inorganic compounds\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

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ISSN: 2414-3146

Crystal structure of the Al8Cr5-type inter­metallic Al7.85Cr5.16

aState Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, People's Republic of China
*Correspondence e-mail: chzfan@ysu.edu.cn

Edited by S. Bernès, Benemérita Universidad Autónoma de Puebla, México (Received 24 December 2019; accepted 27 March 2020; online 9 April 2020)

An aluminium-deficient Al8Cr5-type inter­metallic with formula Al7.85Cr5.16 (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 Al16Cr9.5 and cubic Al8Cr5.

3D view (loading...)
[Scheme 3D1]

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[Bradley, A. J. & Lu, S. S. (1937). Z. Kristallogr. 96, 20-37.]). 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[Bradley, A. J. & Lu, S. S. (1937). Z. Kristallogr. 96, 20-37.]). A high-temperature γ1 phase was also reported to be stable between 1350 and 980°C at the same composition (Bradley & Lu, 1937[Bradley, A. J. & Lu, S. S. (1937). Z. Kristallogr. 96, 20-37.]) 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[Brandon, J. K., Pearson, W. B., Riley, P. W., Chieh, C. & Stokhuyzen, R. (1977). Acta Cryst. B33, 1088-1095.]). 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[Braun, J., Ellner, M. & Predel, B. (1992). J. Alloys Compd. 183, 444-448.]). 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 Al7.85Cr5.16 and hereafter named as γ2′-Al8Cr5 phase, was determined by single-crystal X-ray diffraction measurements.

Fig. 1[link] 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[link]). For simplicity, only two distorted icosa­hedra centred at Wyckoff sites 3a (Cr4, with coordinates 0, 0, z) are illustrated in Fig. 1[link], and the environment of the Cr4 atoms is shown in Fig. 2[link]. 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.

Table 1
Experimental details

Crystal data
Chemical formula Al7.85Cr5.16
Mr 479.72
Crystal system, space group Trigonal, R3m:R
Temperature (K) 296
a (Å) 7.8777 (5), 7.8777 (5)
α (°) 109.566 (2)
V3) 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[Bruker (2015). APEX3, SAINT and SADABS. Bruker AXS Inc., Madison, Wisconsin, USA.])
Tmin, Tmax 0.496, 0.523
No. of measured, independent and observed [I > 2σ(I)] reflections 7330, 576, 547
Rint 0.090
(sin θ/λ)max−1) 0.634
 
Refinement
R[F2 > 2σ(F2)], wR(F2), 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[Bruker (2015). APEX3, SAINT and SADABS. Bruker AXS Inc., Madison, Wisconsin, USA.]), SHELXT2014/5 (Sheldrick, 2015a[Sheldrick, G. M. (2015a). Acta Cryst. A71, 3-8.]), SHELXL2017/1 (Sheldrick, 2015b[Sheldrick, G. M. (2015b). Acta Cryst. C71, 3-8.]), DIAMOND (Brandenburg & Putz, 2017[Brandenburg, K. & Putz, H. (2017). DIAMOND. Crystal Impact GbR, Bonn, Germany.]) and publCIF (Westrip, 2010[Westrip, S. P. (2010). J. Appl. Cryst. 43, 920-925.]).
[Figure 1]
Figure 1
The crystal structure of Al7.85Cr5.16. The icosa­hedra centred on Cr4 are emphasized.
[Figure 2]
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[link], similarly to the building blocks of the I-cell (space group I[\overline{4}]3m) of the γ-brass phase (Pankova et al., 2013[Pankova, A. A., Blatov, V. A., Ilyushin, G. D. & Proserpio, D. M. (2013). Inorg. Chem. 52, 13094-13107.]). According to the topological analysis of the structure model with the `nanocluster' method available in the ToposPro package (Akhmetshina & Blatov, 2017[Akhmetshina, T. G. & Blatov, V. A. (2017). Z. Kristallogr. 232, 497-506.]), 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[link].

[Figure 3]
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]
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[Bradley, A. J. & Lu, S. S. (1937). Z. Kristallogr. 96, 20-37.]), and closely related to the the disordered rhombohedral Al16Cr9.5 phase (Brandon et al., 1977[Brandon, J. K., Pearson, W. B., Riley, P. W., Chieh, C. & Stokhuyzen, R. (1977). Acta Cryst. B33, 1088-1095.]) and the disordered cubic Al8Cr5 phase (Braun et al., 1992[Braun, J., Ellner, M. & Predel, B. (1992). J. Alloys Compd. 183, 444-448.]).

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[Liu, C. & Fan, C. (2018). IUCrData, 3, x180363.]). 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[link] 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 Å2 (Sheldrick, 2015b[Sheldrick, G. M. (2015b). Acta Cryst. C71, 3-8.]). 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[Parsons, S., Flack, H. D. & Wagner, T. (2013). Acta Cryst. B69, 249-259.]), and the batch scale factor converged to x = 0.3 (2).

Structural data


Computing details top

Data collection: APEX3 (Bruker, 2015); cell refinement: APEX3 (Bruker, 2015); data reduction: APEX3 and SAINT (Bruker, 2015); program(s) used to solve structure: SHELXT2014/5 (Sheldrick, 2015a); program(s) used to refine structure: SHELXL2017/1 (Sheldrick, 2015b); molecular graphics: DIAMOND (Brandenburg & Putz, 2017); software used to prepare material for publication: publCIF (Westrip, 2010).

