Coexistence of three types of sodium motion in double molybdate Na₉Sc(MoO₄)₆: ²³Na and ⁴⁵Sc NMR data and ab initio calculations

Abstract

The rechargeable Na-ion batteries attract much attention as an alternative to the widely used but expensive Li-ion batteries. The search for materials with high sodium diffusion is important for the development of solid state electrolytes. We present the results of experimental and ab initio studies of the Na-ion diffusion mechanism in Na₉Sc(MoO₄)₆. The ion conductivity reaches the value of 3.6 × 10⁻² S cm⁻¹ at T ∼ 850 K. The ²³Na and ⁴⁵Sc NMR data reveal the coexistence of three different types of Na-ion motion in the temperature range from 300 to 750 K. They are activated at different temperatures and are characterized by substantially different dynamics parameters. These features are confirmed by ab initio calculations of activation barriers for sodium diffusion along various paths.

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1. Introduction

The renewal of interest in Na-conducting materials, observed in recent years, is mainly due to the wide availability and low price of sodium. Na-ion batteries can compete with lithium-ion batteries in areas where the mass and energy density are not very critical, for example, for large-scale storage of renewable energy sources such as sun, wind, etc.^1–8 Despite the similar electrochemistry for Li and Na, sodium devices have a lower operating voltage and specific capacity than lithium batteries. To compensate for these effects, a viable strategy is to use polymer or inorganic solid electrolytes instead of conventional organic materials. An all-solid battery makes it possible to increase both the battery power and equipment safety due to the thermal stability and the absence of leaks. Besides the well-known sodium-conducting materials based on Na₃PS₄,^9–14 β-Al₂O₃^15–18 and the NASICON-related phosphates,^19–24 which have extremely high Na-ion transport characteristics, other compounds with an “open” crystal structures are also promising. In particular, a sufficiently high Na-ion mobility has been observed in the double molybdates containing both alkali and di- or trivalent cations. In the Na₂MoO₄–R₂(MoO₄)₃ systems the Na_xR_y(MoO₄)_(x+3y)/2 molybdates are known to exist in a wide range of x : y ratios, 1 : 1, 5 : 1, 1 : 5, 3 : 1 and 9 : 1^25–28 and rich variety of structure types. The ionic conductivity in these compounds, σ > 10⁻³ S cm⁻¹ at 670–770 K, and activation energy, E_a ∼ 0.6–0.8 eV^29–34 are comparable with those for Na_2+2xFe_2−x(SO₄)₃³⁵ and exceed the corresponding values for isostructural orthophosphates Na₂M₃(PO₄)₃ with M₃ = GaMn₂, GaCd₂, InMn₂ and FeMnCd (which are characterized by σ ∼ 10⁻⁵ S cm⁻¹ at 673 K and E_a = 0.7–0.8 eV).³⁶

Besides practical promise, these compounds are also interesting for fundamental studies of ion dynamics. In particular, our recent researches revealed a non-uniform sodium motion in Na₉Al(MoO₄)₆.³⁷ At T < 490 K it occurs exclusively through ion jumps within the sublattice of sodium sites located far from polyhedral [Al(MoO₄)₆]⁹⁻ clusters, forming the structure framework. The activation energy for this type of motion E_a ≈ 0.55 eV. The ions located in the vicinity of [Al(MoO₄)₆]⁹⁻ clusters are activated only at T > 490 K with E_a ≈ 0.8 eV. Strong correlation effects in sodium motion are present at low temperatures. They are manifested in a slowdown of ion diffusion and an artificial increase of the E_a value estimated from conductivity data.

A comparison of the electrophysical properties for the NASICON-type compounds^19,38,39 shows that the substitution of the transition element can lead to a drastic change of the ionic conductivity (up to several orders of magnitude). In this regard, it was interesting to study the Na diffusion mechanisms in the compound related to Na₉Al(MoO₄)₆: having similar local structure, but another trivalent cation. As such a related compound, Na₉Sc(MoO₄)₆ was chosen. The basic structural units in Na₉R(MoO₄)₆ (R = Al, Sc) are the isolated polyhedral [R(MoO₄)₆]⁹⁻ clusters composed of the central RO₆ octahedron sharing vertices with six MoO₄ tetrahedra to form an open framework, where the Na⁺ cations are bound to three vertices of the MoO₄ tetrahedra (see Fig. 1).

Fig. 1 Crystal structures of Na₉Al(MoO₄)₆ (a and c) and Na₉Sc(MoO₄)₆ (b and d) reproduced with program VESTA⁴⁰ on structural data from ref. 37 and 41, respectively. Figures a and b represent the [Al/Sc(MoO₄)₆]⁹⁻ clusters forming the structure framework and adjacent NaO₆ octahedra. Figures c and d represent the Na-sites located in the structure cavities between the [Al/Sc(MoO₄)₆]⁹⁻ polyhedra.

The Na₉Sc(MoO₄)₆ is crystallized in a structure with trigonal R3̄ space group.⁴¹ Replacing scandium (r_VI = 0.74 Å⁴²) with aluminum (r_VI = 0.53 Å⁴²) induces the monoclinic distortions (C/2c s.g.) and the “splitting” of sodium sites.³⁷ As a result, there are five nonequivalent structural positions of sodium in Na₉Al(MoO₄)₆, where Na1 and Na2 are located in the vicinity of AlO₆ octahedra, and the Na3, Na4 and Na5 positions are placed in the cavities between the [Al(MoO₄)₆]⁹⁻ polyhedra. Meanwhile, only three types of sodium positions are present in Na₉Sc(MoO₄)₆: the Na1 ions are located near the ScO₆ octahedra, and Na2 and Na3 ones are in between the [Sc(MoO₄)₆]⁹⁻ clusters (Fig. 1). It has to be noted however that these structural changes during the Al → Sc substitution can only be detected by a detailed analysis of electronic diffraction, while the powder X-ray diffraction patterns can also be satisfactory fitted in trigonal R3̄ and/or R3c space groups.³⁷ So, despite the apparent differences in the crystal structures presented on Fig. 1 they are really quite similar.

Here we present the results of detailed studies of the sodium diffusion mechanisms in Na₉Sc(MoO₄)₆ performed at the “atomic level” using the NMR methods and DFT calculations. Differences in the sodium dynamics in comparison with Na₉Al(MoO₄)₆ are discussed.

