Doubly differential cross section for binary encounter with massive projectile and isotropic target velocities

The nonrelativistic doubly differential cross section for electrons ejected from atoms by a massive beam of bare ions is derived in the binary encounter approximation. For cases already published the results sometimes differ by as much as a factor of 80 from those of Bonsen and Vriens, although the agreement should be exact. The case when the struck electron was initially very slow or stationary is also considered. Methods are presented for visualizing the kinematics including questions of minimum and maximum initial electron velocity for a given energy transfer. Also presented is a graphical means of visualizing the origin of the binary peak and of other effects for various atomic numbersZ of the target and various projectile velocities.