RCuMg₂ compounds (R = Dy–Tm, Lu): crystal structure, chemical bonding and magnetic properties

Abstract

The RCuMg₂ intermetallics (R = Dy–Tm, Lu) were found to crystallize in a new structure type (DyCuMg₂, oP32, space group: Pmma, a = 13.5397(4), b = 3.7594(1), c = 13.7985(4) Å). This unique crystal space is characterized by the presence of clusters formed by four Cu-centred trigonal prisms sharing edges to create a cubic-like cavity occupied by a Mg atom; 2D wavy slabs of these star-like fragments and of Mg with a bcc topology are alternated along the c-direction. An architecture based on similar moieties was recognized for the related R₂Cu₂Mg (tP10-Mo₂FeB₂) and RCuMg₄ (oS48-TbCuMg₄) compounds, proposing a generalization scheme. Electronic structure calculations and a chemical bonding analysis in the position space were applied to LuCuMg₂, as a representative. The calculated effective charges indicated that Lu and Cu act as the QTAIM cation and anion, respectively, and the Mg species show a slight positive charge, except that at the centre of the bcc-fragment, showing a tiny negative one. In general, a complex bonding scenario was revealed, dominated by both hetero and homoatomic (within the Mg bcc-slabs) multicentre interactions. The temperature and field dependencies of magnetic susceptibility and specific heat capacity were measured for RCuMg₂. The nonmagnetic LuCuMg₂ phonon reference compound was identified as a Pauli paramagnet. The {Ho, Er, Tm}CuMg₂ compounds ordered antiferromagnetically at critical temperatures between about 5 and 8 K, whereas DyCuMg₂ revealed multiple magnetic transitions. The complexity of the magnetic behaviour of the studied compounds was indicated by the temperature evolution of magnetic entropy.

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1. Introduction

The R–T–Mg alloy systems (R = rare earth metal; T = transition metal) are very rich in ternary intermetallic compounds, covering different stoichiometries and crystal structures^1,2 and representing a wide range of materials for different functional and structural applications.^3–9

Such compounds can be classified in various families, some of which have been the object of long-lasting investigations of our research group.^10–14 Here, we focus our attention on phases characterized by a 1/1 stoichiometric ratio of the R and T components; the Cu-containing ones are shown in Fig. 1 in a Gibbs triangle.

Fig. 1 Distribution and crystal structures of R–Cu–Mg compounds existing along the compositional line with a R/Cu ratio equal to 1/1. The number of representatives is indicated in parentheses. The region of existence of long-period stacking ordered (LPSO) phases known so far is also shown.

The Mg-poorest representatives correspond to the very common 2:2:1 and 1:1:1 stoichiometries and crystallize in the two ubiquitous Mo₂FeB₂ and ZrNiAl prototypes.² The former exists for the entire series of R elements, and the latter is limited to light rare earth metals and yttrium. At Mg contents between about 40 and 70 at%, only a few compounds have been reported: R₅Cu₅Mg₈, R₅Cu₅Mg₁₃ and R₅Cu₅Mg₁₆ are only known for Y, all being own prototypes,¹⁵ and RCuMg₄ exists with two different structures, UCoAl₄-type for R = La and TbCuMg₄-type for R = Y, Tb–Tm.^15–17

The Mg-richest region is the field of existence of a special class of compounds named “long-period stacking ordered (LPSO) phases” after their structural peculiarities. Despite the most studied LPSO phases contain Ni and/or Zn, some studies have been conducted even on Cu analogues, which are located in the region between ∼80 and 92 at% Mg.^18–20 In our recent study on Y–Ni–Mg LPSO phases,²¹ their range of existence was extended down to ∼58 at% Mg; therefore, a similar situation is also not ruled out for Cu-containing systems.

The intermetallic R–Cu–Mg compounds exhibit a variety of magnetic behaviours, including ferromagnetism, antiferromagnetism, mixed or intermediate valence states, and spin fluctuations.^1,3,22,23 For example, {La,Y}₂Cu₂Mg were found to be Pauli paramagnets, whereas Nd₂Cu₂Mg and Pr₂Cu₂Mg ordered ferromagnetically at Curie temperatures of 42 K and 12 K, showing 3.47μ_B and 3.67μ_B magnetic moments, respectively. Interestingly, the Pr₂Cu₂Mg compound revealed an additional magnetic transition when applying an external field of ∼2.5 T.²³ In contrast, Ce₂Cu₂Mg was reported to order antiferromagnetically at T_N = 7.5 K with an effective magnetic moment of 2.58μ_B, in agreement with the trivalent state of cerium ions.²² The increase in the magnetization with the application of an external field up to H ≈ 4 T indicated spin reorientation in this compound. Importantly, it was found that the reported magnetization of ≈0.9μ_B at 5 K and 8 T was considerably smaller than the expected moment of 2.14μ_B for fully saturated Ce³⁺, revealing the splitting of the six fold degenerate ²F_5/2 ground state of the Ce³⁺ ion due to the crystal electric field.²² Among the RCu₄Mg Laves phases, TmCu₄Mg is paramagnetic in the entire temperature range, whereas SmCu₄Mg Van Vleck's paramagnetism down to T_N = 10.8(5) K with an experimental magnetic moment of 0.69(1)μ_B has been reported.³ Moreover, GdCu₄Mg as well as TbCu₄Mg undergo antiferromagnetic transitions at 48 K and 30 K, respectively, together with magnetic moments close to those of trivalent R-ions.³ The recently studied orthorhombic DyCuMg₄ and ErCuMg₄ revealed two subsequent antiferromagnetic transitions at T_N = 21 and 7.9 K as well as a single ordering at T_N = 8.6 K, respectively.¹⁷

In this work, the results of a combined investigation on the structural and magnetic properties of the new RCuMg₂ series of compounds are further complemented with a chemical bonding analysis of LuCuMg₂, which shows a complex bonding scenario.

