  The effect of disorder on lattice vibrational modes has been a topic of
interest for several decades. In this work, we employ a Green's function based
approach, namely the dynamical cluster approximation (DCA), to investigate
phonons in mass disordered systems. Detailed benchmarks with previous exact
calculations are used to validate the method in a wide parameter space. An
extension of the method, namely the typical medium DCA (TMDCA), is used to
study Anderson localization of phonons in three dimensions. We show that, for
binary isotopic disorder, lighter impurities induce localized modes beyond the
bandwidth of the host system, while heavier impurities lead to a partial
localization of the low-frequency acoustic modes. For a uniform (box)
distribution of masses, the physical spectrum is shown to develop long tails
comprising mostly localized modes. The mobility edge separating extended and
localized modes, obtained through the TMDCA, agrees well with results from the
transfer matrix method. A re-entrance behavior of the mobility edge with
increasing disorder is found that is similar to, but somewhat more pronounced
than, the behavior in disordered electronic systems. Our work establishes a new
computational approach, which recovers the thermodynamic limit, is versatile
and computationally inexpensive, to investigate lattice vibrations in
disordered lattice systems.
