The harmonic oscillator supplies a useful reference model. With the same effective bond stiffness and an effective vibrating mass , the mode frequency is proportional to . 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 in ideal carbon-12 material would move toin the corresponding ideal carbon-13 model. The approximately frequency reduction is smaller than the mass increase because frequency depends on the inverse square root of mass.
If stiffness changes too, the ratio instead contains both effects: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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