LA-ICP-MS fission track dating

Basics

LA-ICP-MS FT single-grain age calculation using the ‘zeta’ approach:

\(t = \frac{1}{\lambda}~\ln \left(1+\frac{1}{2} \lambda \zeta_{icp} \frac{N_s}{A_s [ ^{238}U/^xX]} \right)\)

single grain uncertainties:

\(\frac{s[t]}{t} \approx \sqrt{ \left( \frac{s[\zeta_{icp}]}{\zeta_{icp}} \right) ^2 + \left( \frac{s[^{238}U/^xX]}{[^{238}U/^xX]} \right) ^2 + \frac{1}{N_s} }\)

The pooled age is

\(\overline{t} = \frac{1}{\lambda}~\ln \left( 1 + \frac{1}{2} \lambda \zeta_{icp} \frac{\sum N_s}{\sum A_s [^{238}U/^{x}X]} \right)\)

Session ‘zeta’ is given by

\(\zeta_{icp} = \frac{\left( e^{\lambda t} -1 \right) U }{ \frac{1}{2} \lambda \rho_s}\)

and its uncertainties are

\(\frac{s[\zeta_{icp}]}{\zeta_{icp}} = \sqrt{ \frac{1}{N_s} + \left( \frac{s[t]}{t}\right)^2 + \left(\frac{s[^{238}U/^{x}X]}{[^{238}U/^{x}X]}\right)^2}\)

Data import

U [U content] is given as the ratio between 238U/43Ca based on cps (for apatite) or U/Si (zircon). Therefor, we have to rename the columns.

library(readxl)
library(dplyr)

library(laftr)

# load the U concentrations
sample <- read_xlsx("../inst/extdata/example.xlsx", sheet = "sample") |>
  rename(U = UCa, sU = sUCa)

# load the calibration measurements for zeta factor
zeta.measurements <- read_xlsx("../inst/extdata/example.xlsx", sheet = "standard") |>
  rename(U = UCa, sU = sUCa)

Zeta factor of the ICP-session

zeta_ICP() by giving the calibration measurements and the used standard:

session.zeta <- zeta_ICP(zeta.measurements, spotsize = 40, mineral = "apatite", standard = "Dur")

It returns a list with the original data:

head(session.zeta$data) |> as_tibble()

## # A tibble: 6 × 6
##   Grain    Ns UO           U       sU comments
##   <chr> <dbl> <lgl>    <dbl>    <dbl> <lgl>   
## 1 DUR-1     4 NA    0.000496 0.000016 NA      
## 2 DUR-2     3 NA    0.00052  0.000018 NA      
## 3 DUR-3     1 NA    0.000476 0.000014 NA      
## 4 DUR-4     3 NA    0.000465 0.000014 NA      
## 5 DUR-5     6 NA    0.000471 0.000012 NA      
## 6 DUR-6     3 NA    0.000469 0.000014 NA

… the area, track densities, and the individual zetas:

head(session.zeta$results) |> as_tibble()

## # A tibble: 6 × 2
##      rhos   zeta
##     <dbl>  <dbl>
## 1 318310. 0.0982
## 2 238732. 0.137 
## 3  79577. 0.377 
## 4 238732. 0.123 
## 5 477465. 0.0622
## 6 238732. 0.124

… and the final zeta factor (with uncertainty)

session.zeta$zeta |> as_tibble()

## # A tibble: 2 × 1
##    value
##    <dbl>
## 1 0.118 
## 2 0.0118

Age calculation

age_ICP() using the just calculated zeta factor for calibration

result <- age_ICP(sample, spotsize = 40, zeta = session.zeta$zeta)

returns a list containing the original dataset:

head(result$data) |> as_tibble()

## # A tibble: 6 × 5
##   Grain    Ns        U       sU comments    
##   <chr> <dbl>    <dbl>    <dbl> <chr>       
## 1 FCT-1     7 0.000905 0.000016 NA          
## 2 FCT-2     1 0.000828 0.000017 broken grain
## 3 FCT-3     2 0.000774 0.000015 NA          
## 4 FCT-4     1 0.000416 0.000022 NA          
## 5 FCT-5     2 0.000724 0.000019 edge        
## 6 FCT-6     1 0.000122 0.000013 NA

.. the individual ages:

head(result$ages) |> as_tibble()

## # A tibble: 6 × 7
##       t    st st.perc t.max   t.min    rhos zeta.error
##   <dbl> <dbl>   <dbl> <dbl>   <dbl>   <dbl>      <dbl>
## 1 36.3  14.2     39.1  50.5 22.1    557042.     0.0644
## 2  5.68  5.71   101.   11.4 -0.0295  79577.     0.168 
## 3 12.2   8.68    71.4  20.8  3.47   159155.     0.119 
## 4 11.3  11.4    101.   22.7 -0.0720  79577.     0.168 
## 5 13.0   9.28    71.5  22.3  3.71   159155.     0.119 
## 6 38.5  38.9    101.   77.3 -0.408   79577.     0.169

… the pooled age of the sample:

result$age.pooled

##       age       std 
## 15.212652  3.387202

… and the chi-squared statistics:

result$stats

##       chisq probability 
##  42.2710325   0.8545437

Compatibility with IsoplotR

# library(IsoplotR)
data.iso <- as.fissiontracks(sample, spotsize = 40, zeta = session.zeta$zeta)
ages.iso <- as.fissiontracks(result$ages, ages = TRUE)

KDE

using the functions from the IsoplotR package (P. Vermeesch):

IsoplotR::kde(ages.iso[, 1])

or the laftr function geom_aKDE() using adaptive KDE with ggplot2:

library(ggplot2)

data.frame(t = ages.iso[, 1], st = ages.iso[, 2]) |>
  ggplot(aes(x = t, weight = t / st)) +
  geom_aKDE(aes(y = after_stat(scaled)), fill = "steelblue", alpha = .75) +
  geom_histogram(aes(y = after_stat(ncount)), color = "grey", fill = "grey", alpha = .5) + # weighted histogram
  geom_rug(alpha = 0.5)

Radial plot

IsoplotR::radialplot(ages.iso)

Central Age

IsoplotR::central(ages.iso)

## $df
## [1] 52
## 
## $mswd
## [1] 0.7975667
## 
## $p.value
## [1] 0.8545437
## 
## $age
##         t      s[t] 
## 23.968739  1.783239 
## 
## $disp
##            w         s[w] 
## 6.481998e-10 3.893294e+07

Weighted mean

IsoplotR::weightedmean(ages.iso, ranked = TRUE)

Peak fitting

IsoplotR::peakfit(ages.iso)

## $peaks
##              1
## t    23.968739
## s[t]  1.783239
## 
## $props
##      1
## p    1
## s[p] 0
## 
## $L
## [1] -70.7582
## 
## $legend
## expression("Peak 1:" ~ "24.0" %+-% "3.5" ~ "Ma" ~ "(prop=" * 
##     100 * "%)")

Tobias Stephan