Crystal Electric Field Hamiltonian Research Articles

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.

Eigenstate analysis of the crystal electric field at low-symmetry sites: Application for an orthogonal site in the tetragonal crystal Ce2Pd2Pb

We have established an analyzing method to determine eigenstates of $4f$ electrons under the crystal electric field (CEF) at low-symmetry sites. This analysis requires macroscopic physical properties only, namely the saturation moments along several symmetrical directions of the crystal in addition to the magnetic susceptibility and specific heat, instead of the spectrum of the inelastic neutron scattering frequently used for such a complicated circumstance. As a successful case, the eigenstates of a CEF Hamiltonian under an orthogonal point group ${C}_{2v}$ are determined for the $J=\frac{5}{2}$ state of the cerium ion in a tetragonal compound ${\mathrm{Ce}}_{2}{\mathrm{Pd}}_{2}\mathrm{Pb}$. During the analysis, an angular momentum operator along the arbitrary direction is deduced from the space rotation operator. A matrix representation of the projection operator that transforms the basis of wave functions from $|{J}_{z}\ensuremath{\rangle}$ to $|{J}_{x}\ensuremath{\rangle}$ or $|{J}_{y}\ensuremath{\rangle}$ is also shown for $J=\frac{5}{2}$. This analysis reveals the anisotropic moments within highly symmetrical structures, and will contribute to understanding anisotropic field responses in rare-earth compounds.

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.

Neutron scattering measurement of crystalline-electric fields in magnesium rare-earth selenide spinels

The symmetry of local moments plays a defining role in the nature of exotic grounds states stabilized in frustrated magnetic materials. We present inelastic neutron scattering (INS) measurements of the crystal electric field (CEF) excitations in the family of compounds MgRE$_2$Se$_4$ (RE $\in$ $\{$Ho, Tm, Er and Yb$\}$). These compounds form in the spinel structure, with the rare earth ions comprising a highly frustrated pyrochlore sublattice. Within the symmetry constraints of this lattice, we fit both the energies and intensities of observed modes in the INS spectra to determine the most likely CEF Hamiltonian for each material and comment on the ground state wavefunctions in the local electron picture. In this way, we experimentally confirm MgTm$_2$Se$_4$ has a non-magnetic ground state, and MgYb$_2$Se$_4$ has effective $S=\frac{1}{2}$ spins with $g_\parallel = 5.188(79)$ and $g_\perp = 0.923(85)~\mu_B$. The spectrum of MgHo$_2$Se$_4$ indicates a ground state doublet containing Ising spins with $g_\parallel = 2.72(46)$, though low-lying CEF levels are also seen at thermally accessible energies $\delta_E = 0.591(36)$, 0.945(30) and 2.88(7)~meV, which can complicate interpretation. These results are used to comment on measured magnetization data of all compounds, and are compared to published results on the material MgEr$_2$Se$_4$.

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.

Magnetic structures in RNi4B (R=Nd, Tb, Ho, Er)

Neutron diffraction has been performed on R Ni 4 B ( R =Nd, Tb, Ho, Er) polycrystals. The orthorhombic structure for NdNi4B and the CeCo 4 B structure type (hexagonal) for TbNi 4 B and HoNi 4 B are confirmed. Our data also show that this last structure is currently the best approximant for ErNi 4 B. The R Ni 4 B ( R =Nd, Tb, Ho, Er) order ferromagnetically at respectively 11.0, 18.1, 6.2 and 10.0 K. The crystal electric field (CEF) interaction controls the magnetic anisotropy in this series leading to an easy axis ~30 deg above the basal plane in R Ni 4 B ( R =Nd, Tb, Ho) and parallel to the c -axis in ErNi 4 B at 1.6 K. The R Ni 4 B ( R =Nd, Tb, Ho) display a spin re-orientation below T c which arises from a competition between the second order term and the higher order terms of the CEF hamiltonian. Simplified magnetic structure in NdNi 4 B and full magnetic structure in R Ni 4 B ( R =Tb, Ho). Variation with temperature of the easy magnetization axis direction (angle with c -axis) showing a spin re-orientation. • R Ni 4 B ( R =Nd, Tb, Ho, Er) polycrystals were inductively melted. • Neutron diffraction confirms they all order ferromagnetically. • The magnetization easy axis is parallel to the c -axis in ErNi 4 B. • The R Ni 4 B ( R =Nd, Tb, Ho) display a spin re-orientation below T c . • Their easy axis tilts from the c -axis at T c to ~30 deg above the basal plane at 1.5 K.

Magneto-crystalline anisotropy in R 2PdSi 3 (R=Tb, Dy, Ho, Er, Tm) single crystals

We present single crystal AC-susceptibility and magnetisation measurements of R 2PdSi 3 compounds with R=Tb, Dy, Ho, Er, and Tm. The crystal-electric field (CEF) parameter B 2 0 and the value of the two-ion interaction exchange integral J ( Q = 0 ) are evaluated from the anisotropy of the AC-susceptibility measurements in the paramagnetic state. In addition, the magnetic anisotropy is investigated by magnetisation measurements well below the magnetic ordering temperature. Our results yield the first parameter in the CEF Hamiltonian B 2 0 / ( α 〈 r 2 〉 ) to be around −40 meV/Å 2 for R 2PdSi 3. The influence of higher order terms in the CEF Hamiltonian results in a crossing of magnetic hard and easy directions in the paramagnetic state for Dy 2PdSi 3 and Ho 2PdSi 3. The anisotropy due to the CEF is conserved in the ordered state. The observed magnetic properties turn out to be complex due to the delicate interplay of CEF effects and magnetic two-ion interactions.

Investigation of the Crystal Field in PrNi5

AbstractInelastic slow‐neutron scattering experiments are made on polycrystalline samples of intermetallic compounds PrNi5 and LaNi5 over the temperature range 8 to 295 K. The peaks of the inelastic magnetic‐neutron—rare‐earth‐ion scattering resulting from the splitting of the ion ground state into a number of levels by the crystal electric field (CEF) of hexagonal symmetry are identified by temperature and angular dependence of resonance intensities and by comparing PrNi5 spectra with those of LaNi5. On the basis of an effective point charge model for the CEF Hamiltonian with four variable parameters, calculations are made and CEF level scheme is restored. Magnetic contributions to the temperature dependence of heat capacity and magnetic susceptibility of PrNi5 are calculated and compared with available experimental data.