Prediction of thermodynamic stability and limits of isomorphous substitutions in potential high-temperature superconductors of the Pb_10–xCu_x(PO₄)₆O (LK-99) system

Using Urusov’s crystal energy theory of isomorphous miscibility, the mixing energy and critical (maximum) decomposition temperature of solid solutions in the Pb_10–xCu_x(PO₄)₆O system were calculated. Decomposition domes were constructed for the Pb_10–xCu_x(PO₄)₆O system under the approximation of regular and asymmetric regular solid solutions, allowing for graphical determination of decomposition temperature at a given substitution limit or equilibrium substitution limit at a specified temperature, as well as thermodynamically stable regions of the solid solutions. It was shown that isomorphous substitution in the approximation of asymmetric regular solid solutions follows the Goldschmidt polarity rule: a smaller component dissolves in a larger one to a higher limit than in the reverse case. Additionally, our findings align with our established rule, which states that “the decomposition temperature is higher for the component with a smaller substituent structural unit and lower for the larger one”. It is hypothesized that the synthesis of Pb₉Cu[(PO₄)₆O] (LK-99) at temperatures below the calculated values is influenced by sodium cations or other single-charged cations from uncontrolled impurities, which may stabilize the apatite structure at temperatures of 1100–1200 K.