Redetermination of the crystal structure of R5Si4 (R = Pr, Nd) from single-crystal X-ray diffraction data

Yokota, K.; Watanuki, R.; Nakashima, M.; Uehara, M.; Gouchi, J.; Uwatoko, Y.; Umehara, I.

doi:10.1107/S2056989020002789

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Volume 76| Part 4| April 2020| Pages 510-513

https://doi.org/10.1107/S2056989020002789

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Redetermination of the crystal structure of R₅Si₄ (R = Pr, Nd) from single-crystal X-ray diffraction data

Kaori Yokota,^a Ryuta Watanuki,^b ^* Miki Nakashima,^a Masatomo Uehara,^a Jun Gouchi,^c Yoshiya Uwatoko ^c and Izuru Umehara ^a

^aDepartment of Physics, Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan, ^bDepartment of Chemistry, Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan, and ^cThe Institute for Solid State Physics, The University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa, Chiba 277-8581, Japan
^*Correspondence e-mail: watanuki-ryuta-sm@ynu.ac.jp

Edited by P. Roussel, ENSCL, France (Received 20 January 2020; accepted 28 February 2020; online 10 March 2020)

The crystal structures of praseodymium silicide (5/4), Pr₅Si₄, and neodymium silicide (5/4), Nd₅Si₄, were redetermined using high-quality single-crystal X-ray diffraction data. The previous structure reports of Pr₅Si₄ were only based on powder X-ray diffraction data [Smith et al. (1967 ). Acta Cryst. 22 940–943; Yang et al. (2002b ). J. Alloys Compd. 339, 189–194; Yang et al., (2003 ). J. Alloys Compd. 263, 146–153]. On the other hand, the structure of Nd₅Si₄ has been determined from powder data [neutron; Cadogan et al., (2002 ). J. Phys. Condens. Matter, 14, 7191–7200] and X-ray [Smith et al. (1967). Acta Cryst. 22 940–943; Yang et al. (2002b). J. Alloys Compd. 339, 189–194; Yang et al., (2003). J. Alloys Compd. 263, 146–153] and single-crystal data with isotropic atomic displacement parameters [Roger et al., (2006 ). J. Alloys Compd. 415, 73–84]. In addition, the anisotropic atomic displacement parameters for all atomic sites have been determined for the first time. These compounds are confirmed to have the tetragonal Zr₅Si₄-type structure (space group: P4₁2₁2), as reported previously (Smith et al., 1967). The structure is built up by distorted body-centered cubes consisting of Pr(Nd) atoms, which are linked to each other by edge-sharing to form a three-dimensional framework. This framework delimits zigzag channels in which the silicon dimers are situated.

Keywords: crystal structure; rare-earth silicide; chiral structure; chiral magnet; single-crystal growth; X-ray diffraction.

CCDC references: 1974014; 1974013

1. Chemical context

In natural science, there are some essential concepts concerned with symmetry, among which chiral symmetry is one of the fundamentals in all fields of physics, especially magnetism in solid-state materials. A chiral magnet in solids is of great interest in both science and technology. These magnets have been studied for novel phenomena such as chiral magnetic soliton lattices and use in future spintronic devices such as magnetic memories and logic gates. The critical point is that the crystal-structure chirality affects the arrangement of magnetic moments in these materials. The symmetry of crystals plays an important role in the spatial arrangement of the magnetic moments. For example, the intermetallic compound YbNi₃Al₉ has a trigonal ErNi₃Al₉-type structure in space group R32, a member of the Sohncke group (Gladyshevskii et al., 1993 ). This compound exhibits a characteristic helical magnetic structure, reflecting the symmetry of the crystal (Aoki et al., 2018 ). To study magnetism for chiral symmetry, we focused on the intermetallic compound R₅Si₄ (R = Pr and Nd), which has a tetragonal Zr₅Si₄-type crystal structure in the chiral space group P4₁2₁2 (Smith et al., 1967).

Roger et al. (2006) isolated a small single crystal of Nd₅Si₄ by crushing the solidified sample and collected single-crystal X-ray data. Very recently, Sato et al. (2018 ) reported the single-crystal growth and magnetic properties of Ce₅Si₄, which has the same crystal structure as Pr₅Si₄ and Nd₅Si₄. At present, there has only been a report of large-size single-crystal growth for R = Ce, and there are no reports of a large single crystal having been grown successfully for R = Pr or Nd. In particular, for Pr₅Si₄, the crystal-structure analysis is based only on powder XRD data (Yang et al., 2002a ,b,c , 2003; Cadogan et al., 2002; Smith et al., 1967). It is still unknown, however, whether there is a relationship between chiral symmetry and electronic properties, including magnetic ones. In this paper, we report the details of crystallographic studies of single-crystal X-ray analysis of high-quality single-crystalline Pr₅Si₄ and Nd₅Si₄, which are expected to be candidate materials for chiral magnets.

2. Structural commentary

The crystal structures of Pr₅Si₄ and Nd₅Si₄ refined in this study are essentially the same as those determined previously, belonging to chiral space group P4₁2₁2 (No. 92) for R = La, Ce, and Nd (Yang et al., 2002a; Sato et al., 2018). The asymmetric unit of these compounds consists of three Pr (Nd) and two Si atoms. The Pr1(Nd1) atom occupies the Wyckoff 4a site, and the Pr2 (Nd2), Pr3 (Nd3), Si1 and Si2 are located on the general position 8b sites. The principal units in the crystal structures of Pr₅Si₄ and Nd₅Si₄ are illustrated in Fig. 1, and selected bond lengths are given in Tables 1 and 2. The Pr1(Nd1) coordination environment in these compounds can be described as a distorted cube with four Pr2 (Nd2) and four Pr3 (Nd3) [Pr1—Pr2 and Pr1—Pr3 bond lengths ranging from 3.4914 (4) to 3.6423 (3) Å, Pr2—Pr3 bond lengths in the range 3.9156 (3) to 4.0074 (2) Å, Nd1—Nd2 and Nd1—Nd3 bond lengths of 3.4725 (5)–3.6265 (3) Å and Nd2—Nd3 bond lengths of 3.9094 (4)–3.9752 (2) Å]. In addition, the Pr1(Nd1)—Si bonds protruding through the distorted rectangular faces formed by two Pr2 (Nd2) and two Pr3 (Nd3) atoms have Pr1—Si bond lengths ranging from 3.0985 (13) to 3.1780 (13) Å and Nd1—Si bond lengths from 3.0744 (15) to 3.1661 (16) Å. The distorted cubes are connected through common two Pr2—Pr3 (Nd2—Nd3) edges, and Si1 (Si2) atoms form dimers with Si2 (Si1) atoms in the adjacent unit (Fig. 2). The Si1—Si2 bond length in Pr₅Si₄ is 2.4738 (16) Å, and that of Nd₅Si₄ is 2.482 (2) Å. The extended structure is shown in polyhedral representation in Fig. 3. The structure is built up by distorted body-centered cubes consisting of Pr (Nd) atoms, which are linked to each other by edge-sharing to form a three-dimensional framework. This framework delimits zigzag channels oriented along the [100] and [010] directions, in which the Si–Si dimers are situated.

Table 1
Selected bond lengths (Å) for Pr₅Si₄

Pr1—Pr2ⁱ	3.4914 (4)	Pr1—Si1^ix	3.1756 (13)
Pr1—Pr2ⁱⁱ	3.5319 (4)	Pr1—Si2ⁱⁱ	3.1780 (13)
Pr1—Pr2ⁱⁱⁱ	3.5319 (4)	Pr1—Si2ⁱⁱⁱ	3.1780 (13)
Pr1—Pr2^iv	3.4914 (4)	Pr2—Pr2ⁱ	3.9561 (6)
Pr1—Pr3^v	3.6423 (3)	Pr2—Pr3^vii	3.9414 (4)
Pr1—Pr3^vi	3.6423 (3)	Pr2—Pr3^x	3.9717 (3)
Pr1—Si1^vii	3.1756 (13)	Pr2—Pr3^xi	3.9156 (3)
Pr1—Si1^viii	3.0985 (13)	Pr3—Pr3ⁱⁱ	4.0074 (2)
Pr1—Si1	3.0985 (13)	Si1—Si2	2.4738 (16)
Symmetry codes: (i) −y + 1, −x + 1, −z + $[{1\over 2}]$ ; (ii) −y + $[{3\over 2}]$ , x + $[{1\over 2}]$ , z + $[{1\over 4}]$ ; (iii) x + $[{1\over 2}]$ , −y + $[{3\over 2}]$ , −z + $[{3\over 4}]$ ; (iv) −x + 1, −y + 1, z + $[{1\over 2}]$ ; (v) −y + 2, −x + 1, −z + $[{1\over 2}]$ ; (vi) −x + 1, −y + 2, z + $[{1\over 2}]$ ; (vii) −y + $[{3\over 2}]$ , x − $[{1\over 2}]$ , z + $[{1\over 4}]$ ; (viii) y, x, −z + 1; (ix) x − $[{1\over 2}]$ , −y + $[{3\over 2}]$ , −z + $[{3\over 4}]$ .

