Effect of the momentum transfer on the Rayleigh scattering cross section

Abstract

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 $\ensuremath{\tau}$, following the expression $\frac{{S}_{\mathrm{exp}}}{{S}_{\mathrm{theor}}}=(\frac{{\ensuremath{\tau}}_{1}}{\ensuremath{\tau}})(1\ensuremath{-}{e}^{\ensuremath{-}\frac{\ensuremath{\Delta}t}{\ensuremath{\tau}}})$, where ${S}_{\mathrm{exp}}$ and ${S}_{\mathrm{theor}}$ are the experimental and the theoretical cross sections and ${\ensuremath{\tau}}_{1}$ is the classical time constant corresponding to the dipole damping. The time constant $\ensuremath{\tau}$ 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 $\ensuremath{\tau}$ is proportional to ${\ensuremath{\lambda}}^{2}$ (square of the wavelength). The variation of $\ensuremath{\tau}$ for gases of different molecular sizes shows that $\ensuremath{\tau}$ 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 $\ensuremath{\tau}$ is proportional to $sin(\frac{\ensuremath{\theta}}{2})$ (where $\ensuremath{\theta}$ is the scattering angle), i.e., to the momentum transfer. By analogy with a corpuscular collision, dimensional considerations lead to the formula $\ensuremath{\tau}=0.85{(\frac{\ensuremath{\lambda}}{{\ensuremath{\lambda}}_{c}})}^{2}(\frac{a}{c})sin(\frac{\ensuremath{\theta}}{2})$ (${\ensuremath{\lambda}}_{C}$ is the Compton wavelength), which well describes the experimental results.