= One-mode Raman spectrum example
{c}
Choose a vibrational <Raman shift> magnitude $\widetilde\nu_v=1000\,\mathrm{cm^{-1}}$, excitation <vacuum> <wavelength> $\lambda_0=532\,\mathrm{nm}$ and <temperature> $T=300\,\mathrm K$. These are original illustrative inputs, not a measured 1928 <optical spectrum> or a claim about a particular liquid. They give
$$
\begin{aligned}
\nu_v&=c\widetilde\nu_v=29.9792458\,\mathrm{THz},&
h\nu_v&\simeq0.123984\,\mathrm{eV},\\
\widetilde\nu_0&\simeq18796.9925\,\mathrm{cm^{-1}},&
\lambda_{\rm S}&=\frac{1}{\widetilde\nu_0-\widetilde\nu_v}
\simeq561.893\,\mathrm{nm},\\
\lambda_{\rm AS}&=\frac{1}{\widetilde\nu_0+\widetilde\nu_v}
\simeq505.127\,\mathrm{nm}.&&
\end{aligned}
$$
Take reciprocal lengths in consistent units before converting to nanometres. At $300\,\mathrm K$, $r\simeq0.00826337$ and $\overline v\simeq0.00833222$. The predicted line-integrated <radiant intensity> ratio is $0.0126522$, whereas the ideal <photon> count-rate ratio is $0.0113740$. Thus the radiometric ratio is about $1.27$ percent, rather than the $0.826$ percent obtained by using the <Boltzmann factor> alone.
\Image[/raman-vibrational-levels-and-spectrum.png]
{title=Molecular <energy> transfer and an original one-mode <Raman scattering> line model}
{description=Upper panels show initial and final vibrational levels with schematic virtual intermediates. Lower panel uses the stated $1000\,\mathrm{cm^{-1}}$, $532\,\mathrm{nm}$, $300\,\mathrm K$ example. Chosen equal <Gaussian function> linewidths have standard deviation $25\,\mathrm{cm^{-1}}$; the <Rayleigh scattering> line has an arbitrarily chosen integrated area of $20$ relative to a unit Stokes area. The anti-Stokes area follows the line-integrated <radiant intensity> formula. The vertical axis is logarithmic. No line shape, width, relative elastic strength or line position is a historical measurement.}
{source=raman-vibrational-levels-and-spectrum.py}
Back to article page
