Crystal Chemistry of the New Families of Interstitial Compounds

Article pubs.acs.org/IC

Crystal Chemistry of the New Families of Interstitial Compounds R6Mg23C (R = La, Ce, Pr, Nd, Sm, or Gd) and Ce6Mg23Z (Z = C, Si, Ge, Sn, Pb, P, As, or Sb) Federico Wrubl,† Pietro Manfrinetti,*,†,‡ Marcella Pani,†,‡ Pavlo Solokha,† and Adriana Saccone† †

Dipartimento di Chimica e Chimica Industriale, Università di Genova, Via Dodecaneso 31, 16146 Genova, Italy CNR-SPIN, Corso Perrone 24, 16152 Genova, Italy

‡

S Supporting Information *

ABSTRACT: The crystal chemical features of the new series of compounds R6Mg23C with R = La−Sm or Gd and Ce6Mg23Z with Z = C, Si, Ge, Sn, Pb, P, As, or Sb have been studied by means of single-crystal and powder X-ray diffraction techniques. All phases crystallize with the cubic Zr6Zn23Si prototype (cF120, space group Fm3̅m, Z = 4), a filled variant of the Th6Mn23 structure. While no Th6Mn23-type binary rare earth−magnesium compound is known to exist, the addition of a third element Z (only 3 atom %), located into the octahedral cavity of the Th6Mn23 cell (Wyckoff site 4a), stabilizes this structural arrangement and makes possible the formation of the ternary R6Mg23Z compounds. The results of both structural and topological analyses as well as of LMTO electronic structure calculations show that the interstitial element plays a crucial role in the stability of these phases, forming a strongly bonded [R6Z] octahedral moiety spaced by zeolite cage-like [Mg45] clusters. Considering these two building units, the crystal structure of these apparently complex intermetallics can be simplified to the NaCl-type topology. Moreover, a structural relationship between RMg3 and R6Mg23C compounds has been unveiled; the latter can be described as substitutional derivatives of the former. The geometrical distortions and the consequent symmetry reduction that accompany this transformation are explicitly described by means of the Bärnighausen formalism within group theory.

1. INTRODUCTION Intermetallic compounds with complex cubic unit cells adopt a number of quite varied structure types. In descending order of occurrence, they can be listed as it follows:1,2 MgCu2, Cr3Si, Th6Mn23, NaZn13, AuBe5, α-Mn, Ti2Ni, β-Mn, Cu5Zn8, Sm11Cd45, and YCd6 prototypes. R6Mn23 (R = Pr−Er) and R6Fe23 (R = Gd−Tm or Yb) phases are among the numerous compounds that adopt the Th6Mn23 structure type3 (cF116, space group Fm3̅m). The rather large unit cell of Th6Mn23 (a = 12.52 Å; Vmeas = 1963 Å3) hosts one site for R (24e Wyckoff position) and four independent sites for the transition metal (2 × 32f, 24d, and 4b respectively). One important feature of its architecture is the presence of a large octahedral cavity in 4a (r = 0.74 Å for Th6Mn23), potentially able to accommodate guest atoms. According to Structure Tidy,4 the standard presentation of the Th6Mn23 structure is cF116-f 2eda. Under affine normalizers within the group theoretical concept,5 the 4a and 4b Wyckoff sites for the Fm3̅m space group belong to the same Wyckoff set. That is why both cF116-f 2eda and cF116-f 2edb descriptions of Th6Mn23 are equivalent, having the origin shifted by (1/2, 1/2, 1/2). To be consistent with the further crystallographic discussions presented here, the cF116-f 2edb notation was chosen as the basic one and the 4a site considered as interstitial. In the Th6Mn23□ formula, we indicate the unoccupied cavity with an empty square (one cavity per formula, four cavities per © XXXX American Chemical Society

unit cell). A number of other compounds adopt structure types formally derived from the Th6Mn23 prototype. They can be divided into stuffed and nonstuffed compounds, depending on whether the cavity in 4a is occupied. The four Mn positions in Th6Mn23 can in turn be occupied in an orderly fashion by two different species, and the cavity can be occupied by one of the already present elements (homostuffed case) or by a different one (heterostuffed). The following groups can be identified as in the scheme reported in Figure 1: (1) Th6Mn23-type binary compounds, (2) nonstuffed ternary compounds, with nonoccupied 4a cavities (Mg6Cu16Si7□ prototype, where Si occupies positions 24d and 4b),6,7 (3) homostuffed binary compounds, with 4a cavities occupied by one of the two elements (Sc22Ir8),8,9 (4) heterostuffed ternary compounds, Th6Mn23 derivatives with a third element in the 4a cavity (Zr6Zn23Si),10 (5) homostuffed ternary compounds, like case 2 and with the 4a cavity filled by the elements either in 32f or 24d (Ti6M7Al16Al and Ti6Ni16Si7Si),11 and (6) heterostuffed quaternary compounds, like case 2 and with a fourth element in 4a (U6Fe16Si7C).12 Magnesium-based binary compounds crystallizing with the Th6Mn23 type are known to exist only in combination with an alkaline earth metal (Sr or Ba) or with Th. No R6Mg23 compound Received: September 15, 2015

A

DOI: 10.1021/acs.inorgchem.5b02114 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry

Figure 1. Th6Mn23 and related structures. size, and dusty in some cases; they are stable toward air and moisture. During the investigation, alloys Ce6Mg23C, Ce6Mg23Ge, and Ce6Mg23P were subjected to differential thermal analysis (DTA). A small specimen of the alloy (∼0.5 g), prepared and annealed as described above and closed by arc welding into a Mo crucible, was transferred to the DTA equipment and subjected to heating and cooling cycles at rates of 20 and 10 °C/min, respectively (5 °C accuracy). Metallographic specimens were prepared by a standard technique and examined by optical and electron microscopy; on selected alloys, the composition of the phases was determined by means of electron probe microanalysis (EPMA). Xray analysis was conducted on powders and single crystals for selected alloys. The polycrystalline alloys were analyzed by X-ray powder diffraction, using a Guinier camera (with Si as an internal standard; a = 5.4308 Å; Cu Kα1 radiation); powder intensity data for Rietveld refinements were collected using a Philips diffractometer (Bragg− Brentano geometry, Ni-filtered Cu Kα radiation) with 0.02° 2θ steps. The corresponding refinements were accomplished by means of the FULLPROF program.15 Single-crystal intensities were measured for the Ce6Mg23Ge (DTA sample, after annealing at 700 °C for 14 days) and Pr6Mg23C compounds, at room temperature, on a Bruker−Nonius MACH3 diffractometer, equipped with graphite-monochromated Mo Kα radiation. The data collection conditions are summarized in Table 1. A spherical correction accounted for absorption effects. Structure refinements were made by full-matrix least-squares on Fo2 using SHELXL.16 All the structural images were produced by means of DRAWxtl.17 2.2. Electronic Structure Calculations. Electronic structure calculations were performed for some La6 Mg23Z, with Z = C, P, or Sb, and for Ba6 Mg23, using the linear muffin-tin orbital (LMTO) method18−20 in its tight-binding representation21 as implemented in TB-LMTO-ASA version 4.7.22 Basis sets were composed of short-range atom-centered TB-LMTOs with a scalar-relativistic Hamiltonian and the atomic-spheres approximations (ASA). Electronic energies were calculated via density functional theory (DFT), based on the local density approximation (LDA) for the exchange-correlation functional as parametrized by von Barth and Hedin.23 For the La6Mg23C structure, the average ASA radii for the constituents were as follows: 1.98 Å for La, 1.80−1.82 Å for Mg, and