2. Experimental section

2.1. Synthesis and SEM/EDXS characterization

Alloys with the nominal composition R₂₅Cu₂₅Mg₅₀ (R = trivalent rare earth metal) and a total mass of about 0.6 g were prepared by direct synthesis. The pure (>99.9 mass%) metals were weighed in stoichiometric amounts, placed in tantalum crucibles, subsequently sealed by arc welding to avoid Mg loss, and induction-melted under an Ar stream. Melting was repeated three times to ensure homogeneity; then, the furnace was switched off, and the samples were left to cool down to room temperature. Subsequently, crucibles containing the as-cast samples were closed in evacuated silica ampules, annealed for 1 month at 500 °C in a resistance furnace, and then water-quenched. The samples were extracted from the crucible as partially fragmented/powdered, having the look of grey brittle alloys, stable in air for months.

Selected fragments of each alloy were embedded in two-component cold-cast Technovit epoxy resin (ATM GmbH, Germany). Surfaces suitable for metallographic analysis were obtained using the automatic polishing machine Saphir 520 (ATM GmbH, Germany); grinding with SiC abrasive papers and no lubricant was followed by polishing with 6–1 μm size diamond pastes using petroleum ether as a lubricant. Metallographic specimens were made conductive using a thin layer of graphite. A Zeiss Evo 40 scanning electron microscope (SEM) equipped with a dispersive X-ray spectroscopy (EDXS) system (INCA X-ACT) operated by the INCA Energy software (Oxford Instruments, Analytical Ltd, Bucks, U.K.) was used for microstructural analysis and compositional evaluation. The composition of each phase was established as the average of at least five measurements from different points/areas. Quantitative results agreed with the expected compositions within ∼2 at% tolerance. The SEM/EDXS characterization highlighted the existence of new compounds with the R₂₅Cu₂₅Mg₅₀ composition in samples with R = Dy, Ho, Er, Tm, and Lu, and they became the object of this work.

2.2. XRD characterization

A good-quality DyCuMg₂ single crystal, selected from the mechanically crushed alloy, was glued to a glass fibre and mounted on a goniometric head. X-ray diffraction was performed on a three-circle Bruker D8 QUEST diffractometer fitted with a PHOTON III 14 photon-counting detector. The graphite-monochromatized Mo Kα radiation was used. The applied data collection strategy, elaborated using the APEX5 software,²⁴ consisted of four ω-scans and covered the reciprocal space up to a maximum θ of about 31° (resolution of ca. 0.7 Å) with exposures of 5 s per frame. Data reduction was performed using software SAINT²⁵ and XPREP;²⁶ Lorentz, polarization and absorption effects were corrected by SADABS.²⁷ The crystal structure was solved and refined with the aid of SHELXTL.²⁸ The crystal possessed orthorhombic symmetry, and the lattice parameter b (∼3.76 Å) was significantly smaller than a and c (>13 Å). The best structural model was found using the intrinsic phasing method in the Pmma space group (N. 51), representing its own prototype. The unit cell contained 8 formula units with the DyCuMg₂ composition, corresponding to 32 atoms distributed among 10 Wyckoff sites (3 occupied by Dy, 2 by Cu and 5 by Mg atoms). The occupancy of each site was probed to vary, and any hint of a statistical mixture was observed so that the structural model was perfectly stoichiometric. The final anisotropic full-matrix least-squares refinement converged to excellent residuals and a flat difference Fourier map. The anisotropic displacement parameters (ADPs) of all the atoms showed a smooth distribution.

Details of data collection and structure refinement are summarized in Table 1, together with selected crystal data; standardized atomic coordinates, site occupancy factors and ADPs are listed in Table 2. The corresponding CIF file, available as SI, was deposited at the Cambridge Database.

Table 1 Selected crystallographic data and structure refinement parameters for the DyCuMg₂ single crystal

Formula	DyCuMg2
EDXS composition	Dy23.8Cu23.4Mg52.8
Depositing CSD-code	2381656
Formula weight (g mol^−1)	274.66
Space group	Pmma (N.51)
Pearson symbol-prototype, Z	oP32-DyCuMg2, 8
a, Å	13.5397(4)
b, Å	3.7594(1)
c, Å	13.7985(4)
V, Å^3	702.36(3)
Calc. density (g cm^−3)	5.195
Absorption coefficient (μ, mm^−1)	27.23
Theta range (°)	2.3 ≤ θ ≤ 30.46
Index ranges h, k, l	−19 ≤ h ≤ 19
	−5 ≤ k ≤ 5
	−19 ≤ l ≤ 19
Data/parameters	1259/53
GOF	1.18
Rint/Rsym	0.0311/0.015
R1/wR2 (I > 2σ(I))	0.0201/0.0435
R1/wR2 (all data)	0.0232/0.0448
Max diff. peak and hole (e^− Å^−3)	1.023 and −1.689

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Table 2 Standardized atomic coordinates and equivalent (U_eq.) and anisotropic (U_ij) displacement parameters for the DyCuMg₂ single crystal; U₁₂ = U₂₃ = 0

Atom	Site	Atomic coordinates			Ueq. [Å^2]
		x/a	y/b	z/c
Dy1	4j	0.04785(2)	½	0.37604(2)	0.0130(1)
Dy2	2f	¼	½	0.18284(3)	0.0151(1)
Dy3	2f	¼	½	0.57710(3)	0.0162(1)
Cu1	4i	0.09222(5)	0	0.53356(5)	0.0145(1)
Cu2	4i	0.09149(6)	0	0.21873(5)	0.0163(1)
Mg1	4j	0.5098(2)	½	0.1226(2)	0.0181(4)
Mg2	2e	¼	0	0.3788(2)	0.0155(5)
Mg3	4i	0.6245(2)	0	0.2619(2)	0.0186(4)
Mg4	4i	0.1338(2)	0	0.0138(2)	0.0194(4)
Mg5	2f	¼	½	0.8699(2)	0.0211(6)

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Atom	Site	U11 [Å^2]	U22 [Å^2]	U33 [Å^2]	U13 [Å^2]
Dy1	4j	0.01466(12)	0.01320(12)	0.01116(12)	−0.00052(9)
Dy2	2f	0.01411(18)	0.01440(18)	0.01678(18)	0.00000
Dy3	2f	0.01630(19)	0.01507(18)	0.01720(18)	0.00000
Cu1	4i	0.0128(3)	0.0176(3)	0.0132(3)	0.0008(2)
Cu2	4i	0.0164(3)	0.0181(3)	0.0145(3)	−0.0016(3)
Mg1	4j	0.0174(9)	0.0208(10)	0.0160(9)	0.0012(8)
Mg2	2e	0.0138(12)	0.0176(13)	0.0151(13)	0.00000
Mg3	4i	0.0162(10)	0.0248(11)	0.0149(9)	0.0004(8)
Mg4	4i	0.0167(10)	0.0244(11)	0.0171(10)	0.0000(8)
Mg5	2f	0.0156(13)	0.0249(15)	0.0229(15)	0.00000

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No single crystals suitable for structure solution were found in samples with other rare earth components. For all of them, X-ray powder diffraction (XRPD) patterns were recorded on a Rigaku Smartlab diffractometer equipped with a Dtex strip detector operated in the Bragg–Brentano mode (Cu K_α radiation, continuous mode of scanning). The PowderCell²⁹ and CrystalDiffract³⁰ software were used for indexation; the precise lattice parameters of the compounds of interest were calculated by a least square routine.³¹ Rietveld refinements were carried out using Fullprof software;³² refined atomic positions for the studied compounds are available in the SI.

