Analytic expressions for the initial Cooperative Decay Rate and Cooperative Lamb Shift for a spherical sample of two-level atoms

We give the analytic expressions for the initial Cooperative Decay Rate and Cooperative Lamb Shift for a spherical cloud of two-level atoms for the cases of uniform and Gaussian number density distributions. We derive these expressions in both scalar and vector models for the cases when the system's initial polarization is uniform and when it is coherently phased.