Octaaluminium pentachromium top
Crystal data top
Al7.85Cr5.16Dx = 4.248 Mg m3
Mr = 479.72Mo Kα radiation, λ = 0.71073 Å
Trigonal, R3m:RCell parameters from 2208 reflections
a = 7.8777 (5) Åθ = 3.2–26.3°
α = 109.566 (2)°µ = 8.05 mm1
V = 375.01 (7) Å3T = 296 K
Z = 2Graininess, metallic silver
F(000) = 4510.13 × 0.06 × 0.05 mm
Data collection top
Bruker D8 Venture Photon 100 CMOS
diffractometer
547 reflections with I > 2σ(I)
φ and ω scansRint = 0.090
Absorption correction: multi-scan
(SADABS; Bruker, 2015)
θmax = 26.8°, θmin = 3.2°
Tmin = 0.496, Tmax = 0.523h = 99
7330 measured reflectionsk = 99
576 independent reflectionsl = 99
Refinement top
Refinement on F237 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) = 0.178Δρmax = 1.02 e Å3
S = 1.16Δρmin = 1.27 e Å3
576 reflectionsAbsolute structure: Refined as an inversion twin.
53 parametersAbsolute structure parameter: 0.3 (2)
Special details top

Refinement. Refined as a two-component inversion twin

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzUiso*/UeqOcc. (<1)
Al10.2998 (14)0.2998 (14)0.0928 (15)0.0108 (19)0.772
Cr10.2998 (14)0.2998 (14)0.0928 (15)0.0108 (19)0.228
Al21.3096 (13)0.6646 (11)0.6646 (11)0.0117 (19)0.5
Cr21.3096 (13)0.6646 (11)0.6646 (11)0.0117 (19)0.5
Al30.9238 (18)0.6488 (14)0.6488 (14)0.012 (2)0.958
Cr30.9238 (18)0.6488 (14)0.6488 (14)0.012 (2)0.042
Cr40.0434 (12)0.0434 (12)0.0434 (12)0.006 (2)
Cr50.6391 (8)0.3060 (8)0.3060 (8)0.0028 (10)
Cr61.3001 (10)0.9464 (8)0.9464 (8)0.0108 (14)
Cr70.5198 (13)0.5198 (13)0.5198 (13)0.008 (2)
Al40.5500 (17)0.0607 (15)0.2804 (16)0.018 (2)
Al50.0366 (18)0.0366 (18)0.313 (2)0.015 (3)
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Al10.007 (3)0.007 (3)0.019 (5)0.005 (3)0.006 (3)0.006 (3)
Cr10.007 (3)0.007 (3)0.019 (5)0.005 (3)0.006 (3)0.006 (3)
Al20.008 (4)0.008 (3)0.008 (3)0.000 (3)0.000 (3)0.001 (3)
Cr20.008 (4)0.008 (3)0.008 (3)0.000 (3)0.000 (3)0.001 (3)
Al30.013 (5)0.010 (3)0.010 (3)0.007 (3)0.007 (3)0.001 (3)
Cr30.013 (5)0.010 (3)0.010 (3)0.007 (3)0.007 (3)0.001 (3)
Cr40.007 (4)0.007 (4)0.007 (4)0.005 (4)0.005 (4)0.005 (4)
Cr50.001 (2)0.0034 (18)0.0034 (18)0.0001 (17)0.0001 (17)0.002 (2)
Cr60.016 (3)0.008 (3)0.008 (3)0.006 (3)0.006 (3)0.003 (3)
Cr70.006 (3)0.006 (3)0.006 (3)0.000 (4)0.000 (4)0.000 (4)
Al40.024 (5)0.021 (4)0.031 (5)0.016 (3)0.021 (4)0.021 (4)
Al50.015 (4)0.015 (4)0.017 (5)0.004 (5)0.011 (4)0.011 (4)
Geometric parameters (Å, º) top