2.3. Physical property measurements and computational details

The temperature and field dependencies of magnetic susceptibility and specific heat capacity were measured using a vibrating sample magnetometer (VSM) and the specific heat options of DynaCool-12 from Quantum Design, respectively.

Density functional theory (DFT) calculations to determine the electronic band structure and the relative stability of different magnetic phases were performed for DyCuMg₂, HoCuMg₂, and ErCuMg₂ using the linear augmented plane wave (LAPW) method, as implemented in the WIEN2K package.³³ The basis set consisted of spherical harmonics within non-overlapping muffin-tin spheres around the atoms augmented by plane waves in the space outside the muffin-tin spheres and local orbitals for the semi-core states. The core electrons were treated separately by numerically solving the radial Dirac equation in the spherical symmetric part of the potential. The kinetic energy cut-off for the plane-wave basis functions was adjusted using the product of the smallest muffin-tin radius and the largest plane-wave vector and was set here to 7. The PBESOL exchange–correlation functional³⁴ was applied in all calculations. The magnetism was accounted for using spin-dependent electron density in a scalar relativistic approximation for the valence and semi-core electrons.

To save computational time, all symmetry elements were kept. For the ferromagnetic state, the starting densities were created with spin-up for all the rare earth ions. In contrast, the antiferromagnetic state can only be approximated in DFT by a broken symmetry solution; because the R atoms sit in three independent sites (4j, 2f, and 2f), the starting density for the 4f site was assigned spin-up and those for the other two sites as spin-down. Moreover, all atomic positions were further optimized by keeping the unit cell parameters constant for all three compounds.

DFT/PBE³⁵ quantum chemical calculations were performed for LuCuMg₂ by means of the all-electron full-potential local orbital (FPLO) code.³⁶ The compound with R = Lu was selected to avoid partially filled 4f states. The experimentally determined lattice spacings were used, and the atomic positions optimized using as input those refined based on the single-crystal X-ray diffraction data for DyCuMg₂. Relativistic effects were treated at the scalar relativistic level, and the Brillouin zone sampling was conducted with a (4 14 4) k-point mesh. To investigate the chemical bonding by position-space techniques,³⁷ the electron density (ED) and the electron localizability indicator, in its ELI-D^38,39 representation, were calculated using an equidistant grid of about 0.05 Bohr by means of a dedicated module implemented within the FPLO code.⁴⁰ The topological analysis of both fields was performed based on the quantum theory of atoms in molecules (QTAIM)⁴¹ with the DGrid⁴² software. The QTAIM/ELI-D intersection technique⁴³ was employed to evaluate the contribution, in terms of the electronic population, of a QTAIM atom (X) intersecting an ELI-D basin (B_i), quantified with the bond fraction p(B^X_i).^44,45 The ED, the ELI-D and their basins were displayed with the ParaView visualization application thanks to specifically designed plug-ins.⁴⁶

3. Results and discussion

3.1. Crystal structure

The samples of interest turned out to be almost RCuMg₂ single-phase (see Table 3), with small quantities of a secondary compound, with an average composition of ∼16.7 at% R, 16.7 at% Cu, and 66.6 at% Mg, compatible with the RCuMg₄ stoichiometry.

Table 3 Results of the SEM-EDXS and PXRD characterizations of samples with a R₂₅Cu₂₅Mg₅₀ nominal composition

Overall composition (at%)	RCuMg2 composition R; Cu; Mg (at%)	Crystal structure: oP32-DyCuMg2 Lattice parameters (Å)
		a	b	c
1-Dy24.4Cu23.3Mg52.3	23.8; 23.4; 52.8	13.530(5)	3.755(1)	13.793(4)
2-Ho26.8Cu24.0Mg49.2	24.6; 23.6; 51.8	13.506(4)	3.737(1)	13.785(4)
3-Er24.3Cu24.4Mg51.3	24.2; 25.0; 50.8	13.462(3)	3.720(1)	13.781(4)
4-Tm25.6Cu24.3Mg50.1	26.2; 23.8; 50.0	13.452(3)	3.700(1)	13.779(3)
5-Lu24.4Cu25.4Mg50.2	23.5; 24.2; 52.3	13.409(4)	3.6655(9)	13.773(3)

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Powder X-ray diffraction analysis confirmed that the samples are almost single-phase and that all five compounds are isostructural, belonging to the orthorhombic DyCuMg₂ prototype (see Fig. 2).

Fig. 2 Observed (red circles), calculated (black line) and difference (bottom blue line) X-ray powder diffraction patterns of the Dy_25.0Cu_25.0Mg_50.0 sample. Vertical bars indicate the Bragg positions of the DyCuMg₂ phase (R_B = 0.0951; R_F = 0.0494; R_p = 0.198; R_wp = 0.198); the most intense peaks of the secondary phase are indicated with *. The relative shift of experimental diffraction peaks for other RCuMg₂ compounds is shown in the 2θ range from 31 to 42° (inset).

