Kramers Doublet Ground State Research Articles

Magnetism in frustrated Kondo and non-Kondo intermetallics: CeInCu2 versus NdInCu2

Single crystalline intermetallics ${\mathrm{CeInCu}}_{2}$ and ${\mathrm{NdInCu}}_{2}$, where the rare earth ions occupy geometrically frustrated fcc lattice of Heusler-type structure, have been studied comparatively to shed light on the ground-state magnetism derived from Kondo physics and/or spin frustration competing with the Ruderman-Kittel-Kasuya-Yosida interaction. ${\mathrm{CeInCu}}_{2}$ is distinct from ${\mathrm{NdInCu}}_{2}$ due to the significant Kondo effect that is absent in the latter. They order antiferromagnetically at ${T}_{N}\phantom{\rule{0.16em}{0ex}}\ensuremath{\approx}\phantom{\rule{0.16em}{0ex}}1.4$ and 2.0 K with large paramagnetic Curie-Weiss temperature of $\ensuremath{-}35.4$ K and $\ensuremath{-}41.1$ K, respectively. The electronic specific-heat coefficient $\ensuremath{\gamma}$ = $C/T$ ($T\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}0$) = 870 mJ ${\mathrm{mol}}^{\ensuremath{-}1}\phantom{\rule{4pt}{0ex}}{\mathrm{K}}^{\ensuremath{-}2}$ in ${\mathrm{CeInCu}}_{2}$ is significantly enhanced, much larger than 180 mJ ${\mathrm{mol}}^{\ensuremath{-}1}\phantom{\rule{4pt}{0ex}}{\mathrm{K}}^{\ensuremath{-}2}$ of the latter that is also enhanced albeit its non-Kondo nature. For both compounds, the Kadowaki-Woods ratio deviates greatly from the standard value expected from a Kramers doublet ground state, which, however, can be restored if one considers a smaller but still sizable $\ensuremath{\gamma}$ estimated from the paramagnetic state. Likewise, the Kondo temperature of ${\mathrm{CeInCu}}_{2}$ is revised to be ${T}_{K}\phantom{\rule{4pt}{0ex}}\ensuremath{\approx}$ 13 K that is double the literature values, manifesting a major effect of spin frustration to low-temperature thermodynamics beyond the Kondo physics. Similarities between the two compounds is also noticed in their temperature-field phase diagrams in spite of their thermodynamic distinctions in the thermal expansion and the Gr\"uneisen ratio, which are much more sensitive to Kondo hybridization than to geometrical frustration.

Magnetic ordering in Co2+-containing layered double hydroxides via the low-temperature heat capacity and magnetisation study

The low-temperature heat capacity and the magnetisation of Co2+nAl3+ layered double hydroxides (LDH) with the cobalt-to-aluminium ratio n = 2 and 3 and intercalated with different anions have been studied in a wide range of magnetic fields up to 50 kOe. The heat capacity, C(T), was found to demonstrate a Schottky-like anomaly observed as a broad local maximum in the temperature dependence below 10 K. The effect is caused by a splitting of the ground-state Kramers doublet of Co2+ in the internal exchange field and correlates with magnetic ordering in these LDH. In low applied fields, the temperature-dependent dc magnetic susceptibility demonstrates a pronounced rise, which is associated with an onset of magnetic ordering. Both the heat capacity anomaly and the magnetic susceptibility peak are more pronounced for the LDH with n = 2 than for those with n = 3. This feature is associated with an excess of the honeycomb-like CoAl coordination (which corresponds to a 2:1 CoAl ordering) over the statistical cation distribution in Co2Al LDH, while a rather random cobalt-aluminium distribution is typical for Co3Al LDH. The temperature of the Schottky-like anomaly measured in a zero field is independent of the interlayer distance. Application of the magnetic field results in a widening of the anomaly range and a shift to higher temperatures. The observed experimental data are typical for a cluster spin glass ground state.