is formed with the trivalent rare earth metals (R). On the other hand, examples of ternary derivatives are known for La6Mg22Al (Th6Mn23 type), with Al fully occupying the 4b site in an ordered fashion,13 and for La6Mg23Si (Zr6Zn23Si type), with Si filling the 4a cavity.14 These two examples show how the presence of small amounts of a different atomic species (here 4σ(F0) Rint(Fo2) transmission ratio (Tmax/Tmin) absorption correction R1 [F0 > 4σ(F0)] wR2(Fo2) (all data) shift/e.s.d. goodness of fit Δρmax, Δρmin (e Å−3)

Ce6Mg23Ge

Pr6Mg23C

a = 14.597(5) 3110.2(1) 4 2624 10.00 3.145 irregular 0.11 × 0.10 × 0.06 ω−θ 2−32 0 ≤ h ≤ 21 −21 ≤ k ≤ 0 −21 ≤ l ≤ 21 2885 323 253 0.065 1.02 spherical 0.018 0.029 0.000 0.756 +2.28, −1.74

a = 14.491(4) 3043.0(1) 4 2544 9.89 3.092 irregular 0.11 × 0.11 × 0.07 ω−θ 2−32 0 ≤ h ≤ 21 0 ≤ k ≤ 21 −21 ≤ l ≤ 21 2809 316 205 0.074 1.016 spherical 0.050 0.077 0.000 1.529 +5.05, −6.11

1.34 Å for C. For all calculations performed here, the basis sets included 6s/(6p)/5d/4f orbitals for La, 6s/(6p)/5d/(5f) orbitals for Ba, 3s/3p/ (3d) orbitals for Mg, 2s/2p/(3d) orbitals for C, 3s/3p/(3d) orbitals for P, and 5s/5p/(5d)/(4f) orbitals for Sb, with orbitals in parentheses being downfolded.24 To satisfy the LMTO volume criterion, no empty spheres were needed. The k-space integrations were conducted using an improved tetrahedron method25 with a 145-irreducible k-point mesh in the first Brillouin zone. Energies and crystal orbital Hamilton population (COHP) convergence with respect to the number of k points were checked in all calculations. To evaluate various orbital interactions, density of states (DOS), crystal orbital Hamilton population (COHP) curves,25 and integrated COHP values (iCOHPs) were also calculated. All the figures and graphics concerning electron structure calculations were generated by wxDragon.26

3. RESULTS AND DISCUSSION Samples with compositions of R6Mg23C (R = La, Ce, Pr, Nd, Sm, Gd, or Ho), Ce6Mg23Z (Z = B, C, Si, Ge, Sn, Pb, P, As, Sb, or Te), and La6Mg23Z (Z = P or Sb) were prepared and annealed according to the procedure reported in the previous section. No trace of the 6:23:1 phase was detected for the Ho−Mg carbide or for the boron- or tellurium-containing Ce−Mg alloys. All the Guinier powder patterns were indexed on the basis of the cubic Zr6Zn23Si cell. The structure type and stoichiometry (in particular for what concerns the guest atom) were confirmed by the structural refinements. In addition, SEM microscopy and Rietveld analysis revealed that the formation of the 6:23:1 ternary phase is always accompanied by different and, in most cases, nonnegligible amounts of extra phases (binary compounds), as reported in Table 2. In particular, the cubic RMg3 phase (cF16, BiF3 type) is the most frequently encountered. It occurs alone in the R6Mg23C series with La, Ce, and Pr, together with tetragonal Ce5Mg41 type (Nd) or along with cubic GdMg∼5 type (Sm or C


Z

(interstitial)

Te

As Sb

P

Pb

Si Ge Sn

C

B

La6Mg23Sb (94%), LaSb (6%)

La6Mg23P (97%), LaMg3 (3%)

La6Mg23C (67%), LaMg3 (33%)

La

Ce

CeMg3 + Ce5Mg41, no traces of Ce6Mg23Te

Ce6Mg23As (95%), CeAs (5%) Ce6Mg23Sb (79%), CeMg3 (21%)

Ce6Mg23Si (98%), CeMg3 (2%) Ce6Mg23Ge (100%) Ce6Mg23Sn (70%), CeMg3 (11%), Ce2Mg17 (19%) Ce6Mg23Pb (not determined) + CeMg3 (not determined) + unknown Ce6Mg23P (98%), CeMg3 (2%)

CeMg3 + Ce5Mg41, no traces of Ce6Mg23B Ce6Mg23C (96%), CeMg3 (4%) Pr6Mg23C (76%), PrMg3 (24%)

Pr

Nd6Mg23C (64%), NdMg3 (28%), Nd5Mg41 (8%)

Nd

R (rare earth) Gd

Gd6Mg23C (62%), GdMg3 (30%), GdMg∼5 (8%)

combinations to be tested

Sm6Mg23C (63%), SmMg3 (36%), SmMg∼5 (1%)

Sm

Table 2. Overview of the Compositions of the R6Mg23Z Alloys, Obtained by Varying the Rare Earth R and the Interstitial Element Z

no traces of Ho6Mg23C, HoMg2 + Ho5Mg24 + unknown

Ho

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Inorganic Chemistry Table 3. Lattice Parameter a (measured by means of Guinier powder diffraction), Unit Cell Volume Vmeas, and Resulting Volume Contraction ΔVf for the R6Mg23Z Compounds (Zr6Zn23Si type, cF120, Fm3̅m, Z = 4)a

a

compound

a (Å)

Vmeas (Å3)

ΔVf (%)

La6Mg23C Ce6Mg23C Pr6Mg23C Nd6Mg23C Sm6Mg23C Gd6Mg23C Ce6Mg23Si Ce6Mg23Ge Ce6Mg23Sn Ce6Mg23Pb Ce6Mg23P Ce6Mg23As Ce6Mg23Sb La6Mg23P La6Mg23Sb

