Orientation Relationship of Intergrowth Al 2 Fe and Al 5 Fe 2 Intermetallics Determined by Single-Crystal X-ray Diffraction

: Although the Al 2 Fe phase has similar decagonal-like atomic arrangements as that of the orthorhombic Al 5 Fe 2 phase, no evidence for intergrowth samples of Al 2 Fe and Al 5 Fe 2 has been reported. In the present work, the co-existence of Al 2 Fe and Al 5 Fe 2 phases has been discovered from the educts obtained with a nominal atomic ratio of Al:Fe of 2:1 by arc melting. First, single-crystal X-ray diffraction (SXRD) as well as scanning electron microscope (SEM) equipped with energy-dispersive X-ray spectroscopy (EDX) measurements have been utilized to determine the exact crystal structures of both phases, which are refined to be Al 12.48 Fe 6.52 and Al 5.72 Fe 2 , respectively. Second, the orientation relationship between Al 2 Fe and Al 5 Fe 2 has been directly deduced from the SXRD data sets, and the co-existence structure model has been constructed. Finally, four pairs of parallel atomic planes and their unique orientation relations have been determined from the reconstructed reciprocal-space precession images of (0 kl ), ( h 0 l ), and ( hk 0) layers. In addition, one kind of interface atomic structure model is constructed by the orientation relations between two phases, correspondingly.


Introduction
Similar to the Al-Mn system, diverse complex intermetallics in the binary Al-Fe system have been discovered, including quasiperiodic and periodic quasi-crystalline approximate phases [1,2].Among them, the Al 2 Fe and Al 5 Fe 2 phases have been extensively studied [3][4][5][6][7].Studies on the crystal structure of the Al 2 Fe phase can be traced back to half a century ago.In 1973, Corby et al. [8] studied the crystal structure of Al 2 Fe for the first time using the anomalous dispersion method.The refined composition of the phase is Al 11 Fe 7 with space group P1; There are three Al/Fe co-occupying atoms in the unit cell.The lattice parameters are determined to be a = 4.878 (1) Å, b = 6.461 (2) Å, c = 8.800 (3) Å, α = 91.75(5) • , β = 73.27(5) • , γ = 96.89(3) • .In 1978, Bastin et al. [9] investigated the crystal structure of Al 2 Fe using the Weissenberg technique.The authors consider the unit cell of Al 2 Fe can be described as an A-centered pseudo-monoclinic cell with a = 7.594 Å, b = 16.886Å, c = 4.863 Å, α = 89.55• , β = 122.62• , γ = 90.43• .Due to the inconsistency of the above two research results, Chumak et al. [10] have re-determined the crystal structure of Al 2 Fe again by single-crystal X-ray diffraction method in 2010.The refined composition was Al 12.59 Fe 6.41 with space group P1.There are two Al/Fe co-occupying atoms in the unit cell, and the lattice parameters are a = 4.8745 (6) Å, b = 6.4545 (8) Å, c = 8.7361 (10) Å, α = 87.930(9) • , β = 74.396(9) • , γ = 83.062(9) • .For the crystal structure of Al 5 Fe 2 .Schubert et al. [11] first determined the crystal structure of the Al 5 Fe 2 phase by X-ray powder diffraction method in 1953.They consider the phase to have Cmcm space group and consists of three independent atoms, including a vacancy atom Al1 with an occupancy of 0.7.The cell parameters are a = 7.675 Å, b = 6.403Å, c = 4.203 Å, α = 90 • .In 1994, Burkhardt et al. [12] re-determined the fine crystal structure of this phase by the single-crystal X-ray diffraction method.They proposed that this phase has two vacant atoms, namely Al1 and Al2 atoms, with the site of occupation factors (S.O.F) of 0.32 and 0.24, respectively.Furthermore, several studies have shown that the Al 5 Fe 2 phase has an ordered cryogenic phase [13][14][15][16].
As aforementioned, the crystal structure models of Al 2 Fe and Al 5 Fe 2 phases have been studied extensively; however, there are no reports on the co-existence of the two phases.Mihalkovič et al. [17] studied the structure and stability of Al 2 Fe and Al 5 Fe 2 phases by firstprinciples calculations.Hirata et al. [18] studied the crystal structure of Al 2 Fe and Al 5 Fe 2 phases and found that the crystal structure of the two phases has a similar decagonal-like atomic arrangement.Romero-Romero et al. [19] studied the structural stability of Al 2 Fe and Al 5 Fe 2 phases by the high-energy ball-milling method.They found that the Al 2 Fe phase underwent a phase transition and transformed into the Al 5 Fe 2 phase after 10 h of high-energy ball milling.
In the present work, samples with nominal Al 2 Fe composition were prepared using the arc melting method, and the co-existence of the Al 2 Fe phase and Al 5 Fe 2 phase in the educts was discovered by analyzing the SXRD data sets.Second, we solved and refined the crystal structures of the two phases separately and obtained the orientation relationship of the two phases in real space by comparing their orientation matrix in the reciprocal space.Finally, concerning the vitally important role played by interfaces for industrial applications [20,21], the parallel atomic planes between the coexisting phases were analyzed, and one preferential interface was constructed by referring to the reconstructed reciprocal-space precession images.