As expected, a regular peak shift towards higher angles is observed when increasing the atomic number of the R component (see Fig. 2, inset), corresponding to a decrease in the lattice parameters due to the lanthanide contraction. The cell volume is plotted in Fig. 3 as a function of the R³⁺ ionic radius,⁴⁷ showing a perfectly linear trend. The percentage volume contraction is a qualitative indicator of all types of interactions in solids, especially useful when comparing related series of compounds.^48,49 This parameter was calculated for the title compounds, according to the formula , where V_calc = Σ_iN_i × V_i (N_i = number of i-type atoms in the unit cell, V_i = atomic volume of the i-type species taken from ref. 50) and V_meas is the experimentally determined volume. Negative ΔV(%) values (in the range from −2.5 to −3.7%) are obtained for all compounds, decreasing regularly as a function of the R component dimensions (see Fig. 3) and indicating interactions between components somewhat stronger with respect to pure metals.

Fig. 3 Unit cell volume and volume contraction trends for RCuMg₂ compounds.

For comparison, ΔV(%) was also calculated for the complete R₂Cu₂Mg series (R = La, Ce–Nd, Sm, Gd–Lu), considering its structural relationships with RCuMg₂ (see below). For the Dy–Lu subset, similar values and a decreasing trend are obtained (from −2.5 to −4.7%), as can be seen in Fig. S1.

Considering the Dy representative, the minimum interatomic distances in the 1:1:2 structure are Cu–Mg (2.678 Å) and Cu–Dy (2.896 Å), somewhat shorter than the metallic radii sums.⁵¹ Instead, the numerous Mg–Mg contacts range between 3.092 and 3.759 Å, which are common distances for intermetallics. The shortest interatomic distances for all the species are listed in Table 4. The coordination polyhedra of the smallest Cu atoms are three-capped trigonal prisms, with two different compositions Cu@Dy6CuMg2 and Cu@Dy4Mg5. The coordination topology of the different Mg species can be envisaged as derived from a rhombic dodecahedron (CN = 8 + 6), more or less distorted. Visualising a larger portion of the structure, what catches the eye is the presence of clusters formed by four trigonal prisms sharing edges to create a cubic-like cavity occupied by a Mg atom. This star-like structural fragment is typical of the ubiquitous Mo₂FeB₂ structure type, also adopted by R₂Cu₂Mg compounds. This similarity was the starting point of a comparative structural analysis of some compounds existing in these systems (see Fig. 1 and 4).

Fig. 4 Crystal structure relationships between R₂Cu₂Mg, RCuMg₂ and RCuMg₄ compounds. Structures are projected along the shortest lattice vector, and unit cells are shown in blue. As a guide for the eye, one star-like cluster is highlighted by a yellow circle.

Table 4 Interatomic distances (<4 Å) for DyCuMg₂, as obtained from single crystal data

Central atom	Adjacent atoms	Distance (Å)	Central atom	Adjacent atoms	Distance (Å)
Dy1	2× Cu2	2.9316(6)	Mg1	2× Cu2	2.6785(16)
	2× Cu1	2.9357(6)		2× Mg4	3.0923(24)
	2× Cu1	2.9472(6)		2× Mg3	3.1056(24)
	2× Mg2	3.3206(2)		2× Mg4	3.1451(24)
	2× Mg3	3.3849(18)		1× Mg5	3.2542(22)
	1× Mg1	3.5836(21)		1× Mg1	3.3926(29)
	1× Dy1	3.6581(4)		1× Dy1	3.5836(21)
	2× Dy1	3.7594(1)		1× Dy2	3.6144(22)
	1× Dy2	3.8208(4)		2× Mg1	3.7594(1)
	1× Dy3	3.8972(4)	Mg2	2× Cu1	3.0206(21)
Dy2	4× Cu2	2.8956(6)		2× Cu2	3.0797(21)
	2× Mg2	3.2931(23)		2× Dy2	3.2931(23)
	4× Mg4	3.3839(18)		2× Dy3	3.3197(23)
	2× Mg1	3.6144(22)		4× Dy1	3.3206(2)
	2× Dy2	3.7594(1)		2× Mg2	3.7594(1)
	2× Dy1	3.8208(4)	Mg3	1× Cu1	2.8554(22)
Dy3	4× Cu1	2.9083(5)		1× Cu2	2.9846(23)
	2× Mg2	3.3197(23)		2× Mg1	3.1056(24)
	4× Mg3	3.3695(18)		2× Mg5	3.1193(23)
	2× Dy3	3.7594(1)		2× Dy3	3.3695(18)
	2× Dy1	3.8972(4)		2× Dy1	3.3849(18)
Cu1	1× Cu1	2.6635(10)		1× Mg3	3.3985(31)
	1× Mg3	2.8554(22)		2× Mg3	3.7594(1)
	2× Dy3	2.9083(5)		1× Mg4	3.8069(29)
	2× Dy1	2.9357(6)	Mg4	1× Cu2	2.8851(22)
	2× Dy1	2.9472(6)		2× Mg1	3.0923(24)
	1× Mg2	3.0206(21)		2× Mg1	3.1451(24)
	2× Cu1	3.7594(1)		1× Mg4	3.1477(33)
Cu2	2× Mg1	2.6785(16)		2× Mg5	3.1548(25)
	1× Mg4	2.8851(22)		2× Dy2	3.3839(18)
	2× Dy2	2.8956(6)		1× Mg4	3.6421(33)
	2× Dy1	2.9316(6)		2× Mg4	3.7594(1)
	1× Mg3	2.9846(23)		1× Mg3	3.8069(29)
	1× Mg2	3.0797(21)	Mg5	4× Mg3	3.1193(23)
	2× Cu2	3.7594(1)		4× Mg4	3.1548(25)
				2× Mg1	3.2542(22)
				2× Mg5	3.7594(1)

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Because their R/Cu ratios are constant, it is convenient to examine these compounds in the ascending order of the Mg content. In the case of R₂Cu₂Mg, the crystal space is filled by the star-like clusters, with no gaps in between; therefore, Mg atoms are only located at the centre of the clusters.

With increasing Mg concentration, in the RCuMg₂ compounds, 2D wavy arrangements of these building blocks are alternated with Mg strips with a three-atomic-layer thickness. It is interesting to note that the Mg matrix possesses a bcc-like topology, instead of the hcp adopted by the pure element. This topology perfectly matches the clusters’ shape. The observed trend is even more evident when considering the Mg-richer RCuMg₄ compounds. In their structure, the same clusters are further diluted, adopting a 1D distribution in a Mg matrix, which still has a bcc-like topology. Following the proposed description, the dilution of star-like clusters with magnesium can be schematized as follows:

Considering the nature of constituents, the structural peculiarities, and the localization of the R component in common building blocks, both a chemical bonding analysis and a comparison of the magnetic properties of these compounds are discussed below.

