  The band-unfolding method is widely used to calculate the effective band
structures of a disordered system from its supercell model. The unfolded band
structures show the crystallographic symmetry of the underlying structure,
where the difference of chemical components and the local atomic relaxation are
ignored. It has been, however, still difficult to decompose the unfolded band
structures into the modes based on the crystallographic symmetry of the
underlying structure, and therefore detailed analyses of the unfolded band
structures have been restricted. In this study, we develop a procedure to
decompose the unfolded band structures according to the small representations
(SRs) of the little groups. For this purpose, we derive the projection
operators for SRs based on the group representation theory. We also introduce
another type of projection operators for chemical elements, which enables us to
decompose the unfolded band structures into the contributions of different
combinations of chemical elements. Using the current method, we investigate the
phonon band structure of disordered face-centered cubic Cu0.75Au0.25, which has
large variations of atomic masses and force constants among the atomic sites
due to the chemical disorder. We find several peculiar behaviors such as
discontinuous and split branches in the unfolded phonon band structure. They
are found to occur because different combinations of the chemical elements
contribute to different regions of frequency.
