On mechanical twinning in tetragonal lattice

Figures & data

Figure 1. Formation of mechanical twin by shear and definition of elements of twinning.

Figure 2. Crystallography of the most common twinning in fcc, schematised in (110) plane.

Figure 3. (a) Result of shearing by $\overrightarrow{s}=1/6\left[\overline{1}12\right]$ in fct lattice, c/a = 0.8. The same positions of atoms can be reached also by shearing in the opposite direction by the vector $\overrightarrow{s^{\mathrm{\prime }}}$. (b) Result of mirroring the crystal along the $\left(1\overline{1}1\right)$ plane in fct lattice, c/a = 0.8.

Figure 4. Parameters f and f ‘ as a function of ratio r = c/a. Magnitude of the $\left[\overline{1}12\right]$ vector is the unity of the y-axis.

Figure 5. The generalised planar fault energies γ (GPFE) of (a) Al, (b) In, (c) TiAl, (d) Ni₂FeGa as a function of $\left|\overrightarrow{s}\right|/\left|\overrightarrow{b}\right|$ ratio (the ratio of shear displacement $\overrightarrow{s}$ and Burgers vector $\overrightarrow{b}=1/6\left[\overline{1}12\right])$. The structures in each subplot display from left to right the perfect lattice, the intrinsic stacking fault, and two- and three-layer twins. The dashed arrows correspond to the minima of non-optimised GPFE whereas solid arrows represent the minima of fully optimised GPFE. If only solid arrow is shown, both minima coincide.

Table 1. Experimental and ab initio lattice parameters a and c/a and magnitude of Burges vector $\left|\overrightarrow{b}\right|$, shearing vector $\left|\overrightarrow{s}\right|$, ratio $\left|\overrightarrow{s}\right|/\left|\overrightarrow{b}\right|$, and factor f calculated from ab initio lattice parameters. The experimental data were taken from the following works: Al [71]; In [72]; TiAl [73]; Ni₂FeGa [43].

	Experiment		Theory
	a [Å]	c/a	a [Å]	c/a	$\left|\overrightarrow{b}\right|$	$\left|\overrightarrow{s}\right|$	$\left|\overrightarrow{s}\right|/\left|\overrightarrow{b}\right|$	f
Al	4.046	1.00	4.021	1.000	1.650	1.650	1.00	0.1667
In	4.599	1.075	4.677	1.073	2.004	2.339	1.17	0.1973
TiAl	3.997	1.021	3.979	1.025	1.651	1.759	1.06	0.1776
Ni2FeGa	3.810	0.858	3.674	0.954	1.454	1.296	0.87	0.1453

Table 2. Energies of intrinsic stacking fault γ_isf, two-layer twin γ_2t and three-layer twin γ_3t in mJ/m² and corresponding ratio of vectors $\left|\overrightarrow{b}\right|$ and $\left|\overrightarrow{s}\right|$ for non-optimised and fully-optimised structures.

	Non-optimised						Optimised
	γisf	γ2t	γ3t	${\displaystyle \frac{\left|1\overrightarrow{s}\right|}{\left|\overrightarrow{b}\right|}}$	${\displaystyle \frac{\left|2\overrightarrow{s}\right|}{\left|\overrightarrow{b}\right|}}$	${\displaystyle \frac{\left|3\overrightarrow{s}\right|}{\left|\overrightarrow{b}\right|}}$	γisf	γ2t	γ3t	${\displaystyle \frac{\left|1\overrightarrow{s}\right|}{\left|\overrightarrow{b}\right|}}$	${\displaystyle \frac{\left|2\overrightarrow{s}\right|}{\left|\overrightarrow{b}\right|}}$	${\displaystyle \frac{\left|3\overrightarrow{s}\right|}{\left|\overrightarrow{b}\right|}}$
Al	135	152	149	1.00	2.00	3.00	119	131	117	0.94	1.94	2.94
In	42	27	32	1.12	2.38	3.42	40	25	30	1.41	2.51	3.40
TiAl	176	162	165	1.07	2.13	3.20	163	140	136	1.00	2.13	3.20
Ni2FeGa	62	63	86	0.87	1.67	2.38	30	32	40	0.95	1.76	2.64

Figure A1. Geometry of mirror-type twin in fct lattice with lattice constants a and c.

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Data availability

Data will be made available on request.