3.2. Chemical bonding

The calculated electronic density of states (DOS, Fig. 5) for the LuCuMg₂ compound, chosen as a representative, reveals it to be a metallic conductor, as indicated by the non-zero number of states at the Fermi level.

Fig. 5 Total and species-projected electronic density of states (DOS/pDOS) for LuCuMg₂.

The narrow peak around −5 eV corresponds to the filled and localized Lu 4f states, whereas the broader region between approximately −2.5 and −4 eV is mainly contributed to by the filled Cu 3d states. Near the Fermi level (E_F), the DOS is mainly dominated by Mg 3p and Lu 5d states.

To get further insights into the chemical interactions among the constituents, position-space quantum-chemical techniques were selected. The calculated effective charges indicate that Lu and Cu act as the QTAIM cation and anion, respectively, regardless of the occupied Wyckoff position (see Fig. 6).

Fig. 6 Shapes and effective charges of QTAIM atomic basins for LuCuMg₂. Black sticks indicate Mg–Mg contacts within the bcc-like slabs, while the red square and orange triangles highlight the star-like clusters.

Therefore, LuCuMg₂ enriches the family of intermetallics with anionic transition metals, which have recently been increasingly studied owing to their often intriguing structural, physical and chemical properties.^52–59 Focusing on the two copper species, the difference between the charges of Cu1 and Cu2 (−1.74 vs. −2.34) can initially be attributed to the fact that Cu1 has another Cu1 neighbor (d_Cu1−Cu1 = 2.64 Å), as indicated by a large flat basin surface, quite typically observed for homocontacts (orange basin in Fig. 6).^33,60–63 The situation of the Mg atoms is more heterogeneous. In fact, all species are cationic except for Mg5, which shows a small negative charge (yellow basin in Fig. 6). Mg2, the only one not belonging to the bcc-like slab, bears the largest positive charge of +0.90, followed by Mg3, Mg1, and Mg4, all having, in addition to other Mg atoms, both Cu and Lu in their coordination sphere. The trend observed in the magnesium charges can be rationalized by considering the number of more electronegative atoms in its coordination environment, namely, Cu. At this stage of the discussion, the approximation of neglecting Lu is justified by the longer Mg–Lu distances compared to the Mg–Cu ones. Mg2 (+0.90) is coordinated by four Cu atoms, while both Mg3 and Mg1 are coordinated by two Cu atoms and exhibit similar charges (+0.65 and +0.56, respectively). Mg4 (+0.33), in contrast, is coordinated by only one Cu atom. Mg5 has a noticeably different coordination environment, comprising only Mg species. It is located at the centre of the bcc unit, which may cause its low negative charge, the origin of which will be further elucidated through the QTAIM/ELI-D intersection.

The average effective charges of +1.10 for Lu, +0.47 for Mg and −2.04 for Cu suggest classifying LuMg₂Cu as an intermetallic cupride, analogous to the LaMg₂Au auride.¹⁴ Interestingly, a bonding analysis based on orbital-based techniques was recently conducted for YCuMg₄ ⁶⁴ (oS48-TbCuMg₄), which is structurally related to the title compound, as displayed in Fig. 4. In that case, the calculated Löwdin charges indicate that both yttrium and copper act as cations, while the magnesium species get either zero or increasingly negative values as the number of Cu neighbours increases. It is important to highlight that because QTAIM and Löwdin charges are defined on completely different bases, their numerical values may differ significantly and should not be directly compared. Nevertheless, it is highly interesting to observe which overall description of related compounds emerges when different methods are applied. Reimann et al.⁶⁴ claimed that the covalent character of the Mg–Mg bonds in YCuMg₄ is responsible for this trend, although the charge values seem to contradict the electronegativity difference between Cu and Mg. In our case, the charge transfer better follows the electronegativity differences. A reasonable explanation for the low effective charge of Mg with respect to its formal value observed in LuCuMg₂ is the involvement of magnesium in covalent interactions, hinting toward a consistent picture.

Further insights were obtained through a topological analysis of the ELI-D along with the intersection of its basins with the QTAIM ones.

Several ELI-D attractors (maxima) are found in the crystal space around Mg species in the bcc-like slab and in the inner zone of the star-like cluster between Mg2, Lu and Cu species, as indicated by its planar distribution (Fig. 7a) and isosurfaces (Fig. 7b and c).

Fig. 7 ELI-D planar distribution (a) in the (020) and (010) planes together with 1.21 (b) and 1.14 (c) localization domains of ELI-D.

More specifically, ELI-D attractors in the bcc slab located in tetrahedral interstices are pointed out in Fig. 7b by three types of irreducible and one type of reducible localization domain. The latter, located between the Mg4–Mg4 contact, encloses two types of maxima (Fig. 7a in the (010) plane): one in the interstitial tetrahedral Mg4₂Mg5₂ site and the other in the region between Mg4 and neighbouring Cu2 and Lu2. Isosurfaces at lower ELI-D values (Fig. 7c) enable the visualization of attractors within the star-like clusters, while a single reducible domain encloses all maxima within the bcc slab as well as those between the slab and the surrounding Cu and Lu atoms. Focusing on the region of the star-like cluster, an attractor is situated between Mg2 atoms, in a sort of octahedral interstice, surrounded by four Lu and two Mg2 species and further by eight Cu atoms (similar to the octahedral interstices in a fcc lattice, Fig. 7a and c). In this region, numerous maxima of the ELI-D are also located, and the positions of some are indicated by the isosurfaces shown in Fig. 7c. These findings reveal a rather complex overall topology characterized by approximately twenty distinct types of ELI-D attractors in the valence region. In this case, it is convenient to visualize the bonding situation using basin sets. ELI-D basins with attractors higher than 1.06 and having a common isosurface were merged. In this way, only two types of basin sets were obtained: one that surrounds and incorporates the bcc Mg layer (blue in Fig. 8a) and the other shared along [010] between the fcc-like clusters (green in Fig. 8a).

Fig. 8 ELI-D basin sets (a) and bond basins (b, c, d) for LuCuMg₂.

