Crystal Electric Field Parameters Research Articles

CrysFieldExplorer: rapid optimization of the crystal field Hamiltonian

A new approach to the fast optimization of crystal electric field (CEF) parameters to fit experimental data is presented. This approach is implemented in a lightweight Python-based program, CrysFieldExplorer. The main novelty of the method is the development of a unique loss function, referred to as the spectrum characteristic loss (L Spectrum), which is based on the characteristic polynomial of the Hamiltonian matrix. Particle swarm optimization and a covariance matrix adaptation evolution strategy are used to find the minimum of the total loss function. It is demonstrated that CrysFieldExplorer can perform direct fitting of CEF parameters to any experimental data such as a neutron spectrum, susceptibility or magnetization measurements etc. CrysFieldExplorer can handle a large number of non-zero CEF parameters and reveal multiple local and global minimum solutions. Crystal field theory, the loss function, and the implementation and limitations of the program are discussed within the context of two examples.

Single crystal growths and magnetic properties of hexagonal polar semimetals RAuGe (R = Y, Gd–Tm, and Lu)

We study structural and magnetic properties of rare-earth based semimetals RAuGe (R = Y, Gd–Tm, and Lu) using flux-grown single crystals. These compounds belong to the space group P63mc, which is associated with the noncentrosymmetric polar point group 6mm. We confirm the systematic structural evolution at room temperature as a function of ionic radius of rare earths to clarify the isopointal crossover between two polar structures: three-dimensional LiGaGe-type and quasi-two-dimensional NdPtSb-type. Magnetism shows a characteristic anisotropy in reasonable agreement with the crystal electric field (CEF) theory; the easy-plane-type anisotropy for R = Tb and Dy turns into the Ising-type anisotropy for R = Er and Tm. We evaluate the CEF parameters based on the Stevens operators to reasonably reproduce the temperature dependence of magnetic susceptibilities and specific heat for RAuGe (R = Tb–Tm). The estimated energy scale of the Ising gap (∼ 11 meV) in TmAuGe is consistent with an excitation observed in an inelastic neutron scattering experiment. These findings suggest an opportunity for interplay between conduction electrons and nontrivial spin structures in the family of magnetic polar semimetals RAuGe.

Crystal-field Stark effect on the upconversion light emission spectrum of α−NaYF4 nanoparticles doped with Dy3+, Er3+ , or Yb3+

${\mathrm{NaYF}}_{4}$ nanoparticles (NPs) form in two crystal structures, cubic ($\ensuremath{\alpha}$-phase) and hexagonal ($\ensuremath{\beta}$-phase), each one presenting a different crystal electric field (CEF) Stark effect, that affects the upconversion (UC) light emission of the NPs when doped with rare-earth elements. Therefore, the knowledge of the CEF parameters, the wave functions, and energy levels of the rare earth (RE) $J$-multiplet is expected to be of great help for the understanding and improvement of the UC light emission. In this work, $\ensuremath{\alpha}$-phase ${\mathrm{NaYF}}_{4}$ NPs doped with ${\mathrm{Dy}}^{3+}, {\mathrm{Er}}^{3+}$, or ${\mathrm{Yb}}^{3+}$ were investigated by means of magnetization, electron spin resonance (ESR), and optical spectroscopy techniques. Fittings of the temperature- and magnetic-field-dependent magnetization were performed to determine the fourth- and sixth-order cubic CEF parameters, ${B}_{4}$ and ${B}_{6}$. The ground state of ${\mathrm{Er}}^{3+}, {\mathrm{Yb}}^{3+}$, and ${\mathrm{Dy}}^{3+}$ in these $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{NaYF}}_{4}$ NPs was confirmed by low-temperature ESR experiments. The obtained CEF parameters were used to write down a total Hamiltonian that allows to determine the CEF Stark splitting for all energy levels of the REs $4f$ unfilled shell. We give details of how the Stark effect affects the overall energy splitting of the various $J$-multiplets and may explain the fine structure of the UC light emission in these cubic ${\mathrm{NaY}}_{1\ensuremath{-}\ensuremath{\delta}}{\mathrm{RE}}_{\ensuremath{\delta}}{\mathrm{F}}_{4}$ NPs.

