Basal-plane incommensurate phases in hexagonal-close-packed structures

An Ising model with competing interactions is used to study the appearance of incommensurate phases in the basal plane of a hexagonal-close-packed structure. The calculated mean-field phase diagram reveals various $1q$-incommensurate and lock-in phases. The results are applied to explain the basal-plane incommensurate phase in some compounds of the $A^{\prime }{A}^{″}{\mathrm{BX}}_4$ family, like ${\mathrm{K}}_2{\mathrm{MoO}}_4$, ${\mathrm{K}}_2{\mathrm{WO}}_4$, ${\mathrm{Rb}}_2{\mathrm{WO}}_4$, and to describe the sequence of high-temperature phase transitions in other compounds of this family.