Crystal structure of the Al₈Cr₅-type inter­metallic Al_7.85Cr_5.16

An aluminium-deficient γ-brass phase, Al_7.85Cr_5.16, was synthesized by high-pressure sinter­ing and its crystal structure determined.

Abstract

An aluminium-deficient Al₈Cr₅-type inter­metallic with formula Al_7.85Cr_5.16 (octa­aluminium penta­chromium) was uncovered when high-pressure sinter­ing of a mixture with composition Al₁₁Cr₄ was carried out. Structure analysis reveals that there are three co-occupied positions with refined occupancy factors for Al atoms being 0.958, 0.772 and 1/2. The present phase is confirmed to be isotypic with the previously reported rhombohedral Al₈Cr₅ ordered phase [Bradley & Lu (1937). Z. Kristallogr. 96, 20–37] and structurally closely related to the disordered phases of rhombohedral Al₁₆Cr_9.5 and cubic Al₈Cr₅.

Structure description

The γ₂-Al₈Cr₅ phase (hereafter named as the γ₂ phase) was determined to have a γ-brass-like structure by powder diffraction photographs. This phase was found in slowly cooled chromium–aluminium alloys (Bradley & Lu, 1937 ▸). Although the same clusters of 26 atoms are found in the γ₂ phase, the atomic arrangement in the γ₂ phase is much more complex than that of the γ-brass, and results in a rhombohedral rather than cubic symmetry (Bradley & Lu, 1937 ▸). A high-temperature γ₁ phase was also reported to be stable between 1350 and 980°C at the same composition (Bradley & Lu, 1937 ▸) and its structure has been redetermined by single-crystal methods for a sample sintered at 1000°C for 6 h and re-annealed at 1215°C for 287 h (Brandon et al., 1977 ▸). As a result of the close agreement of Brandon’s analysis with that of Bradley & Lu, it was suggested that either the structure of γ₁ and γ₂ are very similar, or that in the former case the crystals decomposed to γ₂ on quenching. In another work, the high-temperature γ₁ phase prepared by splat cooling was reported to be of the same type as Cu₅Zn₈, by using power diffraction data combined with electron diffraction patterns (Braun et al., 1992 ▸). When comparing the three aforementioned models (see Table S1 of the supporting information), it was found that there are one vacancy position and three co-occupied positions in the Brandon model, while all atomic sites are fully occupied in Bradley & Lu’s model. For the convenience of comparison, the cubic Braun model was transformed to the rhombohedral description, and it was found that there are two co-occupied positions. In the study reported herein, the crystal structure of a third type of Al₈Cr₅ phase, with the refined chemical composition Al_7.85Cr_5.16 and hereafter named as γ₂′-Al₈Cr₅ phase, was determined by single-crystal X-ray diffraction measurements.

Fig. 1 ▸ shows the crystal structure of γ₂′-Al₈Cr₅ based on the standardized crystal data in the primitive trigonal setting (see Tables S2 and S3 of the supporting information). There are 78 atoms in the unit cell (a = b = 12.8717 Å, c = 7.8408 Å, α = β = 90°, γ = 120°), whose volume is three times that of the refined model (trigonal cell, rhombohedral axes, see Table 1 ▸). For simplicity, only two distorted icosa­hedra centred at Wyckoff sites 3a (Cr4, with coordinates 0, 0, z) are illustrated in Fig. 1 ▸, and the environment of the Cr4 atoms is shown in Fig. 2 ▸. The twelve vertices include three Al atoms (Al5), three Cr atoms (Cr6) along with six co-occupied Al/Cr sites (Al1/Cr1 and Al3/Cr3), for which the refined site occupancies converged to 0.772 (4) and 0.958 (4) for Al atoms Al1 and Al3.

Figure 1.

The crystal structure of Al_7.85Cr_5.16. The icosa­hedra centred on Cr4 are emphasized.

Table 1. Experimental details.

Crystal data
Chemical formula	Al7.85Cr5.16
M r	479.72
Crystal system, space group	Trigonal, R3m:R
Temperature (K)	296
a (Å)	7.8777 (5), 7.8777 (5)
α (°)	109.566 (2)
V (Å3)	375.01 (7)
Z	2
Radiation type	Mo Kα
μ (mm−1)	8.05
Crystal size (mm)	0.13 × 0.06 × 0.05
Data collection
Diffractometer	Bruker D8 Venture Photon 100 CMOS
Absorption correction	Multi-scan (SADABS; Bruker, 2015 ▸)
T min, T max	0.496, 0.523
No. of measured, independent and observed [I > 2σ(I)] reflections	7330, 576, 547
R int	0.090
(sin θ/λ)max (Å−1)	0.634
Refinement
R[F 2 > 2σ(F 2)], wR(F 2), S	0.068, 0.178, 1.16
No. of reflections	576
No. of parameters	53
No. of restraints	37
Δρmax, Δρmin (e Å−3)	1.02, −1.27
Absolute structure	Refined as an inversion twin.
Absolute structure parameter	0.3 (2)

Computer programs: APEX3 and SAINT (Bruker, 2015 ▸), SHELXT2014/5 (Sheldrick, 2015a ▸), SHELXL2017/1 (Sheldrick, 2015b ▸), DIAMOND (Brandenburg & Putz, 2017 ▸) and publCIF (Westrip, 2010 ▸).

Figure 2.

The environment of the Cr4 atom. Displacement ellipsoids are given at the 99% probability level. [Symmetry codes: (i) y, z, x; (v) y, z, x.]

The principle building blocks in the structure can also be represented by four inter­penetrating distorted icosa­hedra centred at one Cr4 and three Al3/Cr3 atomic sites, as shown in Fig. 3 ▸, similarly to the building blocks of the I-cell (space group I 3m) of the γ-brass phase (Pankova et al., 2013 ▸). According to the topological analysis of the structure model with the ‘nanocluster’ method available in the ToposPro package (Akhmetshina & Blatov, 2017 ▸), these one Cr4 and three Al3/Cr3 sites form an inner tetra­hedron (IT), followed by an outer tetra­hedron (OT), an octa­hedron (OH), whose vertices are projected onto the edges of the outer tetra­hedron, and finally a distorted cubocta­hedron (CO) with vertices located above the edges of the octa­hedron, as illustrated in Fig. 4 ▸.

Figure 3.

