# Impact of Alloying on Stacking Fault Energies in \(\gamma\)-TiAl
## 2 Methods  
### Geometry of Planar Defects in \(\gamma\)-TiAl  
The stacking faults in fcc materials are irregularities of stacking the \((111)\) planes. Let \(A\), \(B\) and \(C\) denote the three configurations of the close-packed \((111)\) planes, being mutually displaced by a vector \(1/6[\bar{2}11]\) in coordinates related to the conventional cubic cell (containing four atoms). The perfect stacking \(\ldots ABCABCABC\ldots\) could change to \(\ldots ABC\underline{A}CBA\ldots\), where the \(\underline{A}\) denotes the position of a twin boundary (a twin mirror plane). Other faults include a missing or an extra plane without a mirror. The former is called intrinsic stacking fault (ISF) and is described by a stacking sequence \(\ldots ABCA.CABC\ldots\), while the latter is called extrinsic stacking fault (ESF) and corresponds to \(\ldots ABCA\underline{C}\underline{B}C\underline{A}C\ldots\).  
The situation is somewhat more complicated for the L1\({}_{0}\) structure as in the case of \(\gamma\)-TiAl. Its lattice is a slightly tetragonally-deformed fcc lattice (\(c/a\approx 1.016\); hence, the \([001]\) direction is not equivalent with \([100]\) or \([010]\) any more). Moreover, the \((001)\) planes are alternatively occupied by Al and Ti atoms. Consequently, out of the three displacement vectors \(\vec{b}_{1}=1/6\langle 211\rangle\), \(\vec{b}_{2}=1/6\langle 121\rangle\) and \(\vec{b}_{3}=1/6\langle 11\bar{2}\rangle\), being equivalent in the fcc structure (and all producing ISF), only \(\vec{b}_{3}\) creates a fault-preserving local chemical neighborhood of the atoms. The resulting fault is called superlattice ISF (SISF). \(\vec{b}_{1}\) and \(\vec{b}_{2}\), in addition to the stacking fault, also alter the chemical occupation of the sites, yielding so-called complex stacking faults (CSFs). Finally, a displacement \(\vec{b}_{4}=\vec{b}_{1}-\vec{b}_{2}=1/2\langle 110\rangle\) results in an undistorted lattice, but altered lattice occupations, so-called anti-phase, and the corresponding fault plane is therefore called the anti-phase boundary (APB).  
SFE expresses the energy difference between the faulted, \(E_{\text{faulted}}\), and perfect, \(E_{\text{perfect}}\), configurations per unit area, \(A\):  
\[\gamma=\frac{E_{\text{faulted}}-E_{\text{perfect}}}{A}. \tag{1}\]  
This energy can be calculated not only for the special translations corresponding to SFs as described above, but for any displacement vector \(\vec{b}\), hence yielding the generalized stacking fault energy (GSFE) surface [19]. In addition, to the actual values of SFEs, the GSFE surface containsalso energy barriers that need to be overcome when an SF is created. In this work, we discuss SFs on the \(\{111\}\) planes; hence, GSFE surfaces are evaluated along \(u\) and \(v\) coordinates decomposing \(\vec{b}=u\frac{1}{2}\langle\tilde{1}10\rvert+v\frac{1}{2}\langle 11\bar{2}\rangle\). It is worth noting that generalization of ESF for the L1\({}_{0}\) structure yields the superlattice extrinsic stacking fault (SESF), which is, however, not compatible with a single shear plane. Since it is not contained by the simple GSFE surface, it will not be discussed any further in this work. For similar reasons, we do not include twin boundaries or APBs on the \(\{010\}\) planes.