The solid solutions CeRu_1–xPd_xSn and CeRh_1–xPd_xSn – Applicability of the ICF model to determine intermediate cerium valencies by comparison with XANES data

2.1 Synthesis

Starting materials for the syntheses of the CeRu_1–xPd_xSn and CeRh_1–xPd_xSn samples were a cerium ingot (Sigma Aldrich), ruthenium and rhodium powder (Allgemeine Pforzheim; Merck), a palladium sheet (Allgussa) and tin granules (Merck), all with a stated purity better than 99.9%. Small pieces of cerium were cut under dried (sodium wire) paraffin oil, washed with dried (sodium wire) n-hexane, and kept under argon in Schlenk tubes prior to the reactions. At first small cold-pressed pellets of ruthenium, respectively rhodium powder were prepared to prohibit splashing during the arc-melting procedure. The remaining elements were then weighed in the ideal stoichiometric ratios in relation to the compositions of these pellets and arc-melted [36] under argon (ca. 800 mbar). The argon was purified over molecular sieves, silica gel and a titanium sponge (900 K). The molten buttons were re-melted three times to ensure homogeneity. Bulk samples have metallic luster while ground polycrystalline powders are dark grey. The samples are air-stable.

2.2 X-ray diffraction

The polycrystalline samples of both solid solutions were characterized by Guinier powder patterns (imaging plate technique, Fujifilm BAS-1800) using CuK_α1 radiation and α-quartz (a = 491.30, c = 540.46 pm) as an internal standard. Standard least-squares refinements led to the hexagonal and orthorhombic lattice parameters listed in Table 1. Comparison of the experimental patterns with calculated [37] ones assured correct indexing.

Irregularly shaped crystals were selected from the CeRu_0.28Pd_0.72Sn and CeRh_0.66Pd_0.34Sn samples by mechanical fragmentation. The crystals were glued to quartz fibers using beeswax and investigated by Laue photographs on a Buerger camera (white molybdenum radiation, image plate technique, Fujifilm, BAS-1800) in order to check crystal quality and suitability for intensity data collection. Single-crystal diffraction intensities were collected at room temperature on a Stoe IPDS-II image plate system (graphite monochromatized Mo radiation; λ = 71.073 pm) in oscillation mode. Numerical absorption corrections were applied to the data sets. Details of the data collections and the structure refinements are listed in Table 2.

2.3 EDX data

The CeRu_0.28Pd_0.72Sn and CeRh_0.66Pd_0.34Sn crystals studied on the diffractometer were analyzed by semiquantitative EDX analysis using a Zeiss EVO MA10 scanning electron microscope with CeO₂, ruthenium, rhodium, palladium, and tin as standards. No impurity elements heavier than sodium (detection limit of the instrument) were observed. The obtained ratios of the transition metals are with respect to standard deviations of the EDX technique in good agreement to the refined ratios.

2.4 Structure refinements

The two data sets showed hexagonal lattices with high Laue symmetry and no further systematic extinctions. Since isotypism with CeRhSn (space group P6̅2m) was already evident from the Guinier powder patterns, we used the atomic parameters of the rhodium compound [38] as starting values. Both structures were refined using jana2006 [39] with anisotropic atomic displacement parameters for all atoms. A severe problem concerns the small difference in scattering power between ruthenium and palladium (2 electrons) and rhodium and palladium (1 electron). In the first cycles we refined the T1 (2d) and T2 (1a) sites exclusively with ruthenium, respectively rhodium and then refined their occupancy parameters. The 2d site of the ruthenium containing crystal showed full palladium occupancy within two standard deviations and the 1a site was refined with Ru/Pd mixing, leading to the data listed in Table 3. In contrast, we obtained no stable refinement with free rhodium/palladium mixing for the rhodium containing crystal. In agreement with the starting composition and the results from EDX analyses of the crystal, we fixed the Rh/Pd ratio at 66/34 in the final cycles. Refinement of the correct absolute structures was ensured through calculation of the Flack parameter [40, 41]. The final difference Fourier syntheses were flat (Table 2). The positional parameters and interatomic distances (exemplarily for CeRh_0.66Pd_0.34Sn in comparison with CeRhSn [38]) are listed in Tables 3 and 4. Further information on the structure refinements is available.^[1]

