Thermodynamics of the incommensurate state in ${\mathrm{Rb}}_2\mathrm{W}{\mathrm{O}}_4:$ The Lifshitz point in $A_2{\mathrm{BX}}_4$ compounds

We consider the evolution of the phase transition from the parent hexagonal phase ${P6\mathrm{}}_3/mmc$ to the orthorhombic phase $\mathrm{Pmcn}$ that occurs in several compounds of the $A_2{\mathrm{BX}}_4$ family and depends on the hcp lattice parameter $c/a.\mathrm{}$ For compounds of ${\mathrm{K}}_2{\mathrm{SO}}_4$ type with $c/a$ larger than the threshold value 1.26 the direct first-order transition $\mathrm{Pmcn}\leftrightarrow {P6\mathrm{}}_3/mmc$ is characterized by a large entropy jump $\sim R\ln 2.$ For compounds ${\mathrm{Rb}}_2{\mathrm{WO}}_4,{\mathrm{K}}_2{\mathrm{MoO}}_4,{\mathrm{K}}_2{\mathrm{WO}}_4$ with $c/a<\mathrm{}1.26\mathrm{}$ this transition occurs via an intermediate incommensurate $\left(\mathrm{Inc}\right)$ phase. Differential scanning calorimetry measurements were performed in ${\mathrm{Rb}}_2{\mathrm{WO}}_4$ to characterize the thermodynamics of the $\mathrm{Pmcn}\leftrightarrow \mathrm{Inc}\leftrightarrow {P6\mathrm{}}_3/mmc$ transitions. It was found that both transitions are again of the first order with entropy jumps $0.2\cdot R\ln 2$ and $0.3\cdot R\ln 2.$ Therefore at $c/a\sim 1.26$ the $A_2{\mathrm{BX}}_4$ compounds reveal an unusual Lifshitz point where three first-order transition lines meet. We propose the coupling of crystal elasticity with ${\mathrm{BX}}_4$ tetrahedra orientation as a possible source of the transitions discontinuity.