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That’s the problem with technologists.You clearly do not know a thing or two about measuring counts such as being able to differentiate between a macroscopic and microscopic system.
Let’s start off with a very simple example, measuring the air pressure in a tyre.
How do you think the air pressure is measured with a gauge, does it measure the average pressure exerted by molecules or the individual effect of molecules striking the inside of the tyre?
Hopefully you would have answered the former which is a macroscopic system defined by a statistical distribution whereas the latter is a microscopic system.
The same principles apply to radiometric dating, you are dealing with a macroscopic system typically composed of trillions of radioactive atoms.
Radioactive decay is probabilistic in nature and not surprisingly follows an exponential statistical distribution given exponential decay is involved.
For very large numbers the distribution follows a Poisson distribution where the probability P(n) of n atoms decaying in time t is given by:
P(n) = [(kNₒt)ⁿexp(-kNₒt)]/n!
Nₒ is population or sample size, k is the decay constant.
Furthermore since radioactive decay is probabilistic, the half-life t₀.₅ = ln(2)/k is the median value for the distribution.
Now for the nonsense in your post, firstly mass spectrometers are highly efficient near 100% for decay counting and even if there was a radiation counting error of say 5% which is treating the system as microscopic and not macroscopic, the exponential decay curve can still be fitted statistically and does not meaningfully alter the half-life calculation.
Secondly it is not the half-life being calibrated but the decay constant.
When calibrating the decay constant for U-Pb as an example, the clock is reset by melting the sample to expel any Pb daughter atoms, mass spectrometers being highly sensitive will detect Pb daughter atoms during the decay process even when the time t is very short.
To give you an idea of what this time frame is and using uranium metal as the calibration sample instead of ores.
Decay constant k of U-238 is 1.55125×10⁻¹⁰ yr⁻¹.
The maximum practical sample weight for a thermal ionization mass spectrometer (TIMS) is ~1 μg and the molar mass of U-238 is 238 gmol⁻¹.
The number of U-238 atoms Nₒ = (1 μg/ gmol⁻¹) x 6.022 x 10²³ atoms/mol ≈ 2.53 x 10¹⁵.
For the detection threshold based on a 1% precision for k, Δk/k ≤ 0.01.
The uncertainty in Pb counts is Δn ≈ √n since the standard deviation σ of a Poisson distribution is √n.
The relative uncertainty is therefore Δn/n=√n /n = 1/√n ≤ 0.01 ⟹ n ≥10⁴ atoms.
The number of atoms n that have decayed in time t is given by the equation;
n = Nₒ(1-exp(-kt)) ≈ k Nₒt where kt << 1.
t = 10⁴ /(2.53 x 10¹⁵)(1.55125×10⁻¹⁰ ) ≈ 0.0255 years ≈ 9.3 days.
Note we don't need to use ridiculously long time frames to calibrate which can also be cross checked with other mass spectrometers.
Get yourself an education on the subject and stop making up rubbish.
Long on narrow analytics.. Believe all they read in a paper without question,
Short on metrology, and accepting the limitations of over simplistic models
It is how they screwed up the dating of the shroud so badly.
they even ignored their own failed equipment validation!
Meanwhile as a professional modeller for a decade l have plenty of experience of having to revise over simplistic assumptions.
ALL i have claimed is there are so many assumptions you cannot be certain of such a tight band.
its an estimate. It might even be a good estimate, but nobody knows.
It’s hardly an earth shattering conclusion. Not least because you cannot disprove it!
It is scientism to presume all that there is, is all you have so far detected.
Or all that happened is restricted to present record or witness.
or that your model includes all there is .
It didnt work out with dark matter very well did it?
Science must ACCEPT uncertainty, and that the only reality is defined by a model. That multiple models exist sometimes in conflict . So scientific modelling is ALWAYS imperfect
Hawking agrees with me, though it tookhim a career to accept what philosophers told him a century before,
I will leave you to your echo chamber of confirmation bias.
( And the attempt to impress with a math formula as an icon. Why?
I don’t need to bother, as I was doing complex numbers before I was 10, messing around with electronics back then., and I’m not seeking to impress anyone, so I don’t need to bolster my ego with formulae! )
See ya around, there are so many interesting bits of science to waste it with those who wont seemibgly read what was already known decades ago about the context of observers and limitations of models. .
its a shame we got off on the wrong foot.
But heh….
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