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Identifiability of stiffness from isotope-resolved Raman shifts asks which stiffness parameters become determined once the moving masses are independently constrained. In the assigned harmonic oscillator model of reduced-mass ambiguity in a Raman stiffness inference,Here $c$ is the speed of light, ${\tilde{\nu }}_i$ the vibrational wavenumber and ${\mu }_i$ the mode's known effective mass. Known masses and identified modes make each effective stiffness identifiable within this model. Unknown masses still permit the simultaneous rescaling $(k_i,{\mu }_i)\mapsto (a{k}_i,a{\mu }_i)$ without changing a frequency; knowing only mass ratios determines only stiffness ratios.

Controlled isotope substitution is useful because it changes nuclear mass while approximately preserving the interatomic force model. In a pure-carbon mode for which every moving mass changes by the same ratio, unchanged stiffness predictsThe mass numbers here are an explicit approximation. They replace a full normal mode calculation only for the stipulated uniform mass scaling.

For an original numerical test, a chosen carbon-12 reference band at $1580\mathrm{c}{\mathrm{m}}^{-1}$ predicts about $1518.01\mathrm{c}{\mathrm{m}}^{-1}$ after uniform carbon-13 replacement with unchanged stiffness. If a hypothetical carbon-13 band is instead $1520\mathrm{c}{\mathrm{m}}^{-1}$, its isotope-corrected Raman stiffness ratio isThe residual inferred change is about $0.262\%$, not the several-percent softening one would infer by ignoring mass. This is a chosen calculation, not a quoted experimental peak pair.

The primary graphene study measured several Raman bands at different carbon-13 concentrations and laser excitation energies; its measured mass-dependent frequencies followed the harmonic-oscillator relation.^[ref] Such a controlled series tests the mass model. It does not by itself rule out changes in strain, temperature or mode mixing in a different sample. Uncertainty in isotope composition, spectral calibration and mode assignment must accompany any small residual stiffness inference.

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.