A systematic study of ²⁵Mg NMR in paramagnetic transition metal oxides: applications to Mg-ion battery materials

Abstract

Rechargeable battery systems based on Mg-ion chemistries are generating significant interest as potential alternatives to Li-ion batteries. Despite the wealth of local structural information that could potentially be gained from Nuclear Magnetic Resonance (NMR) experiments of Mg-ion battery materials, systematic ²⁵Mg solid-state NMR studies have been scarce due to the low natural abundance, low gyromagnetic ratio, and significant quadrupole moment of ²⁵Mg (I = 5/2). This work reports a combined experimental ²⁵Mg NMR and first principles density functional theory (DFT) study of paramagnetic Mg transition metal oxide systems Mg₆MnO₈ and MgCr₂O₄ that serve as model systems for Mg-ion battery cathode materials. Magnetic parameters, hyperfine shifts and quadrupolar parameters were calculated ab initio using hybrid DFT and compared to the experimental values obtained from NMR and magnetic measurements. We show that the rotor assisted population transfer (RAPT) pulse sequence can be used to enhance the signal-to-noise ratio in paramagnetic ²⁵Mg spectra without distortions in the spinning sideband manifold. In addition, the value of the predicted quadrupolar coupling constant of Mg₆MnO₈ was confirmed using the RAPT pulse sequence. We further apply the same methodology to study the NMR spectra of spinel compounds MgV₂O₄ and MgMn₂O₄, candidate cathode materials for Mg-ion batteries.

---

1 Introduction

Mg transition metal (TM) compounds are important materials for future energy technology, as they can potentially be used as cathodes for Mg-ion batteries¹ that offer high volumetric capacity, low cost, and improved safety compared to the prevalent Li-ion batteries.^2,3 In addition, they can, for example, be potentially used as carbon capture materials⁴ and electrochemical catalysts for water oxidation.⁵

Solid-state Nuclear Magnetic Resonance (NMR) spectroscopy is a valuable tool to aid the development of such materials, as it can give information about local disorder which is difficult to analyze with diffraction methods that probe long range order. Solid-state NMR has been successfully used to study a range of technologically important materials, such as battery electrodes, fuel cells, zeolites, and supercapacitors.^6–9 Utilizing the same approach for Mg, however, is challenging because of the low gyromagnetic ratio (approximately 1/20 of ¹H), low natural abundance (10%), and significant quadrupolar moment (I = 5/2, Q = 0.2 barns) of ²⁵Mg. Only recently has the development of sophisticated NMR equipment, such as high-field magnets and magic-angle spinning (MAS) probes, enabled high resolution ²⁵Mg NMR spectra to be acquired.^10,11 With the development and application of novel pulse sequences, such as rotor-assisted population transfer (RAPT)¹² to enhance the sensitivity and magic-angle turning (MAT)¹³ to separate the spinning-sideband manifolds from the isotropic resonances, such studies of challenging quadrupolar nuclei are expected to become more common.

On top of the experimental difficulties described above, studying battery materials with solid-state NMR is often made more complicated by the presence of unpaired electron spins in the systems, resulting in large paramagnetic NMR shifts which are difficult to measure and interpret. Due to the inherent difficulty of ²⁵Mg paramagnetic NMR experiments, computational predictions are invaluable aids for recording and interpreting the spectra.

Recently, Kim et al. reported ²⁵Mg NMR spectra of paramagnetic materials;¹⁴ two ²⁵Mg resonances were seen in the spectrum of MgMn₂O₄, but no detailed investigation was performed to determine the origin of the two peaks. To our knowledge, no previous study of the ²⁵Mg paramagnetic NMR parameters involved both experimental and computational approaches. To address this, we study four Mg TM oxides which have different crystal structures: Mg₆MnO₈ (defect rocksalt), MgCr₂O₄, MgV₂O₄ (cubic normal spinel), and MgMn₂O₄ (tetragonal normal spinel). First, we report the results of their structural and magnetic characterization, which is followed by ²⁵Mg NMR experiments. Ab initio calculations for magnetic data and paramagnetic ²⁵Mg NMR shifts follow. First principles results are compared to experimental spectra acquired using various state-of-the-art methods. Finally, we discuss the observed NMR shifts in terms of Fermi contact interaction and decompose the value of shifts in terms of the individual TM–O–Mg spin transfer pathways.

2 Experimental

2.1 Sample preparation

Mg₆MnO₈, MgCr₂O₄, and MgV₂O₄ samples were prepared via solid-state synthesis from stoichiometric amounts of MgO (Sigma-Aldrich, 99.99%), V₂O₃ (Sigma-Aldrich, 99.7%), Cr₂O₃ (Sigma-Aldrich, 99.99%), and MnO₂ (Sigma-Aldrich, 99.99%). Mixtures were ball milled, pressed into pellets, and calcined according to Table 1. The MgMn₂O₄ sample was synthesized from anhydrous citric acid (Breckland Scientific, 99%), Mg(NO₃)₂·6H₂O (Sigma-Aldrich, 99%), and Mn(NO₃)₂·4H₂O (Sigma-Aldrich, 97%) through a citrate sol–gel method.¹⁵ The organic matter was burned off at 453 K and final calcination was performed at 773 K with an intermediate grinding step. Synthesized samples were checked using a PANalytical Empyrean powder X-ray diffractometer (Cu Kα = 1.5406 Å) and the structures were refined from their diffraction patterns by using the Rietveld method¹⁶ as implemented in the X'pert Highscore Plus software.

Table 1 Synthesis conditions of samples

Sample	Thermal profile	Atmosphere
MgV2O4	1273 K 24 h	5% H2 in Ar
MgCr2O4	1073 K 12 h–1473 K 24 h	Air
Mg6MnO8	1173 K 12 h	Air
MgMn2O4	453 K 12 h–773 K 36 h	Air

---

2.2 Magnetic measurements

Magnetization measurements were performed with Quantum Design Material Property Measurement System (MPMS). Magnetic moments of zero field-cooled samples were measured at temperatures from 2 K up to 301 K to obtain the dependence of magnetic susceptibility χ(T) = dM/dH ≈ M/H on temperature. The Weiss constant Θ was extracted by fitting the Curie–Weiss law χ = C/(T − Θ), where C is the Curie constant. The effective electron magnetic moment μ_eff was extracted from C by the relation where k_B is the Boltzmann constant and N_A is the Avogadro constant. The magnetic exchange coupling constant J₁ between the nearest neighboring spins (S) was calculated from Θ by the mean-field relationwhere z is the number of the nearest neighboring spins.

2.3 ²⁵Mg NMR

All NMR spectra were acquired on a 16.4 T Bruker Avance III spectrometer operating at a Larmor frequency of 42.9 MHz for ²⁵Mg with conventional Bruker 4 mm and 3.2 mm triple resonance MAS low-γ probes. π/2-pulse amplitudes were calibrated on solid MgO, giving 90° pulse amplitudes of 89 kHz and 108 kHz for the 4 mm and 3.2 mm probes, respectively. All shifts were referenced to MgO at 26 ppm.¹⁷ Spectra were acquired with rotor-synchronised spin echo, magic angle turning (MAT),¹⁸ and rotor assisted population transfer (RAPT)¹² pulse sequences as indicated in the ESI† (Section S2). We have used a X–X̄ pulse train to saturate the satellite levels. RAPT modulation frequency ν_m was calculated from DFT values of C_Q. Values of ν_m = 270 kHz (C_Q = 3.6 MHz) and 240 kHz (C_Q = 3.2 MHz) were used for Mg₆MnO₈ and MgMn₂O₄, respectively. Experimental signal enhancement was measured for varying modulation frequencies for Mg₆MnO₈. Fitting of the spectra was performed assuming a central transition lineshape using the Bruker Topspin 3.0 software.