2. Experimental

2.1. Synthesis

Na₉Sc(MoO₄)₆ was synthesized by annealing of stoichiometric mixtures of Na₂MoO₄ and Sc₂(MoO₄)₃.⁴¹ The sintering temperature was raised from 773 to 873 K in steps of 20°, each temperature being held for 12 h, with intermediate grindings. For grinding the furnace was switched off and cooled down to room temperature. At final step the sample was taken out of the hot furnace. A stoichiometric mixtures of Sc₂O₃ and MoO₃ was used for synthesis of simple molybdate Sc₂(MoO₄)₃ in two stages (573–723 K for 25–40 h and 1023 K for 60 h). Anhydrous Na₂MoO₄ was obtained by calcination of the corresponding crystalline dihydrate at 823–873 K. The prepared compounds have been checked by using JCPDS PDF-2 Data Base and do not contain reflections of initial or foreign phases.

2.2. Characterization

The PXRD patterns were obtained by using a D8 ADVANCE Bruker diffractometer (CuKα radiation, λ = 1.5418 Å, reflection geometry). The PXRD data were collected at RT over the 7°–100° 2θ range with steps of 0.02076°. To determine the lattice parameters, the Le Bail decomposition was carried out using the JANA2006 software. Thermoanalytic studies were carried out with a STA 449 F1 Jupiter NETZSCH thermoanalyser (Pt crucible, heating rate of 10 degree min⁻¹ in Ar stream). Ceramic disks for conductivity measurements were prepared by pressing of powders at 1 kbar and sintering at 863 K for 4 h. The densities of the obtained pellets (measured by the Archimedes method using kerosene as a saturant) were typically 90–95% of theoretical ones. The disks had a diameter of 7–8 mm and a thickness of 1–2 mm, they were electroded by painting of colloidal platinum on their large surfaces with subsequent annealing for one hour at 843 K. Electrical conductivity of the samples was determined by impedance spectroscopy in the temperature range of 343–843 K at heating and cooling rates of 2 K min⁻¹ using two-probe measurements in a NorECs ProboStat cell. The signal was monitored with a Novocontrol Beta-N impedance analyzer at applied voltage of 0.5 V at selected frequencies in the range of 0.3 Hz–1 MHz.

2.3. NMR

The static NMR experiments were performed over the temperature range 300–750 K. The experimental data were obtained on an AVANCE III 500WB BRUKER spectrometer in an external magnetic field of 11.74 T. A commercial high-temperature wide-line NMR probe (Bruker Biospin GmbH), including a Pt-wired rf coil and nonmagnetic heater was used to heat a sample in static air atmosphere. The sample was tightly packed inside an open quartz ampule throughout NMR measurements. The static spectra for ²³Na (nuclear spin ²³I = 3/2, quadrupole moment ²³Q = 0.108 barn) were acquired by Fourier transform of both the free induction decay: τ_p-acq, and, in some cases, the spin echo signals: τ_p-t_del-2τ_p-t_del-acq. The pulse duration was in all cases, τ_p = 2 μs, corresponding to the nuclear magnetization tip angle, θ ∼ 60°. The spin–lattice relaxation times for the ²³Na nuclei were measured by the “inversion-recovery” technique: 2τ_p-t_del-τ_p-acq. The ⁴⁵Sc (nuclear spin ⁴⁵I = 7/2, quadrupole moment ⁴⁵Q = −0.22 barn) static NMR spectra were acquired by Fourier transform of free induction decay with pulse duration equal to 3 μs.

The ²³Na MAS NMR spectra were obtained with room temperature bearing gas by using standard Bruker MAS NMR probeheads with 1.3 mm (AVANCE DSX 200 NMR spectrometer, external magnetic field 4.7 T) and 3.2 mm rotors (AVANCE III 500WB spectrometer, 11.7 T), and standard Agilent 4.0 mm MAS Probehead (AGILENT VNMR 400WB spectrometer, 9.4 T).

As for the static regime, the ²³Na MAS NMR spectra were acquired by Fourier transform of free induction decay and/or spin echo signals with exciting pulse equal to 2 μs. The spectra deconvolution was performed by using the DMFit program.⁴³

2.4. Computational details

The density functional theory (DFT) calculations were performed using the Vienna ab initio simulation package (VASP)^44,45 within the projector-augmented wave (PAW) method.⁴⁶ The exchange–correlation functional was described by the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE).⁴⁷ The convergence criterion for the total energy was 0.01 meV within a 400 eV cutoff. Integration in the Brillouin zone was done with a mesh of 4 × 4 × 4 irreducible k-points according to the Monkhorst–Pack scheme.⁴⁸ The lattice parameters were fixed at the experimental values,⁴¹ whereas the atomic positions were relaxed via a conjugate gradient algorithm until the forces on all unconstrained atoms were less than 0.01 eV Å⁻¹.

The tensors of electric field gradient (EFG) at Na nuclei in different positions were calculated using the method of ref. 49. After diagonalization of the EFG tensor, the principal components V_xx, V_yy and V_zz are chosen as |V_zz| > |V_yy| > |V_xx|. The largest principal component V_zz determines the quadrupole frequency ν_Q ≡ ω_Q/2π = 3eQV_zz/2I(2I − 1)h and asymmetry parameter η_Q = (|V_yy| − |V_xx|)/|V_zz|. The values of ν_Q and η were used for an assignment of the experimental NMR lines to the specific sodium sites.

The nudged elastic band (NEB) method was employed to calculate the energy barriers for Na-ion migration. We considered the diffusion mechanism through migration of single vacancy between two neighbor positions of Na ion. For every path, the total energies were calculated for a few intermediate images. The relaxed maximal energy and coordinates during the sodium migration provided the activation barrier and diffusion pathway between the starting and end states, respectively.

3. Results and discussion

3.1. The characterization of Na₉Sc(MoO₄)₄

The Na₉Sc(MoO₄)₆ sample has a white color and melts at 946 K.

The determination of unit cell parameters from PXRD patterns using Le Bail decomposition revealed that all reflections could be indexed in the R3̄ space group with a = 15.0047(1) Å, c = 19.1891(1) Å; values of structural R-factors: R_P = 3.10% and R_wP = 4.37% (Fig. 2). It verifies good phase purity of the powder prepared by solid state synthesis.