Weyl–Kondo semimetal behavior in the chiral structure phase of Ce3Rh4Sn13

The spin dynamics, crystalline-electric-field (CEF) level scheme, specific heat, and x-ray photoemission spectra (XPS) of ${\mathrm{Ce}}_{3}{\mathrm{Rh}}_{4}{\mathrm{Sn}}_{13}$ were investigated, which exhibits anomalous semimetal transport in the chiral crystallographic phase. CEF excitations observed at approximately 7 and 39 meV are consistent with the two inequivalent Ce-ion cites in the chiral structure. Because of broader CEF excitations and a strong $4{f}^{1}$ peak at the Fermi level in the Ce $4f$ on-resonance spectrum, the hybridized Ce $4f$ electrons contribute to the semimetal carriers. In addition, the spin fluctuation associated with the Kramers doublet ground state is characterized by the peak located at 0.15 meV. The electronic state involving the spin fluctuation causes the ${T}^{3}$ behavior of specific heat below 0.6 K, which is attributed to linear dispersion relations of electrons of the Weyl--Kondo semimetal in the chiral-lattice symmetry.

Spin excitations in the quantum dipolar magnet Yb(BaBO3)3

We report results of magnetization, specific-heat and muon-spin relaxation measurements on single crystals of disorder-free Yb$^{3+}$ triangular lattice Yb(BaBO$_3$)$_3$. The magnetization experiments show anisotropic magnetic properties with Curie-Weiss temperatures $\theta_{\perp}=-1.40$~K ($H \perp c$) and $\theta_{\parallel}=-1.16$~K ($H \parallel c$) determined from low temperature data. The absence of both long-range antiferromagnetic order and spin freezing is confirmed down to 0.27 K at zero field. A two-level Schottky anomaly due to the opening of the ground-state Kramers doublet is observed from the low-temperature specific-heat measurements when the applied magnetic fields $\mu_0H >0.7$~T. At zero field, the increase of both $C_{\rm mag}/T$ and the muon spin relaxation rate $\lambda$ below 1~K is due to the electronic spin excitations, which often exist in quantum magnets where dipole-dipole interaction creates an anisotropy of magnetic properties. The spin excitation is also supported by the unusual maximum of field dependence of $\lambda$ due to the field-induced increase of the density of excitations. We argue that dipolar interaction is dominant and induces the spin dynamics in the quantum magnet Yb(BaBO$_3$)$_3$.

SmI3:4f5 honeycomb magnet with spin-orbital entangled Γ7 Kramers doublet

We report magnetic properties of a 4f-honeycomb iodide SmI3 made up of edge-shared network of SmI6 octahedra. High temperature magnetic susceptibility indicates {\Gamma}7 Kramers doublet ground state of Sm3+ (4f5) ions stabilized by the spin-orbit coupling and octahedral crystal electric field, which interact with Sm-I-Sm bond angle nearly 90 degree. Magnetization measurements down to 0.1 K detected antiferromagnetic correlations and an anomaly in the magnetization curve before saturation without a sign of long-range order. Relevance between SmI3 and the antiferromagnetic Kitaev material proposed in the 4f-electron system is discussed.

Simple Realistic Model of Spin Reorientation in 4f-3d Compounds

This is a simple but realistic microscopic theory of spontaneous spin reorientation in rare-earth perovskites, orthoferrites RFeO3 and orthochromites RCrO3, induced by the 4f-3d interaction, namely, the interaction of the well-isolated ground-state Kramers doublet or non-Kramers quasi-doublet of the 4f ion with an effective magnetic field induced by 3d sublattice. Both the temperature and the nature of the spin-reorientation transition are the result of competition between the second- and fourth-order spin anisotropy of the 3d sublattice, the crystal field for 4f ions, and 4f-3d interaction.

Field-induced anomalous magnetic state beyond the magnetically ordered state in the slightly distorted triangular S=12 rare-earth antiferromagnet CeZn3P3