Within these basin sets, two groups of ELI-D multiatomic (atomicity a ≥ 3) bonds can be distinguished: homoatomic, contributed almost exclusively by Mg atoms, and heteroatomic. The blue basin set in Fig. 8a is formed by both homo- and heteroatomic basins, whereas the green one is formed only by heteroatomic basins. Fig. 8b shows the most populated ELI-D basins located within and around the bcc slab. Specifically, homoatomic (Fig. 8b, basins 1–4) and heteroatomic (Fig. 8b, basins 5–7) bonds are displayed in grey and blue, respectively. The four homoatomic ones involve three or four Mg, with the bond population in all cases almost entirely contributed by Mg QTAIM atoms (see Table 5). This is evidenced by the total bond fraction of Mg p(B^Mg_i) ≥ 0.94, resulting in effectively 3- and 4-atomic bonds (3a or 4a; see Table 5).

Table 5 Average electronic populations N̅(B_i) and bond fractions for the ELI-D basins B_i of LuCuMg₂ displayed in Fig. 8. The numbers used to label basins are the same as those shown in Fig. 8b–d

	ELI-D basin (Bi)	Atomicity MgnCukLuh	N̅(Bi)
Fig. 8b	1	Mg4Lu2	0.83	0.99	—	0.01
	2	Mg3Cu1	0.64	0.99	0.01	—
	3	Mg4Cu2Lu2	1.06	0.94	0.03	0.03
	4	Mg3Cu2	0.39	0.94	0.06	—
	5	Mg2Cu2Lu2	2.09	0.37	0.40	0.23
	6	Mg3Lu1	0.75	0.88	—	0.12
	7	Mg1Cu1Lu4	1.90	0.12	0.66	0.22
Fig. 8c	1	Mg2Cu8Lu4	1.13	0.47	0.13	0.40
Fig. 8d	1	Cu2Lu2	0.62	—	0.80	0.20
	2	Cu2Lu3	0.43	—	0.77	0.23
	3	Cu2Lu2	0.50	—	0.83	0.17

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A similar scenario, with many Mg homoatomic interactions, was recently observed in the complex Mg_29−xPt_4+y intermetallic;⁵² interestingly, this is not the case for the LaAuMg₂ compound.¹⁴

Heteroatomic basins 5 and 7 in Fig. 8b are both contributed by all elements, resulting in 6a bonds, while basin 6 is 4a with no contribution from Cu.

At this point, the origin of the slight negative charge on Mg5 may be traced by intersecting the Mg5 QTAIM atom with the adjacent ELI-D basins. The contribution from the Mg5 core basin is 10.09 e⁻, and 1.70 e⁻ and 0.44 e⁻ are contributed by seven homoatomic and two heteroatomic bonds, respectively, yielding, for this species, a total average population of 12.23 e⁻.

The largest and most populated heteroatomic basin in the green basin set is labeled with number 1 in Fig. 8c; this is the bond showing the largest atomicity of 14 (see Table 5), located in the above-mentioned octahedral-like Mg₂Lu₄ interstitial site. It has to be noted that the eight Cu atoms located on the vertices of the fcc cluster contribute just 0.15 electrons, corresponding to a total bond fraction of 0.13 (see Table 5).

Three types of ELI-D attractors not included in the blue and green basin sets (because their values are lower than 1.06) are located in the columns formed by condensed Cu1₄Lu1₂ distorted octahedra. Their basins (1, 2 and 3 in Fig. 8d) are the only ones exclusively contributed by Lu and Cu (see Table 5). These are 4a (basins 1 and 3) and 5a (basin 2) polar bonds, where the main contribution to the population is made by the most electronegative copper atoms; the total bond fraction for copper is 0.77 ≤ p(B^Cu_i) ≤ 0.83. Therefore, no 2a polar R–T bonds indicated by bulges of the copper penultimate shell have been found, contrary to the case of Au–La in LaAuMg₂.¹⁴ This difference between the transition metals of the fourth period and those below is similar to that observed for other ternary rare earth intermetallics.⁶²

In summary, the chemical bonding scenario in LuCuMg₂ is highly heterogeneous, involving both homoatomic and heteroatomic interactions. The overall picture is further complicated by the presence of distinct types of heteroatomic bonds, some involving all three elements, while others occurring exclusively between Lu cations and Cu anions. As suggested by the partitioning of space obtained through basin sets, three distinct regions can be identified. The first one corresponds to the bcc-like Mg slabs and the region between them and the adjacent Cu and Lu atoms. In this area, the bonding scenario is characterized by homoatomic 3a and 4a bonds realized within the Mg slabs, which interact with surrounding Lu and Cu atoms via heteroatomic 4a and 6a interactions. The second region is located at the centre of star-like clusters. This area consists of fcc-like units (8 Cu, 4 Lu, and 2 Mg) condensed along the [010] direction, where multiatomic heteroatomic interactions are realized. The third region comprises the remaining crystal space. Here, heteroatomic interactions are still observed, with atomicities of 4 or 5, but notably, Mg is not involved, marking a clear distinction from the previous regions. Therefore, it can be concluded that LuCuMg₂ exhibits bonding inhomogeneity, a feature also recently observed in Mg_29−xPt_4+y, despite structural differences between the two compounds. These findings provide further evidence of the chemical tendency of Mg to segregate within the crystal space of certain intermetallic phases, resulting in homoatomic bonds. A similar bonding inhomogeneity may be supposed for the LPSO phases as well, with similar structural features.²¹

3.3. Physical properties

The temperature dependencies of the reciprocal magnetic susceptibility χ⁻¹(T) of selected RCuMg₂ compounds are depicted in Fig. 9. They are linear in a broad T-range (20–300 K) and thus fit well to the modified Curie–Weiss (CW) law: χ = C/(T − θ_CW). The deviation from the linearity of χ⁻¹(T) for DyCuMg₂ at much higher (in comparison with other isostructural compounds) T ≈ 80 K can be explained by (i) the enhanced ordering temperature (T_N) and/or (ii) the stronger crystal electric field (CEF) effects. The parameters deduced from CW fits are presented in Table 6. The obtained effective magnetic moments are in good agreement with the theoretically deduced values for R³⁺ ions. The negative θ_CW values hint towards antiferromagnetic interactions in the studied materials. Instead, the positive θ_CW for DyCuMg₂ is due to the narrow T-range used in the fit.

