Cr5B3 with the Shastry–Sutherland lattices

Tokuda, M.; Yubuta, K.; Shishido, T.; Sugiyama, K.

doi:10.1107/S2056989025006437

research communications

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ISSN: 2056-9890

Volume 81| Part 8| August 2025| Pages 748-752

https://doi.org/10.1107/S2056989025006437

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Cr₅B₃ with the Shastry–Sutherland lattices

Makoto Tokuda,^a ^* Kunio Yubuta,^b Toetsu Shishido ^c and Kazumasa Sugiyama ^c

^aInstitute of Industrial Nano materials, Kumamoto University, 2-39-1 Kurokami, Chuo-ku, Kumamoto 860-8555, Japan, ^bInstitute for Aqua Regeneration, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan, and ^cInstitute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
^*Correspondence e-mail: [email protected]

Edited by Y. Ozawa, University of Hyogo, Japan (Received 14 May 2025; accepted 18 July 2025; online 29 July 2025)

The structural parameters of pentachromium triboride, Cr₅B₃, with Shastry–Sutherland lattices were refined based on single-crystal X-ray diffraction data. Cr₅B₃ crystallizes in the space group I4/mcm (No. 140), with the following lattice parameters: a = 5.4728 (1) and c = 10.0794 (2) Å. The present study succeeded in refining the positional and anisotropic atomic displacement parameters of the Cr and B atoms.

Keywords: single-crystal diffraction; crystal structure; boride.

CCDC reference: 2473809

1. Chemical context

Several intermetallic compounds, such as CrB₄ (orthorhombic, Immm), CrB₂ (hexagonal, P6/mmm), Cr₃B₄ (orthorhombic, Immm), CrB (orthorhombic, Cmcm), Cr₅B₃ (tetragonal, I4/mcm), t-Cr₂B (tetragonal, I4/mcm) and o-Cr₂B (orthorhombic, Fddd), exist in the binary Cr–B system (Lundström, 1969 ; Guy & Uraz, 1976 ; Massalski et al., 2016 ). These binary chromium borides have attracted research attention as high-strength materials with excellent thermal, corrosion, and wear resistance. In addition to synthesis techniques (Okada et al., 1996 ; Iizumi et al., 1998 ), their applications in industrial fields have been also developed.

Notable magnetic (Guy, 1976 ; Leyarovska et al., 1979 ), thermal (Leyarovska et al., 1979) and transportation (Cruceanu et al., 1975 ) properties have been reported in more than 130 intermetallic compounds with a Cr₅B₃ prototype (ICSD, 2025 ). Antiferromagnetic ordering occurs in Cr₅B₃ at T_N = 91.5 K (Leyarovska et al., 1979). The magnetic susceptibility and specific heat measurements of Cr₅B₃ suggest that its magnetic behavior originates from antiferromagnetic spin fluctuations in an itinerant electron system, rather than from localized magnetic moments (Leyarovska et al., 1979). The observed effective magnetic moment of 0.02 μ_B per Cr atom indicates a low-spin state with significant d-electron delocalization. These findings imply that the formal oxidation state of Cr lies between +2 and +3. In the Cr₅B₃-type structure, metal atoms locate the two non-equivalent crystallographic sites. A two-dimensional square lattice formed by one of these metal sites can host magnetically frustrated orthogonal dimer systems. This square lattice is classified the Sharstry–Sutherland lattice (SSL; Shastry & Sutherland, 1981 ; Kageyama et al., 1999 ; Siemensmeyer et al., 2008 ; Coleman & Nevidomskyy, 2010 ). In particular, Ce₅Si₃, which has the Cr₅B₃-type structure, features a bilayer system of the SSL layers formed by Ce atoms, with additional interactions from out-of-plane Ce atoms contributing to its complex magnetic properties. (Ueta et al., 2024 ).

The first structural investigation of Cr₅B₃ was conducted using a single crystalline sample (Bertaut & Blum, 1953 ). Both Cr and B atoms in the tetragonal Cr₅B₃ occupy two different Wyckoff sites, namely, 4c (0, 0, 0) and 16l (x, x + $[{1\over 2}]$ , z) denoted as the Cr2 and Cr1 sites, respectively, and 4a (0, 0, 1/4) and 8h (x, x + $[{1\over 2}]$ , 0) denoted as the B1 and B2 sites, respectively. Subsequent structural studies on Cr₅B₃ have been limited to determining the lattice parameters, and the structural parameters have remained unchanged for over half a century (Portnoi et al., 1969 ; von Robitsch, 1974 ; Paradelli & Gianoglio, 1976 ; Hu et al., 2014 ). Furthermore, regarding the atomic displacement parameters (ADPs) Bertaut & Blum (1953) did not determine them. Therefore, refining the anisotropic ADPs is essential for a more accurate structural description. Authors have reported that ADPs provide useful information about the structural properties of several metal boride compounds, such as YCrB₄ (Tokuda et al., 2022 ) and RERh₃B₂ (RE = Pr, Nd, and Sm) (Tokuda et al., 2023 ). In this study, we update the structural parameters of Cr₅B₃ and perform a refinement, including the anisotropic ADPs.