Table 2
Selected bond lengths (Å) for Nd₅Si₄

Nd1—Nd2ⁱ	3.4725 (5)	Nd1—Si1^ix	3.1528 (16)
Nd1—Nd2ⁱⁱ	3.5021 (5)	Nd1—Si2ⁱⁱ	3.1661 (16)
Nd1—Nd2ⁱⁱⁱ	3.5021 (5)	Nd1—Si2ⁱⁱⁱ	3.1661 (16)
Nd1—Nd2^iv	3.4725 (5)	Nd2—Nd2ⁱ	3.9202 (7)
Nd1—Nd3^v	3.6265 (3)	Nd2—Nd3^x	3.9094 (4)
Nd1—Nd3^vi	3.6265 (3)	Nd2—Nd3^xi	3.9378 (4)
Nd1—Si1^vii	3.1528 (16)	Nd2—Nd3^vii	3.9061 (4)
Nd1—Si1^viii	3.0744 (15)	Nd3—Nd3^xii	3.9752 (2)
Nd1—Si1	3.0744 (15)	Si1—Si2	2.482 (2)
Symmetry codes: (i) −y + 1, −x + 1, −z + $[{1\over 2}]$ ; (ii) −y + $[{3\over 2}]$ , x + $[{1\over 2}]$ , z + $[{1\over 4}]$ ; (iii) x + $[{1\over 2}]$ , −y + $[{3\over 2}]$ , −z + $[{3\over 4}]$ ; (iv) −x + 1, −y + 1, z + $[{1\over 2}]$ ; (v) −y + 2, −x + 1, −z + $[{1\over 2}]$ ; (vi) −x + 1, −y + 2, z + $[{1\over 2}]$ ; (vii) −y + $[{3\over 2}]$ , x − $[{1\over 2}]$ , z + $[{1\over 4}]$ ; (viii) y, x, −z + 1; (ix) x − $[{1\over 2}]$ , −y + $[{3\over 2}]$ , −z + $[{3\over 4}]$ .

Figure 1
Principal units in the structure of (a) Pr₅Si₄ and (b) Nd₅Si₄, illustrated using VESTA (Momma & Izumi, 2011

). Displacement ellipsoids are drawn at the 90% probability level. Symmetry codes: (i) −y + 1, −x + 1, −z + $[{1\over 2}]$ ; (ii) −y + $[{3\over 2}]$ , x + $[{1\over 2}]$ , z + $[{1\over 4}]$ ; (iii) x + $[{1\over 2}]$ , −y + $[{3\over 2}]$ , −z + $[{3\over 4}]$ ; (iv) −x + 1, −y + 1, z + $[{1\over 2}]$ ; (v) −y + 2, −x + 1, −z + $[{1\over 2}]$ ; (vi) −x + 1, −y + 2, z + $[{1\over 2}]$ ; (vii) −y + $[{3\over 2}]$ , x − $[{1\over 2}]$ , z + $[{1\over 4}]$ ; (viii) y, x, −z + 1; (ix) x − $[{1\over 2}]$ , −y + $[{3\over 2}]$ , −z + $[{3\over 4}]$ .

Figure 2
Parts of the crystal structure showing five distorted body-centered cubes sharing Pr2—Pr3 edges (polyhedral drawing). Si1 and Si2 atoms form dimers with atoms Si2 and Si1, respectively, of the adjacent unit.

Figure 3
Polyhedral representation of the crystal structure of Pr₅Si₄ showing the Si–Si dimers situated in zigzag channels running along the [100] and [010] directions.

3. Synthesis and crystallization

We have succeeded in growing single-crystalline samples of Pr₅Si₄ for the first time. For Nd₅Si₄, Roger et al. (2006) obtained a very small single crystal, but we have succeeded in growing a large single crystal. These compounds are incongruently melting compounds (Shukla et al., 2009 ), so we synthesized source materials with the non-stoichiometric molar ratio of Pr (Nd):Si of 58:42 in a mono-arc furnace. Each melted button of source materials was turned over and remelted three times to ensure homogeneity. Single crystals of Pr₅Si₄ and Nd₅Si₄ were grown by the Czochralski pulling method in a tetra arc furnace in an argon atmosphere on a water-cooled copper hearth. A tungsten rod was used as a pulling axis with no seed crystal, and after optimizing the initial conditions of the growth, the crystal was pulled at a constant rate of 12 mm hour⁻¹. The sizes of the grown ingots were about 30 mm in length and 5 mm in diameter. The grown single-crystal samples were characterized by powder X-ray diffraction using a Rigaku MiniFlexII diffractometer with Cu Kα radiation. The powder X-ray diffraction peaks can be well indexed based on the tetragonal Zr₅Si₄-type structure. In addition, it has been confirmed that the whole grown crystal is a single grain crystal by means of the back-reflection Laue method.

4. Database survey

A survey of the Inorganic Crystal Structure Database (ICSD; Belsky et al., 2002 ) for Pr₅Si₄ yielded three hits. In all three, it is reported that Pr₅Si₄ has a Zr₅Si₄-type structure (Smith et al., 1967; ICSD 649362; Yang et al., 2002b; ICSD 95099; Yang et al., 2003; ICSD 98352). On the other hand, for Nd₅Si₄, previous reports have shown that Nd₅Si₄ has two types of crystal structure, a Sm₅Ge₄ type (Raman, 1968 ; ICSD 645983; Roger et al., 2006; ICSD 154658 and 154659) and a Zr₅Si₄-type structure (Smith et al., 1967; ICSD 645939; Mokra et al., 1978 ; ICSD 645946; Eremenko et al., 1984 ; ICSD 600990; Yang et al., 2002a; ICSD 94987; Yang et al., 2002c; ICSD 190404; Cadogan et al., 2002; ICSD 190404). Roger et al. (2006) reported that Sm₅Ge₄-type Nd₅Si₄ could be obtained only with the addition of a tiny amount of boron of less than three at.% in the initial mixture, and that when synthesized with Nd and Si alone, Zr₅Si₄-type Nd₅Si₄ was obtained.

5. Refinement

Crystal data, data collection, and structure refinement details are summarized in Table 3. The highest and deepest remaining difference electron density features are located at 0.90 Å from Pr2 and 1.08 Å from Pr3 for Pr₅Si₄, and 0.74 Å from Nd1 and 1.38 Å from Nd2 for Nd₅Si₄. The absolute structures of the samples were well-defined in space group P4₁2₁2 (No. 92), although the bulk samples possibly also contain the other enantiomer; space group P4₃2₁2 (No. 96).

Table 3
Experimental details

	Pr₅Si₄	Nd₅Si₄
Crystal data
M_r	816.91	833.56
Crystal system, space group	Tetragonal, P4₁2₁2	Tetragonal, P4₁2₁2
Temperature (K)	223	223
a, c (Å)	7.9001 (2), 14.9568 (6)	7.8644 (2), 14.8085 (5)
V (Å³)	933.48 (6)	915.89 (6)
Z	4	4
Radiation type	Mo Kα	Mo Kα
μ (mm⁻¹)	26.03	28.27
Crystal size (mm)	0.13 × 0.08 × 0.03	0.12 × 0.09 × 0.07

Data collection
Diffractometer	XtaLAB AFC12 (RINC): Kappa dual offset/far	XtaLAB AFC12 (RINC): Kappa dual offset/far
Absorption correction	Analytical [CrysAlis PRO (Rigaku OD, 2019 $[Rigaku OD (2019). CrysAlis PRO. Oxford Diffraction, Yarnton, England.]$ ) based on Clark & Reid (1995 )]	Analytical [CrysAlis PRO (Rigaku OD, 2019 $[Rigaku OD (2019). CrysAlis PRO. Oxford Diffraction, Yarnton, England.]$ ) based on Clark & Reid (1995)]
T_min, T_max	0.588, 0.830	0.561, 0.702
No. of measured, independent and observed [I > 2σ(I)] reflections	5553, 1260, 1225	6054, 1238, 1203
R_int	0.025	0.032
(sin θ/λ)_max (Å⁻¹)	0.710	0.708

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.013, 0.026, 1.10	0.015, 0.028, 1.09
No. of reflections	1260	1238
No. of parameters	43	43
Δρ_max, Δρ_min (e Å⁻³)	0.69, −0.85	1.10, −0.71
Absolute structure	Flack x determined using 431 quotients [(I⁺)−(I⁻)]/[(I⁺)+(I⁻)] (Parsons et al., 2013 )	Flack x determined using 422 quotients [(I⁺)−(I⁻)]/[(I⁺)+(I⁻)] (Parsons et al., 2013)
Absolute structure parameter	−0.04 (2)	−0.01 (3)
Computer programs: CrysAlis PRO (Rigaku OD, 2019 $[Rigaku OD (2019). CrysAlis PRO. Oxford Diffraction, Yarnton, England.]$ ), SHELXT (Sheldrick, 2015a ), SHELXL (Sheldrick, 2015b ), OLEX2 (Dolomanov et al., 2009 ) and publCIF (Westrip, 2010 ).