Fig. 9 Temperature dependencies of the reciprocal magnetic susceptibility χ⁻¹(T) at μ₀H = 1 T for RCuMg₂ compounds together with Curie–Weiss fits. Inset (a) magnetic susceptibility χ(T) of RCuMg₂ at the antiferromagnetic transitions; (b) magnetic susceptibility χ(T) of LuCuMg₂.

Table 6 Magnetic characteristics of RCuMg₂

Compound	T-range (K)	μ^theoreff (μB)	μ^expeff (μB)	θCW (K)	(K)	(K)	(K)	(K)	(K)
DyCuMg2	80–300	10.65	10.77(9)	3.0(1)	25.5(9)	20.5(9)
HoCuMg2	20–300	10.61	10.71(9)	−0.6(1)	7.5(9)
ErCuMg2	20–300	9.58	9.67(9)	−3.3(1)	5.5(9)		8.6(3)	6.6(3)	5.7(3)
TmCuMg2	20–300	7.56	7.71(9)	−4.1(1)	5.3(9)		6.1(3)	4.8(3)	3.3(3)

---

In agreement with the negative Curie–Weiss temperatures, all studied RCuMg₂ order most probably antiferromagnetically, with their critical temperatures T_N listed in Table 6. This is reflected in the broad cusps clearly visible in the temperature dependencies of magnetic susceptibility χ(T) (Fig. 9, inset a). The magnetic structure of DyCuMg₂ seems to be of a more complex character, because the well pronounced shoulder at becomes clearly visible. Notably, the anomalies in the antiferromagnetic orderings of the isostructural RCuMg₂ are remarkably broadened in χ(T) dependencies; the measurements of the specific heat capacity indicate multiple transitions for them (see discussion below).

The magnetic susceptibility of LuCuMg₂ (Fig. 9, inset b) is positive, it is about 3 orders of magnitude lower than those of other RCuMg₂ and is weakly temperature-dependent at 100 K < T < 300 K [χ₀ = 6.6(1) × 10⁻⁵ emu mol⁻¹, which corresponds to the density of states at the Fermi level N(E_F) = 2.1 state eV⁻¹ f.u.⁻¹, in fair agreement with the specific heat data]. All these features indicate that LuCuMg₂ is a Pauli paramagnet. The strong upturn in χ(T) below ≈50 K is due to a minor paramagnetic impurity (<0.1 vol%), which is not detectable by XRD and EDX.

The temperature dependencies of the specific heat capacity c_p(T) of RCuMg₂ (R = Er, Tm, Lu) under different magnetic fields are depicted in Fig. 10. The anomalies due to the antiferromagnetic ordering for compounds containing magnetic rare-earth elements reveal additional bumps and shoulders, thus indicating multiple transitions. The values at which they occur are presented in Table 6. Interestingly, the best coincidence of is observed with for Er and for Tm. Because all observed anomalies will be suppressed with increasing magnetic field, one can suggest that they have an antiferromagnetic origin. To shed light on the possible magnetic ordering, the total energies of the optimized structural models (standardized relaxed atomic coordinates are presented in Table S1) in ferromagnetic and antiferromagnetic cases were calculated (Table 7). The obtained electronic densities of states (DOS) and band structures are depicted in Fig. S2–S4. As can be seen from the projected DOS, the bands close to the Fermi level are dominated by the corresponding f-electrons of the rare-earth metal.

Fig. 10 Temperature dependencies of the specific heat capacities of RCuMg₂ (R = Er, Tm, Lu) under different magnetic fields. For LuCuMg₂, the fit for the c_p(T) = γT + βT³ + δT⁵ ansatz is shown.

Table 7 Total energies (in Ry) of the optimized structural models for antiferromagnetic (AFM) and ferromagnetic (FM) cases

	DyCuMg2	HoCuMg2	ErCuMg2
AFM	−227 486.29782	−234 847.44499	−242 385.31189
FM	−227 486.22978	−234 847.57351	−242 385.03028

---

In the cases of DyCuMg₂ and ErCuMg₂, the antiferromagnetic solution has a lower energy, whereas for HoCuMg₂, the ferromagnetic solution appears to be more favourable. At first glance, the latter result looks like a discrepancy with experimental data. However, considering the complexity of the magnetic interactions in the title compounds, which are characterized by multiple transitions [the broadened ordering at T_N = 7.5(9) for HoCuMg₂ could be obviously resolved in a c_p(T) dependence], one would need to account for other variants (e.g., canting of the spins and different ferromagnetic arrangements) to speculate about the reliable spin configurations in the studied series. Moreover, it is worth mentioning that starting from the optimized atomic positions of the AFM case and performing a single-point FM calculation (without further geometrical optimization), a total energy slightly below the AFM solution is obtained (Table 7). This suggests that small changes in the atomic positions may affect the results of magnetic behaviour simulations. Obviously, to finally and unambiguously shed light on the nature of the transitions, the complex magnetic structures of RCuMg₂ should be investigated via neutron diffraction.

In the temperature range of 1.8–7 K, the specific heat of the non-magnetic phonon reference compound LuCuMg₂ fits well to the c_p(T) = γT + βT³ + δT⁵ ansatz, with the Sommerfeld coefficient of the electronic contribution (c_el) γ = 6.58(8) mJ mol⁻¹ K⁻² [corresponds to N(E_F) = 2.8 state eV⁻¹ f.u.⁻¹], phononic terms (characteristic of c_ph) β = 0.86(7) mJ mol⁻¹ K⁻⁴ (indicates Debye temperature θ_D = 209 K) and δ = 1.6(7) × 10⁻³ mJ mol⁻¹ K⁻⁴.

Further, we subtracted the specific heat of the phonon reference LuCuMg₂ from that of RCuMg₂ (R = Er, Tm) and obtained magnetic contributions (c_mag) to their c_p(T) (Fig. 11). By calculating c_mag/T and integrating it, we found the temperature evolution of magnetic entropy in the studied compounds (Fig. 11, inset). In both cases, their values are smaller than Rln2 at the antiferromagnetic transitions and thus well below the theoretically expected Rln4 and Rln3 for Er³⁺ and Tm³⁺ ions, respectively. Such a strong discrepancy between these values may indicate that the classical LS coupling scheme is not adequate for RCuMg₂, and one of the possible reasons for this could be the enhanced level of itineracy of 4f electrons in the compounds.

