Nature of $e_g$ Electron Order in La$_{1-x}$Sr$_{1+x}$MnO$_4$

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 ($x 0.5$, 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.