The change of the perturbation caused by excess daughter product can be modeled as proportional to the excess present at any given time compared to the equilibrium value, which is assumed to be 0 — i.e., modern reservoirs should tend towards measuring modern ages. This can be written as the measured unperturbed reservoir age at time t from equation 5, for an apparent initial reservoir age, H( t*R| t s ) (from equation 7), less the amount of excess that has already been removed by the perturbation r( t). The constant of proportionality, β, is the relaxation parameter. ( 11 ) The solution to the above equation yields two contributing relaxation perturbation terms. The first ( ) is related to the relaxation of the initial excess present in the reservoir and the second ( ) is the relaxation of additional excess generated during accelerated nuclear decay. The derivation of these perturbations is derived by solving equation 11 in appendix section B. These perturbations represent additional daughter product incorporated into rock samples which decrease in magnitude in later samples as the magmatic reservoir loses its excess daughter product. ( 12, 13, 14 ) Subtracting the above two perturbation terms and adding the contribution from inheritance ( 0) from the parent reservoir, P, the total accumulated radiometric age of a reservoir at any given time can be written as ( 15 ) Because the inheritance and relaxation of a reservoir system are locale dependent, each reservoir will necessarily have a unique function satisfying the above equation. For the sake of notational clarity, let us introduce the Total Radiometric Response function defined as follows: ( 16 ) where X is the particular magmatic system under consideration. Given that H( t*R| t s ) − H( t) ≥ 0 and − ( t) ≥ 0, then d _ d t ≤ 0, and ( t) is monotonic. Because its contribution to the total radiometric response is either 0 or in the same direction as H( t), and equation 15 covers the entire lifetime of the reservoir and is strictly increasing, H+( X,t), therefore, is a bijection and can be inverted over the lifetime of the reservoir, for a fixed magmatic system, X. The above relaxation relation is fundamental to the response of a magmatic system undergoing AND. The only additional assumption admitted was that the relaxation of daughter products in a reservoir back to equilibrium can only happen at a finite rate. Figure 1 for a hypothetical decay history, illustrates the relationship between the theoretical radiometric age at the true age of closure ( t) arising only from AND ( H( t)), measured radiometric age ( H( t*)) and the associated apparent closure time ( t*), and the total radiometric response ( H+( X,t)) which accounts for extraneous daughter products present in the sample at the true closure time. D. Python Implementation I have developed a Python implementation of the relations defined in this paper at the following link: https://github.com/nmogk/radiometric-tf. The implementation provides a Python package as well as a command line interface that allows a user to load a decay history model file, obtain overall information about its behavior, generate plots of radiometric versus calendar time, and convert between individual radiometric dates and calendar dates, including handling of uncertainties. The command line interface includes help information. Code relating to the decay history function and the command line interface are in the rtf. py file. The rrelax.py file implements the sampling and reservoir relaxation relations and allows modeling of the locale-dependent inputs to total radiometric response, including reservoir relaxation. The repository also includes five model definition files. Three represent the as-published models of Vardiman, Snelling, and Chaffin (2005), Humphreys (2014), and Snelling (2014). Two others are end-member models of AND which have a single pulse during the Flood which include Precambrian radiometric dates. The accuracy of absolute dates produced with the implemented models depends heavily on how well the models of decay history and reservoir paFigure 1. Schematic graph illustrating the relationship between the total radiometric response function, H+( X,t), the decay history function, H( t), and the maximum closure time, t*. Measured Age Closure Time MOGK Disequilibrium Relaxation Following Accelerated Nuclear Decay 2023 ICC 332
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