2.4 Ab initio calculations

Magnetic parameters. Values of Θ were obtained through ab initio methods and also from SQUID magnetometer measurements. As determination of the magnetism is crucial to calculating paramagnetic shifts, magnetic exchange coupling constants J were obtained from DFT calculations. Assuming we have a Ising-type magnetic Hamiltonianwhere E₀ is the energy of hypothetical structure with all magnetic interactions removed, n is the order of interaction (n = 1 for the nearest neighbors, n = 2 for the next nearest neighbors) and S_i,j are the spin numbers of sites i and j. Note that we are explicitly double counting the interactions present by looping over all (i,j) pairs. Here J can be determined by a multivariate linear regression using the DFT energies for supercells with different antiferromagnetic spin configurations. In this definition of H₀, positive J corresponds to a ferromagnetic interaction, and negative J corresponds to an antiferromagnetic interaction. Using this value of J, the Brillouin function for magnetization is solved self-consistently at various temperatures to yield the susceptibility χ(T) as a function of temperature. A Curie–Weiss fit is then performed to obtain the Weiss constant Θ.

NMR parameters. Calculation of the total NMR shift tensor requires consideration of two contributions: the diamagnetic orbital contribution and the paramagnetic contribution due to unpaired electrons. Diamagnetic ²⁵Mg shifts in inorganic solids are known to be around tens of ppm (e.g. MgAl₂O₄δ_iso = 48 ppm).¹⁹ As the diamagnetic contribution is much smaller than the paramagnetic contribution, we have ignored the former in the calculation of shifts.

The paramagnetic contribution to the shift tensor originates from the isotropic A_iso and anisotropic A_aniso components of the electron–nuclear hyperfine coupling tensor. Here the isotropic component is ascribed solely to the Fermi contact (FC) mechanism, which contributes to the isotropic shift. The anisotropic (dipolar) component contributes to the shift anisotropy.

As ²⁵Mg is a quadrupolar nucleus (I = 5/2), the overall observed isotropic shift δ_iso under MAS conditions can be expressed as a sum of Fermi contact component δ_FC and second-order quadrupolar component δ_Q. Paramagnetic NMR parameters were calculated using the scaling approach of Kim et al.²⁰ assuming the FC is the dominant shift mechanism. This method takes A_iso and A_aniso from a ferromagnetic calculation performed at 0 K, and then scales it to paramagnetic case using a Curie–Weiss type factor Φ:

δ_iso = δ_FC + δ_Q (3)

where h is the Planck constant, ν₀ is the nuclear Larmor frequency in Hz, ∣Ψ^α–β_N∣² is the unpaired spin density at the nuclear position, B₀ is the static magnetic field, μ_eff is the effective electronic magnetic moment per TM in Bohr magneton, S is the formal electronic spin of metal species, k_B is the Boltzmann constant, g_e is the free electron g-factor, g_I is the nuclear g-factor, μ_B is the Bohr magneton, T is the temperature of sample under MAS condition, and Θ is the Weiss constant. Here we assumed B₀ = 16.4 T, T = 320 K due to frictional heating at 14 kHz MAS, and (spin-only value).

In the above formalism, shift anisotropy can also be computed with the electron–nuclear dipolar interaction tensor T_ij and scaled in the same way to obtain the anisotropic components of the shift tensor δ_ij:

After this δ_ij can be diagonalized to obtain the principal components δ_ii which can be used to calculate the shift anisotropy. In this study, we employ the convention of Herzfeld and Berger,²¹ where the span Ω and skew κ are used to represent the anisotropy. Following the approach of Middlemiss et al., the contributions of each Fermi contact pathway to the total spin density Δ∣Ψ^α–β_N∣² were also calculated by reversing the direction of electronic spins on each TM site in turn.²² In addition, the total spin density ∣Ψ^α–β_N∣² was calculated from experimental shift by eqn (5) and (7).

The second-order quadrupolar shift δ_Q can be obtained from the quadrupolar frequency ν_Q from the expression²³

where C_Q is the quadrupolar coupling constant, I is the nuclear spin, ω₀ is the nuclear Larmor frequency in rad s⁻¹, and η_Q is the quadrupolar asymmetry parameter.

For systems with a large shift anisotropy tensor and quadrupolar coupling tensor such as Mg₆MnO₈ (see results), the relative orientation of the two tensors is also important in accurate fitting of the spinning sideband manifolds. This can be represented by three Euler angles α, β, and γ, where the angles represent rotations of the electric field gradient principal axes to match the A^aniso_ii principal axes. Hyperfine and electric field gradient tensor principal axes are ordered such that ∣A₂₂∣ ≤ ∣A₁₁∣ ≤ ∣A₃₃∣ and ∣V₂₂∣ ≤ ∣V₁₁∣ ≤ ∣V₃₃∣. The ZYZ convention and counterclockwise (positive) rotation are assumed. The full rotation matrix is shown in Section S4 (ESI†).

DFT calculations. All calculations were performed in CRYSTAL09, a solid-state DFT code using a Gaussian-type basis set to describe core states accurately. CRYSTAL has been used successfully in calculation of ^6/7Li-, ²³Na-, and ³¹P-paramagnetic shifts.^20,22,24 Given that the calculation of paramagnetic shifts depends substantially on the quality of Gaussian basis sets, two types of basis sets were utilized: a smaller basis set (termed BS-I) for geometry optimizations, and a more extended basis set (BS-II) for hyperfine and magnetic single-point calculations. This dual basis set scheme was first reported by Kim et al.²⁰ for paramagnetic shift calculations in periodic systems and is known to work well for ^6/7Li-, ²³Na-, and ³¹P-paramagnetic shifts.^22,24 BS-I sets were taken from solid-state studies of Catti et al.^25–27 and were used unchanged. BS-II sets for metal ions were taken from the Ahlrichs set, with TZDP-derived basis for Mg and DZP-derived sets for Cr and V. Modified IGLO-III sets²⁸ are adopted for O. The choice of BS-II is similar to the previous works on Li transition metal oxides.^20,22,24

All calculations were performed using the spin-polarized B3LYP functional.^29,30 Recent results indicate that using 35% Hartree Fock (HF) exchange gives better results for solid-state magnetic calculations.^31,32 Previous ab initio studies on ⁷Li, ²³Na, and ³¹P paramagnetic shifts show that values obtained using 20% and 35% provide the upper and lower bounds for the experimental shifts.^20,22,24 Hence the original B3LYP (termed ‘Hyb20’) and a modified B3LYP with 35% of HF exchange (termed ‘Hyb35’) were both used in calculations.

Convergence of energy and spin density was checked with the number of sampled points in the reciprocal space, and 4 × 4 × 4 Monkhorst–Pack³³ sampling scheme was used for all systems. Self-consistent field (SCF) cycles were converged to an energy difference of 10⁻⁷ hartree.

Experimental cell geometries were expanded into supercells and were optimized in CRYSTAL09 with all cell symmetries removed, except for the MgV₂O₄ case where the cubic cell was used with the available space group symmetry due to instabilities in the SCF cycles. All geometry optimizations were performed under the CRYSTAL default convergence criteria. Results from DFT calculations are reported in Table 2 and are also discussed in detail in the results section.