Al1—Cr52.611 (7)Cr2—Al5xi2.930 (16)
Al1—Cr5i2.611 (7)Al3—Cr52.572 (12)
Al1—Cr6ii2.634 (7)Al3—Cr62.606 (11)
Al1—Cr6iii2.634 (7)Al3—Cr4xiii2.655 (12)
Al1—Cr42.644 (11)Al3—Cr72.678 (13)
Al1—Al4i2.654 (11)Al3—Al5xiii2.758 (11)
Al1—Al4iv2.654 (11)Al3—Al5xiv2.758 (11)
Al1—Al52.659 (17)Al3—Al4xii2.798 (12)
Al1—Al1i2.665 (16)Al3—Al4ix2.798 (12)
Al1—Al1v2.665 (16)Cr3—Cr52.572 (12)
Al1—Cr72.739 (11)Cr3—Cr62.606 (11)
Cr1—Cr52.611 (7)Cr3—Cr4xiii2.655 (12)
Cr1—Cr5i2.611 (7)Cr3—Cr72.678 (13)
Cr1—Cr6ii2.634 (7)Cr3—Al5xiii2.758 (11)
Cr1—Cr6iii2.634 (7)Cr3—Al5xiv2.758 (11)
Cr1—Cr42.644 (11)Cr3—Al4xii2.798 (12)
Cr1—Al4i2.654 (11)Cr3—Al4ix2.798 (12)
Cr1—Al4iv2.654 (11)Cr4—Al52.616 (10)
Cr1—Al52.659 (17)Cr4—Al5v2.616 (10)
Cr1—Cr72.739 (11)Cr4—Al5i2.616 (10)
Al2—Cr62.604 (9)Cr4—Cr6iii2.761 (7)
Al2—Cr7vi2.647 (9)Cr4—Cr6ii2.761 (7)
Al2—Al4vii2.656 (9)Cr5—Cr72.603 (7)
Al2—Al4viii2.656 (9)Cr5—Al4xii2.620 (10)
Al2—Cr5ix2.754 (6)Cr5—Al4ix2.620 (10)
Al2—Cr5x2.754 (6)Cr5—Al4iv2.652 (9)
Al2—Al5xi2.930 (16)Cr5—Al42.652 (9)
Al2—Al4xii2.996 (10)Cr5—Al5x2.683 (12)
Al2—Al4ix2.996 (10)Cr6—Al5xi2.654 (12)
Cr2—Cr62.604 (9)Cr6—Al4xv2.739 (11)
Cr2—Cr7vi2.647 (9)Cr6—Al4xiii2.739 (11)
Cr2—Al4vii2.656 (9)Cr6—Al5xiv2.849 (8)
Cr2—Al4viii2.656 (9)Al4—Al5x2.693 (12)
Cr2—Cr5ix2.754 (6)Al4—Al4ix2.726 (5)
Cr2—Cr5x2.754 (6)Al4—Al4xvi2.726 (5)
Cr5—Al1—Cr5i110.4 (5)Cr3—Cr5—Cr762.3 (4)
Cr5—Al1—Cr6ii63.75 (14)Al3—Cr5—Cr762.3 (4)
Cr5i—Al1—Cr6ii167.7 (4)Cr3—Cr5—Al1v116.2 (4)
Cr5—Al1—Cr6iii167.7 (4)Al3—Cr5—Al1v116.2 (4)
Cr5i—Al1—Cr6iii63.75 (14)Cr7—Cr5—Al1v63.4 (3)
Cr6ii—Al1—Cr6iii119.6 (5)Cr7—Cr5—Al163.4 (3)
Cr5—Al1—Cr4112.8 (3)Al1v—Cr5—Al161.4 (4)
Cr5i—Al1—Cr4112.8 (3)Cr7—Cr5—Cr163.4 (3)
Cr6ii—Al1—Cr463.1 (2)Cr3—Cr5—Al4xii65.2 (3)
Cr6iii—Al1—Cr463.1 (2)Al3—Cr5—Al4xii65.2 (3)
Cr5—Al1—Al4i125.5 (4)Cr7—Cr5—Al4xii110.8 (3)
Cr5i—Al1—Al4i60.5 (2)Al1v—Cr5—Al4xii168.9 (3)
Cr6ii—Al1—Al4i131.8 (4)Al1—Cr5—Al4xii107.8 (3)
Cr6iii—Al1—Al4i62.4 (2)Cr1—Cr5—Al4xii107.8 (3)
Cr4—Al1—Al4i120.0 (3)Cr3—Cr5—Al4ix65.2 (3)
Cr5—Al1—Al4iv60.5 (2)Al3—Cr5—Al4ix65.2 (3)
Cr5i—Al1—Al4iv125.5 (4)Cr7—Cr5—Al4ix110.8 (3)
Cr6ii—Al1—Al4iv62.4 (2)Al1v—Cr5—Al4ix107.8 (3)
Cr6iii—Al1—Al4iv131.8 (4)Al1—Cr5—Al4ix168.9 (3)
Cr4—Al1—Al4iv120.0 (3)Cr1—Cr5—Al4ix168.9 (3)
Al4i—Al1—Al4iv81.7 (5)Al4xii—Cr5—Al4ix83.0 (5)
Cr5—Al1—Al5124.0 (2)Cr3—Cr5—Al4iv121.2 (3)
Cr5i—Al1—Al5124.0 (2)Al3—Cr5—Al4iv121.2 (3)
Cr6ii—Al1—Al565.1 (3)Cr7—Cr5—Al4iv115.6 (3)