Materials and Methods
High-purity aluminum (2.457 g) and iron (2.543 g) powders, according to the atomic ratio of 2:1, were pressed into blocks and then melted in a vacuum furnace for 4 cycles to ensure a uniform composition.The sintered block was broken into small pieces, and a cuboid-shaped fragment with a size of 0.08 × 0.06 × 0.03 mm 3 was selected and mounted on a thin glass fiber for SXRD measurements.Diffraction measurements were carried out with a four-circle single-crystal X-ray diffractometer (Bruker D8 Venture, Bruker AXS GmbH, Karlsruhe, Germany).Two SEM and EDX tests have been conducted on the sample.In the first test, the electron microscope Hitachi S-3400N type equipped with EDX (EDAX Inc., Mahwah, NJ, USA) was used, and in the second test, the electron microscope ZEISS Sigma 300 type equipped with EDX (Oxford, UK) was used.
All data sets from SXRD are processed by the APEX3 program [22], including indexing, integration, scaling, absorption correction [23], space group determination, structural solving, and refinement [24,25].The structural models are drawn with the Diamond program [26].The building clusters of the studied phases are analyzed by the ToposPro package [27].

Single-Crystal XRD Patterns
The diffraction points in the reciprocal space of the whole sample collected by 10 runs of the single-crystal XRD measurements are illustrated in Figure 1.As can be seen from Figure 1, these diffraction points can be clearly divided into two different data sets, implying two independent different phases.In the following, the two data sets will be analyzed separately in the reciprocal space.It needs to be noted that the single-crystal X-ray diffraction measurements on the sample include a total of 10 runs, and a total of 4793 diffraction points were harvested with the criteria of I/σ(I) equals 3 in the reciprocal space when indexing the phases.Among these 4793 diffraction points, there are 2984 diffraction points belong to the Al 2 Fe phase, 1032 diffraction points belong to the Al 5 Fe 2 phase, and the remaining 777 diffraction points belong to anomalous phases, either an amorphous phase or some very tiny crystalline phases, which cannot be indexed to de-termine a unit cell.We have also conducted a Phi360 test (with the sample rotating around the psi axis) on the sample by the single-crystal X-ray diffraction diffractometer, which is equivalent to XRD powder diffraction (see Figure S1 of the Supplementary Material).
Metals 2024, 14, 337 3 of 17 points belong to the Al2Fe phase, 1032 diffraction points belong to the Al5Fe2 phase, and the remaining 777 diffraction points belong to anomalous phases, either an amorphous phase or some very tiny crystalline phases, which cannot be indexed to determine a unit cell.We have also conducted a Phi360 test (with the sample rotating around the psi axis) on the sample by the single-crystal X-ray diffraction diffractometer, which is equivalent to XRD powder diffraction (see Figure S1 of the Supplementary Material).Figures 2 and 3 illustrate the diffraction patterns of the Al2Fe and Al5Fe2 phases projected in three axes, along with their crystal structures, respectively.It can be seen that the diffraction points in the reciprocal space are arranged quite neatly for both phases.The first data set (indicated as white color in Figure 1) is indexed to be a = 4.86 Å, b = 6.44 Å, c = 8.74 Å, α = 87.88°,β = 74.47°,γ = 83.06°, in accordance with those of the Al2Fe phase as shown in Figure 2. The second data set (indicated as green color in Figure 1) is indexed to be a = 7.63 Å, b = 6.41 Å, c = 4.20 Å, α = β = γ = 90°, in accordance with those of the Al5Fe2 phase as shown in Figure 3.The cell parameters of the Al2Fe and Al5Fe2 phases are detailed in Table 1.    1.

