rameters (inheritance and relaxation) are constrained in the context of the radiometric measurement. E. Conditions for relative dating As previously discussed, in an unperturbed system only subject to the decay history function, which is a bijection, the ability to derive an absolute date from a measurement is guaranteed. Similarly, over the lifetime of a single reservoir (as defined by a single inheritance value and relaxation parameter), the Total Radiometric Response is a bijection which allows for relative dating between samples of the reservoir. However, when comparing samples from different sources with potentially different reservoir relaxation parameters, then direct comparison can only be made if certain conditions hold. Without loss of generality for two sources, A and B, relative dating can be guaranteed for any two samples if the following condition holds: Relative Dating Condition 1. ( 17 ) In the absence of perturbations, H+( X,t) → H( t), then the bijective property of H( t) automatically satisfies condition 1. This condition may or may not be met between different isotope systems in general, as it is unknown at this time if all radioisotope systems had the same acceleration factor functions, even if the condition would otherwise be met. The implication is that the most likely situation for this condition to hold is when comparing radiometric dates from the same magmatic system with the same radioisotope system originating during the same time period A= B. The above condition is not met in the general case for systems with different magmatic histories, even though both are subject to the same decay history function. It can clearly be seen, however, that this condition does hold if it can be established that H+( A,t) = H+( B,t). All is not lost, however. A condition for restricted relative dating applicability can be applied to a single source in relation to another over a restricted interval t0 ≤ t1, t2, ≤ t3 given by Relative Dating Condition 2. ( 18 ) This condition applies when the total radiometric response function associated with reservoir A is always lower than reservoir B, i.e., for any two samples erupted at the same time in history, the measured radiometric age of the one associated with reservoir B is always the higher one. In this case, relative ordering can be maintained in a one-sided fashion. B being older than A does not provide relevant chronological information, but if A measures older, then it is guaranteed to be older than the same numerical measurement of B. The inverse is also true, B measuring younger can guarantee that it is in fact younger. This condition possibly provides the easiest method of establishing its validity during field studies. Finally, even in the case where neither condition will hold for any possible sample time, it is possible that relative ordering of dates will hold if the available samples from each source are sufficiently sparse so as to overcome the maximum size of the ambiguity arising from differential reservoir relaxation parameters. This is the case if, for all samples of A and B adjacent in time, the following holds: Relative Dating Condition 3. ( 19 ) Relatively accurate radiometric dating is necessary, but not sufficient for absolute dating. Relative dating holds, particularly if the relative dating condition 1 is applicable, even without knowledge of the form of the decay history function. If the decay history function can be identified, then absolute dating becomes possible. In the case of a system with inheritance or significant reservoir relaxation, then it is necessary to determine the total radiometric response function for the system to achieve absolute dates. When comparing radiometric dates of samples from two different systems in the general case where inheritance and reservoir relaxation are significant factors, then the total radiometric response function must be known for both systems for both relative and absolute dating. If this is not achievable, then to use relative dating, one of the less stringent conditions listed above must be shown to hold for the systems in question. III. ILLUSTRATIVE EXAMPLE The flexibility and behavior of the above derivations are best exposited with a concrete example. The Python implementation of the above equations was used to produce an example model. The example will assume a radiometric history function of the form: ( 20 ) which encodes a single bell-shaped pulse of accelerated nuclear decay where is the total accumulated age of the pulse, is a parameter controlling the pulse width, and c is the center of maximum acceleration. The example uses the following parameters: = 4 Ga, 0.294 yr, c = 4301.1816YBP (Illustrative model 1) The value of c is 66 ⅓ days prior to the start of the Flood, which for simplicity, is assumed to be 4301YBP. A history is chosen for the example and described in figure 2 which illustrates several different features of the mathematical formalism derived previously. In it, we begin with a primordial magma reservoir which does not have any inherent age signature. AND begins slightly before the Flood. At the start of the Flood, a parent magma reservoir is differentiated from the primordial reservoir with one tenth of the contemporaneous age signature of the primordial reservoir. Midway through the Flood, this parent reservoir is partitioned into two derived reservoirs, A and B. The partitioning is accomplished by distributing all the accumulated daughter from the parent into one or the other. This is seen by the unity sum of emplacement efficiency parameters ( η). Each reservoir has unique relaxation behaviors with A relaxing at a slow rate, and B relaxing at an elevated rate. Each reservoir is regularly sampled and measured. The samples do not relax, but they do continue to accumulate daughter products from AND. At the end of the Flood, reservoir B begins to relax more slowly, though still more quickly than A. The successor B reservoir contains the entire age signature accumulated by B. Samples are again regularly taken from the reservoirs to show the change in radiometric dates MOGK Disequilibrium Relaxation Following Accelerated Nuclear Decay 2023 ICC 333
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