  Synchrotron x-ray scattering measurements of the low-temperature structure of
the single-layer manganese oxide La$_{1-x}$Sr$_{1+x}$MnO$_4$, over the doping
range $0.33 \le x \le 0.67$, indicate the existence of three distinct regions:
a disordered phase ($x < 0.4$), a charge-ordered phase ($x \ge 0.5$), and a
mixed phase ($0.4 \le x < 0.5$). For $x>0.5$, the modulation vector associated
with the charge order is incommensurate with the lattice and depends linearly
on the concentration of $e_g$ electrons. The primary superlattice reflections
are strongly suppressed along the modulation direction and the higher harmonics
are weak, implying the existence of a largely transverse and nearly sinusoidal
structural distortion, consistent with a charge density wave of the $e_g$
electrons.
