  The diatomic linear chain of masses coupled by harmonic springs is a
textboook model for vibrational normal modes (phonons) in crystals. In addition
to propagating acoustic and optic branches, this model is known to support a
``gap mode'' localized at the surface, provided the atom at the surface has
light rather than heavy mass. An elementary argument is given which explains
this mode and provides values for the frequency and localization length. By
reinterpreting this mode in different ways, we obtain the frequency and
localization lengths for three other interesting modes: (1) the surface
vibrational mode of a light mass impurity at the surface of a monatomic chain;
(2) the localized vibrational mode of a stacking fault in a diatomic chain; and
(3) the localized vibrational mode of a light mass impurity in a monatomic
chain.