Crystal-field states and defect levels in candidate quantum spin ice Ce2Hf2O7

We report the synthesis of powder and single-crystal samples of the cerium pyrohafnate and their characterization using neutron diffraction, thermogravimetry and X-ray absorption spectroscopy. We evaluate the amount of non-magnetic Ce$^{4+}$ defects and use this result to interpret the spectrum of crystal-electric field transitions observed using inelastic neutron scattering. The analysis of these single-ion transitions indicates the dipole-octupole nature of the ground state doublet and a significant degree of spin-lattice coupling. The single-ion properties calculated from the crystal-electric field parameters obtained spectroscopically are in good agreement with bulk magnetic susceptibility data down to about 1 K. Below this temperature, the behavior of the magnetic susceptibility indicates a correlated regime without showing any sign of magnetic long-range order or freezing down to 0.08 K. We conclude that Ce$_2$Hf$_2$O$_{7}$ is another candidate to investigate exotic correlated states of quantum matter such as the octupolar quantum spin ice recently argued to exist in the isostructural compounds Ce$_2$Sn$_2$O$_7$ and Ce$_2$Zr$_2$O$_7$.

Quantifying uncertainties in crystal electric field Hamiltonian fits to neutron data

We systematically examine uncertainties from fitting rare earth single-ion crystal electric field (CEF) Hamiltonians to inelastic neutron scattering data. Using pyrochlore and delafossite structures as test cases, we find that uncertainty in CEF parameters can be large despite visually excellent fits. These results show Yb^{3+}3+ compounds have particularly large gg-tensor uncertainty because of the few available peaks. In such cases, additional constraints are necessary for meaningful fits.

Temperature evolution of optical transition 7F6 ⇒ 5D4 in Tb3Ga5O12 and Tb2(MoO4)3 from 4.2 to 300 K

Absorption spectra of Tb3Ga5O12 and Tb2(MoO4)3 are measured in temperature range from 4.2 to 300 K. Two low-energy lines appear above 20 K. This lines are ascribed to the vibronic-assisted optical absorption. Energy of the zero-point vibrations should be taken into account when optical data are used in calculation of the crystal electric field parameters.

The Zeeman effect and CEF parameters of Tb3Ga5O12

The Zeeman effect was observed in Tb3Ga5O12 single crystal for H∥[110]. Obtained results were used to determine the crystal electric field (CEF) parameters of the excited 5D4 multiplet of Tb3+ ion and exclude Γ1 symmetry for the lowest level of 7F6 ground state. The second order CEF parameters of the ground 7F6 multiplet were determined by fitting the magnetization measured for H∥[001] and H∥[110].

PyCrystalField: software for calculation, analysis and fitting of crystal electric field Hamiltonians

PyCrystalField is a Python software package for calculating single-ion crystal electric field (CEF) Hamiltonians. This software can calculate a CEF Hamiltonian ab initio from a point charge model for any transition or rare earth ion in either the J basis or the LS basis, perform symmetry analysis to identify nonzero CEF parameters, calculate the energy spectrum and observables such as neutron spectrum and magnetization, and fit CEF Hamiltonians to any experimental data. The theory, implementation and examples of its use are discussed.

Crystal electric field excitations in the quantum spin liquid candidate NaErS2

The delafossite family of compounds with a triangular lattice of rare earth ions has been recently proposed as a candidate host for quantum spin liquid (QSL) states. To realize QSLs, the crystal-electric-field (CEF) ground state of the rare earth ions should be composed of a doublet that allows sizable quantum tunneling, but till now the knowledge on CEF states in the delafossite compounds is still limited. Here we employ inelastic neutron scattering (INS) to study the CEF transitions in a powder sample of the delafossite NaErS$_2$, where the large total angular momentum $J = 15/2$ of the Er$^{3+} $ ions and the resulting plethora of CEF transitions enable an accurate fit of the CEF parameters. Our study reveals nearly isotropic spins with large $J_z = \pm 1/2$ components for the Er$^{3+}$ CEF ground states, which might facilitate the development of a QSL state. The scaling of the obtained CEF Hamiltonian to different rare earth ions suggests that sizable $J_z = \pm 1/2$ components are generally present in the CEF ground states, supporting the ternary sulfide delafossites as potential QSL hosts.

