Chemical U-Th-Pb geochronology: A precise explicit approximation of the age equation and associated errors

Abstract

In a single decaying system, the age determined from the exponential decay law is directly related to its linear Maclaurin approximation. This relationship can be additively extended to several decaying systems resulting in the same daughter element, by using proportionality functions, thus allowing an explicit formulation of the age as a function of element concentrations. The values of the binary proportionality function for the 238 U-235 U-Pb system and the ternary proportionality function for the 232 Th-238U-235 U-Pb system were determined by iterations of the exponential decay formula up to 4 Ga, with a step of 10 Ma, for a set of 24 different U/Th ratios. From the iteration data, the expressions of the two functions and the associated coefficients were determined by polynomial regression and mathematical programing on conveniently separated time and compositional intervals. Additional time-and composition-dependent age corrections optimized by mathematical programming of the residuals lead to an accuracy of 0.005 Ma of the resulting age. The error propagation can be traced through all the operations defined by explicit formulas according to simple error propagation rules, finally allowing the calculation of the standard error of the result. The formulas and parameters derived can be used in a calculation spreadsheet.

New Insights into the Genesis of Dibrova U-Th-REE Mineral Deposit (West Azov Megablock, Ukraine) Using Monazite Chemistry

“…Even though the EPMA does not measure isotopic ratios, it is possible to date the accessory minerals rich in U and Th based on elemental concentrations (e.g., monazite) [23][24][25][26][27][28][29]. The selected method was explained in detail by [24,30]. This method of "chemical dating" uses the measurement of Pb, Th, and U mass concentrations to allow an age to be calculated from each spot analysis.…”

“…In this research, the chemical ages were calculated using the EPMA data and the procedures of [24,30,32]. Data are provided in the Supplementary Data (Table S1).…”

A Mineralisation Age for the Sediment-Hosted Blackbush Uranium Prospect, North-Eastern Eyre Peninsula, South Australia

“…Following [30], uncertainties used here have been calculated using peak and background counts, corrections for interference, ZAF etc. This can be applied to the age formula of [28], which can be generalised as Pb = f(U,Th,t).…”

“…In practice, all terms are very small compared to σ Pb , and σPb Pb ≈ σt t . Although we did not include decay constant errors, all calculated ages and errors were crosschecked using precise explicit approximations of [30], which includes decay constant errors. Incorporation of systematic decay constant errors into the uncertainty propagation is not typically done for chemical dating via EPMA, predominantly due to the (relatively) imprecise nature of the chemical EPMA dating technique.…”