26-atom γ-brass-type cluster represented as four inter­penetrating distorted icosa­hedra centred at one Cr4 and three Al3/Cr3 sites.

Figure 4.

26-atom γ-brass-type cluster represented as a sequence of polyhedral shells.

The present rhombohedral γ₂′-Al₈Cr₅ phase is thus confirmed to be isotypic to the previously reported ordered Al₈Cr₅ phase (Bradley & Lu, 1937 ▸), and closely related to the the disordered rhombohedral Al₁₆Cr_9.5 phase (Brandon et al., 1977 ▸) and the disordered cubic Al₈Cr₅ phase (Braun et al., 1992 ▸).

Synthesis and crystallization

The high-purity elements Al (indicated purity 99.8%, 0.588 g) and Cr (indicated purity 99.95%, 0.539 g) were mixed uniformly in the stoichiometric ratio 11:4 and thoroughly ground in an agate mortar. The blended powders were then placed in a cemented carbide grinding mould of 5 mm diameter, and pressed into a tablet at about 4 MPa for 5 min. A cylindrical block (5 mm in diameter and 3 mm in height) was obtained without deformations or cracks. Details of the high-pressure sinter­ing experiment using a six-anvil high-temperature high-pressure apparatus can be found elsewhere (Liu & Fan, 2018 ▸). The samples were pressurized up to 5 GPa and heated to 1400°C for 30 minutes, slowly cooled to 660°C and held at this temperature for 2 h, and then rapidly cooled to room temperature by turning off the furnace power. Subsequently, a small amount of powder sample was uniformly placed on the inner wall of a quartz tube, annealed in a vacuum environment, heated to 300°C for 24 h, and then cooled within the furnace. A piece of a single crystal (0.13 × 0.06 × 0.05 mm³) was selected and mounted on a glass fibre for single-crystal X-ray diffraction measurements.

Refinement

Table 1 ▸ shows the details of data collection and structural refinement. Three sites are co-occupied by Al and Cr atoms (Al1/Cr1, Al2/Cr2, Al3/Cr3). Site occupancies were refined, and then fixed to their as-found values, 0.772, 0.5 and 0.958 for Al1, Al2 and Al3, respectively, assuming full occupancy for each site. Atoms sharing the same site were constrained to have the same coordinates and displacement parameters. Moreover, disordered atoms were restrained to be isotropic, with standard deviations of 0.01 Ų (Sheldrick, 2015b ▸). The maximum and minimum residual electron densities in the last difference map are located 1.68 Å from atom Cr3 and 0.36 Å from atom Cr4, respectively. The crystal was considered as a sample twinned by inversion (Parsons et al., 2013 ▸), and the batch scale factor converged to x = 0.3 (2).

Supplementary Material

CCDC reference: 1993113

Additional supporting information: crystallographic information; 3D view; checkCIF report

full crystallographic data

Crystal data

Al7.85Cr5.16	Dx = 4.248 Mg m−3
Mr = 479.72	Mo Kα radiation, λ = 0.71073 Å
Trigonal, R3m:R	Cell parameters from 2208 reflections
a = 7.8777 (5) Å	θ = 3.2–26.3°
α = 109.566 (2)°	µ = 8.05 mm−1
V = 375.01 (7) Å3	T = 296 K
Z = 2	Graininess, metallic silver
F(000) = 451	0.13 × 0.06 × 0.05 mm

Data collection

Bruker D8 Venture Photon 100 CMOS diffractometer	547 reflections with I > 2σ(I)
φ and ω scans	Rint = 0.090
Absorption correction: multi-scan (SADABS; Bruker, 2015)	θmax = 26.8°, θmin = 3.2°
Tmin = 0.496, Tmax = 0.523	h = −9→9
7330 measured reflections	k = −9→9
576 independent reflections	l = −9→9

Refinement

Refinement on F2	37 restraints
Least-squares matrix: full	w = 1/[σ2(Fo2) + (0.0652P)2 + 19.1482P] where P = (Fo2 + 2Fc2)/3
R[F2 > 2σ(F2)] = 0.068	(Δ/σ)max < 0.001
wR(F2) = 0.178	Δρmax = 1.02 e Å−3
S = 1.16	Δρmin = −1.27 e Å−3
576 reflections	Absolute structure: Refined as an inversion twin.
53 parameters	Absolute structure parameter: 0.3 (2)

Special details

Refinement. Refined as a two-component inversion twin

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq	Occ. (<1)
Al1	0.2998 (14)	0.2998 (14)	0.0928 (15)	0.0108 (19)	0.772
Cr1	0.2998 (14)	0.2998 (14)	0.0928 (15)	0.0108 (19)	0.228
Al2	1.3096 (13)	0.6646 (11)	0.6646 (11)	0.0117 (19)	0.5
Cr2	1.3096 (13)	0.6646 (11)	0.6646 (11)	0.0117 (19)	0.5
Al3	0.9238 (18)	0.6488 (14)	0.6488 (14)	0.012 (2)	0.958
Cr3	0.9238 (18)	0.6488 (14)	0.6488 (14)	0.012 (2)	0.042
Cr4	−0.0434 (12)	−0.0434 (12)	−0.0434 (12)	0.006 (2)
Cr5	0.6391 (8)	0.3060 (8)	0.3060 (8)	0.0028 (10)
Cr6	1.3001 (10)	0.9464 (8)	0.9464 (8)	0.0108 (14)
Cr7	0.5198 (13)	0.5198 (13)	0.5198 (13)	0.008 (2)
Al4	0.5500 (17)	−0.0607 (15)	0.2804 (16)	0.018 (2)
Al5	0.0366 (18)	0.0366 (18)	−0.313 (2)	0.015 (3)

Atomic displacement parameters (Ų)

