A new glance on R₂MGe₆ (R = rare earth metal, M = another metal) compounds. An experimental and theoretical study of R₂PdGe₆ germanides

Abstract

The R₂PdGe₆ series (R = rare earth metal) was structurally characterized, and the results achieved were extended for a comprehensive study on R₂MGe₆ (M = another metal) compounds, employing symmetry-based structural rationalization and energy calculations. Directly synthesized R₂PdGe₆ exists for almost all R-components (R = Y, La–Nd, Sm and Gd–Lu) and even if with La is probably metastable. Several single crystal X-ray analyses (R = Y, Ce, Pr, Nd, Er and Lu) indicated oS72-Ce₂(Ga_0.1Ge_0.9)₇ as the correct structure. The alternative In-flux method, once optimized, produced three good quality R₂PdGe₆ single crystals: La₂PdGe₆ and Pr₂PdGe₆ turned out to be mS36-La₂AlGe₆-type non-merohedrally twinned crystals and Yb₂PdGe₆ is of oS72-Ce₂(Ga_0.1Ge_0.9)₇-type. The vacancy ordering phenomenon was considered as a possible cause of the symmetry reduction relations connecting the most frequently reported 2 : 1 : 6 structural models (oS18, oS72 and mS36) with the oS20-SmNiGe₃ aristotype. The detected twin formation is consistent with the symmetry relations, which are discussed even considering the validity of the different structural models. DFT total energy calculations were performed for R₂PdGe₆ (R = Y and La) in the three abovementioned structural models, and for La₂MGe₆ (M = Pt, Cu, Ag and Au) in the oS18 and oS72 modifications. The results indicate that the oS18-Ce₂CuGe₆ structure, prevalently proposed in the literature, is associated with the highest energy and thus it is not likely to be realized in these series. The oS72 and mS36 polytypes are energetically equivalent, and small changes in the synthetic conditions could easily stabilize any of them, in agreement with experimental results obtained by direct and flux syntheses.

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1. Introduction

Numerous germanium-rich compounds of idealized general formula R₂MGe₆ (R = rare earth metal, M = another metal) have been investigated since the eighties of the last century up to the present.¹ Some R₂MGe₆ are off-stoichiometric due to the M/Ge statistical mixture (as in Ce₂(Ga_0.1Ge_0.9)₇) or to partially occupied M sites (as in Dy₂Zn_1−xGe₆ (x ∼ 0.5)²). Scientific interest in these phases arises from the fact that they exist with many different metals M, ranging from the s-block (M = Li and Mg) to the p-block (M = Al and Ga) of the periodic table, including the major part of late transition elements. For this reason, title compounds are well suited for systematic studies on crystal structure–electronic structure–property relationships, aiming to evidence for example the role of both R and M properties (size, electronegativity, valence electrons, etc.) in the chemical bonding scenario, set up by characteristic Ge-based covalent fragments.

A variety of physical properties, including electrical resistivity, magnetic susceptibility, specific heat and thermoelectric power, have already been measured on several R₂MGe₆ representative samples, mostly with transition elements, such as Co, Ni, Pd, Pt and Cu, as M components.^3–9

Many of the investigated compounds exhibit an antiferromagnetic ordering below ∼30 K;^3,4,8 for some Yb₂MGe₆ phases a behaviour characteristic of intermediate valence systems has been reported.^5,9,10

Despite the great amount of experimental work, controversial data exist both on the interpretation of the physical/magnetic properties and on the crystal structures of the R₂MGe₆ compounds. They are generally reported to crystallize in orthorhombic or monoclinic space groups, distributing among the following prototypes: oS18-Ce₂CuGe₆ (space group: Amm2), which is the most represented, oS72-Ce₂(Ga_0.1Ge_0.9)₇ (space group: Cmce) and mS36-La₂AlGe₆ (C2/m). A different orthorhombic structure (space group: Cmmm) has been assigned to a few R₂LiGe₆ (R = La, Ce and Pr) phases, referred to as the oS18-Pr₂LiGe₆ prototype^11,12 and to Yb₂Pd_1.075(1)Ge₆, referred to as the oS20-SmNiGe₃ prototype.¹³

The oS72-Ce₂(Ga_0.1Ge_0.9)₇ structure was found to be the correct one for a number of compounds previously reported as oS18, such as La₂PdGe₆, Dy₂PdGe₆ ^7,14 and Yb₂PdGe₆,^9,15 and for several new 2 : 1 : 6 representative samples, such as La₂MgGe₆ ¹⁶ and R₂ZnGe₆.² Interestingly, along the Zn-containing series of compounds, a structural change was observed from the orthorhombic oS72-Ce₂(Ga_0.1Ge_0.9)₇ structure (R = La–Nd, Sm, Gd and Tb) to the new monoclinic structure mP34-Dy₂Zn_1−xGe₆ (x ∼ 0.5, R = Dy and Ho), which is an ordered superstructure of the La₂AlGe₆ prototype.

Recently, the alternative synthesis in metal flux was employed for Yb₂CuGe₆, and the mS36-La₂AlGe₆ crystal structure was proposed for this compound, instead of the previously reported oS18-Ce₂CuGe₆ type.¹⁷

The abovementioned structural models oS18, oS72 and mS36 are strictly related, as can be pointed out both by means of group–subgroup relations² and by a description based on the linear intergrowth concept.^18,19 As a consequence, the correct model can be assigned only by single crystal X-ray diffraction analysis; powder X-ray diffraction could be sufficient only in the case of high quality patterns of single phase samples, generally quite difficult to obtain.

A correct structural description is of fundamental importance not only to evidence differences, similarities and regularities among the numerous 2 : 1 : 6 compounds in view of further chemical bonding studies, but also to appropriately interpret the results of magnetic characterization.

Taking into account the fact that the R₂PdGe₆ phases are reported to exist for many rare earth metals and that the structures of some of them were recently revised, this 2 : 1 : 6 series of compounds were targeted to perform an accurate structural investigation, mostly based on single crystal analysis.

The challenging experimental work necessary for this achievement was complemented both by a comprehensive crystal structure analysis based on group–subgroup relations and by DFT total energy calculations, in order to generalize the obtained results to the whole R₂MGe₆ family and guide future investigations on new compounds.

2. Experimental section

2.1 Synthesis, microstructure and phase analysis

Different synthetic routes were followed, all starting from the pure components (rare earth metals and In used as a metal solvent were supplied by Newmet Koch, Waltham Abbey, England, palladium and germanium by MaTecK, Jülich, Germany, nominal purities of all metals >99.9 mass %).