Supporting information

CCDC references: 1974014; 1974013

Crystal structure: contains datablocks global, A, B. DOI: https://doi.org/10.1107/S2056989020002789/ru2068sup1.cif

Structure factors: contains datablock A. DOI: https://doi.org/10.1107/S2056989020002789/ru2068Asup2.hkl

Structure factors: contains datablock B. DOI: https://doi.org/10.1107/S2056989020002789/ru2068Bsup3.hkl

checkCIF report

Computing details top

For both structures, data collection: CrysAlis PRO (Rigaku OD, 2019); cell refinement: CrysAlis PRO (Rigaku OD, 2019); data reduction: CrysAlis PRO (Rigaku OD, 2019); program(s) used to solve structure: SHELXT (Sheldrick, 2015a); program(s) used to refine structure: SHELXL (Sheldrick, 2015b); molecular graphics: OLEX2 (Dolomanov et al., 2009); software used to prepare material for publication: publCIF (Westrip, 2010).

Pentapraseodymium tetrasiliside (A) top

Crystal data top

Pr₅Si₄	D_x = 5.813 Mg m⁻³
M_r = 816.91	Mo Kα radiation, λ = 0.71073 Å
Tetragonal, P4₁2₁2	Cell parameters from 4009 reflections
Hall symbol: P 4abw 2nw	θ = 3.8–30.3°
a = 7.9001 (2) Å	µ = 26.03 mm⁻¹
c = 14.9568 (6) Å	T = 223 K
V = 933.48 (6) Å³	Plate, metallic gray
Z = 4	0.13 × 0.08 × 0.03 mm
F(000) = 1404

Data collection top

XtaLAB AFC12 (RINC): Kappa dual offset/far diffractometer	1260 independent reflections
Radiation source: micro-focus sealed X-ray tube	1225 reflections with I > 2σ(I)
Mirror monochromator	R_int = 0.025
Detector resolution: 5.8140 pixels mm^-1	θ_max = 30.3°, θ_min = 3.7°
ω scans	h = −11→11
Absorption correction: analytical [CrysAlisPro (Rigaku OD, 2019) based on Clark & Reid (1995)]	k = −10→10
T_min = 0.588, T_max = 0.830	l = −19→17
5553 measured reflections

Refinement top

Refinement on F²	w = 1/[σ²(F_o²) + (0.0078P)² + 0.0556P] where P = (F_o² + 2F_c²)/3
Least-squares matrix: full	(Δ/σ)_max = 0.001
R[F² > 2σ(F²)] = 0.013	Δρ_max = 0.69 e Å⁻³
wR(F²) = 0.026	Δρ_min = −0.85 e Å⁻³
S = 1.10	Extinction correction: SHELXL (Sheldrick, 2015b), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
1260 reflections	Extinction coefficient: 0.00247 (9)
43 parameters	Absolute structure: Flack x determined using 431 quotients [(I⁺)-(I^-)]/[(I⁺)+(I^-)] (Parsons et al., 2013)
0 restraints	Absolute structure parameter: −0.04 (2)
Primary atom site location: dual

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
Pr1	0.81086 (3)	0.81086 (3)	0.500000	0.00721 (8)
Pr2	0.51578 (3)	0.37028 (3)	0.12478 (2)	0.00578 (7)
Pr3	0.51088 (3)	0.87144 (3)	0.04758 (2)	0.00676 (7)
Si1	0.92537 (16)	0.71000 (16)	0.30916 (8)	0.0069 (2)
Si2	0.69967 (16)	0.66478 (16)	0.19702 (8)	0.0078 (2)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Pr1	0.00720 (10)	0.00720 (10)	0.00723 (16)	−0.00017 (11)	0.00069 (9)	−0.00069 (9)
Pr2	0.00587 (11)	0.00618 (12)	0.00529 (12)	−0.00033 (9)	0.00050 (9)	−0.00083 (8)
Pr3	0.00630 (12)	0.00650 (12)	0.00748 (12)	−0.00045 (7)	−0.00114 (9)	0.00172 (9)
Si1	0.0074 (5)	0.0078 (6)	0.0056 (6)	−0.0011 (4)	−0.0004 (4)	0.0001 (5)
Si2	0.0066 (5)	0.0074 (6)	0.0094 (6)	−0.0018 (4)	−0.0012 (5)	0.0007 (5)

Geometric parameters (Å, º) top

Pr1—Pr2ⁱ	3.4914 (4)	Pr2—Pr3^xi	3.9156 (3)
Pr1—Pr2ⁱⁱ	3.5319 (4)	Pr2—Si1^xi	3.0641 (13)
Pr1—Pr2ⁱⁱⁱ	3.5319 (4)	Pr2—Si1ⁱ	3.1005 (14)
Pr1—Pr2^iv	3.4914 (4)	Pr2—Si1^xii	3.0668 (14)
Pr1—Pr3^v	3.6423 (3)	Pr2—Si2	2.9480 (13)
Pr1—Pr3^vi	3.6423 (3)	Pr2—Si2^xi	2.9737 (13)
Pr1—Si1^vii	3.1756 (13)	Pr2—Si2ⁱ	3.0730 (13)
Pr1—Si1^viii	3.0985 (13)	Pr3—Pr3ⁱⁱ	4.0074 (2)
Pr1—Si1	3.0985 (13)	Pr3—Si1^xiii	3.1554 (13)
Pr1—Si1^ix	3.1756 (13)	Pr3—Si1^xii	3.3434 (12)
Pr1—Si2ⁱⁱ	3.1780 (13)	Pr3—Si1^xiv	3.1957 (12)
Pr1—Si2ⁱⁱⁱ	3.1780 (13)	Pr3—Si2^xiii	3.2566 (12)
Pr2—Pr2ⁱ	3.9561 (6)	Pr3—Si2^xii	3.1708 (12)
Pr2—Pr3^vii	3.9414 (4)	Pr3—Si2	3.1442 (12)
Pr2—Pr3^x	3.9717 (3)	Si1—Si2	2.4738 (16)

