Mean-field and Monte Carlo calculations of the three-dimensional structure factor for ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$

We develop a general mean-field matrix theory for calculating phase boundaries and structure factors for a class of lattice gas Hamiltonians. We investigate the oxygen ordering in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$ by applying the theory and by Monte Carlo simulation using an extension to three dimensions of the two-dimensional anisotropic next nearest neighbor lattice gas model. The calculation of the structure factor in three spatial dimensions on a 256×256×16 system has been implemented on a massively parallel computer, the Connection Machine CM2. This allows for an extremely accurate determination of the line shape of the structure factor in the entire reciprocal space in excellent agreement with the mean-field prediction. We report on the results for an oxygen stoichiometry of x=0.4, x=0.5, and x=0.6 for temperatures between T=450 and 800 K. Using a sum-rule argument we have calculated the fluctuation-renormalized transition temperature into the ortho-II phase, ${\mathit{T}}_{\mathit{c}}$, in good agreement with Monte Carlo simulation results. This allows an analytic description of the dependence of ${\mathit{T}}_{\mathit{c}}$ on the interaction ${\mathit{V}}_4$ in the third dimension and on the oxygen stoichiometry x. An explanation for the experimental observation of the lack of a Bragg peak in the ordered ortho-II phase of ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\mathit{x}}$ is offered in terms of scattering from platelike antiphase oxygen-ordered domains. © 1996 The American Physical Society.