Cr₅B₃-type (called T₂-phase) analogous compounds have been studied using various experimental and theoretical methods, because they have many derivatives of the form M₅XB₂ (M = metal, X = non-metal or semi-metal such as Si, P, and Ge; Dahlqvist & Rosen, 2022 ). The structures of M₅XB₂ and Cr₅B₃ are related to the order–disorder atomic arrangements, in which M₅XB₂ and X atoms solely occupy the 4a site. Recently, several compounds of the form M₅XB₂ have been reported to exhibit superconductivity, such as Ta₅GeB₂ (T_c ∼3.8 K; Hadi et al., 2016 ; Corrêa et al., 2016 ), Mo₅GeB₂ (T_c = 5.8 K; Ruan et al., 2021 ), Mo₅SiB₂ (T_c = 5.8 K; Machado et al., 2011 ), Mo₅PB₂ (T_c = 9 K; McGuire & Parker, 2016 ), and (W, Ta)₅SiB₂ (T_c = 6.5 K; Fukuma et al., 2012 ).

2. Structural commentary

In the Cr₅B₃ structure, the Cr atoms at the Cr2 and Cr1 sites are surrounded by a Cr₈B₆ rhombic dodecahedron and a Cr₁₁B₅ 16-vertex Frank–Kasper cluster, respectively, and those at the B1 and B2 sites are surrounded by a Cr₁₀ bicapped square antiprism and Cr₈B tricapped trigonal prism, respectively (Fig. 1).

Figure 1
(left) Crystal structures of Cr₅B₃. The purple and green displacement ellipsoids correspond to Cr and B atoms, respectively. Displacement ellipsoids are drawn at the 99% probability level. (right) Coordination polyhedra for each site, Cr₁₁B₅ 16-vertex Frank–Kasper cluster around Cr1 on 16l site (z = 0.15), Cr₈B₆ rhombic dodecahedron around Cr2 on 4c site (z = 0), Cr₁₀ bicapped square antiprism around B1 on 4a site (z = 1/4), and Cr₈B tricapped trigonal prism around B2 on 8h sites (z = 0).

The B2—B2 interatomic distances on the z = 0 and 1/2 plane is 1.8168 (16) Å, which is significantly longer than the average B—B covalent bond distances of 1.77 Å in rhombohedral boron (Donohue, 1974 ). This B2 pair (B dimer) serves as a bridging unit between two adjacent Cr–B tricapped trigonal prisms, with each boron atom occupying the center of a respective polyhedron. The Cr—B interatomic distances are in the range of 2.1803 (3)–2.2826 (1) Å (Table 1), which are close to the sum of the Goldschmidt radii (r_Cr = 1.36 Å and r_B = 0.97 Å; Brandes & Brook, 1992 ). The intraplane Cr2⋯Cr2 and intraplane Cr2⋯Cr1 distances are 3.8698 (1) and 2.5072 (1) Å, respectively. The interplane Cr2⋯Cr1 distance is significantly smaller than the sum of the radii of the Cr atoms, and the anisotropic atomic displacement parameters (ADPs) for Cr2 exhibit a larger anisotropy than those of Cr1 (Table 2). The U₃₃ of Cr2 is approximately 1.75 times larger than U₁₁ (= U₂₂), indicating that the displacement ellipsoid of Cr2 is elongated along the c-axis direction. These Cr⋯Cr distances and ADPs of Cr2 indicate a strong correlation between the Cr2 and Cr1 atoms.

Table 1
Selected bond lengths (Å) in Cr₅B₃

Cr1:Cr₁₁B₅ 16-vertex Frank–Kasper
Cr1—B2×2	2.1803 (3)
Cr1—B2×1	2.2015 (6)
Cr1—B1×2	2.2826 (1)
Cr1—Cr2×1	2.4218 (4)
Cr1—Cr2×2	2.5072 (2)
Cr1—Cr1×1	2.6513 (3)
Cr1—Cr1×2	2.8098 (1)
Cr1—Cr1×4	2.8688 (1)
Cr1—Cr1×1	2.9468 (5)

Cr2:Cr₈B₆ rhombic dodecahedron
Cr2—B2×4	2.1903 (4)
Cr2—Cr1×8	2.5072 (1)
Cr2—B1×2	2.5199 (1)

B1:Cr₁₀ bicapped square antiprism
B1—Cr1×8	2.2826 (1)
B1—Cr2×2	2.5199 (1)

B2:Cr₈B tricapped trigonal prism
B2—B2×1	1.8168 (16)
B2—Cr1×4	2.1802 (3)
B2—Cr2×2	2.1903 (4)
B2—Cr1×2	2.2015 (6)

Table 2
Atomic coordinates and anisotropic atomic displacement parameters (10 ³ Å²) for Cr₅B₃

The Cr atoms lie on the Wyckoff site, the 4c site (0, 0, 0) and the 16l site (x, x + $[{1\over 2}]$ , z), and the B atoms occupy the 4a site (0, 0, 1/4) and the 8h site (x, x + $[{1\over 2}]$ , 0). The anisotropic displacement factor exponent takes the form 2π²[(ha*)²U₁₁ + ⋯ + 2hka*b*U₁₂]. U_eq is defined as a third of the trace of the orthogonalized U_ij tensor; U₁₁ = U₂₂, U₁₃ = U₂₃.