	U11	U22	U33	U12	U13	U23
Al1	0.007 (3)	0.007 (3)	0.019 (5)	0.005 (3)	0.006 (3)	0.006 (3)
Cr1	0.007 (3)	0.007 (3)	0.019 (5)	0.005 (3)	0.006 (3)	0.006 (3)
Al2	0.008 (4)	0.008 (3)	0.008 (3)	0.000 (3)	0.000 (3)	0.001 (3)
Cr2	0.008 (4)	0.008 (3)	0.008 (3)	0.000 (3)	0.000 (3)	0.001 (3)
Al3	0.013 (5)	0.010 (3)	0.010 (3)	0.007 (3)	0.007 (3)	−0.001 (3)
Cr3	0.013 (5)	0.010 (3)	0.010 (3)	0.007 (3)	0.007 (3)	−0.001 (3)
Cr4	0.007 (4)	0.007 (4)	0.007 (4)	0.005 (4)	0.005 (4)	0.005 (4)
Cr5	0.001 (2)	0.0034 (18)	0.0034 (18)	0.0001 (17)	0.0001 (17)	0.002 (2)
Cr6	0.016 (3)	0.008 (3)	0.008 (3)	0.006 (3)	0.006 (3)	0.003 (3)
Cr7	0.006 (3)	0.006 (3)	0.006 (3)	0.000 (4)	0.000 (4)	0.000 (4)
Al4	0.024 (5)	0.021 (4)	0.031 (5)	0.016 (3)	0.021 (4)	0.021 (4)
Al5	0.015 (4)	0.015 (4)	0.017 (5)	0.004 (5)	0.011 (4)	0.011 (4)

Geometric parameters (Å, º)

