Character of the structural and magnetic phase transitions in the parent and electron-doped BaFe2As2compounds

We present a combined high-resolution x-ray diffraction and x-ray resonant magnetic scattering study of as-grown BaFe${}_2$As${}_2$. The structural and magnetic transitions must be described as a two-step process. At $T_S$ $=$ 134.5 K we observe the onset of a second-order structural transition from the high-temperature paramagnetic tetragonal structure to a paramagnetic orthorhombic phase, followed by a discontinuous step in the structural order parameter that is coincident with a first-order antiferromagnetic (AFM) transition at $T_N$ $=$ 133.75 K. These data, together with detailed high-resolution x-ray studies of the structural transition in lightly doped Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ and Ba(Fe${}_{1-x}$Rh${}_x$)${}_2$As${}_2$ compounds, show that the structural and AFM transitions do, in fact, occur at slightly different temperatures in the parent BaFe${}_2$As${}_2$ compound, and evolve toward split second-order transitions as the doping concentration is increased. We estimate the composition for the tricritical point for Co doping and employ a mean-field approach to show that our measurements can be explained by the inclusion of an anharmonic term in the elastic free energy and magnetoelastic coupling in the form of an emergent Ising-nematic degree of freedom.