C. Rudowicz

SPIN-HAMILTONIAN FORMALISMS IN ELECTRON MAGNETIC RESONANCE (EMR) AND RELATED SPECTROSCOPIES

The historical development leading to, and the current status of, the spin-Hamiltonian (SH) formalisms, characterizing electron magnetic resonance (EMR), also referred to as electron paramagnetic (or spin) resonance (EPR or ESR), is reviewed. The spin-Hamiltonian concept is briefly outlined to set a framework for definitions of relevant terms. Meanings of the terms which are often confused with each other, e.g. physical versus effective Hamiltonian, real versus effective versus fictitious spin, microscopic SH (MSH), zero-field-splitting (ZFS) Hamiltonian, generalized SH (GSH), and phenomenological SH (PSH), are elucidated. Various general approaches to ‘derive’ MSH as well as to ‘construct’ GSH for transition ions are discussed. This enables clarification of relationship between the ZFS Hamiltonian and (a) the crystal-field (CF), (b) electronic spin-spin, and (c) nuclear-quadrupole Hamiltonians. The inadmissibility of the odd-order ZFS terms (l = 3, 5) is also discussed. A brief general classification of the major operator and parameter notations used in the literature to describe ZFS Hamiltonian is provided. Other important areas relevant to EMR, where the SH concept plays a central role, are outlined.

Crystal field and microscopic spin Hamiltonians approach including spin–spin and spin–other-orbit interactions for d2 and d8 ions at low symmetry C3 symmetry sites: V3+ in Al2O3

Abstract A new module has been developed within the CFA/MSH computer package, which is applicable for d2 and d8 ions at sites of trigonal symmetry type I (C3v,D3,D3d) and type II (C3,C3i), including the ‘imaginary’ CF term. For the first time the spin–spin (SS) and spin–other-orbit (SOO) interactions have also been included in the Hamiltonian. This module enables to study the contributions to the energy levels and the spin Hamiltonian parameters, i.e. zero-field splitting D and g-factors: g∥ and g⊥. The contributions arising from the spin–orbit (SO), SS, and SOO interaction as well as those due to the low symmetry CF effects induced by the distortion angle ϕ, which describes the difference between C3 and C3v symmetry, can be studied. As an application of the new module, calculations have been carried out for V3+(3d2) ions in α-Al2O3 crystal, taking into account for the first time the SS and SOO interactions, and the low symmetry CF effects. The results show that (i) the contributions from the SS and SOO interactions to the energy levels are larger for free V3+ ions than those for V3+ ions in α-Al2O3 crystal, (ii) both the contributions to the SH parameters and the energy levels arising from the SOO interaction are larger than those arising from the SS interaction, (iii) the contributions due to the low symmetry CF effects induced by the distortion angle ϕ are in general significant, (iv) D and g⊥ are sensitive to the distortion angle ϕ, whereas g∥ is insensitive to ϕ, and (v) the influence of the lattice distortions on the spectroscopic properties of V3+ ion in α-Al2O3 is pronounced. It appears important for similar ion-crystal cases to consider the lattice distortions in detailed calculations, which take into account the relevant contributions from the SO, SS and SOO interactions. A good agreement between the theoretical and experimental results has been obtained.

Microscopic study of Cr2+ ion in the quasi-2D mixed system Rb2MnxCr1−xCl4

Abstract The effective point symmetry around Cr 2+ ion in the quasi-two-dimensional (2D) mixed system Rb 2 Mn x Cr 1− x Cl 4 is D 2h but it approximates closely to D 4h . Possible energy-level schemes for Cr 2+ ion in the D 4h as well as D 2h symmetry are investigated using the superposition model in order to discriminate between the two energy-level schemes assigned previously to the ground states ∥ x 2 − y 2 〉 and ∥z 2 〉. The superposition model enables also study of the dependence of the crystal-field parameters and the energy levels on the concentration ( x ) in Rb 2 Mn x Cr 1− x Cl 4 . Using our earlier microscopic relationships between the spin Hamiltonian parameters b k q , g i and the energy level splittings (Δ i ) within the 5 D term for D 4h and D 2h symmetry, values of b 2 0 , b 2 2 , b 4 0 , b 4 2 , b 4 4 and g x , g y , g z are predicted for a wide range of the microscopic parameters involved. The diagrams of the parameters b 2 0 and b 2 2 versus the spin-orbit (λ) and the spin-spin (ϱ) interaction parameters are worked out for a range of values of λ and ϱ specific for Cr 2+ . The dependence of the zero-field splitting parameters b k q on the concentration ( x ) in Rb 2 Mn x Cr 1− x Cl 4 is also studied. Possible sources of the discrepancies between various experimental data for the parameters b 2 0 and b 2 2 are discussed. For the first time, the fourth-order parameters b 4 0 , b 4 2 and b 4 4 are determined. The present comprehensive investigation of the single-ion properties of Cr 2+ ion in Rb 2 Mn x Cl 1− x Cl 4 provides results indispensible for explaining the magnetic anisotropy as well as for predicting the EPR spectra at high frequency and high magnetic field.