Al1—Cr5	2.611 (7)	Cr2—Al5xi	2.930 (16)
Al1—Cr5i	2.611 (7)	Al3—Cr5	2.572 (12)
Al1—Cr6ii	2.634 (7)	Al3—Cr6	2.606 (11)
Al1—Cr6iii	2.634 (7)	Al3—Cr4xiii	2.655 (12)
Al1—Cr4	2.644 (11)	Al3—Cr7	2.678 (13)
Al1—Al4i	2.654 (11)	Al3—Al5xiii	2.758 (11)
Al1—Al4iv	2.654 (11)	Al3—Al5xiv	2.758 (11)
Al1—Al5	2.659 (17)	Al3—Al4xii	2.798 (12)
Al1—Al1i	2.665 (16)	Al3—Al4ix	2.798 (12)
Al1—Al1v	2.665 (16)	Cr3—Cr5	2.572 (12)
Al1—Cr7	2.739 (11)	Cr3—Cr6	2.606 (11)
Cr1—Cr5	2.611 (7)	Cr3—Cr4xiii	2.655 (12)
Cr1—Cr5i	2.611 (7)	Cr3—Cr7	2.678 (13)
Cr1—Cr6ii	2.634 (7)	Cr3—Al5xiii	2.758 (11)
Cr1—Cr6iii	2.634 (7)	Cr3—Al5xiv	2.758 (11)
Cr1—Cr4	2.644 (11)	Cr3—Al4xii	2.798 (12)
Cr1—Al4i	2.654 (11)	Cr3—Al4ix	2.798 (12)
Cr1—Al4iv	2.654 (11)	Cr4—Al5	2.616 (10)
Cr1—Al5	2.659 (17)	Cr4—Al5v	2.616 (10)
Cr1—Cr7	2.739 (11)	Cr4—Al5i	2.616 (10)
Al2—Cr6	2.604 (9)	Cr4—Cr6iii	2.761 (7)
Al2—Cr7vi	2.647 (9)	Cr4—Cr6ii	2.761 (7)
Al2—Al4vii	2.656 (9)	Cr5—Cr7	2.603 (7)
Al2—Al4viii	2.656 (9)	Cr5—Al4xii	2.620 (10)
Al2—Cr5ix	2.754 (6)	Cr5—Al4ix	2.620 (10)
Al2—Cr5x	2.754 (6)	Cr5—Al4iv	2.652 (9)
Al2—Al5xi	2.930 (16)	Cr5—Al4	2.652 (9)
Al2—Al4xii	2.996 (10)	Cr5—Al5x	2.683 (12)
Al2—Al4ix	2.996 (10)	Cr6—Al5xi	2.654 (12)
Cr2—Cr6	2.604 (9)	Cr6—Al4xv	2.739 (11)
Cr2—Cr7vi	2.647 (9)	Cr6—Al4xiii	2.739 (11)
Cr2—Al4vii	2.656 (9)	Cr6—Al5xiv	2.849 (8)
Cr2—Al4viii	2.656 (9)	Al4—Al5x	2.693 (12)
Cr2—Cr5ix	2.754 (6)	Al4—Al4ix	2.726 (5)
Cr2—Cr5x	2.754 (6)	Al4—Al4xvi	2.726 (5)
Cr5—Al1—Cr5i	110.4 (5)	Cr3—Cr5—Cr7	62.3 (4)
Cr5—Al1—Cr6ii	63.75 (14)	Al3—Cr5—Cr7	62.3 (4)
Cr5i—Al1—Cr6ii	167.7 (4)	Cr3—Cr5—Al1v	116.2 (4)
Cr5—Al1—Cr6iii	167.7 (4)	Al3—Cr5—Al1v	116.2 (4)
Cr5i—Al1—Cr6iii	63.75 (14)	Cr7—Cr5—Al1v	63.4 (3)
Cr6ii—Al1—Cr6iii	119.6 (5)	Cr7—Cr5—Al1	63.4 (3)
Cr5—Al1—Cr4	112.8 (3)	Al1v—Cr5—Al1	61.4 (4)
Cr5i—Al1—Cr4	112.8 (3)	Cr7—Cr5—Cr1	63.4 (3)
Cr6ii—Al1—Cr4	63.1 (2)	Cr3—Cr5—Al4xii	65.2 (3)
Cr6iii—Al1—Cr4	63.1 (2)	Al3—Cr5—Al4xii	65.2 (3)
Cr5—Al1—Al4i	125.5 (4)	Cr7—Cr5—Al4xii	110.8 (3)
Cr5i—Al1—Al4i	60.5 (2)	Al1v—Cr5—Al4xii	168.9 (3)
Cr6ii—Al1—Al4i	131.8 (4)	Al1—Cr5—Al4xii	107.8 (3)
Cr6iii—Al1—Al4i	62.4 (2)	Cr1—Cr5—Al4xii	107.8 (3)
Cr4—Al1—Al4i	120.0 (3)	Cr3—Cr5—Al4ix	65.2 (3)
Cr5—Al1—Al4iv	60.5 (2)	Al3—Cr5—Al4ix	65.2 (3)
Cr5i—Al1—Al4iv	125.5 (4)	Cr7—Cr5—Al4ix	110.8 (3)
Cr6ii—Al1—Al4iv	62.4 (2)	Al1v—Cr5—Al4ix	107.8 (3)
Cr6iii—Al1—Al4iv	131.8 (4)	Al1—Cr5—Al4ix	168.9 (3)
Cr4—Al1—Al4iv	120.0 (3)	Cr1—Cr5—Al4ix	168.9 (3)
Al4i—Al1—Al4iv	81.7 (5)	Al4xii—Cr5—Al4ix	83.0 (5)
Cr5—Al1—Al5	124.0 (2)	Cr3—Cr5—Al4iv	121.2 (3)
Cr5i—Al1—Al5	124.0 (2)	Al3—Cr5—Al4iv	121.2 (3)
Cr6ii—Al1—Al5	65.1 (3)	Cr7—Cr5—Al4iv	115.6 (3)
Cr6iii—Al1—Al5	65.1 (3)	Al1v—Cr5—Al4iv	110.7 (4)
Cr4—Al1—Al5	59.1 (3)	Al1—Cr5—Al4iv	60.5 (3)
Al4i—Al1—Al5	76.7 (3)	Cr1—Cr5—Al4iv	60.5 (3)
Al4iv—Al1—Al5	76.7 (3)	Al4xii—Cr5—Al4iv	62.3 (2)
Cr5—Al1—Al1i	108.1 (2)	Al4ix—Cr5—Al4iv	129.1 (2)
Cr5i—Al1—Al1i	59.3 (2)	Cr3—Cr5—Al4	121.2 (3)
Cr6ii—Al1—Al1i	111.0 (2)	Al3—Cr5—Al4	121.2 (3)
Cr6iii—Al1—Al1i	59.6 (2)	Cr7—Cr5—Al4	115.6 (3)
Cr4—Al1—Al1i	59.73 (18)	Al1v—Cr5—Al4	60.5 (3)
Al4i—Al1—Al1i	109.0 (2)	Al1—Cr5—Al4	110.7 (4)
Al4iv—Al1—Al1i	168.2 (3)	Cr1—Cr5—Al4	110.7 (4)
Al5—Al1—Al1i	110.2 (3)	Al4xii—Cr5—Al4	129.1 (2)