Magnetic properties of ${\mathrm{CeZn}}_{3}{\mathrm{P}}_{3}$, which has been believed to have the same hexagonal ${\mathrm{ScAl}}_{3}{\mathrm{C}}_{3}$-type crystal structure as that of a spin dimer system of ${\mathrm{YbAl}}_{3}{\mathrm{C}}_{3}$ at room temperature, have been investigated by magnetization, magnetostriction, specific heat ($C$), and magnetocaloric effect measurements. In this research, we have found that ${\mathrm{CeZn}}_{3}{\mathrm{P}}_{3}$ certainly has a slightly deformed crystal structure from the hexagonal one even at room temperature, which is in contrast to the structural phase transition of ${\mathrm{YbAl}}_{3}{\mathrm{C}}_{3}$ occuring at around 80 K, although the very slight deformation along the $c$ plane, i.e., slightly deformed triangular lattice, and the formation of the multidomain structure are common to both compounds. ${\mathrm{CeZn}}_{3}{\mathrm{P}}_{3}$ is thought to maintain its deformed structure from a high-temperature region above room temperature to an extremely low-temperature region and shows a magnetic order below ${T}_{\mathrm{N}}=0.8$ K. The analysis of the temperature dependence of the magnetic susceptibility ($\ensuremath{\chi}$) and $C$ has revealed that a Kramers doublet ground state with an easy-plane type anisotropy on the $c$ plane is well isolated from the excited states by the crystal field splitting energy of more than 300 K. Above ${T}_{\mathrm{N}}, \ensuremath{\chi}$ makes a broad peak at around 2 K, which certainly originates from the dimer formation due to the slight deformation of the triangular lattice. On the other hand, below ${T}_{\mathrm{N}}$, a magnetic phase diagram reminiscent of a magnetic flower blooming on the $c$ plane was observed, which may have a close relation to a quantum effect of a quasi $S=\frac{1}{2}$ spin system and a contribution of the orbital component. With increasing magnetic field, we have found an anomalous magnetic state beyond the usual magnetically ordered state, where $C/T$ is anomalously enhanced. This anomalous magnetic state is similar to that observed in ${\mathrm{YbAl}}_{3}{\mathrm{C}}_{3}$ induced by the field, although the magnetic ground state in ${\mathrm{YbAl}}_{3}{\mathrm{C}}_{3}$ is a nonmagnetic dimer state different from the normal magnetically ordered state in ${\mathrm{CeZn}}_{3}{\mathrm{P}}_{3}$.

Evidence of discrete energy states and cluster-glass behavior in Sr2−xLaxCoNbO6

We report the detailed analysis of specific heat [C$_{\rm P}$(T)] and ac-susceptibility for magnetically frustrated Sr$_{2-x}$La$_x$CoNbO$_6$ ($x=$ 0--1) double perovskites to understand low temperature complex magnetic interactions and their evolution with $x$. Interestingly, the observed Schottky anomaly in the $x\leqslant$ 0.4 samples shifts gradually towards higher temperature with magnetic field as well as $x$, and the analysis reveal the persistence of the discrete energy states in these samples resulting from the spin-orbit coupling and octahedral distortion. Moreover, the extracted values of Land\'e g--factor indicate the existence of high-spin state Co$^{3+}$ ions close to non-magnetic low-spin state. The specific heat data show the $\lambda$-type anomaly for the $x\geqslant$ 0.6 samples due to evolution of the long range antiferromagnetic ordering. Our analysis of low temperature C$_{\rm P}$(T) data for the $x\geqslant$ 0.6 samples demonstrate the 3D isotropic Heisenberg antiferromagnetic (AFM) interactions and the temperature induced second order AFM--paramagnetic phase transition. More interestingly, we demonstrate the presence of the free Co$^{2+}$ like Kramers doublet ground state in the $x =$1 sample. Further, the ac susceptibility and time evolution of the magnetization data reveal the low temperature cluster-glass like behavior in the $x=$ 0--0.4 samples, where spin-spin correlation strength decreases with $x$.

Quantum Phase Transitions in an Yb-based Semiconductor YbCuS2 with an Effective Spin-1/2 Zigzag Chain

Magnetic properties of an Yb-based semiconductor YbCuS2 with a zigzag chain of Yb3+ ions were studied. The Curie–Weiss behavior of the magnetic susceptibility χ(T) indicates antiferromagnetic interaction and an isolated Kramers doublet ground state with effective spin-1/2. Magnetic specific heat Cm/T exhibits a sharp peak at TO = 0.95 K, where the magnetic entropy is 20% of R ln 2, indicating strong magnetic fluctuations. In magnetic fields, TO is robust for B < 4 T, and it shifts to higher temperatures but reverses downward for B ≥ 7 T. Below TO, a magnetization plateau manifests itself at around 6 T which is followed by transitions at 9.1 and 15.6 T. The phase diagram suggests that nontrivial phases arise from magnetic frustration in the Yb zigzag chain.