The generalization of the extended Stevens operators to higher ranks and spins, and a systematic review of the tables of the tensor operators and their matrix elements

Spherical?(S) and tesseral?(T) tensor operators?(TOs) have been extensively used in, for example, EMR and optical spectroscopy of transition ions. To enable a systematic review of the published tables of the operators and their matrix elements?(MEs) we have generated the relevant tables by computer, using Mathematica programs. Our review reveals several misprints/errors in the major sources of TTOs?the conventional Stevens operators (CSOs?components ) and the extended ones (ESOs?all q) for rank k = 2,4, and?6?as well as of some STOs with . The implications of using incorrect operators and/or MEs for the reliability of EMR-related programs and interpretation of experimental data are discussed. Studies of high-spin complexes like Mn12 (S = 10) and Fe19 (S = 33/2) require operator and ME listings up to k = 2S, which are not presently available. Using the algorithms developed recently by Ryabov, the generalized ESOs and their MEs for arbitrary rank k and spin S are generated by computer, using Mathematica. The extended tables enable simulation of the energy levels for arbitrary spin systems and symmetry cases. Tables are provided for the ESOs not available in the literature, with odd k = 3,5, and?7 for completeness; however, for the newly generalized ESOs with the most useful even rank k = 8,10, and?12 only, in view of the large listings sizes. General source codes for the generation of the ESO listings and their ME tables are available from the authors.

Ligand field analysis of the 3dN ions at orthorhombic or higher symmetry sites

Abstract A computer package has been developed to calculate the energy levels and the ligand field states for a full Hamiltonian including the electrostatic terms, Trees correction, the spin-orbit interaction and the ligand field interactions within the 3dN configuration (N = 2 to 8). The program can deal with various cubic and non-cubic site symmetries of the 3d ion, namely the tetragonal, trigonal and orthorhombic ones. The package may be very useful in analyzing the EPR spectra of 3dN ions and in correlating the ligand field parameters with spin Hamiltonian parameters derived from the orbital singlet states for some F-term (S = 1, 3 2 ) and S-term (S = 5 2 ) ions as well as the 3d4 and 3d6 ions with high-spin S = 2. Numerous examples of these ions in various crystals exist in the literature. Other spin states arising from strong ligand fields can also be dealt with.

Theoretical investigations of the microscopic spin Hamiltonian parameters including the spin-spin and spin-other-orbit interactions for Ni2+(3d8) ions in trigonal crystal fields

The microscopic origin of the spin Hamiltonian (SH) parameters for Ni2+(3d8) ions in a trigonal type I symmetry (C3v,D3d,D3) crystal field (CF) is studied. In addition to the spin–orbit (SO) interaction, we consider also the spin–spin (SS) and spin–other-orbit (SOO) interactions. The relative importance of the four (SO, SS, SOO, and combined SO–SS–SOO) contributions to the SH parameters is investigated using the CFA/MSH package and the complete diagonalization method (CDM). The SO mechanism is dominant for all CF parameter (CFP) ranges studied, except where the contributions DSO to the zero-field splitting (ZFS) parameter D change sign. For the trigonal CFP, due to the other three mechanisms exceeds DSO. Although |DSOO| is quite small, the combined |DSO−SOO| is appreciable. The SO-based perturbation theory (PT) works generally well for the g-factors: and , while it fails for D in the vicinity of vc and for large and v>0. The high percentage discrepancy ratio δD = 2020% for vc indicates unreliability of DSO (in PT). Applications to Ni2+ ions at trigonal symmetry sites in LiNbO3, α-LiIO3, and Al2O3, are provided. The theoretical SH parameters are in good agreement with the experimental data. The low symmetry (C3) effects induced by the angle are tentatively studied, but appear to be quite small.