Cr5—Al1—Al1v	59.3 (2)	Al4ix—Cr5—Al4	62.3 (2)
Cr5i—Al1—Al1v	108.1 (2)	Al4iv—Cr5—Al4	111.7 (5)
Cr6ii—Al1—Al1v	59.6 (2)	Cr3—Cr5—Al5x	128.3 (4)
Cr6iii—Al1—Al1v	111.0 (2)	Al3—Cr5—Al5x	128.3 (4)
Cr4—Al1—Al1v	59.73 (18)	Cr7—Cr5—Al5x	169.4 (4)
Al4i—Al1—Al1v	168.2 (3)	Al1v—Cr5—Al5x	107.8 (3)
Al4iv—Al1—Al1v	109.0 (2)	Al1—Cr5—Al5x	107.8 (3)
Al5—Al1—Al1v	110.2 (3)	Cr1—Cr5—Al5x	107.8 (3)
Al1i—Al1—Al1v	60.000 (1)	Al4xii—Cr5—Al5x	76.8 (3)
Cr5—Al1—Cr7	58.2 (2)	Al4ix—Cr5—Al5x	76.8 (3)
Cr5i—Al1—Cr7	58.2 (2)	Al4iv—Cr5—Al5x	60.6 (3)
Cr6ii—Al1—Cr7	111.2 (3)	Al4—Cr5—Al5x	60.6 (3)
Cr6iii—Al1—Cr7	111.2 (3)	Cr3—Cr5—Al2xvi	64.3 (2)
Cr4—Al1—Cr7	110.2 (4)	Al3—Cr5—Al2xvi	64.3 (2)
Al4i—Al1—Cr7	111.1 (3)	Cr7—Cr5—Al2xvi	59.1 (2)
Al4iv—Al1—Cr7	111.1 (3)	Al1v—Cr5—Al2xvi	60.5 (3)
Al5—Al1—Cr7	169.4 (5)	Al1—Cr5—Al2xvi	110.8 (3)
Al1i—Al1—Cr7	60.89 (18)	Cr1—Cr5—Al2xvi	110.8 (3)
Al1v—Al1—Cr7	60.89 (18)	Al4xii—Cr5—Al2xvi	125.9 (4)
Cr5—Cr1—Cr5i	110.4 (5)	Al4ix—Cr5—Al2xvi	59.2 (3)
Cr5—Cr1—Cr6ii	63.75 (14)	Al4iv—Cr5—Al2xvi	170.8 (4)
Cr5i—Cr1—Cr6ii	167.7 (4)	Al4—Cr5—Al2xvi	67.3 (2)
Cr5—Cr1—Cr6iii	167.7 (4)	Al5x—Cr5—Al2xvi	123.0 (2)
Cr5i—Cr1—Cr6iii	63.75 (14)	Cr2—Cr6—Al1xiii	149.6 (2)
Cr6ii—Cr1—Cr6iii	119.6 (5)	Al2—Cr6—Al1xiii	149.6 (2)
Cr5—Cr1—Cr4	112.8 (3)	Cr3—Cr6—Al1xiii	108.9 (3)
Cr5i—Cr1—Cr4	112.8 (3)	Al3—Cr6—Al1xiii	108.9 (3)
Cr6ii—Cr1—Cr4	63.1 (2)	Al1xiii—Cr6—Al1xiv	60.8 (4)
Cr6iii—Cr1—Cr4	63.1 (2)	Cr2—Cr6—Al5xi	67.7 (4)
Cr5—Cr1—Al4i	125.5 (4)	Al2—Cr6—Al5xi	67.7 (4)
Cr5i—Cr1—Al4i	60.5 (2)	Cr3—Cr6—Al5xi	136.9 (4)
Cr6ii—Cr1—Al4i	131.8 (4)	Al3—Cr6—Al5xi	136.9 (4)
Cr6iii—Cr1—Al4i	62.4 (2)	Al1xiii—Cr6—Al5xi	108.0 (4)
Cr4—Cr1—Al4i	120.0 (3)	Al1xiv—Cr6—Al5xi	108.0 (4)
Cr5—Cr1—Al4iv	60.5 (2)	Al1xiii—Cr6—Al4xv	59.2 (3)
Cr5i—Cr1—Al4iv	125.5 (4)	Al1xiv—Cr6—Al4xv	107.4 (4)
Cr6ii—Cr1—Al4iv	62.4 (2)	Al5xi—Cr6—Al4xv	59.9 (3)
Cr6iii—Cr1—Al4iv	131.8 (4)	Cr2—Cr6—Al4xiii	96.4 (3)
Cr4—Cr1—Al4iv	120.0 (3)	Al2—Cr6—Al4xiii	96.4 (3)
Al4i—Cr1—Al4iv	81.7 (5)	Cr3—Cr6—Al4xiii	126.1 (2)
Cr5—Cr1—Al5	124.0 (2)	Al3—Cr6—Al4xiii	126.1 (2)
Cr5i—Cr1—Al5	124.0 (2)	Al1xiii—Cr6—Al4xiii	107.4 (4)
Cr6ii—Cr1—Al5	65.1 (3)	Al1xiv—Cr6—Al4xiii	59.2 (3)
Cr6iii—Cr1—Al5	65.1 (3)	Al5xi—Cr6—Al4xiii	59.9 (3)
Cr4—Cr1—Al5	59.1 (3)	Al4xv—Cr6—Al4xiii	106.5 (5)
Al4i—Cr1—Al5	76.7 (3)	Al1xiii—Cr6—Cr4xiii	58.6 (2)
Al4iv—Cr1—Al5	76.7 (3)	Al1xiv—Cr6—Cr4xiii	58.6 (2)
Cr5—Cr1—Cr7	58.2 (2)	Al5xi—Cr6—Cr4xiii	163.9 (4)
Cr5i—Cr1—Cr7	58.2 (2)	Al4xv—Cr6—Cr4xiii	113.1 (3)
Cr6ii—Cr1—Cr7	111.2 (3)	Al4xiii—Cr6—Cr4xiii	113.1 (3)
Cr6iii—Cr1—Cr7	111.2 (3)	Cr2—Cr6—Cr5xiii	127.0 (4)
Cr4—Cr1—Cr7	110.2 (4)	Al2—Cr6—Cr5xiii	127.0 (4)
Al4i—Cr1—Cr7	111.1 (3)	Cr3—Cr6—Cr5xiii	163.9 (3)
Al4iv—Cr1—Cr7	111.1 (3)	Al3—Cr6—Cr5xiii	163.9 (3)
Al5—Cr1—Cr7	169.4 (5)	Al1xiii—Cr6—Cr5xiii	57.7 (2)
Cr6—Al2—Cr7vi	150.7 (4)	Al1xiv—Cr6—Cr5xiii	57.7 (2)
Cr6—Al2—Al4vii	71.0 (2)	Al5xi—Cr6—Cr5xiii	59.2 (3)
Cr7vi—Al2—Al4vii	108.3 (2)	Al4xv—Cr6—Cr5xiii	57.6 (2)
Cr6—Al2—Al4viii	71.0 (2)	Al4xiii—Cr6—Cr5xiii	57.6 (3)
Cr7vi—Al2—Al4viii	108.3 (2)	Cr4xiii—Cr6—Cr5xiii	104.7 (3)
Al4vii—Al2—Al4viii	141.3 (4)	Cr2—Cr6—Al5xiv	99.6 (3)
Cr6—Al2—Cr5ix	128.85 (18)	Al2—Cr6—Al5xiv	99.6 (3)
