= Isotope-corrected Raman stiffness ratio
{title2=$K=(\widetilde\nu_2/\widetilde\nu_1)^2\mu_2/\mu_1$}
An \b[<isotope>-corrected Raman stiffness ratio] removes the stipulated mass contribution before interpreting a vibrational frequency change as a stiffness change. For corresponding harmonic modes,
$$
K=\frac{k_2}{k_1}=\left(\frac{\widetilde\nu_2}{\widetilde\nu_1}\right)^2\frac{\mu_2}{\mu_1}.
$$
The <identifiability of stiffness from isotope-resolved Raman shifts> article explains why the effective masses and mode assignment are required. The ratio does not by itself validate those assumptions.
For an original calculation, take hypothetical pure-carbon bands at $1600\,\mathrm{cm^{-1}}$ for <carbon-12> and $1530\,\mathrm{cm^{-1}}$ for <carbon-13>. Uniform <isotope> replacement with the mass-number approximation gives $\mu_2/\mu_1=13/12$ and $K\approx0.99062$. The inferred stiffness reduction is about $0.938\%$, despite the $4.375\%$ decrease in wavenumber. These chosen values are not a published experimental pair.
Uncertainty in the ratio follows from
$$
\delta\log K=2\frac{\delta\widetilde\nu_2}{\widetilde\nu_2}
-2\frac{\delta\widetilde\nu_1}{\widetilde\nu_1}
+\frac{\delta(\mu_2/\mu_1)}{\mu_2/\mu_1}.
$$
With exact masses in the model and independent wavenumber standard uncertainties $0.5\,\mathrm{cm^{-1}}$, the relative standard uncertainty of $K$ is approximately $0.0904\%$. A single common multiplicative spectral-scale error cancels exactly in the frequency ratio. A common additive offset does not: its first-order effect is $2\delta(1/1530-1/1600)$, giving only about $0.00286\%$ relative uncertainty for a $0.5\,\mathrm{cm^{-1}}$ offset uncertainty. The <relative calibration error in an isotope-corrected stiffness ratio> therefore depends on the spectral calibration model and correlations, not just each peak's marginal error bar.<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}
The primary graphene experiment varied <carbon-13> concentration and laser excitation energy and observed Raman-frequency dependence following the harmonic-oscillator mass relation.<Probing carbon isotope effects on the Raman spectra of graphene with different ¹³C concentrations by Bruno R. Carvalho and others>[Abstract, mass dependence of D, G, G\* and 2D peak frequencies]{ref} A residual stiffness comparison additionally needs controlled strain, <temperature> and composition. Mode mixing or uncertain <isotope> distribution can dominate the small residual even when fitted peak centres are precise.
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