relation where i is the place where the atom would sit if it were in equilibrium, mi is the mass of the atom that should sit at i, α is the direction of its displacement, ui,α is the amount of displacement of the atom from i, and are the force constants which come from the crystal potential. The solution to this gives the atomic displacement due to the phonon, which is given by where the atomic position i is described by l, m, and κ, which represent the specific atomic layer, l, the particular unit cell it is in, m, and the position of the atom with respect to its own unit cell, κ. The term x(l,m) is the position of the unit cell with respect to some chosen origin. Normal modes of vibration and types of surfaces phonons Phonons can be labeled by the manner in which the vibrations occur. If the vibration occurs lengthwise in the direction of the wave and involves contraction and relaxation of the lattice, the phonon is called a "longitudinal phonon". Alternatively, the atoms may vibrate side-to-side, perpendicular to wave propagation direction; this is known as a "transverse phonon”. In general, transverse vibrations tend to have smaller frequencies than longitudinal vibrations. The wavelength of the vibration also lends itself to a second label. "Acoustic" branch phonons have a wavelength of vibration that is much bigger than the atomic separation so that the wave travels in the same manner as a sound wave; "optical" phonons can be excited by optical radiation in the infrared wavelength or longer. Phonons take on both labels such that transverse acoustic and optical phonons are denoted TA and TO, respectively; likewise, longitudinal acoustic and optical phonons are denoted LA and LO. The type of surface phonon can be characterized by its dispersion in relation to the bulk phonon modes of the crystal. Surface phonon mode branches may occur in specific parts of the SBZ or encompass it entirely across. These modes can show up both in the bulk phonon dispersion bands as what is known as a resonance or outside these bands as a pure surface phonon mode. Thus surface phonons can be purely surface existing vibrations, or simply the expression of bulk vibrations in the presence of a surface, known as a surface-excess property. A particular mode, the Rayleigh phonon mode, exists across the entire BZ and is known