Effect of the $\alpha -\gamma$ Phase Transition on the Stability of Dislocation Loops in bcc Iron

Body-centered-cubic iron develops an elastic instability, driven by spin fluctuations, near the $\alpha -\gamma$ phase transition temperature $T_c=912°\mathrm{C}$ that is associated with the dramatic reduction of the shear stiffness constant $c^{\prime }=(c_{11}-c_{12})/2$ near $T_c$. This reduction of $c^{\prime }$ has a profound effect on the temperature dependence of the anisotropic elastic self-energies of dislocations in iron. It also affects the relative stability of the $a⟨100⟩$ and $a/2⟨111⟩$ prismatic edge dislocation loops formed during irradiation. The difference between the anisotropic elastic free energies provides the fundamental explanation for the observed dominant occurrence of the $a⟨100⟩$, as opposed to the $a/2⟨111⟩$, Burgers vector configurations of prismatic dislocation loops in iron and iron-based alloys at high temperatures.

Figure 1

The variation of the ratio of the prelogarithmic energy factors for straight edge dislocations with $\mathbf{b}=a[001]$ and $\mathbf{b}=a/2[111]$, for several bcc metals, as we rotate the dislocation line direction vector $\mathbf{t}$ around their respective Burgers vector directions. The experimental values of stiffness constants used in calculations were taken from 3.

Figure 2

The leading prelogarithmic factors (3) in the elastic free energy for straight $\mathbf{b}$, $\mathbf{t}=a/2[111]$, $[11\overline{2}]/\sqrt{6}$; [001], [100]; [001], $[110]/\sqrt{2}$ and $a/2[111]$, $[1\overline{1}0]/\sqrt{2}$ edge dislocations evaluated as a function of temperature $T$ near the iron $\alpha -\gamma$ transition temperature $T_c$. Inset shows the curves for the two lowest energy dislocations on the logarithmic scale. The intersection point between the 001[100] and the $111[11\overline{2}]$ curves corresponds approximately to $820°\mathrm{C}$. The 001[100] and the $111[1\overline{1}0]$ curves intersect at $685°\mathrm{C}$. Elastic constants are described by smooth fits to experimental points [6, 7], with $c^{\prime }\approx 56.8(1-T/T_c{)}^{1/2}\mathrm{GPa}$. The 001[100] curves follow the exact analytical solution (4).

Figure 3

The total energy of various types of dislocation loops at $T=0\mathrm{K}$ found using the magnetic potential [12]. The lines are given by Eq. (5) with $\widehat{F}(\mathbf{t})$ and $F_{\delta }$ calculated using anisotropic elasticity.

Figure 4

The ratio of the total free energies of $P=10\mathrm{nm}$ prismatic dislocation loops with 111 and 001 Burgers vectors plotted as a function of temperature. The two sets of lighter lines correspond to using an $F_c$ found using the empirical potentials for $\alpha$-Fe [12, 13, 14], whereas the darker curves represent the best fit in terms of $F_c$ to experimental observations.