Figure 1

Schematic structure of the bcc lattice, containing (a) three interpenetrating bcc sublattices of octahedral interstices (open circles, triangles, squares) within the Fe host matrix (gray balls), and (b) only one octahedral sublattice occupied by impurity atoms [red (small gray) balls]. Octahedral cages are indicated in either case. Scenario (b) is interpreted as orientational ordering and causes the tetragonal distortions of the Fe host lattice. A fully random distribution of impurities on all three sublattices in (a) results in cubic symmetry. The first five coordination shells for interstitial impurities are indicated by arrows.Reuse & Permissions

Figure 2

Convergence of the equilibrium lattice constant (a), (b) and bulk modulus (c) of Fe with respect to the cutoff energy (a) and the $k$-point mesh density (b), (c). Squares and circles in (b) correspond to plane-wave cutoff energies of 440 and 460 eV, respectively. $a_0^{\mathrm{ref}}$ are the equilibrium lattice constants calculated (a) using a cutoff energy of 520 eV and a $27\times 27\times 27$ $k$-point grid, and (b) using a $48\times 48\times 48$ $k$-point mesh for either cutoff energy. The maximal error in the calculated lattice constant using a $27\times 27\times 27$ $k$-point mesh and a smearing parameter of 0.1 eV is less then $1.5\times {10}^{-4}$ Å. (c) The different symbols correspond to different electronic temperatures in eV, as marked on the graph.Reuse & Permissions

Figure 3

Dependencies of the pairwise chemical contribution to C-C interactions as a function of the C-C separation for supercells of different sizes. Doubling of the supercell of the Fe matrix (from 54 to 128 atoms) dramatically decreases the spurious effect of periodic boundary conditions [note in particular the interatomic distance of $4(a_0/2)$, corresponding to coordination shells 18 and 19].Reuse & Permissions

Figure 4

Convergence of the elastic constants of bcc Fe with respect to the number of $k$ points. The different symbols correspond to electronic temperatures in eV, as marked on the graph. Experimental measurements (Ref. 39) at 4.2 K are indicated by the dashed-dotted red horizontal line.Reuse & Permissions

Figure 5

Total energy of bcc Fe (left-hand axis) as a function of the volume per atom (top axis) and the corresponding external pressure (bottom axis). Circles indicate the results of ab initio calculations. The dotted line was obtained by a least-square fit to the Murnaghan equation of state $E(V)$. The red (solid) line represents the dependence of the lattice parameter (right-hand axis) on the external pressure determined by differentiating the Murnaghan equation of state.Reuse & Permissions

Figure 6

Dependence of the elastic constants of bcc Fe (dots) on the external hydrostatic pressure (bottom axis) and corresponding volume per atom (top axis). The solid lines are fourth-order polynomial fits to the calculated data. Pressure derivatives are listed in Table .Reuse & Permissions

Figure 7

Dependence of the lattice parameters of various Fe-$X$ solid solutions on the impurity concentration under zero external pressure. The dashed lines emphasize the increased tetragonality with increasing concentration and serve as guide to the eye only.Reuse & Permissions

Figure 8

Pressure dependence of $L_{[100]}$ (a), (b) and $L_{[001]}$ (c), (d) for impurity concentrations of 1.818 at. % (circles) and 3.57 at. % (squares). The lines are fourth-order polynomial fits to the calculated data.Reuse & Permissions

Figure 9

Dispersion curves for the eigenvalues, ${\lambda }_{\sigma }(\mathbf{k})$, of the strain-induced interaction matrix of the Fe-C system at zero external pressure.Reuse & Permissions

Figure 10

Dependence of the N-N chemical interaction on external pressure (bottom axis) and corresponding volume per atom (top axis). Only the interaction within the first four shells are shown. Symbols indicate ab initio results and lines represent the third-order polynomial fits.Reuse & Permissions

Figure 11

Dependence of pairwise (a) chemical, (b) strain-induced, and (c) total impurity-impurity interactions as a function of the separation distance. The insets magnify the interactions within shells with higher coordination number. The top axis indicates the coordination shells corresponding to the interaction radii.Reuse & Permissions

Figure 12

Long-range order parameter as a function of temperature assuming a carbon content of 1.13 at. $\%$ in the Fe host matrix. The equilibrium LRO parameter at $T=300$ K is defined by $\delta F/\delta \eta =0$ and indicated by the dotted line. The AB region of $\eta$ values corresponds to the absolute stability of the ordered phase; BC delineates the metastable ordered phase, for which the energy minimum of the ordered phase is higher than the energy minimum of the disordered phase; in the CD region ordered phases are absolutely unstable; DE is the region of metastable stability of the disordered phase. A first-order phase transition occurs at point B at a temperature $T_0$ corresponding to the critical LRO parameter ${\eta }_0$.Reuse & Permissions

Figure 13

Predicted equilibrium phase diagram of Fe-based solid solutions. The order-disorder transformation temperatures ($T_0$) as calculated by Eq. (7) are plotted by solid lines. The dot corresponds to the only available experimental data (Ref. 9) for the dilute Fe-C system. Above $c_{\max }$ the disordered Fe-C phase is thermodynamically unstable. Below temperature $T_{\mathrm{d}}$ the disorder-order transition is kinetically hindered, due to the slow diffusion of the interstitial C atoms. Thus no martensitic transformation is expected below the corresponding concentration $c_{\min }$. Qualitatively similar arguments hold for the Fe-N, Fe-O, and Fe-C-N systems. The dashed-dotted line represents the Fe-C solid solution as described by ab initio determined chemical interactions and a MET-based description of the strain-induced interaction with experimentally obtained input parameters. No ordered, tetragonally distorted phase was found in the low concentration regime of the Fe-B phase diagram.Reuse & Permissions

Figure 14

Pressure dependence of the critical concentration of the order-disorder transition at room temperature.Reuse & Permissions