= Marcos A. Pimenta
{c}
\b[Marcos A. Pimenta] coauthored the 2015 Raman investigation of <isotopic composition> in <graphene>.<Probing carbon isotope effects on the Raman spectra of graphene with different ¹³C concentrations by Bruno R. Carvalho and others>[published byline]{ref}
The <harmonic oscillator> supplies a useful reference model. With the same effective bond stiffness $k$ and an effective vibrating mass $m$, the mode frequency is proportional to $\sqrt{k/m}$. Replacing carbon-12 by <carbon-13> changes mass without changing the number of protons; comparing isotopes can therefore help separate mass effects from changes in the bonding model.
For an original unchanged-stiffness estimate, a mode at $1580\,\mathrm{cm^{-1}}$ in ideal carbon-12 material would move to
$$
\widetilde\nu_{13}=1580\sqrt{\frac{12}{13}}\approx1518.0\,\mathrm{cm^{-1}}
$$
in the corresponding ideal carbon-13 model. The approximately $3.92\%$ frequency reduction is smaller than the $8.33\%$ mass increase because frequency depends on the inverse square root of mass.
If stiffness changes too, the ratio instead contains both effects:
$$
\frac{\nu_2}{\nu_1}=\sqrt{\frac{k_2}{k_1}\frac{m_1}{m_2}}.
$$
A frequency shift alone then cannot identify a stiffness change without composition information. This connects <Raman spectroscopy> to an <inverse problem>: a measured peak position constrains a combination of model parameters. The numerical frequencies above are an illustrative calculation, not a reproduction of Pimenta's experimental data.
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