Samples of about 0.8 g with R_22.2Pd_11.1Ge_66.7 (R = Sc; Y; La–Nd; and Sm–Lu) nominal compositions were prepared by direct synthesis in a resistance furnace. Stoichiometric amounts of components were placed in an arc-sealed Ta crucible in order to prevent oxidation; the constituents of the Eu_22.2Pd_11.1Ge_66.7 sample were weighed and sealed in the crucible inside an inert atmosphere glove box. The Ta crucible was then closed in an evacuated quartz phial to prevent oxidation at high temperature, and finally placed in a resistance furnace, where the following thermal cycle was employed:

A continuous rotation, at a speed of 100 rpm, was applied to the phial during the thermal cycle. These synthetic conditions were chosen with the aim to obtain samples containing crystals of good quality and size, suitable for further structural studies. The synthesized alloys are very brittle and mostly stable in air, except for the Eu_22.2Pd_11.1Ge_66.7 sample. No tantalum contamination of the samples was detected.

After direct synthesis in the resistance furnace, some pieces of the La_22.2Pd_11.1Ge_66.7 sample were annealed at different temperatures. Further attempts to synthesize the La₂PdGe₆ compound were made, by arc melting of components, followed by annealing treatments (see the “Results and discussion” section).

Flux synthesis was also performed for samples containing La, Pr and Yb, using In as a metal solvent. Stoichiometric amounts of R, Pd and Ge, giving the nominal composition of R₂₁Pd₇Ge₇₂, were placed in an arc-sealed Ta crucible with a 1 : 45 molar excess of indium, to obtain a total mass of about 3 g. Then, the Ta crucible, closed in an evacuated quartz phial, was placed in a resistance furnace, and the following thermal cycles were employed:

For the La-containing sample both thermal cycles were tested, but only cycle III was employed for the Pr and Yb-containing samples.

A continuous rotation of the quartz phial during the thermal cycle was employed to favour better dissolution of the constituting elements inside the flux. In all cases, a vertical section of the obtained ingots revealed the presence of large shining crystals, visible to the naked eye, randomly distributed within the flux solidified matrix.

In order to perform metallographic analysis, samples were embedded in a phenolic resin with a carbon filler by using the automatic hot compression mounting press Opal 410 (ATM GmbH, Germany). The samples synthesized by the flux method were analysed prior to product separation: to avoid the metal solvent melting, they were embedded in a cold-curing resin, conductive due to the presence of a copper filler.

The automatic grinding and polishing machine Saphir 520 (ATM GmbH, Germany) was used to obtain smooth alloy surfaces suitable for microscopic examinations. Grinding was performed using SiC papers with grain size decreasing from 600 to 1200 mesh, using running water as a lubricant; for polishing, diamond paste with particle size decreasing from 6 to 1 μm was used with an alcohol based lubricant. Petroleum ether was employed to clean samples ultrasonically after each polishing step.

In the case of flux prepared samples only the 1200 mesh SiC paper was used applying a low pressure, due to In ductility.

After SEM-EDXS analysis, crystals of R₂PdGe₆ compounds (R = La, Pr and Yb) obtained by flux synthesis were extracted from the flux by immersion and sonication of the ingot in glacial acetic acid for about 24 h, allowing indium dissolution. The obtained crystalline product was rinsed with water and dried with acetone. After that, another SEM-EDXS analysis was performed in order to check the goodness of the separation procedure and the quality/composition of the isolated crystals.

Microstructure examination as well as qualitative and quantitative analyses were performed using a scanning electron microscope (SEM) Zeiss Evo 40 (Carl Zeiss SMT Ltd, Cambridge, England) equipped with a Pentafet Link Energy Dispersive X-ray Spectroscopy (EDXS) system managed using INCA Energy software (Oxford Instruments, Analytical Ltd, Bucks, UK). Cobalt standard was used for calibration.

2.2 X-ray diffraction (XRD) analysis on single crystal and powder samples

Single crystals of R₂PdGe₆ compounds prepared by both direct synthesis (R = Y, Ce, Pr, Nd, Er and Lu) and flux synthesis (R = La, Pr and Yb) were selected with the aid of a light optical microscope (Leica DM4000 M, Leica Microsystems Wetzlar GmbH, Wetzlar, Germany) operated in the dark field mode.

A full-sphere dataset was obtained in a routine fashion under ambient conditions on a four-circle Bruker Kappa APEXII CCD area-detector diffractometer equipped with graphite monochromatized Mo Kα (λ = 0.71073 Å) radiation, operating in ω-scan mode. Crystals exhibiting metallic luster, glued on glass fibres, were mounted in a goniometric head and then in the goniostat inside the diffractometer camera. Intensity data were collected over the reciprocal space up to ∼30° in θ with exposures of 20–30 s per frame. Semiempirical absorption corrections based on a multipolar spherical harmonic expansion of equivalent intensities were employed for all data using SADABS/TWINABS software.²⁰

The details of the crystal structure solution and refinement are reported in the “Results and discussion” section. The corresponding CIF files are available in the ESI† and they have also been deposited at Fachinformationszentrum Karlsruhe, 76344 Eggenstein-Leopoldshafen, Germany, with the following depository numbers: CSD 433026 (Y₂PdGe₆), CSD 433022 (La₂PdGe₆, flux), CSD 433081 (Ce₂PdGe₆), CSD 433025 (Pr₂PdGe₆), CSD 433205(Pr₂PdGe₆, flux), CSD 433024 (Nd₂PdGe₆), CSD 433021 (Er₂PdGe₆), CSD 433151 (Yb₂PdGe₆, flux), CSD 433023 (Lu₂PdGe₆).

Selected crystallographic data and structure refinement parameters for the studied single crystals are listed in Tables 1 and 2.

Table 1 Crystallographic data for R₂PdGe₆ (R = Y, Ce, Pr, Nd, Er and Lu) single crystals taken from samples prepared by direct synthesis and experimental details of the structural determination. All compounds are isostructural (space group: Cmce, No. 64; Pearson's symbol-prototype: oS72-Ce₂(Ga_0.1Ge_0.9)₇; Z = 8)

