Ultrasimple calculation of very-low-energy momentum-transferand rotational-excitation cross sections: e-${\mathrm{N}}_2$ scattering

The calculation of electron-molecule cross sections at scattering energies well below 0.1 eV using conventional algorithms for solving the Schrödinger equation is often rendered problematic by severe numerical problems. Here we describe and implement an alternative procedure that combines known analytic properties of the body-frame electron-molecule scattering matrix, as codified in the modified effective range theory, with an analytic correction that imposes physically correct threshold laws. This approach eliminates completely the need for numerically solving the Schrödinger equation at energies below about 0.1 eV. Instead, one uses scattering matrices above this energy to determine parameters for an extrapolation to subthermal energies. We apply this method to the calculation of e-${\mathrm{N}}_2$ momentum-transfer and rotational excitation cross sections from threshold to 1.25 eV. The results resolve a long-standing apparent anomaly in the analysis of experimental data for very low-energy electron scattering from ${\mathrm{N}}_2$. Finally, we use linear regression to present our theoretical results in a user-friendly form.