Cross section measurements at sub-thermal energies

Experiments with merged beams of electrons and cooled molecular ions now extend to relative energies lower than the electron temperature. These experiments measure a velocity-averaged cross section [sigma upsilon(rel)] or rate coefficient. A formula is presented here for calculating the rate coefficient for an E-rel(-n) cross section and a velocity distribution characterized by separate temperatures for thermal motion parallel and perpendicular to the electron-ion relative velocity (e.g., like that of an electron cooler). The exponent n and the two temperatures T-parallel to and T-perpendicular to are left arbitrary. Inspection of plots of the rate coefficient leads to the surprising conclusion that n can be found most readily from data taken in the region E-rel < kT(perpendicular to) where one might expect information loss due to electron thermal energy spread. Some limiting cases are examined. The rate coefficient is found to diverge in the limit E-rel --> 0 and T-parallel to --> 0 when n > 3/2. For the special case n = 1 and T-parallel to = 0 the rate coefficient is given by an error function.