Orientation Relationship of Intergrowth Al₂Fe and Al₅Fe₂ Intermetallics Determined by Single-Crystal X-ray Diffraction

Abstract

Although the Al₂Fe phase has similar decagonal-like atomic arrangements as that of the orthorhombic Al₅Fe₂ phase, no evidence for intergrowth samples of Al₂Fe and Al₅Fe₂ has been reported. In the present work, the co-existence of Al₂Fe and Al₅Fe₂ 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_12.48Fe_6.52 and Al_5.72Fe₂, respectively. Second, the orientation relationship between Al₂Fe and Al₅Fe₂ 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 (0kl), (h0l), and (hk0) layers. In addition, one kind of interface atomic structure model is constructed by the orientation relations between two phases, correspondingly.

1. 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 Al₂Fe and Al₅Fe₂ phases have been extensively studied [3,4,5,6,7]. Studies on the crystal structure of the Al₂Fe phase can be traced back to half a century ago. In 1973, Corby et al. [8] studied the crystal structure of Al₂Fe for the first time using the anomalous dispersion method. The refined composition of the phase is Al₁₁Fe₇ 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 Al₂Fe using the Weissenberg technique. The authors consider the unit cell of Al₂Fe can be described as an A-centered pseudo-monoclinic cell with a = 7.594 Å, b = 16.886 Å, 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 Al₂Fe again by single-crystal X-ray diffraction method in 2010. The refined composition was Al_12.59Fe_6.41 with space group P$\overline{1}$. 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 Al₅Fe₂. Schubert et al. [11] first determined the crystal structure of the Al₅Fe₂ 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 = 6.403 Å, 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 Al₅Fe₂ phase has an ordered cryogenic phase [13,14,15,16].

As aforementioned, the crystal structure models of Al₂Fe and Al₅Fe₂ 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 Al₂Fe and Al₅Fe₂ phases by first-principles calculations. Hirata et al. [18] studied the crystal structure of Al₂Fe and Al₅Fe₂ 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 Al₂Fe and Al₅Fe₂ phases by the high-energy ball-milling method. They found that the Al₂Fe phase underwent a phase transition and transformed into the Al₅Fe₂ phase after 10 h of high-energy ball milling.

In the present work, samples with nominal Al₂Fe composition were prepared using the arc melting method, and the co-existence of the Al₂Fe phase and Al₅Fe₂ 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.

2. 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 mm³ 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].

3. Results

3.1. 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 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).

Figure 2 and Figure 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. Crystallographic and experimental data of Al_12.48Fe_6.52 and Al_5.72Fe₂.

Chemical Formula	Al12.48Fe6.52	Al5.72Fe2
a, b, c/Å	4.8569 (5), 6.4389 (7), 8.7323(10)	7.635 (3), 6.392 (2), 4.2007 (14)
α, β, γ/°	87.873 (4), 74.463 (4), 83.060 (4)	90, 90, 90
V/Å3	261.18 (5)	205.01 (12)
Z	4	4
Space group	P$\overline{1}$	Cmcm
Crystal system	Triclinic	Orthorhombic
Diffractometer	Bruker D8 Venture Photon 100 COMS
Monochromator	Graphite
Tmeas/K	300 (2)
Radiation	Mo-Kα, λ = 0.71073 (Å)
Scan mode	φ and ω scan
Time per step/s	3
Absorption correction	Multi-scan
F (000)	332	253
θ range/°	2.421~27.571	4.158~24.997
μ/mm−1	9.803	8.088
No. measured reflections	7202	283
No. unique reflections	1210	100
No. observed reflections (I > 2σ (I))	907	76
No. reflections used in refinement	1210	100
No. parameters used in refinement	89	19
Reflection range	−6 ≤ h ≤ 6, −8 ≤ k ≤ 8, −11 ≤ l ≤ 11	−6 ≤ h ≤ 9, −6 ≤ k ≤ 7, −4 ≤ l ≤ 4
Rint	0.0957	0.0567
R (σ)	0.0692	0.0915
Final R indices (Fobs > 4σ (Fobs))	R1 = 0.0543, ωR2 = 0.1044	R1 = 0.0363, ωR2 = 0.0844
R indices (all data)	R1 = 0.0797, ωR2 = 0.1044	R1 = 0.0597, ωR2 = 0.0844
Goodness of fit	1.057	1.017

.

3.2. 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 Al_12.48Fe_6.52. The final crystallographic parameter R₁ is 0.0543, ωR₂ is 0.1044 (F_obs > 4σ (F_obs)), 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 Al_5.72Fe₂. The final crystallographic parameter R₁ is 0.0363, ωR₂ 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. Fractional atomic coordinates and equivalent isotropic displacement parameters (Ų) of Al_12.48Fe_6.52.

