Theory for oxygen content and ordering in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathrm{x}}$ in equilibrium with oxygen gas

A model using statistical thermodynamics is presented which quantitatively describes the oxygen content and ordering in ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{6+\mathrm{x}}$. This model is in good agreement with experiments within a broad range of temperature and oxygen partial pressure. In the framework of this model it is found that the site (binding) energy of the oxygen in the lattice is quadratic in the oxygen concentration and linear in the temperature. A possible relation of this dependence to the electronic structure of ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{6+\mathrm{x}}$ is discussed.