ratio between λ( t) and λ0, a particular decay history function might be applicable only to a single radioisotope system. In such a case, then H( t) will be subscripted with the relevant system. The accumulated radiometric age of an object at any particular time, t f, compared to an initial reference time, t i , with t f < t i can be written as ( 5 ) The decay history function tracks the accumulated radiometric age of a particular sample under ideal closed system conditions and a time-varying decay rate. Non-radiogenic perturbations, pn, however, can produce an apparent age, t*, which is functionally indistinguishable from a real age. ( 6 ) Any apparent age as defined above is dependent on both the time of measurement and the magmatic system, which if it contains excess daughter product, will yield an apparent age greater than the actual age. This is locale dependent, so as a point of notation, t* may be subscripted with a capital letter to indicate the source or reservoir in view. Additional notation relating to this is developed later. B. Sampling and mixing relations Objects (rocks, minerals, etc.) that can be radiometrically dated must be samples from a magmatic source (reservoir) which cooled below the closure temperature at the time of sampling t s. The measured age of the sample is reset from the reservoir age by a factor of 0 ≤ η ≤ 1 It is assumed that the system remains closed after this point and accumulates decay products according to the decay history function thereafter. The apparent measured age of the sample can be written in terms of the reservoir apparent age, t*P, and the time of sampling as follows: ( 7 ) The same can also be written in the below alternate form: ( 8 ) This relation provides a general framework for modeling reservoir differentiation from a common source, mixing between multiple reservoir sources, inheritance of radiogenic daughter products into a rock, primordial radiometric signatures, and literal sampling from a rock or lava. In the limits of η → 0 or H( t*P| t s )− H( t s) → 0, then there is no significant inheritance, and t*s| t s = t s. However, for reservoirs with a large radiometric age, significantly older than the time of sampling, then inheritance will be a factor, particularly if H( t*P| t s ) η is greater than the analytical uncertainty of the isotopic measurement. In this case the amount of inheritance is sensitive to the time evolution of the reservoir apparent age. These situations are expected to be common as a result of AND. One strong implication is that inheritance is more likely to be worse for later emplaced samples than earlier emplaced samples during the AND epoch, as H( t s) decreases quickly. After the epoch, the H( t s) remains relatively fixed, and the inheritance behavior is dominated by the reservoir evolution. Reservoirs are themselves sampled from older reservoirs all the way back until one reaches a primordial apparent age, t*Ω. Magma reservoirs have an additional evolution applied to their apparent ages as described in the next section. In general, for workable radioisotope systems, the emplacement efficiency will be small, and therefore inheritance from a generation before the immediate source can be safely ignored, although daughter products may accumulate and persist in a melt. In general, the emplacement efficiency, η, is related to the partition coefficient and is dependent on the pressure, temperature and chemistry during emplacement, so is dependent on the particular magmatic system in view. It is also likely to include a small random component. Partitioning at a particular time can be modeled by a set of samples, each with a separate inheritance parameter, η such that ( 9 ) The inheritance relation above can be generalized to represent the generation of a sample or reservoir as the mixing of two (or more) different parent reservoirs. After all, inheritance can be thought of as the mixture of a mature source with a juvenile source. The inheritance for some sample, C, derived as a mixture of sources A and B with some mixing parameter 0 ≤ α ≤ 1 can be written as ( 10 ) Mixing may be calculated in a chain to produce any arbitrary mixture of multiple sources. A general form to accomplish this in a single calculation is left as an exercise to the reader. The relations derived above implement potential models 1, 3, and 4 proposed for Ar loss from the Cardenas Basalt by Austin and Snelling (1998). C. Reservoir relaxation Consider a homogeneous reservoir of magma which is in communication with its environment and which contains an excess of daughter products arising from a process like crystallization and expulsion of incompatible daughter products or an episode of accelerated nuclear decay. While these daughter products are still in the magma, they are available to be incorporated into samples through inheritance, but over time, the reservoir will release these into the environment. Enhanced inheritance from large excesses of daughter products is a perturbation to the radiometric age of samples taken from the reservoir. Radiogenic products released to the environment decrease this effect, causing an apparent downwards trend of radiometric age of emplaced samples with decreasing closure time that is not directly radiogenic in cause. This excess cryptic inheritance which decreases over time due to the return of the source reservoir to chemical equilibrium is here termed reservoir relaxation. “Relaxation” as a term is used in the fields of chemistry and dynamical systems to refer to a system returning to its equilibrium state after being affected by a perturbation, and I adopt that usage here. MOGK Disequilibrium Relaxation Following Accelerated Nuclear Decay 2023 ICC 331
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