Atom	x	z	U₁₁	U₃₃	U₁₂	U₁₃	U_eq
Cr1	0.17128 (2)	0.14618 (2)	0.00396 (2)	0.00369 (4)	−0.00001 (1)	−0.00022 (1)	0.00285 (3)
Cr2	0.0	0.0	0.00312 (6)	0.00547 (4)	0.0	0.0	0.00391 (2)
B1	0.0	0.25	0.00544 (17)	0.0066 (3)	0.0	0.0	0.00583 (12)
B2	0.38263 (10)	0.0	0.00489 (12)	0.0050 (4)	−0.00013 (15)	0.0	0.00492 (8)

The bonding configurations of boron in metal boride compounds (MxBy) are classified with the metal-to-boron ratio (M:B), and their structural characteristics have been systematically investigated (Rogl & Nowotny, 1978 ). In metal-rich compositions M/B > 1.5, boron typically exists as isolated B or a B–B dimer, occupying localized positions within the metal network. As the metal-to-boron ratio decreases, the bonding motifs transform progressively into mono-periodic chains (e.g., Cr₂AlB₂-type), di-periodic boron layers (e.g., CrB₂-type), and eventually into three-dimensional frameworks constructed from B₆ octahedra or B₁₂ icosahedra (e.g. UB₄-type). This structural diversity plays a crucial role in determining the electronic structure and physical properties of borides. In particular, Cr₅B₃, with a M:B of 5:3 (1.67), is categorized as a metal-rich boride and the boron configurations B1 and B2 sites correspond to ‘isolated B' and ‘B–B dimer', respectively. The slabs of Cr₈ square antiprisms [Cr⋯Cr interatomic distances of 2.42178 (14) and 2.86881 (3) Å] together with the B1 site and Cr square lattice [Cr⋯Cr interatomic distance of 2.50718 (5) Å] with B2 dimers [B2⋯B2 interatomic distance of 1.8168 (16) Å] alternately stack along the c-axis as shown in Fig. 2. The slabs of Cr₈ square antiprisms together with the B1 site connect via edge-sharing. Unlike boron-rich borides, the boron unit, Cr₅B₃, are locally confined, reflecting a characteristic intermetallic bonding framework in which boron can act predominantly as an electron donor.

Figure 2
(left) Cr square lattices with B dimers at z = 0 and z = 1/2, and (right) slabs of Cr₈B square antiprisms around the B1 site between z = 0.15 and z = 0.35.

Fig. 3 depicts the crystal structures of the binary metal borides Cr₅B₃ (a, b) and TmB₄ (c, d) viewed along the c-axis (a, c) and b-axis (b, d). The purple, green and blue spheres indicate the Cr, B, and Tm atoms, respectively. The gray squares in (a, c, e, f, g) are the respective unit cells. Cr lattices [Cr⋯Cr interatomic distance of 2.6513 (1) and 2.8688 (1) Å] at z = 0.15 (Fig. 3e) and z = 0.35 (Fig. 3f) in the slabs of Cr₈ square antiprisms around B1 site of Cr₅B₃ (in the right panel of Fig. 2) are illustrated. The distribution of Cr are clearly demonstrated by square and triangle tilting. This tilting geometrical feature, composed of squares and triangles, is found in the TmB₄ phase (Tm–Tm interatomic distances of 3.635 and 3.729 Å; Fisk et al., 1972 ) (Fig. 3g) belonging to the UB₄-type structure (tetragonal, P4/mbm). Both the Cr₅B₃ and TmB₄ structures feature two-dimensional magnetic layers resembling the Shastry–Sutherland lattice (SSL), characterized by orthogonal dimers and interdimer interactions. However, while the SSL network in TmB₄ is isolated within the structure, the SSL in Cr₅B₃ are embedded in a three-dimensional framework, making the magnetic frustration more complex and less idealized. This tiling structure is known as the SSL (Shastry & Sutherland, 1981). A schematic of the SSL is shown in Fig. 3h. The first-nearest-neighbor (1NN) and second-nearest-neighbor (2NN) pairs are denoted by the gray and black lines, respectively. Frustrated magnetic behavior with magnetic order is expected in Cr₅B₃-type analogous intermetallic compounds consisting of SSL.

Figure 3
Crystal structures of binary metal borides: Cr₅B₃ (a, b) and TmB₄ (c, d) viewed along the c-axis (a, c) and b-axis (b, d) directions. Red, green and blue spheres indicate Cr, B and Tm atoms. Gray squares in (a, c, e, f, g) are respective unit cells. Cr lattices at z = 0.15 (e) and z = 0.35 (f) in Cr₅B₃. (g) Tm lattice at z = 0 in TmB₄. (h) Schematic of the SSL. The 1NN and the 2NN pairs are denoted by the solid and dashed lines, respectively.