Pr2ⁱ—Pr1—Pr2^iv	71.331 (11)	Si2^xi—Pr2—Si2ⁱ	124.47 (4)
Pr2ⁱⁱⁱ—Pr1—Pr2ⁱⁱ	68.120 (11)	Pr1^xvi—Pr3—Pr1^xii	97.461 (6)
Pr2^iv—Pr1—Pr2ⁱⁱ	176.841 (6)	Pr1^xvi—Pr3—Pr2^x	99.525 (8)
Pr2ⁱ—Pr1—Pr2ⁱⁱⁱ	176.841 (6)	Pr1^xii—Pr3—Pr2^xvii	158.527 (9)
Pr2ⁱ—Pr1—Pr2ⁱⁱ	110.343 (4)	Pr1^xvi—Pr3—Pr2^xiii	136.554 (9)
Pr2^iv—Pr1—Pr2ⁱⁱⁱ	110.343 (4)	Pr1^xii—Pr3—Pr2^x	97.821 (7)
Pr2^iv—Pr1—Pr3^v	109.131 (9)	Pr1^xii—Pr3—Pr2^xiii	99.441 (7)
Pr2ⁱⁱⁱ—Pr1—Pr3^vi	74.025 (6)	Pr1^xvi—Pr3—Pr2^xvii	102.639 (8)
Pr2ⁱ—Pr1—Pr3^v	70.258 (6)	Pr1^xvi—Pr3—Pr3ⁱⁱ	98.905 (6)
Pr2ⁱⁱ—Pr1—Pr3^v	74.025 (6)	Pr1^xii—Pr3—Pr3ⁱⁱ	56.190 (4)
Pr2ⁱⁱ—Pr1—Pr3^vi	106.588 (8)	Pr2^xiii—Pr3—Pr2^xvii	60.464 (9)
Pr2ⁱ—Pr1—Pr3^vi	109.131 (9)	Pr2^xvii—Pr3—Pr2^x	86.316 (6)
Pr2^iv—Pr1—Pr3^vi	70.258 (6)	Pr2^xiii—Pr3—Pr2^x	117.252 (7)
Pr2ⁱⁱⁱ—Pr1—Pr3^v	106.588 (8)	Pr2^x—Pr3—Pr3ⁱⁱ	149.882 (8)
Pr3^vi—Pr1—Pr3^v	179.290 (14)	Pr2^xvii—Pr3—Pr3ⁱⁱ	112.494 (8)
Si1—Pr1—Pr2ⁱⁱⁱ	122.55 (3)	Pr2^xiii—Pr3—Pr3ⁱⁱ	60.158 (7)
Si1^viii—Pr1—Pr2ⁱ	127.07 (3)	Si1^xii—Pr3—Pr1^xvi	89.84 (2)
Si1—Pr1—Pr2ⁱⁱ	54.63 (3)	Si1^xiv—Pr3—Pr1^xii	54.18 (2)
Si1^ix—Pr1—Pr2ⁱⁱⁱ	128.63 (2)	Si1^xiii—Pr3—Pr1^xvi	55.14 (2)
Si1^ix—Pr1—Pr2^iv	54.47 (2)	Si1^xii—Pr3—Pr1^xii	51.84 (2)
Si1—Pr1—Pr2ⁱ	55.75 (3)	Si1^xiii—Pr3—Pr1^xii	144.17 (3)
Si1^vii—Pr1—Pr2^iv	54.52 (2)	Si1^xiv—Pr3—Pr1^xvi	53.40 (2)
Si1^viii—Pr1—Pr2ⁱⁱⁱ	54.63 (3)	Si1^xiii—Pr3—Pr2^xiii	89.98 (2)
Si1^ix—Pr1—Pr2ⁱ	54.52 (2)	Si1^xiv—Pr3—Pr2^xiii	108.89 (3)
Si1^vii—Pr1—Pr2ⁱⁱⁱ	124.06 (2)	Si1^xii—Pr3—Pr2^xvii	134.80 (2)
Si1^vii—Pr1—Pr2ⁱ	54.47 (2)	Si1^xiv—Pr3—Pr2^x	129.63 (3)
Si1—Pr1—Pr2^iv	127.07 (3)	Si1^xii—Pr3—Pr2^xiii	131.45 (2)
Si1^viii—Pr1—Pr2ⁱⁱ	122.55 (3)	Si1^xiii—Pr3—Pr2^xvii	50.33 (3)
Si1^viii—Pr1—Pr2^iv	55.75 (3)	Si1^xii—Pr3—Pr2^x	48.60 (2)
Si1^ix—Pr1—Pr2ⁱⁱ	124.06 (2)	Si1^xiii—Pr3—Pr2^x	108.41 (2)
Si1^vii—Pr1—Pr2ⁱⁱ	128.63 (2)	Si1^xiv—Pr3—Pr2^xvii	136.01 (3)
Si1—Pr1—Pr3^vi	124.12 (2)	Si1^xiii—Pr3—Pr3ⁱⁱ	101.66 (2)
Si1^viii—Pr1—Pr3^v	124.12 (2)	Si1^xii—Pr3—Pr3ⁱⁱ	108.02 (2)
Si1^vii—Pr1—Pr3^v	54.62 (2)	Si1^xiv—Pr3—Pr3ⁱⁱ	50.43 (2)
Si1^viii—Pr1—Pr3^vi	55.90 (2)	Si1^xiv—Pr3—Si1^xii	85.949 (17)
Si1^ix—Pr1—Pr3^v	124.76 (3)	Si1^xiii—Pr3—Si1^xii	137.237 (18)
Si1^ix—Pr1—Pr3^vi	54.62 (2)	Si1^xiii—Pr3—Si1^xiv	90.036 (12)
Si1^vii—Pr1—Pr3^vi	124.76 (3)	Si1^xiv—Pr3—Si2^xiii	90.54 (3)
Si1—Pr1—Pr3^v	55.90 (2)	Si1^xiii—Pr3—Si2^xiii	45.35 (3)
Si1—Pr1—Si1^vii	91.45 (4)	Si1^xiii—Pr3—Si2^xii	92.74 (3)
Si1^viii—Pr1—Si1^ix	91.45 (4)	Si2^xii—Pr3—Pr1^xii	87.19 (2)
Si1—Pr1—Si1^ix	90.57 (2)	Si2—Pr3—Pr1^xii	54.53 (2)
Si1^viii—Pr1—Si1^vii	90.57 (2)	Si2—Pr3—Pr1^xvi	145.87 (2)
Si1—Pr1—Si1^viii	177.18 (5)	Si2^xiii—Pr3—Pr1^xii	123.76 (2)
Si1^ix—Pr1—Si1^vii	88.74 (5)	Si2^xiii—Pr3—Pr1^xvi	90.26 (2)
Si1^vii—Pr1—Si2ⁱⁱ	178.20 (3)	Si2^xii—Pr3—Pr1^xvi	55.08 (2)
Si1^ix—Pr1—Si2ⁱⁱ	92.22 (3)	Si2^xiii—Pr3—Pr2^xvii	49.43 (2)
Si1^vii—Pr1—Si2ⁱⁱⁱ	92.22 (3)	Si2^xiii—Pr3—Pr2^xiii	47.45 (2)
Si1—Pr1—Si2ⁱⁱ	90.07 (3)	Si2^xiii—Pr3—Pr2^x	135.71 (2)
Si1^ix—Pr1—Si2ⁱⁱⁱ	178.20 (3)	Si2^xii—Pr3—Pr2^xiii	164.53 (2)
Si1—Pr1—Si2ⁱⁱⁱ	87.88 (3)	Si2—Pr3—Pr2^xiii	48.31 (2)
Si1^viii—Pr1—Si2ⁱⁱⁱ	90.07 (3)	Si2—Pr3—Pr2^xvii	103.99 (2)
Si1^viii—Pr1—Si2ⁱⁱ	87.88 (3)	Si2^xii—Pr3—Pr2^xvii	110.50 (2)
Si2ⁱⁱⁱ—Pr1—Pr2ⁱⁱ	54.20 (2)	Si2—Pr3—Pr2^x	103.11 (2)
Si2ⁱⁱⁱ—Pr1—Pr2^iv	127.29 (2)	Si2^xii—Pr3—Pr2^x	47.59 (2)
Si2ⁱⁱ—Pr1—Pr2ⁱⁱ	51.80 (2)	Si2^xiii—Pr3—Pr3ⁱⁱ	67.57 (2)
Si2ⁱⁱⁱ—Pr1—Pr2ⁱ	125.04 (2)	Si2—Pr3—Pr3ⁱⁱ	50.90 (2)
Si2ⁱⁱ—Pr1—Pr2^iv	125.04 (2)	Si2^xii—Pr3—Pr3ⁱⁱ	133.74 (2)
Si2ⁱⁱ—Pr1—Pr2ⁱⁱⁱ	54.20 (2)	Si2^xii—Pr3—Si1^xiv	86.34 (3)
Si2ⁱⁱⁱ—Pr1—Pr2ⁱⁱⁱ	51.80 (2)	Si2—Pr3—Si1^xii	86.35 (3)
Si2ⁱⁱ—Pr1—Pr2ⁱ	127.29 (2)	Si2^xiii—Pr3—Si1^xii	175.55 (3)
Si2ⁱⁱⁱ—Pr1—Pr3^vi	125.73 (2)	Si2^xii—Pr3—Si1^xii	44.54 (3)
Si2ⁱⁱ—Pr1—Pr3^vi	54.90 (2)	Si2—Pr3—Si1^xiv	92.48 (3)