Table 2 Ab initio calculated and experimentally fitted paramagnetic NMR parameters of Mg₆MnO₈, MgCr₂O₄, MgV₂O₄, and MgMn₂O₄. Hyb20 and Hyb35 refer to the degree of Hartree–Fock exchange energy (see methods). Values marked with asterisks (*) were fixed in the fitting. For MgV₂O₄, the experimental Weiss constant was obtained from literature.⁵⁷ Electric field gradient (EFG) tensor eigenvalues and anisotropic hyperfine tensor components are near zero for MgV₂O₄ and MgCr₂O₄ and Euler angles are not reported for these systems. Due to space limitations, detailed magnetic characterization data are reported in Table S6 (ESI†)

		Mg6MnO8			MgCr2O4			MgV2O4			MgMn2O4
		Hyb20	Hyb35	Expt	Hyb20	Hyb35	Expt	Hyb20	Hyb35	Expt	Hyb20	Hyb35	Expt
Magn.	J 1/K	−0.7	−0.5	−0.7	−19.8	−14.8	−30.4	See text		−50	−45.0	−46.2	—
	Θ/K	−22.5	−16.4	−21.9	−297.3	−222.2	−456.7			−600	−448.3	−466.8	−452.5
NMR	∣Ψ^α–βN∣^2/10^−3 × Bohr^−3	19.5	17.4	17.0	40.4	34.1	37.9	41.2	36.1	36.1	43.0	34.1	34.9
	Δ∣Ψ^α–βN∣^2/10^−3 × Bohr^−3	9.5	8.5	—	3.3	3.0	—	3.4	3.0	—	4.0/2.6	3.3/1.8	—
	δ iso/ppm	3291	3152	2994	3842	3690	2862	2105	1840	1845	3901	3021	3128
	δ FC/ppm	3337	3199	3041	3842	3690	2862	2105	1840	1845	3938	3054	3178
	δ Q/ppm	−46	−47	−47	0	0	0	0	0	0	−37	−33	−50
	Ω/ppm	2110	2297	1930	0	0	840	0	0	672	698	632	1340
	κ	−0.84	−0.82	−0.84	—	—	1.0	—	—	−0.75	−1.0	−1.0	1.0
	C Q/MHz	3.64	3.69	3.70*	0	0	0	0	0	0	3.41	3.22	3.40*
	η Q	0.40	0.38	0.38*	—	—	—	—	—	—	0.0	0.0	1.0*
	α		270°*			—			—			0°*
	β		90°*			—			—			0°*
	γ		90°*			—			—			42°*

---

3 Results and discussion

3.1 Diffraction and magnetic characterization

Defect rocksalt Mg₆MnO₈. First reported in 1954,³⁴ Mg₆MnO₈ has an ordered defect rocksalt structure [Mg²⁺_6/8Mn⁴⁺_1/8□_1/8]O²⁻ with Fm3̄m space group symmetry (Fig. 2). The structure of Mg₆MnO₈ exhibits ordered cation vacancies □, where a vacancy is formed according to the following defect reaction written in Kroger–Vink notation:

The oxygen sublattice is also slightly distorted from the ideal cubic rocksalt structure due to the smaller ionic radius of Mn⁴⁺ (0.53 Å) compared to Mg²⁺ (0.72 Å).³⁵ The Mg site in this structure shows m.mm site symmetry, reflecting the distorted octahedra as opposed to the undistorted octahedra in the parent MgO structure. This should give a nonvanishing quadrupolar coupling constant, C_Q, and quadrupolar asymmetry parameter, η, for ²⁵Mg.

The powder X-ray diffraction pattern of Mg₆MnO₈ and the Rietveld refinement result are shown in Fig. S3 (ESI†). The product is single phase, with a diffraction pattern that matches to the previously reported result of Mg₆MnO₈, showing a cubic Fm3̄m symmetry.³⁴

Transition metal (TM) sites in Mg₆MnO₈ are separated by Mn–O–Mg–O–Mn bond pathways, resulting in an extended superexchange interaction between the unpaired electronic spins of Mn⁴⁺ (Fig. 1a). The nature (ferromagnetic or antiferromagnetic) and magnitude of this interaction can be represented by an exchange coupling parameter J₁, where we are only considering the nearest neighboring interactions (12 for each Mn⁴⁺). At temperatures T > 20 K, the magnetic susceptibility data of Mg₆MnO₈ (Fig. S7, ESI†) show the weak Curie–Weiss paramagnetism, with an antiferromagnetic ordering transition at the Néel temperature of T_N = 5 K. Fitting to the Curie–Weiss law, 35 < T < 300 K, yielded the Weiss constant Θ = −21.9 ± 0.4 K and J₁ = −0.73 ± 0.01 K. Due to the long Mn–Mn distance (6 Å), weak exchange coupling is expected; in this case, the coupling is antiferromagnetic (J < 0) as expected for the extended superexchange interactions.³⁶ The obtained Weiss constant is in excellent agreement with the published experimental result of −20 ± 5 K.³⁷ In addition, the effective electron magnetic moment μ_eff = 3.99 ± 0.01 μ_B obtained from the Curie constant shows good agreement with the spin-only value of 3.87 μ_B.

Fig. 1 ²⁵Mg bond pathway (P) contributions and TM–TM J couplings.

Cubic spinels MgCr₂O₄ and MgV₂O₄. MgCr₂O₄ and MgV₂O₄ both adopt the normal spinel structure with cubic Fd3̄m space group symmetry and Mg²⁺ ions in the tetrahedral (8a) sites, the Cr/V occupying the octahedral (16d) metal sites. The thermodynamic factors controlling normal versus inverse spinel formation are well described in the literature.^38–41 In both these materials, the crystal field stabilization energy (CFSE) of the TM ions are responsible for their preference for occupancy of the 16d site. In the MgV₂O₄ case, the orbital degree of freedom in V³⁺ ion also favors the normal spinel structure.⁴⁰

The powder X-ray diffraction patterns of MgCr₂O₄ and MgV₂O₄ are shown in Fig. S4 and S5 (ESI†), respectively. As expected from the structural discussion above, both spinels show cubic Fd3̄m symmetries and are normal spinels. 15.4 ± 0.4% by weight of unreacted V₂O₃ is also seen in the diffraction pattern. This behavior is known to occur due to the reduction of MgO and sublimation of Mg metal at high temperatures under a reducing atmosphere of H₂.⁴² In contrast, no impurity phase is detected in the MgCr₂O₄ sample.

The TM sites in cubic spinels have six nearest neighboring TM sites connected through 90° TM–O–TM bonds, resulting in magnetic exchange which can be represented with J₁ (Fig. 1b). First we consider the MgCr₂O₄ sample. From the Curie–Weiss fit (100 < T < 301 K) to the experimental magnetic susceptibility data (Fig. S8, ESI†), Néel temperature of T_N = 13 K, Weiss constant of Θ = −456.7 ± 3.4 K, and J₁ = −30.4 ± 0.2 K are obtained. The effective electron magnetic moment μ_eff = 4.25 ± 0.03 μ_B obtained from the Curie constant is slightly larger than the spin-only value of 3.87 μ_B, which originates from the fact that there are still significant short-range fluctuations over the temperature range of fitting (Θ > 300 K). The observed Weiss constant is of the same order of magnitude as the previously reported value of −433 K.⁴³

Fig. 2 Structure of Mg₆MnO₈, shown with oxygen polyhedra around metal ions.

In the case of MgV₂O₄, an experimental measurement of magnetic susceptibility was not performed due to the presence of the V₂O₃ secondary phase. Detailed discussion on the magnetism follows in the DFT section below.