3. Synthesis and crystallization

Cr₅B₃ exhibits incongruent melting behavior during a peritectic reaction at 2247 K (Massalski et al., 2016). Cr₅B₃ was obtained as a by-product of the synthesis of YCrB₄ crystals. The starting materials were Y (99.9%), Cr (99.95%), and B (99.5%). The samples were weighed at an atomic ratio Y:Cr:B = 1:7:4. The mixtures were then melted in an Ar arc melting furnace (ACM-01, Diavac). The resulting button-like product was then turned over and remelted thrice to improve its chemical homogeneity. Single crystals for the X-ray diffraction measurements were obtained from the fractured surface of this button-like product.

4. Refinement

The refinement process was conducted for the space-group type I4/mcm as described by Bertaut & Blum 1953. A correction for isotropic extinction was applied during the least-squares refinement. The final refinements were performed by applying the anisotropic ADPs to each atom. These final refinement results are listed in Table 3. The refinement was successful, with the R factor converging without any problems and no noticeable residuals.

Table 3
Experimental details

Crystal data
Chemical formula	Cr₅B₃
M_r	292.43
Crystal system, space group	Tetragonal, I4/mcm
Temperature (K)	294
a, c (Å)	5.47276 (5), 10.07939 (16)
V (Å³)	301.89 (1)
Z	4
Radiation type	Mo Kα
μ (mm⁻¹)	17.06
Crystal size (mm)	0.06 × 0.04 × 0.03

Data collection
Diffractometer	XtaLAB Synergy, Dualflex, HyPix
Absorption correction	Gaussian (CrysAlis PRO; Rigaku OD, 2019 )
T_min, T_max	0.520, 0.737
No. of measured, independent and observed [I > 2σ(I)] reflections	14650, 726, 687
R_int	0.032
(sin θ/λ)_max (Å⁻¹)	1.273

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.011, 0.024, 1.10
No. of reflections	726
No. of parameters	16
Δρ_max, Δρ_min (e Å⁻³)	0.61, −0.79
Computer programs: CrysAlis PRO (Rigaku OD, 2019), SHELXT (Sheldrick, 2015a ), SHELXL2016/6 (Sheldrick, 2015b ) and VESTA (Momma & Izumi, 2011 ).

Supporting information

CCDC reference: 2473809

Crystal structure: contains datablock I. DOI: https://doi.org/10.1107/S2056989025006437/ox2016sup1.cif

Structure factors: contains datablock I. DOI: https://doi.org/10.1107/S2056989025006437/ox2016Isup3.hkl

checkCIF report

Computing details top

Pentachromium triboride top

Crystal data top

Cr₅B₃	D_x = 6.434 Mg m⁻³
M_r = 292.43	Mo Kα radiation, λ = 0.71073 Å
Tetragonal, I4/mcm	Cell parameters from 8812 reflections
a = 5.47276 (5) Å	θ = 4.0–65.0°
c = 10.07939 (16) Å	µ = 17.06 mm⁻¹
V = 301.89 (1) Å³	T = 294 K
Z = 4	Block, metallic
F(000) = 540	0.06 × 0.04 × 0.03 mm

Data collection top

XtaLAB Synergy, Dualflex, HyPix diffractometer	726 independent reflections
Radiation source: micro-focus sealed X-ray tube, PhotonJet (Mo) X-ray Source	687 reflections with I > 2σ(I)
Mirror monochromator	R_int = 0.032
Detector resolution: 10.0000 pixels mm^-1	θ_max = 64.8°, θ_min = 4.0°
ω scans	h = −13→12
Absorption correction: gaussian (CrysAlisPro; Rigaku OD, 2019)	k = −12→13
T_min = 0.520, T_max = 0.737	l = −25→25
14650 measured reflections

Refinement top

Refinement on F²	0 restraints
Least-squares matrix: full	w = 1/[σ²(F_o²) + (0.0099P)² + 0.1133P] where P = (F_o² + 2F_c²)/3
R[F² > 2σ(F²)] = 0.011	(Δ/σ)_max = 0.001
wR(F²) = 0.024	Δρ_max = 0.61 e Å⁻³
S = 1.10	Δρ_min = −0.79 e Å⁻³
726 reflections	Extinction correction: SHELXL2016/6 (Sheldrick, 2015b), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
16 parameters	Extinction coefficient: 0.0092 (5)

Special details top

Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds involving l.s. planes.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Cr01	0.17128 (2)	0.67128 (2)	0.14618 (2)	0.00387 (2)
Cr02	0.000000	0.000000	0.000000	0.00391 (2)
B01	0.000000	0.000000	0.250000	0.00583 (12)
B02	0.38263 (10)	0.88263 (10)	0.000000	0.00492 (8)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Cr01	0.00396 (2)	0.00396 (2)	0.00369 (2)	−0.00001 (1)	−0.00022 (1)	−0.00022 (1)
Cr02	0.00312 (2)	0.00312 (2)	0.00547 (4)	0.000	0.000	0.000
B01	0.00544 (17)	0.00544 (17)	0.0066 (3)	0.000	0.000	0.000
B02	0.00489 (12)	0.00489 (12)	0.0050 (2)	−0.00013 (15)	0.000	0.000