2.5 X-Ray Absorption Near Edge Structure (XANES)

A few samples of the solid solutions CeRu_1–xPd_xSn and CeRh_1–xPd_xSn have been investigated by X-ray absorption spectra at the CeL_III edge. The data were recorded in transmission mode at the BM01B station (Swiss Norwegian Beamlines, SNBL), ESRF, Grenoble, France. The measurements were performed using a Si(111) double crystal monochromator. The second crystal of the monochromator was detuned by 60% in order to suppress higher harmonic radiation from the synchrotron. The intensities of the incident and transmitted X-rays were monitored with ionization chambers (nitrogen and helium gas filled). All spectra were acquired in a continuous scanning mode from 5680 to 5900 eV, with energy steps of 0.5 eV and 1 s integration time. Each scan was completed after about 7.5 min, 4 scans were collected for each sample and merged.

The samples were ground in an agate mortar under cyclohexane in order to avoid oxidation and then homogeneously mixed with small amounts of cellulose (Sigma-Aldrich) and pressed to pellets. The amount of sample was optimized for transmission measurements. The experiments were performed at ambient conditions. Energy calibration was based on the energy of the first peak of CeO₂ (reference compound) at 5731 eV. The athena software was used for data normalization [42].

2.6 Magnetic measurements

The magnetic susceptibility measurements were carried out on a Quantum Design Physical Property Measurement System using the VSM or the AC-Transport option, respectively. For the magnetic measurements approximately 40 mg pieces of the crushed samples were fixed with kapton foil and attached to the sample holder rod. Magnetic investigations were performed in the temperature range of 2.5–350 K with magnetic flux densities up to 80 kOe (1 kOe = 7.96 × 10⁴ A m^–1).

For the resistivity measurements two different procedures were pursued. Arc-melted buttons were either grinded and pressed to pellets or directly sintered at 500 K. Subsequently, the annealed buttons were polished in a parallel arrangement. To achieve a parallel geometry, the buttons were embedded in a polymethylmethacrylate (PMMA) matrix. After the first polishing step, a self-built device allowed parallel polishing of the second side. The samples were polished until an approximately 1 mm thick specimen remained inside the polymer matrix. The disc shaped samples were removed by dissolving the PMMA matrix in acetone. The resistivity measurements were carried out by the van-der-Pauw technique [43] in AC transport mode. The ACT puck was modified by a van-der-Pauw press contact assembly purchased from Wimbush Science & Technology. The probes are spring contacts, gold plated over nickel and the distances between the pins were set to 2 mm. The resistivity was measured between 2.5 and 300 K and the recorded data of channel 1 and 2 were converted according to the van-der-Pauw equation given in the Quantum Design Application Note 1076-304.

3.1 Crystal chemistry

Depending on the valence electron count (VEC), the stannides of the solid solutions CeT_1–xT′_xSn crystallize with different structure types (see Introduction). The main question concerns a possible change of the structure type through stepwise substitution of the transition metal. Within the solid solution CePt_1–xNi_xSn [26] the VEC remains constant and all members of the solid solution crystallize with the orthorhombic TiNiSi type [44]. This structure type also occurs for the solid solution CeNi_1–xCo_xSn [28] up to x = 0.5, indicating a certain electronic flexibility. A change in structure type arises for CeNi_1–xRh_xSn [27]. The orthorhombic TiNiSi type is retained up to x ≈ 0.5 and then switches to the hexagonal ZrNiAl type [45–47], similar to pure CeRhSn. A double switch in structure type occurs for the recently reported solid solution CeRu_1–xNi_xSn [33]. The monoclinic CeRuSn type allows only for tiny ruthenium-nickel substitution and one readily observes the ZrNiAl type up to x ≈ 0.5. In this range the VEC is close to isotypic CeRhSn. With increasing nickel content, the orthorhombic TiNiSi type forms and the cerium ground state switches from intermediate valent to stable trivalent cerium.

In the present work we studied the Ru/Pd and Rh/Pd substitutions in CeRu_1–xPd_xSn and CeRh_1–xPd_xSn. Again, monoclinic CeRuSn allows almost no ruthenium substitution, and the structure type readily changes to ZrNiAl. In the entire region up to x ≈ 0.8, the cerium atoms remain in an intermediate valence state, followed by a switch to the orthorhombic TiNiSi type and trivalent cerium. The rhodium-based system shows similar behavior (Fig. 1). Each switch in structure type produces a two-phase region, probably due to small immiscibility gaps.