Tetragonal spinel MgMn₂O₄. MgMn₂O₄, which shows a spinel structure with I41/amd symmetry, is different from the above two compounds since the octahedral Mn³⁺ (d⁴) ions cause a Jahn–Teller distortion of the original cubic spinel structure, resulting in a tetragonal spinel. Mn³⁺ ions disproportionate at high temperatures, resulting in antisite defects between the Mn and Mg ions:

Mg²⁺(tet) + 2Mn³⁺(oct) → Mg²⁺(oct) + Mn⁴⁺(oct) + Mn²⁺(tet) (12)

This inversion behavior is known to occur around 1073 K,^44,45 resulting in the following cation distribution:

[Mg²⁺_1−xMn²⁺_x]_tet[Mg²⁺_xMn²⁺_xMn⁴⁺_2−2x]_octO₄ (13)

It is known experimentally that low-temperature synthesis of this material through a coprecipitation or sol–gel method suppresses this inversion, resulting in an ordered normal spinel.^15,46 We also note that this material has been previously shown to be a promising Mg-ion battery cathode material, being able to reversibly de-insert Mg²⁺ ions in an aqueous electrolyte system.¹⁴

As discussed above, we have attempted the synthesis of an ordered, normal MgMn₂O₄ spinel through a citrate sol–gel method at a relatively low temperature of 773 K. The X-ray diffraction pattern of this sample is shown in Fig. S6 (ESI†), alongside the results from the Rietveld refinement. Refinement of the sample shows a tetragonal spinel, with 7.9 ± 0.3% of Mg₆MnO₈ secondary phase by weight. This impurity phase could not be eliminated despite attempts to make a homogeneous mixture of Mg and Mn in the gel, and repeated calcination attempts with intermediate grinding. Previous syntheses of MgMn₂O₄ through sol–gel and solid-state methods have also reported the presence of this secondary phase.^14,15,47 This suggests that excess Mn is present in the tetrahedral (4a) site of the spinel structure in a divalent form, Mn²⁺. From the starting Mg to Mn ratio of 1 : 2, the stoichiometry of this sample is calculated as (Mg²⁺_0.81Mn²⁺_0.19)Mn³⁺₂O₄. However, refinement of the site occupancies shows a composition of (Mg²⁺_0.92±0.02Mn²⁺_0.08±0.02)Mn³⁺₂O₄, which suggests overall Mn deficiency in the starting composition. Due to the low calcination temperature, the peaks are inherently broad and good fit could not be obtained.

Evidence for presence of this tetrahedral Mn²⁺ comes from the magnetic susceptibility data (Fig. S9, ESI†). In an ordered normal MgMn₂O₄ spinel, the tetragonal Jahn–Teller distortion of Mn³⁺ ions results in four nearest neighbor interactions J₁ along the a and b axes and two second nearest neighbor interactions J₂ along the c axis (Fig. 1c). Despite the fact that these values are not straightforward to obtain from experiments without a good model for the magnetism, we should expect a maximum below the magnetic ordering transition in the inverse susceptibility (1/χ) versus temperature plot, since the exchange coupling in the octahedral sublattice is antiferromagnetic. Zero-field cooled susceptibility data, however, clearly shows presence of a ferrimagnetic component with a minimum in the plot. This behavior is similar to Mn₃O₄, where an antiferromagnetic coupling between the tetrahedral Mn²⁺ and octahedral Mn³⁺ causes ferrimagnetic ordering at low temperatures.⁴⁸ In addition, the fitted effective electron magnetic moment μ_eff = 5.93 ± 0.06 μ_B is clearly larger than the spin-only value of 4.90 μ_B for the Mn³⁺ ion (d⁴), suggesting the presence of Mn²⁺ ion (d⁵). The calculated spin-only magnetic moments μ_eff of the (Mg²⁺_0.81Mn²⁺_0.19)Mn³⁺₂O₄ and (Mg²⁺_0.91Mn²⁺_0.09)Mn³⁺₂O₄ (determined from composition and refinement, respectively) are 5.00 and 4.95 μ_B, respectively. As for the MgCr₂O₄ case, the large discrepancy is likely due to the short-range fluctuations at T < Θ.

The magnetic properties of the (Mg_xMn_1−x)Mn₂O₄ solid solution have previously been studied at T < 60 K.⁴⁸ The magnetic susceptibility is shown to be small (<0.1 emu mol⁻¹) and qualitatively similar for MgMn₂O₄ and Mg_0.75Mn_0.25Mn₂O₄ at 60 K. In this work, we are interested in χ at 320 K (MAS frictional heating) and so will approximate our sample to MgMn₂O₄. Finally, we note that the experimental Weiss constant Θ = −452.5 K is consistent with a previously reported value of −500 K,⁴⁹ with the deviation between the two values potentially arising due to the excess Mn as discussed above.

3.2 ²⁵Mg NMR

Mg₆MnO₈. The ²⁵Mg NMR spectrum of Mg₆MnO₈ is shown in Fig. 3a, at 14 kHz MAS. To determine the isotropic shift, a MAT experiment was performed (Fig. 3b) at 20 kHz MAS. The MAT spectrum (performed with the RAPT enhancement scheme as below) clearly shows a single peak with δ_iso = 2960 ppm in the isotropic dimension (ω₂). This clearly shows the applicability of RAPT-enhanced 2-dimensional experiments in paramagnetic systems with poor signal-to-noise ratio. Especially, RAPT-enhanced MAT experiments could be useful in determining isotropic resonances, since the conventional method of performing MAS experiments at two different spin rates (hence different degree of frictional heating) is not straightforward due to the strong temperature dependence of the shift tensor.

Fig. 3 (a) ²⁵Mg spin echo spectrum of Mg₆MnO₈ (14 kHz MAS, 81 712 transients), with fitted central transition lineshape including contributions from both the paramagnetic shift anisotropy and the quadrupolar interaction (parameters are listed in Table 2). (b) Magic angle turning (MAT) spectrum of Mg₆MnO₈ (20 kHz MAS, 128 slices in the F1 dimension with 2.23 μs delay increment, 3072 transients acquired in each slice). RAPT pulses were applied before the MAT pulses to enhance the signal-to-noise ratio.

Determination of quadrupolar parameters from paramagnetic NMR spectra is considered to be difficult, as the characteristic quadrupolar patterns are often smeared out due to paramagnetic broadening of the spectra. We used the RAPT pulse sequence, which was previously used to estimate ²⁷Al quadrupolar parameters in diamagnetic samples,⁵⁰ to estimate the quadrupolar coupling constant, C_Q, of the Mg site in Mg₆MnO₈. For optimum enhancement, the modulation frequency ν_m of saturating pulse trains alternating in phase (+π/2, −π/2) should match the value of ν_Q/2, where ν_Q is the quadrupolar frequency of the observed nucleus given by eqn (9).¹² This condition is given by ν_m = 3C_Q/40 for I = 5/2. By plotting the enhancement in integrated signal intensity versus the offset frequency, we see that the maximum enhancement occurs between ν_m = 250–300 kHz (Fig. 4a). This is consistent with the DFT prediction of ν_m = 277 kHz, or the quadrupolar coupling constant C_Q = 3.69 MHz (Table 2).

Fig. 4 (a) ²⁵Mg spin echo signal intensity of Mg₆MnO₈ with increasing offset frequency of saturating pulse trains in RAPT experiment. (b) Enhancement of spin echo signal intensity using the RAPT pulse. Saturating pulses were applied at modulation frequency of 270 kHz. Both experiments were performed at MAS spin rate of 14 kHz. 1024 transients were acquired in each spectrum.

Starting from these values of δ_iso and C_Q, a spectral fitting was performed with quadrupolar parameters fixed to the Hyb35 calculated values. The MAS lineshape could be fit assuming paramagnetic shift anisotropy and a quadrupolar interaction and the fitted parameters are shown in Table 2. Good fitting was obtained with the quadrupolar parameters from DFT calculations, and no further attempts were made to fit the quadrupolar parameters. The spectrum is dominated by the central transition (transition between spin levels 1/2 ↔ −1/2), which is only affected to second-order by the quadrupolar coupling, and hence (other than the linewidth of the individual peaks within the spinning sideband manifold) is relatively insensitive to the size of the quadrupolar interaction. The paramagnetic shift anisotropy, arising from the dipolar coupling to the Mn⁴⁺ ions, which gives rise to a lineshape identical to that observed from the chemical shift anisotropy, is clearly visible. We note that the fitted δ_iso = 2994 ppm is larger than the δ_iso = 2960 ppm obtained from the MAT experiment above. As the slower MAS rate results in lower sample temperatures, an increase in the paramagnetic shift is expected in the 14 kHz MAS case when compared to 20 kHz MAS spectrum (i.e. the MAT experiment).