Fig. 2 The experimental, calculated, and difference powder X-ray diffraction patterns of Na₉Sc(MoO₄)₆.

3.2. Electric conductivity measurements

The temperature dependence of the conductivity is shown in Fig. 3 in Arrhenius log(σT) − (10³/T) coordinates. Conductivity values are given for the impedance frequency 100 kHz since at this frequency the real part of the impedance practically independent on the AC frequencies and corresponds to the bulk resistance of the sample. The temperature dependency of the conductivity can be divided into two approximately linear portions with different slopes. The activation energies for these linear portions are ^LTE^cond_a ≈ 0.86 eV for the region below 550 K and ^HTE^cond_a ≈ 0.98 eV for the high-temperature one. Conductivity values for Na₉Sc(MoO₄)₆ are as high as 1.6 × 10⁻⁴ S cm⁻¹ at 573 K and 3.63 × 10⁻² S cm⁻¹ at 843 K.

Fig. 3 Temperature dependence of the conductivity measured for Na₉Sc(MoO₄)₆ in the range 343–843 K. The inset shows the Nyquist plot at 573 K.

3.3. ²³Na NMR spectra at room temperature: assignment of the observed NMR signals to the distinct Na positions

The Fig. 4 shows the ²³Na NMR spectral lines obtained for Na₉Sc(MoO₄)₆ in external magnetic fields 11.7 T (Larmor frequency ²³Na, ν₀ ≡ ω₀/2π = 132.29 MHz) and 4.7 T (ω₀/2π = 52.94 MHz). Similar NMR spectra were also recorded in the 9.4 T field (ν₀ = 105.82 MHz, not shown here).

Fig. 4 ²³Na NMR spectra obtained for Na₉Sc(MoO₄)₆ in magnetic fields 11.7 (a and b) and 4.7 (c and d) T in static and MAS modes. The color lines represent the results of spectra deconvolution (see text).

As can be seen a decrease of the external magnetic field strength leads to a substantial broadening of NMR spectrum. Similar effects can be expected when the main factor determining the shape of NMR spectrum is quadrupole interaction. Nuclei with a spin I > 1/2 have a non-spherical charge distribution in the nucleus. This leads to the appearance of a quadrupole moment that interacts with the Electric Field Gradient (EFG) which in its turn is determined by the features of local environment.⁵⁰ The influence of quadrupole interaction is usually described as small corrections to the Zeeman energy within the framework of the perturbation theory. In strong magnetic fields or for nuclei with low Q (such as the ⁷Li), only the first-order effects are usually observed which are manifested in a characteristic splitting of NMR spectrum and appearance (in the case of ²³Na with I = 3/2) of three lines: one line corresponding to the central transition, m_I = −1/2 ↔ +1/2, and two satellite lines (m_I = ±3/2 ↔ ±1/2). For a powder sample, the peaks of satellites are shifted relative to the position of central line at a distance ±1/2ν_Q(1 − η_Q). In weaker magnetic fields or for nuclei with high Q an important role can be played also by the second-order effects those are manifested in the splitting of central line. The second order splitting is proportional to square quadrupole frequency and inversely proportional to the resonance frequency: ν⁽²⁾_Q ∼ ν_Q²/ν₀.^50,51

Interestingly, three ²³Na NMR spectra on Fig. 4: static and MAS spectra acquired at ν₀ = 52.94 MHz and MAS spectrum measured at ν₀ = 132.29 MHz (panel c, d and b, respectively) can be fitted simultaneously taking into account only the second-order quadrupole effects. This “Quad^2nd” model assumes the presence of two spectral components with ν_Q = 1350 ± 50 and 620 ± 30 kHz; and η_Q = 0.30 ± 0.05 and 0.70 ± 0.05 for Line-1 (red line) and Line-2 (blue one), respectively. Nevertheless, the simulation of static spectrum recorded in 11.7 T yields the result far from ideal with the same fitting parameters (see panel a). It suggests that the situation is more complicated and besides quadrupolar interaction there are another factors affecting the ²³Na NMR spectrum in Na₉Sc(MoO₄)₆. This interaction is directly proportional to the magnetic field strength and substantially averaged even at low MAS speeds. The most probable candidate is a chemical shift anisotropy (CSA), reflecting the anisotropic shielding of Na nuclei by surrounding electrons. The parameters of the anisotropy, Δδ, and the asymmetry, η_CS, of chemical shift are determined by the corresponding tensor components: Δδ = δ₃₃ − δ_iso, η_CS = |δ₂₂ − δ₁₁|/|δ₃₃ − δ_iso|, where isotropic shift, δ_iso = (δ₁₁ + δ₂₂ + δ₃₃)/3.^52,53 The best fit is achieved with the values of Δδ equal approximately to 20 ppm and −5 ppm for lines 1 and 2, respectively. The parameter η_CS is found to be in the range of 0–0.3 for Line-1. For Line-2 it was difficult to estimate the η_CS value unambiguously since its influence on the line shape is rather weak. The “Quad^2nd + CSA” model contains another set of parameters which affect the line shape: the Euler angles φ, χ and ψ (in DMFit⁴³ notations) defining the orientation of principal axes of the EFG and CSA tensors. The best result is achieved at χ = 15 ± 5° for both lines (the changes of φ and ψ do not disturb visibly the line shape).

We assign the detected Line-1 and Line-2 to the ²³Na nuclei in the Na1 sites and the Na2 + Na3 sites, respectively. The relative intensities for observed signals (⁽¹⁾I = 0.30–0.35 and ⁽²⁾I = 0.65–0.70) agree well with the occupancies of corresponding positions in the crystal structure.⁴¹ Moreover this assignment is consistent with the results of ab initio calculations. The calculated quadrupole frequencies ν_Q and asymmetry parameters η_Q at ²³Na nuclei in Na₉Sc(MoO₄)₆ are shown in Table 1.

