3.1. Evaluation of the BKD patterns

The microstructures of the samples included in Fig. 1 always appear to be single-phased b.c.c./austenitic at the annealing temperature of 1473 K with grain sizes up to several hundred µm. Upon quenching, martensite was formed to an extent which is strongly composition dependent. In the case of the presently investigated Fe₂₁Mn₅₈Al₁₇Ni₄ alloy, starting from the grain boundaries, the β-manganese phase has formed, presumably during quenching [see Fig. 3(b)]. For more Fe-rich alloys there is the tendency to form a grain-boundary f.c.c. phase instead (see Vollmer et al., 2015), which was avoided in the current work, however, by sufficiently rapid quenching.

Figure 3 Microstructure of the investigated Fe21Mn58Al17Ni4 alloy. (a) SEM-BSE micrograph of the Fe21Mn58Al17Ni4 alloy quenched from 1473 K, depicting former austenite grains transformed during quenching (see text). The cracks were formed during sample preparation. (b) SEM-BSE micrograph [located in the middle of a grain in (a)] of the region of interest with areas (c) and (d) marked (red boxes). (c) EBSD phase map (left) and EBSD orientation map colored after cubic inverse pole figure coloring in the out-of-plane direction (right). The martensite is indexed using an f.c.c. structure (displayed in blue) and the austenite was indexed with a b.c.c. structure (red). Kikuchi patterns of austenite and martensite are marked with `A' and `M', respectively. (d) Electron channeling contrast imaging micrograph of a martensite lamella with visible linear contrasts.

The inner parts of the grains of the quenched Fe₂₁Mn₅₈Al₁₇Ni₄ alloy show pronounced varying backscattered electron (BSE) contrast [Fig. 3(b)]. These chemically homogeneous regions, as checked by EDS analysis, have obviously been transformed to martensite to a large extent, where the contrast variations are attributable to different orientations of the martensite variants in the size range of 10–100 µm. The BKD patterns taken from such regions can be tentatively indexed as either b.c.c. austenite (very small regions) or f.c.c. martensite [Fig. 3(c)]. Thereby, as expected, regions with homogeneous BSE contrast show a constant crystallographic orientation. However, electron channeling contrast imaging (ECCI) reveals that the regions of apparently homogeneous orientation, according to the tentative EBSD indexing, show fine-scale linear contrasts [Fig. 3(c) versus 3(d)], indicating some regular structural inhomogeneity (Zaefferer & Elhami, 2014) within the martensite variants.

In order to analyze the origin of this structural heterogeneity in the martensite implied by the ECCI images, the BKD patterns were analyzed more closely. An acquired BKD pattern of the austenite phase is shown in Fig. 4(a). This was used to determine the pattern center. Fig. 4(b) shows a pattern taken from martensite [identical to Figs. 2(b)–2(d)]. Taking the tentative indexing suggested by the Hough-space-based method (TSL OIM DC 7), it becomes obvious that not all visible and detected bands are accounted for by the bands predicted for the assessed orientation [see Fig. 2(d)]. Note that these bands are narrow and hence cannot be bands with indices higher than those used for indexing in Fig. 2(d).

Figure 4 Kikuchi patterns taken from the quenched Fe21Mn58Al17Ni4 alloy. (a) Austenite BKD pattern taken from the region marked with `A' in Fig. 3(c). (b) Martensite BKD pattern from the region marked with `M' in Fig. 3(c). In the following figures, the martensite pattern is shown with indexed polytype-invariant bands (c), bands of the first (major) orientation (d) and the second (minor) orientation (e), and distortions in the shape of certain bands (f).

Kikuchi bands additional to those expected from the f.c.c. structure may be a sign of a more complicated, polytypic structure [as e.g. encountered by Omori et al. (2013)] or of twinning. To identify the polytype characteristics of the diffracting volume, bands due to the available f.c.c. indexing [Fig. 2(d)] are divided into bands which are (i) polytype invariant {for f.c.c. indexing h + k + l = 3N (Warren, 1990) with integer N taking [111] as the stacking direction, for hexagonal indexing h − k = 3N taking [001] as the stacking direction} and (ii) polytype specific (all others).

The polytype-invariant bands are shown in Fig. 4(c) beside the polytype-specific bands in Fig. 4(d). The band that stems from the lattice plane (111) is more pronounced than other bands stemming from the remaining equivalent lattice planes {111}. Thus, the stacking direction was assumed to be perpendicular to the lattice plane (111).

