Calculate the dispersion relation for longitudinal and transverse phonons along the [100] direction of a fcc crystal whose atoms are joined to their nearest neighbors by springs. Using symmetry arguments, first identify any possible degeneracy. In which other high-symmetry direction does a similar consideration apply? Then, try to describe the phonons such that the equation of motion for a linear chain becomes applicable. For this, it is important to be aware of the effect of the position of the phase planes on the displacements of the atoms. Draw the displacements of the atoms for various phonons at the edge of the Brillouin zone. *Note) High-symmetry directions: directions that involve symmetries such as [100], [110], and [111]