Table 1 Calculated quadrupole frequencies ν_Q and asymmetry parameters η_Q at ²³Na nuclei in Na₉Sc(MoO₄)₆

Site	ν Q (kHz)	η Q
Na1	1798	0.36
Na2	661	0.61
Na3	745	0.77

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As seen, the quadrupole frequencies and asymmetry parameters for Na2 and Na3 sites are very close to each other and we can expect the appearance of two NMR lines with ν_Q ∼ 1800 and 700 kHz; and η_Q ∼0.4 and 0.7 for Na₉Sc(MoO₄)₆. The relative intensities of these two lines should be ⁽¹⁾I = 0.33, ⁽²⁾I = 0.67 as follows from the occupancies of different Na sites.⁴¹

3.4. Temperature behavior of the ²³Na and ⁴⁵Sc NMR spectra: mechanisms of the long-range sodium diffusion

The Fig. 5 shows the evolution of the ²³Na NMR spectra in the Na₉Sc(MoO₄)₆ in temperature range 300–750 K. The Fig. 6 represents the temperature dependencies of characteristic parameters for two spectral components observed in Na₉Sc(MoO₄)₆. The data on central transition linewidth, Δν (a), quadrupole frequency, ν_Q (b), EFG tensor asymmetry, η_Q (c), chemical shift anisotropy, Δδ (d), isotropic shift, δ_iso (e) and the relative intensities for two spectral components (f) are present. To reduce the number of fitting parameters the η_CS values as well as the φ and ψ angles were fixed at zero for all temperatures and for both spectral components, χ = 15 ± 5°.

Fig. 5 ²³Na NMR spectra for Na₉Sc(MoO₄)₆ acquired in static mode at resonance frequency ω₀/2π = 132.29 MHz in temperature range 300–750 K.

Fig. 6 Temperature behavior of the ²³Na NMR spectra parameters for Na₉Sc(MoO₄)₆. Filled red and open blue circles correspond to NMR signals, attributed to Na1 and Na2 + Na3 ions, respectively. Black squares represent the corresponding parameters after the “merging” of two NMR components (see text). The data on dipolar central linewidth, Δν (a), quadrupole frequency, ν_Q (b), EFG tensor asymmetry, η_Q (c), chemical shift anisotropy, Δδ (d), isotropic shift, δ_iso (e), and the relative intensities for two spectral components (f) are present.

The temperature dependencies of NMR spectra parameters are qualitatively similar to those observed earlier for the related molybdate Na₉Al(MoO₄)₆³⁷ and indicate the presence of rather fast Na diffusion in the Na₉Sc(MoO₄)₆. It should be noted however that all “motional” effects in Sc-containing compound are observed at higher temperatures compared to those for Na₉Al(MoO₄)₆ that implies the slower ion diffusion in Na₉Sc(MoO₄)₆. At temperatures above 390 K, the Δν value for a signal corresponding to Na2 + Na3 ions (open blue circles in Fig. 6a) sharply decreases. Such a Δν(T) dependence is typical for the materials with fast ion diffusion.^50,54 The Δν value is affected mainly by the internuclear dipolar interaction. For a pair of interacting nuclei, it depends on the distance between nuclei and the orientation of this pair with respect to the external magnetic field. Atomic jumps lead to changes in both the distances and the orientations. As a result, with increasing temperature and growing ion jump frequency, the dipolar interaction is averaged and a sharp decrease of Δν is expected. The weak decrease of Δν for Na1 ions (filled red circles in Fig. 6a) can be explained as follows: for mobile Na2/Na3 ions themselves, the dipolar interaction is fully averaged (excluding some non-averaged part due to inhomogeneity of the external magnetic field etc.). Meanwhile the ions jumps in the Na2/Na3 sublattice affect only weakly the line width of static nuclei in Na1 sites, because for them the Δν is determined mainly by the ²³Na–⁴⁵Sc interaction (indeed, the nearest neighbor for Na1 is the Sc at a distance r ≈ 3.36 Å). At T > 550 K, the Δν for Na2/Na3 ions increases again. Such a Δν(T) dependence is reminiscent of that predicted for “classical” chemical exchange between two non-equivalent sites: with temperature increasing the NMR signals corresponding to different sites broaden at first, then they are merged into one wide line, then a dynamic narrowing of the merged line is expected.⁵⁰ The parameters of this “merged” line are shown as black squares in Fig. 6. Thus, the experimental data on Δν can be interpreted within the next scenario: at 400 < T < 550 K the Na⁺ ions motion occurs exclusively within the Na2/Na3 positions which form the continuous paths for long-range ion diffusion, while sodium in Na1 sites remains static (on the NMR frequency scale). At T > 550 K the Na1 ions also start to participate in the diffusion processes through the Na1 ↔ Na2/Na3 jumps. It has to be noted that the latter assumption is in agreement with structure considerations: the Na1 ↔ Na2/Na3 jump it is only one possibility for Na1 because direct jump Na1 ↔ Na1 seems to be quite improbable due to too long jump length (r_Na1–Na1 ≈ 5.83 Å).

Temperature behavior of ν_Q for Na2/Na3 ions is consistent with the proposed picture of ion motion. As is expected,^50,55,56 besides the averaging of dipole–dipole interaction the ion jumps in the Na2/Na3 sublattice induce the changes of the EFG tensor components, V_ij, for the corresponding NMR signal, those are reflected in the changes of ν_Q and η_Q values at T > 390 K (see Fig. 6b and c). Meanwhile only monotonous changes of the EFG parameters for spectral component corresponding to Na1 ions are observed at 300 < T < 550 K. Nevertheless, there are some substantial deviations from “classical” behavior expected at two-site exchange. In particular, instead of characteristic coalescence of two spectral components,^55,57–60 at T > 550 K (i.e., with the beginning of Na1 ions motion) the corresponding signal sharply decreases. As a result, this line completely disappears at T ≈ 600 K. The changes of line intensity can be explained by a natural assumption that Na1 ions are involved in the motional process gradually with increasing temperature, thus the decreasing signal corresponds to ions still retained in the Na1 sites (whose fraction decreases with temperature). Concerning the absence of spectra coalescence, we have to take into account that the chemical exchange Na1 ↔ Na2/Na3 can not occur directly, but only through intermediate tetrahedral position Na^int. For these “two-steps” jumps, Na1 ↔ Na^int + Na^int ↔ Na2/Na3, the NMR spectrum transformation can significantly differ from that expected at direct Na1 ↔ Na2/Na3 hopping. It has to be noted that some approaches allowing to predict the NMR spectrum shape at two-site exchange have been developed recently (see for example ref. 61–65). However, even for the simplest cases, they require rather complicated numerical calculations. In our case they already can hardly be applied, moreover, if we take into account that the probabilities for the “first step” (Na1 ↔ Na^int) jump and for the “second step” (Na^int ↔ Na2/Na3) one can be rather different.