Geometric parameters (Å, º) top

Cr01—B02ⁱ	2.1803 (3)	Cr01—Cr01^viii	2.8098 (1)
Cr01—B02ⁱⁱ	2.1803 (3)	Cr01—Cr01^ix	2.8098 (1)
Cr01—B02	2.2015 (6)	Cr01—Cr01^x	2.8688 (1)
Cr01—B01ⁱⁱⁱ	2.2826 (1)	Cr02—B02^xi	2.1903 (4)
Cr01—B01^iv	2.2826 (1)	Cr02—B02ⁱ	2.1903 (4)
Cr01—Cr01^v	2.4218 (1)	Cr02—B02^xii	2.1903 (4)
Cr01—Cr02^vi	2.5072 (1)	Cr02—B02^xiii	2.1903 (4)
Cr01—Cr02^iv	2.5072 (1)	B02—B02^xiv	1.8168 (16)
Cr01—Cr01^vii	2.6513 (1)

B02ⁱ—Cr01—B02ⁱⁱ	49.25 (4)	Cr01ⁱ—Cr02—Cr01^xvi	71.983 (3)
B02ⁱ—Cr01—B02	89.970 (13)	Cr01^vii—Cr02—Cr01^xvi	110.203 (2)
B02ⁱⁱ—Cr01—B02	89.970 (13)	Cr01^xv—Cr02—Cr01^xvi	69.797 (2)
B02ⁱ—Cr01—B01ⁱⁱⁱ	96.830 (19)	Cr01^xiii—Cr02—Cr01^xvi	110.203 (2)
B02ⁱⁱ—Cr01—B01ⁱⁱⁱ	145.619 (18)	B02^xi—Cr02—Cr01^xii	125.194 (13)
B02—Cr01—B01ⁱⁱⁱ	96.230 (7)	B02ⁱ—Cr02—Cr01^xii	54.806 (13)
B02ⁱ—Cr01—B01^iv	145.619 (18)	B02^xii—Cr02—Cr01^xii	55.399 (13)
B02ⁱⁱ—Cr01—B01^iv	96.830 (18)	B02^xiii—Cr02—Cr01^xii	124.601 (13)
B02—Cr01—B01^iv	96.230 (7)	Cr01^xi—Cr02—Cr01^xii	69.797 (2)
B01ⁱⁱⁱ—Cr01—B01^iv	115.923 (3)	Cr01ⁱ—Cr02—Cr01^xii	110.203 (2)
B02ⁱ—Cr01—Cr01^v	152.867 (17)	Cr01^vii—Cr02—Cr01^xii	71.983 (3)
B02ⁱⁱ—Cr01—Cr01^v	152.867 (17)	Cr01^xv—Cr02—Cr01^xii	108.017 (3)
B02—Cr01—Cr01^v	101.803 (14)	Cr01^xiii—Cr02—Cr01^xii	180.0
B01ⁱⁱⁱ—Cr01—Cr01^v	57.961 (2)	Cr01^xvi—Cr02—Cr01^xii	69.797 (2)
B01^iv—Cr01—Cr01^v	57.961 (1)	B02^xi—Cr02—Cr01^xvii	55.399 (13)
B02ⁱ—Cr01—Cr02^vi	55.184 (13)	B02ⁱ—Cr02—Cr01^xvii	124.601 (13)
B02ⁱⁱ—Cr01—Cr02^vi	94.138 (15)	B02^xii—Cr02—Cr01^xvii	54.806 (13)
B02—Cr01—Cr02^vi	54.980 (5)	B02^xiii—Cr02—Cr01^xvii	125.194 (13)
B01ⁱⁱⁱ—Cr01—Cr02^vi	63.279 (1)	Cr01^xi—Cr02—Cr01^xvii	71.983 (3)
B01^iv—Cr01—Cr02^vi	149.178 (3)	Cr01ⁱ—Cr02—Cr01^xvii	108.017 (3)
Cr01^v—Cr01—Cr02^vi	112.681 (3)	Cr01^vii—Cr02—Cr01^xvii	69.797 (2)
B02ⁱ—Cr01—Cr02^iv	94.138 (15)	Cr01^xv—Cr02—Cr01^xvii	110.203 (2)
B02ⁱⁱ—Cr01—Cr02^iv	55.184 (13)	Cr01^xiii—Cr02—Cr01^xvii	69.797 (2)
B02—Cr01—Cr02^iv	54.980 (5)	Cr01^xvi—Cr02—Cr01^xvii	180.0
B01ⁱⁱⁱ—Cr01—Cr02^iv	149.178 (3)	Cr01^xii—Cr02—Cr01^xvii	110.203 (2)
B01^iv—Cr01—Cr02^iv	63.279 (1)	Cr01^xvii—B01—Cr01^viii	149.038 (3)
Cr01^v—Cr01—Cr02^iv	112.681 (3)	Cr01^xvii—B01—Cr01^xiii	77.867 (1)
Cr02^vi—Cr01—Cr02^iv	101.022 (3)	Cr01^viii—B01—Cr01^xiii	131.507 (3)
B02ⁱ—Cr01—Cr01^vii	52.552 (7)	Cr01^xvii—B01—Cr01ⁱⁱⁱ	131.507 (3)
B02ⁱⁱ—Cr01—Cr01^vii	52.552 (7)	Cr01^viii—B01—Cr01ⁱⁱⁱ	77.867 (2)
B02—Cr01—Cr01^vii	137.988 (14)	Cr01^xiii—B01—Cr01ⁱⁱⁱ	64.078 (3)