14.617(9) 14.530(9) 14.491(4) 14.453(4) 14.380(7) 14.320(7) 14.586(9) 14.597(5) 14.629(9) 14.633(8) 14.552(4) 14.573(4) 14.628(3) 14.634(7) 14.711(3)

3123(6) 3068(6) 3043(3) 3019(3) 2974(3) 2936(3) 3103(6) 3110(3) 3131(6) 3133(5) 3082(3) 3095(3) 3130(2) 3134(4) 3184(2)

−1.7 −2.0 −1.3 −0.8 −0.1 1.07 −1.6 −1.5 −1.6 −1.25 −1.1 −1.2 −1.2 −0.7 −0.9

Figure 3. Formation volume contraction ΔVf vs rare earth χR (Pauling electronegativity scale) in the RMg3, R5Mg41, and R6Mg23C series (R = La−Gd).

atom instead, driving to the formation of the Zr6Zn23Si-type arrangement, enhances the overall volume expansion. 3.2. Structural Analysis. Single crystals of Ce6Mg23Ge and Pr6Mg23C, isolated after DTA cycles and subsequent annealing, were selected after a preliminary check in a precession chamber. After unit cell dimensions and space group Fm3m ̅ had been confirmed, reflection intensities were collected. For the fullmatrix least-squares refinement, Zr6Zn23Si-type atomic positions were used as an initial model. Refined atomic coordinates for both crystals are listed in Table 4. For all the other compounds, no suitable crystals were found, and structures were refined by means of the powder Rietveld method (details thereof are provided in the Supporting Information). For the two aforementioned members, which can be considered as representatives of the Ce6Mg23X and R6Mg23C series, good agreement was found between the two methods. 3.2.1. Crystal Structure Description. The Zr6Mg23Si structure was discovered and first described in terms of a filled Th6Mn23 framework by Chen and colleagues;10 the structural arrangement is not easy to visualize, as also mentioned by the authors. An illustration of the Ce6Mg23Ge structure is reported for the sake of clarity in Figure 4, with a simplified version of the original description. Considering that {100} and {200} are mirror planes, one can analyze only half of the structure along any of the crystallographic directions, in terms of a−b sections, between z = 0 and z = 1/2. As one can see, planes at z = 0 and z = 1/2 contain all three atomic species, the intermediate section centered at z = 1/4 contains Mg and Ce, while neighboring portions at z = 0.1−0.2 and z = 0.3−0.4 contain only Mg atoms. The entire unit cell is also depicted in the same Figure 4, viewed in perspective along the three high-symmetry directions. By looking at the ⟨100⟩ render, one can simultaneously see how the different layers shown at the left side, in both [001] and [100] projections, combine and build up the entire unit cell. Semitransparent surfaces of Ge-containing Ce6 octahedra are depicted. The other two perspective views (along ⟨110⟩ and ⟨111⟩) are drawn to help the reader visualize how this interesting structure appears in three-dimensional space. All metal atoms have high coordination numbers, as ususally observed for intermetallic phases. Interatomic distances for Ce6Mg23Ge and Pr6Mg23C are reported in Table 5. Each Ce atom is surrounded by 18 atoms (13 Mg atoms, 4 Ce atoms, and 1 Ge atom), while the four inequivalent Mg atoms have coordination numbers of 12 (Mg1 in 32f and Mg3 in 24d), 13 (Mg2 in 32f), and 14 (Mg4 in 4b). Most Ce−Mg distances are in the narrow range of 3.54−3.70 Å, while a looser contact occurs for the Ce−

For the definition of ΔVf, see Volume Effects.

Figure 2. (a) Unit cell parameters of the R6Mg23C compounds (R = La− Gd) plotted vs the rR3+ ionic radii. (b) Unit cell parameters of the Ce6Mg23Z compounds [(●) Z = C−Pb, and (○) Z = P−Sb] plotted vs the rZ(CN12) radii of the interstitial element.

the R:Mg ratio only, one would expect to find values between the two neighboring curves. The presence of the interstitial carbon E


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Inorganic Chemistry

Table 4. Fractional Atomic Coordinates and Equivalent Displacement Parameters for Ce6Mg23Ge and Pr6Mg23C (Zr6Zn23Si type, Cubic, cF120, Fm3̅m, Z = 4), As Obtained from Single-Crystal X-ray Analysis Ce6Mg23Ge

Pr6Mg23C

atom

Wyckoff site

x

y

z

Ueq (Å2)

Ce Mg1 Mg2 Mg3 Mg4 Ge Pr Mg1 Mg2 Mg3 Mg4 C

24e 32f 32f 24d 4b 4a 24e 32f 32f 24d 4b 4a

0.20535(3) 0.1694(1) 0.3782(1) 0 1 /2 0 0.1894(1) 0.1706(2) 0.3793(2) 0 1 /2 0

0 0.1694(1) 0.3782(1) 1 /4 1 /2 0 0 0.1706(2) 0.3793(2) 1 /4 1 /2 0

0 0.1694(1) 0.3782(1) 1 /4 1 /2 0 0 0.1706(2) 0.3793(2) 1 /4 1 /2 0

0.0143(1) 0.0142(4) 0.0161(4) 0.0151(5) 0.018(1) 0.0217(5) 0.0301(5) 0.010(1) 0.021(1) 0.021(1) 0.023(4) 0.08(3)

Table 5. Interatomic Distances for Ce6Mg23Ge and Pr6Mg23C (Zr6Zn23Si type, cF120, Fm3̅m, Z = 4), As Obtained from Single-Crystal X-ray Analysis Ce6Mg23Ge atom 1 Ce

Mg1

Mg2

Mg3

Mg4 Ge

Pr6Mg23C

atom 2

distance (Å)

Ge 4 Mg1 4 Mg2 4 Mg3 4 Ce Mg4 3 Mg3 3 Mg2 3 Mg1 3 Ce Mg4 3 Mg3 3 Mg1 3 Mg2 3 Ce 4 Mg1 4 Mg2 4 Ce 8 Mg2 6 Ce 6 Ce

2.998(1) 3.536(2) 3.562(1) 3.707(1) 4.240(1) 4.301(1) 2.980(1) 3.203(2) 3.328(4) 3.536(2) 3.078(3) 3.189(1) 3.203(2) 3.555(3) 3.562(1) 2.980(1) 3.189(1) 3.707(1) 3.078(3) 4.301(1) 2.998(1)

atom 1 Pr

Mg1

Mg2

Mg3

Mg4 C

atom 2

distance (Å)