Néel-type antiferromagnetic order and magnetic field–temperature phase diagram in the spin- 12 rare-earth honeycomb compound YbCl3

Most of the searches for Kitaev materials deal with $4d/5d$ magnets with spin-orbit-coupled $J=1/2$ local moments such as iridates and $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. Here we propose the monoclinic ${\mathrm{YbCl}}_{3}$ with a ${\mathrm{Yb}}^{3+}$ honeycomb lattice for the exploration of Kitaev physics. We perform thermodynamic, $ac$ susceptibility, angle-dependent magnetic torque, and neutron diffraction measurements on ${\mathrm{YbCl}}_{3}$ single crystal. We find that the ${\mathrm{Yb}}^{3+}$ ion exhibits a Kramers doublet ground state that gives rise to an effective spin ${J}_{\text{eff}}=1/2$ local moment. The compound exhibits short-range magnetic order below 1.20 K, followed by a long-range N\'eel-type antiferromagnetic order at 0.60 K, below which the ordered ${\mathrm{Yb}}^{3+}$ spins lie in the $ac$ plane with an angle of 16(11)${}^{\ensuremath{\circ}}$ away from the $a$ axis. These orders can be suppressed by in-plane and out-of-plane magnetic fields at around 6 and 10 T, respectively. Moreover, the N\'eel temperature varies nonmonotonically under the out-of-plane magnetic fields, suggesting a reduced spin dimensionality. Together with the strong in-plane magnetic anisotropy and the reduced order moment 0.8(1) ${\ensuremath{\mu}}_{B}$ at 0.25 K, all indicate that ${\mathrm{YbCl}}_{3}$ could be a two-dimensional spin system to proximate the Kitaev physics.

Spin-lattice relaxation processes of transition metal ions in a heavily cobalt doped ZnO: Phonon heating effect

Inversion recovery with electron spin-echo detection has been used to study the electron spin-lattice relaxation rates 1/T1 for transition metal impurities in heavily cobalt-doped hydrothermally grown ZnO single crystals. The relaxation dynamics of Co2+ ions dominates the phonon bottleneck effect in the Orbach-Aminov process, which involves the modulation of the zero-field-splitting tensor. The relaxation mechanism may be treated in terms of phonon heating with fast rate of energy pump from Co2+ spins into the lattice phonon modes. The measurements reveal the cross-relaxation process in which the single Co2+ ions cross-relax to exchange-coupled clusters of cobalt ions. The higher temperature relaxation of Co2+ indicates an additional Orbach-Aminov process via a state of ∼226 cm−1 above the ground state Kramers doublet. It is shown that in this system Co2+ ions play a role of the rapidly relaxing centers, strongly mediating the spin-lattice relaxation of the other transition metal impurities, such as Mn2+ and Fe3+.

Incommensurate Magnetism Near Quantum Criticality in CeNiAsO.

We report the discovery of incommensurate magnetism near quantum criticality in CeNiAsO through neutron scattering and zero field muon spin rotation. For T<T_{N1}=8.7(3) K, a second order phase transition yields an incommensurate spin density with a wave vector k=(0.44(4),0,0). For T<T_{N2}=7.6(3) K, we find coplanar commensurate order with a moment of 0.37(5)μ_{B}, reduced to 30% of the saturation moment of the |±1/2⟩ Kramers doublet ground state, which we establish through inelastic neutron scattering. Muon spin rotation in CeNiAs_{1-x}P_{x}O shows the commensurate order only exists for x≤0.1 so we infer the transition at x_{c}=0.4(1) is between an incommensurate longitudinal spin density wave and a paramagnetic Fermi liquid.

The crystal-field states and the magnetism in the Kondo-lattice antiferromagnet CeRh[formula omitted]Si[formula omitted