The Refinement of Al2Fe Phase and Al5Fe2 Phase
Detailed crystal data, data collection, and structure refinement details are summarized in Table 1.For the Al2Fe phase, as the residual electron density around the Fe4 atom is too large, thus it is designated as a disordered atom, and the PART instruction (means to divide disordered atoms into two or more groups, each representing a disordered component) is used to separate it to Fe4A atom and Al4B atom.The S.O.F for Fe4A and Al4B atoms is 0.758 and 0.242, respectively.All the remaining atoms are completely occupied, and the chemical formula was refined to Al12.48Fe6.52.The final crystallographic parameter R1 is 0.0543, ωR2 is 0.1044 (Fobs > 4σ (Fobs)), and goodness of fit S is 1.057.During the structural refinement of Al5Fe2 phase, it is found that it is more reasonable for Al1 and Al2 atoms to be refined in the form of vacancy atoms with the S.O.F of the Al1 atom to be 0.50 while that of the Al2 atom to be 0.18, resulting the final refined chemical formula to be Al5.72Fe2.The final crystallographic parameter R1 is 0.0363, ωR2 is 0.0844 (Fobs > 4σ (Fobs)), and goodness of fit S is 1.017.All the parameters meet the requirements of international crystallography for the rationalization of the crystal structure.

