Vibrational excitation of molecules by resonance scattering of electrons

The excitation of vibrational energy levels of molecules in collisions with electrons of energy in the range of a few electron volts is studied. The assumption is made that the prominent peaks in the scattering cross-sections which are sometimes observed may be interpreted as resonances of the incident electron in the potential well represented by the molecule, the width of the peak being the inverse of the lifetime of a short-lived negative ion. The scattering is treated by a modification of the Kapur-Peierls formalism originally developed for resonant scattering in nuclear physics. Equations are derived for the scattering amplitudes to the different vibrational states of the target molecule, and solved analytically for a number of extreme cases characterized by the relative magnitude of the lifetime of the negative ion and the periods of the molecular vibrations, and by the strength of the coupling between the vibrations and the motion of the incident electron. The molecular rotations and the Coriolis interaction are neglected during the lifetime of the negative ion, and arguments are given to justify these approximations. The theory is applied to the scattering of electrons from nitrogen molecules, where the observed partial inelastic cross-sections show a complicated oscillatory behaviour as functions of the energy. This is reminiscent of that predicted analytically for certain of the extreme cases, but since the parameters required do not fall into any of the extreme categories, the equations for the scattering amplitudes are solved numerically. The salient features of the observed cross-sections are accounted for, although there is room for improvement in the details. The parameters required are a width of the electronic resonance of 0.20 eV (which is almost identical with the 0.3 eV of the spacing of the vibrational energy levels of the molecule) and a value of 12 x 108 eV/cm for the slope of the potential felt by the nuclei due to the extra electron when they are at their normal separation.