Si2ⁱⁱ—Pr1—Pr3^v	125.73 (2)	Si2—Pr3—Si1^xiii	136.40 (3)
Si2ⁱⁱⁱ—Pr1—Pr3^v	54.90 (2)	Si2—Pr3—Si2^xiii	91.079 (17)
Si2ⁱⁱⁱ—Pr1—Si2ⁱⁱ	86.85 (5)	Si2^xii—Pr3—Si2^xiii	138.014 (18)
Pr1^xv—Pr2—Pr1^xii	103.710 (6)	Si2—Pr3—Si2^xii	130.86 (2)
Pr1^xii—Pr2—Pr2ⁱ	55.940 (5)	Pr1—Si1—Pr1^xvii	123.37 (4)
Pr1^xv—Pr2—Pr2ⁱ	140.135 (10)	Pr1—Si1—Pr2ⁱ	68.56 (3)
Pr1^xv—Pr2—Pr3^vii	102.847 (8)	Pr1—Si1—Pr3ⁱⁱ	70.13 (2)
Pr1^xv—Pr2—Pr3^x	106.888 (8)	Pr1^xvii—Si1—Pr3ⁱⁱ	134.03 (4)
Pr1^xii—Pr2—Pr3^xi	106.191 (8)	Pr1^xvii—Si1—Pr3^v	71.14 (3)
Pr1^xii—Pr2—Pr3^vii	105.646 (8)	Pr1—Si1—Pr3^v	70.70 (3)
Pr1^xii—Pr2—Pr3^x	117.969 (7)	Pr1—Si1—Pr3^xi	136.94 (4)
Pr1^xv—Pr2—Pr3^xi	149.920 (8)	Pr2^xiii—Si1—Pr1^xvii	68.02 (3)
Pr2ⁱ—Pr2—Pr3^x	112.953 (9)	Pr2ⁱⁱ—Si1—Pr1	69.90 (3)
Pr3^vii—Pr2—Pr2ⁱ	59.445 (6)	Pr2^xiii—Si1—Pr1	139.92 (4)
Pr3^xi—Pr2—Pr2ⁱ	60.092 (6)	Pr2ⁱⁱ—Si1—Pr1^xvii	67.99 (3)
Pr3^xi—Pr2—Pr3^x	61.068 (5)	Pr2ⁱ—Si1—Pr1^xvii	141.38 (4)
Pr3^xi—Pr2—Pr3^vii	65.880 (8)	Pr2^xiii—Si1—Pr2ⁱ	129.66 (4)
Pr3^vii—Pr2—Pr3^x	117.851 (7)	Pr2ⁱⁱ—Si1—Pr2ⁱ	138.39 (4)
Si1^xi—Pr2—Pr1^xv	57.51 (2)	Pr2^xiii—Si1—Pr2ⁱⁱ	83.22 (3)
Si1ⁱ—Pr2—Pr1^xii	96.00 (2)	Pr2^xiii—Si1—Pr3^xi	82.63 (3)
Si1^xii—Pr2—Pr1^xii	55.47 (2)	Pr2^xiii—Si1—Pr3ⁱⁱ	76.47 (3)
Si1^xii—Pr2—Pr1^xv	57.49 (2)	Pr2ⁱⁱ—Si1—Pr3^xi	138.21 (4)
Si1ⁱ—Pr2—Pr1^xv	55.69 (2)	Pr2ⁱⁱ—Si1—Pr3^v	87.20 (3)
Si1^xi—Pr2—Pr1^xii	147.18 (3)	Pr2ⁱⁱ—Si1—Pr3ⁱⁱ	79.97 (3)
Si1^xii—Pr2—Pr2ⁱ	111.25 (2)	Pr2ⁱ—Si1—Pr3^xi	78.10 (3)
Si1^xi—Pr2—Pr2ⁱ	155.46 (3)	Pr2ⁱ—Si1—Pr3ⁱⁱ	84.10 (3)
Si1ⁱ—Pr2—Pr2ⁱ	90.03 (3)	Pr2ⁱ—Si1—Pr3^v	81.41 (3)
Si1^xi—Pr2—Pr3^vii	104.88 (3)	Pr2^xiii—Si1—Pr3^v	138.72 (4)
Si1^xi—Pr2—Pr3^x	54.93 (2)	Pr3^xi—Si1—Pr1^xvii	70.24 (3)
Si1ⁱ—Pr2—Pr3^x	145.54 (2)	Pr3^xi—Si1—Pr3^v	78.24 (3)
Si1ⁱ—Pr2—Pr3^vii	51.57 (2)	Pr3^xi—Si1—Pr3ⁱⁱ	133.50 (4)
Si1^xi—Pr2—Pr3^xi	97.03 (3)	Pr3^v—Si1—Pr3ⁱⁱ	140.82 (4)
Si1^xii—Pr2—Pr3^x	101.67 (2)	Si2—Si1—Pr1	116.82 (5)
Si1^xii—Pr2—Pr3^xi	147.77 (2)	Si2—Si1—Pr1^xvii	119.66 (5)
Si1ⁱ—Pr2—Pr3^xi	117.17 (2)	Si2—Si1—Pr2^xiii	63.92 (4)
Si1^xii—Pr2—Pr3^vii	140.10 (2)	Si2—Si1—Pr2ⁱⁱ	135.25 (5)
Si1^xii—Pr2—Si1ⁱ	92.601 (18)	Si2—Si1—Pr2ⁱ	65.79 (4)
Si1^xi—Pr2—Si1ⁱ	93.56 (3)	Si2—Si1—Pr3^v	137.54 (6)
Si1^xi—Pr2—Si1^xii	92.85 (4)	Si2—Si1—Pr3ⁱⁱ	64.03 (4)
Si1^xi—Pr2—Si2ⁱ	140.79 (3)	Si2—Si1—Pr3^xi	69.49 (4)
Si1^xii—Pr2—Si2ⁱ	90.38 (3)	Pr1^xii—Si2—Pr3^xi	135.69 (4)
Si2ⁱ—Pr2—Pr1^xv	92.68 (2)	Pr2—Si2—Pr1^xii	70.30 (3)
Si2—Pr2—Pr1^xii	57.90 (2)	Pr2^xiii—Si2—Pr1^xii	141.97 (4)
Si2^xi—Pr2—Pr1^xii	158.04 (2)	Pr2ⁱ—Si2—Pr1^xii	68.78 (3)
Si2ⁱ—Pr2—Pr1^xii	57.02 (2)	Pr2—Si2—Pr2^xiii	132.15 (4)
Si2—Pr2—Pr1^xv	152.08 (2)	Pr2—Si2—Pr2ⁱ	82.12 (3)
Si2^xi—Pr2—Pr1^xv	98.16 (2)	Pr2^xiii—Si2—Pr2ⁱ	134.64 (4)
Si2^xi—Pr2—Pr2ⁱ	107.11 (2)	Pr2^xiii—Si2—Pr3ⁱⁱ	80.47 (3)
Si2—Pr2—Pr2ⁱ	50.30 (2)	Pr2^xiii—Si2—Pr3^xi	82.34 (3)
Si2ⁱ—Pr2—Pr2ⁱ	47.57 (2)	Pr2—Si2—Pr3ⁱⁱ	140.09 (4)
Si2ⁱ—Pr2—Pr3^vii	53.61 (2)	Pr2ⁱ—Si2—Pr3ⁱⁱ	87.54 (3)
Si2^xi—Pr2—Pr3^x	51.94 (2)	Pr2—Si2—Pr3^xi	78.08 (3)
Si2^xi—Pr2—Pr3^xi	52.15 (2)	Pr2ⁱ—Si2—Pr3^xi	76.97 (3)
Si2^xi—Pr2—Pr3^vii	70.89 (2)	Pr2^xiii—Si2—Pr3	79.53 (3)
Si2ⁱ—Pr2—Pr3^xi	100.75 (2)	Pr2—Si2—Pr3	85.15 (3)
Si2—Pr2—Pr3^xi	54.47 (2)	Pr2ⁱ—Si2—Pr3	140.56 (4)
Si2ⁱ—Pr2—Pr3^x	160.33 (3)	Pr3—Si2—Pr1^xii	71.78 (3)
Si2—Pr2—Pr3^x	70.64 (3)	Pr3ⁱⁱ—Si2—Pr1^xii	70.02 (3)
Si2—Pr2—Pr3^vii	102.52 (2)	Pr3—Si2—Pr3^xi	136.07 (4)
Si2—Pr2—Si1^xii	95.16 (3)	Pr3—Si2—Pr3ⁱⁱ	78.78 (3)
Si2—Pr2—Si1ⁱ	139.57 (4)	Pr3ⁱⁱ—Si2—Pr3^xi	136.58 (4)
Si2ⁱ—Pr2—Si1ⁱ	47.24 (3)	Si1—Si2—Pr1^xii	121.26 (5)
Si2^xi—Pr2—Si1^xii	140.11 (3)	Si1—Si2—Pr2ⁱ	66.96 (4)
Si2^xi—Pr2—Si1ⁱ	97.97 (3)	Si1—Si2—Pr2^xiii	67.74 (4)
Si2^xi—Pr2—Si1^xi	48.35 (3)	Si1—Si2—Pr2	135.86 (5)
Si2—Pr2—Si1^xi	125.48 (3)	Si1—Si2—Pr3ⁱⁱ	71.43 (4)
Si2—Pr2—Si2^xi	100.995 (18)	Si1—Si2—Pr3	138.49 (5)
Si2—Pr2—Si2ⁱ	93.03 (4)	Si1—Si2—Pr3^xi	65.16 (4)