Cr7vi—Al2—Cr5ix	57.6 (2)	Cr3—Cr6—Al5xiv	60.5 (3)
Al4vii—Al2—Cr5ix	159.6 (4)	Al3—Cr6—Al5xiv	60.5 (3)
Al4viii—Al2—Cr5ix	57.9 (2)	Al1xiii—Cr6—Al5xiv	105.5 (3)
Cr6—Al2—Cr5x	128.85 (18)	Al1xiv—Cr6—Al5xiv	57.8 (3)
Cr7vi—Al2—Cr5x	57.6 (2)	Al5xi—Cr6—Al5xiv	127.5 (2)
Al4vii—Al2—Cr5x	57.9 (2)	Al4xv—Cr6—Al5xiv	164.0 (4)
Al4viii—Al2—Cr5x	159.6 (4)	Al4xiii—Cr6—Al5xiv	72.2 (3)
Cr5ix—Al2—Cr5x	102.2 (4)	Cr4xiii—Cr6—Al5xiv	55.6 (2)
Cr6—Al2—Al5xi	56.9 (3)	Cr5xiii—Cr6—Al5xiv	111.8 (3)
Cr7vi—Al2—Al5xi	93.8 (4)	Cr5—Cr7—Cr5v	110.9 (2)
Al4vii—Al2—Al5xi	82.8 (3)	Cr5—Cr7—Cr5i	110.9 (2)
Al4viii—Al2—Al5xi	82.8 (3)	Cr5v—Cr7—Cr5i	110.9 (2)
Cr5ix—Al2—Al5xi	111.2 (3)	Cr5—Cr7—Al2xviii	166.5 (5)
Cr5x—Al2—Al5xi	111.2 (3)	Cr5v—Cr7—Al2xviii	63.27 (8)
Cr6—Al2—Al4xii	99.9 (3)	Cr5i—Cr7—Al2xviii	63.27 (8)
Cr7vi—Al2—Al4xii	103.8 (3)	Cr5—Cr7—Al2xix	63.27 (8)
Al4vii—Al2—Al4xii	57.30 (18)	Cr5v—Cr7—Al2xix	166.5 (5)
Al4viii—Al2—Al4xii	123.5 (4)	Cr5i—Cr7—Al2xix	63.27 (8)
Cr5ix—Al2—Al4xii	108.9 (3)	Cr5—Cr7—Al2xvi	63.27 (8)
Cr5x—Al2—Al4xii	54.7 (2)	Cr5v—Cr7—Al2xvi	63.27 (8)
Al5xi—Al2—Al4xii	139.6 (3)	Cr5i—Cr7—Al2xvi	166.5 (5)
Cr6—Al2—Al4ix	99.9 (3)	Al2xviii—Cr7—Al2xvi	119.39 (7)
Cr7vi—Al2—Al4ix	103.8 (3)	Al2xix—Cr7—Al2xvi	119.39 (7)
Al4vii—Al2—Al4ix	123.5 (4)	Cr5—Cr7—Al3	58.3 (2)
Al4viii—Al2—Al4ix	57.30 (18)	Cr5v—Cr7—Al3	124.25 (17)
Cr5ix—Al2—Al4ix	54.7 (2)	Cr5i—Cr7—Al3	124.25 (17)
Cr5x—Al2—Al4ix	108.9 (3)	Cr5—Cr7—Cr3	58.3 (2)
Al5xi—Al2—Al4ix	139.6 (3)	Cr5v—Cr7—Cr3	124.25 (17)
Al4xii—Al2—Al4ix	70.8 (4)	Cr5i—Cr7—Cr3	124.25 (17)
Cr6—Cr2—Cr7vi	150.7 (4)	Cr5—Cr7—Al3i	124.25 (17)
Cr6—Cr2—Al4vii	71.0 (2)	Cr5v—Cr7—Al3i	124.25 (17)
Cr7vi—Cr2—Al4vii	108.3 (2)	Cr5i—Cr7—Al3i	58.3 (2)
Cr6—Cr2—Al4viii	71.0 (2)	Al2xviii—Cr7—Al3i	64.46 (17)
Cr7vi—Cr2—Al4viii	108.3 (2)	Al2xix—Cr7—Al3i	64.46 (17)
Al4vii—Cr2—Al4viii	141.3 (4)	Al2xvi—Cr7—Al3i	135.3 (5)
Cr6—Cr2—Cr5ix	128.85 (18)	Al3—Cr7—Al3i	82.8 (4)
Cr7vi—Cr2—Cr5ix	57.6 (2)	Cr3—Cr7—Al3i	82.8 (4)
Al4vii—Cr2—Cr5ix	159.6 (4)	Cr5—Cr7—Al3v	124.25 (17)
Al4viii—Cr2—Cr5ix	57.9 (2)	Cr5v—Cr7—Al3v	58.3 (2)
Cr6—Cr2—Cr5x	128.85 (18)	Cr5i—Cr7—Al3v	124.25 (17)
Cr7vi—Cr2—Cr5x	57.6 (2)	Cr5—Cr7—Al1v	58.5 (2)
Al4vii—Cr2—Cr5x	57.9 (2)	Cr5v—Cr7—Al1v	58.5 (2)
Al4viii—Cr2—Cr5x	159.6 (4)	Cr5i—Cr7—Al1v	106.2 (4)
Cr5ix—Cr2—Cr5x	102.2 (4)	Al2xviii—Cr7—Al1v	110.2 (3)
Cr6—Cr2—Al5xi	56.9 (3)	Al2xix—Cr7—Al1v	110.2 (3)
Cr7vi—Cr2—Al5xi	93.8 (4)	Al2xvi—Cr7—Al1v	60.3 (3)
Al4vii—Cr2—Al5xi	82.8 (3)	Al3—Cr7—Al1v	108.7 (3)
Al4viii—Cr2—Al5xi	82.8 (3)	Cr3—Cr7—Al1v	108.7 (3)
Cr5ix—Cr2—Al5xi	111.2 (3)	Al3i—Cr7—Al1v	164.4 (4)
Cr5x—Cr2—Al5xi	111.2 (3)	Al3v—Cr7—Al1v	108.7 (3)
Cr5—Al3—Cr6	157.4 (4)	Cr5—Cr7—Al1i	106.2 (4)
Cr5—Al3—Cr4xiii	139.3 (5)	Cr5v—Cr7—Al1i	58.5 (2)
Cr6—Al3—Cr4xiii	63.3 (3)	Cr5i—Cr7—Al1i	58.5 (2)
Cr5—Al3—Cr7	59.4 (3)	Al2xviii—Cr7—Al1i	60.3 (3)
Cr6—Al3—Cr7	143.2 (4)	Al2xix—Cr7—Al1i	110.2 (3)
Cr4xiii—Al3—Cr7	79.9 (4)	Al2xvi—Cr7—Al1i	110.2 (3)
Cr5—Al3—Al5xiii	123.2 (3)	Al3—Cr7—Al1i	164.4 (4)
Cr6—Al3—Al5xiii	64.1 (3)	Cr3—Cr7—Al1i	164.4 (4)
Cr4xiii—Al3—Al5xiii	57.8 (3)	Al3i—Cr7—Al1i	108.7 (3)
Cr7—Al3—Al5xiii	97.1 (3)	Al3v—Cr7—Al1i	108.7 (3)
Cr5—Al3—Al5xiv	123.2 (3)	Al1v—Cr7—Al1i	58.2 (4)