Label	Site	x	y	z	Occ.	Ueq
Fe1	2i	0.1425 (3)	0.15847(19)	0.41690 (14)	1	0.0066 (3)
Fe2	2i	0.2296 (3)	0.3508 (2)	0.87452 (15)	1	0.0102 (3)
Fe3	1a	0.000000	0.000000	0.000000	1	0.0083 (4)
Fe4A	2i	0.1640 (3)	0.4744 (2)	0.59039 (18)	0.758 (12)	0.0115 (5)
Al4B	2i	0.1640 (3)	0.4744 (2)	0.59039 (18)	0.242 (12)	0.0115 (5)
Al1	2i	0.4915 (6)	0.0106 (4)	0.1650 (3)	1	0.0129 (6)
Al2	2i	0.0430 (6)	0.1158 (5)	0.7096 (3)	1	0.0135 (6)
Al3	2i	0.5994 (6)	0.1853 (4)	0.5253 (3)	1	0.0140 (6)
Al4	2i	0.0150 (6)	0.2924 (4)	0.1691 (3)	1	0.0123 (6)
Al5	2i	0.3101 (5)	0.6635 (4)	0.0349 (3)	1	0.0079 (6)
Al6	2i	0.4198 (6)	0.4451 (4)	0.2983 (3)	1	0.0102 (6)

shows detailed information about the atomic occupancy of the Al_12.48Fe_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 Al₂Fe phase as refined to be Al_12.59Fe_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 Al₂Fe 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. Fractional atomic coordinates and equivalent isotropic displacement parameters (Ų) of Al_5.72Fe₂ phase.

Label	Site	x	y	z	Occ.	Ueq
Fe1	4c	0.000000	0.8279 (4)	0.250000	1	0.0081 (8)
Al1	4b	0.000000	0.500000	0.000000	0.50 (10)	0.03 (2)
Al2	8f	0.000000	0.537 (6)	0.81 (4)	0.18 (5)	0.03 (2)
Al3	8g	0.1888 (4)	0.1454 (6)	0.250000	1	0.0158 (12)

shows detailed information on the Al_5.72Fe₂ 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 Al_5.4Fe₂ obtained by Schubert et al. [11], the present model contains an additional Al atom at the 8f position. Compared with the Al_5.6Fe₂ determined by Burkhardt et al. in 1994 [12], Al1 and Al2 atoms in the present Al_5.72Fe₂ phase have slightly different S.O.F. The occupancy of Al atoms at 4b and 8f site in the previous Al_5.6Fe₂ phase is 0.32 and 0.24, while the occupancy of Al atoms at 4b and 8f in the Al_5.72Fe₂ 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 Al_5.6Fe₂ phase as the basic unit. As the difference in the occupancy of vacant Al atoms in the present Al_5.72Fe₂ phase and the previous Al_5.6Fe₂ phase, 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_5.72Fe₂ phase. Figure 6a shows the 2 × 2 × 2 supercell of Al_5.72Fe₂ 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_5.72Fe₂ 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 Al_5.72Fe₂ 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 Al_12.48Fe_6.52 can also be described by icosahedron. Such common structural features could be the reason for their growing together.

3.3. Structure Models for Intergrowth Al_12.48Fe_6.52 Phase and Al_5.72Fe₂ Phase in Real Space

In the above section, we have explained the crystal structure of the Al_12.48Fe_6.52 phase and Al_5.72Fe₂ 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_12.48Fe_6.52 phase and Al_5.72Fe₂ 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 Al_12.48Fe_6.52 and Al_5.72Fe₂ 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 Al_12.48Fe_6.52, and the right shows the unit cell of Al_5.72Fe₂. It is interesting to find that the angle between the crystal plane of Al_12.48Fe_6.52 (001) and the crystal plane of Al_5.72Fe₂ (100) is 63.35°.

3.4. Interfaces between Al_12.48Fe_6.52 Phase and Al_5.72Fe₂ Phases

In the previous section, we obtained the oriented structural models of Al_12.48Fe_6.52 and Al_5.72Fe₂. 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 Al_12.48Fe_6.52 phase, while Figure 9d–f represent the precession images of the (0kl), (h0l) and (hk0) planes from the Al_5.72Fe₂ phase. In Figure 9a–c, the green and blue circles represent the crystal planes of the Al_12.48Fe_6.52 phase and the Al_5.72Fe₂ 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 Al_5.72Fe₂ phase and the Al_12.48Fe_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_12.48Fe_6.52 and Al_5.72Fe₂ 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 Al_12.48Fe_6.52 phase and the (hk0) planes of the Al_5.72Fe₂ phase are shown in

Table 4. Four crystallographic orientation relationships at the interface of Al_12.48Fe_6.52 and Al_5.72Fe₂.