The Refinement of Al 2 Fe Phase and Al 5 Fe 2 Phase
Detailed crystal data, data collection, and structure refinement details are summarized in Table 1.For the Al 2 Fe phase, as the residual electron density around the Fe4 atom is too large, thus it is designated as a disordered atom, and the PART instruction (means to divide disordered atoms into two or more groups, each representing a disordered component) is used to separate it to Fe4A atom and Al4B atom.The S.O.F for Fe4A and Al4B atoms is 0.758 and 0.242, respectively.All the remaining atoms are completely occupied, and the chemical formula was refined to Al 12.48 Fe 6.52 .The final crystallographic parameter R 1 is 0.0543, ωR 2 is 0.1044 (F obs > 4σ (F obs )), and goodness of fit S is 1.057.During the structural refinement of Al 5 Fe 2 phase, it is found that it is more reasonable for Al1 and Al2 atoms to be refined in the form of vacancy atoms with the S.O.F of the Al1 atom to be 0.50 while that of the Al2 atom to be 0.18, resulting the final refined chemical formula to be Al 5.72 Fe 2 .The final crystallographic parameter R 1 is 0.0363, ωR 2 is 0.0844 (F obs > 4σ (F obs )), and goodness of fit S is 1.017.All the parameters meet the requirements of international crystallography for the rationalization of the crystal structure.
Table 2 shows detailed information about the atomic occupancy of the Al 12.48 Fe 6.52 phase, where U eq is the equivalent isotropic temperature factor and Occ. is the site of occupation factors of atoms.It can be seen that there are two disordered (co-occupying) atoms in this structure, named the Fe4A atom and the Al4B atom.These two atoms occupy the same position with S.O.F of 0.758 and 0.242 for Fe4A and Al4B, respectively.When comparing the present refined crystal structure model with the previously reported Al 2 Fe phase as refined to be Al 12.59 Fe 6.41 determined in 2010 [10], one can find that they agree with each other quite well except for the slightly different S.O.F for the co-occupied position.In the previous model, the S.O.F is 0.705 and 0.295 for Fe4A and Al4B, respectively, while in the present one, the ratio of S.O.F between Fe4A and Al4B is much closer to 3:1.In the following, the building units of the Al 2 Fe phase have been analyzed by applying the nanocluster method as integrated into the ToposPro software (Version 5.5.2.0, programed by V. A. Blatov and A. P. Shevchenko).One finds that the crystal structure model can be described by two cluster types: Fe3 (1) (1@12) and Al6 (1) (1@13).Among them, the Fe3 (1) (1@12) cluster is an icosahedral cluster with the Fe3 atom as the center, and the Al6 (1) (1@13) is a 21-dihedral cluster with an Al6 atom located at the center.Figure 4a shows the centers of the aforementioned different clusters in the unit cell.As shown in Figure 4b, the Al6 (1) (1@13) clusters are connected with Fe3 (1) (1@12) clusters either by common vertex, edge, or plane, while two Al6 (1) (1@13) clusters are connected with a common plane.Two Fe3 (1) (1@12) clusters along the a-axis are also connected by a common edge.Figure 5 shows the environments of Fe3 and Al6 atoms.It can be seen that the Fe3 atom is surrounded by 12 atoms while the Al6 atom is surrounded by 13 atoms.
Table 3 shows detailed information on the Al 5.72 Fe 2 phase.There are two vacancy atoms, namely the Al1 atom and the Al2 atom, and the occupying factor of the Al1 atom is 0.50 (10), and that of the Al2 atom is 0.18 (5).By comparing the present work with the Al 5.4 Fe 2 obtained by Schubert et al. [11], the present model contains an additional Al atom at the 8f position.Compared with the Al 5.6 Fe 2 determined by Burkhardt et al. in 1994 [12], Al1 and Al2 atoms in the present Al 5.72 Fe 2 phase have slightly different S.O.F.The occupancy of Al atoms at 4b and 8f site in the previous Al 5. 6 Fe 2 phase is 0.32 and 0.24, while the occupancy of Al atoms at 4b and 8f in the Al 5.72 Fe 2 phase is 0.50 and 0.18, respectively, in the present work.It is also necessary to note that Okamoto et al. [14] have constructed Al 8 Fe 3 and Al 7 Fe 3 phases with the Al 5.6 Fe 2 phase as the basic unit.As the difference in the occupancy of vacant Al atoms in the present Al 5.72 Fe 2 phase and the previous Al 5. 6 Fe 2 phase, it is believed that this may lead to the emergence of some modified Al 8 Fe 3 and Al 7 Fe 3 structure models.(See Supplementary Materials for details).Table 3 shows detailed information on the Al5.72Fe2 phase.There are two vacancy atoms, namely the Al1 atom and the Al2 atom, and the occupying factor of the Al1 atom is 0.50 (10), and that of the Al2 atom is 0.18 (5).By comparing the present work with the Al5.4Fe2 obtained by Schubert et al. [11], the present model contains an additional Al atom at the 8f position.Compared with the Al5.6Fe2 determined by Burkhardt et al. in 1994 [12], Al1 and Al2 atoms in the present Al5.72Fe2 phase have slightly different S.O.F.The occupancy of Al atoms at 4b and 8f site in the previous Al5.6Fe2phase is 0.32 and 0.24, while the occupancy of Al atoms at 4b and 8f in the Al5.72Fe2 phase is 0.50 and 0.18, respectively, in the present work.It is also necessary to note that Okamoto et al. [14] have constructed Al8Fe3 and Al7Fe3 phases with the Al5.6Fe2 phase as the basic unit.As the difference in the occupancy of vacant Al atoms in the present Al5.72Fe2 phase and the previous Al5.6Fe2phase, it is believed that this may lead to the emergence of some modified Al8Fe3 and Al7Fe3 structure models.(See Supplementary Materials for details.)    Table 3 shows detailed information on the Al5.72Fe2 phase.There are two vacancy atoms, namely the Al1 atom and the Al2 atom, and the occupying factor of the Al1 atom is 0.50 (10), and that of the Al2 atom is 0.18 (5).By comparing the present work with the Al5.4Fe2 obtained by Schubert et al. [11], the present model contains an additional Al atom at the 8f position.Compared with the Al5.6Fe2 determined by Burkhardt et al. in 1994 [12], Al1 and Al2 atoms in the present Al5.72Fe2 phase have slightly different S.O.F.The occupancy of Al atoms at 4b and 8f site in the previous Al5.6Fe2phase is 0.32 and 0.24, while the occupancy of Al atoms at 4b and 8f in the Al5.72Fe2 phase is 0.50 and 0.18, respectively, in the present work.It is also necessary to note that Okamoto et al. [14] have constructed Al8Fe3 and Al7Fe3 phases with the Al5.6Fe2 phase as the basic unit.As the difference in the occupancy of vacant Al atoms in the present Al5.72Fe2 phase and the previous Al5.6Fe2phase, it is believed that this may lead to the emergence of some modified Al8Fe3 and Al7Fe3 structure models.(See Supplementary Materials for details.)Then, we focus on the crystallographic feature of the Al 5.72 Fe 2 phase.Figure 6a shows the 2 × 2 × 2 supercell of Al 5.72 Fe 2 projected along the [001] direction.The Al1 and Al2 atoms (designated in green and pink color, respectively) are surrounded by a hole composed of eight Al3 atoms and two Fe1 atoms (designated in blue and orange color, respectively).The distance between Al1 and Al2 is only 0.7982 Å, confirming that both have to be partially occupied atoms.Figure 6b shows a projection of the Al 5.72 Fe 2 supercell along the direction [100], where one can see that the Al1 and Al2 atoms are alternatively distributed along the c-axis.
(designated in green and pink color, respectively) are surrounded by a hole composed of eight Al3 atoms and two Fe1 atoms (designated in blue and orange color, respectively).The distance between Al1 and Al2 is only 0.7982 Å, confirming that both have to be partially occupied atoms.Figure 6b shows a projection of the Al5.72Fe2 supercell along the direction [100], where one can see that the Al1 and Al2 atoms are alternatively distributed along the c-axis.It was found that the unit cell of Al5.72Fe2 is composed of four twisted icosahedrons, as shown in Figure 7.The icosahedron takes an Al1 atom as its center, and each icosahedron is connected by a common edge.As described in Section 3.1, the structural model of Al12.48Fe6.52 can also be described by icosahedron.Such common structural features could be the reason for their growing together.It was found that the unit cell of Al 5.72 Fe 2 is composed of four twisted icosahedrons, as shown in Figure 7.The icosahedron takes an Al1 atom as its center, and each icosahedron is connected by a common edge.As described in Section 3.1, the structural model of Al 12.48 Fe 6.52 can also be described by icosahedron.Such common structural features could be the reason for their growing together.
(designated in green and pink color, respectively) are surrounded by a hole composed of eight Al3 atoms and two Fe1 atoms (designated in blue and orange color, respectively).The distance between Al1 and Al2 is only 0.7982 Å, confirming that both have to be partially occupied atoms.Figure 6b shows a projection of the Al5.72Fe2 supercell along the direction [100], where one can see that the Al1 and Al2 atoms are alternatively distributed along the c-axis.It was found that the unit cell of Al5.72Fe2 is composed of four twisted icosahedrons, as shown in Figure 7.The icosahedron takes an Al1 atom as its center, and each icosahedron is connected by a common edge.As described in Section 3.1, the structural model of Al12.48Fe6.52 can also be described by icosahedron.Such common structural features could be the reason for their growing together.In the above section, we have explained the crystal structure of the Al 12.48 Fe 6.52 phase and Al 5.72 Fe 2 phase, respectively.In this section, the orientation model of real space will be constructed through the orientation matrix of these two phases in reciprocal space.Please refer to Appendix A for the specific construction method of structure models for intergrowth Al 12.48 Fe 6.52 phase and Al 5.72 Fe 2 phase in real space.First, the orientation matrix of two phases in reciprocal space is obtained by APEX3 software (v2018.1-0),and then the orientation relationship of two phases in real space is obtained by the basic correspondence between reciprocal space and real space [28], and the orientation model of two-phase single cell edges in real space is obtained.As shown in Figure 8a, the black and red frames show the cell edges of the Al 12.48 Fe 6.52 and Al 5.72 Fe 2 phases, respectively.Finally, the final orientation model is obtained by adding atoms to the two-phase cell edges.growth Al12.48Fe6.52phase and Al5.72Fe2 phase in real space.First, the orientation matrix of two phases in reciprocal space is obtained by APEX3 software (v2018.1−0),and then the orientation relationship of two phases in real space is obtained by the basic correspondence between reciprocal space and real space [28], and the orientation model of two-phase single cell edges in real space is obtained.As shown in Figure 8a, the black and red frames show the cell edges of the Al12.48Fe6.52 and Al5.72Fe2 phases, respectively.Finally, the final orientation model is obtained by adding atoms to the two-phase cell edges.In Figure 8b, the left shows the unit cell of Al12.48Fe6.52,and the right shows the unit cell of Al5.72Fe2.It is interesting to find that the angle between the crystal plane of Al12.48Fe6.52(001) and the crystal plane of Al5.72Fe2 (100) is 63.35°.