Cr6—Al3—Al5xiv	64.1 (3)	Cr5xvi—Al4—Cr5	144.8 (4)
Cr4xiii—Al3—Al5xiv	57.8 (3)	Cr5xvi—Al4—Al1v	86.3 (4)
Cr7—Al3—Al5xiv	97.1 (3)	Cr5—Al4—Al1v	59.0 (3)
Al5xiii—Al3—Al5xiv	109.5 (6)	Cr5xvi—Al4—Al2xx	62.9 (2)
Cr5—Al3—Al4xii	58.2 (3)	Cr5—Al4—Al2xx	150.4 (5)
Cr6—Al3—Al4xii	105.2 (4)	Al1v—Al4—Al2xx	148.8 (4)
Cr4xiii—Al3—Al4xii	141.1 (3)	Cr5xvi—Al4—Al5x	145.4 (6)
Cr7—Al3—Al4xii	103.4 (4)	Cr5—Al4—Al5x	60.2 (3)
Al5xiii—Al3—Al4xii	83.5 (3)	Al1v—Al4—Al5x	106.3 (3)
Al5xiv—Al3—Al4xii	154.2 (5)	Al2xx—Al4—Al5x	102.4 (4)
Cr5—Al3—Al4ix	58.2 (3)	Cr5xvi—Al4—Al4ix	134.5 (5)
Cr6—Al3—Al4ix	105.2 (4)	Cr5—Al4—Al4ix	58.3 (4)
Cr4xiii—Al3—Al4ix	141.1 (3)	Al1v—Al4—Al4ix	103.5 (4)
Cr7—Al3—Al4ix	103.4 (4)	Al2xx—Al4—Al4ix	95.5 (4)
Al5xiii—Al3—Al4ix	154.2 (5)	Al5x—Al4—Al4ix	74.9 (5)
Al5xiv—Al3—Al4ix	83.5 (3)	Cr5xvi—Al4—Al4xvi	59.4 (3)
Al4xii—Al3—Al4ix	76.6 (5)	Cr5—Al4—Al4xvi	128.9 (4)
Cr5—Cr3—Cr6	157.4 (4)	Al1v—Al4—Al4xvi	102.1 (5)
Cr5—Cr3—Cr4xiii	139.3 (5)	Al2xx—Al4—Al4xvi	67.6 (4)
Cr6—Cr3—Cr4xiii	63.3 (3)	Al5x—Al4—Al4xvi	86.1 (5)
Cr5—Cr3—Cr7	59.4 (3)	Al4ix—Al4—Al4xvi	151.5 (6)
Cr6—Cr3—Cr7	143.2 (4)	Cr5xvi—Al4—Cr6ii	106.6 (5)
Cr4xiii—Cr3—Cr7	79.9 (4)	Cr5—Al4—Cr6ii	61.8 (2)
Cr5—Cr3—Al5xiii	123.2 (3)	Al1v—Al4—Cr6ii	58.4 (3)
Cr6—Cr3—Al5xiii	64.1 (3)	Al2xx—Al4—Cr6ii	132.5 (4)
Cr4xiii—Cr3—Al5xiii	57.8 (3)	Al5x—Al4—Cr6ii	58.5 (3)
Cr7—Cr3—Al5xiii	97.1 (3)	Al4ix—Al4—Cr6ii	116.5 (5)
Cr5—Cr3—Al5xiv	123.2 (3)	Al4xvi—Al4—Cr6ii	68.0 (3)
Cr6—Cr3—Al5xiv	64.1 (3)	Cr5xvi—Al4—Al3xvi	56.6 (3)
Cr4xiii—Cr3—Al5xiv	57.8 (3)	Cr5—Al4—Al3xvi	118.0 (5)
Cr7—Cr3—Al5xiv	97.1 (3)	Al1v—Al4—Al3xvi	97.2 (4)
Al5xiii—Cr3—Al5xiv	109.5 (6)	Al2xx—Al4—Al3xvi	62.7 (3)
Cr5—Cr3—Al4xii	58.2 (3)	Al5x—Al4—Al3xvi	147.5 (5)
Cr6—Cr3—Al4xii	105.2 (4)	Al4ix—Al4—Al3xvi	78.0 (4)
Cr4xiii—Cr3—Al4xii	141.1 (3)	Al4xvi—Al4—Al3xvi	110.9 (4)
Cr7—Cr3—Al4xii	103.4 (4)	Cr6ii—Al4—Al3xvi	153.0 (5)
Al5xiii—Cr3—Al4xii	83.5 (3)	Cr5xvi—Al4—Al2xvi	99.8 (4)
Al5xiv—Cr3—Al4xii	154.2 (5)	Cr5—Al4—Al2xvi	58.0 (2)
Cr5—Cr3—Al4ix	58.2 (3)	Al1v—Al4—Al2xvi	56.9 (3)
Cr6—Cr3—Al4ix	105.2 (4)	Al2xx—Al4—Al2xvi	120.5 (4)
Cr4xiii—Cr3—Al4ix	141.1 (3)	Al5x—Al4—Al2xvi	114.1 (4)
Cr7—Cr3—Al4ix	103.4 (4)	Al4ix—Al4—Al2xvi	55.1 (2)
Al5xiii—Cr3—Al4ix	154.2 (5)	Al4xvi—Al4—Al2xvi	153.3 (5)
Al5xiv—Cr3—Al4ix	83.5 (3)	Cr6ii—Al4—Al2xvi	106.8 (3)
Al4xii—Cr3—Al4ix	76.6 (5)	Al3xvi—Al4—Al2xvi	61.3 (4)
Al5—Cr4—Al5v	118.81 (14)	Cr4—Al5—Cr6xxi	152.4 (6)
Al5—Cr4—Al5i	118.81 (14)	Cr4—Al5—Cr1	60.2 (4)
Al5v—Cr4—Al5i	118.81 (14)	Cr6xxi—Al5—Cr1	147.5 (5)
Al5—Cr4—Cr1	60.7 (4)	Cr4—Al5—Al1	60.2 (4)
Al5v—Cr4—Cr1	112.2 (3)	Cr6xxi—Al5—Al1	147.5 (5)
Al5i—Cr4—Cr1	112.2 (3)	Cr4—Al5—Cr5xix	145.1 (6)
Al5—Cr4—Al1	60.7 (4)	Cr6xxi—Al5—Cr5xix	62.5 (3)
Al5v—Cr4—Al1	112.2 (3)	Cr1—Al5—Cr5xix	84.9 (5)
Al5i—Cr4—Al1	112.2 (3)	Al1—Al5—Cr5xix	84.9 (5)
Al5—Cr4—Al1i	112.2 (3)	Cr4—Al5—Al4xxii	125.1 (3)
Al5v—Cr4—Al1i	112.2 (3)	Cr6xxi—Al5—Al4xxii	61.6 (3)
Al5i—Cr4—Al1i	60.7 (4)	Al1—Al5—Al4xxii	102.9 (4)
Cr1—Cr4—Al1i	60.5 (4)	Cr5xix—Al5—Al4xxii	59.1 (3)
Al1—Cr4—Al1i	60.5 (4)	Cr4—Al5—Al4xix	125.1 (3)
Al5—Cr4—Al1v	112.2 (3)	Cr6xxi—Al5—Al4xix	61.6 (3)
Al5v—Cr4—Al1v	60.7 (4)	Cr1—Al5—Al4xix	102.9 (4)
Al5i—Cr4—Al1v	112.2 (3)	Al1—Al5—Al4xix	102.9 (4)