C 4 Mg1 4 Mg2 4 Mg3 4 Pr Mg4 3 Mg3 3 Mg2 3 Mg1 3 Pr Mg4 3 Mg3 3 Mg1 3 Mg2 3 Pr 4 Mg1 4 Mg2 4 Pr 8 Mg2 6 Pr 6 Pr

2.745(2) 3.507(4) 3.700(2) 3.728(1) 3.882(1) 4.501(1) 2.960(1) 3.192(5) 3.255(8) 3.507(4) 3.030(6) 3.175(2) 3.192(5) 3.499(7) 3.700(2) 2.960(1) 3.175(2) 3.728(1) 3.030(6) 4.501(1) 2.745(2)

Table 6. Germanium Coordination Numbers (CNGe) and Shortest Ce−Ge Distances in Binary Ce−Ge Compounds

Figure 4. Zr6Zn23Si-type structure of Ce6Mg23Ge. a−b plane content (left panels), at different z portions (z = 0, 0.1−0.2, 0.2−0.3, 0.3−0.4, and 0.5), viewed along [001] and [100]. Three perspective unit cell views (right panels), along the ⟨100⟩, ⟨110⟩, and ⟨111⟩ directions. Gecontaining Ce6 octahedra are highlighted.

Mg4 pair (d = 4.30 Å). These values are quite comparable with those observed in CeMg3 (3.23−3.72 Å) and Ce5Mg41 (3.40− 4.17 Å). With regard to Ce−Ce coordination distances, it should be underlined that they are rather large, the shortest Ce−Ce distance (dmin Ce−Ce = 4.24 Å) being more than twice the Ce metallic

compd

type

CNGe

min dCe−Ge (Å)

ref

Ce3Ge Ce5Ge3 Ce4Ge3 Ce5Ge4 CeGe Ce3Ge5 CeGe2

Th3P Mn5Si3 anti Th3P4 Sm5Ge4 FeB Y3Ge5 ThSi2

9 11 8 9 9 10 9

3.104 3.102 3.190 2.980 3.120 2.990 3.160

33 34 34 34 34 35 34

radius [rCN12(Ce) = 1. 836 Å]. However, according to the maximum-gap rule,32 Ce atoms should be included in the polyhedron, which would otherwise appear to be rather incomplete. As a comparison, in all the other Ce−Mg phases (CeMg3, CeMg5, Ce5Mg41, Ce2Mg17, and CeMg12), the minimal F


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Inorganic Chemistry

CN12 Table 7. Minimal R−Z Distances (dmin , Calculated Radii and Volumes of the R−Z), Rare Earth Metallic Radii for Coordination 12 rR min CN12 CN12 Z Element (rZ = dR−Z − rR , VZ), Compared with the CN12 Z Volumes (VZ ), and Relative Volume Changes in Passing from CN12 to CN6 in the Octahedral Environment [ΔVZ = (VZ − VCN12 )/VCN12 ]) for the R6Mg23Z Compounds Z Z

compd

dmin R−Z (Å)

rCN12 (Å) R

rZ (Å)

VZ (Å3)

VCN12 (Å3) Z

ΔVZ (%)

dmin R−R/2 (Å)

La6Mg23C Ce6Mg23C Pr6Mg23C Nd6Mg23C Sm6Mg23C Gd6Mg23C Ce6Mg23C Ce6Mg23Si Ce6Mg23Ge Ce6Mg23Sn Ce6Mg23P Ce6Mg23As Ce6Mg23Sb La6Mg23P La6Mg23Sb

2.805(3) 2.784(3) 2.768(2) 2.757(2) 2.742(3) 2.719(6) 2.784(3) 3.010(3) 2.994(2) 3.185(3) 2.934(2) 2.999(2) 3.175(2) 2.967(3) 3.196(2)

1.877 1.825 1.828 1.821 1.802 1.802 1.825 1.825 1.825 1.825 1.825 1.825 1.825 1.877 1.877

0.928(4) 0.959(4) 0.940(3) 0.936(3) 0.940(4) 0.917(7) 0.959(4) 1.185(4) 1.169(3) 1.360(4) 1.109(3) 1.174(3) 1.350(3) 1.090(4) 1.319(3)

3.35(5) 3.70(5) 3.48(3) 3.44(3) 3.48(5) 3.23(7) 3.70(5) 6.97(7) 6.69(6) 10.54(9) 5.71(4) 6.77(5) 10.32(6) 5.42(6) 9.62(7)

3.22 3.22 3.22 3.22 3.22 3.22 3.22 9.61 10.75 17.91 8.78 11.25 16.84 8.78 16.84

4.0 14.9 8.1 6.7 8.0 0.4 14.9 −27.5 −38.7 −41.1 −35.0 −39.8 −38.7 −38.3 −42.9

1.983(2) 1.969(2) 1.957(1) 1.950(1) 1.939(2) 1.923(4) 1.969(2) 2.128(2) 2.117(2) 2.252(2) 2.075(1) 2.120(1) 2.245(1) 2.098(2) 2.260(2)

Table 8. Comparison of Octahedral Cavities and Carbon Dimensions in R6Mg23C and Some Other Ce, Pr, and Nd Compoundsa Ce compd R6Mg23C R5Ge3 R5Ge3C R15Ge9C R3Sn R3SnC

ddiag RR

(Å)

5.569 5.357 − 5.327 4.929 5.101

Pr rC/cav (Å) 0.959 0.87 0.87 0.837 0.640 0.726

ddiag RR

Nd rC/cav (Å)

(Å)

5.536 5.320 − 5.249 4.99 5.060

0.940 0.832 0.87 0.797 0.667 0.702

ddiag RR

(Å)

5.338 − 5.171 5.002 5.028

rC/cav (Å)

ref

0.936 0.848 0.865 0.764 0.68 0.693

this work 38 38 39 40, 41 40, 41

a

The shortest octahedron square base diagonal, ddiag RR , is given. For non-purely trivalent Ce, the 1.825 Å value reported in ref 37 was used (valence = 3.1).