Empirical formula	Y2PdGe6	Ce2PdGe6	Pr2PdGe6	Nd2PdGe6	Er2PdGe6	Lu2PdGe6
EDXS data	Y21.0Pd10.8Ge68.2	Ce22.2Pd10.8Ge67.0	Pr20.2Pd11.4Ge68.4	Nd21.8Pd11.2Ge67.0	Er22.7Pd11.0Ge66.3	Lu23.8Pd10.6Ge65.6
M w, [g mol^−1]	719.76	822.18	823.76	830.42	876.46	891.88
a [Å]	8.1703(5)	8.3548(4)	8.3129(5)	8.2831(4)	8.1122(6)	8.0725(8)
b [Å]	8.0451(5)	8.1774(4)	8.1540(5)	8.1323(4)	8.0068(6)	7.9791(8)
c [Å]	21.558(1)	22.0272(9)	21.996(1)	21.915(1)	21.399(2)	21.317(2)
V [Å^3]	1417.1(2)	1504.9(1)	1491.0(2)	1476.2(1)	1389.9(2)	1373.0(2)
Calc. density [g cm^−3]	6.748	7.258	7.340	7.473	8.377	8.629
Abs. coeff. (μ), mm^−1	43.6	37.7	38.9	40.1	51.8	56.8
Unique reflections	1028	1308	1296	1307	1206	1225
Reflections I > 2σ(I)	884	1072	870	848	1020	874
R sigma	0.0233	0.0084	0.0270	0.0263	0.0133	0.0284
Data/parameters	884/50	1072/50	870/50	848/50	1020/50	874/50
GOF on F^2 (S)	1.166	1.543	1.059	1.002	1.326	0.912
R indices [I > 2σ(I)]	R 1 = 0.0232; wR2 = 0.0642	R 1 = 0.0208; wR2 = 0.0364	R 1 = 0.0197; wR2 = 0.0372	R 1 = 0.0165; wR2 = 0.0299	R 1 = 0.0202; wR2 = 0.0331	R 1 = 0.0224; wR2 = 0.0557
R indices [all data]	R 1 = 0.0282; wR2 = 0.0667	R 1 = 0.0260; wR2 = 0.0379	R 1 = 0.0366; wR2 = 0.0421	R 1 = 0.0351; wR2 = 0.0354	R 1 = 0.0264; wR2 = 0.0343	R 1 = 0.0399; wR2 = 0.0634
Δρfin (max/min), [e Å^−3]	1.31/−1.36	1.03/−1.29	1.01/−1.68	1.26/−1.05	1.09/−1.25	1.77/−1.63

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Table 2 Crystallographic data for R₂PdGe₆ (R = La, Pr and Yb) single crystals taken from samples prepared by flux-synthesis and experimental details of the structural determination

Empirical formula	La2PdGe6	Pr2PdGe6	Yb2PdGe6
EDXS data	La22.4Pd10.5Ge67.1	Pr20.6Pd11.3Ge68.1	Yb23.2Pd10.9Ge65.9
M w, [g mol^−1]	819.76	823.76	888.02
Space group (no.)	C2/m (12)	C2/m (12)	Cmce (64)
Pearson symbol-prototype, Z	mS36-La2AlGe6, 4	mS36-La2AlGe6, 4	oS72-Ce2(Ga0.1Ge0.9)7, 8
a [Å]	8.2163(8)	8.157(2)	8.1428(3)
b [Å]	8.4161(9)	8.328(2)	7.9807(3)
c [Å]	11.294(1)	11.195(2)	21.8331(9)
β (°)	100.477(1)	100.391(3)	90
V [Å^3]	768.0(1)	748.1(3)	1418.83(9)
Calc. density [g cm^−3]	7.090	7.314	8.314
abs coeff. (μ), mm^−1	36.2	38.7	53.5
Twin law	[1 0 0 0 −1 0 −1/2 0 −1]	[1 0 0 0 −1 0 −1/2 0 −1]	—
k(BASF)	0.414(5)	0.423(5)	—
Unique reflections	1083	1262	1242
Reflections I > 2σ(I)	1030	1062	1144
R sigma	0.0218	0.0298	0.0149
Data/parameters	1030/50	1062/50	1144/50
GOF on F^2 (S)	1.200	1.453	1.428
R indices [I > 2σ(I)]	R 1 = 0.0242; wR2 = 0.0773	R 1 = 0.0357; wR2 = 0.1166	R 1 = 0.0231; wR2 = 0.0496
R indices [all data]	R 1 = 0.0254; wR2 = 0.0778	R 1 = 0.0501 wR2 = 0.1209	R 1 = 0.0255; wR2 = 0.0506
Δρfin (max/min), [e Å^−3]	2.33/−2.61	4.35/−5.29	1.92/−1.78

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X-ray powder diffraction (XRPD) was performed on all samples by using a diffractometer Philips X'Pert MPD (Cu Kα radiation, λ = 1.5406 Å, graphite crystal monochromator, scintillation detector, step mode of scanning), with a θ:2θ Bragg–Brentano geometry. Measured powder patterns were collected in the 10°–100°2θ range, with a scanning step of ca. 0.02° with a time per step varying from 10 to 20 s. Phase identification was performed with the help of the software PowderCell.²¹

2.3 Differential thermal analysis (DTA)

Differential thermal analysis (DTA) was performed on a few La–Pd–Ge samples, in order to gain insights into the formation conditions of the La₂PdGe₆ compound. Measurements were carried out using a LABSYS EVO (SETARAM Instrumentation, Caluire, France) equipped with type S (Pt–PtRh 10%) thermocouples, in the temperature range of 25–1100 °C using custom-made tantalum crucibles. The sample crucible was loaded with about 200 mg of the alloy to analyze, closed with a cap, and subsequently arc-sealed under an inert atmosphere after cooling it in liquid nitrogen, so as to avoid undesired reactions. A crucible of the same weight as the sample container was used as the reference. The DTA curves were recorded under a continuous flow of argon (20 mL min⁻¹) to avoid oxidation of the crucibles at high temperatures. Different thermal cycles were employed depending on the sample. The obtained thermograms were evaluated with the software Calisto, supplied by SETARAM with the DTA equipment. Prior and after the DTA experiments the samples were characterized by SEM/EDXS and/or XRPD.

2.4 Computational details

DFT total energy calculations were performed for R₂PdGe₆ (R = Y and La) in the three structural modifications oS18, oS72 and mS36, and for La₂MGe₆ (M = Pt, Cu, Ag and Au) in the two orthorhombic modifications, by means of the plane wave pseudopotential code QUANTUM-ESPRESSO.²² The PBE functional²³ for the exchange and correlation energy was used. Ultrasoft pseudopotentials,²⁴ available in the “GBRV” open-source library,²⁵ were employed for M, La and Y, whereas for Ge a norm-conserving pseudopotential, including the 4s and 4p valence orbitals, was used. The semicore states 4p for Pd, 5p for Pt, 3s and 3p for Cu, 4s and 4p for Ag and Y and 5s and 5p for La were treated as valence electrons. The Brillouin zone was sampled within uniform grids generated with different k-points for the three polymorphs: 12 × 12 × 2 for the oS18 modification, 6 × 6 × 2 for oS72 and 6 × 6 × 4 for mS36. The plane-wave and density cut-off were set to 45 Ry and 450 Ry, respectively. The orbital occupancies at the Fermi level were treated with a Gaussian smearing of 0.01 Ry.