Another possible explanation for the disappearance of Line-1 could be the presence of some phase transition at 550–600 K resulting in that the local surrounding of sodium ions in Na1 sites becomes similar to that for Na2/Na3. To trace possible structure transformations we performed also the ⁴⁵Sc NMR spectra measurements in the temperature range 300–750 K. In contrast to the Na sites (18f in R3̄ structure) the point symmetry of the Sc site (6c) contains a 3-fold axis. It applies the strict constraints on the EFG and CSA tensors: they must have axial symmetry (i.e., η_CS = η_Q = 0) and, in addition, their principal axes must be collinear to this 3-fold axis (i.e., φ = χ = ψ = 0°). However, with these constraints we could not find the ν_Q and Δδ values allowing to fit the experimental spectra. The better result can be achieved either with the parameters φ = χ = ψ = 0° and η_Q = 0, η_CS = 1, or with η_CS = η_Q = 0 and φ = ψ = 0°, χ = 90°. Although the point symmetry plays as it is considered a decisive role, the deviations in both the orientations of principal axes and the axiality of CSA and EFG tensors have been observed in some cases (see, for example, ref. 66 and 67). On the one hand, local symmetry can be distorted due to deviations of the bond lengths/angles from their ideal values. On another hand the electron shielding and especially EFG on the nucleus-probe can be strongly affected by the second and even third coordination spheres. The environment of Sc³⁺ ions in Na₉Sc(MoO₄)₆ is characterized by the “isotropic” first coordination sphere (containing 3 × O²⁻ ions at a distance r = 2.09 Å and 3 × O²⁻ at r = 2.10 Å), and rather “anisotropic” second one with 3 × Mo⁶⁺ (r = 3.49 Å), 3 × Mo⁶⁺ (r = 3.66 Å), and 1 × Na1⁺ (r = 3.37 Å). Thus, as a tentative hypothesis we can infer that the observed features are determined by these peculiarities of local Sc surrounding. Nevertheless we leave a detailed discussion of this issue beyond the scope of present article. It is rather a question of “NMR crystallography” and other studies are required to clarify it.

The Fig. 7a represents the results of ⁴⁵Sc NMR spectra simulation. For all spectra deconvolutions the parameters of φ = χ = ψ = 0°, η_Q = 0 and η_CSA = 1, the estimates of dipolar line width, Δν, quadrupole frequency, ν_Q, anisotropy of chemical shift, Δδ, and isotropic shift value, δ_iso, are presented in Fig. 7b–e.

Fig. 7 ⁴⁵Sc NMR spectra acquired for Na₉Sc(MoO₄)₆ at resonance frequency ω₀/2π = 121.49 MHz in temperature range 300–750 K (a), temperature behavior of the central line width, Δν (b), quadrupole frequency, ν_Q (c), anisotropy of chemical shift, Δδ (d), and isotropic shift value, δ_iso (e).

As can be seen, the parameters of CSA and EFG on the ⁴⁵Sc nuclei show only monotonous changes with temperature indicating the absence of any structure transformations in the entire temperature range 300–750 K (excepting, of course, uniform lattice expansion). Meantime, the Δν(T) dependence reveals two characteristic “steps” at T ≈ 365 and 585 K which are most likely induced by the averaging of ²³Na–⁴⁵Sc internuclear dipolar interaction due to the activation at these temperatures of Na2/Na3 and Na1 ions, respectively. Thus, the ⁴⁵Sc NMR data confirm our conclusions on the mechanisms of sodium diffusion in the Na₉Sc(MoO₄)₆ compound.

The features of local structure allow us to explain why the Na2 and Na3 ions are activated at substantially lower temperature while the Na1 ions remain static up to 550 K (although, considering the jump length as a main criterion we should get the opposite situation because the interatomic distances r_Na1–Na3 ≈ 3.65 Å, r_Na1–Na2 ≈ 3.69 Å are shorter than r_Na2–Na2 ≈ 3.85 Å and r_Na3–Na3 ≈ 4.07 Å). We have to take into account the fact that the probability of ion jump is determined not only by the jump length directly but also by other factors such as the type of adjacent cation (that defines the force of coulombic attraction) and the size of the “saddle point” (triangular oxygen window that has to be passed for ion release from a distinct site).^38,68–70 In some cases even more tiny features such as the “geometry” of saddle point can play an important role and determine the mechanisms of ion diffusion.⁶⁰ From this point of view the “immobility” of Na1 ions at T < 550 K seems to be quite natural. Indeed, the jump Na1 → Na3 which is the most expected from interatomic distance is restricted most likely due to a very small square of a triangle face: the corresponding O–O distances for this face are only 3.528, 3.257 and 2.921 Å. The jump Na1 → Na2 seems to be more probable from “geometrical” point of view: the corresponding O–O distances are equal to 3.257, 3.835 and 4.463 Å. However, the adjacent tetrahedral site for this jump is very close to highly charged Mo⁶⁺ ion (r < 2 Å).

Besides the clarifying of ion transport mechanisms, the temperature evolution of NMR spectrum allows us to estimate the parameters of ion diffusion, in particular, the characteristic ion jump frequency, τ_d⁻¹. As it is expected,⁵⁰ the motional narrowing occurs at a temperature where the characteristic frequency of ion jumps exceeds the “rigid lattice” (i.e., in the absence of ion motion) line width: τ_d⁻¹ ∼ 2πΔν_RL. Taking into account the value of Δν_RL ≈ 1.2 kHz (see Fig. 6a) we can estimate τ_d⁻¹ ∼ 10⁴ s⁻¹ at T ≈ 400 K for Na-ions jumps within the Na2/Na3 sublattice. Similarly, for the “two-site exchange” it is expected⁷¹ that in the point of spectra coalescence the τ_d⁻¹ ∼ |ω_Q1⁽²⁾_RL − ω_Q2⁽²⁾_RL|. Thus, for Na1 ↔ Na2/Na3 jumps we can estimate the τ_d⁻¹ ∼ 10⁴ s⁻¹ at T ≈ 600 K. The analysis of Δν(T) allows also to estimate roughly the activation energy for ion motion, E_a. Several approaches have been developed for such an analysis (see for example, ref. 72–74). The simplest one, proposed by Waugh and Fedin⁷⁴ suggests that:

E_a (meV) = 1.617T₀ (K), (1)

where T₀ is the temperature of the motional line narrowing onset. Estimating the T₀ ≈ 400 and 600 K from the data on Δν for ²³Na and ⁴⁵Sc we can evaluate E_a ≈ 0.65 and 0.95 eV for sodium jumps within Na2/Na3 sites and for activation of Na1 ions, respectively.

