  We have used an atomistic {\it ab initio} approach with no adjustable
parameters to compute the lattice thermal conductivity of Si$_{0.5}$Ge$_{0.5}$
with a low concentration of embedded Si or Ge nanoparticles of diameters up to
4.4 nm. Through exact Green's function calculation of the nanoparticle
scattering rates, we find that embedding Ge nanoparticles in
$\text{Si}_{0.5}\text{Ge}_{0.5}$ provides 20% lower thermal conductivities than
embedding Si nanoparticles. This contrasts with the Born approximation which
predicts an equal amount of reduction for the two cases, irrespective of the
sign of the mass difference. Despite these differences, we find that the Born
approximation still performs remarkably well, and it permits investigation of
larger nanoparticle sizes, up to 60 nm in diameter, not feasible with the exact
approach.
