Phonons and elasticity of cementite through the Curie temperature

Phonon partial densities of states (pDOS) of ${}^{57}\mathrm{Fe}_3\mathrm{C}$ were measured from cryogenic temperatures through the Curie transition at 460 K using nuclear resonant inelastic x-ray scattering. The cementite pDOS reveal that low-energy acoustic phonons shift to higher energies (stiffen) with temperature before the magnetic transition. This unexpected stiffening suggests strongly nonharmonic vibrational behavior that impacts the thermodynamics and elastic properties of cementite. Density functional theory calculations reproduced the anomalous stiffening observed experimentally in cementite by accounting for phonon-phonon interactions at finite temperatures. The calculations show that the low-energy acoustic phonon branches with polarizations along the [010] direction are largely responsible for the anomalous thermal stiffening. The effect was further localized to the motions of the ${\mathrm{Fe}}_{\mathrm{II}}$ site within the orthorhombic structure, which participates disproportionately in the anomalous phonon stiffening.

Figure 1

Structure of cementite. Carbon atoms are black, ${\mathrm{Fe}}_{\mathrm{I}}$ atoms ($4c$ sites) are gold, and ${\mathrm{Fe}}_{\mathrm{II}}$ atoms ($8d$ sites) are red. Bonds with lengths below 2.5 Å  are illustrated including first-nearest-neighbor Fe-C bonds comprising the trigonal prisms. The shortest Fe-Fe interatomic distance is shown (red) along with an Fe-C bond that bridges the layers of stacked trigonal prisms along the $b$ axis (red/black). Figure was rendered using vesta package [6].

Figure 2

Normalized ${}^{57}\mathrm{Fe}$ pDOS extracted from NRIXS measurements at various temperatures. Statistical uncertainties are shown as shaded regions.

Figure 3

${}^{57}\mathrm{Fe}$ pDOS from NRIXS at 14 K, compared with results from DFT. The experimental pDOS is shown in black, where the shaded region is representative of the experimental uncertainties. The DFT Fe pDOS is shown in green, along with the C pDOS (gray dashed). The energies of the DFT calculated pDOS have been scaled to match the experimental mean energies.

Figure 4

Mean energy of the ${}^{57}\mathrm{Fe}$ pDOS from NRIXS measurements with temperature (points). The thick dashed line is the mean energy calculated from a Grüneisen parameter quasiharmonic model. The thin dashed line is the Fe pDOS mean energy from quasiharmonic DFT lattice dynamics.

Figure 5

Nonharmonic contributions to the Fe partial vibrational entropy, from high temperature NRIXS (points), CVFT calculations (open diamonds), and several QH models.

Figure 6

${}^{57}\mathrm{Fe}$ pDOS compared at pairs of temperatures. Phonon difference spectra are shown beneath the phonon pDOS. The inset shows the low-energy region. Statistical uncertainties are shown as shaded regions. Panel (a) shows changes over the ferromagnetic temperatures. Panel (b) shows changes above $T_{\mathrm{C}}$. Panel (c) compares the spectra from the highest and lowest temperatures.

Figure 7

Fe partial phonon DOS from CVFT calculations, convolved with an experimental resolution function, are shown in panel (a). The difference spectra are beneath, and the inset shows the changes at low energies. Panel (b) shows the corresponding phonon dispersions along high symmetry directions at 0 K (black) and 400 K (red).

Figure 8

Fe pDOS from CVFT calculations, convolved with an experimental resolution function, and projected onto the two distinct iron sites. Difference spectra are beneath. The inset shows the low-energy region. Panel (a) shows the ${\mathrm{Fe}}_{\mathrm{I}}$ sites (four atoms/unit cell). Panel (b) shows the ${\mathrm{Fe}}_{\mathrm{II}}$ sites (eight atoms/unit cell).

Figure 9

Thermal trends of elastic moduli obtained using the self-consistent method with the QH and CVFT calculations.