3.5. ²³Na spin–lattice relaxation rate: localized sodium jumps

The eqn (1) yields usually only rough estimates of E_a.⁷⁵ The well known NMR technique allowing to evaluate the ion transport parameters more precisely is the spin–lattice relaxation rate, T₁⁻¹, measurements. The Fig. 8 represents temperature behavior of the ²³Na relaxation rates measured for Na₉Sc(MoO₄)₆ over the temperature range 300–750 K in the magnetic field of 11.7 T. For the entire temperature range, the nuclear magnetization recovery after inverting pulse was characterized by nonexponential behavior and at least two relaxation components were required for the experimental data approximation: M_z(t) = M_z,eq − [M_z,eq − M_z(0)](c_s exp(−t/T_1S) + c_f exp(−t/T_1F)), where M_z,eq is the equilibrium value of nuclear magnetization, T_1F and T_1S are the fast and slow spin–lattice relaxation components, respectively, the values of c_s and c_f determine the fractions of corresponding components (c_s + c_f = 1). As in the case of Na₉Al(MoO₄)₆³⁷ the appearance of two “branches” of spin–lattice relaxation for ²³Na nuclei (I = 3/2) is most likely due to dominant contribution of the quadrupole mechanisms in nuclear relaxation (see, for example, ref. 76 and 77). For the entire range of temperatures the value of c_s = 0.6 ± 0.1. The final estimates of T_1F⁻¹ and T_1S⁻¹ were obtained with fixed c_s = c_f = 0.5, that is expected for the case when only central transition is excited and detected.⁷⁸

Fig. 8 Semi-logarithmic Arrhenius plot of the ²³Na spin–lattice relaxation rate for Na₉Sc(MoO₄)₆. The filled and empty circles correspond to the slow (T_1S⁻¹) and fast (T_1F⁻¹) relaxation components, respectively. The green line corresponds to fit of experimental data on T_1F⁻¹ by eqn (5). Red and blue solid lines are the contributions from high- and low-temperature T_1F⁻¹ peaks (see text). The inset shows the values of τ_d⁻¹ = 10⁹ s⁻¹ estimated from (T₁⁻¹)_HTmax (red circle) and (T₁⁻¹)_LTmax (blue circle), as well as τ_d⁻¹ = 10⁴ s⁻¹ (black square and triangle) which are expected from the ²³Na Δν(T) and ν_Q(T) data for Na2/Na3 and Na1 jumps, respectively. The red and blue solid lines correspond to (τ^HT_d)⁻¹ and (τ^LT_d)⁻¹ values obtained from eqn (5), black dashed line – the estimates of τ_d⁻¹ for Na1 ↔ Na2/Na3 jumps from eqn (4) (see text).

The main feature of the T_1F⁻¹ temperature dependence in Na₉Sc(MoO₄)₆ is the maximum at T_max ≈ 730 K. The appearance of a maximum on the T₁⁻¹(T) dependence is typical for materials with fast ion diffusion. The simplest model describing the dynamic contribution to T₁⁻¹ was proposed by Bloembergen, Purcell and Pound.⁷⁹ Within the framework of BPP model, the T₁⁻¹(T) dependence is determined by the expression:

T₁⁻¹(T) ∝ C{J⁽¹⁾(ω₀) + 4J⁽²⁾(2ω₀)}, (2)

where J⁽ⁿ⁾(nω₀) is the spectral density function that is the result of Fourier transform of the correlation function G(t) = exp(−t/τ_c). The parameter C defines the amplitude of the T₁⁻¹ maximum and depends on the nature of interaction inducing spin–lattice relaxation.^50,54 As in the case of Na₉Al(MoO₄)₆,³⁷ we restricted our further considerations exclusively to the quadrupole interaction. The value Δν_RL ∼ 1 kHz yields for the second moment 〈Δω²〉 ∼ 10⁷–10⁸ s⁻². Thus, the maximum value of T₁⁻¹ that can be induced by dipolar internuclear interaction does not exceed ∼1 s⁻¹,^50,54 which is much lower than the observed values of about 3 × 10² s⁻¹ (see Fig. 8). The τ_c is the correlation time, in our case, it can be taken to be equal to residence time of ion at a distinct site, τ_c = τ_d. In its turn, the τ_d(T) dependence follows the Arrhenius law:

τ_d = τ_d0 exp(E_a/k_BT), (4)

where τ_d0 is the mean residence time at an infinite temperature (the τ_d0⁻¹ value is of the same order of magnitude as the phonon frequency: ∼10¹¹–10¹⁴ s⁻¹). The BPP model assumes the appearance of symmetric peak on the ln T₁⁻¹vs. T⁻¹ dependence. The maximum of relaxation rate is observed at the temperature T_max where the ion jump frequency becomes comparable with the resonance frequency: τ_d⁻¹ ∼ ω₀ (i.e., reaches ∼10⁹ s⁻¹ in our case). The slopes of the T₁⁻¹ peak at high- and low-temperature limits are equal to E_a/k_B and −E_a/k_B, respectively.

Nevertheless, as can be seen from Fig. 8 the T₁⁻¹vs. T⁻¹ dependence for Na₉Sc(MoO₄)₆ significantly deviates from the expected asymptotic behavior due to the presence of an additional low-temperature maximum at T_max ≈ 500 K. Similar dependences with two peaks of T₁⁻¹ were observed earlier for some lithium and proton-conducting materials and have been interpreted in the assumption of a coexistence of different types of atomic motion with highly differing jump frequencies and activation energies.^80–82 For this case the experimental data on T₁⁻¹ can be fitted by the modified equation:

T₁⁻¹(T) ∝ C₁{J⁽¹⁾_HT(ω₀) + 4J⁽²⁾_HT(2ω₀)} + C₂{J⁽¹⁾_LT(ω₀) + 4J⁽²⁾_LT(2ω₀)}, (5)

whereand

τ^HT_d = τ^HT_d0 exp(E^HT_a/k_BT), τ^LT_d = τ^LT_d0 exp(E^LT_a/k_BT). (7)

