Ab initio calculation of the solution enthalpies of substitutional and interstitial impurities in paramagnetic fcc Fe

In the framework of disordered local moment approach by using magnetic sampling method, we suggested a model that takes into account the magnetic disorder in paramagnetic Fe with point defects. We calculate solution enthalpies of substitutional (Nb, V) and interstitial (C, N) impurities in paramagnetic face-centered cubic Fe and obtain results that are in agreement with available experimental data. It is found that both interstitial and substitutional atoms may favor the local magnetic polarization of the Fe host around the impurities by decreasing the potential energy of the system. The possibility of a formation of predominantly ferromagnetic Fe clusters around carbon in the temperature range of overcooled austenite is discussed.

Figure 1

Calculated (calc.) and experimental (exp.) [36, 37] ($T$ ∼ 1100 K) lattice constants as a function of the carbon content in fcc Fe.

Figure 2

The average distortion ${\overline{R}}^i$ (%) of host atoms in the first five coordination shells around C (red squares), N (blue triangles), V (black circles), and Nb (green diamonds) impurities in paramagnetic fcc Fe.

Figure 3

Local magnetic moments ${\mu }_i$ (in ${\mu }_B$) induced at the impurity atoms plotted as a function of the net magnetic moment $\sum {\mu }^{\mathrm{Fe}}\left(\mathrm{in}{\mu }_B\right)$ of Fe atoms located in the first coordination shell of (a) C, (b) N, (c) V, and (d) Nb impurities. Red triangles and black squares denote calculations with and without local lattice distortions around the impurities, respectively.

Figure 4

Mean absolute value of magnetic moments on Fe atoms as a function of coordination shell $i$ around (a) C, (b) N, (c) V, and (d) Nb impurities. Averaging is done over the MSM-SQS realizations used in our simulations. Calculated mean magnetic moment for Fe atoms in pure paramagnetic fcc iron with experimental lattice parameter $a$ = 3.64 Å is shown with horizontal dashed line. Red triangles and black squares denote calculations with and without local lattice distortions around the impurities, respectively.

Figure 5

Calculated site-projected DOS of impurity atoms (C, N, V, and Nb; black dotted, two-dashed lines) and NN Fe atoms with different values of magnetic moments (Fe in 1 c.s.; blue dashed lines and cyan dashed-dotted lines). For comparison, the DOS of iron atoms located in the fifth coordination shell of the impurity (Fe in 5 c.s.; red lines) are also included. Values in parentheses correspond to the magnetic moments of the corresponding atoms.

Figure 6

Calculated absolute values of magnetic moments of all Fe atoms located in the first coordination shell of C, N, V, and Nb impurities. Red squares and green circles denote calculations with and without local lattice distortions around the impurities, respectively.

Figure 7

Calculated energies of MSM supercells with different impurity positions versus the total magnetic moment of Fe atoms located in the first coordination sphere of C, N, V, and Nb impurities. Red squares and blue circles correspond to calculations with and without local lattice distortions around the impurities, respectively.

Figure 8

Solubility products of VC, VN, NbC, and NbN in paramagnetic $\gamma$-Fe. Experimental data are taken from ${}^{\mathrm{a}}$Ref. [58], ${}^{\mathrm{b}}$Ref. [12], ${}^{\mathrm{c}}$Ref. [59], ${}^{\mathrm{d}}$Ref. [60], ${}^{\mathrm{e}}$Ref. [61], ${}^{\mathrm{f}}$Ref. [62], and ${}^{\mathrm{g}}$Ref. [63].

Figure 9

The total magnetic moment ($\Sigma {\mu }^{\mathrm{Fe}},\mathrm{in}{\mu }_B$) of Fe NN sites to the impurity as a function of the total magnetic moment of the same Fe atoms in the host SQS supercell. The color map shows variations of $H_{\mathrm{sol}}$ (in electron volts) for each MSM realization of the impure supercells. It smoothly ranges from black (minimum of $H_{\mathrm{sol}}$, in electron volts), through violet and brown, to yellow (maximum of $H_{\mathrm{sol}}$, in electron volts). The first and second columns correspond to calculations without and with local lattice distortions around the impurities, respectively.

Figure 10

Calculated average pair exchange parameter $\langle J^{\text{Fe-Fe}}\rangle$ in fcc Fe in a static DLM state for first, second, and third coordination shells as a function of the lattice parameter. The EMTO-CPA method has been employed for the calculations. The inset shows the relative atomic displacements $R^{\mathrm{Fe}-\mathrm{Fe}}$ (%) between the iron atoms Fe-Fe within the Fe NN cluster for some Fe-$X$ realizations.