= Reference-versus-sample allocation in repeated isotope-ratio measurements
{title2=$n_s/n_r=(s/r)\sqrt{c_r/c_s}$}
\b[Reference-versus-sample allocation in repeated isotope-ratio measurements] distributes a finite measurement budget between sample readings and the reference used to calibrate them. Suppose independent sample and reference readings have first-order gain-error <variances> $s^2$ and $r^2$. Averaging $n_s$ sample readings and $n_r$ reference readings gives log stiffness-ratio <variance>
$$
V=4\left(\frac{s^2}{n_s}+\frac{r^2}{n_r}+s_c^2\right),
$$
where $s_c^2$ is the <variance> of a differential calibration error shared by all ratio estimates. The <shared-reference floor in averaged isotope stiffness ratios> explains why that last component does not average down. <Covariance> propagation must reflect which observations and calibration influences are actually shared.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref}
Let one sample reading cost $c_s$ budget units and one reference reading cost $c_r$, with $c_sn_s+c_rn_r=T$. Treating counts continuously for the initial optimization, a constraint multiplier gives
$$
\frac{n_s}{n_r}=\frac{s}{r}\sqrt{\frac{c_r}{c_s}},\qquad
V_{\min}=\frac{4(s\sqrt{c_s}+r\sqrt{c_r})^2}{T}+4s_c^2.
$$
The counts are proportional to standard uncertainty divided by the square root of cost, rather than simply to each <variance>. A budget-independent common component changes the attainable floor but not this optimum ratio under the stated model.
For an original example, take equal costs, $s=0.3\%$, $r=0.1\%$ and 40 total readings. The optimum is 30 sample and ten reference readings. Without a common component, the predicted log-ratio standard uncertainty is about $0.1265\%$, compared with $0.1414\%$ for twenty of each. A common standard uncertainty $s_c=0.05\%$ raises those values to about $0.1612\%$ and $0.1732\%$, respectively.
An <integer allocation near a continuous isotope-reference optimum> must compare feasible neighboring count choices and any acquisition-order constraints. More reference readings help only to the extent that their errors are independent; repeatedly measuring an unchanged unknown calibration bias does not create independent references. If reference drift depends on elapsed time, a single constant cost and common-<variance> term may be inadequate.
In practice, record which reference observations calibrate which sample spectra and account for any correlated peak fitting, isotope-composition uncertainty and spectral offsets. The <variance> and cost model above is a chosen allocation problem, not an experimentally established precision specification for an isotope measurement system.
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