We have theoretically analysed temperature dependence of the specific heat at temperatures below 50 K of a Kondo-lattice antiferromagnet CeRh2Si2. We have reproduced experimentally-observed λ-type peak at the magnetic-ordering temperature TN of 36.5 K as originating from the splitting of the Kramers-doublet ground state of the Ce3+ ion. We have derived a set of crystal-field parameters, which, apart of the reproduction of crystal-field excitations of 30 and 52 meV, of the temperature dependence of the specific heat and of the paramagnetic susceptibility, reproduces a value of the magnetic moment, of 1.60 μB. Our results for the temperature dependence of the specific heat and of the entropy are physically adequate and substantially improved compared to theoretical descriptions reported in a recent publication in Nature Comm. (7 (2016) 11029).A good reproduction of the physical magnetic and electronic properties indicates that 99±1% of Ce ions are in the 4f1 configuration. A large value of the entropy removed in the magnetic transition, very close to Rln2, indicates that the hybridization effects and a broadening of crystal-field states in CeRh2Si2 are very small. The Quantum Atomistic Solid-State Theory (QUASST) approach can be used for the theoretical description of magnetic and electronic properties of other Ce, rare-earth and actinides compounds.

Kramers doublet ground state in topological Kondo insulators

We consider the simplest variant of a Kondo insulator where a doublet of localized $f$-electrons hybridizes with spin-degenerate conduction electrons. We analyse the symmetries of $f$-orbitals involved in the hybridization and point out that the effective four-band model of such systems is adiabatically connected with one that possesses chiral symmetry, which provides further descriptions of clean Kondo insulators. We obtain general conditions for the appearance of topological non-trivial states and discuss implications for rare-earth based compounds. As an example, we derive the full phase diagram of tetragonal Kondo insulators. We show that the chiral symmetry leads to a non-trivial topological phase when the band-width of conduction electrons sets the largest energy scale, and a new weak topological phase appears as function of the normalized distance between bands' centers.

A new β-diketonate Dy(III) single‒ion magnet featuring multiple magnetic relaxation processes

A β-diketonate mononuclear dysprosium compound, [Dy(TFNB)3(bpy)] (1) (TFNB = 4,4,4-trifluoro-1-(2-naphthyl)-1,3-butanedione, bpy = 2,2′-bipyridine), has been prepared and structurally and magnetically characterized. X-ray crystallographic analysis reveals that 1 contains two crystallographically equivalent Dy(III) ions of which the eight-coordinate geometries uniformly behave as distorted square antiprismatic configurations (D4d). Magnetic investigations demonstrate that 1 displays dual relaxation processes of SMMs behavior with the effective barrier (ΔE/kB) of 23.44 K under 1200 Oe DC field, corresponding to the coexistence of two metal centers in the structure of the compound. The comparative studies of some Dy(III)-based SIMs with TFNB ligand have been conducted as well. Ab initio studies demonstrate that the Kramers doublet ground state is predominantly axial with the gz tensors of two Dy(III) fragments matching the Ising-limit factor (20) anticipated for the pure MJ = ±15/2 state.

Understanding the magnetism in noncentrosymmetric CeIrGe3 : Muon spin relaxation and neutron scattering studies

The magnetic properties of a pressure induced noncentrosymmetric heavy-fermion superconductor CeIrGe$_3$ have been investigated by muon spin relaxation ($\mu$SR), powder neutron diffraction (ND) and inelastic neutron scattering (INS) techniques at ambient pressure. For completeness we have also measured the ac magnetic susceptibility $\chi_{\rm ac}(T)$, dc magnetic susceptibility $\chi(T)$, dc isothermal magnetization $M(H)$ and heat capacity $C_{\rm p}(T,H)$ down to 2 K. CeIrGe$_{3}$ is known to exhibit pressure induced superconductivity ($T_{\rm c}\approx 1.5$ K) at a pressure of 20 GPa and antiferromagnetic ordering at 8.7 K, 4.7 K and 0.7 K at ambient pressure. Our $\chi_{\rm ac}(T)$ and $\chi(T)$ data show an additional anomaly near 6.2 K which is also captured in $C_{\rm p}(T)$ data. From $\chi_{\rm ac}(T)$, $\chi(T)$ and $C_{\rm p}(T)$ measurements we infer three antiferromagnetic transitions above 2 K at $T_{\rm N1}= 8.5$ K, $T_{\rm N2}= 6.0$ K and $T_{\rm N3}= 4.6$ K. Our $\mu$SR study also confirms the presence of three transitions through the observation of one frequency for $T_{\rm N2} < T\leq T_{\rm N1}$, two frequencies for $T_{\rm N3} < T\leq T_{\rm N2}$ and three frequencies for $T\leq T_{\rm N3}$ in the oscillatory asymmetry. The ND data reveal an incommensurate nature of the magnetic ordering at $T=7$ K with the propagation vector k = (0,0,0.688(3)), and a commensurate magnetic structure at $T=1.5$ K with the propagation vector locked to the value k = (0,0,2/3) and magnetic moments oriented along the $c$ axis. The commensurate structure couples a macroscopic ferromagnetic component, resulting in a strong dependence of the lock-in transition temperature on external magnetic field. The INS data show two well defined crystal electric field (CEF) excitations arising from the CEF-split Kramers doublet ground state of Ce$^{3+}$.