Symmetry codes: (i) −y+1, −x+1, −z+1/2; (ii) −y+3/2, x+1/2, z+1/4; (iii) x+1/2, −y+3/2, −z+3/4; (iv) −x+1, −y+1, z+1/2; (v) −y+2, −x+1, −z+1/2; (vi) −x+1, −y+2, z+1/2; (vii) −y+3/2, x−1/2, z+1/4; (viii) y, x, −z+1; (ix) x−1/2, −y+3/2, −z+3/4; (x) y, x, −z; (xi) −x+3/2, y−1/2, −z+1/4; (xii) y−1/2, −x+3/2, z−1/4; (xiii) −x+3/2, y+1/2, −z+1/4; (xiv) −y+1, −x+2, −z+1/2; (xv) −x+1, −y+1, z−1/2; (xvi) −x+1, −y+2, z−1/2; (xvii) y+1/2, −x+3/2, z−1/4.

Pentaneodymium tetrasiliside (B) top

Crystal data top

Nd₅Si₄	D_x = 6.045 Mg m⁻³
M_r = 833.56	Mo Kα radiation, λ = 0.71073 Å
Tetragonal, P4₁2₁2	Cell parameters from 3985 reflections
Hall symbol: P 4abw 2nw	θ = 3.8–30.4°
a = 7.8644 (2) Å	µ = 28.27 mm⁻¹
c = 14.8085 (5) Å	T = 223 K
V = 915.89 (6) Å³	Plate, metallic gray
Z = 4	0.12 × 0.09 × 0.07 mm
F(000) = 1424

Data collection top

XtaLAB AFC12 (RINC): Kappa dual offset/far diffractometer	1238 independent reflections
Radiation source: micro-focus sealed X-ray tube	1203 reflections with I > 2σ(I)
Mirror monochromator	R_int = 0.032
Detector resolution: 5.8140 pixels mm^-1	θ_max = 30.2°, θ_min = 3.7°
ω scans	h = −11→10
Absorption correction: analytical [CrysAlisPro (Rigaku OD, 2019) based on Clark & Reid (1995)]	k = −10→10
T_min = 0.561, T_max = 0.702	l = −20→19
6054 measured reflections

Refinement top

Refinement on F²	w = 1/[σ²(F_o²) + 0.4858P] where P = (F_o² + 2F_c²)/3
Least-squares matrix: full	(Δ/σ)_max = 0.001
R[F² > 2σ(F²)] = 0.015	Δρ_max = 1.10 e Å⁻³
wR(F²) = 0.028	Δρ_min = −0.71 e Å⁻³
S = 1.09	Extinction correction: SHELXL (Sheldrick, 2015b), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
1238 reflections	Extinction coefficient: 0.00090 (6)
43 parameters	Absolute structure: Flack x determined using 422 quotients [(I⁺)-(I^-)]/[(I⁺)+(I^-)] (Parsons et al., 2013)
0 restraints	Absolute structure parameter: −0.01 (3)
Primary atom site location: dual

Special details top

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

	x	y	z	U_iso*/U_eq
Nd1	0.81184 (4)	0.81184 (4)	0.500000	0.00637 (9)
Nd2	0.51490 (3)	0.36933 (4)	0.12498 (2)	0.00518 (7)
Nd3	0.51034 (4)	0.87021 (4)	0.04676 (2)	0.00649 (7)
Si1	0.9274 (2)	0.7098 (2)	0.30921 (10)	0.0060 (3)
Si2	0.7003 (2)	0.6636 (2)	0.19543 (10)	0.0078 (3)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Nd1	0.00630 (11)	0.00630 (11)	0.00652 (17)	−0.00026 (13)	0.00064 (11)	−0.00064 (11)
Nd2	0.00538 (14)	0.00534 (14)	0.00483 (12)	0.00002 (12)	0.00035 (11)	−0.00097 (10)
Nd3	0.00617 (15)	0.00631 (14)	0.00700 (13)	−0.00027 (9)	−0.00098 (11)	0.00152 (11)
Si1	0.0060 (7)	0.0066 (7)	0.0054 (6)	−0.0007 (5)	0.0001 (5)	0.0001 (6)
Si2	0.0069 (7)	0.0073 (7)	0.0093 (6)	−0.0016 (5)	−0.0015 (6)	0.0017 (6)

Geometric parameters (Å, º) top

Nd1—Nd2ⁱ	3.4725 (5)	Nd2—Nd3^vii	3.9061 (4)
Nd1—Nd2ⁱⁱ	3.5021 (5)	Nd2—Si1^x	3.0366 (16)
Nd1—Nd2ⁱⁱⁱ	3.5021 (5)	Nd2—Si1ⁱ	3.0848 (17)
Nd1—Nd2^iv	3.4725 (5)	Nd2—Si1^xii	3.0436 (17)
Nd1—Nd3^v	3.6265 (3)	Nd2—Si2	2.9272 (17)
Nd1—Nd3^vi	3.6265 (3)	Nd2—Si2^x	2.9533 (17)
Nd1—Si1^vii	3.1528 (16)	Nd2—Si2ⁱ	3.0565 (15)
Nd1—Si1^viii	3.0744 (15)	Nd3—Nd3^xii	3.9752 (2)
Nd1—Si1	3.0744 (15)	Nd3—Si1^xiii	3.1359 (15)
Nd1—Si1^ix	3.1528 (16)	Nd3—Si1^xii	3.3315 (14)
Nd1—Si2ⁱⁱ	3.1661 (16)	Nd3—Si1^xiv	3.1748 (15)
Nd1—Si2ⁱⁱⁱ	3.1661 (16)	Nd3—Si2^xiii	3.2425 (15)
Nd2—Nd2ⁱ	3.9202 (7)	Nd3—Si2^xii	3.1619 (16)
Nd2—Nd3^x	3.9094 (4)	Nd3—Si2	3.1177 (14)
Nd2—Nd3^xi	3.9378 (4)	Si1—Si2	2.482 (2)