From the RAPT pulse sequence, a maximum signal-to-noise enhancement by a factor of close to 2 is observed (Fig. 4b). Under ideal conditions, the RAPT pulse sequence should result in an enhancement factor of I + 1/2 = 3 for ²⁵Mg. However, due to the difficulty of saturating multiple satellite levels in a polycrystalline I = 5/2 system, it is not uncommon to observe enhancement factors lower than the theoretical maximum (for instance, an enhancement factor of 2 is reported for polycrystalline ²⁷Al (I = 5/2) spectrum in diamagnetic albite NaAlSi₃O₈).^12,51 Despite the added difficulty of enhanced relaxation of the spin polarization in paramagnetic materials, the observed enhancement factor of 2 clearly shows that the RAPT pulse sequence can be used for significant improvement in signal-to-noise ratio in paramagnetic materials.

In addition to the RAPT pulse sequence, we have also attempted a Double Frequency Sweep (DFS) enhancement scheme⁵² also frequently used for quadrupolar nuclei. Only a signal-to-noise enhancement factor of around 1.5 is observed (Fig. S2, ESI†), as opposed to 2 times enhancement in RAPT. This could be explained in terms of faster paramagnetic relaxation effects present in the sample, as DFS pulses are usually longer (around 2000 μs) to satisfy adiabaticity, whereas RAPT pulses are shorter (around 110 μs optimized for Mg₆MnO₈).

Based on the DFT (see below) and ²⁵Mg NMR results of Mg₆MnO₈, we propose that Mg₆MnO₈ can be conveniently used as a model compound for paramagnetic ²⁵Mg NMR studies owing to a number of attractive properties: (1) it has six Mg atoms per formula unit and good NMR sensitivity, allowing signals to be observed with only 1024 scans at 16.4 T field; (2) it has nonvanishing quadrupolar coupling, meaning that we can study the quadrupolar behavior of Mg atoms, which is likely to be the case for many technologically relevant materials showing non-zero quadrupolar couplings; (3) the d³ electron configuration of Mn⁴⁺ in an octahedral environment eliminates the need to consider spin–orbit coupling effects in ab initio calculations to a good approximation; (4) the large distance (6 Å) between Mn atoms makes it a weak, well-defined paramagnet, which reduces the possible error in the paramagnetic scaling approach as taken in this work; (5) it is easy to prepare as a pure phase through solid-state reaction of corresponding oxides in air, although previous works have reported a sol–gel route⁵³ or calcination of carbonates under oxygen atmosphere.³⁴ It is also interesting to note that this compound was theoretically predicted to be a good carbon capture and storage material through a large-scale screening study.⁵⁴

MgCr₂O₄ and MgV₂O₄. These samples have spinel-type structures with Mg atoms sitting on the tetrahedral (8a) site on the spinel lattice. Due to this tetrahedral coordination of Mg sites, the quadrupolar coupling constants C_Q are expected to be zero for both structures. The observed spectra of MgCr₂O₄ and MgV₂O₄ are shown in Fig. 5. Experimentally, the vanishingly small C_Q is confirmed by the sharp isotropic peak that does not exhibit typical quadrupolar lineshape under MAS conditions.

Fig. 5 ²⁵Mg spin echo spectrum of (a) MgCr₂O₄ and (b) MgV₂O₄, both at a MAS spin rate of 10 kHz. 20 480 and 505 600 transients were acquired in each case. The broad feature seen around 2700 ppm is due to probe background.

Tetrahedral coordination also dictates that the anisotropic (dipolar) electron–nuclear spin interaction be zero. Despite this, small nonvanishing dipolar interactions were observed (Ω = 840 ppm for MgCr₂O₄ and 672 ppm for MgV₂O₄). This discrepancy may be attributed to (i) the bulk magnetic susceptibility effect which originates from inhomogeneous magnetic field due to random crystallite orientations, as noted in previous paramagnetic MAS NMR works,^55,56 or (ii) local defects present in the open spinel structure, resulting in breaking of the local symmetry. With the current data, it is difficult to ascertain which of these is going to be dominant; it is likely that both are responsible for the sideband intensities.

MgMn₂O₄. The ²⁵Mg NMR spectrum of MgMn₂O₄ sample is shown in Fig. 6a and the existence of the Mg₆MnO₈ secondary phase is clearly seen in the NMR spectrum. Fitted parameters (Table 2) reveal that nonzero anisotropic components in the hyperfine coupling tensor are present. As the tetragonal distortion reduces the tetrahedral symmetry of the Mg sites, we should compare the anisotropy to the DFT value, which will be discussed in detail in the DFT section below. In addition, the RAPT enhancement scheme (Fig. 6b) also shows around twicefold increase in intensity without any distortion to the sideband manifold, the offset used in the sequence again being chosen on the basis of the DFT-calculated C_Q value. This clearly demonstrates the utility of using RAPT pulses to investigate paramagnetic ²⁵Mg NMR samples.

Fig. 6 ²⁵Mg spin echo spectrum of MgMn₂O₄ at a MAS spin rate of 14 kHz. Mg₆MnO₈ secondary phase signals are shown with asterisks (*). 3 550 928 transients were acquired. (a) Comparison between experimental and fitted spectrum. Fitted parameters are listed in Table 2. (b) Enhancement of spin echo signal intensity with RAPT pulse sequence. RAPT modulation pulses were applied at 240 kHz. 1024 000 transients were acquired in each spectrum.

We now compare this result to the work of Kim et al., who have recently reported a ²⁵Mg NMR spectrum of this compound.¹⁴ Their reported NMR spectrum at a MAS frequency of 24 kHz shows two peaks at 2980 and 2850 ppm, which they assigned to MgMn₂O₄, the two peaks being assigned to discontinuities of a second-order quadrupolar lineshape with a C_Q of 5.4 MHz. Considering the inverse temperature dependence of Fermi contact shifts (eqn (6)), the MgMn₂O₄ resonance frequency should occur at higher shifts in the case of the 14 kHz MAS used here. For example, assuming representative rotor temperatures of 340 K and 320 K for 24 kHz and 14 kHz MAS (from our previous temperature calibration measurements), respectively, and using experimental value of Θ, our observed shift of δ_iso = 3128 ppm at 14 kHz MAS translates to δ_iso = 3047 ppm at 24 kHz MAS for our MAS probes (eqn (3)). Thus, we assign their 2980 ppm peak to the Mg in MgMn₂O₄ structure, the lower shift originating from a lower sample temperature and/or slight differences in magnetic properties between samples. Considering the twicefold (i.e. the maximum attainable enhancement observed in the Mg₆MnO₈ case) enhancement in signal intensity using the RAPT pulse parameters calculated from C_Q = 3.2 MHz, the extra peak at 2850 ppm does not appear to arise from a MAS quadrupolar lineshape with C_Q of 5.4 MHz as suggested in their paper. Alternatively, we suggest that these peaks represent Mg sites in different coordination environments to the Mn in different oxidation states, arising from the antisite defect (eqn (12)). We tentatively assign the peak at 2850 ppm to Mg neighboring one Mn⁴⁺ and eleven Mn³⁺, as opposed to the 2980 ppm peak where the Mg is neighboring twelve Mn³⁺. Detailed investigation on the nature of this site is ongoing.

3.3 DFT calculation of NMR and magnetic parameters

We now discuss the ab initio results on the compounds studied above. DFT calculated values of magnetic parameters and paramagnetic NMR shifts are shown in Table 2.