On the standardization of crystal-field parameters and the multiple correlated fitting technique: Applications to rare-earth compounds

Abstract This work investigates the crystal-field parameter (CFP) sets for rare-earth (RE) ions doped at orthorhombic and/or lower symmetry, i.e. monoclinic or triclinic sites. Two important questions are addressed: (1) How do you compare CFP sets reported by different authors when there can be as many as six numerically different equivalent sets? and (2) How do you distinguish between global and local minima in the multi-parameter fittings? To answer the first question we propose to adopt the standardization of CFP sets. The orthorhombic and/or monoclinic standardization has been carried out for several illustrative CFP sets, including the contributions from various mechanisms, for NdF 3 , RE 2 Te 4 O 11 , and RE 3+ in garnet materials. It is shown that adopting a well-defined standardization convention enables meaningful comparison of CFP sets taken from various sources. The analysis of literature data reveals also cases when the CF parameters may be misinterpreted due to the choice of a different representation in the multi-parameter fittings. To answer the second question we propose the multiple correlated fitting technique (MCFT). Using experimental data for Nd 3+ in YAG and LaF 3 , MCFT is employed to probe for more reliable CFP sets.

Relationship between oxygen defects and the photoluminescence property of ZnO nanoparticles: A spectroscopic view

The present paper deals with the synthesis and characterization of ZnO nanoparticles obtained by coprecipitation method at three different pH conditions. Samples characterizations aimed at understanding their spectroscopic properties are carried out using a variety of experimental techniques. X-ray diffraction and transmission electron microscopic studies show significant increase in the particle size of the synthesized ZnO nanoparticles ranging from 3–40 nm with increasing pH values. Absorption spectra show particle size dependent blueshift in the energy band, which may be ascribed to quantum confinement effect. Fourier transform infrared (FTIR) spectroscopy reveals enhancement in the surface defects of the synthesized ZnO nanoparticles with increasing pH values. Electron paramagnetic resonance studies at room temperature (300 K) and several liquid helium temperatures (including the lowest temperature 13 K for all samples) indicate the presence of singly ionized oxygen vacancy defects (VO+) and O2− super...

On the relations between the zero-field splitting parameters in the extended Stevens operator notation and the conventional ones used in EMR for orthorhombic and lower symmetry

Electron magnetic resonance (EMR) studies of paramagnetic species with the spin S ≥ 1 at orthorhombic symmetry sites require an axial zero-field splitting (ZFS) parameter and a rhombic one of the second order (k = 2), whereas at triclinic sites all five ZFS (k = 2) parameters are expressed in the crystallographic axis system. For the spin S ≥ 2 also the higher-order ZFS terms must be considered. In the principal axis system, instead of the five ZFS (k = 2) parameters, the two principal ZFS values can be used, as for orthorhombic symmetry; however, then the orientation of the principal axes with respect to the crystallographic axis system must be provided. Recently three serious cases of incorrect relations between the extended Stevens ZFS parameters and the conventional ones have been identified in the literature. The first case concerns a controversy concerning the second-order rhombic ZFS parameters and was found to have lead to misinterpretation, in a review article, of several values of either E or b22 published earlier. The second case concerns the set of five relations between the extended Stevens ZFS parameters bkq and the conventional ones Dij for triclinic symmetry, four of which turn out to be incorrect. The third case concerns the omission of the scaling factors fk for the extended Stevens ZFS parameters bkq. In all cases the incorrect relations in question have been published in spite of the earlier existence of the correct relations in the literature. The incorrect relations are likely to lead to further misinterpretation of the published values of the ZFS parameters for orthorhombic and lower symmetry. The purpose of this paper is to make the spectroscopists working in the area of EMR (including EPR and ESR) and related spectroscopies aware of the problem and to reduce proliferation of the incorrect relations.

Superposition model and crystal-field analysis of the 4A2 and a2E states of Cr3+ ions at C3 sites in LiNbO3

The energy levels and wavefunctions including the two lowest-lying levels, namely 4A2 and a2E, for which reliable experimental data exist for Cr3+ ions at C3 symmetry sites in LiNbO3, are calculated using the complete matrix diagonalization method within the 3d3 configuration. The Hamiltonian considered includes the electrostatic term, the Trees correction, the spin-orbit interaction and the crystal-field interaction. The role of the additional low-symmetry crystal-field term B4-3O4-3 (in the Stevens operator notation), neglected in the C3v approximation used so far in the literature, is studied. The superposition model is developed for 3d3 ions at C3 symmetry sites and applied to study the site occupancy of Cr3+ in LiNbO3. Analysis of the optical data indicates that Cr3+ ions substitute at Nb sites and Li sites simultaneously. The present considerations offer an improvement over the earlier approximations using C3v symmetry only. The zero-field splitting predicted by the crystal-field calculations for Cr3+ at the Nb site matches the experimental value from EPR studies very well. This is contrary to the earlier prediction by the superposition model analysis of the spin-Hamiltonian parameters indicating that the zero-field splitting for Cr3+ ions at Li sites matches the experimental zero-field splitting better than that for Cr3+ at Nb sites. Since the present calculations involve fitting not only the zero-field splitting but also the energies of the a2E state, the present predictions may be more reliable than the previous predictions.