	[uvw] Al12.48Fe6.52//[uvw] Al5.72Fe2	(hkl) Al12.48Fe6.52//(hkl) Al5.72Fe2
OR1	[010] Al12.48Fe6.52$//[3\stackrel{\mathrm{-}}{7}$26] Al5.72Fe2	$(308){\mathrm{Al}}_{12.48}{\mathrm{Fe}}_{6.52}//(\stackrel{\mathrm{-}}{7}\stackrel{\mathrm{-}}{3}$0) Al5.72Fe2
OR2	[010] Al12.48Fe6.52$//[3\stackrel{\mathrm{-}}{7}$26] Al5.72Fe2	$(308) {\mathrm{Al}}_{12.48}{\mathrm{Fe}}_{6.52}//(\stackrel{\mathrm{-}}{7}\stackrel{\mathrm{-}}{3}$0) Al5.72Fe2
OR3	[001] Al12.48Fe6.52//[794] Al5.72Fe2	$(350) {\mathrm{Al}}_{12.48}{\mathrm{Fe}}_{6.52}//(\stackrel{\mathrm{-}}{4}$$4\stackrel{\mathrm{-}}{2}$) Al5.72Fe2
OR4	$[5\stackrel{\mathrm{-}}{9}\stackrel{\mathrm{-}}{2}$] Al12.48Fe6.52//[001] Al5.72Fe2	$(313) {\mathrm{Al}}_{12.48}{\mathrm{Fe}}_{6.52}//(\stackrel{\mathrm{-}}{5}$10) Al5.72Fe2

.

As mentioned above, we have identified four crystallographic orientation relationships between the Al_12.48Fe_6.52 and Al_5.72Fe₂ 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. Orientation relationships and corresponding conversion matrices between Al_12.48Fe_6.52 and Al_5.72Fe₂ interfaces.

	Orientation Relationship	Conversion Matrix B	Conversion Matrix A
OR1	$(0\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{2}$) Al12.48Fe6.52//(111) Al5.72Fe2 [100]${\mathrm{Al}}_{12.48}{\mathrm{Fe}}_{6.52}//[\stackrel{\mathrm{-}}{2}$11] Al5.72Fe2	$\left(\begin{array}{ccc}1.60& -0.34& 0.44\\ -0.44& -0.93& 0.06\\ -0.24& 0.01& -0.48\end{array}\right)$	$\left(\begin{array}{ccc}0.65& -0.32& -0.32\\ -0.23& -0.97& 0.09\\ 0.57& -0.41& -2.37\end{array}\right)$
OR2	$(308) {\mathrm{Al}}_{12.48}{\mathrm{Fe}}_{6.52}//(\stackrel{\mathrm{-}}{7}\stackrel{\mathrm{-}}{3}$0) Al5.72Fe2 [010] ${\mathrm{Al}}_2\mathrm{Fe}//[3\stackrel{\mathrm{-}}{7}$26] Al5.72Fe2	$\left(\begin{array}{ccc}-1.23& 0.76& 0.35\\ 0.38& -0.40& 0.57\\ -0.39& -0.65& -0.13\end{array}\right)$	$\left(\begin{array}{ccc}-0.54& 0.22& 0.51\\ 0.16& -0.35& 1.39\\ -0.73& -1.06& -0.26\end{array}\right)$
OR3	$(350) {\mathrm{Al}}_{12.48}{\mathrm{Fe}}_{6.52}//(\stackrel{\mathrm{-}}{4}$$4\stackrel{\mathrm{-}}{2}$) Al5.72Fe2 [001] Al12.48Fe6.52//[794] Al5.72Fe2	$\left(\begin{array}{ccc}0.10& 0.33& -0.90\\ -0.87& 0.61& 0.14\\ 0.61& 0.43& 0.25\end{array}\right)$	$\left(\begin{array}{ccc}0.12& 0.38& -0.96\\ -0.60& 0.74& 0.20\\ 0.76& 0.98& 0.44\end{array}\right)$
OR4	$(313) {\mathrm{Al}}_{12.48}{\mathrm{Fe}}_{6.52}//(\stackrel{\mathrm{-}}{5}$10) Al5.72Fe2 $[5\stackrel{\mathrm{-}}{9}\stackrel{\mathrm{-}}{2}$] Al12.48Fe6.52//[001] Al5.72Fe2	$\left(\begin{array}{ccc}-1.14& 0.83& 0.38\\ -0.04& 0.35& -0.61\\ -0.45& -0.63& -0.15\end{array}\right)$	$\left(\begin{array}{ccc}-0.55& 0.35& 0.23\\ -0.15& 0.43& -1.38\\ -0.81& -0.90& -0.46\end{array}\right)$

