Configurational thermodynamics of the Fe-Cr $\sigma$ phase

Configurational thermodynamics of the Fe-Cr sigma phase is investigated on the basis of an Ising-type configurational Hamiltonian and a single-site mean-field model for the free energy. The parameters of the statistical models are obtained from efficient first-principles calculations using different computational techniques. We demonstrate that the effective pair and multisite interactions in the $\sigma$ phase are relatively small, which allows using a simplified model for distributing Fe and Cr atoms among sublattices. We also show that this system exhibits a nontrivial magnetic behavior at high temperatures, which affects the site occupation by Fe and Cr atoms. The structural variation (volume and $c/a$) that might be present due to neutron irradiation and thermal expansion can lead to an additional atomic redistribution.

Figure 1

The tetragonal unit cell of the Fe-Cr $\sigma$-phase structure with five crystallographically inequivalent sites (A,B,C,D,E).

Figure 2

Radii of coordination shells (in units of the Wigner-Seitz radii of the corresponding structures) of the five sublattices of the $\sigma$ phase (A,B,C,D,E), bcc and fcc structures versus the coordination shell number.

Figure 3

One-electron contribution to the effective pair interactions obtained by the generalized perturbation method (GPM).

Figure 4

Total density of states (DOSs) of the Fe${}_{0.5}$Cr${}_{0.5}$ $\sigma$ phase obtained in the LSGF supercell and CPA calculations.

Figure 5

Local magnetic moments of Fe in the supercells in the LSGF calculation, average magnitudes in two supercells (SC1 and SC2), and the CPA result.

Figure 6

The Cr site occupancies of the (A,B,C,D,E) sublattices versus temperature for $x_{Cr}=0.50$ calculated for the experimental lattice parameters $V_{\exp }$ = $0.35271$ nm${}^3$ and ${(c/a)}_{\exp }$ = 0.518 in different magnetic states: (a) nonmagnetic state (NM), (b) disordered local moment state (DLM), (c) disordered local moment state including longitudinal spin fluctuations (LSFs). The full symbols denote the calculated results, and the empty symbols the available experimental data.[2, 4] The vertical lines indicate the temperature range of the $\sigma$-phase formation in the Fe-Cr alloy.

Figure 7

The Cr site occupancies of the (A,B,C,D,E) sublattices for four different sets of structural parameters given in Table . In parenthesis we specify the element which predominately occupies the given sublattice $\alpha$ = (A,B,C,D,E).

Figure 8

The Cr site occupancies of the (A,B,C,D,E) sublattices versus temperature including all effects mentioned. The empty symbols denote the available experimental data.[2, 4]