Mg₆MnO₈. Good agreements between the ab initio values of NMR and magnetic parameters are seen, with the experimental shift lying between the Hyb20 and Hyb35 calculated values. In particular, the anisotropy parameter Ω and κ shows good agreement, despite the difficulties in obtaining good anisotropy parameters from powder MAS patterns in paramagnetic systems. Considering the possible origins of the discrepancies as discussed above, the dilute nature of paramagnetic spins in the Mg₆MnO₈ structure and close-packed rocksalt type structural arrangement are likely to be responsible for this good agreement.

MgCr₂O₄ and MgV₂O₄. The exchange coupling constant J₁ for MgCr₂O₄ determined from DFT (−19.8 K for Hyb20 and −14.8 K for Hyb35) and experiment (−30.4 K) shows that the magnitude of this exchange is strong and antiferromagnetic as expected. The experimentally measured and calculated J₁ values, while of the same order of magnitude, differ by around a factor of 2. The differences between the two values are ascribed to the difficulties in modelling the frustrated pyrochlore lattice with a simple Ising-type spin model at 0 K with DFT. A Heisenberg-type spin model would be more appropriate, but this is not supported by the current version of CRYSTAL. Consistent with this proposal, a previous plane-wave DFT calculation on this compound using the Heisenberg model has yielded a value close to the experimental value, J₁ = −26.5 K.⁵⁸ However, we note that the isotropic shift calculated using the DFT-predicted spin density ∣Ψ^α–β_N∣² and the experimental Weiss constant Θ = −456.7 K is close to the ab initio value (3053 ppm, 2576 ppm, and 2862 ppm for Hyb20, Hyb35, and experimental, respectively). This shows that accurate determination of magnetism is key to ab initio calculation of paramagnetic shifts.

For MgV₂O₄, an ab initio prediction of magnetic parameters could not be made due to SCF instabilities in the supercell structure. However, a reasonable estimate of the paramagnetic shift could still be made from a previous experimental measurement of the Weiss constant Θ = −600 K.⁵⁷ Despite the fact that spin–orbit interactions at the d² center may influence the shift, calculations involving explicit spin–orbit coupling Hamiltonian could not be performed in the present version of CRYSTAL. Despite this limitation, the shift values obtained without any correction for spin–orbit coupling shows a close match with the experimental result. Mulliken spin population analysis (Table S1, ESI†) and spin density maps (Fig. 7d) of the converged wavefunction shows equal occupation of the three t_2g orbitals d_xy, d_yz, and d_zx (which is expected, since we have preserved the cubic symmetry of the experimental cell). This is likely to arise from the fact that the spin–orbit coupling is relatively weak for V³⁺ ion (the spin–orbit coupling constant is λ = 104 cm⁻¹ for free ion, 95 cm⁻¹ when doped into Al₂O₃).⁵⁹ Hence the energy levels split by spin–orbit coupling are equally occupied. It is thought that the ab initio result reflects this average occupation in effect, producing results close to the experimental values.

Fig. 7 3-Dimensional spin density maps. Yellow denotes positive spin; blue denotes negative spin. Bounding box refers to the unit cell and solid lines connecting atoms refer to the metal–oxygen bonds. Bond pathway contributions P to the spin density are shown in red.

Finally, we note that in both cases DFT predicts zero paramagnetic shift anisotropy, which reflects the tetrahedral symmetry of Mg sites. As discussed above, this discrepancy is attributed to both the bulk susceptibility effect and local defects in the open spinel structures.

MgMn₂O₄. As values of J₁ and J₂ cannot be obtained from experiments without a good model to fit the susceptibility data, they were calculated from DFT. DFT Values of J₁ and J₂ for MgMn₂O₄ differ roughly by an order of magnitude (J₁ = −45.0 K, J₂ = −5.5 K for Hyb20 and J₁ = −46.2 K, J₂ = −6.1 K for Hyb35). As magnitudes of exchange interactions are known to be very sensitive to bond lengths, ∣J₁∣ > ∣J₂∣ as expected. In this case, the tetragonal distortion effectively reduces the frustration and the main J₁ exchange occurs along the Mn chain (see Fig. 7f), with J₂ perturbations between the chains. This explains the good agreement of DFT calculated Weiss constant Θ to the experimental value (−448.3, −466.8, and −452.5 K for Hyb20, Hyb35, and experimental, respectively).

Paramagnetic NMR shifts calculated using these values also show good agreement with the experimental value, where the experimental shift is between the Hyb20 and Hyb35 calculated values. However, again we see a discrepancy in the anisotropic shift parameter, similar to the result for other spinels MgCr₂O₄ and MgV₂O₄. The discrepancy is also attributed to the inhomogeneous magnetic field created by random crystallite orientations and local defects. These effects are more significant in the MgMn₂O₄ sample, as the low-temperature preparation condition results in (i) smaller crystallites, which increases the magnetic field inhomogeneity, and (ii) higher concentrations of defects present in the sample.

3.4 Shift mechanism and Fermi contact pathways

We now turn our attention to factors determining these large paramagnetic shifts and how they might be rationalized in terms of electronic structures. Paramagnetic NMR shifts in various Li transition metal oxides have been ascribed to a Fermi contact (FC) mechanism.⁶⁰ In this framework, a superexchange-like mechanism operates between the TM t_2g/e_g and the Li s orbital, resulting in small amounts of paramagnetic electron spins on the Li nucleus. As a similar kind of mechanism is expected to be responsible for paramagnetic Mg shifts, understanding the shift mechanism is important for rationalizing both the sign and magnitude of shifts.

As shown in eqn (4), factors contributing to the Fermi contact shift can be separated into the spin density transfer ∣Ψ^α–β_N∣² (in units of Bohr⁻³), and electron paramagnetism. Considering the expression for Θ, we can observe that metal ions with larger S are expected to give larger Fermi contact shifts, due to the μ_eff factor. This factor only depends on the formal spin of metal ions involved. On the other hand, the Fermi contact spin density transfer depends heavily on the geometry of the TM–O–Mg pathway. Here we can evaluate the contributions of each FC pathway P as Δ∣Ψ^α–β_N∣², which sums up to ∣Ψ^α–β_N∣². This is important since pathways with similar bond lengths and angles should contribute similar amounts of spin density to the observed nucleus. With sufficient amount of experimental and ab initio data, a database can be constructed which can give approximate shifts for novel structures without the need for further first principles calculations.

3-Dimensional maps of electron spin density obtained from DFT wavefunctions are shown in Fig. 7. From the spin density maps, it is clear that a delocalization mechanism involving the t_2g orbitals is in operation for all four compounds, resulting in positive spin density ∣Ψ^α–β_N∣² on the nucleus. This is particularly evident in the Mg₆MnO₈ case, where a 95° interaction along the TM–O–Mg pathway (P₁ in Fig. 1a) results in p orbitals with positive spin density pointing directly towards the Mg site (Fig. 7a). We also note that p orbitals with negative spin density all point towards the vacancy site, which would result in a negative shift mediated by a polarization mechanism, had this site been occupied. Also large differences in spin densities are observed between the two crystallographically distinct oxygen sites O1 and O2, where essentially no spin is observed on the O2 site. This originates from the fact that O2 is only surrounded by Mg in its first coordination shell due to the ordered cation arrangements in the defect rocksalt structure (Fig. 7b).

The situation in spinels is more complex, as the TM–O–Mg pathway is no longer near 90°. Here, both delocalization (t_2g–p_π–s) and polarization (e_g–p_σ–s) mechanisms are likely to operate along the P₁ pathway (Fig. 1b), resulting in transfer of positive and negative spins on Mg, respectively. Hybridization of the oxygen orbital is evident, as the main lobe points towards the Mg atom in the case of cubic spinels MgCr₂O₄ and MgV₂O₄ (Fig. 7c and d, respectively). In both cases where we have no ambiguity in the t_2g orbital occupation, it is clear that the delocalization mechanism is dominant, resulting in positive spin density on Mg.