B01ⁱⁱⁱ—Cr01—Cr01^vii	105.481 (2)	Cr01^xvii—B01—Cr01^xviii	64.078 (3)
B01^iv—Cr01—Cr01^vii	105.481 (2)	Cr01^viii—B01—Cr01^xviii	125.425 (3)
Cr01^v—Cr01—Cr01^vii	120.209 (3)	Cr01^xiii—B01—Cr01^xviii	75.975 (4)
Cr02^vi—Cr01—Cr01^vii	104.063 (2)	Cr01ⁱⁱⁱ—B01—Cr01^xviii	77.867 (1)
Cr02^iv—Cr01—Cr01^vii	104.063 (2)	Cr01^xvii—B01—Cr01ⁱ	125.425 (3)
B02ⁱ—Cr01—Cr01^viii	91.142 (10)	Cr01^viii—B01—Cr01ⁱ	64.078 (3)
B02ⁱⁱ—Cr01—Cr01^viii	114.398 (7)	Cr01^xiii—B01—Cr01ⁱ	77.867 (2)
B02—Cr01—Cr01^viii	148.111 (6)	Cr01ⁱⁱⁱ—B01—Cr01ⁱ	75.975 (4)
B01ⁱⁱⁱ—Cr01—Cr01^viii	52.013 (2)	Cr01^xviii—B01—Cr01ⁱ	149.038 (3)
B01^iv—Cr01—Cr01^viii	100.625 (3)	Cr01^xvii—B01—Cr01^xix	75.975 (4)
Cr01^v—Cr01—Cr01^viii	66.026 (4)	Cr01^viii—B01—Cr01^xix	77.867 (1)
Cr02^vi—Cr01—Cr01^viii	100.852 (2)	Cr01^xiii—B01—Cr01^xix	149.038 (3)
Cr02^iv—Cr01—Cr01^viii	156.417 (2)	Cr01ⁱⁱⁱ—B01—Cr01^xix	125.425 (3)
Cr01^vii—Cr01—Cr01^viii	61.849 (1)	Cr01^xviii—B01—Cr01^xix	77.867 (1)
B02ⁱ—Cr01—Cr01^ix	114.398 (7)	Cr01ⁱ—B01—Cr01^xix	131.507 (3)
B02ⁱⁱ—Cr01—Cr01^ix	91.142 (10)	Cr01^xvii—B01—Cr01^vii	77.867 (2)
B02—Cr01—Cr01^ix	148.111 (6)	Cr01^viii—B01—Cr01^vii	75.975 (4)
B01ⁱⁱⁱ—Cr01—Cr01^ix	100.625 (3)	Cr01^xiii—B01—Cr01^vii	125.425 (3)
B01^iv—Cr01—Cr01^ix	52.013 (2)	Cr01ⁱⁱⁱ—B01—Cr01^vii	149.038 (3)
Cr01^v—Cr01—Cr01^ix	66.026 (4)	Cr01^xviii—B01—Cr01^vii	131.507 (3)
Cr02^vi—Cr01—Cr01^ix	156.417 (2)	Cr01ⁱ—B01—Cr01^vii	77.867 (2)
Cr02^iv—Cr01—Cr01^ix	100.852 (2)	Cr01^xix—B01—Cr01^vii	64.078 (3)
Cr01^vii—Cr01—Cr01^ix	61.849 (2)	Cr01^xvii—B01—Cr02^xx	117.287 (2)
Cr01^viii—Cr01—Cr01^ix	56.302 (3)	Cr01^viii—B01—Cr02^xx	62.713 (2)
B02ⁱ—Cr01—Cr01^x	110.285 (13)	Cr01^xiii—B01—Cr02^xx	117.287 (2)
B02ⁱⁱ—Cr01—Cr01^x	137.034 (5)	Cr01ⁱⁱⁱ—B01—Cr02^xx	62.713 (2)
B02—Cr01—Cr01^x	48.782 (11)	Cr01^xviii—B01—Cr02^xx	62.713 (2)
B01ⁱⁱⁱ—Cr01—Cr01^x	51.067 (1)	Cr01ⁱ—B01—Cr02^xx	117.287 (2)
B01^iv—Cr01—Cr01^x	98.917 (3)	Cr01^xix—B01—Cr02^xx	62.713 (2)
Cr01^v—Cr01—Cr01^x	63.500 (3)	Cr01^vii—B01—Cr02^xx	117.287 (2)
Cr02^vi—Cr01—Cr01^x	55.102 (1)	Cr01^xvii—B01—Cr02	62.713 (2)
Cr02^iv—Cr01—Cr01^x	98.112 (3)	Cr01^viii—B01—Cr02	117.287 (2)
Cr01^vii—Cr01—Cr01^x	152.478 (2)	Cr01^xiii—B01—Cr02	62.713 (2)
Cr01^viii—Cr01—Cr01^x	101.560 (1)	Cr01ⁱⁱⁱ—B01—Cr02	117.287 (2)
Cr01^ix—Cr01—Cr01^x	129.526 (3)	Cr01^xviii—B01—Cr02	117.287 (2)
B02^xi—Cr02—B02ⁱ	180.0	Cr01ⁱ—B01—Cr02	62.713 (2)
B02^xi—Cr02—B02^xii	90.0	Cr01^xix—B01—Cr02	117.287 (2)
B02ⁱ—Cr02—B02^xii	90.0	Cr01^vii—B01—Cr02	62.713 (2)