fact, in the Ce−Ge binary system,2,36 germanium atoms always have higher coordination numbers, ranging from 8 to 11. As listed in Table 6, dmin Ge−Ce values are larger and comparable with the sum of the radii, the exceptions being Ce5Ge4 and Ce3Ge5, which show similar values. 3.2.2. Trends in Structural Parameters. The term interstitial refers to the fact that large rare earth atoms can be sometimes arranged in space in such a way that some voids, or interstices, appear and they are usually able to accommodate small elements like H, B, C, and N. In the R6Mg23Z families of compounds, not only small but also medium and large elements are accommodated in the octahedral cavity. It is thus interesting to focus on the aspect that is more influenced by the insertion of the varius Z atoms, namely the dimension of the Z environment itself, formally considered as an interstitial cavity (R 6 octahedron). Relevant parameters useful for a geometrical analysis of the interstitial cavity in the studied compounds are listed in Table 7: minimal R−Z distances (dmin R−Z), calculated radii and volumes of Z elements in the R6Mg23Z compounds (rZ and VZ, respectively) in ),37 and the relative comparison with the CN12 Z volumes (VCN12 Z volume change (ΔVZ) in passing from CN12 to CN6 in the R6Mg23Z octahedral environment. Carbon has a large radius (rC̅ = 0.937 Å) showing only slight changes in passing from La to Gd. An analysis of carbon dimensions in interstitial cavities, reported in Table 8, showed that this is one of the largest values ever encountered. Interstitial carbon can either shrink (R15Ge9C), enlarge (R3SnC), or leave the octahedral cavity unchanged (R5Ge3C), but the cavity radius is always significantly smaller

Figure 5. Atomic volumes of the Z atoms in the octahedral environment ), in [VZ (●)] as a function of the CN12 atomic volume (VCN12 Z comparison with the same CN12 volume (○, straight line) and with the smallest atomic volume in binary Ce−Z compounds.

Ce−Ce distance exceeds 5.2 Å, which clearly indicates that Ce atoms are not directly bound to each other. Germanium atoms, like all interstitial Z elements in the series, occupy the 4a site, octahedrally coordinated by six Ce atoms. Like for the Si−Zr distance in Zr6Zn23Si,10 here too it can be noticed that dmin Ge−Ce is rather short, being 2.998(1) Å smaller than the sum of the CN12 radii [rCN12(Ce) + rCN12(Ge) = 3.194 Å]. This could be related to the small coordination number of the site and a strong bonding interaction between the constituents. In G


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Inorganic Chemistry

Figure 8. (a) Mg4 (4b Wyckoff position) coordination polyhedron, corresponding to the first inner shell of the cluster in panel b. (b) Rhombic dodecahedron cluster, centered in Mg4. Mg−Mg bonds are highlighted with thick blue lines. (c) fcc lattice, with the origin translated by (1/2, 1/2, 1/2). The cuboctahedron coordination polyhedron (CN = 12) is depicted. Each sphere represents the cluster in panel b.

described as a standard octahedral environment; this is confirmed by the good agreement with the minimal R−Z distances found in binary Ce−Z compounds (Table S15). As proof, Figure 5 shows the volume of the Z element in the compound (VZ) as a function of the corresponding CN12 atomic volume (VCN12 ), in comparison with the smallest atomic volume Z found in binary Ce−Z compounds. As a reference value, the CN12 volume itself is also reported (○). The R atoms are moved apart from each other by the Z element, so that they are not in contact and the octahedron is larger than in the geometrically ideal case.42 Moreover, it should be noted that the compounds described here form with a volume expansion that is larger than what could be reasonably expected on the basis of the binary Ce−Mg compositional ratio. In addition, we have previously accounted for the fact that the Th6Mn23-type structure is formed by the larger alkaline earths strontium and barium with magnesium; we could then tentatively assign the specific effect of enlarging the R 6 octahedron to the presence of the Z element. This may contribute to favor the formation of the Th6Mg23-type framework. From another perspective, it is also worth noting that the Mg framework is able to respond to the dimensional variation of the octahedral moiety with Z. We could account for this by looking more closely at the interactions between R6Z and the Mg matrix visible in Figure 6a. It has been built up by considering that each of the six Ce atoms (vertices of the inner octahedron) has its own coordination sphere of 13 Mg atoms (4 Mg1, 4 Mg2, 4 Mg3, and 1 Mg4) and four Ce atoms belonging to the same R6Z and forming a square centered around the interstitial atom (Figure 6b). Mg−Ce connections, colored red in Figure 6a, involve Mg1 and Mg3 atoms in contact with more than one Ce atom belonging to the same octahedron. Twenty Mg1 and Mg3 atoms around R6Z form the cuboctahedron depicted in Figure 6c. Via addition of the next neighbor Mg2 atoms to this primary cage, forming regular square motives, Figure 6d is finally revealed. Because this rather complicated environment is formed by a combination of Ce shells, it is sufficient to focus on a single Ce coordination polyhedron to gain some insight into the whole shell. It is therefore interesting to compare the set of Ce−X distances (X being each of the atoms forming the polyhedron) of all the Ce6Mg23Z compounds with those of the other known R6Mg23 compounds, i.e., with Ba6Mg23, Sr6Mg23,43 and Th6Mg23.44 The following structural observations, although apparently timeconsuming, help us to understand how the same structure type

Figure 6. (a) Coordination superpolyhedron of the Ce6Z moiety (octahedron surface in transparent red), built as a convolution of six Ce coordination polyhedra (one is shown with a transparent gray surface). Mg−Ce connections colored red identify Mg atoms bound to multiple Ce atoms belonging to the same octahedron. The surface of one Ce coordination polyhedron is drawn. (b) Ce coordination polyhedron. (c) Cuboctahedron formed by the closest Mg1 and Mg3 atoms (red connections highlighting its shape). (d) Cuboctahedron, surrounded by the nearest neighbor Mg2 atoms, forming squares around it. Mg4 atoms have been removed from panels c and d for the sake of clarity.

Figure 7. dCe−X (X = Ce, Mg1, Mg2, Mg3, or Mg4) as a function of the metallic radius (CN12) of the Z element in the Ce6Z moiety. Gray bars on the right side represent the ranges of the corresponding dR−X values (with X as per the central symbol) observed for the known Ba6Mg23 (upper limit) and Th6Mg23 (lower limit) compounds. dCe−X points marked with a star fall outside this range.

than that of R6Mg23C. The same applies to all members of the Ce6Mg23Z series. This geometrical evidence suggests that what we have termed cavity in our analysis, in these compounds would be better H


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Inorganic Chemistry

Figure 9. (a) Mg45 cluster, centered in Mg4 (4b Wyckoff position), isolated from the whole cluster shown in Figure 8b. External Mg bonds are depicted as thick blue lines while internal bonds as thin gray lines. Symmetrically different Mg species are indicated. Faces fa and f b are highlighted, as a guide for comments given in the text. (b) One section of the unit cell, centered at z = 1/2, resulting in a NaCl-type arrangement of the Mg45 cluster and the R6Z octahedron. On the left of the plane, the cuboctahedral R6Z environment is highlighted. (c) NaCl-type structure. Blue and red spheres represent the Mg45 cluster and the R6Z octahedron, respectively.

