= Separate spectral-calibration gains in isotope stiffness inference
{title2=$u^2(K)/K^2\simeq4(u_1^2+u_2^2-2\rho u_1u_2)$}
\b[Separate spectral-calibration gains in <isotope> stiffness inference] arise when the two Raman spectra retain different multiplicative calibration errors. Let the corrected wavenumbers satisfy $\widehat\nu_i=(1+g_i)\nu_i$. For $K=(\mu_2/\mu_1)(\nu_2/\nu_1)^2$, small residual gains give
$$
\delta\log K\simeq2(g_2-g_1).
$$
The <relative calibration error in an isotope-corrected stiffness ratio> article treats the accompanying offset terms. Here the important quantity is the gain difference, not either gain alone.
If the two residual gains have standard uncertainties $u_1,u_2$ and correlation $\rho$, their contribution is
$$
\frac{u^2(K)}{K^2}\simeq4(u_1^2+u_2^2-2\rho u_1u_2).
$$
This follows by applying the full <covariance> propagation rule.<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} A shared scale component cancels; independent session errors do not. The <covariance of differential spectral-calibration gains> therefore determines whether combining calibrations improves the ratio.
For an original example, let $u_1=u_2=0.2\%$. Independent calibrations give relative uncertainty about $0.566\%$ in $K$. A justified correlation $\rho=0.9$ reduces it to about $0.179\%$; a correlation $-0.9$ instead gives about $0.780\%$. These are different calibration models, not alternative ways of adding the same independent errors.
A practical comparison can measure the same known reference line during each sample session. If raw readings are $y_i=c_i\nu_i$ and reference readings $r_i=c_i\nu_{\rm ref}$, a purely multiplicative model gives
$$
K=\frac{\mu_2}{\mu_1}\left(\frac{y_2/r_2}{y_1/r_1}\right)^2.
$$
Both session gains cancel, but reference-reading noise, offsets and drift between reference and sample readings remain. Repeated readings do not automatically reveal an unknown common bias.
The primary graphene <isotope> experiment compared Raman bands over <isotope> concentrations and excitation energies, finding 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} The calibration model above concerns precision of the inferred residual stiffness ratio; it does not validate mode assignment or the effective-mass approximation.
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