  A semiclassical impact parameter version of the continuum distorted
wave-Eikonal initial state theory is developed to study the differential
ionization of $Li$ atoms in collisions with $Li^{2+}$ ions. Both post and prior
forms of the transition amplitude are considered. The fully differential cross
sections are calculated for the lithium targets in their ground and their first
excited states and for the projectile ions at 16~MeV impact energy. The role of
the internuclear interaction as well as the significance of the post-prior
discrepancy in the ejected electron spectra are investigated. The obtained
results for ejection of the electron into the azimuthal plane are compared with
the recent measurements and with their corresponding values obtained using a
fully quantum mechanical version of the theory. In most of the cases, the
consistency of the present approach with the experimental and the quantum
theoretical data is reasonable. However, for $2p$-state ionization, in the
cases where no experimental data exist, there is a considerable difference
between the two theoretical approaches. This difference is questionable and
further experiments are needed to judge which theory makes a more accurate
description of the collision dynamics.