3.2.3. Topological Analysis. The idea to simplify complex cubic structures as an intergrowth of nested polyhedra units was introduced in the 1980s.45,46 According to this geometric conception, Th6Mn23 (or its stuffed variant) could not be described unambiguously. To provide an alternative description of the complicated cubic structure of the R6Mg23Z phases, the idea of nanocluster search in intermetallic compounds introduced recently47,48 and further analysis of their topological arrangement were applied by using the ToposPro program.49 The program output provides evidence that the whole structure can be built up by a unique primary two-shell nanocluster, repeated according to the fcc lattice scheme. This atomic cluster, centered on the Mg4 atom [4b Wyckoff position in (1/2, 1/2, 1/2), coordination polyhedron reported in Figure 8a], has the shape of a truncated rhombic dodecahedron and contains all the atomic species of the structure (Figure 8b). All the atoms lying on the face of one cluster are shared with another adjacent cluster. Each face is perpendicular to the direction joining two clusters, so that a Wigner−Seitz-type relationship holds between the shape of the cluster (rhombic dodecahedron) and the cuboctahedral coordination polyhedron of each fcc lattice point (Figure 8c). To improve our understanding of the whole structure and its structural relationships, it may be useful to remove the interstitial atoms (green) and the R atoms (red) from the rhombic dodecahedron by considering that the latter are part of a R6Z octahedron, which in turn we consider to be a fundamental structural motive. As a result, a new reduced arrangement is obtained, which clearly appears to be a complex Mg45 cluster (Figure 9a). The whole structure can be obtained by alternating in the three directions the Mg45 cluster and the R6Z octahedron, in a NaCl-like arrangement (Figure 9b,c). In the section of the unit cell, centered at z = 1/2, it can be noted that Mg forms a complex homogeneous network of Mg1 and Mg3 atoms, presenting large and regular zeolite-like voids (gray cuboctahe-

can appear in different systems, and also help to infer that this arrangement is robust and can adapt to dimensional variations of its structural motives. Figure 7 shows the various dCe−X values (X = Ce, Mg1, Mg2, Mg3, or Mg4) as a function of the metallic radius (CN12) of the Z element in Ce6Z. Gray bars on the right show the range of the corresponding dR−X values (X indicated by the central symbol) observed for the known Ba6Mg23 (upper limit) and Th6Mg23 (lower limit), with values for Sr6Mg23 falling within this range; dCe−X points marked with a star fall outside the corresponding range. By looking at the trends, we find it appears that the distance with the nearest Mg3 atoms, forming the vertices of the cuboctahedron (Figure 6c), slightly decreases and falls within the range of the R6Mg23 phases. Because the dimensions of R6Z increase with Z (see Table 7), the cuboctahedron is also slightly enlarged [as the Mg3−Mg3 distance edge (trend not shown)]. This is reflected in the increase in the a parameter with an increase in rZ (see Figure 2b); it is obvious that a linear relationship holds between the growth of dMg3−Mg3 and the a parameter, because Mg3 occupies the fixed 24d Wyckoff position in (0, 1/4, 1/4). Ce−Mg1 distances (●; Mg1 atoms are placed in the centers of the cuboctahedron triangular faces) also tend to lie in the corresponding range. When the value falls outside of this range, this is compensated by the dCe−Mg2 value (○) that remains in range and vice versa. The Ce−Ce distance (□) grows with rZ, while Ce−Mg4 and Ce−Mg2 distances (△ and □, respectively) follow an opposite trend. With the larger Sn and Sb atoms, the larger octahedron is compensated by shorter Ce−Mg4 and Ce− Mg2 distances. The opposite is observed for carbon, where a smaller octahedron is compensated by longer Ce−Mg4 and Ce− Mg2 distances. From this analysis, we can conclude that the increase in the R6Z dimensions is well tolerated by the Mg framework through small adjustments of the interatomic distances. I


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Inorganic Chemistry

Figure 11. Comparison of the [MgMg8Ce6] and [CCe6Mg8] closest environments in (a) CeMg3 and (b) Ce6Mg23C structures.

the clearer rationalization of this rather complicated cubic structure, to give it a meaningful crystallochemical interpretation. 3.2.4. Symmetry Relationships Accompanying the Formation of Interstitial R 6Mg 23 Z Phases. As previously mentioned, the formation of the R6Mg23Z phases is always accompanied by the occurrence of the corresponding BiF3-type RMg3 phases (Table 2). The thermodynamic stability of the 6:23:1 phase appears to decrease along the series, as can be inferred both from the lower yield and from its nonoccurrence starting from Ho forward. It is worth recalling here that the 1:3 phase exists in all the R−Mg binary systems (where R is a trivalent rare earth element from La to Dy); for the first four members of the series (La, Ce, Pr, and Nd), its formation is congruent-type, while it occurs peritectically for the others (Sm, Gd, Tb, and Dy). The formation temperature decreases almost regularly from 798 °C (La) to 520 °C (Dy), thus again indicating a lower stability along the 1:3 series. On the other hand, in the case of divalent Eu, the 1:4 stoichiometry is observed instead of the 1:3 one. Analogously, in the binary A−Mg systems formed with divalent alkaline earth metals (A = Sr or Ba), the 1:3 phase does not exist while the Mg-richer phase A6Mg23 is found to occur, the composition of which is very close to the 1:4 composition. Such features triggered our interest in looking for more detailed structural relationships between the RMg3 and R6Mg23Z phases. Crystallographic group theory was applied to disclose the relationships between the structures. First, the metrical relationship between unit cell dimensions of RMg3 and R6Mg23Z must be highlighted; i.e., aR6Mg23Z ≅ 2aRMg3. This is valuable information in searching for group−subgroup relationships.51 The symmetry relationships between them were formalized in the Bärnighausen-tree form (Figure 10). Evolution of the Wyckoff positions of the involved structural models was checked with the aid of the ToposPro program.49 This presentation is the most concise way to depict the volume effects that take place and to explain the crystallochemical peculiarities of the 6:23:1 structure. As shown in Figure 10, two main steps are needed to obtain the R6Mg23Z model starting from the RMg3 one. The first is a klassengleiche reduction of index 4 (k4) giving rise to the Pm3̅m model, cP16, corresponding to the Fe13Ge3 structure type. The second step is the doubling of the translation vector through a klassengleiche reduction (k2) and a subsequent (1/2, 1/2, 1/2) origin shift. The obtained cF128-f 2edcba structural model is a hypothetical intermediate, having no real representatives. In the R6Mg23Z structure, the 8c site remains vacant and the 4a positions, originally occupied by Mg atoms in RMg3, are substituted with Z atoms. In the A6Mg23 structural model (Th6Mg23 type), the 4a site is also vacant. The presence of

