Inelastic scattering of electrons by metastable helium: First Born and Glauber cross sections for $2{}_{}{}^3{}_{}{}^{}S-3\mathrm{}{}_{}{}^3{}_{}{}^{}S$ excitation

We calculate the differential and the total cross sections for the direct excitation of the $3{}_{}{}^3{}_{}{}^{}S$ state of helium from the metastable $2{}_{}{}^3{}_{}{}^{}S$ state by electron impact using the Glauber and Born methods. For the Glauber cross-section calculation we use the analytic method of Thomas and Chan to reduce the Glauber amplitude for charged-particle-neutral-atom collisions to a one-dimensional integral representation involving modified Lommel functions. The calculations are made for incident electron energies between 3 and 1000 eV, and for two different sets of $3{}_{}{}^3{}_{}{}^{}S$ wave functions. The results of the differential and the total excitation cross sections are compared with all other available theoretical calculations.