3. Results and discussion

3.1 The formation of R₂PdGe₆ germanides along the R series

The results of SEM/EDXS characterization confirm the presence of the R₂PdGe₆ phase in most of the samples prepared by direct synthesis (see Table 3).

Table 3 Results of SEM/EDXS characterization of the R_22.2Pd_11.1Ge_66.7 samples prepared by direct synthesis in the resistance furnace using cycle I and of the R₂₁Pd₇Ge₇₂ samples prepared by flux synthesis using cycle III (indicated with f in the sample code)

Sample code (R)	R2PdGe6 phase composition [EDXS, at%] R; Pd; Ge	Other known detected phases
a Samples where single crystals were taken.
Sc	–	Sc(Pd,Ge)2−x	Ge	ScPdGe
Y^a	21.0; 10.8; 68.2	YPdGe2
La	22.4; 9.9; 67.7	La(Pd,Ge)2−x	Ge	LaPdGe3
La,^af	22.4; 10.5; 67.1	La(Pd,Ge)2−x	Ge
Ce^a	22.5; 10.8; 66.7		Ge	Ce2Pd3Ge5
Pr^a	20.2; 11.4; 68.4		Ge	Pr2Pd3Ge5
Pr,^af	20.7; 11.0; 68.3	Pr(Pd,Ge)2−x	Ge
Nd^a	21.8; 11.2; 67.0	NdPdGe2	Ge	Nd2Pd3Ge5
Sm	21.5; 11.1; 67.4		Ge	Sm2Pd3Ge5
Eu	–	Eu(Pd,Ge)2−x	Ge	EuPdGe3
Gd	22.2; 11.4; 66.4	Gd(Pd,Ge)2−x
Tb	22.2; 11.1; 66.7	TbPdGe2
Dy	22.7; 10.9; 66.4	DyPdGe2
Ho	22.3; 11.3; 66.4	Ho(Pd,Ge)2−x
Er^a	22.7; 11.0; 66.3	Er(Pd,Ge)2−x
Tm	23.0; 11.1; 65.9	Tm(Pd,Ge)2−x
Yb	23.1; 10.7; 66.2	Yb(Pd,Ge)2−x
Yb,^af	23.2; 10.9; 65.9	YbGe2−x
Lu^a	23.8; 10.6; 65.6	Lu(Pd,Ge)2−x	Ge

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According to the SEM micrographs, under the applied experimental conditions for direct synthesis, R₂PdGe₆ is generally the highest yield phase. As it is common for alloys prepared by slow cooling from liquid, many secondary phases were detected, among which R(Pd,Ge)_2−x, R₂Pd₃Ge₅, RPdGe₂ and Ge are the most common. Representative microstructures are available in the ESI.†

In samples with R = Sc and Eu no traces of the R₂PdGe₆ phase were detected: in both cases three-phase alloys were obtained, in which an already known ternary compound (ScPdGe or EuPdGe₃) coexists with Ge and the binary RGe_2−x phase dissolving a small amount of Pd.

In the La_22.2Pd_11.1Ge_66.7 sample prepared by direct synthesis employing cycle I (sample code “La” in Table 3), only a small amount of La₂PdGe₆ was found, in the form of a thin border around big LaGe_2−x crystals (see Fig. 1a). From such a sample, it was not possible to isolate suitable single crystals. In the literature, this phase has been reported to crystallize in the oS18-Ce₂CuGe₆ ³ or in the oS72-Ce₂(Ga_0.1Ge_0.9)₇ model,⁴ the latter with no complete structural data available. For this reason, further attempts were made to synthesize a La₂PdGe₆ sample convenient for structural resolution, which are briefly accounted for in the following.

Fig. 1 Micrographs (SEM-BSE mode) of representative samples of La_22.2Pd_11.1Ge_66.7 nominal composition (a–d) and of La₂₁Pd₇Ge₇₂ nominal composition (e, f) obtained under the following experimental conditions: (a) synthesis in the resistance furnace, cycle I; (b) arc melting; (c, d) arc melting followed by one DTA cycle; (e) synthesis in the resistance furnace with In flux, cycle II; (f) synthesis in the resistance furnace with In flux, cycle III.

The as-cast samples with nominal composition La_22.2Pd_11.1Ge_66.7 did not show any trace of La₂PdGe₆, independently using the melting method (arc or induction furnace). These samples are characterized by a clear microstructure, where crystals of La(Pd,Ge)_2−x and LaPdGe₃ coexist with an eutectic structure containing Ge (see Fig. 1b). Annealing at 700 °C of an as-cast sample for 2 weeks, performed on the basis of literature data,⁴ did not succeed in obtaining the desired compound; what is more, the same annealing treatment carried out on the sample (La) prepared using cycle I caused the thin La₂PdGe₆ border to disappear.

At this point, it became clear that La₂PdGe₆ behaves somewhat differently from the other rare earth homologues, and DTA measurements were performed in order to gain insights into its temperature formation.

The DTA curve obtained at a heating rate of 5 °C min⁻¹ (T_max = 1100 °C), is shown in Fig. S2 (ESI†). Four thermal effects were detected at the following temperatures (onset): 800 °C, 886 °C, 927 °C and 1027 °C. The same thermal effects are recorded on cooling. The peak at 800 °C is in good agreement with the temperature of the binary eutectic equilibrium L → (Ge) + LaGe_2−x reported at T = 810 °C.²⁶ However, the other peaks are not easily interpretable. After this thermal cycle the presence of a small amount of La₂PdGe₆ was indeed detected by SEM-EDXS analysis in the form of a border between La(Pd,Ge)_2−x and LaPdGe₃ (see Fig. 1c). In a restricted region of the sample after DTA, bigger crystals of La₂PdGe₆ are present enclosing a brighter core of LaPdGe₃ (see Fig. 1d). The measured global composition of this region is La₂₁Pd₇Ge₇₂. Aiming to obtain a higher yield of the desired compound, different annealing treatments (lasting for one month each) were performed at 830 °C, 890 °C and 1000 °C: temperatures were chosen slightly below the recorded thermal effects. Unfortunately, no traces of 2 : 1 : 6 were found after all these treatments. From the gathered results, it was concluded that La₂PdGe₆ is probably a metastable phase which is likely to form only in small amounts during relatively slow cooling.

