Fig. 4.

d-form factor DW: predominance and robustness. (A) Power spectral-density (PSD) Fourier transforms of R(r) measured only at the O_x/O_y sites yielding |$\left({\tilde{O}}_x\left(\mathit{q}\right)-{\tilde{O}}_y\left(\mathit{q}\right)\right)/2$|². This provides the measure of relative strength of the d-form factor in the DW. (B) Measured PSD is plotted along the dashed line through Q in A and shows the d-form factor component predominates greatly. The measured ratios within the DW peaks surrounding Q are D/S > 5 and D/S′ > 12. (C) ${\tilde{O}}_x\left(\mathit{q}\right)\equiv Re{\tilde{O}}_x\left(\mathit{q}\right)+iIm{\tilde{O}}_x\left(\mathit{q}\right)$ compared with ${\tilde{O}}_y\left(\mathit{q}\right)\equiv Re{\tilde{O}}_y\left(\mathit{q}\right)+iIm{\tilde{O}}_y\left(\mathit{q}\right)$ for each of a series of representative q within the DW peaks surrounding Q. This shows how, wherever the CuO₂ unit cell resides in the disordered DW (Fig. 2A), the relative phase between the O_x and O_y sites is very close to π whereas the differences in magnitudes are close to zero. (D) Two-axis histogram of difference in normalized magnitude (vertical) and phase (horizontal) between all pairs ${\tilde{O}}_x\left(\mathit{r},\mathit{Q}\right)$ and ${\tilde{O}}_y\left(\mathit{r},\mathit{Q}\right)$ that are obtained by Fourier filtration of O_x(r) and O_y(r) to retain only q ∼ Q (SI Text, section 5). This represents the measured distribution of amplitude difference, and phase difference, between each pair of O_x/O_y sites everywhere in the DW. It demonstrates directly that their relative phase is always close to π and that their magnitude differences are always close to zero. (E) Measured R(r) images of local electronic structure patterns that commonly occur in BSCCO and NaCCOC (11). The Cu and O_x sites (labeled by solid and dashed arrows, respectively) were determined independently and directly from topographic images (11). (F) (Left) dFF-DW model with Q = (0.25,0) and amplitude maximum on the central O_x site (dashed arrow). (Right) The calculated density-wave pattern from this model. Therefore, a dFF-DW model with this particular spatial phase provides an excellent explanation for the observed density patterns shown in E and reported previously in refs. 11–13 and 32.