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 numbers Z of the target and various projectile velocities.When fast ions pass through matter the electrons ejected from the atoms are called delta rays. The distributions in energy and angle of these delta rays is an interesting subject with a long history. The detailed understanding of these doubly differential cross sections (ddcs) provides information about atomic structure as well as the mechanism of the reaction. The integral over angle gives the singly differential cross section (sdes). The integral of the sdcs over all positive total energies of the outgoing electron gives the cross section for ionization leading to X rays and the Auger processes. The most detailed information is contained in the ddcs. The relative importance of Coulomb ionization, charge transfer and shell interpenetration effects depend on the relation of the ion velocity, the average velocity of the bound electrons, the ionic charge, and the emission angle of the electron. The Coulomb ionization is most important for a bare ion at relatively large velocity, and the experimental ddcs for low Z targets is sometimes well described by Born approximation [-1]. The experimental sdcs is also fairly well described by a much simpler analysis [2] which is classical rather than quantal. The fact that the classical and the quantal derivations of the Rutherford scattering cross section give identical results is a key to this agreement. The classical model [3] of the ddcs which has provided some reasonable agreements with experiment is that of Bonsen and Vriens (BV). In this model the ion and a bound electron undergo Rutherford scattering. After energy is transfered to the electron it emerges from the atom with a final kinetic energy reduced by its ionization potential. The role of the atom's nucleus and other electrons is simply to provide an initial momentum distribution of the target electron, and one must integrate over this momentum distribution to obtain the ddcs. At least 19 publications [1,[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] have involved this BV model of the ddcs in their analysis. The advantage of the BV model is that the calculational complexity is much reduced, compared to Born approximation. Consequently the Born approximation for ddcs has been used almost exclusively for light targets. As mentioned above, the classical approach a...