  We extend a semi-analytical model that includes the finite nuclear thickness
to calculate the energy density $\epsilon(t)$ and conserved-charge densities
including the net-baryon density $n_{\rm _B}(t)$ produced at
mid-spacetime-rapidity in central Au+Au collisions. Assuming the formation of a
quark-gluon plasma with an ideal gas equation of state of either quantum or
Boltzmann statistics or with a lattice QCD-based equation of state, we extract
the temperature $T$ and chemical potentials $\mu_{\rm _B}$, $\mu_{\rm _Q}$ and
$\mu_{\rm _S}$ as functions of time. This then allows us to semi-analytically
calculate the $T-\mu_{\rm _B}$ trajectory of relativistic nuclear collisions in
the QCD phase diagram, which should benefit the studies of high density physics
including the search for the critical end point. This model is also useful for
exploring the trajectories in the more general $T-\mu_{\rm _B}-\mu_{\rm
_Q}-\mu_{\rm _S}$ QCD phase space.