A similar analysis could be done for tetragonal spinel MgMn₂O₄ where we have a Jahn–Teller distorted Mn³⁺ ion (d⁴), but the d orbital occupancies need to be determined. A close examination of the spin density map (Fig. 7e and f) and Mulliken spin population analysis (Table S1, ESI†) reveals that the valence electron configuration for Mn is , showing occupation typical of a positive Jahn–Teller elongation. This has important implications for the FC mechanism, as we have two distinct FC pathways P₁ and P₂ as a result of the Jahn–Teller distortion. P₁ lies on the crystallographic a,b-plane and P₂ points along the c-axis in the crystal structure (Fig. 1c). In both cases, the occupied d_z² orbital on Mn³⁺ is likely to contribute positive spin density to the oxygen along the c-axis (delocalization), whereas the empty d_x²−y² orbital will contribute negative spin density to the oxygen along the a-axis (polarization). Again, examination of the spin density map reveals that the dominant shift mechanism is also the delocalization mechanism in both cases, with the oxygen p_z orbital playing the most significant role. A large contribution from the d_z²–p_σ–s pathway is evident. A noticeable negative spin density is observed on oxygen positions (resulting from polarization along the e_g–p_σ pathway), although this does not contribute significantly to the shift.

By obtaining the spin density contributions Δ∣Ψ^α–β_N∣² from each FC pathways (P), magnitudes of the observed shifts could be rationalized. Comparing the first-order contribution P₁ of two d³ ions, Cr³⁺ in MgCr₂O₄ and Mn⁴⁺ in Mg₆MnO₈ (Table 2), a significant decrease in spin density transfer could be observed in the former case. In Mg₆MnO₈, each Mg atom is bonded to two Mn⁴⁺ ions via two 95° TM–O–Mg bonds. In terms of spin densities, pathway decomposition shows that each Mn⁴⁺ contributes approximately P₁ = 9.5 × 10⁻³ Bohr⁻³ (Hyb20) or 8.5 × 10⁻³ Bohr⁻³ (Hyb35) to Mg. In MgCr₂O₄ where each Mg is bonded to 12Cr³⁺, each Cr³⁺ contributes approximately P₁ = 3.3 × 10⁻³ Bohr⁻³ (Hyb20) or 3.0 × 10⁻³ Bohr⁻³ (Hyb35). This is rationalized by the differences in bond angles: as shown by Carlier et al., 90° interactions between the TM t_2g and observed nucleus result in positive spin transfer via a delocalization mechanism, whereas 180° interactions result in negative spin transfer via a polarization mechanism.⁶⁰ In Mg₆MnO₈, near 90° interactions dictate that the delocalization mechanism should be the dominant spin transfer mechanism, whereas in MgCr₂O₄ both delocalization and polarization mechanism should be in operation due to 121° bond angle. The polarization mechanism partially cancels the delocalization mechanism, resulting in small spin density transfer in MgCr₂O₄.

In MgMn₂O₄, the Jahn–Teller elongation dictates that c > a. Due to the occupation of the d_z² orbital and bond elongation, we expect P₁ > P₂. Values of P₁ and P₂ cannot be gained from experiments and need to be determined ab initio. The DFT calculation shows that the first contribution is roughly 1.5 times larger in terms of spin density contribution (P₁ = 4.0 × 10⁻³versus P₂ = 2.6 × 10⁻³ Bohr⁻³ for Hyb20 and P₁ = 3.3 × 10⁻³versus P₂ = 1.8 × 10⁻³ Bohr⁻³ for Hyb35, respectively). As for the case of magnetic exchange coupling (see above), the magnitude of the FC interaction is very sensitive to bond distances, which explains the difference.

4 Conclusion

Paramagnetic ²⁵Mg NMR can be a useful tool to study Mg TM compounds, which finds a variety of uses in technologically important materials. However, due to the low sensitivity and paramagnetic nature of ²⁵Mg centers in such compounds, solid-state NMR has been considered difficult and no systematic study is reported to date.

In this work, we have successfully combined DFT studies and experimental NMR techniques, such as RAPT and MAT to predict and obtain the paramagnetic NMR spectra of three Mg TM oxides. Experimentally, it is demonstrated that signal-to-noise enhancement is possible in paramagnetic Mg systems with RAPT pulses, allowing us to perform a 2-D MAT experiment on samples with natural abundance ²⁵Mg. While MAT is not strictly necessary to obtain the isotropic shift in the Mg₆MnO₈ case, we note that MAT experiments were used in the ²³Na NMR of related Na-ion battery materials to successfully deconvolute the complex spectra comprising multiple sites and sidebands.²⁴ DFT calculation of ²⁵Mg NMR parameters using the paramagnetic scaling approach is demonstrated with good agreement with the experiment. Comparison of the ²⁵Mg shifts in spinel and Mg₆MnO₈ compounds shows that TM–O–Mg 90° interaction results in stronger Fermi contact spin transfer, as expected from previous studies.

For spinel compounds MgCr₂O₄ and MgV₂O₄, comparison of the experimental results (NMR, SQUID magnetometry) and DFT calculations show that NMR anisotropy parameters are difficult to determine from the experiments due to the bulk magnetic susceptibility effects. In addition, DFT prediction of magnetism is difficult due to the magnetic frustration of these structures. Despite these difficulties, we show that reasonable predictions of ²⁵Mg shifts can be made with the DFT values of spin density and the Weiss constant measured from SQUID magnetometry.

With the aid of DFT calculations and RAPT pulse sequence as shown above, it is possible to predict and acquire paramagnetic ²⁵Mg spectra with signal-to-noise enhancement factors of 2. Good agreements are observed between the DFT predicted values of magnetism and NMR shifts, with the exception of NMR anisotropy parameters. This approach enables us to record and interpret the paramagnetic ²⁵Mg spectra of other Mg compounds, which are more difficult to understand without the aid of DFT calculation. Work is in progress on the local structural characterization of electrochemically cycled cathode materials.

Acknowledgements

Via our membership of the UK's HEC Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202), this work used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk). Research was also carried out at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-AC02-98CH10886. J. L. acknowledges Trinity College Cambridge for the graduate studentship. I. D. S. acknowledges the Geoffrey Moorhouse Gibson Studentship from Trinity College Cambridge. A. J. P. acknowledges the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the U.S. Department of Energy under Contract DE-AC02-05CH11231, under the Batteries for Advanced Transportation Technologies (BATT) Program Subcontract 7057154. We thank Dr Derek Middlemiss for helpful discussions and providing a copy of the mean field magnetism code, and Dr Dinu Iuga for helpful advices on DFS experiments.