Cr1—Cr4—Al1v	60.5 (4)	Cr5xix—Al5—Al4xix	59.1 (3)
Al1—Cr4—Al1v	60.5 (4)	Al4xxii—Al5—Al4xix	109.2 (5)
Al1i—Cr4—Al1v	60.5 (4)	Cr4—Al5—Al3iii	59.1 (3)
Al5—Cr4—Al3iii	63.1 (2)	Cr6xxi—Al5—Al3iii	101.0 (4)
Al5v—Cr4—Al3iii	134.0 (6)	Cr1—Al5—Al3iii	103.8 (4)
Al5i—Cr4—Al3iii	63.1 (2)	Al1—Al5—Al3iii	103.8 (4)
Cr1—Cr4—Al3iii	107.2 (3)	Cr5xix—Al5—Al3iii	138.3 (3)
Al1—Cr4—Al3iii	107.2 (3)	Al4xxii—Al5—Al3iii	149.3 (6)
Al1i—Cr4—Al3iii	107.2 (3)	Al4xix—Al5—Al3iii	79.2 (3)
Al1v—Cr4—Al3iii	165.2 (4)	Cr4—Al5—Al3ii	59.1 (3)
Al5—Cr4—Al3xvii	134.0 (6)	Cr6xxi—Al5—Al3ii	101.0 (4)
Al5v—Cr4—Al3xvii	63.1 (2)	Cr1—Al5—Al3ii	103.8 (4)
Al5i—Cr4—Al3xvii	63.1 (2)	Al1—Al5—Al3ii	103.8 (4)
Cr1—Cr4—Al3xvii	165.2 (4)	Cr5xix—Al5—Al3ii	138.3 (3)
Al1—Cr4—Al3xvii	165.2 (4)	Al4xxii—Al5—Al3ii	79.2 (3)
Al1i—Cr4—Al3xvii	107.2 (3)	Al4xix—Al5—Al3ii	149.3 (6)
Al1v—Cr4—Al3xvii	107.2 (3)	Al3iii—Al5—Al3ii	79.9 (6)
Al3iii—Cr4—Al3xvii	83.6 (4)	Cr4—Al5—Cr6ii	60.5 (3)
Al5—Cr4—Al3ii	63.1 (2)	Cr6xxi—Al5—Cr6ii	126.6 (2)
Al5v—Cr4—Al3ii	63.1 (2)	Cr1—Al5—Cr6ii	57.0 (2)
Al5i—Cr4—Al3ii	134.0 (6)	Al1—Al5—Cr6ii	57.0 (2)
Cr1—Cr4—Al3ii	107.2 (3)	Cr5xix—Al5—Cr6ii	101.9 (3)
Al1—Cr4—Al3ii	107.2 (3)	Al4xxii—Al5—Cr6ii	66.8 (2)
Al1i—Cr4—Al3ii	165.2 (4)	Al4xix—Al5—Cr6ii	155.4 (6)
Al1v—Cr4—Al3ii	107.2 (3)	Al3iii—Al5—Cr6ii	117.2 (4)
Al3iii—Cr4—Al3ii	83.6 (4)	Al3ii—Al5—Cr6ii	55.4 (2)
Al3xvii—Cr4—Al3ii	83.6 (4)	Cr4—Al5—Cr6iii	60.5 (3)
Al5—Cr4—Cr6iii	63.93 (12)	Cr6xxi—Al5—Cr6iii	126.6 (2)
Al5v—Cr4—Cr6iii	168.5 (6)	Cr1—Al5—Cr6iii	57.0 (2)
Al5i—Cr4—Cr6iii	63.93 (12)	Al1—Al5—Cr6iii	57.0 (2)
Cr1—Cr4—Cr6iii	58.27 (18)	Cr5xix—Al5—Cr6iii	101.9 (3)
Al1—Cr4—Cr6iii	58.27 (18)	Al4xxii—Al5—Cr6iii	155.4 (6)
Al1i—Cr4—Cr6iii	58.27 (18)	Al4xix—Al5—Cr6iii	66.8 (2)
Al1v—Cr4—Cr6iii	107.8 (4)	Al3iii—Al5—Cr6iii	55.4 (2)
Al3iii—Cr4—Cr6iii	57.5 (3)	Al3ii—Al5—Cr6iii	117.2 (4)
Al3xvii—Cr4—Cr6iii	124.18 (16)	Cr6ii—Al5—Cr6iii	106.1 (4)
Al3ii—Cr4—Cr6iii	124.18 (16)	Cr4—Al5—Al2xxi	97.0 (4)
Al5—Cr4—Cr6ii	63.93 (12)	Cr6xxi—Al5—Al2xxi	55.3 (3)
Al5v—Cr4—Cr6ii	63.93 (12)	Cr1—Al5—Al2xxi	157.2 (5)
Al5i—Cr4—Cr6ii	168.5 (6)	Al1—Al5—Al2xxi	157.2 (5)
Cr1—Cr4—Cr6ii	58.27 (18)	Cr5xix—Al5—Al2xxi	117.9 (4)
Al1—Cr4—Cr6ii	58.27 (18)	Al4xxii—Al5—Al2xxi	90.1 (4)
Al1i—Cr4—Cr6ii	107.8 (4)	Al4xix—Al5—Al2xxi	90.1 (4)
Al1v—Cr4—Cr6ii	58.27 (18)	Al3iii—Al5—Al2xxi	59.8 (3)
Al3iii—Cr4—Cr6ii	124.18 (16)	Al3ii—Al5—Al2xxi	59.8 (3)
Al3xvii—Cr4—Cr6ii	124.18 (16)	Cr6ii—Al5—Al2xxi	113.8 (3)
Al3ii—Cr4—Cr6ii	57.5 (3)	Cr6iii—Al5—Al2xxi	113.8 (3)
Cr6iii—Cr4—Cr6ii	111.1 (2)

Symmetry codes: (i) z, x, y; (ii) x−1, y−1, z−1; (iii) z−1, x−1, y−1; (iv) x, z, y; (v) y, z, x; (vi) x+1, y, z; (vii) z+1, y+1, x; (viii) z+1, x, y+1; (ix) y+1, z, x; (x) z+1, x, y; (xi) z+2, x+1, y+1; (xii) y+1, x, z; (xiii) x+1, y+1, z+1; (xiv) y+1, z+1, x+1; (xv) x+1, z+1, y+1; (xvi) z, x−1, y; (xvii) y−1, z−1, x−1; (xviii) x−1, y, z; (xix) y, z, x−1; (xx) y, z−1, x−1; (xxi) y−1, z−1, x−2; (xxii) z, y, x−1.

Funding Statement

Funding for this research was provided by: Research Foundation of Education Bureau of Hebei Province (grant No. ZD2018069); The National Natural Science Foundation of China (grant No. 51771165).