Concerning Order and Disorder in the Ensemble of Cu-O Chain Fragments in Oxygen Deficient Planes of Y-Ba-Cu-O

In connection with numerous X-ray and neutron investigations of some high temperature superconductors (YBa$_2$Cu$_3$O$_{6+x}$ and related compounds) a non-trivial part of the structure factor, coming from partly disordered Cu-O-$\dots$-O-Cu chain fragments, situated within basal planes, CuO$_x$, can be a subject of theoretical interest. Closely connected to such a diffusive part of the structure factor are the correlation lengths, which are also available in neutron and X-ray diffraction studies and depend on a degree of oxygen disorder in a basal plane. The quantitative measure of such a disorder can be associated with temperature of a sample anneal, $T_q$, at which oxygen in a basal plane remains frozen-in high temperature equilibrium after a fast quench of a sample to room or lower temperature. The structure factor evolution with $x$ is vizualized in figures after the numerical calculations. The theoretical approach employed in the paper has been developed for the orthorhombic state of YBCO.