It is known that alternative synthetic routes may help to produce metastable compounds; among them, the flux method was targeted, taking into account both literature data²⁷ and our previous results on related R₂Pd₃Ge₅ compounds.²⁸ As a metal solvent In was chosen considering its ability to dissolve Ge, rare earth and transition metals, without forming In–Ge binary compounds, allowing the formation of ternary phases.^27,29 Thus, several attempts were made varying both the nominal composition and the thermal cycle. Good results were obtained starting from the nominal composition La₂₁Pd₇Ge₇₂: the 2 : 1 : 6 phase was obtained using both cycle II and cycle III (see Fig. 1e and f). After cycle II, however, La₂PdGe₆ is almost always found as a grey border of big La(Pd,Ge)_2−x crystals and only a few single phase crystals were detected. On the contrary, after cycle III, characterized by a lower maximum temperature (750 °C instead of 1000 °C), many 2 : 1 : 6 single crystals were obtained, allowing further crystal structure determination after their separation from the In-flux (sample indicated by code “La,^af” in Table 3). More details on all samples prepared with La can be found in the ESI.†

Unexpectedly, the flux synthesized La₂PdGe₆ turned out to be monoclinic, differently from all the other series of samples prepared by direct synthesis (see section 3.2): for this reason the flux method (cycle III) was tested also on Pr₂₁Pd₇Ge₇₂ and Yb₂₁Pd₇Ge₇₂ samples. In both cases R₂PdGe₆ phases were detected and analysed by X-ray diffraction (see Table 3); particularly good quality big crystals were obtained in the case of R = Yb (see Fig. 2).

Fig. 2 SEM-BSE micrographs for the Yb₂PdGe₆ single crystal before (a) and after (b) the selective In oxidation.

3.2 Crystal structure of R₂PdGe₆ (R = Y, La (flux), Ce, Pr, Pr (flux), Nd, Er, Yb(flux) and Lu)

To definitively establish the crystal structure of R₂PdGe₆ only the X-ray powder diffraction data are not sufficient. In fact, differences are hardly visible in the calculated diffraction patterns of the orthorhombic and the monoclinic models (see the ESI†). Additionally, the recognition of the correct model is hampered by the fact that samples are usually multiphase giving strong peak overlaps. Hence, single crystal X-ray diffraction is the fundamental method used for the structural investigations conducted here.

For single crystals of R₂PdGe₆ (R = Y, Ce, Pr, Nd, Er, Yb(f) and Lu) compounds (see. Table 1) the cell indexation was straightforward giving an orthorhombic C-centered cell (only h + k = 2n reflections were observed). The analysis of systematic extinctions suggested Cc2e (no. 41) and Cmce (no. 64) as the possible space groups. An almost complete structural model was obtained in the Cmce space group in a few iteration cycles by using the charge-flipping algorithm implemented in JANA2006.³⁰ In this model the rare earth atoms are situated in a 16g general site and palladium occupies the 8f site, while all the other positions are assigned to the lighter germanium atoms. The obtained models have the oS72 Pearson symbol and correspond to the R₂PdGe₆ stoichiometry, satisfactorily matching with the EDXS microprobe analysis data.

The further structure refinements were carried out by full-matrix least-squares methods on |F²| using the SHELX programs.³¹ The site occupancy factors of all species were checked for deficiency, in separate cycles of refinement, obtaining values very close to unity. At this point neither the deficiency nor statistical mixture was considered and stoichiometric R₂PdGe₆ models were further anisotropically refined giving acceptable residuals and flat difference Fourier maps. The results indicate that the Y, Ce, Pr, Er and Lu containing compounds prepared by direct synthesis are isopointal with the Ce₂(Ga_0.1Ge_0.9)₇ prototype. The same is true also for the Yb containing compound prepared by flux synthesis (the image of this crystal is reported in Fig. 2).

The atomic positions of the R₂PdGe₆ compounds are listed in Table 4 together with the equivalent isotropic displacement parameters. The positions were standardized with the Structure Tidy program.³² The interatomic distances are available in the ESI.†

Table 4 Atomic coordinates and equivalent isotropic displacement parameters (Ų) for the studied R₂PdGe₆ single crystals with the oS72-Ce₂(Ga_0.1Ge_0.9)₇ crystal structure

Atom (site)	Atomic param.	R = Y	R = Ce	R = Pr	R = Nd	R = Er	R = Yb (f)	R = Lu
R	x/a	0.25102(5)	0.25042(2)	0.25041(2)	0.25049(2)	0.25120(2)	0.24952(3)	0.25159(3)
(16g)	y/b	0.37584(5)	0.37513(5)	0.3753(1)	0.3751(1)	0.37605(3)	0.37604(3)	0.37592(4)
	z/c	0.08125(2)	0.08270(2)	0.082540(9)	0.082326(8)	0.080953(8)	0.08257(2)	0.08075(1)
	U eq (Å^2)	0.0065(1)	0.00492(6)	0.00671(6)	0.00527(5)	0.00541(6)	0.00529(8)	0.0072(1)
Ge1	x/a	0	0	0	0	0	0	0
(8f)	y/b	0.13181(8)	0.1285(1)	0.1286(2)	0.1291(3)	0.1332(1)	0.1304(1)	0.1345(1)
	z/c	0.02868(3)	0.03066(2)	0.03042(3)	0.03023(2)	0.02798(3)	0.02917(4)	0.02720(5)
	U eq (Å^2)	0.0073(2)	0.0063(1)	0.0084(1)	0.0065(1)	0.0063(1)	0.0062(1)	0.0086(2)
Ge2	x/a	0	0	0	0	0	0	0
(8f)	y/b	0.11933(8)	0.1217(1)	0.1213(2)	0.1213(3)	0.1185(1)	0.1205(1)	0.1177(1)
	z/c	0.45753(3)	0.46255(2)	0.46216(3)	0.46146(2)	0.45619(3)	0.46051(4)	0.45459(5)
	U eq (Å^2)	0.0083(2)	0.0075(1)	0.0089(1)	0.0077(1)	0.0076(1)	0.0082(2)	0.0097(2)
Ge3	x/a	0	0	0	0	0	0	0
(8f)	y/b	0.40519(8)	0.40516(7)	0.40475(8)	0.40492(8)	0.40524(8)	0.4047(1)	0.4052(1)
	z/c	0.19344(3)	0.19462(3)	0.19441(3)	0.19422(3)	0.19305(3)	0.19403(4)	0.19299(5)
	U eq (Å^2)	0.0073(2)	0.0076(1)	0.0089(2)	0.0072(2)	0.0062(1)	0.0048(1)	0.0085(2)
Ge4	x/a	0	0	0	0	0	0	0
(8f)	y/b	0.34733(8)	0.34524(7)	0.34608(8)	0.34617(8)	0.34763(8)	0.3476(1)	0.3484(1)
	z/c	0.30752(3)	0.30553(3)	0.30572(3)	0.30599(3)	0.30810(3)	0.30685(4)	0.30862(5)
	U eq (Å^2)	0.0075(2)	0.0077(1)	0.0086(2)	0.0069(2)	0.0067(1)	0.0060(2)	0.0079(2)
Ge5	x/a	0.27688(6)	0.27767(5)	0.27735(4)	0.27729(4)	0.27700(5)	0.27568(7)	0.27669(8)
(16g)	y/b	0.12631(5)	0.12531(8)	0.12540(1)	0.1254(2)	0.12658(7)	0.12614(7)	0.1267(1)
	z/c	0.19308(2)	0.19461(2)	0.19442(2)	0.19416(2)	0.19259(2)	0.19379(2)	0.19237(3)
	U eq (Å^2)	0.0073(1)	0.00764(8)	0.00864(9)	0.00686(8)	0.00625(9)	0.0055(1)	0.0082(2)
Pd	x/a	0	0	0	0	0	0	0
(8f)	y/b	0.12736(5)	0.12526(6)	0.12555(6)	0.12572(6)	0.12762(6)	0.12666(7)	0.12826(9)
	z/c	0.14223(2)	0.14568(2)	0.14496(2)	0.14453(2)	0.14153(2)	0.14132(2)	0.14062(3)
	U eq (Å^2)	0.0060(1)	0.00664(8)	0.00765(9)	0.00585(8)	0.00504(8)	0.0039(1)	0.0068(2)