Cr6iii—Al1—Al565.1 (3)Al1v—Cr5—Al4iv110.7 (4)
Cr4—Al1—Al559.1 (3)Al1—Cr5—Al4iv60.5 (3)
Al4i—Al1—Al576.7 (3)Cr1—Cr5—Al4iv60.5 (3)
Al4iv—Al1—Al576.7 (3)Al4xii—Cr5—Al4iv62.3 (2)
Cr5—Al1—Al1i108.1 (2)Al4ix—Cr5—Al4iv129.1 (2)
Cr5i—Al1—Al1i59.3 (2)Cr3—Cr5—Al4121.2 (3)
Cr6ii—Al1—Al1i111.0 (2)Al3—Cr5—Al4121.2 (3)
Cr6iii—Al1—Al1i59.6 (2)Cr7—Cr5—Al4115.6 (3)
Cr4—Al1—Al1i59.73 (18)Al1v—Cr5—Al460.5 (3)
Al4i—Al1—Al1i109.0 (2)Al1—Cr5—Al4110.7 (4)
Al4iv—Al1—Al1i168.2 (3)Cr1—Cr5—Al4110.7 (4)
Al5—Al1—Al1i110.2 (3)Al4xii—Cr5—Al4129.1 (2)
Cr5—Al1—Al1v59.3 (2)Al4ix—Cr5—Al462.3 (2)
Cr5i—Al1—Al1v108.1 (2)Al4iv—Cr5—Al4111.7 (5)
Cr6ii—Al1—Al1v59.6 (2)Cr3—Cr5—Al5x128.3 (4)
Cr6iii—Al1—Al1v111.0 (2)Al3—Cr5—Al5x128.3 (4)
Cr4—Al1—Al1v59.73 (18)Cr7—Cr5—Al5x169.4 (4)
Al4i—Al1—Al1v168.2 (3)Al1v—Cr5—Al5x107.8 (3)
Al4iv—Al1—Al1v109.0 (2)Al1—Cr5—Al5x107.8 (3)
Al5—Al1—Al1v110.2 (3)Cr1—Cr5—Al5x107.8 (3)
Al1i—Al1—Al1v60.000 (1)Al4xii—Cr5—Al5x76.8 (3)
Cr5—Al1—Cr758.2 (2)Al4ix—Cr5—Al5x76.8 (3)
Cr5i—Al1—Cr758.2 (2)Al4iv—Cr5—Al5x60.6 (3)
Cr6ii—Al1—Cr7111.2 (3)Al4—Cr5—Al5x60.6 (3)
Cr6iii—Al1—Cr7111.2 (3)Cr3—Cr5—Al2xvi64.3 (2)
Cr4—Al1—Cr7110.2 (4)Al3—Cr5—Al2xvi64.3 (2)
Al4i—Al1—Cr7111.1 (3)Cr7—Cr5—Al2xvi59.1 (2)
Al4iv—Al1—Cr7111.1 (3)Al1v—Cr5—Al2xvi60.5 (3)
Al5—Al1—Cr7169.4 (5)Al1—Cr5—Al2xvi110.8 (3)
Al1i—Al1—Cr760.89 (18)Cr1—Cr5—Al2xvi110.8 (3)
Al1v—Al1—Cr760.89 (18)Al4xii—Cr5—Al2xvi125.9 (4)
Cr5—Cr1—Cr5i110.4 (5)Al4ix—Cr5—Al2xvi59.2 (3)
Cr5—Cr1—Cr6ii63.75 (14)Al4iv—Cr5—Al2xvi170.8 (4)
Cr5i—Cr1—Cr6ii167.7 (4)Al4—Cr5—Al2xvi67.3 (2)
Cr5—Cr1—Cr6iii167.7 (4)Al5x—Cr5—Al2xvi123.0 (2)
Cr5i—Cr1—Cr6iii63.75 (14)Cr2—Cr6—Al1xiii149.6 (2)
Cr6ii—Cr1—Cr6iii119.6 (5)Al2—Cr6—Al1xiii149.6 (2)
Cr5—Cr1—Cr4112.8 (3)Cr3—Cr6—Al1xiii108.9 (3)
Cr5i—Cr1—Cr4112.8 (3)Al3—Cr6—Al1xiii108.9 (3)
Cr6ii—Cr1—Cr463.1 (2)Al1xiii—Cr6—Al1xiv60.8 (4)
Cr6iii—Cr1—Cr463.1 (2)Cr2—Cr6—Al5xi67.7 (4)
Cr5—Cr1—Al4i125.5 (4)Al2—Cr6—Al5xi67.7 (4)
Cr5i—Cr1—Al4i60.5 (2)Cr3—Cr6—Al5xi136.9 (4)
Cr6ii—Cr1—Al4i131.8 (4)Al3—Cr6—Al5xi136.9 (4)
Cr6iii—Cr1—Al4i62.4 (2)Al1xiii—Cr6—Al5xi108.0 (4)
Cr4—Cr1—Al4i120.0 (3)Al1xiv—Cr6—Al5xi108.0 (4)
Cr5—Cr1—Al4iv60.5 (2)Al1xiii—Cr6—Al4xv59.2 (3)
Cr5i—Cr1—Al4iv125.5 (4)Al1xiv—Cr6—Al4xv107.4 (4)
Cr6ii—Cr1—Al4iv62.4 (2)Al5xi—Cr6—Al4xv59.9 (3)
Cr6iii—Cr1—Al4iv131.8 (4)Cr2—Cr6—Al4xiii96.4 (3)
Cr4—Cr1—Al4iv120.0 (3)Al2—Cr6—Al4xiii96.4 (3)
Al4i—Cr1—Al4iv81.7 (5)Cr3—Cr6—Al4xiii126.1 (2)
Cr5—Cr1—Al5124.0 (2)Al3—Cr6—Al4xiii126.1 (2)
