arXiv2026
Isotopic-abundances of any multi-(N)-isotopic-element (E) are interdependent and therefore needed to be measured indirectly from their rather accurately-measurable (N-1) abundance-ratios. ... Even, corresponding-atomic-weight (A_E) is determined by translating the estimated-ratios into the desired-estimate via concerned-input(s)-output-system-specific-relationship (SSR). However, how good the output-estimates y_d(s) and a_E should represent their desired-true-values Y_d(s) and A_E be indicated by the respective uncertainties e_d^Y (s) and e_E^A only; i.e. correct-knowledge of even the output-uncertainties be indispensable for the desired-indirect-measurements. Usually, by assuming the lab-established-(input)-uncertainty(s) as u_m(s) to be accidental-in-nature, a measure of any-corresponding-non-correlated-variable-specific-output-uncertainty is obtained by the popular-error-propagation-law. And, any-correlated-variable as the isotopic-abundance-Y_d-specific-output-uncertainty e_d^Y is evaluated by incorporating additional-(correlation)-correction-factor(s) in the error-propagation-formula, i.e. correlation is being assumed as an output-uncertainty-governing-parameter. However, it is here clarified that: (i) the output-uncertainty e_d^Y (or even e_E^A) should, like the corresponding-output-estimate y_d (or a_E) itself, be independent of whether the isotopic-abundances are correlated; and: (ii) any output-uncertainty e should actually be the corresponding-SSR-governed-systematic-parameter, irrespective of whether the input-uncertainty(s) u_m (s) be purely-random-in-nature. That-is, like the non-correlated-variables-specific-SSRs discussed elsewhere, any isotopic-variable-specific-SSR should be bracketed with certain input(s)-to-output-variation-parameter(s) [MF]_m^d (s) which really prefix(s), for given the input-uncertainty(s) u_m (s), the corresponding-output-uncertainty ...