---

The La₂PdGe₆ crystal selected for X-ray analysis, taken from a flux synthesized sample, is an example of a non-merohedral twin composed of two domains of comparable dimensions. Normally, twins of such a type have problems in the preliminary stages of cell indexing (leading to unexpectedly high values of unit cell parameters) and space group determination (showing inconsistency with any known space group systematic absence).^33,34 Instead, in our case the unit cell indexing was straightforward and the following possible space groups were suggested for the C-centered monoclinic cell: C2 (no. 5), Cm (no. 8) and C2/m (no. 12). The lowest combined figure of merit was associated with the only centrosymmetric C2/m space group. These data strongly hint that the studied crystal is isostructural with the monoclinic mS36-La₂AlGe₆ prototype. A preliminary structural model, obtained by direct methods as implemented in WinGx,³⁵ contained 1 La, 5 Ge and 1 Pd, giving the correct La₂PdGe₆ composition. Nevertheless, the isotropic thermal parameter values were not coherent for the different types of atoms and three additional intense peaks close to some of the Ge-positions were present in the difference Fourier maps at a distance of ∼0.05 nm. The latter three sites have no physical sense if completely occupied and therefore, the sum of occupations for each pair of very close Ge-sites was restrained to be unity in further cycles of least squares refinement. However, the refinement sticks at R₁/wR₂ of 0.05/0.17 with a noisy Fourier map give unreasonable thermal parameters when anisotropically refined. Even less chemically sound structural models were obtained with the possible non-centrosymmetric C2 and Cm space groups. At this point, more careful analysis of diffraction spots in reciprocal space was performed with RLATT and CELL_NOW.³⁶ In fact, a regular spatial distribution of extra-peaks was revealed with respect to those associated with the mS36-like monoclinic cell. All of them could be satisfactorily indexed with a twice as big monoclinic base centered unit cell with a ∼ 8.2 Å, b ∼ 8.4 Å, c ∼ 22.6 Å, and β ∼ 100.5°. The structural model deduced in the C2/m space group by the charge-flipping algorithm³⁰ became less disordered, since it contains only one partially occupied Ge position capping the distorted corrugated Ge fragment.² Even if Ge-rich compounds are frequently off-stoichiometric or disordered,¹ residuals remain unsatisfactory (0.09/0.14) with senseless anisotropic thermal parameters. For this reason, more attention was dedicated to the indexing procedure. According to a detailed output of CELL_NOW, one of the possible interpretations of the observed diffraction peak distribution is considering them originating from different domains of a non-merohedrically twinned crystal. In this case, the metric of the single domain remains the same as for the mS36 model. All the extra-peaks, instead, are due to the presence of the second domain, related to the first one by a 180° rotation twin law .

Consequently, the collected dataset was re-integrated assuming the presence of two domains and the hkl5 file for refinement was prepared by TWINABS.³⁶ After that, excellent residuals were obtained (see Table 5) for the ordered La₂PdGe₆ model with the mS36-type structure. At the final cycles, this model was refined with anisotropic thermal displacement parameters for all atom sites. The refined fractional contribution k of the second domain is ca. 0.4.

Table 5 Atomic coordinates and equivalent isotropic displacement parameters (Ų) for the La₂PdGe₆ and Pr₂PdGe₆ single crystals obtained by flux synthesis

R2PdGe6 analogue	Atomic param.	R (8j)	Ge1 (4i)	Ge2 (4i)	Ge3 (4i)	Ge4 (4i)	Ge5 (8j)	Pd (4i)
R = La	x/a	0.0840(1)	0.1438(3)	0.3585(3)	0.0577(2)	0.4970(2)	0.2773(3)	0.1985(2)
	y/b	0.24980(6)	0	0	0	0	0.22206(8)	0
	z/c	0.33354(3)	0.5622(1)	0.4281(1)	0.1097(1)	0.1093(1)	0.10943(6)	0.79356(6)
	U eq (Å^2)	0.0045(1)	0.0058(2)	0.0071(2)	0.0092(4)	0.0090(4)	0.0083(2)	0.0064(2)
R = Pr	x/a	0.0832(3)	0.1453(6)	0.3580(6)	0.0573(3)	0.4981(3)	0.2778(4)	0.1979(2)
	y/b	0.24962(7)	0	0	0	0	0.2227(1)	0
	z/c	0.33491(4)	0.5609(2)	0.4243(1)	0.1115(2)	0.1112(2)	0.11130(9)	0.7899(1)
	U eq (Å^2)	0.0047(2)	0.0057(3)	0.0075(3)	0.0068(5)	0.0070(5)	0.0066(2)	0.0056(2)

---

The orientation of twin domains and the corresponding reciprocal plots of the completely overlapped and non-overlapped hkl zones are shown in Fig. 3. Keeping in mind the twin law, it becomes clear that only reflections with h = 2n are affected by the twinning (i.e. they are completely overlapped). Considering also the presence of the only h + k = 2n reflections for the C-centered lattice, it is revealed that half of all measured intensities are affected by twinning. The intensity difference between overlapped/non-overlapped reflections is evident from the corresponding precession photos of the h2l and h1l zones of reciprocal space shown in Fig. 3.