Cr5i—Cr1—Al5124.0 (2)Al1xiii—Cr6—Al4xiii107.4 (4)
Cr6ii—Cr1—Al565.1 (3)Al1xiv—Cr6—Al4xiii59.2 (3)
Cr6iii—Cr1—Al565.1 (3)Al5xi—Cr6—Al4xiii59.9 (3)
Cr4—Cr1—Al559.1 (3)Al4xv—Cr6—Al4xiii106.5 (5)
Al4i—Cr1—Al576.7 (3)Al1xiii—Cr6—Cr4xiii58.6 (2)
Al4iv—Cr1—Al576.7 (3)Al1xiv—Cr6—Cr4xiii58.6 (2)
Cr5—Cr1—Cr758.2 (2)Al5xi—Cr6—Cr4xiii163.9 (4)
Cr5i—Cr1—Cr758.2 (2)Al4xv—Cr6—Cr4xiii113.1 (3)
Cr6ii—Cr1—Cr7111.2 (3)Al4xiii—Cr6—Cr4xiii113.1 (3)
Cr6iii—Cr1—Cr7111.2 (3)Cr2—Cr6—Cr5xiii127.0 (4)
Cr4—Cr1—Cr7110.2 (4)Al2—Cr6—Cr5xiii127.0 (4)
Al4i—Cr1—Cr7111.1 (3)Cr3—Cr6—Cr5xiii163.9 (3)
Al4iv—Cr1—Cr7111.1 (3)Al3—Cr6—Cr5xiii163.9 (3)
Al5—Cr1—Cr7169.4 (5)Al1xiii—Cr6—Cr5xiii57.7 (2)
Cr6—Al2—Cr7vi150.7 (4)Al1xiv—Cr6—Cr5xiii57.7 (2)
Cr6—Al2—Al4vii71.0 (2)Al5xi—Cr6—Cr5xiii59.2 (3)
Cr7vi—Al2—Al4vii108.3 (2)Al4xv—Cr6—Cr5xiii57.6 (2)
Cr6—Al2—Al4viii71.0 (2)Al4xiii—Cr6—Cr5xiii57.6 (3)
Cr7vi—Al2—Al4viii108.3 (2)Cr4xiii—Cr6—Cr5xiii104.7 (3)
Al4vii—Al2—Al4viii141.3 (4)Cr2—Cr6—Al5xiv99.6 (3)
Cr6—Al2—Cr5ix128.85 (18)Al2—Cr6—Al5xiv99.6 (3)
Cr7vi—Al2—Cr5ix57.6 (2)Cr3—Cr6—Al5xiv60.5 (3)
Al4vii—Al2—Cr5ix159.6 (4)Al3—Cr6—Al5xiv60.5 (3)
Al4viii—Al2—Cr5ix57.9 (2)Al1xiii—Cr6—Al5xiv105.5 (3)
Cr6—Al2—Cr5x128.85 (18)Al1xiv—Cr6—Al5xiv57.8 (3)
Cr7vi—Al2—Cr5x57.6 (2)Al5xi—Cr6—Al5xiv127.5 (2)
Al4vii—Al2—Cr5x57.9 (2)Al4xv—Cr6—Al5xiv164.0 (4)
Al4viii—Al2—Cr5x159.6 (4)Al4xiii—Cr6—Al5xiv72.2 (3)
Cr5ix—Al2—Cr5x102.2 (4)Cr4xiii—Cr6—Al5xiv55.6 (2)
Cr6—Al2—Al5xi56.9 (3)Cr5xiii—Cr6—Al5xiv111.8 (3)
Cr7vi—Al2—Al5xi93.8 (4)Cr5—Cr7—Cr5v110.9 (2)
Al4vii—Al2—Al5xi82.8 (3)Cr5—Cr7—Cr5i110.9 (2)
Al4viii—Al2—Al5xi82.8 (3)Cr5v—Cr7—Cr5i110.9 (2)
Cr5ix—Al2—Al5xi111.2 (3)Cr5—Cr7—Al2xviii166.5 (5)
Cr5x—Al2—Al5xi111.2 (3)Cr5v—Cr7—Al2xviii63.27 (8)
Cr6—Al2—Al4xii99.9 (3)Cr5i—Cr7—Al2xviii63.27 (8)
Cr7vi—Al2—Al4xii103.8 (3)Cr5—Cr7—Al2xix63.27 (8)
Al4vii—Al2—Al4xii57.30 (18)Cr5v—Cr7—Al2xix166.5 (5)
Al4viii—Al2—Al4xii123.5 (4)Cr5i—Cr7—Al2xix63.27 (8)
Cr5ix—Al2—Al4xii108.9 (3)Cr5—Cr7—Al2xvi63.27 (8)
Cr5x—Al2—Al4xii54.7 (2)Cr5v—Cr7—Al2xvi63.27 (8)
Al5xi—Al2—Al4xii139.6 (3)Cr5i—Cr7—Al2xvi166.5 (5)
Cr6—Al2—Al4ix99.9 (3)Al2xviii—Cr7—Al2xvi119.39 (7)
Cr7vi—Al2—Al4ix103.8 (3)Al2xix—Cr7—Al2xvi119.39 (7)
Al4vii—Al2—Al4ix123.5 (4)Cr5—Cr7—Al358.3 (2)
Al4viii—Al2—Al4ix57.30 (18)Cr5v—Cr7—Al3124.25 (17)
Cr5ix—Al2—Al4ix54.7 (2)Cr5i—Cr7—Al3124.25 (17)
Cr5x—Al2—Al4ix108.9 (3)Cr5—Cr7—Cr358.3 (2)
Al5xi—Al2—Al4ix139.6 (3)Cr5v—Cr7—Cr3124.25 (17)