Magnetic transition due to the inter-singlet spin-exchange interaction and elastic softening by the interplay of electric quadrupoles in the distorted kagome lattice antiferromagnet Tb3Ru4Al12

The distorted kagome lattice antiferromagnet ${\mathrm{Tb}}_{3}{\mathrm{Ru}}_{4}{\mathrm{Al}}_{12}$ with a hexagonal structure has the N\'eel temperature ${T}_{\mathrm{N}}=22$ K. To clarify the $4f$-electronic state and an influence of electric quadrupoles in ${\mathrm{Tb}}_{3}{\mathrm{Ru}}_{4}{\mathrm{Al}}_{12}$, ultrasonic measurements on a single-crystalline sample at zero magnetic field and under fields were carried. A characteristic elastic softening of the transverse modulus ${C}_{66}$ originating from a quadrupole interaction was found. The crystal electric field parameters were determined to reproduce ${C}_{66}$, magnetic susceptibilities, and magnetization curves. The obtained level scheme is that the ground and first excited states are singlets, despite the existence of both the magnetic transition and the quadrupole interaction, indicating that ${\mathrm{Tb}}_{3}{\mathrm{Ru}}_{4}{\mathrm{Al}}_{12}$ is a curious compound. The positive sign of the quadrupole-quadrupole coupling constant for ${C}_{66}$ indicates a ferroquadrupolar-type interaction of the electric quadrupole ${O}_{xy}$ or ${O}_{2}^{2}$. The anisotropic magnetic field dependencies of ${T}_{\mathrm{N}}$ in the field along [100] and [001] were also clarified.

Multiphase competition in the quantum XY pyrochlore antiferromagnet CdYb2Se4 : Zero and applied magnetic field study

We study magnetic behaviour of the Yb$^{3+}$ ions on a frustrated pyrochlore lattice in the spinel {\CYS}. The crystal-electric field parameters deduced from high-energy inelastic neutron scattering reveal well-isolated ytterbium ground state doublet with a weakly Ising character. Magnetic order studied by powder neutron diffraction evolves from the $XY$-type antiferromagnetic $\Gamma_5$ state to a splayed ice-like ferromagnet (both with k=0) in applied magnetic field with $B_c$=3 T. Low-energy inelastic neutron scattering identifies weakly dispersive magnetic bands around 0.72 meV starting at $\mid\bf{Q}\mid$ = 1.1 \AA$^{-1}$~ at zero field, which diminish with field and vanish above 3 T. We explain the observed magnetic behaviour in framework of the nearest-neighbour anisotropic exchange model for effective $S=1/2$ Kramers doublets on the pyrochlore lattice. The estimated exchanges position the {\CYS} spinel close to the phase boundary between the $\Gamma_5$ and splayed ferromagnet states, similar to the Yb-pyrochlores suggesting an important role of the competition between these phases.

Determining the local low-energy excitations in the Kondo semimetalCeRu4Sn6using resonant inelastic x-ray scattering

We have investigated the local low-energy excitations in CeRu$_4$Sn$_6$, a material discussed recently in the framework of strongly correlated Weyl semimetals, by means of Ce $M_5$ resonant inelastic x-ray scattering (RIXS). The availability of both $^2$F$_\frac{5}{2}$ and $^2$F$_\frac{7}{2}$ excitations of the Ce $4f^1$ configuration in the spectra allows for the determination of the crystal-electric field parameters that explain quantitatively the temperature dependence and anisotropy of the magnetic susceptibility. The absence of an azimuthal dependence in the spectra indicates that all crystal-electric field states are close to being rotational symmetric. We show further that the non-negligible impact of the $\check A_6^0$ parameter on the ground state of CeRu$_4$Sn$_6$ leads to a reduction of the magnetic moment due to multiplet intermixing. The RIXS results are consistent with inelastic neutron scattering (INS) data and are compared to the predictions from \textsl{ab-initio} based electronic structure calculations.

Features of magnetization behavior in the rare-earth intermetallic compound (Nd0.5Ho0.5)2Fe14B

The crystal-electric field parameters are determined for the (Nd0.5Ho0.5)2Fe14B compound by analyzing experimental magnetization curves obtained in magnetic fields up to 60 T. The values of the crystal-field parameters B20, B40, B60, B44, B64 are 56.3, −73.2, −10.74, −8.9, 0 cm−1 for Nd3+ ion and 312.38, −176.78, 89.2, −88.43, 0 cm−1 for Ho3+ ion. The transition from the ferri-to the field-induced ferromagnetic state has been studied in detail.

