We analytically study the nonlinear localized gap modes in a one-dimensiona
l atomic chain with uniform atomic mass but two periodically alternating fo
rce constants between the nearest neighbors by means of a quasi-continuum a
pproximation. This model simulates a row of atoms in the (111) direction of
a diamond-structure type of crystals or molecular crystals with alternatin
g double and single bonds. For this lattice system, rye find that the harmo
nic plus quartic anharmonic terms of inter-site potential produce a new typ
e of nonlinear localized gap modes with a slightly asymmetry distribution o
f atomic displacements. These localized gap modes are somewhat different fr
om widely studied localized gap modes with a symmetry atomic displacement d
istribution in diatomic ion lattices.