Nd2ⁱ—Nd1—Nd2^iv	71.144 (14)	Si2^x—Nd2—Si2ⁱ	123.86 (5)
Nd2ⁱⁱⁱ—Nd1—Nd2ⁱⁱ	68.070 (13)	Nd1^xvi—Nd3—Nd1^xii	97.355 (7)
Nd2^iv—Nd1—Nd2ⁱⁱ	177.032 (7)	Nd1^xvi—Nd3—Nd2^xi	99.740 (9)
Nd2ⁱ—Nd1—Nd2ⁱⁱⁱ	177.032 (7)	Nd1^xii—Nd3—Nd2^xiii	99.322 (9)
Nd2ⁱ—Nd1—Nd2ⁱⁱ	110.453 (5)	Nd1^xvi—Nd3—Nd2^xvii	102.777 (9)
Nd2^iv—Nd1—Nd2ⁱⁱⁱ	110.453 (5)	Nd1^xii—Nd3—Nd2^xi	97.621 (8)
Nd2^iv—Nd1—Nd3^v	70.374 (8)	Nd1^xii—Nd3—Nd2^xvii	158.298 (11)
Nd2ⁱⁱⁱ—Nd1—Nd3^vi	106.665 (10)	Nd1^xvi—Nd3—Nd2^xiii	136.478 (12)
Nd2ⁱ—Nd1—Nd3^v	108.985 (11)	Nd1^xvi—Nd3—Nd3^xii	57.694 (6)
Nd2ⁱⁱ—Nd1—Nd3^v	106.665 (10)	Nd1^xii—Nd3—Nd3^xii	137.385 (10)
Nd2ⁱⁱ—Nd1—Nd3^vi	73.978 (8)	Nd2^xvii—Nd3—Nd2^xiii	60.212 (10)
Nd2ⁱ—Nd1—Nd3^vi	70.374 (8)	Nd2^xiii—Nd3—Nd2^xi	117.333 (8)
Nd2^iv—Nd1—Nd3^vi	108.985 (11)	Nd2^xvii—Nd3—Nd2^xi	86.968 (8)
Nd2ⁱⁱⁱ—Nd1—Nd3^v	73.978 (8)	Nd2^xi—Nd3—Nd3^xii	59.212 (7)
Nd3^vi—Nd1—Nd3^v	179.256 (17)	Nd2^xiii—Nd3—Nd3^xii	122.770 (9)
Si1—Nd1—Nd2ⁱⁱⁱ	122.55 (4)	Nd2^xvii—Nd3—Nd3^xii	62.562 (7)
Si1^viii—Nd1—Nd2ⁱ	126.97 (4)	Si1^xii—Nd3—Nd1^xvi	89.95 (3)
Si1—Nd1—Nd2ⁱⁱ	54.67 (3)	Si1^xiv—Nd3—Nd1^xii	54.16 (3)
Si1^ix—Nd1—Nd2ⁱⁱⁱ	128.53 (3)	Si1^xiii—Nd3—Nd1^xvi	55.00 (3)
Si1^ix—Nd1—Nd2^iv	54.30 (3)	Si1^xii—Nd3—Nd1^xii	51.72 (3)
Si1—Nd1—Nd2ⁱ	55.82 (3)	Si1^xiii—Nd3—Nd1^xii	143.78 (3)
Si1^vii—Nd1—Nd2^iv	54.44 (3)	Si1^xiv—Nd3—Nd1^xvi	53.25 (3)
Si1^viii—Nd1—Nd2ⁱⁱⁱ	54.67 (3)	Si1^xiii—Nd3—Nd2^xvii	50.52 (3)
Si1^ix—Nd1—Nd2ⁱ	54.44 (3)	Si1^xiv—Nd3—Nd2^xvii	135.69 (3)
Si1^vii—Nd1—Nd2ⁱⁱⁱ	124.35 (3)	Si1^xii—Nd3—Nd2^xiii	131.30 (3)
Si1^vii—Nd1—Nd2ⁱ	54.30 (3)	Si1^xiv—Nd3—Nd2^xi	129.50 (3)
Si1—Nd1—Nd2^iv	126.97 (4)	Si1^xii—Nd3—Nd2^xvii	135.35 (3)
Si1^viii—Nd1—Nd2ⁱⁱ	122.55 (4)	Si1^xiii—Nd3—Nd2^xiii	90.04 (3)
Si1^viii—Nd1—Nd2^iv	55.82 (3)	Si1^xii—Nd3—Nd2^xi	48.50 (3)
Si1^ix—Nd1—Nd2ⁱⁱ	124.35 (3)	Si1^xiii—Nd3—Nd2^xi	108.97 (3)
Si1^vii—Nd1—Nd2ⁱⁱ	128.53 (3)	Si1^xiv—Nd3—Nd2^xiii	108.81 (3)
Si1—Nd1—Nd3^vi	55.83 (3)	Si1^xiii—Nd3—Nd3^xii	51.39 (3)
Si1^viii—Nd1—Nd3^v	55.83 (3)	Si1^xii—Nd3—Nd3^xii	91.19 (3)
Si1^vii—Nd1—Nd3^v	124.78 (3)	Si1^xiv—Nd3—Nd3^xii	110.85 (3)
Si1^viii—Nd1—Nd3^vi	124.19 (3)	Si1^xiv—Nd3—Si1^xii	85.88 (2)
Si1^ix—Nd1—Nd3^v	54.57 (3)	Si1^xiii—Nd3—Si1^xii	137.46 (2)
Si1^ix—Nd1—Nd3^vi	124.78 (3)	Si1^xiii—Nd3—Si1^xiv	89.678 (15)
Si1^vii—Nd1—Nd3^vi	54.57 (3)	Si1^xiv—Nd3—Si2^xiii	90.11 (4)
Si1—Nd1—Nd3^v	124.19 (3)	Si1^xiii—Nd3—Si2^xiii	45.76 (4)
Si1—Nd1—Si1^vii	91.21 (4)	Si1^xiii—Nd3—Si2^xii	92.67 (4)
Si1^viii—Nd1—Si1^ix	91.21 (4)	Si2^xii—Nd3—Nd1^xii	87.36 (3)
Si1—Nd1—Si1^ix	90.79 (3)	Si2—Nd3—Nd1^xii	54.76 (3)
Si1^viii—Nd1—Si1^vii	90.79 (3)	Si2—Nd3—Nd1^xvi	146.00 (3)
Si1—Nd1—Si1^viii	177.21 (7)	Si2^xiii—Nd3—Nd1^xii	123.04 (3)
Si1^ix—Nd1—Si1^vii	88.51 (6)	Si2^xiii—Nd3—Nd1^xvi	90.38 (3)
Si1^vii—Nd1—Si2ⁱⁱ	178.58 (4)	Si2^xii—Nd3—Nd1^xvi	55.09 (3)
Si1^ix—Nd1—Si2ⁱⁱ	92.27 (4)	Si2^xiii—Nd3—Nd2^xiii	47.19 (3)
Si1^vii—Nd1—Si2ⁱⁱⁱ	92.27 (4)	Si2^xiii—Nd3—Nd2^xvii	49.57 (3)
Si1—Nd1—Si2ⁱⁱ	89.96 (4)	Si2^xiii—Nd3—Nd2^xi	136.52 (3)
Si1^ix—Nd1—Si2ⁱⁱⁱ	178.58 (4)	Si2^xii—Nd3—Nd2^xvii	110.79 (3)
Si1—Nd1—Si2ⁱⁱⁱ	88.01 (4)	Si2—Nd3—Nd2^xvii	103.54 (3)
Si1^viii—Nd1—Si2ⁱⁱⁱ	89.96 (4)	Si2—Nd3—Nd2^xiii	48.09 (3)
Si1^viii—Nd1—Si2ⁱⁱ	88.01 (4)	Si2^xii—Nd3—Nd2^xiii	164.61 (3)
Si2ⁱⁱⁱ—Nd1—Nd2ⁱⁱ	54.28 (3)	Si2—Nd3—Nd2^xi	102.77 (3)
Si2ⁱⁱⁱ—Nd1—Nd2^iv	127.09 (3)	Si2^xii—Nd3—Nd2^xi	47.63 (3)
Si2ⁱⁱ—Nd1—Nd2ⁱⁱ	51.78 (3)	Si2^xiii—Nd3—Nd3^xii	93.42 (3)
Si2ⁱⁱⁱ—Nd1—Nd2ⁱ	125.25 (3)	Si2—Nd3—Nd3^xii	156.02 (3)
Si2ⁱⁱ—Nd1—Nd2^iv	125.25 (3)	Si2^xii—Nd3—Nd3^xii	50.23 (3)
Si2ⁱⁱ—Nd1—Nd2ⁱⁱⁱ	54.28 (3)	Si2^xii—Nd3—Si1^xiv	86.36 (4)
Si2ⁱⁱⁱ—Nd1—Nd2ⁱⁱⁱ	51.78 (3)	Si2—Nd3—Si1^xii	86.26 (4)
Si2ⁱⁱ—Nd1—Nd2ⁱ	127.09 (3)	Si2^xiii—Nd3—Si1^xii	174.74 (4)
Si2ⁱⁱⁱ—Nd1—Nd3^vi	54.98 (3)	Si2^xii—Nd3—Si1^xii	44.85 (4)
Si2ⁱⁱ—Nd1—Nd3^vi	125.68 (3)	Si2—Nd3—Si1^xiv	92.77 (4)
Si2ⁱⁱ—Nd1—Nd3^v	54.98 (3)	Si2—Nd3—Si1^xiii	136.26 (4)
Si2ⁱⁱⁱ—Nd1—Nd3^v	125.68 (3)	Si2—Nd3—Si2^xiii	90.54 (2)
Si2ⁱⁱⁱ—Nd1—Si2ⁱⁱ	86.97 (6)	Si2^xii—Nd3—Si2^xiii	138.34 (2)
Nd1^xv—Nd2—Nd1^xii	103.774 (8)	Si2—Nd3—Si2^xii	131.07 (3)
Nd1^xii—Nd2—Nd2ⁱ	55.965 (6)	Nd1—Si1—Nd1^xvii	123.58 (5)
Nd1^xv—Nd2—Nd2ⁱ	140.405 (13)	Nd1—Si1—Nd2ⁱ	68.64 (3)
Nd1^xv—Nd2—Nd3^x	149.935 (9)	Nd1—Si1—Nd3ⁱⁱ	70.00 (3)
Nd1^xv—Nd2—Nd3^xi	107.244 (10)	Nd1^xvii—Si1—Nd3ⁱⁱ	134.15 (5)
Nd1^xii—Nd2—Nd3^vii	106.182 (10)	Nd1^xvii—Si1—Nd3^vi	71.13 (3)
Nd1^xii—Nd2—Nd3^x	106.111 (9)	Nd1—Si1—Nd3^vi	70.93 (3)
Nd1^xii—Nd2—Nd3^xi	117.604 (9)	Nd1—Si1—Nd3^x	136.75 (6)
Nd1^xv—Nd2—Nd3^vii	102.810 (9)	Nd2^xiii—Si1—Nd1^xvii	68.23 (3)
Nd2ⁱ—Nd2—Nd3^xi	112.321 (11)	Nd2ⁱⁱ—Si1—Nd1	69.84 (3)
Nd3^x—Nd2—Nd2ⁱ	59.853 (7)	Nd2^xiii—Si1—Nd1	139.62 (5)
Nd3^vii—Nd2—Nd2ⁱ	59.937 (8)	Nd2ⁱⁱ—Si1—Nd1^xvii	68.14 (4)
Nd3^vii—Nd2—Nd3^xi	117.388 (8)	Nd2ⁱ—Si1—Nd1^xvii	141.48 (5)
Nd3^vii—Nd2—Nd3^x	65.627 (10)	Nd2^xiii—Si1—Nd2ⁱ	129.31 (5)
Nd3^x—Nd2—Nd3^xi	60.872 (7)	Nd2ⁱⁱ—Si1—Nd2ⁱ	138.41 (5)
Si1^x—Nd2—Nd1^xv	57.47 (3)	Nd2^xiii—Si1—Nd2ⁱⁱ	83.28 (4)
Si1ⁱ—Nd2—Nd1^xii	96.51 (3)	Nd2^xiii—Si1—Nd3^x	83.05 (4)
Si1^xii—Nd2—Nd1^xii	55.49 (3)	Nd2^xiii—Si1—Nd3ⁱⁱ	76.24 (3)
Si1^xii—Nd2—Nd1^xv	57.42 (3)	Nd2ⁱⁱ—Si1—Nd3^x	138.55 (5)
Si1ⁱ—Nd2—Nd1^xv	55.54 (3)	Nd2ⁱⁱ—Si1—Nd3^vi	87.22 (4)
Si1^x—Nd2—Nd1^xii	147.02 (3)	Nd2ⁱⁱ—Si1—Nd3ⁱⁱ	80.06 (4)
Si1^xii—Nd2—Nd2ⁱ	111.31 (3)	Nd2ⁱ—Si1—Nd3^x	77.79 (4)
Si1^x—Nd2—Nd2ⁱ	155.46 (3)	Nd2ⁱ—Si1—Nd3ⁱⁱ	83.83 (4)
Si1ⁱ—Nd2—Nd2ⁱ	90.59 (3)	Nd2ⁱ—Si1—Nd3^vi	81.65 (4)
Si1^x—Nd2—Nd3^x	97.14 (3)	Nd2^xiii—Si1—Nd3^vi	138.92 (5)
Si1^x—Nd2—Nd3^xi	55.26 (3)	Nd3^x—Si1—Nd1^xvii	70.43 (3)
Si1ⁱ—Nd2—Nd3^xi	145.44 (3)	Nd3^x—Si1—Nd3^vi	78.09 (3)
Si1ⁱ—Nd2—Nd3^x	117.05 (3)	Nd3^x—Si1—Nd3ⁱⁱ	133.33 (5)
Si1^x—Nd2—Nd3^vii	104.58 (3)	Nd3^vi—Si1—Nd3ⁱⁱ	140.92 (5)
Si1^xii—Nd2—Nd3^xi	101.82 (3)	Si2—Si1—Nd1	116.71 (6)
Si1^xii—Nd2—Nd3^vii	140.39 (3)	Si2—Si1—Nd1^xvii	119.56 (6)
Si1ⁱ—Nd2—Nd3^vii	51.69 (3)	Si2—Si1—Nd2^xiii	63.78 (5)
Si1^xii—Nd2—Nd3^x	147.88 (3)	Si2—Si1—Nd2ⁱⁱ	135.29 (6)
Si1^xii—Nd2—Si1ⁱ	92.69 (2)	Si2—Si1—Nd2ⁱ	65.57 (5)
Si1^x—Nd2—Si1ⁱ	93.26 (4)	Si2—Si1—Nd3^vi	137.48 (7)
Si1^x—Nd2—Si1^xii	92.72 (5)	Si2—Si1—Nd3ⁱⁱ	63.95 (4)
Si1^x—Nd2—Si2ⁱ	140.91 (4)	Si2—Si1—Nd3^x	69.39 (5)
Si1^xii—Nd2—Si2ⁱ	90.60 (4)	Nd1^xii—Si2—Nd3^x	135.36 (6)
Si2ⁱ—Nd2—Nd1^xv	92.89 (3)	Nd2—Si2—Nd1^xii	70.04 (4)
Si2—Nd2—Nd1^xii	58.18 (3)	Nd2^xiii—Si2—Nd1^xii	142.04 (5)
Si2^x—Nd2—Nd1^xii	157.63 (3)	Nd2ⁱ—Si2—Nd1^xii	68.47 (3)
Si2ⁱ—Nd2—Nd1^xii	57.24 (3)	Nd2—Si2—Nd2^xiii	133.24 (5)
Si2—Nd2—Nd1^xv	151.99 (3)	Nd2—Si2—Nd2ⁱ	81.83 (4)
Si2^x—Nd2—Nd1^xv	98.52 (3)	Nd2^xiii—Si2—Nd2ⁱ	134.00 (5)
Si2^x—Nd2—Nd2ⁱ	106.53 (3)	Nd2^xiii—Si2—Nd3ⁱⁱ	80.09 (4)
Si2—Nd2—Nd2ⁱ	50.51 (3)	Nd2^xiii—Si2—Nd3^x	82.54 (4)
Si2ⁱ—Nd2—Nd2ⁱ	47.66 (3)	Nd2—Si2—Nd3ⁱⁱ	139.78 (6)
Si2ⁱ—Nd2—Nd3^x	100.42 (3)	Nd2ⁱ—Si2—Nd3ⁱⁱ	87.23 (4)
Si2^x—Nd2—Nd3^xi	52.28 (3)	Nd2—Si2—Nd3^x	78.46 (4)
Si2^x—Nd2—Nd3^vii	70.03 (3)	Nd2ⁱ—Si2—Nd3^x	76.59 (3)
Si2^x—Nd2—Nd3^x	51.78 (3)	Nd2^xiii—Si2—Nd3	80.12 (4)
Si2ⁱ—Nd2—Nd3^vii	53.85 (3)	Nd2—Si2—Nd3	85.51 (4)
Si2—Nd2—Nd3^vii	102.94 (3)	Nd2ⁱ—Si2—Nd3	140.17 (6)
Si2ⁱ—Nd2—Nd3^xi	159.76 (3)	Nd3—Si2—Nd1^xii	71.70 (3)
Si2—Nd2—Nd3^xi	69.76 (3)	Nd3ⁱⁱ—Si2—Nd1^xii	69.93 (3)
Si2—Nd2—Nd3^x	54.35 (3)	Nd3—Si2—Nd3^x	137.09 (5)
Si2—Nd2—Si1^xii	95.25 (4)	Nd3—Si2—Nd3ⁱⁱ	78.55 (4)
Si2—Nd2—Si1ⁱ	140.41 (4)	Nd3ⁱⁱ—Si2—Nd3^x	136.04 (5)
Si2ⁱ—Nd2—Si1ⁱ	47.67 (4)	Si1—Si2—Nd1^xii	120.80 (6)
Si2^x—Nd2—Si1^xii	140.61 (4)	Si1—Si2—Nd2ⁱ	66.76 (5)
Si2^x—Nd2—Si1ⁱ	97.55 (4)	Si1—Si2—Nd2^xiii	67.28 (5)
Si2^x—Nd2—Si1^x	48.93 (4)	Si1—Si2—Nd2	135.73 (6)
Si2—Nd2—Si1^x	124.91 (4)	Si1—Si2—Nd3ⁱⁱ	71.20 (5)
Si2—Nd2—Si2^x	100.44 (2)	Si1—Si2—Nd3	138.40 (7)
Si2—Nd2—Si2ⁱ	93.45 (5)	Si1—Si2—Nd3^x	64.85 (5)