Fig. 3 Upper part: Reciprocal orientation of twin domains together with theoretical reciprocal plots of h2l (totally overlapped) and h1l (non-overlapped) zones generated by XPREP.²⁰ Colors of domains I (red) and II (green) are the same as those of the corresponding hkl reflections. For clarity, relationships between direct/reciprocal lattice vector lengths are not considered. Lower part: Experimental precession photos of h2l and h1l zones.

The crystal structure of Pr₂PdGe₆ isolated from In flux was solved in the same way as for La₂PdGe₆, just described. Its structure turned out to be monoclinic mS36, with the same twinning law and with a similar volume ratio to the twinned domains (see Table 2 for more details).

The unit cell volumes of the studied compounds were plotted as a function of the trivalent rare earth metal radii³⁷ (see Fig. 4). The monoclinic cell volume was doubled in order to compare the different related structures.² Only the lattice parameters obtained from single crystal X-ray diffraction were considered.

Fig. 4 Normalized cell volumes (from single crystal data) of R₂PdGe₆ compounds as a function of the R³⁺ ionic radius. The cell volume of Dy₂PdGe₆ was taken as per ref. 14.

In agreement with the lanthanide contraction, a linear decreasing trend is observed, suggesting a similar chemical bonding scenario for all the studied germanides. The literature data on Dy₂PdGe₆ ¹⁴ fit quite well with the general trend. The most significant deviation is observed for Yb₂PdGe₆; this result is in good agreement with a recent magnetic investigation⁸ revealing for Yb a behavior typical of dynamic intermediate valence systems.

The crystal structure of the R₂PdGe₆ (R = Sm, Gd, Tb, Ho and Tm) compounds was not studied by single crystal X-ray diffraction. The powder diffraction patterns, recorded on the corresponding samples, can be satisfactorily indexed to both oS72 and mS36 structures. However, considering the results obtained on the directly synthesized samples with early and late rare earth metals, it is reasonable to suggest that under the same conditions even the abovementioned R₂PdGe₆ phases are of the oS72-Ce₂(Ga_0.1Ge_0.9)₇ type.

A similar rationalization could not be made for the crystal structure of the 2 : 1 : 6 germanides synthesized by the flux method since for Pr₂PdGe₆ a different structure was stabilized, instead for Yb₂PdGe₆ the same orthorhombic modification is formed.

3.3 From R₂PdGe₆ to R₂MGe₆ compounds: crystallochemical and DFT analyses targeting the correct structural model

From the results described above, it is clear that the oS18-Ce₂CuGe₆ structure is never realized in the R₂PdGe₆ series. Within this family, direct synthesis always produces compounds belonging to the oS72-Ce₂(Ga_0.1Ge_0.9)₇ structure type, except for La₂PdGe₆, whose structure under these conditions was not definitely confirmed. The In-flux synthetic route used in this study gave origin to mS36-La₂AlGe₆ twinned crystals of La₂PdGe₆ and Pr₂PdGe₆, whereas Yb₂PdGe₆ crystals grown from the same metal solvent are orthorhombic oS72. These results are coherent with recent structural studies on 2 : 1 : 6 germanides assigning to them the oS72^7,9,14,15 or the mS36 model¹⁷ instead of the previously proposed oS18.

In view of the following discussion it is convenient to distribute all the known 2 : 1 : 6 models within a concise scheme (Bärnighausen tree) starting from the oS20-SmNiGe₃ aristotype. When considering the vacancy ordering phenomenon^16,38,39 causing symmetry reduction, each structural model finds its location on a separate branch. To capture the changes taking place, it is sufficient to follow the M–Ge sub-lattice distortions as depicted in Fig. 5 (R atom positions are negligibly affected by the symmetry reduction).

Fig. 5 Bärnighausen tree relating to the SmNiGe₃ aristotype and its orthorhombic and monoclinic vacancy variants. The type and indexes of the symmetry reductions are indicated. For clarity, only M–Ge frameworks are shown for each structural model; Ge–Ge contacts are shown in red, M–Ge interactions by a dotted line.

From the crystal chemical point of view, the oS18 model is somewhat suspicious:

- it contains many independent crystallographic Ge sites, not in agreement with the symmetry principle;⁴⁰

- the inhomogeneous vacancy distribution implies that corrugated Ge layers are linked to the zig-zag germanium chains through bridging M atoms only on one side (see Fig. 5).

To corroborate this idea arising from symmetry considerations, the DFT total energy values were calculated for La₂PdGe₆ and Y₂PdGe₆ with the oS18, oS72 and mS36 structures. The structural data available in the literature or obtained during this work were used as the starting point for geometric relaxation. In the case of Y₂PdGe₆, for which the mS36 model has never been reported, the cell dimensions and atomic positions of the La-analogue were chosen. The choice of these rare earths allows avoiding the presence of highly correlated electrons lying in partially filled f states. The relaxed structures, obtained at the end of the variable-cell calculations, showed lattice constants about 2% larger with respect to the literature data and results presented in this work, as expected when using the PBE functional (see Table 6).

Table 6 Experimental and calculated parameters for R₂PdGe₆ (R = La and Y) in the oS18, oS72 and mS36 modifications. If not specified, data were obtained in this work

Compound		Experimental			Calculated
		oS18	oS72	mS36	oS18	oS72	mS36
Y2PdGe6	a (Å)	4.0790(4)	8.1703(5)		4.1502	8.3117	8.2181
	b (Å)	4.0168(5)	8.0451(5)		4.0802	8.2239	8.3140
	c (Å)	21.525(2)	21.558(2)		22.212	21.857	11.123
	β (°)						100.5
	V (Å^3)	352.7(1)	1417.1(2)		376.14	1494.0	747.16
	Energy (eV per atom)				−691.647	−691.713	−691.713
	Ref.	3
La2PdGe6	a (Å)	4.2117(3)	8.430(1)	8.2163(8)	4.2914	8.5552	8.3910
	b (Å)	4.1100(3)	8.2180(7)	8.4161(9)	4.1893	8.3902	8.5554
	c (Å)	22.265(5)	22.192(3)	11.294(1)	22.913	22.642	11.515
	β (°)			100.5(1)			100.5
	V (Å^3)	385.4(1)	1537.4(4)	768.0(1)	411.92	1625.2	812.81
	Energy (eV per atom)				−721.181	−721.244	−721.244
	Ref.	3	4

---

On the basis of the obtained results the oS18 structural model is the worst: its total energy is higher by 0.063 eV per atom for La₂PdGe₆ and by 0.066 eV per atom in the case of Y₂PdGe₆ with respect to both oS72 and mS36 models.

Calculations for the oS18 and oS72 models were performed on other La₂MGe₆ analogues (M = Pt, Cu, Ag and Au). As can be seen from the plot shown in Fig. 6, the highest energy value corresponds always to the oS18 model (ΔE(eV per at) = E_oS72 − E_oS18, is always negative).