Site-specific magnetic anisotropies in R2Fe14B systems

The local magnetic anisotropy of R ions in R2Fe14B (R = Dy, Ho) systems is studied based on a microscopic effective spin model constructed from the information obtained by using first-principles calculations. By taking into account up to 6-th order crystal electric field parameters, the model satisfactory describes the observed magnetization curves and the temperature dependence of anisotropy constants. We found that at low temperatures, the noncollinear structure appears in the Ho2Fe14B system reflecting the local magnetic anisotropy.

Dipolar Spin Ice States with a Fast Monopole Hopping Rate in CdEr_{2}X_{4} (X=Se, S).

Excitations in a spin ice behave as magnetic monopoles, and their population and mobility control the dynamics of a spin ice at low temperature. CdEr_{2}Se_{4} is reported to have the Pauling entropy characteristic of a spin ice, but its dynamics are three orders of magnitude faster than the canonical spin ice Dy_{2}Ti_{2}O_{7}. In this Letter we use diffuse neutron scattering to show that both CdEr_{2}Se_{4} and CdEr_{2}S_{4} support a dipolar spin ice state-the host phase for a Coulomb gas of emergent magnetic monopoles. These Coulomb gases have similar parameters to those in Dy_{2}Ti_{2}O_{7}, i.e., dilute and uncorrelated, and so cannot provide three orders faster dynamics through a larger monopole population alone. We investigate the monopole dynamics using ac susceptometry and neutron spin echo spectroscopy, and verify the crystal electric field Hamiltonian of the Er^{3+} ions using inelastic neutron scattering. A quantitative calculation of the monopole hopping rate using our Coulomb gas and crystal electric field parameters shows that the fast dynamics in CdEr_{2}X_{4} (X=Se, S) are primarily due to much faster monopole hopping. Our work suggests that CdEr_{2}X_{4} offer the possibility to study alternative spin ice ground states and dynamics, with equilibration possible at much lower temperatures than the rare earth pyrochlore examples.

ThMn12-type phases for magnets with low rare-earth content: Crystal-field analysis of the full magnetization process

Rare-earth (R)-iron alloys are a backbone of permanent magnets. Recent increase in price of rare earths has pushed the industry to seek ways to reduce the R-content in the hard magnetic materials. For this reason strong magnets with the ThMn12 type of structure came into focus. Functional properties of R(Fe,T)12 (T-element stabilizes the structure) compounds or their interstitially modified derivatives, R(Fe,T)12-X (X is an atom of hydrogen or nitrogen) are determined by the crystal-electric-field (CEF) and exchange interaction (EI) parameters. We have calculated the parameters using high-field magnetization data. We choose the ferrimagnetic Tm-containing compounds, which are most sensitive to magnetic field and demonstrate that TmFe11Ti-H reaches the ferromagnetic state in the magnetic field of 52 T. Knowledge of exact CEF and EI parameters and their variation in the compounds modified by the interstitial atoms is a cornerstone of the quest for hard magnetic materials with low rare-earth content.

Single crystal growth and anisotropic magnetic properties of HoAl2Ge2

We have grown a single crystal of HoAl2Ge2, which crystallizes in the hexagonal CaAl2Si2 type structure with Ho ions in the trigonal coordination in the ab plane. The data obtained from the bulk measurement techniques of magnetization, heat capacity and transport reveal that HoAl2Ge2 orders antiferromagnetically at TN ∼6.5 K. The susceptibility below TN and isothermal magnetization at 2 K indicate the ab plane as the easy plane of magnetization. Heat capacity data reveal a prominent Schottky anomaly with a broad peak centered around 25 K, suggesting a relatively low crystal electric field (CEF) splitting. The electrical resistivity reveals the occurrence of a superzone gap below TN. The point charge model of the CEF is applied to the magnetization and the heat capacity data. While a good fit to the paramagnetic susceptibility is obtained, the CEF parameters do not provide a satisfactory fit to the isothermal magnetization at 2 K and the Schottky anomaly.