Interfaces between Al12.48Fe6.52 Phase and Al5.72Fe2 Phases
In the previous section, we obtained the oriented structural models of Al12.48Fe6.52 and Al5.72Fe2.However, the orientation of the interfaces between the two phases and the arrangement of atoms inside the interfaces are still elusive.In this section, we will focus on solving such issues by investigating the synthesized precession images from the SXRD data sets, as shown in Figure 9. Figure 9a-c represent the precession images of the (0kl), (h0l), and (hk0) planes from the Al12.48Fe6.52phase, while Figure 9d-f represent the precession images of the (0kl), (h0l) and (hk0) planes from the Al5.72Fe2 phase.In Figure 9a-c, the green and blue circles represent the crystal planes of the Al12.48Fe6.52phase and the Al5.72Fe2 phase, respectively.The precession images are constructed with a thickness of 0.05 Å −1 and a resolution of 0.80 Å.While in Figure 9d-f, the green and blue circles represent the crystal planes of the Al5.72Fe2 phase and the Al12.48Fe6.52phase, respectively, the precession images are constructed with a thickness of 0.03 Å −1 and a resolution of 0.80 Å.It needs to be emphasized that the "reconstructed precession images" are obtained by the APEX3 program, where the Synthesize Precession Images plug-in provides an undistorted view of layers of the reciprocal lattice.It generates simulated precession images by finding the appropriate pixels in a series of frames.
The orientation relationship of the Al12.48Fe6.52 and Al5.72Fe2 phases expressed by a pair of crystal planes can be observed directly from Figure 9.Those diffraction points from the two phases overlap, which means the crystal planes they represent are parallel with each other.To summarize, four orientation relationships named OR1, OR2, OR3, and OR4 can be obtained by analyzing the (0kl), (h0l), and (hk0) planes of the Al12.48Fe6.52phase and the (hk0) planes of the Al5.72Fe2 phase are shown in Table 4.In the previous section, we obtained the oriented structural models of Al 12.48 Fe 6.52 and Al 5.72 Fe 2 .However, the orientation of the interfaces between the two phases and the arrangement of atoms inside the interfaces are still elusive.In this section, we will focus on solving such issues by investigating the synthesized precession images from the SXRD data sets, as shown in Figure 9. Figure 9a-c represent the precession images of the (0kl), (h0l), and (hk0) planes from the Al 12.48 Fe 6.52 phase, while Figure 9d-f represent the precession images of the (0kl), (h0l) and (hk0) planes from the Al 5.72 Fe 2 phase.In Figure 9a-c, the green and blue circles represent the crystal planes of the Al 12.48 Fe 6.52 phase and the Al 5.72 Fe 2 phase, respectively.The precession images are constructed with a thickness of 0.05 Å −1 and a resolution of 0.80 Å.While in Figure 9d-f, the green and blue circles represent the crystal planes of the Al 5.72 Fe 2 phase and the Al 12.48 Fe 6.52 phase, respectively, the precession images are constructed with a thickness of 0.03 Å −1 and a resolution of 0.80 Å.It needs to be emphasized that the "reconstructed precession images" are obtained by the APEX3 program, where the Synthesize Precession Images plug-in provides an undistorted view of layers of the reciprocal lattice.It generates simulated precession images by finding the appropriate pixels in a series of frames.
The orientation relationship of the Al 12.48 Fe 6.52 and Al 5.72 Fe 2 phases expressed by a pair of crystal planes can be observed directly from Figure 9.Those diffraction points from the two phases overlap, which means the crystal planes they represent are parallel with each other.To summarize, four orientation relationships named OR1, OR2, OR3, and OR4 can be obtained by analyzing the (0kl), (h0l), and (hk0) planes of the Al 12.48 Fe 6.52 phase and the (hk0) planes of the Al 5.72 Fe 2 phase are shown in Table 4.As mentioned above, we have identified four crystallographic orientation relationships between the Al12.48Fe6.52 and Al5.72Fe2 phases from Figure 9.According to the symmetry principle of crystallography, there are usually multiple variants corresponding to a set of experimentally determined orientation relationships.It is necessary to judge if the four crystallographic orientation relationships observed in this experiment are equivalent.In the following, the matrix method is used to analyze and discuss the experimental results.
A detailed explanation of the matrix method can be found in Appendix B. Through this method, we obtain the conversion matrix between the four orientation relationships, as shown in the following Table 5, where matrix B represents the conversion matrix between crystal directions and matrix A represents the conversion matrix between crystal planes.The absolute values of the elements in the conversion matrices corresponding to the four orientation relationships are different, so it is confirmed that they are four independent orientation relationships.As mentioned above, we have identified four crystallographic orientation relationships between the Al 12.48 Fe 6.52 and Al 5.72 Fe 2 phases from Figure 9.According to the symmetry principle of crystallography, there are usually multiple variants corresponding to a set of experimentally determined orientation relationships.It is necessary to judge if the four crystallographic orientation relationships observed in this experiment are equivalent.In the following, the matrix method is used to analyze and discuss the experimental results.
A detailed explanation of the matrix method can be found in Appendix B. Through this method, we obtain the conversion matrix between the four orientation relationships, as shown in the following Table 5, where matrix B represents the conversion matrix between crystal directions and matrix A represents the conversion matrix between crystal planes.The absolute values of the elements in the conversion matrices corresponding to the four orientation relationships are different, so it is confirmed that they are four independent orientation relationships.
Furthermore, a preliminary interface model of these interface relationships was built.