Fig. 11 Temperature dependencies of the magnetic specific heat of RCuMg₂ (R = Er, Tm) obtained after the subtraction of phonon reference LuCuMg₂. Inset: Temperature evolution of the entropy near magnetic transitions.

The specific heat of LuCuMg₂ in the c_p/T ³(T)-presentation (Fig. 12) reveals a well-pronounced maximum centred at T_f ≈ 12 K. This is a signature of the strong contribution of the Einstein optical modes to the specific heat.^65,66 One of the possible reasons for such behaviour is the so-called ‘rattling’ effect arising from the scattering of the phonons on weakly bound and thus strongly vibrating atoms. The enhanced value of the thermal displacement parameter for the Mg5-atom in the 2f site (Table 2), which is obviously because this atom centres an enlarged [Dy8Cu4] cuboctahedron (Fig. 4), suggests that it is a possible candidate for this phenomenon. However, to confirm this assumption, the temperature evolution of its thermal displacement parameter should be studied.

Fig. 12 Temperature dependence of the specific heat of LuCuMg₂ in the c_p/T ³(T) presentation together with the fit to eqn (1) (red line) as well as separate Debye (eqn (2)), Einstein (eqn (3)) and electronic contributions.

Despite the lack of solid evidence for ‘rattling’ in LuCuMg₂, we tried to describe its c_p/T ³(T) using the model given in eqn (1). For this purpose, the heavier Lu and Cu atoms were assumed to form a framework providing six Debye modes (N_D1 = 6), whereas the lighter magnesium atoms were expected to be responsible for other ones (N_D2). Thus, having 16 Mg atoms in the unit cell, from which 2 were ‘rattlers’, N_D2 and N_E were expected to be 5.25 and 0.75, respectively. Hence, the total number of modes should be N_tot = N_D1 + N_D2 + N_E = 12. Further, the fit to eqn (1) with two Debye (eqn (2)) and one Einstein (eqn (3)) contributions resulted in N_D1 = 9.6(5), N_D2 = 3.2(2), and N_E = 0.2(1), with the corresponding values of the characteristic temperatures of θ_D1 = 342(9), θ_D2 = 143(3), and θ_E = 57(1). The Sommerfeld coefficient [γ = 6.8(2) mJ mol⁻¹ K⁻²] deduced from such a fit was found to perfectly agree with the value obtained above.

Obviously, the calculated number of modes as well as N_tot = 13 somewhat deviate from the theoretical expectations. All these results clearly evidence the complexity of the phononic spectrum of LuCuMg₂, which is poorly described by the simplified model given with eqn (1).

4. Conclusions

A combined study on the five {Dy, Ho, Er, Tm, Lu}CuMg₂ new compounds was conducted following three main lines: crystal structure, chemical bonding and magnetic properties.

These phases, existing only with heavy rare earth metals, represent a new structure type, where characteristic Mg-centred star-like clusters are condensed in 2D wavy fragments spaced by Mg slabs with a bcc topology. The same building principle was recognized for other R_xCu_xMg_y compounds, emphasizing the correlation between their composition and the constitutive clusters’ “aggregation” into the bcc-Mg matrix. Following this idea, the structural generalization can probably be extended to similar compounds containing different R and/or T components.

When looking at the structural peculiarities of the numerous R–T–Mg intermetallics, it becomes evident that there exist “preferred” fragments (depending on the compositional region) serving as common building moieties of the corresponding compounds. Among others, this is the case for the Mg-rich LPSO phases, featuring Mg@R₆T₈ clusters embedded in a magnesium matrix including layers of “pure” hcp-Mg.²¹ Despite the differences in the cluster composition/geometry and in the topology of the Mg matrix, the architectures of LPSO and R_xCu_xMg_y show interesting similarities, including the occurrence of pure magnesium fragments. This opens new perspectives for broadening the characterization of these two families of compounds, the former being especially studied with respect to mechanical properties and the latter with respect to physical and magnetic behaviour. The electronic structure and bonding analyses performed on the representative LuCuMg₂ compound confirm its metallic character and reveal a complex bonding scenario, dominated by both hetero- and homoatomic interactions, the latter involving exclusively Mg atoms within the bcc-like slab. The bonding landscape is highly heterogeneous, with three distinct spatial regions identified: (i) the Mg slabs and their interface with Lu and Cu atoms, where homoatomic (among Mg species) and heteroatomic interactions coexist; (ii) the central star-like clusters composed of fcc-like units (8 Cu, 4 Lu, and 2 Mg), where multiatomic heteroatomic bonding occurs; and (iii) the remaining crystal space exclusively composed by Lu and Cu atoms, where heteroatomic interactions persist without Mg involvement. The negative effective charge on copper allows us to include this compound in the family of intermetallics with anionic transition metals.

The magnetic susceptibility measurements indicate that RCuMg₂ (R = Dy, Ho, Er, Tm) follow the Curie–Weiss law in the 20–300 K range, with effective magnetic moments almost perfectly coinciding with the theoretically calculated ones for the trivalent R-ions. The nonmagnetic LuCuMg₂ phonon reference compound was identified as a Pauli paramagnet. {Ho, Er, Tm}CuMg₂ order antiferromagnetically at critical temperatures between about 5 and 8 K, whereas DyCuMg₂ reveals multiple magnetic transitions. The temperature evolution of the magnetic entropy deduced from the analysis of specific heat is well below the values predicted by the classical LS coupling scheme, indicating the complexity of the magnetic behaviour of the studied compounds.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this article have been included as part of the supplementary information (SI). Supplementary information: unit cell volume and volume contraction trends for R₂Cu₂Mg compounds; refined and optimized atomic coordinates for the studied compounds; total and atomic resolved electronic density of states calculated for {Dy, Er, Ho}CuMg₂; electronic band structures for ferromagnetic and antiferromagnetic {Dy, Er, Ho}CuMg₂. See DOI: https://doi.org/10.1039/d5dt01632k.

CCDC 2381656 (DyCuMg₂) contains the supplementary crystallographic data for this paper.⁶⁷

Acknowledgements

The DynaCool-12 system was acquired within the Deutsche Forschungsgemeinschaft (DFG) Project No. 422219907.

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