_AliquotSize: Estimate the amount of grains on an aliquot

Description

Estimate the number of grains on an aliquot. Alternatively, the packing density of an aliquot is computed.

Usage

calc_AliquotSize(grain.size, sample.diameter, packing.density = 0.65, 
    grains.counted)

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 mean packing density. If packing.density = "inf" a hexagonal structure on an infinite plane with a packing density of $0.906\ldots$ is assumed.

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

Value

• Returns terminal output. In addition a list is returned containing the following element:
• results
  data frame with calculation results.

Details

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 function 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 (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The default value for packing.density is 0.65, which is the mean of empirical values determined by Heer et al. 2012 and unpublished data from the cologne luminescence laboratory.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.

References

Duller, G.A.T., 2008. Single-grain optical dating of Quaternary sediments: why aliquot size matters in luminescence dating. Boreas 37, pp. 589-612. Heer, A.J., Adamiec, G., Moska, P., 2012. How many grains are there on a single aliquot?. Ancient TL, 30, pp. 9-16. Further reading Chang, H.-C., Wang, L.-C., 2010. A simple proof of Thue's Theorem on Circle Packing. http://arxiv.org/pdf/1009.4322v1.pdf, 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, pp. 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, pp. 474-481.

Examples

## Estimate the amount of grains on a small aliquot
calc_AliquotSize(grain.size = 125, sample.diameter = 1) 

## Calculate the mean packing density of large aliquots
calc_AliquotSize(grain.size = 125, sample.diameter = 8, 
                 grains.counted = c(2525,2312,2880))