= Ariete Righi
{c}
\b[Ariete Righi] coauthored the 2015 study comparing <Raman spectra> of <graphene> with different carbon-isotope concentrations.<Probing carbon isotope effects on the Raman spectra of graphene with different ¹³C concentrations by Bruno R. Carvalho and others>[published byline]{ref}
An original random-mass model explains why isotopic mixture and average mass are different quantities. Let each carbon site independently have mass $12u$ with probability $1-x$ and $13u$ with probability $x$. The mean mass is $(12+x)u$, whereas the mass variance is $x(1-x)u^2$. A dimensionless relative variance is therefore
$$
g(x)=\frac{x(1-x)}{(12+x)^2}.
$$
It vanishes in either pure-isotope limit, despite their different mean masses. Thus uniform replacement can change a mode frequency while eliminating this particular source of random mass variation.
Differentiation gives $g'(x)=(12-25x)/(12+x)^3$, so this model's maximum occurs at $x=12/25=0.48$. At $x=1/2$, $g=0.0016$; the nearby maximum is about $0.001603$. These are properties of the stipulated independent-site model, not measured linewidths.
Disorder-induced scattering additionally depends on the vibrational states and their interactions. The variance expression alone does not predict a complete <phonon> lifetime or line shape, and a non-random spatial isotope arrangement can require a different model. The example separates composition, disorder and spectral interpretation without assigning an undocumented individual role in the collaborative experiment.
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