  Double perovskites like Sr$_2$FeMoO$_6$ are materials with half-metallic
ground states and ferrimagnetic T$_{\rm{c}}$'s well above room temperature.
This paper is the second of our comprehensive theory for half metallic double
perovskites. Here we derive an effective Hamiltonian for the Fe core spins by
"integrating out" the itinerant Mo electrons and obtain an unusual double
square-root form of the spin-spin interaction. We validate the classical spin
Hamiltonian by comparing its results with those of the full quantum treatment
presented in the companion paper "Theory of Half-Metallic Double Perovskites I:
Double Exchange Mechanism". We then use the effective Hamiltonian to compute
magnetic properties as a function of temperature and disorder and discuss the
effect of excess Mo, excess Fe, and anti-site disorder on the magnetization and
T$_{\rm{c}}$. We conclude with a proposal to increase T$_{\rm{c}}$ without
sacrificing carrier polarization.