Symmetry codes: (i) −y+1, −x+1, −z+1/2; (ii) −y+3/2, x+1/2, z+1/4; (iii) x+1/2, −y+3/2, −z+3/4; (iv) −x+1, −y+1, z+1/2; (v) −x+1, −y+2, z+1/2; (vi) −y+2, −x+1, −z+1/2; (vii) −y+3/2, x−1/2, z+1/4; (viii) y, x, −z+1; (ix) x−1/2, −y+3/2, −z+3/4; (x) −x+3/2, y−1/2, −z+1/4; (xi) y, x, −z; (xii) y−1/2, −x+3/2, z−1/4; (xiii) −x+3/2, y+1/2, −z+1/4; (xiv) −y+1, −x+2, −z+1/2; (xv) −x+1, −y+1, z−1/2; (xvi) −x+1, −y+2, z−1/2; (xvii) y+1/2, −x+3/2, z−1/4.

Selected bond lengths (Å) for Pr₅Si₄ top

Pr1—Pr2ⁱ	3.4914 (4)	Pr1—Si1^ix	3.1756 (13)
Pr1—Pr2ⁱⁱ	3.5319 (4)	Pr1—Si2ⁱⁱ	3.1780 (13)
Pr1—Pr2ⁱⁱⁱ	3.5319 (4)	Pr1—Si2ⁱⁱⁱ	3.1780 (13)
Pr1—Pr2^iv	3.4914 (4)	Pr2—Pr2ⁱ	3.9561 (6)
Pr1—Pr3^v	3.6423 (3)	Pr2—Pr3^vii	3.9414 (4)
Pr1—Pr3^vi	3.6423 (3)	Pr2—Pr3^x	3.9717 (3)
Pr1—Si1^vii	3.1756 (13)	Pr2—Pr3^xi	3.9156 (3)
Pr1—Si1^viii	3.0985 (13)	Pr3—Pr3ⁱⁱ	4.0074 (2)
Pr1—Si1	3.0985 (13)	Si1—Si2	2.4738 (16)

Symmetry codes: (i) -y+1, -x+1, -z+1/2; (ii) -y+3/2, x+1/2, z+1/4; (iii) x+1/2, -y+3/2, -z+3/4; (iv) -x+1, -y+1, z+1/2; (v) -y+2, -x+1, -z+1/2; (vi) -x+1, -y+2, z+1/2; (vii) -y+3/2, x-1/2, z+1/4; (viii) y, x, -z+1; (ix) x-1/2, -y+3/2, -z+3/4.

Selected bond lengths (Å) for Nd₅Si₄ top

Nd1—Nd2ⁱ	3.4725 (5)	Nd1—Si1^ix	3.1528 (16)
Nd1—Nd2ⁱⁱ	3.5021 (5)	Nd1—Si2ⁱⁱ	3.1661 (16)
Nd1—Nd2ⁱⁱⁱ	3.5021 (5)	Nd1—Si2ⁱⁱⁱ	3.1661 (16)
Nd1—Nd2^iv	3.4725 (5)	Nd2—Nd2ⁱ	3.9202 (7)
Nd1—Nd3^v	3.6265 (3)	Nd2—Nd3^x	3.9094 (4)
Nd1—Nd3^vi	3.6265 (3)	Nd2—Nd3^xi	3.9378 (4)
Nd1—Si1^vii	3.1528 (16)	Nd2—Nd3^vii	3.9061 (4)
Nd1—Si1^viii	3.0744 (15)	Nd3—Nd3^xii	3.9752 (2)
Nd1—Si1	3.0744 (15)	Si1—Si2	2.482 (2)

Acknowledgements

The authors gratefully thank the Instrumental Analysis Center of Yokohama National University for providing access to the single-crystal X-ray diffractometer.

Funding information

Funding for this research was provided by: JSPS Grant-in-Aid for Scientific Research (KAKENHI) (grant No. 18K049922 and 18K03536).

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