Fig. 6 ΔE vs. nature of late transition element M for La₂MGe₆.

The mS36 model was not considered for energy calculations of La₂MGe₆ because the results on La₂PdGe₆ and Y₂PdGe₆ (Table 6) do not show any energy difference between oS72 and mS36. In fact, they can be considered as two different polytypes of 2 : 1 : 6 composition. They both consist of geometrically equivalent layers of defective BaAl₄ (of R□MGe₂ composition), AlB₂ and α-Po type slabs stacked linearly along the c direction.^2,41 These layers are also energetically equivalent, and consequently no strong preference for one of the polytypes can be envisaged. Frequently, the structure of such compounds is sensitive to the crystallization conditions and small fluctuations of these may reverse the energetic preferences and at the same time give origin to stacking faults, twinnings or non-periodic structures; as an example the families of SiC, ZnS, CdI₂, micas, etc. can be cited.⁴²

These findings do not preclude the possibility that new polytypes exist for this numerous group of compounds. One can suppose that modulated structures may form within this family. The non-periodicity could arise from a vacancy ordering or Ge covalent fragment distortions with a periodicity different from that of lattice. This phenomenon was addressed in a recent paper on structurally related germanides.⁴³

Some more aspects associated with the symmetry reduction scheme can be highlighted. The symmetry reduction path conducive to the mS36 model contains a translationengleiche transformation of index 2 that is in perfect agreement with the fact that non-merohedral two domain twins are formed. The presence of klassengleiche relations, instead, should be at the origin of antiphase domains in both oS72 and mS36 polytypes. These domains are not detectable by conventional X-ray diffraction techniques; however, the quality and dimensions of single crystals obtained by the flux method are well promising for further transmission electron microscopic investigations to achieve this goal.

It remains, however, unclear if the oS20 aristotype has any physical meaning. Further investigations should be performed aiming to clarify if elevated temperature/pressure conditions may stabilize the hypothetical oS20-RPdGe₃. If yes, the found non-merohedral twins have been developed by a phase transition in the solid state. Otherwise, their formation took place during the growth of the crystal.

4. Conclusions

The R₂PdGe₆ series of compounds was targeted with the aim to elucidate more on the formation/crystal structure of the R₂MGe₆ family (R = rare earth metal, M = another metal), combining the experimental results with structure analysis and energy calculations. This study was motivated by the controversial and sometimes erroneous structural data available for the title compounds, possibly also affecting the interpretation of the numerous magnetic measurements performed by several authors.

The R₂PdGe₆ phase was detected in all samples prepared by direct synthesis, except for R = Sc and Eu.

Single crystal X-ray analyses conducted on several representative samples (R = Y, Ce, Pr, Nd, Er and Lu) indicated that oS72-Ce₂(Ga_0.1Ge_0.9)₇ is the correct structural model, as previously reported for the Dy and Yb analogues. The same crystal structure is suggested for R = Sm, Gd, Tb, Ho and Tm on the basis of powder X-ray diffraction measurements and behavior regularities along the rare-earth series.

Different annealing treatments along with thermal analysis investigations were performed with the aim to obtain good crystals or a sufficiently high yield of La₂PdGe₆; the obtained results suggest that this is a metastable phase likely to form in a small amount during slow cooling treatments.

Therefore, the flux synthesis, able to stabilize metastable phases, was explored, using In as a metal solvent. The good quality La₂PdGe₆ crystals obtained after optimizing this method are non-merohedrally twins of the mS36-La₂AlGe₆ structure. Monoclinic twins of two equally big domains of the same morphology were obtained for the flux-synthesized Pr₂PdGe₆ compound. This result suggests that also La₂PdGe₆ obtained from direct synthesis might be of the oS72 orthorhombic structure. Instead, for the heavy rare earth representative Yb₂PdGe₆ the flux does not stabilize the monoclinic structure, and the oS72 model remains the correct one.

Taking into account the considerable amount of data on 2 : 1 : 6 germanides, some structure-rationalizing idea was pursued. Considering the vacancy ordering phenomenon to be the key-factor for structural changes, a compact Bärnighausen tree was constructed, with rigorous group–subgroup relations between the oS20-SmNiGe₃ aristotype and the three possible derivatives oS72, mS36 and oS18: their structural models are localized on separate branches of the tree, representing different symmetry reduction paths. The presence of a t2 reduction step on the path bringing to the mS36 model is coherent with the formation of twinned crystals.

Both from our experience and literature data,^44,45 binary and ternary germanides are particularly prone to geminate, and this possibility should be carefully investigated during structural solution. For this reason these phases are also suitable for studies targeted to better understand the origin, formation conditions and type of twinning in intermetallics. At present, we are further studying twinned crystals of germanides, where twinning seems to be related to vacancy ordering phenomena.

The same phenomena can lead to modulated structures, as already found for binary RGe_2−x ⁴⁶ and ternary germanides,⁴⁶ representing another reason for interest in the studied compounds.

From the presented symmetry reduction scheme the symmetry discrepancy became obvious between the oS18 (2^nd order derivative of the aristotype) and both the oS72 and mS36 models (4^th order derivatives), highlighting the poor crystal chemical reliability of the structure mostly reported in the literature and already corrected for several R₂MGe₆ compounds.

Aiming to confirm this idea and to discard energetically less probable modifications, DFT total energy calculations were performed for R₂PdGe₆ (R = Y and La) in the three abovementioned structural models, and for La₂MGe₆ (M = Pt, Cu, Ag and Au) in the oS18 and oS72 modifications. Structure optimization was performed for all calculations. Considering the total energy highest values as well as the crystallochemical factors, the oS18-Ce₂CuGe₆ model is less probable. It was also concluded that the oS72 and mS36 models, containing geometrically equivalent fragments, are energetically equivalent polytypes. In fact, they are sensitive to the crystallization conditions, as it became clear from the results of the flux synthesis. This approach turned out to be an effective alternative method to prepare these compounds.

The mostly structural study presented in this paper constitutes a good and essential basis for further investigations concerning the chemical bonding for R₂MGe₆, which is already one of the objects of our research work.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors thank D. Ceresoli (CNR-ISTM, Milano) for support in DFT calculations. DMP acknowledges the Ministry of Education and Science of Russia (grant 14.B25.31.0005).

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Footnote

† Electronic supplementary information (ESI) available: X-ray crystallographic files in CIF format; back scattered electron images of alloy surfaces; summary of attempts to synthesize the La₂PdGe₆ compound; calculated X-ray powder patterns; interatomic distances obtained by single crystal X-ray analysis. See DOI: 10.1039/c7dt02686b

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