  In this work, we revisit the classic model of diatomic chain with cubic
nonlinearity and investigate the formation mechanism of nonlinear localized
time-periodic solutions (breathers) with frequencies exited the spectral bands.
First we employ the long-chain limit to obtain estimates of linear eigenstates,
especially those with frequencies near the band edges. As the strength of
nonlinearity grows, some frequencies can cross the band edges to turn isolated
while the corresponding states gradually change from non-localized to
localized. Based on the estimates of linear eigenstates, we derive analytical
approximations of these nonlinear states and prove their validity for
frequencies within the bands. Moreover, the process of states growing localized
are illustrated in both analytical and numerical approaches. Although here we
place emphasis on nonlinear middle-localized states with the most generic
boundary conditions, the results can also be extended to a wider range of
localized states including edge states.