Crystal-field effects in Er3+ - and Yb3+ -doped hexagonal NaYF4 nanoparticles

Since the up-conversion phenomenon in rare-earths (REs) doped ${\mathrm{NaYF}}_{4}$ is strongly affected by the crystal electric field (CF), determining the CF parameters, wave functions, and scheme of the energy levels of the RE $J$ multiplets could be crucial to improve and tune the up-conversion efficiency. In this work, the temperature and magnetic field dependent magnetization of ${\mathrm{NaY}}_{1\ensuremath{-}x}{[\mathrm{Er}(\mathrm{Yb})]}_{x}{\mathrm{F}}_{4}$ hexagonal nanoparticles (NPs) is reported. The data were best fit using the appropriated CF Hamiltonian for the $J=15/2\phantom{\rule{0.16em}{0ex}}(J=7/2)$ ground state multiplet of ${\mathrm{Er}}^{3+}\phantom{\rule{0.16em}{0ex}}({\mathrm{Yb}}^{3+}$) ions. The ${B}_{2}^{0}, {B}_{4}^{0}, {B}_{6}^{0}$, and ${B}_{6}^{6}$ CF parameters were considered in the Hamiltonian for RE ions located at the hexagonal ${\mathrm{C}}_{3h}$ point symmetry site of the ${\mathrm{NaYF}}_{4}$ host lattice. These results allowed us to predict an overall CF splitting of $\ensuremath{\sim}214$ ($\ensuremath{\sim}356$ K) for ${\mathrm{Er}}^{3+}$ (${\mathrm{Yb}}^{3+}$) and the wave functions and their energy levels for the $J=15/2$ ($J=7/2$) ground state multiplet which are in good agreement with the low temperature electron spin resonance experiments. Besides, our measurements allowed us to calculate all the excited CF $J$ multiplets that yield to a good estimation of the up-conversion light emission linewidth. The nonlinear optical light emission of the studied ${\mathrm{NaY}}_{1\ensuremath{-}x}{[\mathrm{Er}(\mathrm{Yb})]}_{x}{\mathrm{F}}_{4}$ hexagonal NPs was also compared with the most efficient up-conversion codoped ${\mathrm{NaY}}_{1\ensuremath{-}x\ensuremath{-}y}{\mathrm{Er}}_{x}{\mathrm{Yb}}_{y}{\mathrm{F}}_{4}$ hexagonal NPs.

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.

Physical properties of single crystallineRMg2Cu9(R=Y,Ce−Nd,Gd−Dy,Yb)and the search for in-plane magnetic anisotropy in hexagonal systems

Single crystals of $R$Mg$_{2}$Cu$_{9}$ ($R$=Y, Ce-Nd, Gd-Dy, Yb) were grown using a high-temperature solution growth technique and were characterized by measurements of room-temperature x-ray diffraction, temperature-dependent specific heat and temperature-, field-dependent resistivity and anisotropic magnetization. YMg$_{2}$Cu$_{9}$ is a non-local-moment-bearing metal with an electronic specific heat coefficient, $\gamma \sim$ 15 mJ/mol K$^2$. Yb is divalent and basically non-moment bearing in YbMg$_{2}$Cu$_{9}$. Ce is trivalent in CeMg$_{2}$Cu$_{9}$ with two magnetic transitions being observed at 2.1 K and 1.5 K. PrMg$_{2}$Cu$_{9}$ does not exhibit any magnetic phase transition down to 0.5 K. The other members being studied ($R$=Nd, Gd-Dy) all exhibits antiferromagnetic transitions at low-temperatures ranging from 3.2 K for NdMg$_{2}$Cu$_{9}$ to 11.9 K for TbMg$_{2}$Cu$_{9}$. Whereas GdMg$_{2}$Cu$_{9}$ is isotropic in its paramagnetic state due to zero angular momentum ($L$=0), all the other local-moment-bearing members manifest an anisotropic, planar magnetization in their paramagnetic states. To further study this planar anisotropy, detailed angular-dependent magnetization was carried out on magnetically diluted (Y$_{0.99}$Tb$_{0.01}$)Mg$_{2}$Cu$_{9}$ and (Y$_{0.99}$Dy$_{0.01}$)Mg$_{2}$Cu$_{9}$. Despite the strong, planar magnetization anisotropy, the in-plane magnetic anisotropy is weak and field-dependent. A set of crystal electric field parameters are proposed to explain the observed magnetic anisotropy.