Al4xii—Al2—Al4ix70.8 (4)Cr5i—Cr7—Cr3124.25 (17)
Cr6—Cr2—Cr7vi150.7 (4)Cr5—Cr7—Al3i124.25 (17)
Cr6—Cr2—Al4vii71.0 (2)Cr5v—Cr7—Al3i124.25 (17)
Cr7vi—Cr2—Al4vii108.3 (2)Cr5i—Cr7—Al3i58.3 (2)
Cr6—Cr2—Al4viii71.0 (2)Al2xviii—Cr7—Al3i64.46 (17)
Cr7vi—Cr2—Al4viii108.3 (2)Al2xix—Cr7—Al3i64.46 (17)
Al4vii—Cr2—Al4viii141.3 (4)Al2xvi—Cr7—Al3i135.3 (5)
Cr6—Cr2—Cr5ix128.85 (18)Al3—Cr7—Al3i82.8 (4)
Cr7vi—Cr2—Cr5ix57.6 (2)Cr3—Cr7—Al3i82.8 (4)
Al4vii—Cr2—Cr5ix159.6 (4)Cr5—Cr7—Al3v124.25 (17)
Al4viii—Cr2—Cr5ix57.9 (2)Cr5v—Cr7—Al3v58.3 (2)
Cr6—Cr2—Cr5x128.85 (18)Cr5i—Cr7—Al3v124.25 (17)
Cr7vi—Cr2—Cr5x57.6 (2)Cr5—Cr7—Al1v58.5 (2)
Al4vii—Cr2—Cr5x57.9 (2)Cr5v—Cr7—Al1v58.5 (2)
Al4viii—Cr2—Cr5x159.6 (4)Cr5i—Cr7—Al1v106.2 (4)
Cr5ix—Cr2—Cr5x102.2 (4)Al2xviii—Cr7—Al1v110.2 (3)
Cr6—Cr2—Al5xi56.9 (3)Al2xix—Cr7—Al1v110.2 (3)
Cr7vi—Cr2—Al5xi93.8 (4)Al2xvi—Cr7—Al1v60.3 (3)
Al4vii—Cr2—Al5xi82.8 (3)Al3—Cr7—Al1v108.7 (3)
Al4viii—Cr2—Al5xi82.8 (3)Cr3—Cr7—Al1v108.7 (3)
Cr5ix—Cr2—Al5xi111.2 (3)Al3i—Cr7—Al1v164.4 (4)
Cr5x—Cr2—Al5xi111.2 (3)Al3v—Cr7—Al1v108.7 (3)
Cr5—Al3—Cr6157.4 (4)Cr5—Cr7—Al1i106.2 (4)
Cr5—Al3—Cr4xiii139.3 (5)Cr5v—Cr7—Al1i58.5 (2)
Cr6—Al3—Cr4xiii63.3 (3)Cr5i—Cr7—Al1i58.5 (2)
Cr5—Al3—Cr759.4 (3)Al2xviii—Cr7—Al1i60.3 (3)
Cr6—Al3—Cr7143.2 (4)Al2xix—Cr7—Al1i110.2 (3)
Cr4xiii—Al3—Cr779.9 (4)Al2xvi—Cr7—Al1i110.2 (3)
Cr5—Al3—Al5xiii123.2 (3)Al3—Cr7—Al1i164.4 (4)
Cr6—Al3—Al5xiii64.1 (3)Cr3—Cr7—Al1i164.4 (4)
Cr4xiii—Al3—Al5xiii57.8 (3)Al3i—Cr7—Al1i108.7 (3)
Cr7—Al3—Al5xiii97.1 (3)Al3v—Cr7—Al1i108.7 (3)
Cr5—Al3—Al5xiv123.2 (3)Al1v—Cr7—Al1i58.2 (4)
Cr6—Al3—Al5xiv64.1 (3)Cr5xvi—Al4—Cr5144.8 (4)
Cr4xiii—Al3—Al5xiv57.8 (3)Cr5xvi—Al4—Al1v86.3 (4)
Cr7—Al3—Al5xiv97.1 (3)Cr5—Al4—Al1v59.0 (3)
Al5xiii—Al3—Al5xiv109.5 (6)Cr5xvi—Al4—Al2xx62.9 (2)
Cr5—Al3—Al4xii58.2 (3)Cr5—Al4—Al2xx150.4 (5)
Cr6—Al3—Al4xii105.2 (4)Al1v—Al4—Al2xx148.8 (4)
Cr4xiii—Al3—Al4xii141.1 (3)Cr5xvi—Al4—Al5x145.4 (6)
Cr7—Al3—Al4xii103.4 (4)Cr5—Al4—Al5x60.2 (3)
Al5xiii—Al3—Al4xii83.5 (3)Al1v—Al4—Al5x106.3 (3)
Al5xiv—Al3—Al4xii154.2 (5)Al2xx—Al4—Al5x102.4 (4)
Cr5—Al3—Al4ix58.2 (3)Cr5xvi—Al4—Al4ix134.5 (5)
Cr6—Al3—Al4ix105.2 (4)Cr5—Al4—Al4ix58.3 (4)
Cr4xiii—Al3—Al4ix141.1 (3)Al1v—Al4—Al4ix103.5 (4)
Cr7—Al3—Al4ix103.4 (4)Al2xx—Al4—Al4ix95.5 (4)
Al5xiii—Al3—Al4ix154.2 (5)Al5x—Al4—Al4ix74.9 (5)
Al5xiv—Al3—Al4ix83.5 (3)Cr5xvi—Al4—Al4xvi59.4 (3)
Al4xii—Al3—Al4ix76.6 (5)Cr5—Al4—Al4xvi128.9 (4)
Cr5—Cr3—Cr6157.4 (4)Al1v—Al4—Al4xvi102.1 (5)
