Angular Dependence of the Vibrational and Rotational Excitations Seen in Photoelectron Spectroscopy

The vibrational and rotational excitations seen in photoelectron spectroscopy are shown to have intensities proportional to $k^2{\left(\frac{m}{M}\right)}^2[\alpha +(\beta -\alpha \left){sin}^2\Theta \right]$, where $\overrightarrow{\mathrm{k}}$ is the wave vector of the incident radiation, $m$ is the mass of the excited nucleus, $M$ is the total mass of the molecule, and $\Theta$ is the angle between the wave vector of the ejected electron and the polarization direction of the radiation. The $\alpha$ and $\beta$ are absolute squares of the nuclear transition matrix elements. This angular dependence is similar in form to the dependence of the photoelectric cross section which is $\alpha +(\beta -\alpha ){cos}^2\Theta$, where $\alpha$ and $\beta$ are the absolute squares of the transition matrix elements for the photoejection of an electron.