Mechanism for the $\alpha \to \epsilon$ phase transition in iron

The mechanism of the $\alpha -\epsilon$ transition in iron is reconsidered. A path in the Burgers description of the bcc/hcp transition, different from those previously considered, is proposed. It relies on the assumption that shear and shuffle are decoupled and require some peculiar magnetic order, different from that of $\alpha$ and $\epsilon$ phases as found in density-functional theory. Finally, we put forward an original mechanism for this transition, based on the successive shuffle motion of layers, which is akin to a nucleation-propagation process rather than to some uniform motion.

Figure 1

Scheme of the ${\left(001\right)}_{\mathrm{hcp}}\parallel {\left(110\right)}_{\mathrm{bbc}}$ plane in the bcc configuration. The vectors ${\vec{R}}_1,{\vec{R}}_2$, and ${\vec{R}}_3$ define our simulation cell. The red atoms and blue atoms lie in two different planes (see Fig. 2). ${\vec{R}}_2$ is perpendicular to ${\vec{R}}_1$ and ${\vec{R}}_3$, and $R_2=4c$. The shear is defined by $\epsilon =(2\sin \alpha -\frac{2}{\sqrt{3}})/(1-\frac{2}{\sqrt{3}})$. In bcc, $2\sin \alpha =\frac{2}{\sqrt{3}}$, and $\epsilon =0$. In hcp, $2\sin \alpha =1$, thus, $\epsilon =1$. From this definition, conservation of volume ($R_2{R}_1{R}_3\sin \alpha =V$), conservation of $\frac{c}{a}(=\frac{R_2}{4R_3}=\sqrt{\frac{8}{3}})$, and the geometric relation $2R_3\cos \alpha =R_1$, give ${\vec{R}}_1,{\vec{R}}_2$, and ${\vec{R}}_3$ as a function of the value of the shear.

Figure 2

Sketch of possible paths for the bcc/hcp transition in the shear shuffle plane. TS is the transition state, and the black line corresponds to the lowest pass to go from the bcc valley to the hcp valley. The red and green lines are the transition paths that are studied in Fig. 4 (see also text). ${\epsilon }_1=0.5$, and ${\epsilon }_3=0.875$.

Figure 3

(a) Total energy for the simultaneous shuffle $S_1$ computed for FM, AFMI, AFMII,[31] and nonmagnetic (NM) configurations. S stands for simultaneous, and the index 1 refers to the shear ${\epsilon }_1$ (see Fig. 2). The energy is given for a supercell of eight atoms to allow for an easy comparison with Fig. 4. (b) Stacking of the layers during the transition. In black, the A and $\overline{\mathrm{A}}$ layers. In orange, the B and $\overline{\mathrm{B}}$ layers. The black (respectively, orange) color, thus, indicates that the first shell of neighbors is bcc- (respectively, hcp-) like. In the inset, the evolution of the magnetic moment for the eight atoms during the FM $\to$ AFMI transition.

Figure 4

(a) Comparison of energies for three different shuffling mechanisms (for a shear of ${\epsilon }_1=0.5$). (${\mathrm{S}}_1$) is the simultaneous shuffling of all atoms. (${\mathrm{CfS}}_1$) stands for “consecutive from simultaneous,” a hypothetical mechanism that corresponds to consecutive shuffles within an assumption of independent layers (interactions neglected), computed with the energies of mechanism ${\mathrm{S}}_1$ (see text). (${\mathrm{C}}_1$) is the true energy computed when layers shuffle in a consecutive way (interactions included). $E_i$ and $E_a$, respectively, represent the interface energy and the activation energy for the first shuffle (see text). (${\mathrm{C}}_3$) is the energy for the consecutive shuffling of the atoms for a shear of ${\epsilon }_3=0.875$. (b) Layer stacking during consecutive shuffling: in blue, layers in which the first atomic shell is intermediate between the bcc and the hcp environments. Arrows indicate the sign of the magnetic moment in each layer as described in Fig. 5.

Figure 5

Evolution of the local magnetic moment for the eight atoms of the supercell as a function of the reaction coordinate $\rho$. The black, orange, and blue colors approximately depict the bcc, hcp, and mixed bcc/hcp values for the magnetic moment. The atom numbering is related to Fig. 4. Atoms 1, 3, 5, 7 belong to the A layers, and atoms 2, 4, 6, 8 are successively moving from an $\overline{\mathrm{A}}$-type layer to a $\overline{\mathrm{B}}$-type one.