Cr5—Cr3—Cr4xiii139.3 (5)Al2xx—Al4—Al4xvi67.6 (4)
Cr6—Cr3—Cr4xiii63.3 (3)Al5x—Al4—Al4xvi86.1 (5)
Cr5—Cr3—Cr759.4 (3)Al4ix—Al4—Al4xvi151.5 (6)
Cr6—Cr3—Cr7143.2 (4)Cr5xvi—Al4—Cr6ii106.6 (5)
Cr4xiii—Cr3—Cr779.9 (4)Cr5—Al4—Cr6ii61.8 (2)
Cr5—Cr3—Al5xiii123.2 (3)Al1v—Al4—Cr6ii58.4 (3)
Cr6—Cr3—Al5xiii64.1 (3)Al2xx—Al4—Cr6ii132.5 (4)
Cr4xiii—Cr3—Al5xiii57.8 (3)Al5x—Al4—Cr6ii58.5 (3)
Cr7—Cr3—Al5xiii97.1 (3)Al4ix—Al4—Cr6ii116.5 (5)
Cr5—Cr3—Al5xiv123.2 (3)Al4xvi—Al4—Cr6ii68.0 (3)
Cr6—Cr3—Al5xiv64.1 (3)Cr5xvi—Al4—Al3xvi56.6 (3)
Cr4xiii—Cr3—Al5xiv57.8 (3)Cr5—Al4—Al3xvi118.0 (5)
Cr7—Cr3—Al5xiv97.1 (3)Al1v—Al4—Al3xvi97.2 (4)
Al5xiii—Cr3—Al5xiv109.5 (6)Al2xx—Al4—Al3xvi62.7 (3)
Cr5—Cr3—Al4xii58.2 (3)Al5x—Al4—Al3xvi147.5 (5)
Cr6—Cr3—Al4xii105.2 (4)Al4ix—Al4—Al3xvi78.0 (4)
Cr4xiii—Cr3—Al4xii141.1 (3)Al4xvi—Al4—Al3xvi110.9 (4)
Cr7—Cr3—Al4xii103.4 (4)Cr6ii—Al4—Al3xvi153.0 (5)
Al5xiii—Cr3—Al4xii83.5 (3)Cr5xvi—Al4—Al2xvi99.8 (4)
Al5xiv—Cr3—Al4xii154.2 (5)Cr5—Al4—Al2xvi58.0 (2)
Cr5—Cr3—Al4ix58.2 (3)Al1v—Al4—Al2xvi56.9 (3)
Cr6—Cr3—Al4ix105.2 (4)Al2xx—Al4—Al2xvi120.5 (4)
Cr4xiii—Cr3—Al4ix141.1 (3)Al5x—Al4—Al2xvi114.1 (4)
Cr7—Cr3—Al4ix103.4 (4)Al4ix—Al4—Al2xvi55.1 (2)
Al5xiii—Cr3—Al4ix154.2 (5)Al4xvi—Al4—Al2xvi153.3 (5)
Al5xiv—Cr3—Al4ix83.5 (3)Cr6ii—Al4—Al2xvi106.8 (3)
Al4xii—Cr3—Al4ix76.6 (5)Al3xvi—Al4—Al2xvi61.3 (4)
Al5—Cr4—Al5v118.81 (14)Cr4—Al5—Cr6xxi152.4 (6)
Al5—Cr4—Al5i118.81 (14)Cr4—Al5—Cr160.2 (4)
Al5v—Cr4—Al5i118.81 (14)Cr6xxi—Al5—Cr1147.5 (5)
Al5—Cr4—Cr160.7 (4)Cr4—Al5—Al160.2 (4)
Al5v—Cr4—Cr1112.2 (3)Cr6xxi—Al5—Al1147.5 (5)
Al5i—Cr4—Cr1112.2 (3)Cr4—Al5—Cr5xix145.1 (6)
Al5—Cr4—Al160.7 (4)Cr6xxi—Al5—Cr5xix62.5 (3)
Al5v—Cr4—Al1112.2 (3)Cr1—Al5—Cr5xix84.9 (5)
Al5i—Cr4—Al1112.2 (3)Al1—Al5—Cr5xix84.9 (5)
Al5—Cr4—Al1i112.2 (3)Cr4—Al5—Al4xxii125.1 (3)
Al5v—Cr4—Al1i112.2 (3)Cr6xxi—Al5—Al4xxii61.6 (3)
Al5i—Cr4—Al1i60.7 (4)Al1—Al5—Al4xxii102.9 (4)
Cr1—Cr4—Al1i60.5 (4)Cr5xix—Al5—Al4xxii59.1 (3)
Al1—Cr4—Al1i60.5 (4)Cr4—Al5—Al4xix125.1 (3)
Al5—Cr4—Al1v112.2 (3)Cr6xxi—Al5—Al4xix61.6 (3)
Al5v—Cr4—Al1v60.7 (4)Cr1—Al5—Al4xix102.9 (4)
Al5i—Cr4—Al1v112.2 (3)Al1—Al5—Al4xix102.9 (4)
Cr1—Cr4—Al1v60.5 (4)Cr5xix—Al5—Al4xix59.1 (3)
Al1—Cr4—Al1v60.5 (4)Al4xxii—Al5—Al4xix109.2 (5)
Al1i—Cr4—Al1v60.5 (4)Cr4—Al5—Al3iii59.1 (3)
Al5—Cr4—Al3iii63.1 (2)Cr6xxi—Al5—Al3iii101.0 (4)
Al5v—Cr4—Al3iii134.0 (6)Cr1—Al5—Al3iii103.8 (4)
Al5i—Cr4—Al3iii63.1 (2)Al1—Al5—Al3iii103.8 (4)
Cr1—Cr4—Al3iii107.2 (3)Cr5xix—Al5—Al3iii138.3 (3)
