As a follow-on to the RATE study, Snelling (2014) did a thorough review of meteorite dating. Though the results of previous reports on meteorite dating consistently yielded ages of 4.5 Ga, Snelling reported that some of the evidences which were associated with AND in the RATE report, particularly the systematic discordance data were not found. So, Snelling proposed that the meteorites, and the earth by extension, had an initial primordial fine mixture of parent and daughter isotopes that produced a characteristic radiometric age signature. In this case, the Flood pulse and gradual ramp-down again is retained, but there is no AND during Creation or the Antediluvian period. Since the RATE study, other authors have continued to develop models of radiometric change which do not rely on AND as a primary mechanism. Hung (2008) proposed a sophisticated solute transport model for radiometric change in the U-Pb system undergoing open system behavior. Hung’s proposed model is not an applicable explanation for all old radiometric dates, particularly since techniques have been developed to probe the applicability of the assumptions of secular equilibrium and closed system behavior—though these are seldom used in practice. Nevertheless, transport phenomena and Hung’s model are likely to have increased applicability following an episode of AND when daughter products are exceedingly abundant. This line of research merits further development. Diffusion along grain boundaries and within the crystal lattice redistributes excess daughter products, especially Ar (Pickles et al., 1997; Reddy et al., 1996). Diffusion controls the closure temperature and in complex systems can be a lower temperature than previously assumed (Nteme et al., 2022). The RATE study raised numerous new research questions that were not originally addressed in the report or follow-on research by the authors (Froede and Akridge, 2012; Oard, 2013). This paper seeks mathematically to model one such open question on the evolution of magmatic systems that have been subjected to an episode of AND and the consequences for radiometric analyses. Reed and Froede (2010) made a call for an absolute correction factor between radiometric dates and biblical history. They posed several poignant questions regarding how such conversions might be constructed from observations, how they might be validated, and how discrepancies among different radioisotope methods might be handled. I hope that this paper adequately addresses such concerns. II. DERIVATIONS A. The decay equation with variable decay constant I begin by deriving the decay equation without assuming a time-invariant decay rate, as well as developing a formalism for describing episodes of AND. The result is fairly intuitive and matches the usage of the RATE authors, though that study did not provide a derivation. The following derivation also provides a basis for subsequent derivations made in this paper. First let us assume that the number of parent radioactive atoms, P, in a given rock sample decreases over time at a rate proportional to P. Let us also assume that each decay simultaneously adds a daughter atom to the total number of daughter atoms, D, in the sample. The potentially time variable fraction of radioactive atoms decaying per unit time is defined as λ( t). Conventionally, λ is assumed to be constant, and we will derive that as a special case. In all cases t without decoration represents an independent variable. Variables which indicate a particular value of t will include a lower-case or numerical subscript. ( 1 ) Under standard assumptions, λ( t) = λ 0 is a constant, and the solution to the equation can be written as ( 2 ) which is known as the dating equation. To account for time variable decay we define an acceleration factor, Ξ( t), as ( 3 ) where λ 0 is a reference decay rate. Because the component decay rates are positive, Ξ( t) must be strictly positive. Additionally assume that Ξ( t) is constructed such that it is piece-wise defined, and integrable over the entire interval. Conventionally, the quantity t m above, which assumes a constant decay probability, is understood to represent the radiometric age of a rock sample. However, since λ is a function of t and not constant, t m is similarly a function of t. Let us introduce a function, H( t), with units of time, that encodes the nuclear decay history. This function which we shall refer to as the Decay History function (also known as the radiometric transfer function) is given as ( 4 ) The variable t represents an age before present, and so is 0 at the present time and increases looking backward in time. Since Ξ is dimensionless, H( t) has units of time. It represents what we shall refer to as radiometric age. The derivation of equations 2, 3 and 4 can be found in appendix section A. By virtue of definition, H( t) is continuous and strictly increasing with the fixed point H(0) ≡ 0. Helpfully, H( a) <H( b) ⇔ a < b, and the function covers every value of both the real time and the radiometric time, therefore H( t) is a bijection, which is a necessary condition for valid relative ordering of radiometric dates. Furthermore, this implies that an inverse to the history function exists, in principle, which can produce absolute dates from radiometric measurements: Theorem 1. Given a radiometric history function H( t)=∫0t Ξ ( τ)d τ defined over all t such that Ξ( τ) is strictly positive and piece-wise integrable over τ >0, then there exists an inverse, H ˗1 such that H˗1( H( t)) = t. As long as H( t) is defined continuously for every possible value of t, then it will be unique for a given Ξ( t) regardless of any discontinuous jumps. When attempting to estimate the form of H( t) from discrete measurements, some ambiguity between measurements arises. I address this situation later. Because of the dependence of H on the MOGK Disequilibrium Relaxation Following Accelerated Nuclear Decay 2023 ICC 330
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