Order and disorder in the ensemble of Cu-O chain fragments in oxygen-deficient planes of ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$

In connection with numerous x-ray- and neutron-diffraction investigations of some high-temperature superconductors (${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{6+\mathit{x}}$ and related compounds), a nontrivial part of the structure factor, coming from partly disordered Cu-O-...-O-Cu chain fragments situated within the ${\mathrm{CuO}}_{\mathit{x}}$ basal planes, 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 x-ray- and neutron-diffraction studies and which depend on the degree of oxygen disorder in a basal plane. The quantitative measure of such disorder can be associated with the temperature of a sample anneal ${\mathit{T}}_{\mathit{q}}$ at which the oxygen in the basal plane remains frozen in a high-temperature equilibrium state after a fast quench of the sample to room or lower temperature. The theoretical approach, giving some analytical predictions, is based on numerical calculations. The main goal is to show how the structure factor changes with oxygen content x. Also illustrated are some other physical quantities, i.e., the concentration of holes leaving ${\mathrm{CuO}}_{\mathit{x}}$ planes, the concentration of differently coordinated copper atoms, the number of paramagnetically active Cu-O-... chain fragments, etc. The theoretical approach employed in the paper has been developed for the orthorhombic state of ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{6+\mathit{x}}$ (YBCO).