Mixed valency, hole concentration, and $T_c$ in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathrm{x}}$

Combining the assumptions that in orthorhombic ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{6+\mathrm{x}}$ oxygen is removed from the CuO chains one chain at a time and that the Cu atoms in the twofold coordinated sites of the empty chains have a valence of +1 leads to the prediction that the number of ${\mathrm{Cu}}^{1+}$ ions per unit cell is equal to 1-x. X-ray absorption measurements at the Cu K edge in both oriented and unoriented quenched samples with 0<x≤1 confirm this prediction. Charge neutrality then requires that the number of O 2p holes per unit cell be equal to x. It follows that the superconducting transition temperature $T_c$ should be proportional to x (rather than 2x-1), as observed in our quenched samples. Thus, the CuO chains, though not required for superconductivity, act as an electron reservoir which serves to maximize $T_c$ when oxygen is removed. The nonlinear variation of $T_c$ observed in annealed samples is explainable in terms of spinodal decomposition, as first suggested by Khachaturyan and Morris.