Quasi-3D ordered lattice modulations in a bilayer ruthenate with no long-range order

Bulk measurements reveal disorder-induced unconventional quantum critical behaviors in (Sr$_{1-x}$Ca$_x$)$_3$Ru$_2$O$_7$ (SCRO) compounds, in particular, near $x = 0.3$. Here we report X-ray scattering studies on SCRO with $x =0.3$, as well as those for the end members. We find that at $x = 0.3$ robust 2-unit-cell periodic lattice modulations exist that are characterized by $(\frac12,0,0)$ and $(0,\frac12,0)$, respectively, even at room temperature. These modulations are transversely polarized and quasi-3D ordered in that they are fully coherent in the basal plane with {\bf c} axis correlations at least one unit cell in extent. These modulations are due to correlated displacements of the O atoms. The displacement pattern is consistent with $t_{2g}$-modes of distortion of RuO$_6$ octahedra, signifying the presence of lattice and orbital correlations, although no long-range magnetic or orbital order is present. These modulations are absent in the end members.