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=== Third assumption: understanding of the rate of decay ===
 
=== Third assumption: understanding of the rate of decay ===
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This one is also easy.
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This one is also easy. The assumption is that the rates of decay of the isotopes involved are reasonably accurately known, and that they are constant over long periods of time.  Fortunately, this is the case.  Radioactivity has been studied in great detail for over a century.  The physical constants (nucleon masses, fine structure constant, etc.) involved in radioactive decay are well characterized, and the processes are well understood.  Careful astronomical observations show that the constants have not changed significantly in billions of years—spectral lines from distant galaxies would have shifted perceptibly if these constants had changed.  In some cases radioactive decay itself can be observed and measured in distant galaxies when a supernova explodes and ejects unstable nuclei.
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Another assumption is that the rate of decay is constant over long periods of time.  Radiometric dating requires that the decay rates of the isotopes involved be accurately known, and that there is confidence that these decay rates are constant.  Fortunately, this is the case.  The physical constants (nucleon masses, fine structure constant) involved in radioactive decay are well characterized, and the processes are well understood.  Careful astronomical observations show that the constants have not changed significantly in billions of years—spectral lines from distant galaxies would have shifted perceptibly if these constants had changed.  In some cases radioactive decay itself can be observed and measured in distant galaxies when a supernova explodes and ejects unstable nuclei.
      
There are a few effects that can alter radioactive half-lives, but they are mostly well understood, and in any case would not materially affect the radiometric dating results.  That is, the analysis of the isotopic and chemical composition of the sample has far greater uncertainty than any uncertainty in the decay rate itself.
 
There are a few effects that can alter radioactive half-lives, but they are mostly well understood, and in any case would not materially affect the radiometric dating results.  That is, the analysis of the isotopic and chemical composition of the sample has far greater uncertainty than any uncertainty in the decay rate itself.
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