The fit of experimental data on T_1F⁻¹ by eqn (5) is shown in Fig. 8 as a solid green line. The red and blue lines correspond to the contributions from high- and low-temperature T_1F⁻¹ peaks, respectively. The fit for both T_1F⁻¹ and T_1S⁻¹ components yield the parameters: E^HT_a = 0.70 ± 0.02 eV, E^LT_a = 0.45 ± 0.02 eV, (τ^HT_d0)⁻¹ ≈ 5 × 10¹³ s⁻¹, (τ^LT_d0)⁻¹ ≈ 3 × 10¹³ s⁻¹. The inset in Fig. 8 shows the temperature dependencies of (τ^HT_d)⁻¹ and (τ^LT_d)⁻¹ obtained from eqn (5): solid red and blue lines. Red and blue circles correspond to the values τ_d⁻¹ = 10⁹ s⁻¹ that are expected from the condition τ_d⁻¹ ∼ ω₀ at T_max ≈ 730 and 500 K of high- and low-temperature T₁⁻¹ peaks, respectively. Black square and triangle correspond to τ_d⁻¹ = 10⁴ s⁻¹ at T ≈ 400 K and 600 K estimated from the ²³Na Δν(T) and ν_Q(T) data for Na2/Na3 and Na1 jumps, respectively. The inset in Fig. 8 contains also the τ_d⁻¹(T⁻¹) dependence predicted for Na1 ↔ Na2/Na3 jumps (black dashed line) that was obtained from Arrhenius law assuming τ_d0⁻¹ = 3–5 × 10¹³ s⁻¹, i.e., the same order of magnitude as for the Na2/Na3 jumps. In this case the eqn (4) yields the value of E_a = 0.95–1.05 eV, that is close to the estimates of E_a obtained from the ⁴⁵Sc NMR spectra analysis with eqn (1).

As can be seen, the high-temperature maximum of T₁⁻¹ at T_max ≈ 730 K and the ²³Na NMR line narrowing observed at T ≥ 390 K are induced by one motional process, namely by the long-range Na diffusion within the Na2/Na3 sublattice. The maximum of T₁⁻¹ induced by the Na1 ↔ Na2/Na3 jumps should be observed at T_max ∼ 1300 K. The most interesting feature of the T₁⁻¹(T) data is the low-temperature spin–lattice relaxation maximum at T_max ≈ 500 K. It must be obviously induced by some much faster motional process. The most probable candidate for this fast ion motion is the localized “back and forth” jumps Na3 ↔ Na2. Indeed, the crystal structure of Na₉Sc(MoO₄)₆ implies the formation of adjacent Na3O₆ and Na2O₆ octahedra (see Fig. 1). These sites have the shortest interatomic distance, r_Na2–Na3 = 3.027 Å. Moreover, these Na3O₆ and Na2O₆ octahedra share faces, not edges, so this jump can occur directly without any intermediate position. Furthermore, this pair of ions is distant from other available sites at r_Na2–Na2 = 3.847 Å and/or r_Na3–Na3 = 4.074 Å. So, it seems to be quite reasonable to suppose the realization of localized jumps within this Na3–Na2 pair in the temperature range where the heat energy is already enough for these short “back and forth” jumps but not yet enough for the long-range diffusion onset. These considerations are consistent with the results of ab initio calculations.

3.6. Ab initio calculations: activation energies and most probable paths for sodium motion

The activation barriers for sodium diffusion calculated with the NEB method are shown in Table 2. Our results indicate that the Na-ion diffusion extends across a 3D network of migration pathways (Fig. 9) with activation barriers from 0.1 to 0.8 eV. The lowest barrier of 0.12 eV corresponds to the shortest path (3.03 Å) between the Na2 and Na3 sites. This migration is almost not blocked by oxygen atoms all the way and therefore the Na2–Na3 path is linear, as seen in Fig. 9. The linear paths for the Na2–Na2 and Na3–Na3 migrations are impossible due to the very small distances (1.3–1.5 Å) to the nearest oxygen atoms. As a result, these jumps occur through the curved trajectories, which provide the barriers of 0.59 and 0.53 eV, respectively. The Na1–Na3 and Na1–Na2 migrations (path ∼3.7 Å) have the highest energy barriers (0.69 and 0.84 eV) among the considered sodium hoppings.

Table 2 The activation energies (E_a, eV) and path distance (R_Na–Na, Å) for Na migration in Na₉Sc(MoO₄)₆

Path	R Na–Na	E a, eV
Na2–Na3	3.04	0.12
Na3–Na3	4.05	0.53
Na2–Na2	3.93	0.59
Na1–Na3	3.78	0.69
Na1–Na2	3.84	0.84

---

Fig. 9 Calculated diffusion paths of sodium jumps in Na₉Sc(MoO₄)₆: Na1 ↔ Na2/Na3 (yellow circles), Na2 ↔ Na2 and Na3 ↔ Na3 (purple circles), Na2 ↔ Na3 (black circles).

4. Discussion

Based on NMR data and the DFT calculations, we propose the following mechanism of sodium diffusion in Na₉Sc(MoO₄)₆. With increasing temperature, the localized Na3 ↔ Na2 jumps are activated first. The characteristic frequency for these jumps reaches τ_d⁻¹ ∼ 10⁹ s⁻¹ at T ≈ 500 K and activation energy for these local jumps is E_a ≈ 0.45 eV. At temperatures above 390 K, the Na2 ↔ Na2 and Na3 ↔ Na3 jumps are activated, that leads to the long-range sodium diffusion. The characteristic frequency for these jumps is τ_d⁻¹ ∼ 10⁴ s⁻¹ at T ≈ 400 K and ∼ 10⁹ s⁻¹ at T ≈ 730 K with activation energy E_a ≈ 0.8 eV (remind that from conductivity data we estimated ^LTE^cond_a = 0.86 eV, NMR data yield E_a = 0.65–0.72 eV, ab initio calculations predict a similar trend for jumps within Na2/Na3 sublattice). Above 550 K, the Na1 ions also start to participate in diffusion processes through the Na1 → Na2/Na3 jumps. The characteristics of these jumps: τ_d⁻¹ ∼ 10⁴ s⁻¹ at T ≈ 600 K, E_a = 0.95–1 eV. This scenario allows to explain the σ(T) dependence (Fig. 3) with characteristic bend at T ≈ 590 K. Moreover, the measured σ values are consistent with our estimates of τ_d⁻¹. Indeed, these “macro” and “atomic-scale” data can be roughly compared using the well-known Nernst–Einstein and Einstein–Smoluchowski equations: σ = NqD/k_BT and D = l²/6t_d, where D is the diffusion coefficient, N is the concentration of ions in a cubic centimeter of matter (∼10²²), q is the ion charge, and l is the average jump length. Taking into account the characteristic frequencies of ion jumps in the Na2/Na3 sublattice, which are responsible for the main Na diffusion mechanism in Na₉Sc(MoO₄)₆, we can estimate σ ≈ 1.2 × 10⁻⁷ and ≈6.6 × 10⁻³ S cm⁻¹ at T ≈ 400 and 730 K, respectively. These values are close to the experimental data (Fig. 3).