, 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. The OR1 orientation relationship: [100] _{Al12.48Fe6.52}//[$\stackrel{\mathrm{-}}{2}$11] _Al5.72Fe2, (0$\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{2}$) _{Al12.48Fe6.52}//(111) _Al5.72Fe2 is used as an example. For the (0$\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{2}$) _{Al12.48Fe6.52} surface model, as shown in Figure 10a, the u and v directions are parallel to [22$\stackrel{\mathrm{-}}{1}$] and [100], respectively. At this time, the lattice parameters of the surface of Al_12.48Fe_6.52 (0$\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{2}$) are u = 17.685 Å, v = 4.857 Å, θ = 59.669°. For the surface model of (111) _Al5.72Fe2, as shown in Figure 10b, the u and v directions are parallel to [1$\stackrel{\mathrm{-}}{3}$2] and [2$\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{1}$], respectively. The lattice parameters of the surface of (111) _Al5.72Fe2 are as follows: u = 22.284 Å, v = 17.079Å, θ = 57.611°. As the mismatch between Al_12.48Fe_6.52 (0$\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{2}$) surface and Al_5.72Fe₂ (111) surface is very large in the u and v directions. It is necessary to build a model of the supercell interface to satisfy the periodic boundary conditions. Since it is difficult to eliminate the mismatch in the u and v directions when constructing very large supercells, relatively small mismatches can be constructed by 5 (u) × 7 (v) (0$\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{2}$) Al_12.48Fe_6.52 and 4 (u) × 2 (v) (111) Al_5.72Fe₂ surface models. The mismatches in the u and v directions of the two surfaces are δ (u) = 0.804% and δ (v) = 0.468%, respectively. The lattice parameters of Al_12.48Fe_6.52 and Al_5.72Fe₂ on both sides of the interface are averaged by applying a certain strain to eliminate the two-phase lattice parameters of the interface. The atomic interface model of Al_12.48Fe_6.52(0$\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{2}$)/Al_5.72Fe₂(111) is shown in Figure 10c, red dot lines represent the dividing line between the Al_12.48Fe_6.52 (0$\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{2}$) surface and Al_5.72Fe₂ (111) surface. It can be seen that the atomic arrangement law on the surface of Al_12.48Fe_6.52(0$\stackrel{\mathrm{-}}{1}\stackrel{\mathrm{-}}{2}$) is that two layers of Fe atoms and one layer of Al atoms are alternately arranged, and the atomic arrangement law on the surface of Al_5.72Fe₂(111) is that one layer of Al atoms and one layer of Fe atoms are alternately arranged.

4. Conclusions

In summary, two typical phases, Al_12.48Fe_6.52 and Al_5.72Fe₂ 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 Al_12.48Fe_6.52 (space group P$\overline{1}$) with cell parameters: a = 4.8569 (5) Å, b = 6.4389 (7) Å, c = 8.7323 (10) Å, α = 87.873 (4)°, β = 74.463 (4)°, γ = 83.060 (4)°. There are two co-occupying atoms, namely Fe4A and Al4B, with S.O.F refined to be 0.758 and 0.242, respectively, in accordance with the previously refined model Al_12.59Fe_6.41 determined in 2010. The second phase known as Al₅Fe₂ is refined to be Al_5.72Fe₂ (space group Cmcm) with cell parameters: a = 7.635 (3) Å, b = 6.392 (2) Å, c = 4.2007 (10) Å, α = β = γ = 90°. There are two vacancy atoms, namely Al1 and Al2, with S.O.F refined to be 0.5 and 0.18, respectively. Meanwhile, topological analysis reveals that both phases are composed of 4–8 twisted icosahedrons as structural building units.

Furthermore, crystal structure models for intergrowth Al_12.48Fe_6.52 and Al_5.72Fe₂ phases have been obtained from the SXRD data sets in reciprocal space as well as the refined model in real space. It needs to be emphasized that the angle between the crystal plane of Al_12.48Fe_6.52 (001) and the crystal plane of Al_5.72Fe₂ (100) is 63.35°.

Finally, the orientation relationships of interfaces between the two phases are obtained by investigating the synthesized precession planes from the SXRD data sets. Three orientation relationships named OR1, OR2, OR3, and one named OR4 have been obtained by analyzing the (0kl), (h0l), and (hk0) planes of the Al_12.48Fe_6.52 phase and the (hk0) planes of the Al_5.72Fe₂ phase, respectively. The arrangement of atoms inside the interfaces has been illustrated in a preliminary interface model by taking OR1 as an example.

The present works report a protocol for analyzing the detailed crystal structures of double intergrowth phases and investigating their orientation relationships of interfaces between intergrowth phases, which provide an alternative approach for such investigations besides advanced transmission electron microscopy (TEM) [6,7,20,21] and electron back-scattered diffraction (EBSD) techniques [29,30] and will definitely stimulate further related jobs on intergrowth samples which frequently showing up in the complex metallic alloys.