B02^xi—Cr02—B02^xiii	90.0	Cr02^xx—B01—Cr02	180.0
B02ⁱ—Cr02—B02^xiii	90.0	B02^xiv—B02—Cr01^x	65.376 (19)
B02^xii—Cr02—B02^xiii	180.00 (3)	B02^xiv—B02—Cr01^xxi	65.376 (19)
B02^xi—Cr02—Cr01^xi	55.399 (13)	Cr01^x—B02—Cr01^xxi	74.894 (14)
B02ⁱ—Cr02—Cr01^xi	124.601 (13)	B02^xiv—B02—Cr01^xxii	65.376 (19)
B02^xii—Cr02—Cr01^xi	54.806 (13)	Cr01^x—B02—Cr01^xxii	130.75 (4)
B02^xiii—Cr02—Cr01^xi	125.194 (13)	Cr01^xxi—B02—Cr01^xxii	85.031 (16)
B02^xi—Cr02—Cr01ⁱ	124.601 (13)	B02^xiv—B02—Cr01^xxiii	65.376 (19)
B02ⁱ—Cr02—Cr01ⁱ	55.399 (13)	Cr01^x—B02—Cr01^xxiii	85.031 (16)
B02^xii—Cr02—Cr01ⁱ	125.194 (13)	Cr01^xxi—B02—Cr01^xxiii	130.75 (4)
B02^xiii—Cr02—Cr01ⁱ	54.806 (13)	Cr01^xxii—B02—Cr01^xxiii	74.894 (14)
Cr01^xi—Cr02—Cr01ⁱ	180.0	B02^xiv—B02—Cr02^iv	117.946 (18)
B02^xi—Cr02—Cr01^vii	125.194 (13)	Cr01^x—B02—Cr02^iv	137.087 (3)
B02ⁱ—Cr02—Cr01^vii	54.806 (13)	Cr01^xxi—B02—Cr02^iv	70.010 (2)
B02^xii—Cr02—Cr01^vii	55.399 (13)	Cr01^xxii—B02—Cr02^iv	70.010 (2)
B02^xiii—Cr02—Cr01^vii	124.601 (13)	Cr01^xxiii—B02—Cr02^iv	137.087 (3)
Cr01^xi—Cr02—Cr01^vii	110.203 (2)	B02^xiv—B02—Cr02^vi	117.946 (18)
Cr01ⁱ—Cr02—Cr01^vii	69.797 (2)	Cr01^x—B02—Cr02^vi	70.010 (2)
B02^xi—Cr02—Cr01^xv	54.806 (13)	Cr01^xxi—B02—Cr02^vi	137.087 (3)
B02ⁱ—Cr02—Cr01^xv	125.194 (13)	Cr01^xxii—B02—Cr02^vi	137.087 (3)
B02^xii—Cr02—Cr01^xv	124.601 (13)	Cr01^xxiii—B02—Cr02^vi	70.010 (2)
B02^xiii—Cr02—Cr01^xv	55.399 (13)	Cr02^iv—B02—Cr02^vi	124.11 (4)
Cr01^xi—Cr02—Cr01^xv	69.797 (2)	B02^xiv—B02—Cr01	137.989 (14)
Cr01ⁱ—Cr02—Cr01^xv	110.203 (2)	Cr01^x—B02—Cr01	81.796 (6)
Cr01^vii—Cr02—Cr01^xv	180.0	Cr01^xxi—B02—Cr01	81.796 (6)
B02^xi—Cr02—Cr01^xiii	54.806 (13)	Cr01^xxii—B02—Cr01	139.630 (18)
B02ⁱ—Cr02—Cr01^xiii	125.194 (13)	Cr01^xxiii—B02—Cr01	139.630 (18)
B02^xii—Cr02—Cr01^xiii	124.601 (13)	Cr02^iv—B02—Cr01	69.622 (17)
B02^xiii—Cr02—Cr01^xiii	55.399 (13)	Cr02^vi—B02—Cr01	69.622 (17)
Cr01^xi—Cr02—Cr01^xiii	110.203 (2)	B02^xiv—B02—Cr01^xxiv	137.989 (14)
Cr01ⁱ—Cr02—Cr01^xiii	69.797 (2)	Cr01^x—B02—Cr01^xxiv	139.630 (18)
Cr01^vii—Cr02—Cr01^xiii	108.017 (3)	Cr01^xxi—B02—Cr01^xxiv	139.630 (18)
Cr01^xv—Cr02—Cr01^xiii	71.983 (3)	Cr01^xxii—B02—Cr01^xxiv	81.796 (6)
B02^xi—Cr02—Cr01^xvi	124.601 (13)	Cr01^xxiii—B02—Cr01^xxiv	81.796 (6)
B02ⁱ—Cr02—Cr01^xvi	55.399 (13)	Cr02^iv—B02—Cr01^xxiv	69.622 (17)
B02^xii—Cr02—Cr01^xvi	125.194 (13)	Cr02^vi—B02—Cr01^xxiv	69.622 (17)
B02^xiii—Cr02—Cr01^xvi	54.806 (13)	Cr01—B02—Cr01^xxiv	84.02 (3)
Cr01^xi—Cr02—Cr01^xvi	108.017 (3)