Fig. 1:

Development of the unit cell volume per formula unit in the solid solutions CeRu_1–xPd_xSn and CeRh_1–xPd_xSn as a function of the palladium content. The development of the unit cell volumes of the phases with ZrNiAl type structure is presented by a line of the best fit for Vegard type behavior. Data points due to powder or single crystal diffraction and literature are drawn as black squares, red circles and blue triangles, respectively.

In the ZrNiAl type regime, substitution of ruthenium (124 pm) and rhodium (125 pm) by the slightly larger palladium (128 pm) atoms [48] leads to an increase of both the a and c lattice parameters (Fig. 2). A small excess increase of the unit cell volume results from the Ru/Pd and Rh/Pd disorder. This results in different bond lengths and hampers perfect long-range order. Table 4 lists the interatomic distances of ternary CeRhSn along with data for the palladium substituted stannide CeRh_0.66Pd_0.34Sn, showing an almost isotropic increase of the bond lengths.

Fig. 2:

Development of the a and c lattice parameters in the solid solutions CeRu_1–xPd_xSn (black) and CeRh_1–xPd_xSn (red) with ZrNiAl type structure as a function of the palladium content.

Finally we point to the coordination of the transition metal atoms (Fig. 3) in the different structure types. The coordination is always derived from a trigonal prism. These prisms are ideal in the hexagonal ZrNiAl type phases while small distortions occur in the orthorhombic TiNiSi and in the monoclinic CeRuSn type. Only CeRuSn exhibits additional weak Ru–Ru bonding, while there are no close T–T contacts in the orthorhombic and hexagonal phases.

Fig. 3:

The near neighbor coordination of the transition metal atoms in the ZrNiAl type (top), CeRuSn type (bottom, left and middle) and TiNiSi type (bottom, right) phases CeTSn. Cerium, transition metal and tin atoms are drawn as medium grey, black filled and open circles, respectively.

The structure types TiNiSi and ZrNiAl are well known basic structure types for intermetallic phases, both with several hundred representatives [49]. For further crystal chemical details we refer to review articles [6, 50–52]. The main topic of the present contribution concerns the course of the cerium valence as a function of the transition metal substitution, studied spectroscopically as well as by temperature dependent magnetic susceptibility measurements (vide infra).

3.2 X-Ray Absorption Near Edge Structure (XANES)

Figures 4 and 5 display the results of XANES measurements of the CeL_III edge absorption spectra of the solid solutions CeRu_1–xPd_xSn and CeRh_1–xPd_xSn, respectively. Except for CeRu_0.1Pd_0.9Sn, which crystallizes in the TiNiSi type structure (dashed line), all investigated compounds exhibit the ZrNiAl type structure (solid lines). All spectra have in common a main white line at about 5726 eV that can be attributed to the configuration of Ce³⁺. Compounds with high ruthenium or rhodium contents exhibit an additional peak approximately 10 eV above the main white line that can be ascribed to the Ce⁴⁺ electronic configuration. This characteristic double peak feature reflects a mixed valence state in these compounds as already described in the literature [31–34]. In addition, a systematic development of the relative intensity of the Ce³⁺ and Ce⁴⁺ lines as a function of the VEC can be observed. In general, a higher VEC leads to a decrease of the Ce⁴⁺ peak.

Fig. 4:

Normalized Ce L_III spectra of samples from the solid solution CeRu_1–xPd_xSn for different palladium contents. The dashed line belongs to the compound that exhibits the TiNiSi type, whereas the solid lines illustrate samples with ZrNiAl type structure. The arrows serve as a guide to the eye to underline the development of the cerium valences.

Fig. 5:

Normalized Ce L_III spectra of samples from the solid solution CeRh_1–xPd_xSn for different palladium contents. The arrows serve as a guide to the eye to underline the development of the cerium valences.

