Effect of the momentum transfer on the Rayleigh scattering cross section

The Rayleigh scattering cross section is measured with a short-duration pulse of laser light in different gases. It depends on the pulse duration, involving a time constant $\tau$, following the expression $\frac{S_{\exp }}{S_{\mathrm{theor}}}=\left(\frac{{\tau }_1}{\tau }\right)(1-e^{-\frac{\Delta t}{\tau }})$, where $S_{\exp }$ and $S_{\mathrm{theor}}$ are the experimental and the theoretical cross sections and ${\tau }_1$ is the classical time constant corresponding to the dipole damping. The time constant $\tau$ was determined with a ruby laser whose pulse half-width was varied between 6 and 200 nsec. Using a tunable dye laser and its nitrogen laser pump, the authors have found that $\tau$ is proportional to ${\lambda }^2$ (square of the wavelength). The variation of $\tau$ for gases of different molecular sizes shows that $\tau$ is proportional to their diameter $a$. The angular distribution of the scattered light has been determined and found to be favored in the forward direction. The constant $\tau$ is proportional to $sin\left(\frac{\theta }{2}\right)$ (where $\theta$ is the scattering angle), i.e., to the momentum transfer. By analogy with a corpuscular collision, dimensional considerations lead to the formula $\tau =0.85\mathrm{}{\left(\frac{\lambda }{{\lambda }_c}\right)}^2\left(\frac{a}{c}\right)sin\left(\frac{\theta }{2}\right)$ (${\lambda }_C$ is the Compton wavelength), which well describes the experimental results.