Correction: Magnetic properties of the seven-coordinated nanoporous framework material Co(bpy)1.5(NO3)2 (bpy = 4,4'-bipyridine).

Correction for 'Magnetic properties of the seven-coordinated nanoporous framework material Co(bpy)1.5(NO3)2 (bpy = 4,4'-bipyridine)' by Elena Bartolomé et al., Dalton Trans., 2012, 41, 10382-10389.

Magnetic ground state of the Ising-like antiferromagnet DyScO3

We report the low temperature magnetic properties of the DyScO$_3$ perovskite, which were characterized by means of single crystal and powder neutron scattering, and by magnetization measurements. Below $T_{\mathrm{N}}=3.15$ K, Dy$^{3+}$ moments form an antiferromagnetic structure with an easy axis of magnetization lying in the $ab$-plane. The magnetic moments are inclined at an angle of $\sim\pm{28}^{\circ}$ to the $b$-axis. We show that the ground state Kramers doublet of Dy$^{3+}$ is made up of primarily $|\pm 15/2\rangle$ eigenvectors and well separated by crystal field from the first excited state at $E_1=24.9$ meV. This leads to an extreme Ising single-ion anisotropy, $M_{\perp}/M_{\|}\sim{0.05}$. The transverse magnetic fluctuations, which are proportional to $M^{2}_{\perp}/M^{2}_{\|}$, are suppressed and only moment fluctuations along the local Ising direction are allowed. We also found that the Dy-Dy dipolar interactions along the crystallographic $c$-axis are 2-4 times larger than in-plane interactions.

Magnetic properties of rare-earth sulfide YbAgS2

We have succeeded in synthesizing single-phase polycrystalline samples of YbAgS2 belonging to the tetragonal system with space group I41md. YbAgS2 shows an antiferromagnetic transition at TN=6.6K. The effective magnetic moment is in good agreement with the theoretical value for Yb3+ free ion. A broad anomaly is observed just above TN in the temperature dependence of magnetic susceptibility. The entropy released at TN is only about half of Rln2 expected for a Kramers doublet ground state. We consider that these phenomena are due to the existence of short-range magnetic correlations rather than the partial screening of the Yb moments by conduction electrons via the Kondo effect.

Muon spin relaxation and inelastic neutron scattering investigations of the all-in/all-out antiferromagnetNd2Hf2O7

${\mathrm{Nd}}_{2}{\mathrm{Hf}}_{2}{\mathrm{O}}_{7}$, belonging to the family of geometrically frustrated cubic rare-earth pyrochlore oxides, was recently identified to order antiferromagnetically below ${T}_{\mathrm{N}}\ensuremath{\approx}0.55$ K with an all-in/all-out arrangement of ${\mathrm{Nd}}^{3+}$ moments, however, with a much reduced ordered state moment. Herein, we investigate the spin dynamics and crystal-field states of ${\mathrm{Nd}}_{2}{\mathrm{Hf}}_{2}{\mathrm{O}}_{7}$ using muon spin relaxation $(\ensuremath{\mu}\mathrm{SR})$ and inelastic neutron scattering (INS) measurements. Our $\ensuremath{\mu}\mathrm{SR}$ study confirms the long-range magnetic ordering and shows evidence for coexisting persistent dynamic spin fluctuations deep inside the ordered state down to 42 mK. The INS data show the crystal electric field (CEF) excitations due to the transitions both within the ground-state multiplet and to the first excited state multiplet. The INS data are analyzed by a model based on CEF and crystal-field states are determined. Strong Ising-type anisotropy is inferred from the ground-state wave function. The CEF parameters indicate the CEF-split Kramers doublet ground state of ${\mathrm{Nd}}^{3+}$ to be consistent with the dipolar-octupolar character.