References

1. E. Levi, Y. Gofer and D. Aurbach, Chem. Mater., 2010, 22, 860.
2. P. Novák, R. Imhof and O. Haas, Electrochim. Acta, 1999, 45, 351.
3. D. Aurbach, Y. Gofer, Z. Lu, A. Schechter, O. Chusid, H. Gizbar, Y. Cohen, V. Ashkenazi, M. Moshkovich and R. Turgeman, J. Power Sources, 2001, 97–98, 28.
4. D. M. D'Alessandro, B. Smit and J. R. Long, Angew. Chem., Int. Ed., 2010, 49, 6058.
5. N. Garg, Menaka, K. V. Ramanujachary, S. E. Lofland and A. K. Ganguli, J. Solid State Chem., 2013, 197, 392.
6. C. P. Grey and N. Dupré, Chem. Rev., 2004, 104, 4493.
7. F. Blanc, M. Leskes and C. P. Grey, Acc. Chem. Res., 2013, 46, 1952.
8. L. Mafra, J. A. Vidal-Moya and T. Blasco, Annu. Rep. NMR Spectrosc., 2012, 77, 259.
9. J. Klinowski, Annu. Rev. Mater. Sci., 1988, 18, 189.
10. J. C. C. Freitas and M. E. Smith, Annu. Rep. NMR Spectrosc., 2012, 75, 25.
11. I. L. Moudrakovski, Annu. Rep. NMR Spectrosc., 2013, 79, 129.
12. Z. Yao, H.-T. Kwak, D. Sakellariou, L. Emsley and P. J. Grandinetti, Chem. Phys. Lett., 2000, 327, 85.
13. J. Hu, D. Alderman, C. Ye, R. Pugmire and D. Grant, J. Magn. Reson., Ser. A, 1993, 105, 82–87.
14. C. Kim, P. J. Phillips, B. Key, T. Yi, D. Nordlund, Y.-S. Yu, R. D. Bayliss, S.-D. Han, M. He, Z. Zhang, A. K. Burrell, R. F. Klie and J. Cabana, Adv. Mater., 2015, 27, 3377.
15. U. Maitra, B. S. Naidu, A. Govindaraj and C. N. R. Rao, Proc. Natl. Acad. Sci. U. S. A., 2013, 110, 11704.
16. H. M. Rietveld, J. Appl. Crystallogr., 1969, 2, 65.
17. K. J. D. Mackenzie and M. E. Smith, Multinuclear Solid-State NMR of Inorganic Materials, Elsevier, 2002, vol. 6, pp. 461–532.
18. I. Hung and Z. Gan, Chem. Phys. Lett., 2010, 496, 162.
19. P. J. Pallister, I. L. Moudrakovski and J. A. Ripmeester, Phys. Chem. Chem. Phys., 2009, 11, 11487.
20. J. Kim, D. S. Middlemiss, N. A. Chernova, B. Y. X. Zhu, C. Masquelier and C. P. Grey, J. Am. Chem. Soc., 2010, 132, 16825.
21. J. Herzfeld and A. E. Berger, J. Chem. Phys., 1980, 73, 6021.
22. D. S. Middlemiss, A. J. Ilott, R. J. Clément, F. C. Strobridge and C. P. Grey, Chem. Mater., 2013, 25, 1723.
23. D. Freude, J. Haase, J. Klinowski, T. Carpenter and G. Ronikier, Chem. Phys. Lett., 1985, 119, 365.
24. R. J. Clément, A. J. Pell, D. S. Middlemiss, F. C. Strobridge, J. K. Miller, M. S. Whittingham, L. Emsley, C. P. Grey and G. Pintacuda, J. Am. Chem. Soc., 2012, 134, 17178.
25. M. Catti, G. Valerio, R. Dovesi and M. Causa, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 49, 179.
26. M. Catti, G. Sandrone, G. Valerio and R. Dovesi, J. Phys. Chem. Solids, 1996, 57, 1735.
27. M. Catti, G. Sandrone and R. Dovesi, Phys. Rev. B: Condens. Matter Mater. Phys., 1997, 55, 16122.
28. W. Kutzelnigg, U. Fleischer and M. Schindler, NMR-Basic Principles and Progress, Springer-Verlag, 1990, p. 165.
29. A. D. Becke, J. Chem. Phys., 1993, 98, 5648.
30. P. J. Stephens, F. J. Devlin, C. F. Chabalowski and M. J. Frisch, J. Phys. Chem., 1994, 98, 11623.
31. X. Feng and N. Harrison, Phys. Rev. B: Condens. Matter Mater. Phys., 2004, 70, 092402.
32. I. d. P. R. Moreira, F. Illas and R. Martin, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 65, 155102.
33. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Solid State, 1976, 13, 5188.
34. J. S. Kasper and J. S. Prener, Acta Crystallogr., 1954, 7, 246.
35. R. D. Shannon, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr., 1976, 32, 751.
36. J. Goodenough, Magnetism and the Chemical Bond, Wiley, 1963.
37. P. Porta and M. Valigi, J. Solid State Chem., 1973, 6, 344.
38. H. O'Neill and A. Navrotsky, Am. Mineral., 1983, 68, 181.
39. A. Navrotsky and O. Kleppa, J. Inorg. Nucl. Chem., 1967, 29, 2701.
40. K. E. Sickafus, J. M. Wills and N. W. Grimes, J. Am. Ceram. Soc., 1999, 82, 3279.
41. E. J. W. Verwey and E. L. Heilmann, J. Chem. Phys., 1947, 15, 174.
42. A. T. M. N. Islam, E. M. Wheeler, M. Reehuis, K. Siemensmeyer, M. Tovar, B. Klemke, K. Kiefer, A. H. Hill and B. Lake, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 85, 024203.
43. S. E. Dutton, Q. Huang, O. Tchernyshyov, C. L. Broholm and R. J. Cava, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 83, 064407.
44. R. Manaila and P. Pausescu, Phys. Status Solidi B, 1964, 6, 101.
45. R. Manaila and P. Pausescu, Phys. Status Solidi B, 1965, 9, 385.
46. S. Yagi, Y. Ichikawa, I. Yamada, T. Doi, T. Ichitsubo and E. Matsubara, Jpn. J. Appl. Phys., 2013, 52, 025501.
47. M. Cabello, R. Alcantara, F. Nacimiento, G. Ortiz, P. Lavela and J. L. Tirado, CrystEngComm, 2015, 17, 8728.
48. P. Ghigna, R. De Renzi, M. Mozzati, A. Lascialfari, G. Allodi, M. Bimbi, C. Mazzoli, L. Malavasi and C. Azzoni, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 184402.
49. K. Muramori and S. Miyahara, J. Phys. Soc. Jpn., 1960, 15, 1906.
50. S. Prasad, H.-T. Kwak, T. Clark and P. J. Grandinetti, J. Am. Chem. Soc., 2002, 124, 4964.
51. H. T. Kwak, S. Prasad, T. Clark and P. J. Grandinetti, Solid State Nucl. Magn. Reson., 2003, 24, 71.
52. A. Kentgens and R. Verhagen, Chem. Phys. Lett., 1999, 300, 435.
53. H. Taguchi and M. Nagao, J. Mater. Sci. Lett., 1991, 10, 658.
54. M. T. Dunstan, A. Jain, W. Liu, S. P. Ong, T. Liu, J. Lee, K. A. Persson, S. A. Scott, J. S. Dennis and C. P. Grey, Energy Environ. Sci., 2016, 9, 1346.
55. C. P. Grey, C. M. Dobson and A. K. Cheetham, J. Magn. Reson., 1992, 98, 414.
56. M. Stoll and T. Majors, J. Magn. Reson., 1982, 46, 283.
57. H. Mamiya and M. Onoda, Solid State Commun., 1995, 95, 217.
58. H. J. Xiang, E. J. Kan, S.-H. Wei, M.-H. Whangbo and X. G. Gong, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 84, 224429.
59. W. H. Brumage, C. R. Quade and C. C. Lin, Phys. Rev., 1963, 131, 949.
60. D. Carlier, M. Ménétrier, C. Grey, C. Delmas and G. Ceder, Phys. Rev. B: Condens. Matter Mater. Phys., 2003, 67, 174103.

---

Footnotes

† Electronic supplementary information (ESI) available: Detailed first principles calculation, experimental characterization data, and magnetic measurements. See DOI: 10.1039/c6cp06338a

‡ Current address: Department of Materials and Environmental Chemistry, Stockholm University, Svante Arrhenius Väg 16 C, SE-106 91 Stockholm, Sweden.

---