Structural phase transitions and oxygen-oxygen interaction energies in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$

We investigate the effects of varying the interaction constants in a lattice-gas model for oxygen ordering in the high-temperature superconductor ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$. A detailed ground-state calculation yields seven topologically distinct ground-state diagrams, which are realized for different values of the interactions. For interactions that produce the experimentally observed ordered phases, we present transfer-matrix finite-size scaling (TMFSS) results which, together with the gound-state diagram, can be used to approximate the global shape of the finite-temperature phase diagram. The effective oxygen-oxygen interactions in ${\mathrm{YBa}}_2$Cu ${}_3{}^{}{}_{}{}^{}\mathrm{O}_{6+\mathit{x}}^{}{}_{}{}^{}$ are estimated by a nonlinear least-squares fit to newly available experimental data of the numerical lattice-gas phase diagram obtained by the TMFSS method with strip widths 8 and 12. An interesting feature of the fitted phase diagram is that a narrow region of the ortho-1/2 phase is found to extend between the tetragonal and ortho-1/4 phases at low oxygen concentration. The fitted interactions and the resulting phase diagram are compared with those recently obtained by Sterne and Wille, using the linear muffin-tin orbital method. We find the reasonable agreement achieved between the effective interactions predicted by these two fundamentally different methods highly encouraging.