Symmetry codes: (i) −y+1, x, z; (ii) y−1, −x+1, −z; (iii) −x+1/2, −y+1/2, −z+1/2; (iv) x, y+1, z; (v) −x+1/2, −y+3/2, −z+1/2; (vi) −x+1/2, y+1/2, −z; (vii) −x, −y+1, z; (viii) y−1/2, −x+1/2, −z+1/2; (ix) −y+1/2, x+1/2, −z+1/2; (x) y, −x+1, z; (xi) y−1, −x, −z; (xii) −x, −y+1, −z; (xiii) x, y−1, z; (xiv) −x+1, −y+2, −z; (xv) x, y−1, −z; (xvi) −y+1, x, −z; (xvii) y−1, −x, z; (xviii) −y+1/2, x−1/2, −z+1/2; (xix) x−1/2, y−1/2, −z+1/2; (xx) −x, y, −z+1/2; (xxi) −y+1, x+1, z; (xxii) −y+1, x+1, −z; (xxiii) y, −x+1, −z; (xxiv) x, y, −z.

Selected bond lengths (Å) in Cr₅B₃ top

Cr1:Cr₁₁B₅ 16-vertex Frank–Kasper
Cr1—B2×2	2.1803 (3)
Cr1—B2×1	2.2015 (6)
Cr1—B1×2	2.2826 (1)
Cr1—Cr2×1	2.4218 (4)
Cr1—Cr2×2	2.5072 (2)
Cr1—Cr1×1	2.6513 (3)
Cr1—Cr1×2	2.8098 (1)
Cr1—Cr1×4	2.8688 (1)
Cr1—Cr1×1	2.9468 (5)

Cr2:Cr₈B₆ rhombic dodecahedron
Cr2—B2×4	2.1903 (4)
Cr2—Cr1×8	2.5072 (1)
Cr2—B1×2	2.5199 (1)

B1:Cr₁₀ bicapped square antiprism
B1—Cr1×8	2.2826 (1)
B1—Cr2×2	2.5199 (1)

B2:Cr₈B tricapped trigonal prism
B2—B2×1	1.8168 (16)
B2—Cr1×4	2.1802 (3)
B2—Cr2×2	2.1903 (4)
B2—Cr1×2	2.2015 (6)

Atomic coordinates and anisotropic atomic displacement parameters (10 ³ Å²) for Cr₅B₃ top

The Cr atoms lie on the Wyckoff site, the 4c site (0, 0, 0) and the 16l site (x, x+1/2, z), and the B atoms occupy the 4a site (0, 0, 1/4) and the 8h site (x, x + 1/2, 0). The anisotropic displacement factor exponent takes the form 2π²[(ha*)²U₁₁ + ··· + 2hka*b*U₁₂]. U_eq is defined as a third of the trace of the orthogonalized U_ij tensor; U₁₁ = U₂₂, U₁₃ = U₂₃.

Atom	x	z	U₁₁	U₃₃	U₁₂	U₁₃	U_eq
Cr1	0.17128 (2)	0.14618 (2)	0.00396 (2)	0.00369 (4)	-0.00001 (1)	-0.00022 (1)	0.00285 (3)
Cr2	0.0	0.0	0.00312 (6)	0.00547 (4)	0.0	0.0	0.00391 (2)
B1	0.0	0.25	0.00544 (17)	0.0066 (3)	0.0	0.0	0.00583 (12)
B2	0.38263 (10)	0.0	0.00489 (12)	0.0050 (4)	-0.00013 (15)	0.0	0.00492 (8)

Funding information

We gratefully acknowledge the support from JSPS KAKENHI (grant Nos. JP19K05643, JP20H00189 and JP23K04373) and the GIMRT program (Nos. 202111-RDKGE-0002, 202211-RDKGE-0008, and 202311-RDKGE-0001) at the Institute for Materials Research, Tohoku University, Japan.

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