Abstract
The diatomic linear chain of masses coupled by harmonic springs is a textbook 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 length 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.
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