Conclusions
In summary, two typical phases, Al12.48Fe6.52 and Al5.72Fe2 in the Al-Fe, are discovered to be co-existence in the form of single crystals with size of tens of micromeres, which has been confirmed by SXRD by combing SEM/EDX analysis.The first phase, known as Al2Fe, is refined to be Al12.48Fe6.52(space group P1) with cell parameters: a = 4.8569 ( 5 Through the orientation matrix of Al 12.48 Fe 6.52 and Al 5.72 Fe 2 phases in real space, the comprehensive models of Al 12.48 Fe 6.52 and Al 5.72 Fe 2 described with cell edges in real space can be constructed, as shown in Figure 8a of the main text.Now, we can add the specific atoms for both phases to the orientation models described with cell edges by acknowledging the experimental orientation matrix and the Crystallographic Information File (CIF) related orientation matrix.First, the positions of atoms of the Al 5.72 Fe 2 phase in real space are introduced.We named the experimental orientation matrix of the phase in the real space as matrix B. The CIF-related orientation matrix corresponding to Al Then, the cartesian coordinates of the atoms in the CIF of Al 5.72 Fe 2 are multiplied by the matrix C, resulting in the coordinate positions of the atoms of the Al 5.72 Fe 2 phase in real space.Second, the coordinate positions of the atoms of the Al 12.48 Fe 6.52 phase in real space are also obtained in the same way.Finally, the comprehensive oriented structural models of Al 12.48 Fe 6.52 and Al 5.72 Fe 2 phases in real space are obtained, as shown in Figure 8b of the main text.

