Geological Model Age from Lead Isotope Analysis

Compute geological model age (T) and U/Pb (mu) and Th/U (kappa) ratios from lead isotopic measurements.

Usage

pb_age(x, y, z, ...)

# S4 method for class 'numeric,numeric,numeric'
pb_age(
  x,
  y,
  z,
  t0 = 3.8,
  x_star = 18.75,
  y_star = 15.63,
  z_star = 38.86,
  mu = 9.66,
  kappa = 3.9,
  th232 = 0.049475,
  u238 = 0.155125,
  u235 = 0.98485,
  u238_235 = 137.79,
  tolerance = sqrt(.Machine$double.eps)
)

# S4 method for class 'list,missing,missing'
pb_age(
  x,
  t0 = 3.8,
  x_star = 18.75,
  y_star = 15.63,
  z_star = 38.86,
  mu = 9.66,
  kappa = 3.9,
  th232 = 0.049475,
  u238 = 0.155125,
  u235 = 0.98485,
  u238_235 = 137.79,
  tolerance = sqrt(.Machine$double.eps)
)

Arguments

x: A numeric vector of 206Pb/204Pb ratios. If y and z are missing, must be a list (or a data.frame) with numeric components (columns) x, y and z.
y: A numeric vector of 207Pb/204Pb ratios. If missing, an attempt is made to interpret x in a suitable way.
z: A numeric vector of 208Pb/204Pb ratios. If missing, an attempt is made to interpret x in a suitable way.
...: Currently not used.
t0: A numeric value giving the time of the second stage of the reference model.
x_star: A numeric value giving the 206Pb/204Pb ratio at \(t = 0\).
y_star: A numeric value giving the 207Pb/204Pb ratio at \(t = 0\).
z_star: A numeric value giving the 208Pb/204Pb ratio at \(t = 0\).
mu: A numeric value giving the 238U/204Pb ratio of the reference model.
kappa: A numeric value giving the 232Th/238U ratio of the reference model.
th232: A numeric value giving the decay constants of 232Th.
u238: A numeric value giving the decay constants of 238U.
u235: A numeric value giving the decay constants of 235U.
u238_235: A numeric value giving the actual 238U/235U ratio.
tolerance: A numeric value specifying the tolerance (stopping criteria for the Newton–Raphson method).

Value

A four columns data.frame:

`age`	Geological model age (in Ma)
`mu`	238U/204Pb ratio
`kappa`	232Th/238U ratio
`f`

Note

Reference values from Albarede & Juteau (1984).

References

Albarède, F., Desaulty, A.-M. & Blichert-Toft, J. (2012). A Geological Perspective on the Use of Pb Isotopes in Archaeometry. Archaeometry, 54: 853-867. doi:10.1111/j.1475-4754.2011.00653.x .

Albarède, F. & Juteau, M. (1984). Unscrambling the Lead Model Ages. Geochimica et Cosmochimica Acta, 48(1): 207-12. doi:10.1016/0016-7037(84)90364-8 .

Allègre, C. (2005). Géologie isotopique. Belin sup. Paris: Belin.

Author

N. Frerebeau, F. Albarede (original Matlab code)

Examples


Pb <- data.frame(
  x = c(18.23247, 18.22936, 18.23102), # Pb206/Pb204
  y = c(15.65199, 15.65216, 15.65097), # Pb207/Pb204
  z = c(38.5167, 38.51516, 38.51601)   # Pb208/Pb204
)

## Reference values from Albarede & Juteau (1984)
pb_age(Pb)
#>        age       mu    kappa
#> 1 428.3959 9.859142 4.074891
#> 2 430.9796 9.860659 4.076292
#> 3 427.6003 9.855397 4.075178

## Reference values from Albarede et al. (2012)
pb_age(
  Pb,
  t0 = 4.43,
  x_star = 18.75, y_star = 15.63, z_star = 38.83,
  mu = 9.66, kappa = 3.90, th232 = 0.049475,
  u238 = 0.155125, u235 = 0.98485, u238_235 = 137.79
)
#>        age       mu    kappa
#> 1 394.0812 9.758829 4.055318
#> 2 396.4376 9.759567 4.056451
#> 3 393.9155 9.756973 4.055625