Influence of spin fluctuations on structural phase transitions of iron

The effect of spin fluctuations on the $\alpha$ (bcc)-$\gamma$ (fcc)-$\delta$ (bcc) structural phase transitions in iron is investigated with a tight-binding (TB) model. The orthogonal $d$-valent TB model is combined with thermodynamic integration, spin-space averaging, and Hamiltonian Monte Carlo to compute the temperature-dependent free-energy difference between bcc and fcc iron. We demonstrate that the TB model captures experimentally observed phonon spectra of bcc iron at elevated temperatures. Our calculations show that spin fluctuations are crucial for both the $\alpha \text{−}\gamma$ and $\gamma \text{−}\delta$ phase transitions but they enter through different mechanisms. Spin fluctuations impact the $\alpha \text{−}\gamma$ phase transition mainly via the magnetic/electronic free-energy difference between bcc and fcc iron. The $\gamma \text{−}\delta$ phase transition, in contrast, is influenced by spin fluctuations only indirectly via the spin-lattice coupling. Combining the two mechanisms, we obtain both the $\alpha \text{−}\gamma$ and $\gamma \text{−}\delta$ phase transitions with our TB model. The calculated transition temperatures are in very good agreement with experimental values.

Figure 1

Unit cells of the bct, bcc, and fcc lattices along the Bain path with the order parameter $k=c/a$.

Figure 2

Phonon spectrum of bcc iron at magnetic temperatures of (a) 773 and (b) 1743 K computed in this work (lines) and observed by experiment [30] (dots).

Figure 3

Calculated vibrational free-energy difference between fcc and bcc phases of iron as a function of temperature obtained with SSA forces. Negative values correspond to stable fcc.

Figure 4

Calculated magnetic/electronic free-energy profiles along the Bain path at different temperatures.

Figure 5

Calculated magnetic/electronic free-energy difference between fcc and bcc phases of iron as a function of temperature.

Figure 6

Histograms of the electronic density of states of bcc and fcc iron at different magnetic temperatures. The electronic density of states of 1000 magnetic configurations were calculated at each temperature.

Figure 7

Comparison of the electronic (orange), vibrational (blue), and total free-energy differences (green) between bcc and fcc iron plotted as a function of temperature. For comparison we also include calphad calculations [32] with the SGTE database [33] (black). The gray box in the total free-energy difference panel is the zoom region for Fig. 8.

Figure 8

Computed total free-energy difference near phase transitions (zoom of gray box in Fig. 7) and comparison to calphad calculations [32] with the SGTE database [33].