# Impact of Alloying on Stacking Fault Energies in \(\gamma\)-TiAl
## 3 Results and Discussion  
### SFE in \(\gamma\)-TiAl  
The calculated SFEs are summarized in Table 1. In addition to the method based on tilting the supercell vectors, also a complementary method based on displacing a rigid block of the supercell [13; 14] was employed. The differences are negligible. Similarly, our calculations predict only a small decrease of the SFE values (below 6%) when the local density approximation (LDA) is used instead of GGA. Test calculations with respect to the supercell size also suggest that the used models are large enough to yield converged results. The values calculated here lie in the range of previously published predictions based on DFT. Consequently, the GGA-PW91 xc potential together with the tilted supercell geometry were used for all subsequently discussed results. Figure 2 shows the calculated GSFE surface with its profiles along dissociation paths for \(\frac{1}{2}\langle 01\bar{1}\rangle\) and \(\frac{1}{2}\langle 11\bar{2}\rangle\) superdislocations [1]:  
\[\langle 01\bar{1}\rangle \rightarrow \frac{1}{6}\langle 11\bar{2}\rangle+\text{SISF}+\frac{1}{6}\langle 1 \bar{2}\bar{1}\rangle+\text{APB}+\frac{1}{2}\langle 01\bar{1}\rangle\;, \tag{2}\] \[\frac{1}{2}\langle 11\bar{2}\rangle \rightarrow \frac{1}{6}\langle 11\bar{2}\rangle+\text{SISF}+\frac{1}{6} \langle 2\bar{1}\bar{1}\rangle+\text{APB}+\frac{1}{6}\langle 11\bar{2}\rangle+ \text{CSF}+\frac{1}{6}\langle\bar{1}2\bar{1}\rangle\;. \tag{3}\]  
The APB turns out to be an unstable fault at 0 K as is does not correspond to a local minimum on the GSFE surface. This is in agreement with previous reports [10; 14]. On the contrary, both SISF and CSF are stable; in order to produce them, however, barriers of \(\approx\)335 mJ/m\({}^{2}\) and \(\approx\)550 mJ/m\({}^{2}\), respectively, must be overcome when starting from a perfect stacking. We note, that these barriers \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **APB** & **CSF** & **SISF** & **Note** \\ \hline \multirow{4}{*}{present work} & 717 & 415 & 188 & GGA-PW91, VASP, tilted supercells \\  & 635 & 370 & 173 & GGA-PW91, VASP, tilted supercells, fully relaxed \\  & 711 & 414 & 188 & GGA-PW91, VASP, displaced supercells \\  & 694 & 392 & 179 & LDA, VASP, tilted supercells \\ \hline \multirow{2}{*}{[10]} & 710 & 314 & 134 & LDA, FP-LMTO, tilted supercells(?) \\  & 756 & 420 & 184 & LDA, FP-LAPW, tilted supercells \\
[12] & 499 & 329 & 137 & GGA-PW91, CASTEP, tilted supercells \\
[13] & 355 & 184 & GGA-PW91, VASP, displaced supercells \\
[14] & 663 & 400 & 170 & GGA-PW91, VASP, displaced supercells \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculated SF energies (in m)/m\({}^{2}\)) compared with available literature data for \(\gamma\)-TiAl. Differences in calculation methods are noted. correspond to the GSFE energy profiles along rigid pathways (as visualized in Figure 2), without attempting to estimate exact lowest-energy pathways. Even more importantly, it becomes apparent that the displacement vectors, \(\vec{b}\), of the planar defects (i.e., positions of the local minima on the GSFE surface) do not correspond with the geometrically-determined ones based on the hard-sphere model. In particular, the CSF is shifted by approximately \(-0.018\langle 112\rangle\) (\(0.11(\vec{b}_{\rm{APB}}-\vec{b}_{\rm{CSF}})\)), which is clearly visible also in the profile in Figure 2a. This actually lowers the \(\gamma_{\rm{CSF}}\) from \(415\,\rm{mJ/m^{2}}\) down to \(383\,\rm{mJ/m^{2}}\). Such an effect with similar magnitudes of displacements was previously predicted [10; 11]. During the construction of the GSFE surface, the atomic coordinates in the \(\{111\}\) plane were fixed, while a relaxation along the \(\langle 111\rangle\) direction was allowed. Nonetheless, the SF energy can be decreased even more by fully relaxing the atomic positions once the geometry of the faulted material is trapped near the local energy minimum.