For an estimation of the cerium valence of the mixed valence compounds, fittings of the experimental data were performed with a combination of Lorentzians and arctangent functions. The Athena software was used for these purposes [42]. The ratio of the fractions of Ce³⁺ and Ce⁴⁺ ions was estimated from the ratio of the areas of the two Lorentzian curves. This procedure has been applied in previous works [31–34]. The results of this analysis are reported in Table 5 for both solid solutions and are in accordance with the tendencies described above. No second peak that would indicate Ce⁴⁺ was observed for CeRu_0.1Pd_0.9Sn, which exhibits the TiNiSi type, and for CeRh_1–xPd_xSn with x ≥ 0.5. Thus fitting of the magnetic susceptibility with a Curie-Weiss law should at least be possible for these samples, as discussed in more detail in the next part.

3.3 Magnetic properties

Magnetic measurements were performed for all phase pure samples that could be obtained with the ZrNiAl type structure in order to analyze the influence of the VEC. Additionally, for CeRu_0.1Pd_0.9Sn (TiNiSi type structure) no magnetic ordering down to 2.5 K could be observed and the fitting results obtained with the Curie-Weiss law are in good accordance with a trivalent state of cerium (Table 6). The temperature dependence of the reciprocal magnetic susceptibility (χ^–1 data) of CeRu_1–xPd_xSn and CeRh_1–xPd_xSn is depicted in Figs. 6 and 7, respectively. All samples were investigated in the temperature range of 2.5–350 K with a magnetic field of 10 kOe. No magnetic ordering down to 2.5 K could be observed for any of these compounds in low-field studies. Remarkable is the clear tendency of decreasing χ^–1 values with increasing VEC. For CeRu_0.1Pd_0.9Sn a steep increase of χ^–1 can be observed until a negative curvature leads to an almost temperature independent region above 200 K. However, the samples with a higher VEC exhibit a more and more linear increase like it is expected for Curie paramagnetism, which is caused by trivalent cerium. Higher values of the reciprocal susceptibility can only be explained by partially tetravalent cerium.

Fig. 6:

Temperature dependence of the reciprocal magnetic susceptibility χ^–1 of the solid solution CeRu_1–xPd_xSn with 0.1 ≤ x ≤ 0.6 measured with a magnetic field of 10 kOe.

Fig. 7:

Temperature dependence of the reciprocal magnetic susceptibility χ^–1 of the solid solution CeRh_1–xPd_xSn with 0 ≤ x ≤ 0.7 measured with a magnetic field of 10 kOe.

In both solid solutions an increase of the amount of trivalent cerium with the increasing valence electron count can be observed. Consequently, the key question is which theoretical model is the best to reasonably describe the data. The most common way to fit the magnetic susceptibility χ in dependence of the temperature T is the Curie-Weiss law.

(1)χ=CT−θp (1)

C is the Curie constant and θ_p the Weiss constant. This equation neglects the Pauli-paramagnetic contribution of the conduction electrons, which is considered in a modified version by a temperature independent coefficient χ₀.

(2)χ=CT−θp+χ0 (2)

Due to this additional coefficient it is possible to fit a slightly curved development of the reciprocal susceptibility. However, both Curie-Weiss laws do not consider the temperature dependency of the effective magnetic moment, which is the only possibility to explain any decreasing χ^–1 value. Such a case can only be described by a higher population of the Ce³⁺ state at high temperatures in comparison to lower ones. Sales and Wohlleben developed a model (Interconfiguration fluctuation model, ICF) addressing a mixed-valent behavior of rare earth atoms [35].

(3)χ(T)=(N3kB)[μ12υ(T)+μ22(1−υ(T))T+Tsf]+n[CT]+χ0 (3)

(4)υ(T)=2J1+1(2J1+1)+(2J2+1)exp(−EexkB(T+Tsf)) (4)