The mechanism of the long-range sodium diffusion is similar to that revealed in the related Na₉Al(MoO₄)₆. With increasing temperature, Na-ions located far from the [R(MoO₄)₆]⁹⁻ (R = Al, Sc) clusters are activated first, while ions located in the vicinity of the [R(MoO₄)₆]⁹⁻ polyhedra start to participate in diffusion processes at significantly higher temperatures. Nevertheless, there are substantial differences in the Na dynamics: first, all motional effects are detected by NMR at lower (on ∼100 degrees) temperatures in Na₉Al(MoO₄)₆ and estimates of E_a for different ion jumps yield lower values (on ∼0.2 eV).³⁷ Second, in contrast to Na₉Sc(MoO₄)₆, the localized sodium jumps in Na₉Al(MoO₄)₆ are not observed in explicit form. Some deviations from the linear behavior of ln T₁⁻¹(T¹) at T ∼ 360 K (see Fig. 10 in ref. 37) can be interpreted by the presence of additional low-temperature peak of T₁⁻¹. Nevertheless, these deviations are rather small, so it is difficult to conclude whether this is the real presence of an additional peak of relaxation rate or just the errors in the T₁⁻¹ measurements. Moreover, even if we assume the presence of local Na jumps in Na₉Al(MoO₄)₆, the amplitude of the corresponding T₁⁻¹ maximum is very low reflecting low fraction of sodium participating in this localized motion. The features of sodium dynamics correlate with the features of crystal structures for Na₉Sc(MoO₄)₆ and Na₉Al(MoO₄)₆. Indeed, Na₉Al(MoO₄)₆ also contains the pairs of adjacent octahedra with a common face: Na3O₆–Na5O₆. These sites are even closer to each other (r_Na3–Na5 = 2.86 Å) than the corresponding sites in Na₉Sc(MoO₄)₆. However, due to very large distortions of the Na3O₆ and Na5O₆ octahedra³⁷ the O–O distances at the common face are rather short (4.16, 2.93, and 2.95 Å), so the probability of direct Na3 ↔ Na5 jump should be rather low. The intermediate tetrahedral site must be involved for this jump, which elongates the total path and increases the total value of E_a. Meanwhile, other distances in the sublattice of the Na3–Na5 positions (r_Na4–Na4 = 3.29 Å, r_Na3–Na4 = 3.56 Å, r_Na4–Na5 = 3.88 Å) are significantly shorter than the corresponding values in Sc-containing compound: r_Na2–Na2 = 3.847 Å, r_Na3–Na3 = 4.074 Å. Thus, it becomes clear that there are no structural prerequisites for the realization of localized ion motion in Na₉Al(MoO₄)₆, meanwhile the overall ion dynamics should be higher than that in Na₉Sc(MoO₄)₆.

Note, that the correlation effects in sodium motion that were found in Na₉Al(MoO₄)₆³⁷ were not observed in Na₉Sc(MoO₄)₆. For Na₉Al(MoO₄)₆ we assumed that the correlation effects slowing down the Na motion at T < 575 K are induced by the interaction between the neighboring mobile ions Na3–Na5 and the “static” Na1, Na2 ones.³⁷ Due to similar interatomic distances, we expected to observe similar effects also in Na₉Sc(MoO₄)₆. However, the characteristic features,^{37,57,83–86} such as the bending on the σ(T) dependence with a higher slope in the low-temperature region and a lower slope at higher temperatures and/or a large difference between the values of E^NMR_a and E^cond_a was not found in Na₉Sc(MoO₄)₆. Thus, the question on the origin of the correlation effects in sodium motion in Na₉M³⁺(MoO₄)₆ (R = Al, Sc) remains open so far.

5. Conclusions

Summarizing our results we can make some general conclusions on the mechanisms of ion transport in Na₉Sc(MoO₄)₆. The total conductivity was found to reach 3.6 × 10⁻² S cm⁻¹ at 843 K. The mechanisms of ion transport are rather complicated: the coexistence of at least three different types of Na-ions motion has been found in the temperature range 300–750 K. With increasing temperature the localized ion jumps within the pairs of adjacent sites Na2 ↔ Na3 located far from the [Sc(MoO₄)₆]⁹⁻ clusters are activated first. These local jumps are characterized by the activation energy E_a ≈ 0.45 eV and jump frequency τ_d⁻¹ ∼ 10⁹ s⁻¹ already at T ≈ 500 K. At temperatures higher 390 K the jumps between the Na2 ↔ Na2 and Na3 ↔ Na3 sites belonging to different pairs are activated leading to the appearance of long-range sodium diffusion. The characteristic frequency for these jumps is about 10⁴ s⁻¹ at T ≈ 400 K and ∼10⁹ s⁻¹ at T ≈ 730 K with activation energy E_a ≈ 0.8 eV. Finally, at T > 550 K the ions in Na1-sites placed in the vicinity of [Sc(MoO₄)₆]⁹⁻ clusters also become involved in diffusion processes through the Na1 → Na2/Na3 jumps. The characteristics of these jumps: τ_d⁻¹ ∼ 10⁴ s⁻¹ at T ≈ 600 K, E_a = 0.95–1 eV.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The research was carried out within the state assignment of Ministry of Science and Higher Education of the Russian Federation (Themes No. AAAA-A19-119012990095-0, A19-119031890025-9 and 0339-2016-0007), supported in part by the Russian Foundation for Basic Research (Projects No. 16-03-00164, 17-03-00333).

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