 Separate spectral-calibration gains in isotope stiffness inference

Separate spectral-calibration gains in isotope stiffness inference arise when the two Raman spectra retain different multiplicative calibration errors. Let the corrected wavenumbers satisfy ${\hat{\nu }}_i=(1+g_i){\nu }_i$. For $K=({\mu }_2/{\mu }_1)({\nu }_2/{\nu }_1{)}^2$, small residual gains giveThe 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 isThis follows by applying the full covariance propagation rule.^[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 $r_i=c_i{\nu }_{\mathrm{ref}}$, a purely multiplicative model givesBoth 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.^[ref] The calibration model above concerns precision of the inferred residual stiffness ratio; it does not validate mode assignment or the effective-mass approximation.