In these expressions: (i) E_ex = E(Ce³⁺) – E(Ce⁴⁺) is the energy gap between the ground states of each cerium configuration; (ii) T_sf the quantum spin fluctuation temperature; (iii) ν(T) the fractional occupancy of the Ce⁴⁺ state (valence = 3 + ν(T)); (iv) J₁, J₂, μ₁ and μ₂, respectively, the quantum numbers and effective magnetic moments corresponding to the two levels (J₁ = μ₁ = 0 for Ce⁴⁺; J₂ = 5/2 and μ₂ = 2.54 μ_B for Ce³⁺); (v) n the proportion of stable Ce³⁺ (C = 0.807 emu K mol^–1) impurity, and (vi) χ₀ the temperature independent part of magnetism. Characteristic features of samples with such a behavior are small susceptibility values (of the order of 10^–3 emu per mol Ce atom) and even more important is a characteristic broad peak [53]. Such a peak is caused by a faster population of the state with higher energy (in this case Ce³⁺) in comparison to the decrease of the susceptibility. Consequently, the absence of a broad peak, as observed in the investigated compounds, does not clearly exclude the usability of the ICF model. Simply a smaller population of the Ce³⁺ state has to be fulfilled and a situation like this should possibly be described as intermediate-valent instead of interconfiguration fluctuating. Nevertheless the ICF model is the only one of the three ones mentioned above that considers Ce³⁺ as well as Ce⁴⁺ states. A more detailed description of this model can be found in the literature [35, 53, 54].

Referring to the results from XANES it is known that the ruthenium and rhodium rich compounds show significant amounts of tetravalent cerium, while in the palladium rich ones the cerium atoms are almost or totally trivalent. As a consequence, fittings with all mentioned laws have been obtained. The results achieved with the ICF model are summarized in Table 7, while results of the Curie-Weiss fits are listed in Table 6. No parameters obtained from fitting with the modified Curie-Weiss law are mentioned because no physically meaningful results could be achieved. All attempts have led to very small effective magnetic moments, e.g., for CeRu_0.5Pd_0.5Sn a cerium valence of 3.44 could be calculated, which is not compatible with the XANES and susceptibility results, though very good R² values have been obtained for the fitting procedure. For four compounds both fitting parameters are listed in Tables 6 and 7 (CeRu_0.4Pd_0.6Sn and CeRh_1–xPd_xSn with x = 0–0.2). Cerium valencies calculated with the ICF model are in the range of 3.11–3.13 for these four compounds, while Curie-Weiss fitting leads to values between 3.02 and 3.05. In comparison to the XANES results that show cerium valences in the range of 3.09–3.11, clearly the ICF model shows the better consensus. For CeRh_0.7Pd_0.3Sn a cerium valence of 3.11 and 3.05 was determined by XANES and Curie-Weiss fitting, respectively. Unfortunately, no fitting with the ICF model was possible without obtaining a negative value for E_ex which is synonymous with the physically impossible situation that the Ce³⁺ has a lower energy than the Ce⁴⁺ state. Nevertheless, the results obtained with the Curie-Weiss fit are only slightly deviating and are in even better compliance for higher rhodium contents. For comparison of the susceptibility fitting and the XANES results the cerium valences at 300 K are plotted in Fig. 8 for CeRu_1–xPd_xSn and in Fig. 9 for CeRh_1–xPd_xSn. One important remark for the comparison of cerium valencies detected by susceptibility measurements and XANES is the maximum value of about 3.5 in CeO₂ determined by XANES [55], while magnetic measurements clearly prove diamagnetism.

Fig. 8:

The cerium valence of CeRu_1–xPd_xSn at 300 K in dependence of the palladium content. The data calculated by the ICF model are compared with those determined by XANES.

Fig. 9:

The cerium valence of CeRh_1–xPd_xSn at 300 K in dependence of the palladium content. The data calculated by the ICF model are compared with those determined by XANES.

In Fig. 10 the calculated cerium valencies with the ICF model are plotted in dependence of the temperature. These results are achieved by insertion of the fitting parameters in eq. 4. It is obvious that the temperature dependency of the cerium valence becomes less with increasing palladium content. CeRu_0.9Pd_0.1Sn exhibits a change of the valence of approximately 0.1 in the investigated temperature range, while it is constant for CeRu_0.4Pd_0.6Sn. This might be important evidence that Curie-Weiss fitting also becomes useful for these compounds because no change in the cerium valence leads to a linear increase of the inverse susceptibility. The remaining tetravalent character should result in a reduced effective magnetic moment. However, the fitting with the ICF model seems to be the better way for this compound as discussed above. Two general points concerning the fitting with the ICF model should be finally discussed. The first one concerns the high values for n, the proportion of stable Ce³⁺, which was originally introduced as “impurity term”. As discussed above, less valence change with temperature is observed and the “impurity term” is just a simplified form of the Curie-Weiss law. Consequently, an increasing n underlines again the more sensible applicability of the Curie-Weiss law. Consequently, n cannot anymore be neglected for the calculation of the cerium valence and is considered for all compounds. The second point concerns the fixed fitting parameters during the last refinement cycles. In general, the temperature independent part χ₀ and the amount of stable Ce³⁺n were fixed to prevent over-determination. Due to the increasing importance of n, this parameter was refined, while the quantum spin fluctuation temperature T_sf was fixed.

