Description Usage Arguments Details Value Function version How to cite Author(s) References Examples

View source: R/calc_AliquotSize.R

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

1 2 3 4 5 6 7 8 9 | ```
calc_AliquotSize(
grain.size,
sample.diameter,
packing.density = 0.65,
MC = TRUE,
grains.counted,
plot = TRUE,
...
)
``` |

`grain.size` |
numeric ( |

`sample.diameter` |
numeric ( |

`packing.density` |
numeric ( |

`MC` |
logical ( |

`grains.counted` |
numeric ( |

`plot` |
logical ( |

`...` |
further arguments to pass ( |

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

The following function is used to estimate the number of grains `n`

:

*n = (π*x^2)/(π*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 (2012) and unpublished data from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 *π/√12 = 0.9068…* 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
*μ_x, μ_y, μ_d* and standard deviations *σ_x, σ_y, σ_d*.

For the mean grain size random samples are taken first from
*N(μ_y, σ_y)*, where *μ_y = mean.grain.size* and
*σ_y = (max.grain.size-min.grain.size)/4* so that 95\
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 *μ = 0.65*
(or provided value) and *σ = 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
*μ = sample.diameter* and *σ = 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.

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.

0.31

Burow, C., 2020. calc_AliquotSize(): Estimate the amount of grains on an aliquot. Function version 0.31. In: Kreutzer, S., Burow, C., Dietze, M., Fuchs, M.C., Schmidt, C., Fischer, M., Friedrich, J., 2020. Luminescence: Comprehensive Luminescence Dating Data Analysis. R package version 0.9.7. https://CRAN.R-project.org/package=Luminescence

Christoph Burow, University of Cologne (Germany) , RLum Developer Team

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. 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, 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.

1 2 3 4 5 6 | ```
## 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)
``` |

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