Absence of long-ranged charge order in rm Na_{x}Ca_{2-x}CuO₂Cl₂ at x=0.08

A periodic $4a \times 4a$ density of states (DOS) modulation (a "checkerboard pattern") was observed with STM in $\rm Na_{x}Ca_{2-x}CuO_2Cl_2$ (NCCOC) [T. Hanaguri \emph{et al.}, Nature \textbf{430}, 1001 (2004)]. Its periodicity is the same as that of the "stripe" charge order observed with neutron scattering in $\rm La_{1.875}Ba_{0.125}CuO_4$ (LBCO) [J. M. Tranquada \emph{et al.}, Nature \textbf{429}, 534 (2004)] and $\rm La_{1.48}Nd_{0.4}Sr_{0.12}CuO_4$ (LNSCO) [J. M. Tranquada \emph{et al.}, Phys. Rev. B \textbf{54}, 7489 (1996)]. An obvious question is whether the "stripes" are actually "checkers". Unfortunately, because NCCOC samples are small and LBCO samples do not cleave, neutron and STM measurements cannot be carried out on the same system. To determine the relationship between stripes and checkers we used resonant soft x-ray scattering (RSXS), previously applied to LBCO [P. Abbamonte \emph{et al.}, Nature Physics \textbf{1}, 155 (2005)], to study single crystals of NCCOC. No evidence was seen for a $4a \times 4a$ DOS modulation, indicating that the checkerboard effect is not directly related to the stripe modulation in LBCO. Our measurements suggest either glassy electronic behavior or the existence of a surface-nucleated phase transition in NCCOC [S. E. Brown \emph{et al.}, Phys. Rev. B \textbf{71}, 224512 (2005)].