Fig. 10:

Calculated cerium valence for the solid solution CeRu_1–xPd_xSn with 0.1 ≤ x ≤ 0.6 in the temperature range from 5 to 350 K by applying equation 2 of the ICF model [35].

3.4 Electrical properties

In order to avoid data overlap, only selected resistivity data of both solid solutions are shown in Figs. 11 and 12. All remaining compounds are totally in line with the behavior and the tendencies that will be discussed in this part. As described in the experimental section, two different ways for the sample preparation were chosen. To achieve a better comparability and to neglect the influence of grain boundaries, relative resistivities and no absolute values are discussed here. For the solid solution CeRu_1–xPd_xSn the sample preparation via sintered tablets was favored, while for the CeRh_1–xPd_xSn one polished arc-melted buttons turned out to be the best opportunity.

Fig. 11:

Temperature dependence of the reduced electrical resistivity of samples from the solid solution CeRu_1–xPd_xSn with x = 0.1, 0.3, 0.5 and 0.7.

Fig. 12:

Temperature dependence of the reduced electrical resistivity of CeRhSn and samples from the solid solution CeRh_1–xPd_xSn with x = 0.1, 0.3, 0.5 and 0.7.

Starting with the sample with the lowest VEC (CeRu_0.9Pd_0.1Sn), metallic behavior is observed. At low temperatures saturation due to the residual resistivity, and a linear increase of ρ(T) can be observed with higher temperatures. Nevertheless, two small anomalies should be discussed. A slightly curved character instead of a totally linear increase and a weak increase instead of linear saturation are observed. Both characteristics are more evident for the samples with higher palladium contents. The slightly curved character turns into a weak and broad maximum around 75–100 K. This would be even more pronounced if the lattice contribution could be subtracted as often described in literature [56–58]. In literature such a behavior is often attributed to crystal field (CF) effects, as for example reported for CeAl₂ and CeAl₃ ([57, 58] and references therein).

While all results shown for this solid solution exhibit an increase of ρ(T) with increasing temperature, for CeRu_0.5Pd_0.5Sn it decreases logarithmically. This is an interesting parallel to CeRhSn, which exhibits the same VEC. This decrease at high temperatures is a characteristic of the Kondo effect. Nevertheless, the most prominent feature of a Kondo system, the rapid decrease at low temperatures due to the formation of a spin-compensated state (Kondo state) [59, 60], is not observed for any of the reported compounds. The electrical resistivity of CeRhSn has already been discussed in detail in the literature and is in accordance with the observed result. It is described as Kondo-intermediate valence behavior [18, 56] due to the features discussed above. At temperatures above 50 K an almost identical development of CeRu_0.5Pd_0.5Sn and CeRhSn is observed, whereas below 50 K both compounds behave totally different. While for CeRhSn a linear and steep decrease is obtained, for CeRu_0.5Pd_0.5Sn there is an increase. Such an increase has for example been reported for CeNiSn below 6 K and was related to the opening of a pseudo gap [61]. Nevertheless, further investigations on different single crystals proved that the increase of the resistivity at lower temperatures is caused by impurities, and CeNiSn exhibits metallic behavior [62]. According to the investigations on CeNiSn the observed increase below 25 K is most likely due to intrinsic Ce³⁺ impurities. These are caused by the increasing palladium content and find also expression in higher amounts of stable Ce³⁺ (n) in the ICF fitting (vide supra).

One particularity is the lower ρ(T)/ρ(270 K) value at low temperatures (e.g., 10 K) of CeRu_0.3Pd_0.7Sn in comparison with the remaining members of the solid solution. In general, an opposite trend can be observed. Taking into account that the solid solution contains rhodium instead of ruthenium, this feature seems to be consistent. Except for CeRhSn, CeRh_0.9Pd_0.1Sn exhibits the highest ρ(T)/ρ(270 K) value at 10 K, and a continuous decrease with higher palladium content is observed indicating an increasing metallic character.