Figure 10. (a) Group−subgroup relationships in the Bärnighausen formalism for the RMg3, R6Mg23Z, and A6Mg23 structures. Indices of the klassengleiche (k) transitions, shifts in origin, Wyckoff site positions splitting schemes, and unit cell transformations are given. (b) Unit cell projections of RMg3 and of R6Mg23Z are shown with respect to the proposed transformations. Homo- and heterostuffed variants of these families are discussed in the Introduction.

dron highlighted in Figure 9b). Inside the Mg45 cluster, analogous voids are filled by a Mg9 body-centered cube, formed by 8 Mg2 atoms and 1 Mg4 atom. It is just the stacking of these clusters that ultimately forms the whole framework,50 and this latter hosts in turn the R6Z octahedra. To improve the visualization of theses features, the central sphere in the NaCl arrangement (blue), bonded with its 12 homologous nearest neighbors, can help as a reference. This new perspective allows J


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Inorganic Chemistry

Figure 12. Total and projected DOS for (a) Ba6Mg23 and (b) La6Mg23C (EF is set at 0 eV).

Ce6Mg23C, the CCe6 octahedron is drastically decreased (dC−Ce = 2.797 Å); as a result, the Mg8 cube is noticeably enlarged (dC−Mg = 4.346 Å). The chemical importance of the R6Z moiety as a stabilizing factor will be discussed in the next paragraph. 3.3. Electronic Structure Analysis of R6Mg23Z. Symmetry relations discussed above motivated us to conduct a comparative chemical bonding analysis for 6:23 binary and 6:23:1 ternary families of compounds. The difference between the title R6Mg23Z and A6Mg23 (A = Ba or Sr) compounds is the empty interstitial 4a position of the latter. Moreover, R6IIIMg23 compounds (R = rare earth) were never reported, and therefore, they can be considered nonstable under standard conditions. The introduction of the interstitial atom confers stability to this atomic arrangement. With these observations in mind, key information should be sought in the comparative analysis of the electronic structures of the aforementioned compounds: Ba6Mg23 and La6Mg23C. Ba was chosen because it differs from La for only one electron and La in turn because it is a

different sets of vacancies explains the different chemical compositions of the compounds in focus (see the Introduction). Unit cell projections of RMg3 and R6Mg23Z compounds are shown in Figure 10b with respect to the described transformations. The mechanism of the proposed structural change also allows us to compare the values of the refined fractional positions of atomic species in R6Mg23Z with respect to the ideal values in RMg3 (namely, cF120 and cF128 models in Figure 10). From that, it is evident that R (24e) and Mg2 (32f) sites suffer the largest absolute shift. This can be illustrated by depicting the closest atomic environments of the Mg atom in the parent binary CeMg3 compound and the corresponding C position in Ce6Mg23C (see Figure 11). The former coordination polyhedron is a regular rhombic dodecahedron in which Ce atoms form a regular octahedron. The distance between the central Mg atom and the six Ce atoms (dMg−Ce) is 3.721 Å, while with the eight cube-forming Mg atoms, dMg−Mg is 3.223 Å. On the other hand, in K


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Inorganic Chemistry

Figure 13. Crystal orbital Hamilton populations (-COHP) per bond for La6Mg23C, as obtained from LMTO calculations.

Table 9. Integrated Crystal Orbital Hamilton Populations (-iCOHP, electronvolts per bond per cell) at EF for the Strongest Contacts within the First Coordination Spheres in La6Mg23C and Ba6Mg23 (d < 4 Å) atom 1

atom 2

d (Å)

-iCOHP (eV)

atom 1

atom 2

d (Å)

-iCOHP (eV)

Mg2

Mg4 3 Mg3 3 Mg1 3 Mg2 3 La 4 Mg1 4 Mg2 4 La

3.013(3) 3.219(3) 3.232(6) 3.479(5) 3.700(4) 2.990(5) 3.219(3) 3.752(2)

0.93 0.72 0.57 0.57 0.44 0.83 0.72 0.47

Mg4

8 Mg2

3.013(3)

0.93

Mg2

Mg4 3 Mg3 3 Mg1 3 Mg2 3 Ba

3.214 3.272 3.320 3.711 3.738

0.96 0.61 0.71 0.41 0.40

Mg3

4 Mg1 4 Mg2 4 La 8 Mg2

3.119 3.320 3.869 3.214

0.85 0.71 0.35 0.96

La6Mg23C La

C6 4 Mg1 4 Mg2 4 Mg3 4 La 3 Mg3 3 Mg1 3 Mg2 3 La

2.806(2) 3.608(5) 3.700(4) 3.752(2) 3.968(2) 2.990(5) 3.138(6) 3.232(6) 3.608(5)

1.74 0.58 0.44 0.47 0.39 0.83 0.55 0.57 0.58

C

6 La

2.806(2)

1.74

Ba

4 Mg1 4 Mg2 4 Mg3 4 Ba3

3.738 3.848 3.869 4.367

0.40 0.43 0.35 0.26

Mg1

3 Mg3 3 Mg1 3 Mg2 3 Ba

3.098 3.119 3.272 3.848

0.68 0.85 0.61 0.43

Mg1

Mg3

Ba6Mg23

Mg4

nonmagnetic rare earth: f elements are difficult to handle in the performed calculations. With the aim of exploring the Ba6Mg23 chemical bonding scenario, electronic structure calculations were performed on the model published by Wang.43 Total and projected densities of states (DOS) of Ba6Mg23, calculated by the TB-LMTO-ASA method, are shown in Figure 12b (Fermi level at 0 eV). The presence of electron states on the Fermi level indicates metallic character for this compound. Similar contributions to the total DOS from Mg p orbitals and Ba d orbitals (not shown in the figure) highlight that these interactions are of great importance for this compound. The electronic structure of La6Mg23C was analogously computed (the experimental cell parameters are listed in Table 3, while atomic positions are available as Supporting

Information). Total and projected densities of states of La6Mg23C are shown in Figure 12b. Fermi level EF lies in a continuous DOS region, indicating metallic character for the title compound, like for Ba6Mg23. DOS in the valence band appears to be significantly dominated by magnesium, this element contributing to a widespread area of states, from −6 eV to the Fermi level. It has to be noted that both s and p Mg orbitals contribute to the bonding; La behaves as a typical electropositive element, participating with its s and d orbitals, as recently reported for other cases52,53 and as expected from a chemical perspective. The empty 4f states provide a wide prominent peak in the conduction band from 1.5 to 3.0 eV. Also, from the projected DOS, one can note the very similar shape of La and C state contributions (5d and 2p orbitals, respectively) in the range L