Appendix B
The matrix method is to find out the conversion relationship between the crystal plane index and the crystal direction index between two phases mathematically, i.e., to find the transformation matrix B and A of each kind of orientation relation.If the absolute values of the nine elements in the transformation matrix are identical, it can be judged that the orientation relation of the two phases belongs to the same type even though the positive/negative signs and the arrangement order and position are different.The crystal plane (hkl) can be represented by a reciprocal vector in the reciprocal space.G * hkl (=ha * 1 + ka * 2 + la * 3 ).The normal direction of the crystal plane (hkl) is the positive vector and can be expressed I uvw (=ua 1 + va 2 + wa 3 ).The relationship between G * hkl (=ha * 1 + ka * 2 + la * 3 ) and I uvw (=ua 1 + va 2 + wa 3 ) is as follows: In the general crystallographic study, the orientation relationship between two phases is always expressed in the form of [u ′ 2 v ′ 2 w ′ 2 ]//[u 2 v 2 w 2 ], (h ′ 1 k ′ 1 l ′ 1 )//(h 1 k 1 l 1 ).The second group of crystal plane parallelism (h ′ 2 k ′ 2 l ′ 2 )//(h 2 k 2 l 2 ) can be obtained according to the crystal direction parallelism and Formula (A9).According to (h ′ 1 k ′ 1 l ′ 1 )//(h 1 k 1 l 1 ) and Formula (A10), one can obtain the second crystal orientation parallel relationship [u , the (h 3 k 3 l 3 ) can be obtained by [u 1 v 1 w 1 ] × [u 2 v 2 w 2 ].Therefore, three groups of crystal plane parallelism between the two phases can be obtained.
The above orientation relationship can be expressed in the following matrix form: where B is the conversion matrix, A and B are transposed inverse matrices of each other, and the expression of the conversion matrix B is as follows: where d 1 , d 2 , d 3 are the interplanar spacing of (h 1 k 1 l 1 ), (h 2 k 2 l 2 ) and (h 3 k 3 l 3 ); ) and (h ′ 3 k ′ 3 l ′ 3 ).For the Al 12.48 Fe 6.52 /Al 5.72 Fe 2 phase interfaces reported in the present work, when substituting the lattice constants of these two phases into the formula, the crystal-to-plane conversion matrix of Al 12.48 Fe 6.52 and Al 5.72 Fe 2 can be obtained as follows: GAl 12.48 Fe

Figure 1 .
Figure 1.Diffraction points in the reciprocal space of the whole sample projected along different directions.The data set in white and green color indicates the Al2Fe phase and the Al5Fe2 phase, respectively: (a) Projection along the c*-axis of the Al2Fe phase; (b) Projection along the b*-axis of the Al5Fe2 phase.

