 Relative calibration error in an isotope-corrected stiffness ratio

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,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 becomeswith 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 ${\nu }_1=1000\mathrm{c}{\mathrm{m}}^{-1}$ and ${\nu }_2=900\mathrm{c}{\mathrm{m}}^{-1}$. A differential gain $g_2-g_1=0.001$ creates about $0.20\%$ relative error in $K$. A common offset $b_c=1\mathrm{c}{\mathrm{m}}^{-1}$ 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 $2\sqrt{2}\times 0.1\%\approx 0.2828\%$.

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.^[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.^[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.