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Inorganic Chemistry of −3 to −1 eV. This finding indicates that a strong interaction between the two species exists; this is in agreement with the observed reciprocal environments and the interatomic distances (i.e., CLa6 octahedra). Furthermore, to compare the interatomic orbital interactions in La6Mg23C and Ba6Mg23, crystal orbital Hamilton population analyses have been performed (see the -COHP curves in Figure 13 for La6Mg23C). Positive values in the -COHP curves represent bonding interactions and negative values antibonding ones; in our case, all the interactions are of the bonding type all over the VB region. The weighted and integrated crystal orbital Hamilton population (-iCOHP) values for near neighbor contacts are summarized in Table 9. The strongest bonds belong to polar La− C interactions (1.74 eV per bond per cell); from these data, one can easily conclude that the addition of carbon plays a key role in La6Mg23C stabilization. Because Mg and C atoms are spatially separated by La, Mg−C interactions are weak and negative in sign, i.e., of antibonding character. Mg−Mg contacts, common to both phases, show similar values; the stronger Mg2−Mg4 interaction is consistent with the atomic arrangement, Mg4 being located at the center of the cluster with the Mg2 atoms forming its nearest cubic shell. It has to be noted that the calculated Mg− Mg -iCOHP value for elementary hcp magnesium (0.69 eV) is noticeably weaker than that for the strongest Mg4−Mg2 interaction (0.93 eV), being instead comparable with values of all the other Mg−Mg interactions calculated for this structure. Summarizing the difference in the chemical bonding between La6Mg23C and Ba6Mg23 compounds, we want to emphasize that each La atom contributes with its three valence electrons to form the ternary compound while each Ba contributes with only two. Electronegative carbon (and analogously all the other interstitial elements in the R6Mg23Z series), located at the center of the La6 octahedron, is likely able to attract and polarize on itself part of the excess electronic charge delivered by the six La atoms. In the absence of C, the structure does not tolerate the charge in excess. The positively charged [La6C]δ+ units can be then counterbalanced by [Mg45]δ− clusters, closely resembling those found in Ba6Mg23. From this rather qualitative picture, it can be envisaged that the ability of the interstitial atom to stabilize part of the excess charge appears to be the key factor in determining the formation of the R6Mg23Z compounds. As a final remark, we can cite how the La−Z interaction changes in passing to the other La6Mg23Z analogues. The -iCOHP values for Z = C, P, and Sb are 1.74, 1.63, and 1.50 eV, respectively. They increase with Z electronegativity: the more electronegative Z is, the stronger the interactions between these two species.

With a change in the interstitial element Z inside the R6 octahedra, the interatomic contacts dR−X (X = R and Mg) adjust so that shortening one of them is compensated by lengthening of the other. (4) The results of electron structure calculations qualitatively confirm the role of the octahedral moiety in stabilizing the 6:23:1 structure with respect to the 6:23 one. The complex crystal structure of the 6:23:1 compounds can be conveniently described in a simplified way, as built up by two fundamental blocks: strongly bonded [R6Z]δ+ octahedral moieties (formed by R atoms and centered by Z) and [Mg45]δ− polyatomic clusters. Considering them as structural blocks, this complex crystal structure can be simply described by the NaCl-like topology. Moreover, a group−subgroup relationship (Bärninghausen formalism) is established to exist between the 6:23:1 and 1:3 phases (the formation of the latter always is in competition with the formation of the ternary phase); the addition of the interstitial atoms induces vacancy formation and lowering of symmetry with respect to the RMg3 parent type. The geometrical distortion accompanying this substitution finds sound explanation from both topological and crystal chemical points of view.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.5b02114. Rietveld plots and corresponding tables with refined coordinates and a table containing the shortest distances in binary Ce−Z compounds (PDF) Listings of CIF files (ZIP)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: chimfi[email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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4. CONCLUSIONS A group of novel interstitial intermetallics with a 6:23:1 stoichiometry was discovered along the two series R6Mg23C with R = La−Sm or Gd and Ce6Mg23Z with Z = C, Si, Ge, Sn, Pb, P, As, or Sb. All these compounds are isostructural and belong to the cubic Zr6Zn23Si prototype, with cF120 Pearson’s code and Fm3m ̅ space group. The observation that they are stable for a large number of different combinations of the constituent atoms prompted us to conduct a detailed crystal chemical analysis, which revealed several interesting features and regularities. (1) The lattice parameter varies according to the lanthanide contraction and also regularly increases with an increase in Z dimensions. (2) The formation of the phase occurs with negative volume effects, i.e., with an overall volume expansion, which is quite unusual. (3) M


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(45) Chabot, B.; Cenzual, K.; Parthé, E. Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. 1981, 37, 6−11. (46) Parthé, E.; Gelato, L.; Chabot, B.; Penzo, M.; Cenzual, K.; Gladyshevskii, R. TYPIX Standardized data and crystal chemical characterization of inorganic structure types; Springer-Verlag: Weinheim, Germany, 1994. (47) Blatov, V. A.; Ilyushin, G. D.; Proserpio, D. M. Inorg. Chem. 2010, 49, 1811−1818. (48) Blatov, V. A.; Ilyushin, G. D.; Proserpio, D. M. Inorg. Chem. 2011, 50, 5714−5724. (49) Blatov, V.; Shevchenko, A.; Proserpio, D. Cryst. Growth Des. 2014, 14, 3576−3586. (50) As a proof of concept, we find it is useful to link the stoichiometric coefficient of the Mg45 cluster (45) with that of the R6Mg23Z formula (23). Each of the eight fa planes of a Mg45 cluster (Figure 9), formed by 3 Mg1 atoms (total of 24 Mg1 atoms), is part of a tetrahedron shared by four adjacent clusters and is thus part of three clusters; Mg3 atoms in the middle of 12 f b faces (total of 12 Mg3 atoms) are shared by two clusters only. Mg9 cubes (8Mg2 atoms and 1Mg4 atom) are not shared. Therefore, it holds that 8Mg2 + 1Mg4 + 12Mg3/2 + (8 × 3Mg1)/3 = 23 Mg atoms. (51) Müller, U. Symmetry relationships between crystal structures: Applications of crystallographic group theory in crystal chemistry; Oxford University Press: Oxford, U.K., 2013; pp 320. (52) Solokha, P.; De Negri, S.; Skrobanska, M.; Saccone, A.; Pavlyuk, V.; Proserpio, D. M. Inorg. Chem. 2012, 51, 207−214. (53) Solokha, P.; De Negri, S.; Proserpio, D. M.; Blatov, V. A.; Saccone, A. Inorg. Chem. 2015, 54, 2411−2424.

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