Ariete Righi

Ariete Righi coauthored the 2015 study comparing Raman spectra of graphene with different carbon-isotope concentrations.^[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 thereforeIt 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 $g^{\prime }(x)=(12-25x)/(12+x{)}^3$, 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.