Al1—Cr4—Al3iii107.2 (3)Al4xxii—Al5—Al3iii149.3 (6)
Al1i—Cr4—Al3iii107.2 (3)Al4xix—Al5—Al3iii79.2 (3)
Al1v—Cr4—Al3iii165.2 (4)Cr4—Al5—Al3ii59.1 (3)
Al5—Cr4—Al3xvii134.0 (6)Cr6xxi—Al5—Al3ii101.0 (4)
Al5v—Cr4—Al3xvii63.1 (2)Cr1—Al5—Al3ii103.8 (4)
Al5i—Cr4—Al3xvii63.1 (2)Al1—Al5—Al3ii103.8 (4)
Cr1—Cr4—Al3xvii165.2 (4)Cr5xix—Al5—Al3ii138.3 (3)
Al1—Cr4—Al3xvii165.2 (4)Al4xxii—Al5—Al3ii79.2 (3)
Al1i—Cr4—Al3xvii107.2 (3)Al4xix—Al5—Al3ii149.3 (6)
Al1v—Cr4—Al3xvii107.2 (3)Al3iii—Al5—Al3ii79.9 (6)
Al3iii—Cr4—Al3xvii83.6 (4)Cr4—Al5—Cr6ii60.5 (3)
Al5—Cr4—Al3ii63.1 (2)Cr6xxi—Al5—Cr6ii126.6 (2)
Al5v—Cr4—Al3ii63.1 (2)Cr1—Al5—Cr6ii57.0 (2)
Al5i—Cr4—Al3ii134.0 (6)Al1—Al5—Cr6ii57.0 (2)
Cr1—Cr4—Al3ii107.2 (3)Cr5xix—Al5—Cr6ii101.9 (3)
Al1—Cr4—Al3ii107.2 (3)Al4xxii—Al5—Cr6ii66.8 (2)
Al1i—Cr4—Al3ii165.2 (4)Al4xix—Al5—Cr6ii155.4 (6)
Al1v—Cr4—Al3ii107.2 (3)Al3iii—Al5—Cr6ii117.2 (4)
Al3iii—Cr4—Al3ii83.6 (4)Al3ii—Al5—Cr6ii55.4 (2)
Al3xvii—Cr4—Al3ii83.6 (4)Cr4—Al5—Cr6iii60.5 (3)
Al5—Cr4—Cr6iii63.93 (12)Cr6xxi—Al5—Cr6iii126.6 (2)
Al5v—Cr4—Cr6iii168.5 (6)Cr1—Al5—Cr6iii57.0 (2)
Al5i—Cr4—Cr6iii63.93 (12)Al1—Al5—Cr6iii57.0 (2)
Cr1—Cr4—Cr6iii58.27 (18)Cr5xix—Al5—Cr6iii101.9 (3)
Al1—Cr4—Cr6iii58.27 (18)Al4xxii—Al5—Cr6iii155.4 (6)
Al1i—Cr4—Cr6iii58.27 (18)Al4xix—Al5—Cr6iii66.8 (2)
Al1v—Cr4—Cr6iii107.8 (4)Al3iii—Al5—Cr6iii55.4 (2)
Al3iii—Cr4—Cr6iii57.5 (3)Al3ii—Al5—Cr6iii117.2 (4)
Al3xvii—Cr4—Cr6iii124.18 (16)Cr6ii—Al5—Cr6iii106.1 (4)
Al3ii—Cr4—Cr6iii124.18 (16)Cr4—Al5—Al2xxi97.0 (4)
Al5—Cr4—Cr6ii63.93 (12)Cr6xxi—Al5—Al2xxi55.3 (3)
Al5v—Cr4—Cr6ii63.93 (12)Cr1—Al5—Al2xxi157.2 (5)
Al5i—Cr4—Cr6ii168.5 (6)Al1—Al5—Al2xxi157.2 (5)
Cr1—Cr4—Cr6ii58.27 (18)Cr5xix—Al5—Al2xxi117.9 (4)
Al1—Cr4—Cr6ii58.27 (18)Al4xxii—Al5—Al2xxi90.1 (4)
Al1i—Cr4—Cr6ii107.8 (4)Al4xix—Al5—Al2xxi90.1 (4)
Al1v—Cr4—Cr6ii58.27 (18)Al3iii—Al5—Al2xxi59.8 (3)
Al3iii—Cr4—Cr6ii124.18 (16)Al3ii—Al5—Al2xxi59.8 (3)
Al3xvii—Cr4—Cr6ii124.18 (16)Cr6ii—Al5—Al2xxi113.8 (3)
Al3ii—Cr4—Cr6ii57.5 (3)Cr6iii—Al5—Al2xxi113.8 (3)
Cr6iii—Cr4—Cr6ii111.1 (2)
Symmetry codes: (i) z, x, y; (ii) x1, y1, z1; (iii) z1, x1, y1; (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, x1, y; (xvii) y1, z1, x1; (xviii) x1, y, z; (xix) y, z, x1; (xx) y, z1, x1; (xxi) y1, z1, x2; (xxii) z, y, x1.
 

Funding information

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).

References

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