For the identified stacking direction [111], in Fig. 5 simulated Kikuchi patterns are shown for a series of different polytypes and of twinned f.c.c. structures. Obviously, the more complex polytypes 21, 53 and 52 produce Kikuchi patterns with polytype-specific bands that are significantly different from the experimentally observed ones [compare Fig. 4(d)]. Among the considered structures the simple Σ3 twin exhibits the best agreement with the experimental pattern. The second orientation introduced by such a Σ3 twin boundary within the (111) plane can be crystallographically described either by a rotation of the crystal lattice of 60° around the direction [111] or by a rotation of 70.5° around the direction . To achieve the simulated Σ3 twin pattern, the intensities of two dynamically simulated Kikuchi patterns of corresponding orientations were arithmetically averaged (see Fig. 5).² That pattern accounts reasonably well for the polytype-specific bands [Fig. 4(d)] due to the Hough-based f.c.c. orientation, but also for the further bands that are otherwise unaccounted for [Fig. 4(e)]. These are the polytype-specific bands of the second (twin) orientation, whereas the polytype-invariant bands due to the second (twin) orientation should exactly overlap according to the simulated pattern. It appears that the bands due to the first orientation assigned by the Hough-based indexing [Fig. 2(d)] are more pronounced than those of the second orientation, implying differing volume fractions of the specific orientations within the excitation volume of the electron beam. Accordingly, these two orientations are referred to as the `majority' and `minority' orientations.

Figure 5 Measured BKD pattern (data), and dynamical simulation of different close-packed polytype structures labeled by their Zhdanov symbol. Moreover, superpositions of the dynamically simulated pattern of the f.c.c. structure (f.c.c. twin) and of a tetragonally distorted variant (f.c.t. twin, hence deviating from the close-packed structure principle; see text) are shown.

Analysis of the experimental BKD pattern in terms of mutual presence of f.c.c. martensite with a majority and a minority orientation has been done separately within the TSL OIM DC 7 software. First, only the automatically detected Hough peaks due to the polytype-invariant bands as well as the (strong) polytype-specific bands due to majority orientation were chosen as input to determine the majority f.c.c. orientation. In order to enforce indexing based on the remaining weaker bands of the minority orientation, band orientations were indicated manually within the TSL OIM DC 7 software based on the experimental pattern. These bands could be indexed with a different, minority f.c.c. orientation. The orientation relationship between majority and minority orientations thus determined agrees well with the (Σ3) twin orientation relationship.

The co-existence of two different orientations in the excitation volume of the electron beam is the reason for the varying quality of f.c.c. indexing, as the acquisition program uses a Hough-based algorithm to index the BKD patterns. For this, a distinct number of bands is detected, starting with the band of the highest intensity (Winkelmann et al., 2007). For patterns with good pattern quality, nearly all bands used originate from the same (major) orientation because enough bands are detected. For poor pattern quality, most of the broad bands are not detected, causing the detected bands to be a mixture of just the bands with low hkl values of both diffracting orientations. This also explains the high success rate of patterns indexed with one single f.c.c. orientation in the case of good pattern quality and a martensite orientation showing a low number of minor bands.

Nevertheless, there exist some minor discrepancies between the experimental pattern and the simulated f.c.c. twin. As highlighted in Fig. 4(f), the polytype-invariant bands show triangular features, implying that the polytype-invariant bands that originate from the major and minor orientations do not coincide entirely. Since this splitting of the bands is not observed for the simulated patterns due to the f.c.c. twin, this model does not completely account for the observed diffraction patterns. A typical observed feature in martensites, which can also account for the splitting of the observed bands, is the tetragonal distortion of the unit cell. Tetragonality is a common type of distortion encountered for many martensite crystal structures resulting from a b.c.c. → f.c.c. or f.c.c. → b.c.c. transformation for various reasons (see e.g. Christian, 1992) and, hence, this is an obvious possibility.

To assess such a tetragonal distortion of the diffracting crystal structure, the experimental BKD pattern in Fig. 4(b) was compared with different dynamical simulations of tetragonally distorted f.c.c. (f.c.t.) structure³ obtained using the Bruker DynamicS program (Winkelmann et al., 2007). To determine the tetragonality, the simulated patterns were compared with the experimental pattern while refining the orientation for the simulated patterns upon maximizing the cross-correlation coefficient (CC) (Britton & Hickey, 2018). To keep the procedure simple, only the major orientation has been used for the simulations. As the absolute lattice parameters only affect the band widths of the simulated BKD pattern, the important parameter to vary is the ratio of the tetragonal lattice parameters c/a (referring to a face-centered tetragonal unit cell). Fig. 6 shows the maximized CC for the different tested c/a ratios. The best agreement was found for c/a = 0.96.

