= Relative calibration error in an isotope-corrected stiffness ratio
{title2=$\delta\log K=\delta\log m+2\delta\nu_2/\nu_2-2\delta\nu_1/\nu_1$}
\b[Relative calibration error in an <isotope>-corrected stiffness ratio] is the fractional error remaining when spectral calibration affects the two compared bands differently. Write the model of <isotope-corrected Raman stiffness ratio> as $K=m(\nu_2/\nu_1)^2$, where $m=\mu_2/\mu_1$ and the positive $\nu_i$ denote vibrational wavenumbers. For small errors,
$$
\delta\log K=\delta\log m+2\frac{\delta\nu_2}{\nu_2}-2\frac{\delta\nu_1}{\nu_1}.
$$
The mass correction does not remove differential spectral calibration error.
A useful calibration model is $\delta\nu_i=(g_c+g_i)\nu_i+b_c+b_i$, with common and separate gain errors $g_c,g_i$ and offsets $b_c,b_i$. Its spectral contribution becomes
$$
\delta\log K=2(g_2-g_1)+2b_c(1/\nu_2-1/\nu_1)
+2(b_2/\nu_2-b_1/\nu_1),
$$
with the mass term added if uncertain. A single multiplicative scale cancels exactly when no additive error is present. Separate gains do not: comparing two spectra calibrated independently can lose that cancellation.
For an original example, take $\nu_1=1000\,\mathrm{cm^{-1}}$ and $\nu_2=900\,\mathrm{cm^{-1}}$. A differential gain $g_2-g_1=0.001$ creates about $0.20\%$ relative error in $K$. A common offset $b_c=1\,\mathrm{cm^{-1}}$ instead contributes about $0.0222\%$. If the separate gains each have independent $0.1\%$ standard uncertainty, their combined contribution to the relative standard uncertainty of $K$ is $2\sqrt2\times0.1\%\approx0.2828\%$.
The <separate spectral-calibration gains in isotope stiffness inference> must therefore be distinguished from a shared reference error. <Covariance> propagation keeps precisely those shared and separate influences.<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} Recording both samples with the same validated spectral scale can help, but drift or changed dispersion can still produce differential errors.
The primary graphene <isotope> experiment measured Raman bands over <isotope> concentrations and laser excitation energies and found the harmonic-oscillator mass dependence.<Probing carbon isotope effects on the Raman spectra of graphene with different ¹³C concentrations by Bruno R. Carvalho and others>[Abstract, isotope-mass dependence of Raman frequencies]{ref} Small residual stiffness changes require the additional calibration and mode-model checks above; the numerical errors here are chosen examples, not quoted experimental uncertainties.
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