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The covariance of differential spectral-calibration gains describes the joint errors of gain differences used in several spectral ratios. Let the calibration-error vector be $\mathbf{g}$ with covariance ${\Sigma }_g$, and form differences $\mathbf{d}=B\mathbf{g}$. ThenThis covariance identity is exact for the linear differences; no normal-distribution assumption is required. The first-order relation between gain differences and isotope stiffness ratios is given in separate spectral-calibration gains in isotope stiffness inference.^[ref]

For an original shared-reference model, take three gain errors $g_i=c+e_i$, where $c$ is a common scale error and the $e_i$ are independent residual errors with variances $s_i^2$. Two sample spectra are compared with the same reference spectrum:The common scale cancels, but the residual reference error remains in both differences:Cancellation of one common influence therefore does not establish independent comparisons.

Choose all three residual standard uncertainties as $0.1\%$. Then ${\Sigma }_d=10^{-6}\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$, with correlation $1/2$ between the differences. Even a much larger common scale uncertainty cancels in this stipulated model. Each inferred log stiffness ratio has error approximately twice its gain difference. Averaging the two log-ratio estimates gives variance $6\times 10^{-6}$ and standard uncertainty about $0.245\%$. Incorrectly treating them as independent would give only $0.200\%$.

This shared-reference floor in averaged isotope stiffness ratios matters when many sample spectra reuse one calibrated reference. Independent extra sample readings reduce their own errors, while a reference component shared by all comparisons remains. Assess covariance from the actual reference reuse, calibration drift and repeated acquisition protocol. The matrix calculation concerns multiplicative calibration errors; uncertain peak assignment, isotope composition and additive spectral offsets need their own contributions.

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.