Crystal structure determination of Hägg carbide, χ-Fe₅C₂ by first-principles calculations and Rietveld refinement

Abstract

X-ray powder-diffraction data recorded using different wave lengths as well as neutron powder diffraction data on Hägg carbide, χ-Fe₅C₂, were evaluated by Rietveld or Pawley refinements, respectively. Likewise, employing different starting models, first-principles calculations using density functional theory (DFT) involving structure optimisation with respect to energy were performed for χ-Fe₅C₂. The results from diffraction and DFT imply a crystal structure having a monoclinic C2/c symmetry with a quite regular (monocapped) trigonal-prismatic coordination of C by Fe atoms. The anisotropy of the microstrain broadening observed in the powder-diffraction patterns agrees with the anisotropy of the reciprocal Young’s module obtained from elastic constants calculated by DFT. The anisotropic microstrain broadening can to some degree, be modelled allowing for a triclinic distortion of the metric of χ-Fe₅C₂ (deviation of the lattice angle γ from 90°) involving reflection spitting, which mimics the hkl-dependently broadened reflections. This distortion corresponds to the most compliant shear direction of the monoclinic χ-Fe₅C₂. The anisotropic microstrain broadening results from microstress induced e.g. by anisotropic thermal expansion inducing misfit between the grains, in association with the intrinsic anisotropic elastic compliance of χ-Fe₅C₂. This anisotropic microstrain broadening was likely the origin of previous proposals of triclinic P-1 space-group symmetry for the crystal structure of χ-Fe₅C₂, which is rejected in the present work.