Figure 6 Cross-correlation coefficient of the dynamically simulated BKD pattern in comparison with the measured pattern for different c/a ratios. The simulation was performed only for the major orientation of the pattern. The maximum CC indicates the best agreement.

Comparison of experimental data and the superposition of the dynamically simulated f.c.t. structure shows qualitatively a good fit. Since the majority variant was assumed to be tetragonal, it was taken as likely that the minority variant also shows the same tetragonal distortion. As the DynamicS program was used to analyze the tetragonal distortion, evaluation based on manually indicated bands was continued using this software, equivalently to what has been described above for the f.c.c. structure: the bands due to the majority orientation were selected on the basis of the automatically detected Hough peaks, those due to the minority orientation were selected manually. (Use of the cross-correlation coefficient for the described purpose was not feasible because it was not possible to enforce indexing of the minority component in this way without falling into the deeper minimum due to the bands of the majority orientation.)

The major and minor orientations obtained from indexing imply the following orientation relationship: and or (Fig. 7), as well as and etc., with maj and min referring to majority and minority, respectively. The experimentally observed orientation determined in the described way deviates only by 0.21° from this ideal orientation. The quality of the orientation determination using the DynamicS software is visible in Fig. 8. The red bands in this figure indicate the input bands for the orientation determination and the blue bands indicate the respective bands of the fitted orientation. The blue bands seem to correspond very well with the underlying measured pattern. A detailed analysis of the fit of the simulated structures with the measured BKD pattern is shown in Appendix A.

Figure 7 Schematic representation of the f.c.t. twin with the vertical (111) twinning plane shown along the common direction.

Figure 8 Input bands (red) used for the determination of the major and minor f.c.t. orientation in the DynamicS software. The blue bands are the respective simulated bands for the orientation.

The orientation relationship ensures the absence of misfit and hence full coherency, if the (111 )_fct^maj/min planes are also habit planes. The parallelism of some of the mentioned directions in Fig. 4 is further emphasized in Fig. 9, which reconstructs a larger part of the Kikuchi sphere on the basis of a series of patterns obtained for different rotation angles of the sample around its surface normal in the vicinity of the electron beam. By using this procedure (Fischer et al., 2019) it is possible to show a larger number of zone axes of a BKD sphere with a low symmetry, maintaining nearly the same diffracting volume. With this procedure it can be shown that the twinned f.c.t. structure describes not only the hitherto shown BKD pattern from Fig. 4 very well but also a large part of the BKD sphere [compare measured data in Fig. 9(a) and dynamical simulation in Fig. 9(b)]. In Fig. 9 the corresponding 〈110〉 zone axis is marked in blue.

Figure 9 Comparison of experimental data with simulated patterns. (a) Measured BKD pattern of the same martensite grain, generated by rotation of the sample. The green sphere indicates the theoretical size of the BKD sphere. (b) Superposition of dynamically simulated BKD spheres for a twinned f.c.t. martensite. Zone axes are numbered and colored according to the measured pattern. The indexing is with respect to the major orientation.

3.2. Implications for f.c.c. martensite in Fe–Mn–Al–Ni alloys

The present results imply that, for the investigated alloy composition, thermally generated martensite is a tetragonally distorted and twinned f.c.c. polytype. Tentative analysis of Kikuchi patterns of other compositions from thermally generated martensite implies the presence of these structural features over a large range of composition (see Fig. 1). The results presented here for the structure of the martensite, however, differ from the 53 polytype deduced by Omori et al. (2012) based on selected-area electron diffraction of martensite generated in Fe_43.5Mn₃₄Al₁₅Ni_7.5 and from the apparently single-orientation f.c.c. structure as reported by Vallejos et al. (2018).

Both the short-range periodic faulting present in a 53 polytype and the present type of twinning definitely correspond to a mode of lattice-invariant shear, ensuring that martensite forms by an invariant plane strain (Zhang & Kelly, 2009). This indicates that the mode of lattice-invariant shear may vary in this type of alloy. As it concerns the observed tetragonality, in the field of martensites not containing interstitials, this is frequently a consequence of inherited order from the austenitic state (Christian, 1992). All this insight should be included in predictions of crystallographic and microstructural features (like habit planes) of the martensites in Fe–Mn–Al–Ni alloys.