  Using general properties of the crystal site representation normal mode
matrix, we provide some very simple bounds on localized modes in simple,
body-centered and face-centered cubic crystals with substitutional point
defects. We derive a trace condition constraint on the net change in crystal
eigenfrequencies caused by the introduction of a defect, with the condition
being a completely general one which holds for any combination of central and
non-central crystal force-constants and for all-neighbor interactions. Using
this condition we show that the sufficient condition for producing localized
modes in an arbitrary cubic crystal by a mass change at the defect site is that
the defect mass be less than one half of that of the host atom mass which it
replaces, and that the sufficient condition for producing localized modes in an
arbitrary cubic crystal by force-constant changes alone is that the defect site
self force-constant be greater than twice that of the pure crystal self
force-constant of the host atom which it replaces.
