  The excitation energy and the nuclear density at the time of breakup are
extracted for the $\alpha + ^{197}Au$ reaction at beam energies of 1 and 3.6
GeV/nucleon. These quantities are calculated from the average relative velocity
of intermediate mass fragments (IMF) at large correlation angles as a function
of the multiplicity of IMFs using a statistical model coupled with many-body
Coulomb trajectory calculations. The Coulomb component $\vec{v}_{c}$ and
thermal component $\vec{v}_{0}$ are found to depend oppositely on the
excitation energy, IMFs multiplicity, and freeze-out density. These
dependencies allow the determination of both the volume and the mean excitation
energy at the time of breakup. It is found that the volume remained constant as
the beam energy was increased, with a breakup density of about $\rho_{0}/7$,
but that the excitation energy increased $25\%$ to about 5.5 MeV/nucleon.