Figure 1 .
Figure 1.Diffraction points in the reciprocal space of the whole sample projected along different directions.The data set in white and green color indicates the Al 2 Fe phase and the Al 5 Fe 2 phase, respectively: (a) Projection along the c*-axis of the Al 2 Fe phase; (b) Projection along the b*-axis of the Al 5 Fe 2 phase.

Figures 2 and 3
Figures 2 and 3 illustrate the diffraction patterns of the Al 2 Fe and Al 5 Fe 2 phases projected in three axes, along with their crystal structures, respectively.It can be seen that the diffraction points in the reciprocal space are arranged quite neatly for both phases.The first data set (indicated as white color in Figure 1) is indexed to be a = 4.86 Å, b = 6.44 Å, c = 8.74 Å, α = 87.88• , β = 74.47• , γ = 83.06• , in accordance with those of the Al 2 Fe phase as shown in Figure 2. The second data set (indicated as green color in Figure 1) is indexed to be a = 7.63 Å, b = 6.41 Å, c = 4.20 Å, α = β = γ = 90 • , in accordance with those of the Al 5 Fe 2 phase as shown in Figure 3.The cell parameters of the Al 2 Fe and Al 5 Fe 2 phases are detailed in Table1.

Figure 2 .
Figure 2. The diffraction patterns of the Al2Fe and its crystal structure: (a) Projection along the a*-axis; (b) Projection along the b*-axis; (c) Projection along the c*-axis; (d) Crystal structure of Al2Fe phase.Figure 2. The diffraction patterns of the Al 2 Fe and its crystal structure: (a) Projection along the a*-axis; (b) Projection along the b*-axis; (c) Projection along the c*-axis; (d) Crystal structure of Al 2 Fe phase.

Figure 2 .
Figure 2. The diffraction patterns of the Al2Fe and its crystal structure: (a) Projection along the a*-axis; (b) Projection along the b*-axis; (c) Projection along the c*-axis; (d) Crystal structure of Al2Fe phase.Figure 2. The diffraction patterns of the Al 2 Fe and its crystal structure: (a) Projection along the a*-axis; (b) Projection along the b*-axis; (c) Projection along the c*-axis; (d) Crystal structure of Al 2 Fe phase.

Figure 3 .
Figure 3.The diffraction patterns of the Al5Fe2 and its crystal structure: (a) Projection along the a*-axis; (b) Projection along the b*-axis; (c) Projection along the c*-axis; (d) Crystal structure of Al5Fe2 phase.

Figure 3 .
Figure 3.The diffraction patterns of the Al 5 Fe 2 and its crystal structure: (a) Projection along the a*-axis; (b) Projection along the b*-axis; (c) Projection along the c*-axis; (d) Crystal structure of Al 5 Fe 2 phase.

Figure 4 .
Figure 4. Cluster assembly in Al2Fe phase cell: (a) The central atom of the cluster; (b) Cluster assembly model.

Figure 4 .
Figure 4. Cluster assembly in Al 2 Fe phase cell: (a) The central atom of the cluster; (b) Cluster assembly model.

Figure 4 .
Figure 4. Cluster assembly in Al2Fe phase cell: (a) The central atom of the cluster; (b) Cluster assembly model.

Figure 7 .
Figure 7. Cluster assembly in Al5.72Fe2 phase cell: (a) The central atom of the cluster; (b) Cluster assembly model.

Figure 7 .
Figure 7. Cluster assembly in Al5.72Fe2 phase cell: (a) The central atom of the cluster; (b) Cluster assembly model.

Figure 7 .
Figure 7. Cluster assembly in Al 5.72 Fe 2 phase cell: (a) The central atom of the cluster; (b) Cluster assembly model.

3. 3 .
Structure Models for Intergrowth Al 12.48 Fe 6.52 Phase and Al 5.72 Fe 2 Phase in Real Space

In
Figure 8b, the left shows the unit cell of Al 12.48 Fe 6.52 , and the right shows the unit cell of Al 5.72 Fe 2 .It is interesting to find that the angle between the crystal plane of Al 12.48 Fe 6.52 (001) and the crystal plane of Al 5.72 Fe 2 (100) is 63.35 • .

Figure 8 .
Figure 8. Oriented structural models described with cell edges (a) and unit cell (b) for Al 12.48 Fe 6.52 and Al 5.72 Fe 2 phases.

Table 1 .
Crystallographic and experimental data of Al 12.48 Fe 6.52 and Al 5.72 Fe 2 .

Table 4 .
Four crystallographic orientation relationships at the interface of Al 12.48 Fe 6.52 and Al 5.72 Fe 2 .